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Finite dimensional approximation and convergence in the estimation of the distribution of, and input to, random abstract parabolic systems with application to the deconvolution of blood/breath al...
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Finite dimensional approximation and convergence in the estimation of the distribution of, and input to, random abstract parabolic systems with application to the deconvolution of blood/breath al...

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Content Finite Dimensional Approximation and Convergence in the Estimation of the Distribution of, and Input to, Random Abstract Parabolic Systems with Application to the Deconvolution of Blood/Breath Alcohol Concentration from Biosensor Measured Transdermal Alcohol Level by Melike Sirlanci Tuysuzoglu A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Appied Mathematics) August 2018 Copyright 2018 Melike Sirlanci Tuysuzoglu Dedication To my family ii Acknowledgements Foremost, I would like to express my sincere gratitude to my advisor Prof. I. Gary Rosen for the continuous support of my PhD study and research, for his patience, motivation, enthusiasm, and immense knowledge. His guidance helped me in all the time of research and writing of this thesis. He was not only a great advisor for my research but also a great mentor to navigate me in every area of academic life. I could not have imagined having a better advisor and mentor for my PhD study. Besides my advisor, I would like to thank the rest of my thesis committee, Dr. Susan E. Luczak, Prof. Chunming Wang, and Prof. Larry Goldstein for their valuable comments on this project. I thank Dr. Susan E. Luczak at the University of Southern California for allowing me to use the data collected for a research which was supported under the grant R01 AA11257. Also, I thank Dr. Catharine Fairbairn at the University of Illinois, Urbana Champaign for allowing me to use the data collected for a research supported under the grant R01 AA25969. I would like to thank the National Institute of Health for the generous support under the grants R21 AA17711 and R01 AA26368. I thank my friends at USC, S ebnem Baran, G uher C amlyurt, Emre Demirkaya, _ Ilknur E gilmez, Ozlem Ejder, Duygu Kaba, Ezgi Kantarc O guz, Can Ozan O guz, Enes  Ozel, and Fanhui Xu for their friendship, support, and for all the fun we have had in the last ve years. I would like to thank my parents Meral S rlanc and Mehmet S rlanc for supporting me to follow my dreams, encouraging me to pursue higher education, and their belief in my success. iii I would also like to thank my brother, Melih S rlanc for his endless support and help literally whenever I need. Last but not the least, I would like to thank my spouse Onur T uys uzo glu for being my inspiration to go for a PhD education at rst place, disregarding his own career to be with me, and dealing with all the stressful times along the way. iv Table of Contents Dedication ii Acknowledgements iii List Of Tables vii List Of Figures ix Abstract xi Chapter 1: Introduction 1 Chapter 2: A Mathematical Model of Transdermal Alcohol Transport 8 Chapter 3: Preliminaries 13 Chapter 4: Parameter Estimation Problem 18 4.1 Estimation of Random Discrete Time Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Abstract Parabolic Systems with Unbounded Input and Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2.1 The Discrete Time Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2.2 Systems with Boundary Input . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.3 Random Regularly Dissipative Operators and Their Associated Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.4 Approximation and Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.4.1 The Estimation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.4.2 A Version of the Trotter-Kato Semigroup Approximation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.4.3 Application to the Density Estimation Problem . . . . . . . . . . . . . . . . 38 Chapter 5: Deconvolution Problem 44 5.1 Input Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.2 Approximation and Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.3 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Chapter 6: Application of Methods to the Alcohol Biosensor Problem 54 v Chapter 7: Numerical Results 64 7.1 Computational Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 7.2 Numerical Results with Simulated Data . . . . . . . . . . . . . . . . . . . . . . . . 69 7.2.1 One-Parameter Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 7.2.2 Two-Parameter Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 7.3 Numerical Results with Alcohol Biosensor Data . . . . . . . . . . . . . . . . . . . . 76 7.3.1 Dataset 1 - Single Subject with Multiple Drinking Episodes . . . . . . . . . 77 7.3.2 Dataset 2 - Multiple Subjects each with Single Drinking Episode . . . . . . 84 7.3.3 Dataset 3 - Multiple Subjects each with Single Drinking Episode . . . . . . 92 Chapter 8: Discussion and Concluding Remarks 98 Reference List 101 vi List Of Tables 7.1 Convergence results for Examples 6.1, 6.2 and 6.3; estimation of the parameters in truncated uniform, exponential and normal distributions . . . . . . . . . . . . . . . 72 7.2 Convergence results for Example 6.4; estimation of the parameters in truncated bivariate normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7.3 Peak BrAC for each drinking episode in dataset 1 obtained for total population and stratied population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.4 Time of peak BrAC for each drinking episode in dataset 1 obtained for total population and stratied population . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.5 Area under the BrAC curve for each drinking episode in dataset 1 obtained for total population and stratied population . . . . . . . . . . . . . . . . . . . . . . . . . . 84 7.6 Absorption rate for each drinking episode in dataset 1 obtained for total population and stratied population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 7.7 Elimination rate for each drinking episode in dataset 1 obtained for total population and stratied population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.8 Peak BrAC for each drinking episode in dataset 2 obtained for total and stratied population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.9 Time of peak BrAC for each drinking episode in dataset 2 obtained for total and stratied population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7.10 Area under the BrAC curve for each drinking episode in dataset 2 obtained for total and stratied population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7.11 Absorption rate for each drinking episode in dataset 2 obtained for total and stratied population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7.12 Elimination rate for each drinking episode in dataset 2 obtained for total and stratied population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.13 Peak BrAC for each drinking episode in dataset 3 . . . . . . . . . . . . . . . . . . . 96 vii 7.14 Time of peak BrAC for each drinking episode in dataset 3 . . . . . . . . . . . . . . 96 7.15 Area under the BrAC curve for each drinking episode in dataset 3 . . . . . . . . . 96 7.16 Absorption rate for each drinking episode in dataset 3 . . . . . . . . . . . . . . . . 96 7.17 Elimination rate for each drinking episode in dataset 3 . . . . . . . . . . . . . . . . 97 viii List Of Figures 7.1 One-parameter model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 7.2 Estimation results for one-parameter model . . . . . . . . . . . . . . . . . . . . . . 72 7.3 Estimated probability density function and 75% credible band for forward model . 75 7.4 Alcohol Biosensor Device: The WrisTAS TM 7 . . . . . . . . . . . . . . . . . . . . . 77 7.5 BrAC and TAC measurements for dataset 1 . . . . . . . . . . . . . . . . . . . . . . 78 7.6 Optimal pdfs obtained by total and stratied populations for dataset 1 . . . . . . . 78 7.7 Estimation results for the training drinking episodes by assuming u is a function of only t using total population for dataset 1 . . . . . . . . . . . . . . . . . . . . . . . 79 7.8 Estimation results for the training drinking episodes by assuming u is a function of both t andq using total population for dataset 1 . . . . . . . . . . . . . . . . . . . 80 7.9 Estimation results for the training drinking episodes by assuming u is a function of both t andq after stratication for dataset 1 . . . . . . . . . . . . . . . . . . . . . 80 7.10 Estimation results for the cross validation drinking episodes obtained by assuming u is a function of only t using total population, and assuming u is a function of both t andq using total and stratied population for dataset 1 . . . . . . . . . . . 81 7.11 Convolution kernel, Estimated BrAC, and pdf of estimated BrAC obtained for dataset 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 7.12 Optimal pdfs and scatter plots for q samples obtained by total and stratied populations for dataset 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.13 Estimation results for the training drinking episodes using total population for dataset 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7.14 Estimation results for the training drinking episodes using stratied population for dataset 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 ix 7.15 Estimation results for the testing drinking episodes using total and stratied popu- lation for dataset 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.16 Alcohol Biosensor Device: The SCRAM . . . . . . . . . . . . . . . . . . . . . . . . 92 7.17 Optimal pdf obtained for dataset 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.18 Scatter plot of deterministically obtained parameter estimates for the 18 datasets considered in the dataset 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.19 Estimation results for the training drinking episodes by assuming u is a function of both t andq for dataset 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7.20 Estimation results for the training drinking episodes by assuming u is a function of only t for dataset 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7.21 Estimation results for the cross validation drinking episodes obtained by assuming u is a function of only t, and assuming u is a function of both t andq for dataset 3 95 7.22 Convolution kernel, Estimated BrAC, and pdf of estimated BrAC For dataset 3 . . 95 x Abstract Being motivated by an alcohol biosensor problem, the input to a random parabolic partial dierential equation is estimated given the output based on population data. The problem is parsed into two subproblems: a parameter estimation problem and a deconvolution problem. Because of having various sources of uncertainty, the parameters in the mathematical model are assumed to be random and dened in terms of a joint density. In this case, estimating the distribution is equivalent to estimating the joint distribution function. A nite dimensional abstract approximation and convergence theory is developed for estimation of the distribution of random parameters in innite dimensional discrete time linear systems with dynamics described by regularly dissipative operators and involving, in general, unbounded input and output operators. By taking expectations, the system is re-cast as an equivalent abstract parabolic system in a Gelfand triple of Bochner spaces wherein the random parameters become new space-like variables. Estimating their distribution is now analogous to estimating a spatially varying coecient in a standard deterministic parabolic system. The estimation problems are approximated by a sequence of nite dimensional problems. Convergence is established using a state space-varying version of the Trotter-Kato semigroup approximation theorem. The resulting system, which is formulated based on the estimated distribution of the random parameters, is referred to as a population model. Once the forward population model has been identied or trained based on a sample from the population, the resulting distribution can then be used to deconvolve the input signal from the observed output signal formulated as either a quadratic programming or linear quadratic tracking problem. In addition, our approach allows for the direct computation of corresponding credible xi bands without simulation. When applied to the alcohol biosensor problem, the model takes the form of a diusion equation with the input, which is on the boundary of the domain, being the blood or breath alcohol concentration (BAC/BrAC), and the output, also on the boundary, being the transdermal alcohol concentration (TAC). All the assumptions used to construct our theoretical framework are shown to be satised by the model for the alcohol biosensor problem. To assess the eciency of the approach, numerical results obtained by working with simulated data for a number of examples involving the estimation of exponential families of densities for random parameters in a diusion equation with boundary input and output are presented and discussed. In addition, the technique is used to estimate bivariate normal distributions and deconvolve BAC/BrAC from TAC based on data from a population that consists of multiple drinking episodes from a single subject and two populations consisting of single drinking episodes from multiple subjects. xii Chapter 1 Introduction The ideas and results presented here are motivated by the following problem involving the calibration of a transdermal alcohol biosensor. There has been a lot of research on how alcohol concentrations relate to drinking motives, physical responses to alcohol, behaviors, decision-making, and negative consequences over the course of a drinking episode and across drinking episodes. For many alcohol research areas and clinical treatments, nearly-continuous monitoring of quantitative alcohol levels of patients in naturalistic setting would provide crucial knowledge and would be helpful for eliminating the limitations of research or treatments due to lack of such information. It would also further the understanding of the dierences between social and problem drinkers, the progression of drinking (e.g. initiation, harmful use, dependence), the symptomatology of dependence (e.g., sensitization, craving, tolerance, withdrawal), the evaluation of clinical interventions, and the development of prevention programs. Unfortunately, there is currently no procedure that provides ecient information about alcohol consumption of individuals from continuously collected data in naturalistic settings. Among currently available methods, the most reliable information can be obtained from blood or urine samples, but it is certain that collecting this kind of sample is not practical at all. On the other hand, the data collected by breath analyzers or drink diaries is less reliable when collected in the eld by non-expert users. There is the possibility that a breath analyzer reading being too high due to mouth alcohol or too low due to not taking deep lung 1 breaths, while drink diaries might be inaccurate due to not knowing the alcohol content of a drink or the amount consumed. Moreover, it is also possible that there are large time gaps between assessment points. Recently, new technology has produced biosensor devices that measure transdermal alcohol concentration (TAC), the amount of alcohol diusing through the skin and have the potential for passively collecting quantitative levels of alcohol. However, due to a lot of physiological and environmental factors, there is no direct quantitative correlation in terms of an inverison formula between TAC and BAC/BrAC. Because of this reason, so far, it has been impossible to estimate BAC/BrAC (eBAC/eBrAC) accurately based on TAC data and so, TAC data has only been used for abstinence monitoring. On the other hand, BAC and BrAC are currently and have historically been the standard measures of intoxication among alcohol researchers and clinicians, as well as in the legal system. Unfortunately, since there is currently no well-established method for producing reliable estimates of BAC/BrAC from TAC data, alcohol researchers and clinicians have been slow to accept TAC devices as a fundamental tool that can aid them in carrying out their studies. Therefore, estimating BAC/BrAC from TAC data is one way to make more use of such practically acquirable data. Absence of an obvious relationship between TAC and BAC/BrAC makes the estimation even harder. Indeed, unlike a breath analyzer, which relies on a relatively simple model from basic chemistry (i.e., Henry's Law) for the exchange of gases between circulating pulmonary blood and alveolar air (see [21]) that has been found to be reasonably robust accross the population, the transport and ltering of alcohol by the skin is physiologically more complex and is aected by a number of factors that dier across individuals (e.g., skin layer thickness, porosity and tortuosity, etc.) and even drinking episodes within individuals (e.g., body and ambient temperature, skin hydration, vasodilation). Also note that TAC readings may highly depend on the device that is used to collect the data, and unfortunately this situation poses another obstacle for the estimation problem. Nevertheless, depending on the complexity of the model, even if the 2 device works perfectly in measuring TAC, the raw TAC data will never consistently map directly onto BAC/BrAC across individuals and drinking episodes. Over the past decade, researchers worked on developing methods to address this conversion problem and produce reliable quantitative estimates of BAC/BrAC from TAC data (see [11, 12, 13, 14, 28, 34]). Among these approaches, we will have a closer look at the ones developed by Rosen et al. since they form the basis for the method developed in this thesis. In [11, 28], authors take a strictly deterministic approach to converting TAC to eBAC/eBrAC. They t rst principles physics-based models in the form of a distributed parameter system with unbounded input and output, using individual calibration data to capture the dynamics of the forward process, the propagation of alcohol from the blood, through the skin, and its measurement by the sensor (i.e. the forward model). In the second phase of processing, the model is inverted to produce eBAC/eBrAC based on the TAC. More specically, they model the transport of ethanol molecules from the blood through the various layers of the skin and its measurement by the biosensor in the form of TAC. This linear BAC/BrAC-to-TAC forward mathematical model takes the form of either a two parameter parabolic diusion equation or a three parameter nite speed of propagation hyperbolic telegraph equation with appropriate (unbounded with respect to the traditional state space for systems of this form) boundary conditions involving the BAC/BrAC input and producing simulated TAC in the form of an associated (also unbounded) output operator. The model then yields an impulse response function, or convolution lter, that completely characterizes the forward model. This forward model, together with an inversion process in the form of a non-negatively constrained quadratic programming problem can then be considered to be a blind deconvolution scheme which produces eBAC/eBrAC from the TAC eld data for that person-device pair. This procedure has produced good results (e.g. [11, 15, 28]), and has even already been used in actual alcohol related consumption and behavioral studies. Indeed, in [17] the authors use TAC sensors together with the algorithms in [28] and the corresponding software to convert the TAC data collected in the eld to equivalent BAC or BrAC in order to investigate the relationship between 3 social familiarity and alcohol reward in naturalistic drinking settings and compare it to alcohol reward observed in laboratory drinking studies. However, the results of these preliminary studies using this technology also indicate that some of the dynamics of the system are not being captured by the models. In addition, the calibration protocol has limitations, including that the procedure for collecting the individual calibration data is burdensome for both researchers and participants, and that it is not always feasible to conduct (e.g., for clinicians and lay individuals who do not have access to an alcohol administration laboratory or with patients who are trying to abstain). These drawbacks signicantly reduce the feasibility of using these devices. Thus, we have been investigating ways to eliminate the need to calibrate the sensor's data analysis system to each individual, each sensor, and across varying ambient environmental conditions, by better capturing any un-modeled dynamics of the system. In order to eliminate the calibration phase, we have been investigating the use of our rst principles physics/physiological based models to describe the dynamics common to the entire population (interpreted broadly to include all individuals, devices and environmental conditions) and to then attribute all un-modeled sources of uncertainty (primarily due to variations in physiology, hardware, and the environment) observed in individual data to combined random eects. We refer to this as a population model; it takes the form of our earlier deterministic transport models, but with the parameters now being random variables whose distributions are to be estimated based on aggregate population data. Thus, based on earlier results (see, for example, [11, 28]), the basic modeling assumption here is that there is an underlying single mathematical framework that describes that portion of the system dynamics that is common to all individuals, environmental conditions and devices in the population (e.g., the physics-based model for the transport of ethanol from the blood, through the skin and measurement by the sensor), but that individual members of the population exhibit variation in the parameters appearing in the model (e.g., the rate at which the alcohol is transported, evaporates, etc.). We achieve this by now allowing our models to be based on random partial dierential equations wherein the parameters 4 in the model are now random variables. Consequently, instead of tting the model directly to individual data, it is now t to aggregate population data with the t model describing the mean behavior of the population with estimates for the distributions of the parameters obtained instead of point estimates for the parameters. In this way, instead of considering solutions to the model equations or any other quantities derived from those solutions (e.g. their inversion producing eBrAC from TAC), the model can now be used to produce means and associated condence or credible sets or bands. The partial dierential operators that appear in our ethanol transport models are based on diusion, or Fick's law, and take the form of abstract parabolic or hyperbolic operators with damping. When formulated abstractly in a Gelfand triple setting, these operators are examples of what are known as regularly dissipative operators and can often be shown to generate analytic semigroups. We focus on the development of an abstract approximation framework and convergence theory for the estimation of random parameters in systems with dynamics described by these classes of operators based upon aggregate population data. The parameter estimation problem is formulated in much the same way as it is in standard linear regression. That is, each data point is assumed to be an observation of the mean population behavior plus random error. We then formulate the estimation problem as an optimization problem over the space of feasible distributions for the random parameters. The objective of the optimization problem is to minimize prediction error in the form of the dierence between observed TAC and the expectation of the output of the model. We then demonstrate the convergence of the solutions to a sequence of nite dimensional approximating estimation/optimization problems to the solution of the original innite dimensional problem. Our general approach relies heavily on two relatively recent papers: (1) Banks' and Thompson's [4] framework for the estimation of probability measures in random abstract evolution equations and the convergence of nite dimensional approximations in the Prohorov metric and (2) Gittelson's, Andreev's, and Schwab's [19] theory for random abstract parabolic partial dierential equations with dynamics dened in terms of bounded and coercive sesquilinear forms. This latter 5 approach is novel in the way that it treats the random variable as another space-like independent variable. In this way, nite dimensional approximation is handled in the same way that it is for the standard deterministic space variables. We use the framework in [19] together with the generation and approximation results from linear semigroup theory, (i.e. the Hille-Yosida-Phillips and Trotter Kato theorems) to establish that the sucient conditions for a Banks and Thompson-like convergence result is satised in the case of regularly dissipative systems with random parameters whose distributions can be described by appropriately parameterized probability density functions. These theoretical results allow us to develop computational algorithms that have rigorously been shown to converge, and that yield numerical approximations to the desired distributions. After estimating the distribution of the random parameters, we form our population model based on this estimation, and formulate the input estimation or deconvolution problem in a similar fashion as in the parameter estimation problem. We state the output as a function of the unknown input based on our population model and minimize the sum of the dierence between this quantity and given any arbitrary output data. In addition, we include a regularization term to prevent estimated input from having extreme magnitudes and oscillations. Solution of this problem requires a nite dimensional approximation, too. Thus, we state and prove theorems indicating the hypotheses to have the desired approximation and convergence results for the deconvolution problem. Finally, in addition to estimating the BrAC based on the TAC, borrowing terminology from Bayesian theory, we want to use the distribution of the random parameters to obtain credible bands for the estimated BrAC. That is, 100 percent credible band is a region surrounding the estimated BrAC signal for which the probability that the true BrAC signal lies in that region is at least . An outline of the remainder of this thesis is as follows. In Chapter 2, we derive our mathematical model for the alcohol biosensor problem. In Chapter 3, we include some preliminary results; we state all the denitions and theorems that we use in the remainder of this thesis. In Chapter 4, we formulate the parameter estimation problem, and state and prove theorems related to it. Then, in Chapter 5, we present analogous results for the deconvolution problem. That is, we formulate the 6 input estimation problem, and state and prove the corresponding theorems. In Chapter 6, we show that all the assumptions we made in Chapters 4 and 5 are satised for the alcohol biosensor model presented in Chapter 2. Our numerical results are in Chapter 7 and in Chapter 8, we present some analysis, discussion and concluding remarks. 7 Chapter 2 A Mathematical Model of Transdermal Alcohol Transport In this chapter, we derive the mathematical system of equations that we use to describe the transport of ethanol through the skin based on the rst principles physical modelling. Then, we formulate the parameter estimation and deconvolution problems for this particular model. Let t be the temporal variable and be the spatial variable. Let '(t;) represent the concentration of ethanol in units of moles/cm 2 in the epidermal layer of the skin at time t seconds and depth cm. The following equation models the diusion of ethanol through the epidermal layer of skin which does not contain any blood vessels. @' @t (t;) =D @ 2 ' @ 2 (t;); 0<<L; t> 0; (2.1) where D is the diusivity coecient and in units of cm 2 =sec and L represents the thickness of the epidermal layer. Since ethanol molecules moves from an area of higher concentration to an area of lower concentration, we assume that the amount of ethanol passing through the boundary, = 0 and =L, is proportional to the amount of ethanol of the area with higher ethanol concentration. So, we use the following two equations to describe the evaporation of ethanol on the skin surface, 8 = 0 and the transfer of ethanol from the dermal layer of the skin which is the layer with blood vessels to the epidermal layer. D @' @ (t; 0) ='(t; 0); t> 0; (2.2) D @' @ (t;L) =u(t); t> 0; (2.3) where > 0 and > 0 are the proportionality constants in the units of cm=sec andmoles=(cm secBAC=BrACunits) respectively, since the inut u(t) is in BAC=BrAC units denoting the ethanol concentration in the blood or exhaled breath. Due to the assumption that there is no ethanol in the epidermal layer at time t = 0, we have the following initial condition '(0;) = 0; 0<<L: (2.4) Finally, we assume that the alcohol biosensor measurement is proportional to the ethanol amount on the skin surface at any time, t. Since y(t) is measured in TACunits, letting > 0 to be the proportionality constant in (TACunitscm 2 )=moles, we use the following observation equation to describe this assumption, y(t) = '(t; 0); t> 0: (2.5) Now, by re-scaling the time and space variables as ^ ==L, ^ t =t= with =L=, dening new parameters ^ D =D=(L), ^ = ( )=, and letting ^ '( ^ t; ^ ) = '(t;), one can easily show that the following system is equivalent to the system (2.1)-(2.5). @ ^ ' @ ^ t ( ^ t; ^ ) = ^ D @ 2 ^ ' @^ 2 ( ^ t; ^ ); 0< ^ < 1; ^ t> 0; (2.6) ^ D @ ^ ' @^ ( ^ t; 0) = ^ '( ^ t; 0); ^ t> 0; (2.7) 9 ^ D @ ^ ' @^ ( ^ t; 1) = ^ u( ^ t); ^ t> 0; (2.8) ^ '(0; ^ ) = 0; 0< ^ < 1; (2.9) y( ^ t) = ^ '( ^ t; 0); ^ t> 0: (2.10) Now, instead of ve parameters D, L, , , as in the original system (2.1)-(2.5), we have only three parameters , ^ D, and ^ . In addition, we see from the equations (2.6)-(2.10) that aects only the time scale, and for a given u(t), it is possible to alter ^ D and ^ in a way that the system produces the same y(t). Thus, without loss of generality, we can set = 1, and obtain t = ^ t which means that we have now only two parameters, ^ D and ^ . Denoting these parameters byq = [q 1 ;q 2 ], where q 1 = ^ D, q 2 = ^ and using ' and instead of ^ ' and ^ for simplicity, the resulting system can be stated as @' @t (t;) =q 1 @ 2 ' @ 2 (t;); 0<< 1; t> 0; (2.11) q 1 @' @ (t; 0) ='(t; 0); t> 0; (2.12) q 1 @' @ (t; 1) =q 2 u(t); t> 0; (2.13) '(0;) = 0; 0<< 1; (2.14) y(t) ='(t; 0); t> 0; (2.15) which is an initial-boundary value problem for one dimensional diusion equation with input and output on the boundary. Assume that we have the training dataf~ u j;i g i1; j=0;i=1 ,f~ y j;i g i; j=1;i=1 that we use to estimate the disrtibution of the random parameters. We assume that we can observe the average behavior of the population at any time. So, to formulate the estimation problem, rst we obtain the discrete-time version of the system (2.11)-(2.15) and obtain the corresponding average behavior of the population. To do this, we use the dataf~ u j;i g i1; j=0;i=1 and obtain the corresponding output 10 f y j;i g i; j=1;i=1 which are some functions of the parameters q = [q 1 ;q 2 ] implying them to be random, too. Then, we will integrate each of them against the true unknown pdf f 0 to obtain the desired corresponding average behavior. Let us represent the solution of the system by ' i (t j ;) that is obtained by usingf~ u l;i g j1 l=0 . So, y j;i = Z Q ' i (t j ; 0)f 0 (q)dq: (2.16) Then, we follow a least-squares approach and formulate the estimation problem as f = arg min f2F(Q) X i=1 i X j=0 j~ y j;i y j;i j 2 ; (2.17) whereF(Q) is the admissible set of pdfs dened on Q. To solve this problem in practice, we need nite dimensional approximation whose details can be found in Chapter 4. Then, by using the estimated pdf, we form our population model which does not include the random parameters q = [q 1 ;q 2 ] and can be thought of as a model describing the average behaviour of the process for the whole population. Then, we use this model to formulate the deconvolution problem. Assume that we are given only some outputf^ y k g K k=1 of the system. We write the output of the system by using the discrete-time population model as a function of the unknown input u =fu j g K1 j=0 and denote it byfy k (u)g K k=1 . Then, we formulate the deconvolution problem as u = arg min u2U K X k=1 jy k (u) ^ y k j 2 +jjujj 2 ; (2.18) whereU is a compact set of the admissible inputs andjjjj is standing for regularization. The details for the deconvolution problem can be found in Chapter 5. As it is mentioned in Chapter 1, there are many uncertainty sources such as physiological factors of individuals, environmental conditions, biosensor being used that aect the TAC measurements. 11 The rationale behind having parameters to be random is not only to account for all these factors and still use a simple rst principles physical modeling, but also to by-pass the calibration phase which is inevitable when a deterministic approach is taken. However, with this probabilistic approach, the parameter estimation problem which is performed oine, takes the place of the calibration phase of a deterministic approach. Moreover, we can use the population model that is formed based on the estimated distribution of the parameters to estimate BAC/BrAC of an individual along with credible bands given their TAC, even though some BAC/BrAC-TAC data belong to that particular individual was not included in the training data. Therefore, the framework we present here both eliminates the necessity of a calibration phase and accounts for all factors introducing uncertainty to the system. 12 Chapter 3 Preliminaries In this chapter, we state the denitions and theorems that we use in the remainder of this thesis. The proofs for semigroup theory can be found in [5, 20, 26, 33], for Sobolev spaces in [1, 16], and for spline polynomials in [29]. Let X be a real or complex Banach space andfT (t) : 0t<1g be a family of operators in L(X), the bounded linear operators from X to X. Denition 3.1. The familyfT(t) : 0t<1g is called a strongly continuous or C 0 -semigroup in X if (i) T (0) =I, (ii) T (t +s) =T (t)T (s) for every t;s 0, (iii) for every x2X, the mapping t!T (t)x is continuous on [0;1). If only (i) and (ii) above are satised, we callfT (t) : 0t<1g a semigroup. Denition 3.2. The family T (t) is called an analytic semigroup in X if it is a C 0 -semigroup and, for each x2X, t!T (t)x is real analytic on (0;1). Denition 3.3. Let T (t) be a C 0 -semigroup. The innitesimal generator A of T (t) is dened by dom(A) =fx2X : lim t!0 (T (t)xx)=t existsg and 13 Ax = lim t!0 (T (t)xx)=t for x2dom(A). Theorem 3.1. For each C 0 -semigroup T (t) there exist real numbers M 1 and !2R such that jT (t)jMe !t for t 0. It is written that A2 G(M;!) when a linear operator A is the innitesimal generator of a C 0 -semigroup satisfying such an exponential growth. For M = 1 and ! = 0, A2G(1; 0) is the inniteasimal generator of the contraction semigroup, i.e.,jT (t)xT (t)yjjxyj. Theorem 3.2. Let T (t) be a C 0 -semigroup with innitesimal generator A. Then, T (t)x = lim n!1 I t n A n x for each x2X and the limit is uniform in t on compact subsets of [0;1). Theorem 3.3. Let T(t) be a C 0 -semigroup with innitesimal generators A. Then for each x2dom(A), the map t7!T (t)x is continuously dierentiable on (0;1), T (t)x2dom(A) for each t, and d dt T (t)x =AT (t)x =T (t)Ax: Denition 3.4. Let X be a normed space and let A :dom(A)!X be a linear operator with dom(A)X. The resolvent set (A) of A is the set of all 2C for which the range of the linear transformation IA :dom(A)!X is dense and and IA has a continuous inverse on this set. For any 2(A), the resolvent operator is denoted by R (A) = (IA) 1 . Theorem 3.4. Let A2G(M;!) be the innitesimal generator for T (t). Then, R (A)x = Z 1 0 e t T (t)x dt for any wirh Re>! and all x2X. 14 Theorem 3.5 (Hille-Yosida). For M 1, !2R, we have A2G(M;!) if and only if 1. A is closed and densely dened, i.e. dom(A) =X. 2. For real >!, we have 2(A) and R (A) satises jR (A) n j M (!) n ; n = 1; 2;:::: Denition 3.5. A linear operatorA inX is called dissipative if for everyx2dom(A) there exists an element f x 2ff2X :jfj 2 =jxj 2 =f(x)g such that Ref x (Ax) 0. Theorem 3.6 (Lumer-Philips). Suppose A is a linear operator in a Hilbert space X. 1. If A is densely dened, A!I is dissipative for some real !, and R( 0 IA) =X for some 0 with Re 0 >!, then A2G(1;!). 2. If A2G(1;!), then A is densely dened, A!I is dissipative and R( 0 IA) =X for all 0 with Re 0 >!. Denition 3.6. Let V be a complex vector space. The map :VV !C is called sesquilinear if 1. (u 1 +u 2 ;v 1 +v 2 ) =(u 1 ;v 1 ) +(u 1 ;v 2 ) +(u 2 ;v 1 ) +(u 2 ;v 2 ), 2. (u 1 ;v 1 ) = (u 1 v 1 ), for all u 1 ;u 2 ;v 1 ;v 2 2V and all ;2C. Denition 3.7. Lax-Milgram Let V ,!H ,!V be a Gelfand triple. Let :VV !C be a sesquilinear form satisfying 1. j(; )j jj V j j V for all and in V , 2. there exists a constant > 0 such thatj(;)jjj 2 V . 15 Then,A :V !V given by(; ) =<A; > V ;V is a linear (topological) isomorphism between V and V . Moreover, A 1 is continuous from V to V withjA 1 j L(V ;V ) 1 . Theorem 3.7 (Trotter-Kato). Let X and X N be Hilbert spaces such that X N X. Let P N : X!X N be an orthogonal projection of X onto X N . Assume P N x!x as N!1 for all x2X. Let A N , A be innitesimal generators of C 0 semigroups S N (t), S(t) on X N , X, respectively satisfying 1. there exists M, ! such thatjS N (t)jMe !t for each N, 2. there existsD dense inX such that for some, (IA)D is dense inX andA N P N x!Ax for all x2D. Then, dor each x2X, S N (t)x!S(t)x uniformly in t on compact intervals [0;T ]. Theorem 3.8. Replace (2) in the above theorem by (3) dened as 3. There exists 2(A)\ 1 N=1 (A N ) with Re()>! so that R (A N )P N x!R (A)x for each x2X. Then, under this and the remaining conditions in the above theorem, the conclusions also hold. Theorem 3.9 (Morrey's Inequality). Assume n < p1. Then there exists a constant C, depending only on p and n, such that jjujj C 0; (R n )Cjjujj W 1;p (R n ) for all u2C 1 (R n ), where := 1n=p. Theorem 3.10 (Young's Inequality). Let a and b be nonnegative real numbers and p;q2 (0;1) such that 1 p + 1 q = 1. Then, ab a p p + b q q : 16 For p =q = 2, we have ab a 2 2 + b 2 2 ; and for an > 0 we have the so called \Peter-Paul Inequality" ab a 2 2 + b 2 2 : Theorem 3.11 (Cauchy-Schwarz Inequality). Given u;v2 V an inner product space with the inner product <;>, we have j<u;v>jjjujjjjvjj; where the normjjjj is induced from the inner product <;>. Theorem 3.12 (Schmidt Inequality). If p n (x) is a polynomial of degree n = 1; 2; 3, then Z b a [Dp n (x)] 2 dx 4k n (ba) 2 Z b a [p n (x)] 2 dx; where k 1 = 3, k 2 = 15, and k 3 = 1 2 (45 + p 1605) 42:6. 17 Chapter 4 Parameter Estimation Problem In this chapter, rst we state parameter estimation problem in much the same way as it is in standard linear regression in a general setting. That is, each data point is an observation of the mean population behavior plus random error. We then formulate the estimation problem as an optimization problem over the space of feasible distributions for the random parameters. The objective of the optimization problem is to minimize prediction error in the form of the dierence between observed data and the expectation of the output of the model. Since the underlying state equation is innite dimensional, in practice, solving the corresponding estimation problem requires nite dimensional approximation. Naturally, this requires us to prove that under appropriate conditions the solutions of the nite dimensional approximating problems converge to the solution of the original innite dimensional problem. After proving a theorem indicating the assumptions for such a convergence result, we show that there exists a broad class of problems for which the dynamics dened in terms of bounded and coercive sesquilinear forms in a Gelfand triple setting satises the hypothesis of that theorem. Also, the partial dierential operators that appear in our model take the form of abstract parabolic or hyperbolic operators with damping. When formulated abstractly in a Gelfand triple setting, these operators are examples of what are known as regularly dissipative operators and can often be shown to generate holomorphic semigroups. 18 4.1 Estimation of Random Discrete Time Dynamical Systems We consider the family of (in general, innite dimensional) discrete or sampled time initial value problems given by x j+1;i =g(t j ;x j;i ;u i ;q); j = 0;:::;n i ; i = 1; 2;:::;m; (4.1) x 0;i =x 0;i (q); i = 1; 2;:::;m; (4.2) describing the dynamics of a process common to the entire population, where for j = 0;:::;n i , t j = j, with > 0 the length of the sampling interval. In addition, we assume that we can observe some function of the solutions of (4.1)-(4.2), x j;i , as given by the output equation y j;i =y(t j ;x 0;i ;u i ;q) =C(x j;i ;x 0;i ;u i ;q); j = 0;:::;n i ; i = 1; 2;:::;m: (4.3) In equations (4.1)-(4.3), we assumeq2Q, whereQ is a subspace ofR q and compact with respect to some metric d Q . Here, Q is the set of admissible parameters, and the values of the parameters are specic to each individual in the population. Therefore, assuming that the parameters are random variables, the objective is to estimate their (joint) distribution based on the data sampled from the population. For this purpose, we assume that the distribution of these random vectors is described by the joint pdf f 0 2F(Q), whereF(Q) represents a set of feasible pdfs on Q. In order to state the statistical model that is used as the basis for the estimation of the distribution of the random parameters, we assume that the observed data points can be represented 19 by the mean output of the model plus random error. Thus, we assume that we have random observations of the process given by a random array with components V j;i =E f0 [y j;i ] +" j;i ; j = 0;:::;n i ; i = 1;:::;m; (4.4) where in (4.4), " j;i ; j = 0;:::;n i ; i = 1;:::;m, represent measurement noise and are assumed to be independent and identically distributed with mean 0 and common variance 2 . For f2F(Q), dene v i (t j ;f) =E f [y(t j ;x 0;i ;u i ;q)] = Z Q C(x j;i ;x 0;i ;u i ;q)f(q)dq; (4.5) the mean behavior at time t j , j = 0;:::;n i , if qf. The estimation problem is to estimate the pdf,f 0 , using a least squares approach, the estimator is given by f = arg min f2F(Q) J(f;V ) = arg min f2F(Q) m X i=1 ni X j=0 (V j;i v i (t j ;f)) 2 ; (4.6) with the corresponding point estimate, ^ f , taking the form ^ f = arg min f2F(Q) J(f;V ) = arg min f2F(Q) m X i=1 ni X j=0 (v j;i v i (t j ;f)) 2 ; (4.7) where the v i (t j ;f) are as given in (4.5) and v j;i are the realizations of the random observations V j;i , j = 0; 1;:::;n i , i = 1; 2;:::;m. Solving the optimization problem given in (4.6) typically requires nite dimensional approxi- mation of the dynamical system given in (4.1)-(4.2), and the parametrization of the feasible set of pdfs, F(Q). Indeed, we assume that the set of pdfs, F(Q), is parametrized by a vector of 20 parameters 2 , where R r is a set of feasible parameters. In this case, we denote the set of pdfs byF (Q). We approximate the estimation problem given in (4.6) by a sequence of nite dimensional estimation problems by replacing v i (t j ;f) with a nite dimensional approximation v N i (t j ;f). We obtain f N = arg min f2F (Q) J N (f;V ) = arg min f2F (Q) m X i=1 ni X j=0 (V j;i v N i (t j ;f)) 2 ; (4.8) with the corresponding point estimate ^ f N given by ^ f N = arg min f2F (Q) J N (f;V ) = arg min f2F (Q) m X i=1 ni X j=0 (v j;i v N i (t j ;f)) 2 : (4.9) We note that ultimately, we will want to dispense with the assumption thatF(Q) is parame- trized by the nite dimensional parameter 2 and actually estimate the shape of f directly. In this case,F(Q) will also have to be approximated or discretized with the level, or dimension of the parametrization having to grow in order to establish convergence. Under our current assumption on the set of admissible pdfs, analogous to theorem 5.1 in [4], we have the following convergence result for the f N 's. Theorem 4.1. Let R r be compact. If A. The maps on , 7!f(q;), for almost every q2Q, and 7!J N (f(;);V ), for all N and f2F (Q) are continuous, B. For any sequence of densities f N 2F (Q) with lim N!1 f N (q) = f(q), a.e. q2 Q, for somef2F (Q), we havev N i (t j ;f N ) converging tov i (t j ;f) for alli2f1;:::;mg andj2f0;:::;n i g as N;k!1, 21 then it will follow that there exist solutions f N to the estimation problems overF (Q), given in (4.8), and there exists a subsequence of the f N 's that converges to a solution f of the estimation problem overF (Q) given in (4.6). Proof. Finding the solution to the problem in (4.8) is equivalent to nding the parameters 2 such that J N (f;V ) is minimized. Since is a compact set and the map ! J N (f(;);V ) is continuous for all N by (A), a solution f N to the estimation problem (4.8) overF (Q) exists. Next, letff N gF (Q) be any sequence with lim N!1 f N (q) = f(q), a.e. q2 Q for some f2F (Q) and consider that jJ N (f N ;V )J(f;V )j =j m X i=1 ni X j=0 (V j;i v N i (t j ;f N )) 2 m X i=1 ni X j=0 (V j;i v i (t j ;f)) 2 j m X i=1 ni X j=0 j2V j;i v i (t j ;f) +v N i (t j ;f N )j jv i (t j ;f)v N i (t j ;f N )j M m X i=1 ni X j=0 jv i (t j ;f)v N i (t j ;f N )j; for some M > 0, since v i (t j ;f) and v N i (t j ;f) are uniformly bounded for all i2f1;:::;mg and j2f0;:::;n i g and f2F (Q) as a result of the convergence assumption in (B). Then, again by (B), we obtain J N (f N ;V )!J(f;V ); (4.10) as N!1. On the other hand, since f N =f(; N ), where N 2 , is the minimizer of J N (f;V ), we have J N (f N ;V )J N (g;V ); (4.11) 22 for all g =g(;)2F (Q) and N = 1; 2;:::. Sincef N g , compact, there exists a subsequence N k with N k ! as N k !1. Thus, taking the limit as N k !1 in (4.11) with N replaced by N k , and using (4.10), we obtain J(f ;V )J(f;V ); (4.12) for all g2F (Q), where f =f(; ). Thus, (4.12) implies that f is a solution of estimation problem given in (4.6) overF (Q). 4.2 Abstract Parabolic Systems with Unbounded Input and Output Let V and H be Hilbert spaces with V ,!H, i.e. V is continuously and densely embedded in H. By identifying H with its dual H , we obtain the Gelfand triple V ,!H ,!V . Let <;> H denote the H inner product andjj H ,jjjj V denote norms on H and V , respectively. For q2Q, let a(q;;) :VV !C be a sesquilinear form that has the following properties. Assumption 4.1. (Boundesness) There exists a constant 0 > 0 such thatja(q; 1 ; 2 )j 0 jj 1 jj V jj 2 jj V , 1 ; 2 2V , q2Q. Assumption 4.2. (Coercivity) There exist constants 0 2 R and 0 > 0 such that a(q; ; ) + 0 j j 2 H 0 jj jj 2 V , 2V , q2Q. Assumption 4.3. (Measurability) For all 1 ; 2 2V , the map q7!a(q; 1 ; 2 ) is measurable on Q with respect to all measures dened in terms of the densities inF (Q), where R r is the set of feasible parameters. Assumption 4.4. (Continuous Dependence) For all 1 ; 2 2V andq;q 0 2Q, we haveja(q; 1 ; 2 ) a(q 0 ; 1 ; 2 )jd Q (q;q 0 )jj 1 jj V jj 2 jj V . 23 Assume further thatb(q);c(q) are respectively and dimensional vectors inV with the maps q7!<b(q); > V ;V and q7!<c(q);'> V ;V measurable on Q for ;'2V , where <;> V ;V denotes the duality pairing between V and V . We consider the system which is written in weak form as h _ x; i V ;V +a(q;x; ) =hb(q); i V ;V u; 2V; x(0) =x 0 2H; y(t) = Z T 0 w(t;s) c(q);x (t) (s) V ;V ds; (4.13) where T > 0, ' (t) (s) = '(s) [0;t] (s), s2 [0;T], and w(t;s) is a positive weight function. For u2L 2 ([0;T ];R ), it can be shown that (4.13) has a unique solution (see [23, 33]) x2W (0;T ) := f : 2 L 2 ([0;T];V ); _ 2 L 2 ([0;T];V )g C([0;T];H) which depends continuously on u2 L 2 ([0;T ];R ). It follows that y2L 2 ([0;T ];R ). Forq2Q, under the Assumptions 4.1 and 4.2, the sesquilinear form a(q;;) denes a bounded linear operator A(q) :V !V by <A(q) 1 ; 2 > V ;V =a(q; 1 ; 2 ) where 1 ; 2 2V . It can be shown further that (see [5, 6, 33])A(q) restricted to the setDom(A(q)) =f2V :A(q)2Hg is the innitesimal generator of a holomorphic or analytic semigroup of bounded linear operators on H. Moreover, this semigroup can be restricted to be a holomorphic semigroup on V and extended to be a holomorphic semigroup on V by appropriately restricting or extending the domain, Dom(A(q)), of the operator A(q) (see[6, 33]). For q2 Q, dene the operators B(q) : R ! V byhB(q)u; i V ;V =hb(q); i V ;V u and C(q) : L 2 ([0;T];V )! R by C(q)' = R T 0 w(t;s)hc(q);'(s)i V ;V ds, for u2 R , 2 V , and '2L 2 ([0;T ];V ), and rewrite the system in (4.13) as 24 _ x(t) =A(q)x(t) +B(q)u(t); x(0) =x 0 ; y(t) =C(q)x (t) ; t> 0: (4.14) The mild solution of (4.14) is given by the variation of constants formula as x(t;q) =e A(q)t x 0 + Z t 0 e A(q)(ts) B(q)u(s)ds; t 0: (4.15) Moreover, since the semigroupfe A(q)t :t 0g is analytic it follows that y(t;q) =C(q)x (t) (q) = Z T 0 c(q);x (t) (s;q) V ;V ds; t 0: (4.16) is well dened. 4.2.1 The Discrete Time Formulation Now let > 0 be a sampling time and consider zero-order hold inputs of the form u(t) = u j , t2 [j; (j + 1)), j = 0; 1; 2;::: . Setting x j =x(j), for j = 0; 1; 2;:::, (4.15) yields that x j+1 = ^ A(q)x j + ^ B(q)u j ; y j = ^ C(q)x j ; j = 0; 1; 2;::: (4.17) where now we letx 0 2V . Here, again by the properties of the analytic semigroup (see [26, 33]), we havefe A(q)t :t 0g, x j 2V , ^ A(q) =e A(q) 2L(V;V ) and ^ B(q) = R 0 e A(q)s B(q)ds2L(R ;V ). 25 The operator ^ C(q) appearing in (4.17) is dened by recalling (4.16). By choosing the weight function as w(t;s) = 1 [j;(j+1)) (s), we set ^ C(q)x j =C(q)x j [j;(j+1)) from which we obtain ^ C(q)x j =<c(q);x j > V ;V : (4.18) Now, in light of Assumption 4.2 by making the change of variables z(t) =e 0t x(t) and v(t) = e 0t u(t), without loss of generality we may assume that the operator A(q) is invertible with bounded inverse. Thus we have that ^ B(q) = R 0 e A(q)s B(q)ds = A(q) 1 e A(q)s B(q) 0 = ( ^ A(q) I)A(q) 1 B(q)2L(R ;V ). It follows that the recurrence given in (4.17) is a recurrence in V with ^ A(q)2L(V;V ), ^ B(q)2L(R ;V ), and ^ C(q)2L(V;R ). 4.2.2 Systems with Boundary Input Of primary interest to us here are systems of the form (4.13) or (4.14) where the input u is on the boundary of the spatial domain. The theory developed in [10] and [27] tells us how in this case to dene the input operator B(q) and the notion of a mild solution upon which our approach is based. Let W be a Hilbert space which is densely and continuously embedded in H. Let (q)2L(W;H) and (q)2L(W;R ) and assume that Dom(A(q))N((q))W, (q) is surjective and (q) = A(q) on Dom(A(q)). We then consider the system with input on the boundary given by _ x(t) = (q)x(t); (q)x(t) =u(t); t> 0; y(t) =C(q)x (t) ; t> 0; x(0) =x 0 : (4.19) 26 In [10], Curtain and Salamon dene a solution to the system (4.19) for the case where u2 C([0;T];R ) and x 0 2W with (q)x 0 =u(0), to be a function x2C([0;T];W)\C 1 ([0;T];H) that saties (4.19) at every t2 (0;T). The operator A(q) densely dened implies that it has an adjoint operatorA(q) :Dom(A(q) )H!H which is also densely dened and closed. Dening Z to be the Hilbert spaceDom(A(q) ) endowed with the graph Hilbert space norm associated with A(q) , Z will be continuously and densely embedded in H. So, the Gelfand triple Z ,!H ,!Z is obtained where Z =Z represents the dual space of Z . By denition A(q) 2L(Z ;H) and therefore,A(q)2L(H;Z). It follows that the semigroupfe A(q)t :t 0g can be uniquely extended to a holomorphic semigroup on Z with innitesimal generator A(q) :HZ!Z, the extension A(q) to H dened via the duality pairing < A(q) ; > Z;Z =< ;A(q) > H , for 2 H, and 2Z =Dom(A(q) ). For each q2Q, let + (q)2L(R ;W) be any right inverse of (q)2L(W;R ), and dene the operatorB(q)2L(R ;Z) byB(q) = ((q)A(q)) + (q). It is not dicult to show that B(q) is well dened (i.e. it does not depend on the particular choice of the right inverse + (q)). Then for any x 0 2H and u2L 2 ([0;T ];R ), the mild solution, x2C([0;T ];Z), of the initial boundary value problem in (4.19) is the Z-valued function given by x(t) =e A(q)t x 0 + Z t 0 e A(q)(ts) B(q)u(s)ds; t 0: (4.20) It is shown in [10] that if (4.19) has a solution, then it is given by (4.20) where x2C([0;T ];H)\ H 1 ((0;T);Z) and moreover, we have that the estimate given byj R t 0 e A(q)(ts) B(q)u(s)dsj H kjjujj L2([0;T];R ) holds. We note that if in fact we have that WV , which is often the case (for example, in a one dimensional diusion equation with either Neumann or Robin boundary input (see our examples in Chapter 7 below), but may not be the case if, for example, the boundary input is Dirichlet), then in the above formulation we may take Z = V and Z = V . In this case it will follow 27 that B(q) = ((q)A(q)) + (q)2L(R ;V ) and consequently that the theory presented at the beginning of Section 4.2, and in particular, the discrete time theory presented in Section 4.2.1, applies. For ease of exposition, we will assume that this is indeed the case for what follows below. We note that all the results continue to follow in the more general case where Z =Dom(A(q) ). It then follows that ^ A(q) =e A(q) 2L(V;V ) and that ^ B(q) = R 0 e A(q)s dsB(q)2L(R ;V ) and therefore that ^ B(q) = Z 0 e A(q)s B(q)ds =A(q) 1 e A(q)s B(q) 0 = ( ^ A(q)I)A(q) 1 B(q); (4.21) and ^ C(q)2L(V;R ) as in (4.18). Note that now we have ^ B(q) = (I ^ A(q)) + (q) + Z 0 e A(q)s ds(q) + (q)2L(R ;V ); (4.22) and if + (q) can be chosen so that R( + (q))2N((q)), then the expression in (4.22) becomes ^ B(q) = (I ^ A(q)) + (q). Then, if x 0 = 02H, y i is given by y i = i1 X j=0 ^ C(q) ^ A(q) ij1 ^ B(q)u i = i1 X j=0 K i;j u j ; i = 1; 2;:::; (4.23) where the operator K i;j = ^ C(q) ^ A(q) ij1 (I ^ A(q)) + (q) appearing in (4.23) is the gain that represents the contribution of the j th input channel to the i th output channel. 28 4.3 Random Regularly Dissipative Operators and Their Associated Semigroups In this section, we summarize the key ideas from the framework developed in [19] and [30] which are central to our approach. We assume thatq is a p-dimensional random vector whose support is Q p i=1 [a i ;b i ] where1 < < a i < b i < <1 for all i = 1; 2;:::;p. Letting ~ a = [a i ] p i=1 , ~ b = [b i ] p i=1 and let R r for some r be closed and bounded. We assume that the distribution of q can be represented by a measure =(~ a; ~ b; ~ ), where ~ 2 . Let a(q;;) be a sesquilinear form satisfying Assumptions 4.1-4.3 given in Section 4.2, where the assumed measurability is with respect to all of the measures =(~ a; ~ b; ~ ). Dene the Bochner spacesV =L 2 (Q;V ) andH :=L 2 (Q;H). The assumptions from Section 4.2 on the spaces V and H guarantee that the spaces V,H and V form the Gelfand triple V,!H,!V (see [19]) whereV is identied with L 2 (Q;V ) andH is identied with its dual H =L 2 (Q;H ). For~ a = [a i ] p i=1 , ~ b = [b i ] p i=1 satisfying1 < < a i < b i < <1 for all i = 1; 2;:::;p, and ~ 2 , set = (~ a; ~ b; ~ ). Then we dene the()-averaged sesquilinear formsa(;;) :VV!C (note, the spacesV,H, andV now of course depend on , but our notation here we will not explicitly show this dependence unless clarity demands it) by a(;'; ) = Z Q a(q;'(q); (q))d(q;) =E () [a(q;'(q); (q))]; (4.24) where '; 2V and = (~ a; ~ b; ~ ). 29 Now, by using the Assumptions 4.1-4.3 and the Cauchy-Schwarz Inequality, consider for any '; 2V that ja(;'; )j Z Q ja(q;'(q); (q))jd(q;) 0 Z Q jj'(q)jj V jj (q)jj V d(q;) 0 Z Q jj'(q)jj 2 V d(q;) 1=2 Z Q jj (q)jj 2 V d(q;) 1=2 = 0 jj'jj V jj jj V ; (4.25) and a(; ; ) + 0 j j 2 H = Z Q a(q; (q); (q))d(q;) + 0 Z Q j (q)j 2 H d(q;) = Z Q a(q; (q); (q)) + 0 j (q)j 2 H d(q;) 0 Z Q jj (q)jj 2 V d(q;) = 0 jj jj 2 V ; (4.26) which shows thata(;;) is also a bounded and coercive sesquilinear form onVV. Consequently, this sesquilinear form denes a bounded linear map A() :V!V by <A()'; > V ;V = a(;'; ) which when appropriately restricted or extended is the innitesimal generator of analytic semigroups of bounded linear operatorsfe A()t :t 0g onV,H andV (see [5, 6, 33]). We assume the positive weight functionw(t;s) is chosen so that the mapsq7!<b(q); > V ;V and q7! R T 0 w(t;s)<c(q);'(s)> V ;V ds are()-measurable for any 2V and'2L 2 ([0;T ];V) and thatjjb(q)jj V ,jjc(q)jj V are uniformly bounded for a.e. q2Q. We then deneB() :R !V andC() :L 2 ([0;T ];V)!R by B()u = Z Q hb(q); i V ;V d(q;)u; (4.27) 30 C()' = Z Q Z T 0 w(t;s)hc(q);'(s;q)i V ;V d(q;); (4.28) for u2R , 2V, and '2L 2 ([0;T ];V). With the denitions (4.24) - (4.28) of the operators A, B, and C, consider the abstract evolution system given by _ x(t) =A()x(t) +B()u(t); x(0) =x 0 2H; y(t) =C()x (t) ; t> 0; (4.29) where ' (t) (s) ='(s) [0;t] (s). The mild solution of (4.29) is then given by x(t) =T(t;)x 0 + Z t 0 T(ts;)B()u(s)ds; t 0; (4.30) whereT(t;) =fe A()t :t 0g is the analytic semigroup generated by the operatorA(). From (4.29) and (4.30), it follows that y(t) =C()T(t;)x 0 + Z t 0 C()T(ts;)B()u(s)ds; t 0: (4.31) As in Section 4.2, we obtain a discrete or sampled time version of (4.29). Now letx 0 2V , let > 0 be the sampling time, and consider zero-order hold inputs of the form u(t) =u j , t2 [j; (j + 1)), j = 0; 1; 2;:::. Settingx j =x(j) andy j =y(j), j = 0; 1; 2;:::, (4.30) and (4.31) yield x j+1 = ^ A()x j + ^ B()u j ; y j = ^ C()x j ; j = 0; 1; 2;:::; (4.32) with x 0 2 V and ^ A() = T(;)2 L(V;V), ^ B() = R 0 T(s;)B()ds2 L(R ;V), and by choosing w(t;s) = 1 [j;(j+1)) (s), we get ^ C()x j = R Q < c(q);x j > V ;V dq where ^ C()2 31 L(L 2 ([0;T ];V);R ). Note that the operators ^ A() and ^ B() are bounded sincefT(t;) :t 0g is an analytic semigroup onV,H, andV (see [5, 6, 23, 33]). IfA() : Dom(A())V ! V has bounded inverse, then ^ B() = R 0 T(s;)B()ds = ^ A() 1 T(s;)B() 0 = ( ^ A() I)A() 1 B()2L(R;V). Also, it follows thatq j 2V, j = 0; 1; 2;:::, (x 0 = 0, by assumption), and moreover, by Assumption 4.4, it is not dicult to show (using the Trotter-Kato theorem, see [20, 33]) thatx j 2C(Q;V ), j = 0; 1; 2;::: . It is shown in [19] and [30] that the solutions of systems (4.14) and (4.29) and hence (4.17) and (4.32) agree for -a.e. q2Q. It follows that y(t) =C()x(t) =E () [y(t;q)] =E () [C(q)x(t;q)]; 8t 0; (4.33) and hence, from (4.33), that y j = ^ C()x j =E () [y j (q)] =E () [ ^ C(q)x j (q)]; (4.34) where in (4.34) E () denotes expectation with respect to the measure (). 4.4 Approximation and Convergence In this section, we can now formally state our estimation problem and the sequence of nite dimensional approximating problems. We also state and prove a convergence theorem. 32 4.4.1 The Estimation Problem Assume that data of the form (f~ u j;i g ni1 j=0 ;f~ y j;i g ni j=0 ) m i=1 , has been given. Determine = (~ a ; ~ b ; ~ )2 , a compact subset of R 2p R 2p+r , ~ a = [a i ] p i=1 , ~ b = [b i ] p i=1 , which minimizes J() = m X i=1 ni X j=0 jy j;i (f~ u k;i g ni1 k=0 ;) ~ y j;i j 2 (4.35) where for i = 1; 2;:::;m, y j;i (f~ u k;i g ni1 k=0 ;) is given by (4.32) with u j = ~ u j;i , j = 0;:::;n i , i = 1; 2;:::;m, and (4.34). Recalling the assumption that for i 2 f1; 2;:::;pg, 1 < a i < b i < 1, let Q = Q p i=1 [ ; ]. Let = ([ ] p i=1 ; [ ] p i=1 ; ~ )2 , H =L 2 ( ) ( Q;H) and V =L 2 ( ) ( Q;V ). Then, for N = 1; 2;:::. Let~ a N = [a N i ] p i=1 , ~ b N = [b N i ] p i=1 be such that1< a N i <b N i <1, and let N = (~ a N ; ~ b N ; ~ N )2 . Set Q N = Q p i=1 [a N i ;b N i ],H N = L 2 ( N ) (Q N ;H),V N = L 2 ( N ) (Q N ;V ) and letW N be a nite dimensional subspace ofV N . LetI N : H!H N (or V!V N ) be a linear map dened byI N ( ) = j Q N for any 2 H (or 2 V), letP N :H N !W N denote the orthogonal projection ofH N ontoW N , and deneJ N : H!W N byJ N =P N I N . In addition, henceforth we assume that for2 , the probability measures(;) are continuous; that is (;)f(), where f(;) is a joint density for the random vectorq. Noting that in this formulation,W N is neither a subspace of H nor V, we dene the operators A N () onW N to be what are essentially the restrictions ofA() to the spacesW N . More precisely, we set A N ()v N ;w N =a(;v N ;w N ) = Z Q N a(q;v N (q);w N (q))d(q;) = Z Q N a(q;v N (q);w N (q))f(q;)dq; (4.36) where v N ;w N 2W N . 33 Dene the operatorsB N () :R!W N andC() N :W N !R by B N ()u = Z Q N b(q);v N (q) V ;V f(q;)dq u; (4.37) C N ()v N = Z Q N c(q);v N (q) V ;V f(q;)dq; (4.38) where v N 2W N , and u2R. With these denitions, we can now state the nite dimensional approximating problems. Assume that data of the form (f~ u j;i g ni1 j=0 ;f~ y j;i g ni j=0 ) m i=1 , has been given. Determine N = (~ a N ; ~ b N ; ~ N )2 , a compact subset of R 2p R 2p+r , ~ a N = [a N i ] p i=1 , ~ b N = [b N i ] p i=1 , which minimizes J N () = m X i=1 ni X j=0 jy N j;i (f~ u k;i g ni1 k=0 ;) ~ y j;i j 2 ; (4.39) where in (4.39), for i = 1; 2;:::;m,y N j;i (f~ u k;i g ni1 k=0 ;) = ^ C() N x N j;i is given by (4.32) and (4.34) with u j = ~ u j;i , j = 0;:::;n i , i = 1; 2;:::;m, x j replaced by x N j;i 2 W N , ^ A() replaced by ^ A N () =T N (;) =e A N () 2L(W N ;W N ), ^ B() replaced by ^ B N () = R 0 e A N ()s B N ()ds2 L(R;W N ), ^ C() replaced by ^ C N ()2L(W N ;R), andx 0;i replaced byx N 0;i =J N x 0;i 2W N . It follows that for i = 1; 2;:::;m, x N j+1;i = ^ A N ()x N j;i + ^ B N ()~ u j;i ; y N j;i =C N ()x N j;i j = 0; 1; 2;:::; (4.40) with the operatorsA N (),B N (), andC N () appearing in (4.40) are as they have been dened above using (4.36)-(4.38). In the following sections we prove that there exists a subsequence of solutions to the sequence of approximating problems that converges to the solution of our original estimation/optimization problem. 34 4.4.2 A Version of the Trotter-Kato Semigroup Approximation Theorem Our convergence proof is based on a version of the Trotter-Kato semigroup approximation theorem ([5, 20, 26]) that does not require the approximating spaces to be subspaces of the underlying innite dimensional state space. Banks, Burns and Cli [3] proved just such a result but unfortunately they do not state their hypotheses in terms of resolvent convergence which is what we require here. Consequently we establish the result in its requisite form here. Let ^ H be a Hilbert space with normjj and letf ^ H N g be a sequence of Hilbert spaces, each equipped with normjj N . Assume that for each N2 N, ^ W N is a closed (nite dimensional) subspace of ^ H N . Assume that the operators ^ A on ^ H, and for each N 2 N, ^ A N on ^ W N , are in G(M; 0 ) with M and 0 independent of N; that is they are the innitesimal generators of C 0 -semigroups ^ S(t) on ^ H and ^ S N (t), on ^ W N , respectively, that are uniformly (uniformly in N) exponentially bounded. (We note that if ^ A is obtained from a bounded and coercive sesquilinear form and the ^ W N 's are subspaces with ^ A N dened as the restrictions of ^ A to ^ W N , then this latter assumption is easily veried [5, 6]). Theorem 4.2. Let ^ H, ^ H N , and ^ W N be Hilbert spaces as dened above. LetI N : ^ H! ^ H N be an operator such that Im(I N ) = ^ H N andjI N zj N jzj. LetP N : ^ H N ! ^ W N be the canonical orthogonal projection of ^ H N onto ^ W N and deneJ N :=P N I N . Let ^ A2G(M; 0 ) on ^ H, and ^ A N 2G(M; 0 ) on ^ W N . Suppose that for some 0 , jJ N R ( ^ A)zR ( ^ A N )J N zj N ! 0; as N!1; (4.41) 35 for every z2 ^ H, where R ( ^ A) = (I ^ A) 1 and R ( ^ A N ) = (I ^ A N ) 1 denote respectively the resolvent operators of ^ A and ^ A N at . Then jJ N ^ S(t)z ^ S N (t)J N zj N ! 0; as N!1; (4.42) in ^ H N , for every z2 ^ H uniformly in t on compact t-intervals. Proof. First, note that for any operator B, if R (B) is the corresponding resolvent operator, then we have BR (B) =R (B)I from the denition of the resolvent operator. Now, for ease of exposition and without loss of generality, let 0 = 0. Then, since ^ S(t)R ( ^ A) and ^ S N (t)R ( ^ A N ) are both strongly dierentiable int, by using the above identity for the resolvent operator, we have d dt ^ S(t)R ( ^ A) = ^ A ^ S(t)R ( ^ A) = ^ S(t) ^ AR ( ^ A) = ^ S(t)[R ( ^ A)I]: (4.43) In addition, if T(t) represents the corresponding semigroup operator for B, we know that B commutes with both R (B) and T(t). Also, R (B) and S(t) commutes with each other, as well. Now, using these facts besides the above identity for resolvent operators, analogous to (4.43) , we obtain d ds [ ^ S N (ts)R ( ^ A N )J N ^ S(s)R ( ^ A)] = ^ S N (ts)[J N R ( ^ A)R ( ^ A N )J N ] ^ S(s): (4.44) Then, since ^ S N (ts)R ( ^ A N )J N ^ S(s)R ( ^ A)j s=t s=0 =R ( ^ A N )[J N ^ S(t) ^ S N (t)J N ]R ( ^ A); (4.45) 36 (4.44) and (4.45) imply that R ( ^ A N )[J N ^ S(t) ^ S N (t)J N ]R ( ^ A) = Z t 0 ^ S N (ts)[J N R ( ^ A)R ( ^ A N )J N ] ^ S(s)ds: (4.46) Equation (4.46) andj ^ S N (ts)j N M (recall 0 = 0), for any u2 ^ H, yield jR ( ^ A N )[J N ^ S(t) ^ S N (t)J N ]R ( ^ A)uj N M Z t 0 j[J N R ( ^ A)R ( ^ A N )J N ] ^ S(s)uj N ds: (4.47) By (4.41), we know that the integrand in (4.47) converges to 0 for a xed s, and also since jJ N j N 1 by its denition and using bounds for resolvent operators as shown in [33] and the fact thatj ^ S(s)jM (becaue of choosing 0 = 0), we have j[J N R ( ^ A)R ( ^ A N )J N ] ^ S(s)uj N 2M 1 j ^ S(s)uj 2M 1 M juj; (4.48) whereM 1 = 1 +M= 0 with 0 is as in (4.26). Therefore, by the Lebesque Dominated Convergence Theorem, the right-hand side of (4.46) converges to 0 asN!1, where the convergence is uniform in t on compact t-intervals. Letting v =R ( ^ A)u, and using the fact that Dom( ^ A) is dense in ^ H, we have that jR ( ^ A N )[J N ^ S(t) ^ S N (t)J N ]vj N ! 0; as N!1; (4.49) for all v2 ^ H. Then, sincej ^ S N (t)jM, for all N, (4.41) implies that jR ( ^ A N ) ^ S N (t)J N v ^ S N (t)J N R ( ^ A)vj N =j ^ S N (t)[R ( ^ A N )J N vJ N R ( ^ A)v]j N ! 0; (4.50) 37 and similarly,j ^ S ( t)jM and (4.41) imply that jR ( ^ A N )J N ^ S(t)vJ N ^ S(t)R ( ^ A)vj N =j[R ( ^ A N )J N J N R ( ^ A)] ^ S(t)vj N ! 0: (4.51) Combining (4.50), (4.51), and the triangle inequality we get jR ( ^ A N )[ ^ S N (t)J N J N ^ S(t)]v + [J N ^ S(t) ^ S N (t)J N ]R ( ^ A)vj N ! 0; (4.52) as N!1. Then, because of (4.49), and again by the triangle inequality, we obtain that j(J N ^ S(t) ^ S N (t)J N )R ( ^ A)vj N ! 0; as N!1: (4.53) Letting w =R ( ^ A)v, we have w2Dom( ^ A 2 ); and since Dom( ^ A 2 ) is dense in ^ H, it follows from (4.41), (4.52) and (4.53) that j ^ S N (t)J N zJ N ^ S(t)zj N ! 0; as N!1; (4.54) for all z2 ^ H uniformly in t on compact t-intervals. 4.4.3 Application to the Density Estimation Problem Letf N g and 2 be such that f N (q)! f(q), for almost every q2 Q, where f N (q) = f(q; N ) and f(q) =f(q;). Let H, V,H N ,V N ,W N ,I N : H!H N ,P N :H N !W N , and J N : H!W N be as they were dened earlier. SetA =A() and consider it to be an operator on H and V by extendingf(;), which is dened on Q, to Q by setting it equal to zero on QnQ and letA N =A N ( N ). Then it follows from Assumptions 4.1-4.3 thatA is in G(M; 0 ) on H andA N is in G(M; 0 ) onH N with M and 0 independent of N. 38 In the statement of Theorem 4.2, set ^ H = H, ^ H N =H N , ^ W N =W N , P N =J N , ^ A =A, and ^ A N =A N . To apply Theorem 4.2 and conclude that in this case, (4.42) holds, we need only verify (4.41). In order to do this, we require the following two additional assumptions Assumption 4.5. There exist positive real numbers and such that for any 2 , we have 0< f(q;)<1 for ()-a.e. q2 Q. Assumption 4.6. For all w2 V, there exists u N 2W N such thatjju N I N wjj V N ! 0 as N!1. We are now able to prove the following theorem. Theorem 4.3. Let Assumptions 4.1 - 4.6 be satised and letf N g;2 be such that f N (q)! f(q), for almost every q 2 Q, where f N (q) = f(q; N ) and f(q) = f(q;). Then, with the denitions above, the conditions of Theorem 4.2 (and in particular the resolvent convergence specied in (4.41)) are satised. Consequently, it follows that jjT N (t; N )J N zJ N T(t;)zjj H N! 0; as N!1; (4.55) for every z2 H, uniformly in t on compact t-intervals where T N =fT N (t; N ) : t 0g is the semigroup onH N given byT N (t; N ) =e A N t =e A N ( N )t andT =fT(t;) :t 0g is the semigroup onH and H given byT(t;) =e At =e A()t . Proof. First, note that if we can show resolvent convergence for everyz2 V, then since V is dense in H, andJ N R 0 (A) andR 0 (A N )J N are uniformly bounded, the desired resolvent convergence for every z2 H will have been demonstrated. In what follows, for any = (~ a; ~ b; ~ )2 , f(;) is dened on Q = Q p i=1 [a i ;b i ], but it can be extended to be dened on Q by setting it equal to zero on QnQ. We will use this fact frequently below without further remark. Let z2 V and dene w =R 0 (A)z, and w N =R 0 (A N )J N z. Suppose also that u N 2W N be as in Assumption 4.6 for w =R 0 (A)z. 39 Then, by triangle inequality, we have jjJ N ww N jj V NjjJ N wu N +u N w N jj V N jjJ N wu N jj V N +jju N w N jj V N: (4.56) Let us show that the rst term on the right-hand side of (4.56) goes to 0 as N!1. Keep in mind that we haveV N ,!H N ,! (V N ) and the embedding maps are uniformly bounded in N, i.e.,jj (V N ) k 1 jj H N andjj H Nk 2 jjjj V N . Let v N 2V N and consider by using Riesz Representation Theorem that jjP N v N jj 2 V N = P N v N ;P N v N V N jP N v N j (V N ) jjP N v N jj V N k 1 jP N v N j H NjjP N v N jj V N k 1 jv N j H NjjP N v N jj V N k 1 k 2 jjv N jj V NjjP N v N jj V N; (4.57) which implies thatjjP N v N jj V N k 1 k 2 jjv N jj V N and hencejjP N jj V N k 1 k 2 for every N2 N. Now, by using this result we obtain jjJ N wu N jj V N =jjP N I N wP N u N jj V Nk 1 k 2 jjI N wu N jj V N; (4.58) and by Assumption 4.6, we getjjJ N wu N jj V N! 0 as N!1. Now, it remains to show that jju N w N jj V N! 0 as N!1. Let z N =w N u N . Then, since w N 2W N V N , a( N ;w N ;z N ) = A N w N ;z N H N = ( 0 IA N )R 0 (A N )J N z;z N H N 0 w N ;z N H N = J N z;z N H N 0 w N ;z N H N : (4.59) 40 Also, since w2Dom(A), a(;w;I N + z N ) = D Aw;I N + z N E H = D ( 0 IA)R 0 (A)z;I N + z N E H 0 D w;I N + z N E H = D z;I N + z N E H 0 D w;I N + z N E H ; (4.60) whereI N + denotes the Moore-Penrose generalized inverse [8] ofI N . We note that for 2W N , I N + is the function in H that agrees with on Q N and is zero on QnQ N . Then, from (4.59) and (4.60), we obtain a( N ;w N ;z N )a(;w;I N + z N ) = J N z;z N H N 0 w N ;z N H N D z;I N + z N E H + 0 D w;I N + z N E H : (4.61) Recalling Assumptions 4.1 and 4.2 for the form a(;;) on V V , let ~ 0 , ~ 0 , ~ 0 denote the boundedness and coercivity coecients for the forms a(;;). Then, using boundedness and coercivity of the sesquilinear form, Assumptions 4.5 and 4.6, the Peter-Paul Inequality, and the continuous embeddings of the space V in the space H (i.e. that there exist a constant k such that jj H kjjjj V ) and (4.61), for any "> 0, we obtain ~ 0 jjz N jj 2 V Na( N ;z N ;z N ) + ~ 0 jz N j 2 H N =a( N ;w N ;z N )a( N ;u N ;z N ) + ~ 0 jz N j 2 H N =a( N ;w N ;z N )a(;w;I N + z N ) +a(;w;I N + z N )a( N ;u N ;z N ) + ~ 0 jz N j 2 H N = J N z;z N H N ~ 0 w N ;z N H N D z;I N + z N E H + ~ 0 D w;I N + z N E H + Z Q (a(q;w;I N + z N )f(q)a(q;I N + u N ;I N + z N )f N (q))dq + ~ 0 jz N j 2 H N = Z Q D z;I N + z N E H (f N (q)f(q))dq 41 + ~ 0 Z Q ( D w;I N + z N E H f(q) D I N + u N ;I N + z N E H f N (q))dq + Z Q (a(q;w;I N + z N )f(q)a(q;I N + u N ;I N + z N )f N (q))dq = Z Q D z;I N + z N E H (f N (q)f(q))dq + ~ 0 Z Q D w;I N + z N E H (f(q)f N (q))dq + ~ 0 Z Q D wI N + u N ;I N + z N E H f N (q)dq + Z Q a(q;w;I N + z N )(f(q)f N (q))dq + Z Q a(q;wI N + u N ;I N + z N )f N (q)dq Z Q jzj H jI N + z N j H jf N (q)f(q)jdq + ~ 0 Z Q jwj H jI N + z N j H jf(q)f N (q)jdq + ~ 0 Z Q jwI N + u N j H jI N + z N j H f N (q)dq + ~ 0 Z Q jjwjj V jjI N + z N jj V jf(q)f N (q)jdq + ~ 0 Z Q jjwI N + u N jj V jjI N + z N jj V f N (q)dq "k 2 2 Z Q N jjz N jj 2 V f N (q)dq + 1 2" Z Q jzj 2 H jf N (q)f(q)j 2 dq + ~ 0 "k 2 2 Z Q N jjz N jj 2 V f N (q)dq + ~ 0 k 2 2" Z Q jjwjj 2 V jf N (q)f(q)j 2 dq + ~ 0 "k 2 2 Z Q N jjz N jj 2 V f N (q)dq + ~ 0 k 2 2" Z Q N jjI N wu N jj 2 V f N (q)dq + ~ 0 " 2 Z Q N jjz N jj 2 V f N (q)dq + ~ 0 2" Z Q jjwjj 2 V jf N (q)f(q)j 2 dq + ~ 0 " 2 Z Q N jjz N jj 2 V f N (q)dq + ~ 0 2" Z Q N jjI N wu N jj 2 V f N (q)dq: (4.62) Then, letting ~ c = ~ 0 " 2 k 2 (1 + ~ 0 + ~ 0 ) + ~ 0 (1 + ) , it follows from (4.62) that ~ cjjz N jj 2 V N 1 2" Z Q jzj 2 H jf N (q)f(q)j 2 dq + ~ 0 k 2 + ~ 0 2" Z Q N jjI N wu N jj 2 V f N (q)dq + ~ 0 k 2 + ~ 0 2" Z Q jjwjj 2 V jf N (q)f(q)j 2 dq: (4.63) 42 Choosing " positive, but suciently small in (4.63), it follows from Assumption 4.6 and the hypotheses of the theorem that jjw N u N jj V N =jjz N jj V N! 0 as N!1: (4.64) Thus (4.64) together with (4.56), and Assumption 4.6 yield resolvent convergence and the theorem is proved. We note that in the proof of Theorem 4.3 we are in fact able to establish resolvent convergence in theV N norm. Consequently we may conclude that the semigroup convergence in (4.55) is in theV N norm as well. Moreover, it is not dicult to establish the following corollary to Theorem 4.3. Corollary 4.1. Under the same hypotheses of Theorem 4.3, we have jjx N j;i ( N )J N x j;i ()jj V N! 0; as N!1; jy N j;i ( N )y j;i ()j R ! 0; as N!1; (4.65) for everyi = 1; 2;:::;m, uniformly inj, forj = 0; 1; 2;:::;n i , wherex N j;i ( N ) andy N j;i ( N ) are given in (4.40) andx j;i () andy j;i () are given in (4.32). The assumption that the feasible parameter set is closed and bounded in R 2p+r , together with (4.65) in the statement of Corollary 4.1 and Theorem 4.1 then yield the following result. Theorem 4.4. If, in addition to Assumptions 4.1-4.6, we assume that the maps 7!f(q;) from to R are continuous for ()-a.e. q2 Q, then each of the approximating estimation problems admits a solution, N . Moreover, the sequencef N g has a convergent subsequence,f N k g with N k ! as N k !1 and is a solution to the original estimation problem. 43 Chapter 5 Deconvolution Problem In this chapter, we formulate the problem of estimating or deconvolving the input to the population model, (4.29) (or in discrete time, (4.32)) as a constrained, regularized optimization problem that ultimately takes the form of a quadratic programming problem, or a linear quadratic tracking problem (see, for example, [23]), albeit with a performance index whose evaluation requires the solution of an innite dimensional state equation. In practice of course, as was the case in the distribution estimation problem discussed in the previous chapter, we solve a nite dimensional approximation. In the rst section here, we prove the convergence of solutions of nite dimensional approximating problems to the solution of the original innite dimensional estimation problem. In the second section, we discuss how the approximating problems are regularized and how optimal values of the regularization parameters or weights can be determined. In that section we also discuss how credible bands for the estimated or deconvolved input signal can be obtained based on the distribution of the random parameters in the population model. 5.1 Input Estimation Let us denote the parameters that determine the distribution ofq by, where = (~ a; ~ b; ~ ), optimally t parameters by , where = (~ a ; ~ b ; ~ ), and the support of the random parameters determined by the optimally t parameters, , byQ . We then consider the population model (4.32) in which 44 the inputfu j g is obtained by zero-order hold sampling a continuous time signal. That is, we assume that the input to the population model is given byfu j g withu j =u(j)2L 2 ( ) (Q ), where > 0 is the length of the sampling interval andu is a continuous time signal that is at least continuous (to allow for sampling) on an interval [0;T ]. In our framework, we rst use this model with the training data to estimate the distribution of the random parameters (i.e. to obtain ). Then, we seek an estimate for the input based on this model and the estimated distribution of the random parameters. We consider our input estimate,u, to be a random variable being a function of the random parameters,q, as well as a function of time. Therefore, letu2 S(0;T), where S(0;T) =H 1 (0;T;L 2 ( ) (Q )) and letU be a compact subset of S(0;T). The input estimation problem is then given by u = arg min u2U J(u) = arg min u2U K X k=1 jy k (u) ^ y k j 2 +jjujj 2 S(0;T) ; (5.1) where the termjjujj 2 S(0;T) is related to the regularization withjjjj S(0;T) a norm on the space S(0;T ), the details of which will be discussed later, and y k (u) = k1 X j=0 hh kj ( );u j i L 2 ( ) (Q ) ; k = 1; 2;:::;K; (5.2) withu j =u(j;q), j = 0; 1;:::;K, h l ( ) = ^ C( ) ^ A( ) l1 ^ B( )2 L 2 ( ) (Q ) = L 2 ( ) (Q ), l = 1; 2;:::;K where ^ C( ) =C( )2L(V;R), ^ A( ) = e A( ) 2L(V ;V), and ^ B( ) = R 0 e A( )s B( )ds =A( ) 1 ( ^ A( )I)B( )2L(L 2 ( ) (Q );V). We note that the coercivity assumption implies without loss of generality (via a standard change of variable) that we may assume that the operatorA( ) has a bounded inverse,A( ) 1 2L(V ;V). It is also important to indicate here that due to the assumption thatu is also a function ofq, we consider the operator B( ) now to be dened from L 2 ( ) (Q ) intoV as 45 hB( )u; i = Z Q hB(q)u(q); (q)i V ;V f(q; )dq: (5.3) 5.2 Approximation and Convergence Solving the problem stated in (5.1) requires nite dimensional approximation. Let N;M;L be multi-indices dened byN = (n;m 1 ;m 2 ),M = (m;m 1 ;m 2 ), andL = (N;M). Note that whenever we use the notationN,M, orL!1, we mean all components of these multi-indices go to innity. For eachM, letU M be a closed subset ofU which is contained in a nite dimensional subspace of S(0;T ). The setsU M are, of course, therefore compact. We require the following approximation assumption on the subsetsU M , or more typically, the nite dimensional spaces they are contained in,S M S(0;T ). Assumption 5.1. For eachu2U, there exists a sequencefu M g withu M 2U M , such that jju M ujj S(0;T) ! 0; as M!1: (5.4) Let the Bochner spacesV =L 2 ( ) (Q ;V ) andH =L 2 ( ) (Q ;H) which in the usual manner, form the Gelfand tripleV ,!H ,!V , be as they were dened in Section 4.3. For each N let W N denote a nite dimensional subspace ofV and letP N H :H!W N denote the orthogonal projection ofH ontoW N . We require the assumption that the subspacesW N have the following approximation property. Assumption 5.2. For eachv2V jjP N H vvjj V ! 0; as N!1: (5.5) 46 We note that it can often be shown using the Schmidt inequality (see [29]) that spline-based subspacesW N typically satisfy (5.5). Dene the operatorsA N ( ) :W N !W N by A N ( )v N ;w N =a(v N ;w N ) = Z Q a(q;v N (q);w N (q))dF (q) = Z Q a(q;v N (q);w N (q))f(q;~ a ; ~ b ; ~ )dq; (5.6) wherev N ;w N 2W N . We note that as is the case with the operatorA( ), coercivity implies that without loss of generality, the operatorsA N ( ) given in (5.6) are invertible with inverses that are uniformly bounded in N. Moreover, once again, as a result of coercivity, standard estimates can be used to show that jjA N ( ) 1 P N H 'P N H A( ) 1 'jj V ! 0; '2H (5.7) as N !1 (see, for example, [5, 6, 31]). It then follows from the Trotter-Kato semigroup approximation theorem [5, 6, 26] that jj ^ A N ( )P N H 'P N H ^ A( )'jj V ! 0; '2H (5.8) as N!1. We now state the nite dimensional approximating deconvolution problems as follows u L = arg min u2U M J L (u) = arg min u2U M K X k=1 y L k (u) ^ y k 2 +jjujj 2 S(0;T) ; (5.9) 47 where y L k (u) = k1 X j=0 h N kj ( );u j L 2 ( ) (Q ) ; k = 1; 2;:::;K; (5.10) withu2U M ,u j =u(j)2L 2 ( ) (Q ), j = 0; 1;:::;K, h N l ( ) = ^ C N ( ) ^ A N ( ) l1 ^ B N ( )2 L 2 ( ) (Q ) =L 2 ( ) (Q ), l = 1; 2;:::;K, ^ C N ( ) =C( ), ^ A N ( ) =e A N ( ) 2L(W N ;W N ), ^ B N ( ) = R 0 e A N ( )s B N ( )ds =A N ( ) 1 ( ^ A N ( )I)B N ( ) withB N ( ) = P N H B( )2 L(L 2 ( ) (Q );W N ), the operator B( ) given by (5.3), and P N H being the extension of the orthogonal projection ofH onto W N to V . Note that the existence of the extension of the orthogonal projection operator can be proved by using the fact thatH is a dense subset ofV and the Riesz Representation Theorem. Now, consider the following lemma which we need to prove our convergence theorem. Lemma 5.1. Forl = 1; 2;:::;K, let the bounded linear functionalsh l ( ) andh N l ( ) onL 2 ( ) (Q ) be given by h l ( ) =C( ) ^ A( ) l1 A( ) 1 ( ^ A( )I)B( ) and h N l ( ) =C( ) ^ A N ( ) l1 A N ( ) 1 ( ^ A N ( )I)B N ( ) =C( ) ^ A N ( ) l1 A N ( ) 1 ( ^ A N ( ) I) P N H B( ), respectively. Then, under Assumption 5.2, we have that h N l ( ) converges weakly (i.e. pointwise) to h l ( ) in L 2 ( ) (Q ) (and therefore, their respective Riesz representers in L 2 ( ) (Q ) converge weakly in L 2 ( ) (Q ) as well) as N!1, uniformly in l on any nite set of indices. That is j h N l ( );w L 2 ( ) (Q ) hh l ( );wi L 2 ( ) (Q ) j! 0; (5.11) as N !1, for all w2 L 2 ( ) (Q ), l = 1; 2;:::;K, uniformly in l on any nite set of indices. Moreover, iffu M g is a sequence in L 2 ( ) (Q ) withu M !u2L 2 ( ) (Q ) as M!1, we have that h N l ( );u M !hh l ( );ui as L!1, uniformly in l on any nite set of indices. 48 Proof. The last claim in the statement of the Lemma of course follows immediately from the weak convergence in (5.11). To establish (5.11), we note that in light of the approximation assumption, Assumption 5.2, on the nite dimensional subspacesW N , (5.5), and the strong convergence in (5.7) and (5.8), the result will follow if we can show that the operators P N H converge strongly to the identity on the spaceV . But this follows easily from the approximation condition (5.5) and standard density arguments. Now, we are ready to state and prove the the theorem regarding convergence of the solutions of approximating problems to the solution of the main problem. Theorem 5.1. For each L, the nite dimensional approximating optimization problem given in (5.9) admits a solution denoted by u L . Moreover, under Assumptions 5.1 and 5.2 there exists a subsequence offu L g,fu L k gfu L g, with u L k !u as k!1, with u a solution to the innite dimensional estimation problem given in (5.1). Proof. The existence of a solution to each of the nite dimensional approximating optimization problems given in (5.9) follows from (5.10) and therefore the continuity ofJ L , and the compactness ofU M . Now letfu M g be a convergent sequence inU S(0;T) withu M 2U M andjju M ujj S(0;T) ! 0 as M !1, u2U. Then if u M;j =u M (j)2 L 2 ( ) (Q ) and u j =u(j)2 L 2 ( ) (Q ), j = 1; 2;:::;K, it follows thatjju M;j u j jj L 2 ( ) (Q ) ! 0 as M!1, j = 1; 2;:::;K. It follows from the lemma that j h N l ( );u M;j hh l ( );u j ij! 0; as L = (N;M)!1; j;l = 1; 2;:::;K: (5.12) It follows from (5.12) that J L (u M )!J(u) as L!1. Letfu L g U M U be a sequence of solutions to the nite dimensional approximating optimization problems given in (5.9). The compactness ofU implies that there exists a subsequence 49 offu L g,fu L k gfu L g, withu L k !u for someu 2U as k!1. Then for anyu2U it follows that J(u ) =J( lim k!1 u L k ) = lim k!1 J L k (u L k ) lim k!1 J L k (u M k ) =J( lim k!1 u M k ) =J(u); (5.13) where L k = (N k ;M k ), and the sequencefu M k gU withu M k 2U M k are the approximations to u guaranteed to exist by the approximation assumption on the subsetsU M given in Assumption 5.1, equation (5.4), and thus,u is a solution of the estimation problem stated in (5.1). We note rst that the above theorem continues to hold if S(0;T) is chosen as S(0;T) = L 2 (0;T;L 2 ( ) (Q )). In addition, it is not dicult to see that the way we have formulated the deconvolution problem and its nite dimensional approximations given in (5.1) and (5.9), respectively, the resulting optimization problems take the form of a linear quadratic tracking problem where the systems to be controlled are, respectively, the population model, (4.32), and its nite dimensional approximation, (4.40). Consequently, it becomes possible to adapt some ideas from [2] to obtain a somewhat dierent and in some ways stronger convergence result than that given in Theorem 5.1 above. Now letU be a closed and convex subset ofS(0;T ) and for eachM, letU M be a closed convex subset ofU which is contained in a nite dimensional subspace of S(0;T). Note that the maps u7!J(u) andu7!J L (u) fromU into R and fromU M into R, respectively, are strictly convex. Consequently solutionsu andu L to the optimization problems (5.1) and (5.9), respectively, exist and are unique. Moreover, the sequencefjju L jj S(0;T) g is bounded. If not, there would exist a subsequence,fu L k g withjju L k jj S(0;T) !1, and therefore that J L k (u L k )!1. But then this would contradict the fact that for any u2U, we have that J L k (u L k )J L k (u M k )!J(u)<1 (5.14) 50 where L k = (N k ;M k ), and the sequencefu M k gU withu M k 2U M k are the approximations to u guaranteed to exist by the approximation assumption on the subsetsU M given in Assumption 5.1, equation (5.4). Then since U was assumed to be closed and convex, it is weakly closed. Therefore, there exist a weakly convergent subsequence,fu L k g offu L g withu L k *u , for some u 2U. In addition, the convexity and (lower semi-) continuity of J and J L yield that J and J L are weakly sequentially lower semi-continuous. Then, Assumptions 5.1 and 5.2, Lemma 5.1 together with the weak lower semicontinuity of J and J L imply that J(u ) =J(w lim k!1 u L k ) lim inf k!1 J(u L k ) = lim inf k!1 J L k (u L k ) lim sup k!1 J L k (u L k ) lim sup k!1 J L k (u M k ) =J( lim k!1 u M k ) =J(u); (5.15) where L k = (N k ;M k ), wlim represents the weak limit, and the sequencefu M k gU with u M k 2U M k are the approximations tou guaranteed to exist by the Assumption 5.1 and hence that,u is the solution of the estimation problem stated in (5.1). Finally we note that the strict convexity of J(u) and J L (u) imply that the sequence itself,fu L g must in fact converge weakly to the unique solution,u , of the estimation problem stated in (5.1), and because the norm in S(0;T ) is bounded above by J, that the sequencefu L g itself must in fact converge strongly, or in the norm in S(0;T ), to the unique solution,u , of the estimation problem stated in (5.1). 5.3 Regularization The cost functionals dened in (5.1) and (5.9) include regularization terms in the form of the square of the norm on S(0;T ),jjjj S(0;T) , which we take to be given by R(r 1 ;r 2 ) =jjujj S(0;T) = r 1 Z T 0 jju(t)jj 2 L 2 ( ) (Q ) dt +r 2 Z T 0 jj _ u(t)jj 2 L 2 ( ) (Q ) dt ! 1=2 ; (5.16) 51 where r 1 ;r 2 > 0 are regularization parameters in the form of nonnegative weights. Choosing regularization parameters can entail a mix of art and science and we choser 1 andr 2 as the solution of the following optimization problem based on the original training data used to estimate the distribution of the random parameters in the model. Recall the training data from Chapter 4 consisting of data sets (~ u i ; ~ y i ) i=1 = f~ u j;i g i1 j=0 ;f~ y j;i g i j=1 i=1 : (5.17) After using the training data to estimate the distribution of the random parameters, we use the approach discussed in Section 5.1 together with the training TAC observations,f~ y j;i g i j=1 , given in (5.17) to estimate the BrAC inputsf~ u j;i g i1 j=0 also given in (5.17). Denote these estimates by f ~ u L;j;i g i1 j=0 . We then use these estimates of the input together with the approximating population model and (5.2), to obtain estimates for the corresponding TAC,f~ y L;j;i g i j=1 . Note that both thef ~ u L;j;i g i1 j=0 and thef~ y L;j;i g i j=1 are functions of the weights, r 1 and r 2 appearing in the cost functional for the input estimation problem. We then dene the performance index J(r 1 ;r 2 ) = X i=1 i X j=1 j u L;j1;i ~ u j1;i j 2 +j~ y L;j;i ~ y j;i j 2 ; (5.18) where u L;j;i =E ( ) [ ~ u L;j;i ], and choose the regularization parameters, r 1 and r 2 , to use in our scheme as (r 1 ;r 2 ) = arg min (r1;r2)2R +2 J(r 1 ;r 2 ); (5.19) where J(r 1 ;r 2 ) is given by (5.18). We note that since the approximating optimization problems given in (5.9) are in fact constrained (i.e. we want all the components of U M k to be nonnegative) linear problems, in practice, we solve them as quadratic programming problems. Moreover, in nite dimensions, the two integrals in the 52 expression for the regularization term,R(r 1 ;r 2 ), given in (5.16), become quadratic forms in the vectors U M k with the two positive denite symmetric matrices, denoted by Q M 1 and Q M 2 , having entries that are the easily computed L 2 ( ) (Q ) inner products of the basis elements for the q dependencies of the elements inS M . By appropriately placing the matrix representations for the approximating ltersfh L l ( )g in the block matrix H L , the approximating optimization problem given in (5.9) now takes the equivalent form u L = arg min U M J L (u) = arg min U M 2 6 6 4 H L (r 1 Q M 1 +r 2 Q M 2 ) 1 2 3 7 7 5 U M Y M 2 R K+Kmm 1 m 2 ; (5.20) where (r 1 ;r 2 ) comes from (5.19),U M is theKmm 1 m 2 dimensional column vector of the coecients of theu2U M , and Y M is the K +Kmm 1 m 2 dimensional column vector consisting of the K TAC data pointsf^ y k g followed by Kmm 1 m 2 zeros. 53 Chapter 6 Application of Methods to the Alcohol Biosensor Problem In this chapter, we show that the model (2.11)-(2.15) satisfy all the assumptions that we made through presenting our theoretical framework. We start with showing it satises the hypotheses (A)-(C) of Theorem 4.1. We assume that the random parameterq = [q 1 ;q 2 ] has truncated bivariate normal distribution and as we stated in Section 4.3, there exist some ; such that < a i < b i < for i = 1; 2. So, the pdf of the truncated bivariate normal distribution, f(q;) is a continuous function of its parameters, , for every q2Q. By the same reason, the cost functional J N (f(;);V ) is also a continuous function of the parameters for all N2N and f2F (Q). Now, we show that hypothesis (B) of Theorem 4.1 is satised. LetW =H 2 (0; 1),V =H 1 (0; 1), and H =L 2 (0; 1) so that we have the Gelfand triple as H 1 (0; 1),!L 2 (0; 1),!H 1 (0; 1) where H 1 (0; 1) =V . Denoting the derivative of ' with respect to time t by _ ' and with respect to the space variable by ' 0 , for '2W and 2V , we can write the weak form of (2.11) as < _ '; > V =<q 1 ' 00 ; > V : (6.1) 54 By using the boundary conditions (2.12) and (2.13), consider that < _ '; > V = Z 1 0 q 1 ' 00 d =q 1 ' 0 1 0 Z 1 0 q 1 ' 0 0 d =q 1 ' 0 (t; 1) (t; 1)q 1 ' 0 (t; 0) (t; 0) Z 1 0 q 1 ' 0 0 d =q 2 (t; 1)u(t)'(t; 0) (t; 0) Z 1 0 q 1 ' 0 0 d: (6.2) Let a(q;'; ) =q 1 R 1 0 ' 0 0 d +'(0) (0) and <b(q); > V ;V =q 2 (1). Now, let us show that the sesquilinear form satises the Assumptions 4.1-4.4. Let '; 2V be arbitrary but xed. Consider by using Cauchy-Schwarz Inequality that ja(q;'; )j'(0) (0) +q 1 Z 1 0 ' 0 0 d j'(0) (0)j +q 1 Z 1 0 j' 0 j 2 d 1=2Z 1 0 j 0 j 2 d 1=2 j'(0) (0)j +q 1 jj'jj V jj jj V : (6.3) Since both ' and are continuous functions on [0; 1], by using Morrey's Inequality (with n = 1, p = 2, and = 1=2), there exists 0 > 0 such thatj'(0) (0)j 0 jj'jj V jj jj V . Thus, we can choose 0 > 0 such that ja(q;'; )j 0 jj'jj V jj jj V ; (6.4) which shows the boundedness of the sesquilinear form. Then, for 2V and 0 > 0, consider a(q; ; ) + 0 j j 2 H = (0) 2 +q 1 Z 1 0 j 0 j 2 d + 0 Z 1 0 j j 2 d: (6.5) 55 Then, since QR 2 is compact we can nd 0 > 0 such that a(q; ; ) + 0 j j 2 H 0 Z 1 0 j 0 j 2 d + Z 1 0 j j 2 d (6.6) = 0 jj jj 2 V : (6.7) showing the coercivity of the sesquilinear form. In this case, the sesquilinear form denes a bounded and linear operator A(q) : V ! V by < A(q)'; >= '(0) (0) q 1 R 1 0 ' 0 0 d with Dom(A(q)) = f' 2 W with q 1 ' 0 (0) = '(0) and q 1 ' 0 (1) =q 2 u(t)g. Then letting R T 0 w(t;s)<c(q);'(s)> V ;V ds = R T 0 w(t;s)'(s; 0)ds with = 0, we can write this model in the form of (4.14). Let us also show that for every '; 2V the map q7!a(q;'; ) is measurable with respect to all measures dened in terms of the densities inF (Q), where R R is the set of admissible parameters. Let an arbitrary but xed such measure be represented by a probability density function f(q;). Showing the sesquilinear form is measurable with respect to this measure is equivalent to showing thata(;'; )f(;) is Lebesque measurable as a function ofq for xed' and . In this case, by its denition, the sesquilinear form is a continuous function of q and f(;) is a measurable function by being a probability density function. Thus, the function a(;'; )f(;) is integrable overQ which implies thata(;'; ) is measurable with respect to the probability measure () for any'; 2V . Note that similar arguments imply also that the maps q7!<b(q); > V ;V and q7! R T 0 w(t;s)<c(q);'(s)> V ;V ds are measurable with respect to the probability measure () for 2V , '2L 2 ([0;T ];V ) with an appropriate choice of positive weight function w(t;s). Let us choosed Q to be the Euclidean distance onR. Then, by using Cauchy-Schwarz Inequality, consider for 1 ; 2 2V and q;q 0 2Q that 56 ja(q; 1 ; 2 )a(q 0 ; 1 ; 2 )j = q 1 Z 1 0 0 1 0 2 d + 1 (0) 2 (0)q 0 1 Z 1 0 0 1 0 2 d 1 (0) 2 (0) = (q 1 q 0 1 ) Z 1 0 0 1 0 2 d jq 1 q 0 1 j Z 1 0 0 1 0 2 d jq 1 q 0 1 j Z 1 0 j 0 1 j 2 d 1=2Z 1 0 j 0 2 j 2 d 1=2 =jq 1 q 0 1 jjj 1 jj V jj 2 jj V : (6.8) which shows that Assumption 4.4 is satised, as well. Since the random parameter vectorq = [q 1 ;q 2 ] has truncated bivariate normal distribution for the alcohol biosensor problem, with this choice of the joint probability density function we will have a continuous function ofq, denoted byf(q;; ), which takes a positive value for everyq2Q. Therefore there exists m;M2 R such that 0 < m < f(q;) < M <1 for every q2 Q which shows that our choice of probability density functions for this problem satises even a stronger condition than Assumption 4.5. To show that the Assumption 4.6 is satised, we will rstly state our choice of nite dimensional approximating functionsu N 2W N for each N2N. As we will express in more detail in Chapter 7, we use linear B-spline polynomialsf' n j g n j=0 for -discretization, dened with respect to the uniform meshfj=ng n j=0 on the interval [0; 1]. Dene the operator I n :V N !V N as I n v(q;) = n X i=0 v(q; i )' n i (): (6.9) Let Q N = [a N 1 ;b N 1 ] [a N 2 ;b N 2 ] andf mi i;j (q i )g mi j=1 be the 0 th order B-splines with respect to the uniform mesh on the interval [a N i ;b N i ], fori = 1; 2. So, we haveW N =spanf N i;j;k g,i = 0; 1; 2;:::;n, j = 1; 2;:::;m 1 , k = 1; 2;:::;m 2 where N i;j;k (;q 1 ;q 2 ) = ' n i () m1 1;j (q 1 ) m2 2;k (q 2 ), 2 [0; 1], q 1 2 57 [a 1 ;b 1 ], q 2 2 [a 2 ;b 2 ] with dim(W N ) = (n + 1)m 1 m 2 . For simplicity, let m j (q) = m1 1;j1 (q 1 ) m2 2;j2 (q 2 ) withj = (j 1 1)m 2 +j 2 andR j represent thej th rectangle on which m j (q) is 1 and 0 elsewhere, for j = 1; 2;:::;m 1 m 2 . Also, letp j = RR Rj f(q; N )dq. Now, dene the operatorP m1;m2 :H N !O N m1;m2 withO N m1;m2 =f'2H N :'(q;) = P m1m2 j=1 ' j () m j (q); where ' j ()2H for each jgH N as P m1m2 v(q;) = m1m2 X j=1 v j () m j (q); (6.10) where v j () = 1 p j ZZ Rj v(^ q;)f(^ q; N )d^ q: (6.11) With these denitions, we have P m1;m2 I n :V N !W N dened as P m1m2 I n v(q;) = m1m2 X j=1 1 p j ZZ Rj n X i=0 v(^ q; i )' n i () ! f(^ q; N )d^ q m j (q): (6.12) Let us now show thatO N m1;m2 is a closed subspace ofH N . Letfv k g 1 k=1 O N m1;m2 be a convergent sequence which implies thatfv k g is a Cauchy sequence inH N . Let v k = P m1m2 j=1 ' n;k j () m j (q). So, for each j = 1; 2;:::;m 1 m 2 ,f' n;k j g 1 k=1 is also a Cauchy sequence in H. Since H is a Hilbert space, for each j = 1; 2;:::;m 1 m 2 , there exists ' n j 2 H such thatj' n;k j ' n j j H ! 0 as k!1. Then, let v(q;) = P m1m2 j=1 ' n j () m j (q)2O N m1;m2 and consider 58 jv k vj H N =j m1m2 X j=1 (' n;k j ' n j ) m j j H N m1m2 X k=1 ZZ Rj j' n;k j ' n j j H f(q; N )dq: (6.13) Since each term on the right-hand side of (6.13) converges to 0 as k!1, we have shown that v2O N m1;m2 is the limit offv k g, proving thatO N m1;m2 H N is closed. Now, we show that P m1;m2 dened in (6.10)-(6.11) is an orthogonal projection operator from H N toO N m1;m2 . Let v2H N and w2O N m1;m2 be arbitrary. Consider that <P m1;m2 vv;w> H N = ZZ Q N Z 1 0 0 @ m1m2 X j=1 v j () m j (q) 1 A m1m2 X k=1 w k () m k (q) ! d f(q; N )dq ZZ Q N Z 1 0 v(q;) m1m2 X k=1 w k () m k (q) ! d f(q; N )dq = Z 1 0 m1m2 X j=1 m1m2 X k=1 ZZ Q N v j ()w k () m j (q) m k (q)f(q; N )dqd Z 1 0 m1m2 X k=1 ZZ Q N v(q;)w k () m k (q)f(q; N )dqd = Z 1 0 m1m2 X j=1 p j v j ()w j ()d Z 1 0 m1m2 X k=1 p k v k ()w k ()d = 0: (6.14) So, we havejP m1m2 j H N 1. Let us also show that P m1m2 is a bounded operator from V N to O N m1m2 . Consider by using Riesz Representation Theorem andjj (V N ) k 1 jj H N , jj H Nk 2 jjjj V N for some k 1 ;k 2 > 0 that 59 jjP m1m2 vjj 2 V N =<P m1m2 v;P m1m2 v> V N jP m1m2 vj (V N ) jjP m1m2 vjj V N k 1 jP m1m2 vj H NjjP m1m2 vjj V N k 1 jvj H NjjP m1m2 vjj V N k 1 k 2 jjvjj V NjjP m1m2 vjj V N (6.15) which showsjjP m1m2 vjj V Nk 1 k 2 jvjj V N meaning thatjjP m1m2 jj V N <c for some c> 0. Now, let us choose any v2C( Q;H 2 (0; 1)) and consider that jjI N vP m1m2 I n I N vjj V NjjI N vP m1m2 I N vjj V N +jjP m1m2 I N vP m1m2 I n I N vjj V N: (6.16) Then, observe jjI N vP m1m2 I N vjj 2 V N = ZZ Q N I N vP m1m2 I N v 2 V f(q; N )dq = ZZ Q N I N v(q;) m1m2 X j1 1 p j ZZ Rj I N v(^ q;)f(^ q; N )d^ q m j (q) 2 V f(q; N )dq = ZZ Q N m1m2 X j=1 1 p j ZZ Rj (I N v(q;)I N v(^ q;))f(^ q; N )d^ q m j (q) 2 V f(q; N )dq ZZ Q N 0 @ m1m2 X j=1 1 p j ZZ Rj I N v(q;)I N v(^ q;) V f(^ q; N )d^ q m j (q) 1 A 2 f(q; N )dq (6.17) 60 By the continuity of v(q;) on compact sets, given "> 0, we can nd m 1 2N and m 2 2N so thatjI N v(q;)I N v(^ q;)j<"=2. Then, continuing from the last line of (6.17), we obtain jjI N vP m1m2 I N vjj 2 V N ZZ Q N 0 @ " 2 m1m2 X j=1 m j (q) 1 A 2 f(q; N )dq< " 2 4 : (6.18) Now, let us consider the second term on the right-hand side of (6.16) by using the fact that jjP m1m2 jj V Nc for some c> 0, jjP m1m2 I N vP m1m2 I n I N vjj 2 V Nc 2 jjI N vI n I N vjj 2 V N =c 2 ZZ Q N I N vI n I N v 2 V f(q; N )dq: (6.19) By using Theorem 3.5 in [32] and continuous dependence of functionsv(q;) onq, one can show that for given "> 0, there exists n 0 2 N such thatjI N vI n I N vj V <"=2c for all n>n 0 . So, continuing from the last line of (6.19), we obtain that jjP m1m2 I N vP m1m2 I n I N vjj 2 V Nc 2 ZZ Q N " 2 4c 2 f(q; N )dq " 2 4 : (6.20) Then, combining (6.16), (6.18), and (6.20), we conclude that for given " > 0, there exists N 0 = (n 0 ;m 0 1 ;m 0 2 ) such thatjjvP m1m2 I n vjj V N <" for all (n;m 1 ;m 2 )>N 0 . Then, let ^ v2 V be arbitrary. Then, for any "> 0, by density of C( Q;H 2 (0; 1)) in V, there exists v2C( Q;H 2 (0; 1)) such thatjjI N ^ vI N vjj V N"=3. Consider 61 jjI N ^ vP m1m2 I n I N ^ vjj V NjjI N ^ vI N vjj V N +jjI N vP m1m2 I n I N vjj V N +jjP m1m2 I n I N vP m1m2 I n I N ^ vjj V N: (6.21) Sincev2C( Q;H 2 (0; 1)), we can chooseN large enough so thatjjI N vP m1m2 I N vjj V N"=3 by our previous argument. Finally, sincejjP m1m2 jj V N <c, for some c> 0, we obtain jjP m1m2 I n I N vP m1m2 I n I N ^ vjj V NcjjI n I N vI n I N ^ vjj V N (6.22) Now, by adding and subtracting I N v and I N ^ v to the argument inside the norm on the right-hand side of (6.22), and using the arguments in (6.19) one can show by choosing N large enough thatjjI n I N vI n I N ^ vjj V N < "=3c. Thus, we conclude that for any ^ v2 V and given "> 0, there exists N 0 such that letting u N =P m1m2 I n I N ^ w, we obtain thatjj^ vu N jj V N <" which shows that Assumption 4.6 is satised. At this point, we have shown that alcohol biosensor model satises all the assumptions we stated for the parameter estimation problem. It remains to show it also satises our assumptions for the deconvolution problem. For Assumption 5.1, we use a very similar argument that we used for Assumption 4.6. For this purpose, let us dene the corresponding operators I m and P m1m2 on H 1 ([0;T];L ( ) (Q )) as follows. I m u(t;q) = m X i=1 u(t i ;q) i (t) (6.23) P m1m2 u(t;q) = m1m2 X j=1 1 p j ZZ Rj u(t; ^ q)f(^ q;)d^ q m j (q): (6.24) 62 Then letting u M =P m1m2 I m u(t;q) = m1m2 X j=1 1 p j ZZ Rj m X i=1 u(t i ; ^ q) i (t)f(^ q;)d^ q m j (q) = m1m2 X j=1 m X i=1 1 p j ZZ Rj u(t j ; ^ q)f(^ q;)d^ q i (t) m j (q); (6.25) and repeating the same arguments in (6.17) and (6.19), one can prove that given "> 0, there exists M = (m;m 1 ;m 2 ) such thatjjuu M jj S(0;T) ! 0 as M!1 as stated in Assumption 5.1. Finally we consider Assumption 5.2. By using Riesz Representation Theorem andH,!V (i.e.,jj V kjj H for some constant k> 0) as in (6.15), we obtain jjP N H vvjj V kjjP N H vvjj H : (6.26) Then, let I n and P m1m2 be as in (6.9) and (6.10), but dened onH this time. The argument that we used for showing Assumption 4.6 is satised implies here that for anyv2H,jjP m1m2 I n v vjj H ! 0 as N = (n;m 1 ;m 2 )!1. However, sinceP N H :H!W N is the orthogonal projection ofH ontoW N , we have jjP N H vvjj H jjP m1m2 I n vvjj H ; (6.27) which implies thatjjP N H vvjj H ! 0 as N = (n;m 1 ;m 2 )!1 concluding the proof of Assumption 5.2. 63 Chapter 7 Numerical Results 7.1 Computational Considerations The approximating optimization problems for parameter estimation are solved numerically by using an iterative gradient-based scheme. Once a basis for the spaceW N is chosen, matrix forms of the operators ^ A N , ^ B N , and ^ C N can be computed. The gradient of J N (), with respect to the 2p +r parameters in can be computed accurately (in fact exactly with the exception of nite precision arithmetic round-o) eciently (which is especially important if the dimension of the approximating system (4.40) and/or the number of parameters is large) using the adjoint method (see [22]). For each i = 1;:::;m, set v N i;j = [2( ^ C N x N i;j ~ y i;j ); 0;:::; 0] T 2R K N , j = 0;:::; i where K N is the number of basis elements forW N . The adjoint systems are dened to be z N i;j1 = [ ^ A N ] T z N i;j +v N i;j1 : (7.1) 64 The gradient of J N at = (~ a; ~ b; ~ ) can then be computed from ~ rJ N () = m X i=1 ni X j=1 [z N i;j ] T @ ^ A N @ x N i;j1 (A N ) 1 @A N @ (A N ) 1 ( ^ A N I) @ ^ B N @ ~ u i;j1 @ ^ A N @ ^ B N ~ u i;j1 ( ^ A N I) @ ^ B N @ ~ u i;j1 + m X i=1 ni X j=0 y N j ~ y i;j T@ ^ C N @ x N i;j : (7.2) Using (7.1) and (7.2) to compute the gradient requires the calculation of the tensor @ ^ A N @ . This can be done using the sensitivity equations. For t 0 set N (t) =e A N (t) from which dierentiation yields _ N (t) =A N N (t); N (0) =I: (7.3) Then, setting N (t) = @ N (t) @ , dierentiating (7.3) with respect to , and interchanging the order of dierentiation, we obtain _ N (t) =A N N (t) + @A N @ N (t); N (0) = 0: (7.4) Combining (7.3) and (7.4), and solving the resulting system, we obtain 2 6 6 4 N (t) N (t) 3 7 7 5 =exp 2 6 6 4 A N @A N =@ 0 A N 3 7 7 5 2 6 6 4 0 I 3 7 7 5 (7.5) Setting t = in (7.5), we obtain that @ ^ A N @ = N (). 65 To illustrate our approach, we consider the case of a one dimensional heat/diusion equation on the interval [0; 1] with random (thermal) diusivity and two dierent sets of boundary conditions. Consider the partial dierential equation, boundary conditions and output operator given by @x @t (t;) =q 1 @ 2 x @ 2 (t;); 0<< 1; t> 0; (7.6) D x(t;) :=x(t; 0) = 0; t> 0; (7.7) R x(t;) :=q 1 @x @ (t; 0)x(t; 0) = 0; t> 0; (7.8) 1 x(t;) := q 1 q 2 @x @ (t; 1) =u(t) t> 0; (7.9) x(0;) = 0; 0<< 1; (7.10) y(t) =x(t; 0 ); t> 0; (7.11) where 0 < 0 < 1: In the examples below, we consider the parameterized family of probability density functions dened as follows. Denition 7.1. Let '(q;), q2R n be a member in an exponential family [9], and let denote its cumulative distribution function. Let represent a vector of parameters, and let DR n be a bounded region to which ' will be restricted. Then dene D () = R D '(q;)dq. Then the family of pdfs, f(;) given by f(q;) = '(q;) D (q) D () = 1 D () h(q)c()exp k X i=1 w i ()t i (q) ! D (q) where the parameters include the parameters ~ and parameters~ a and ~ b to describe the domain D, is called a truncated exponential family. It is clear that this family of densities satises Assumption 5.1 and the hypotheses of Theorem 4.2. 66 We show our numerical results for various cases and consider two separate but similar models. One of them has only one random parameter and the other has two random parameters. We use the model with one random parameter to show our parameter estimation results with simulated data. But the other model which has two random parameters is be used to show both the parameter estimation results and the input estimation results. The state,x j , j = 0; 1;::: in the population model described by system (4.32), is a function of the distributed variables, , and the random parameters q 1 and q 2 . It is these dependencies that are discretized in our framework. Recall that the state space for the approximating systems (4.40) is the nite dimensional subspaceW N ofV. Recall also that these spaces must satisfy the approximation property given in Assumption 5.2. We construct the spacesW N using linear B-splinesf' n j g n j=0 dened with respect to the uniform meshfj=ng n j=0 on the interval [0; 1] and use the standard 0 th order (i.e. piecewise constants) B- splinesf m1 1;j g m1 j=1 andf m2 2;j g m2 j=1 dened with respect to the uniform meshfa N i +(b N i a N i )j=m i g mi j=0 on the intervals [a N i ;b N i ] for i = 1; 2, respectively. Then, letting N = (n;m 1 ;m 2 ) as before, and letting s denote the multi-index s = (j;j 1 ;j 2 ), we use tensor products to denef N s g N s as N s = ' n j m1 1;j1 m2 2;j2 and set W N = spanf N s g N S . So, we assume that the nite dimensional approximation ofx j as in (4.32) has the form x j (;q) = n X i=0 m1 X j1=1 m2 X j2=1 x N i;j1;j2 ' m i () m1 1;j1 (q 1 ) m2 2;j2 (q 2 ); (7.12) for every j = 0; 1;::: . Note that results from [29] and Chapter 6 can be used to establish Assumption 5.2 with this choice of basis elements. The input signal,u, is a function of time, t, and also the random parameters q 1 and q 2 and is an element of the space S(0;T ) dened in Chapter 5. To construct nite dimensional subspaces, S M , of S(0;T) that have the approximation property given in equation (5.4) of Assumption 5.1 (see [29]), we again use standard linear B-spline polynomialsf m j g m j=1 dened with respect to the 67 usual uniform mesh,fjT=mg m j=0 , on the interval [0;T ] to discretize the time dependency. We use the same 0 th order B-splinesf m1 1;j g m1 j=1 ,f m2 2;j g m2 j=1 for q 1 and q 2 with respect to corresponding uniform mesh on the intervals [a 1 ;b 1 ] and [a 2 ;b 2 ], respectively, to discretize the dependence on the random parameters where [a 1 ;b 1 ] and [a 2 ;b 2 ] represent the optimal support obtained by solving the parameter estimation problem. LettingM = (m;m 1 ;m 2 ) andp = (i;j 1 ;j 2 ) be the multi-index, we again use tensor products to serve as the basis for the approximating subspaces. We dene f M p g M p as M p = m i m1 1;j1 m2 2;j2 and setS M = spanf M p g M p . In this way, any u2S M can be written as u(t;q) = m X i=1 m1 X j1=1 m2 X j2=1 u i;j1;j2 m i (t) m1 1;j1 (q 1 ) m2 2;j2 (q 2 ): (7.13) Now that the bases for the approximating spaces have been chosen for both problems, we can obtain the matrix representations of the operators for both problems. To generalize, letL = (N;M) and use multi-index even if the corresponding matrix depend only on either N or M. However, the dimensions of the matrices will be explained below for each problem. Now, we write the matrix-vector version of the system (4.40) as follows M L X L k+1 =K L X L k +B L U L k ; Y L k =C L X L k ; k = 0; 1;:::;K; (7.14) where the vectors X N k are coecients of the basis elements, the vectors U M k represent inputs, and the vectors Y L k represent outputs. For the parameter estimation problem, we have training data that contains both the input and output measurements. So, U L k and Y L k will be known scalars for each k = 0; 1;::;K. For generality, considering the model with two random parameters,M L andK L will be (n+1m 1 m 2 )(n+1)m 1 m 2 matrices. B L and C L will be (n + 1)m 1 m 2 1 and 1 (n + 1)m 1 m 2 vectors, respectively. X L k will be an (n + 1)m 1 m 2 1 dimensional vector containing the coecientsx N i;j1;j2 in (7.12). When we 68 use the system (7.14) for the deconvolution problem, Y L k will be known output measurements, but U L k will be unknown inputs to be estimated. So, U L k will be an mm 1 m 2 1 dimensional vector containing the coecientsu i;j1;j2 in (7.13). However, as we dened the corresponding operator in (5.3), we will have B L as an (n + 1)m 1 m 2 mm 1 m 2 matrix. All the other matrices M L , K L and vectors C L , X L will have the same representation and hence dimension as in the parameter estimation problem. The detailed representations of these matrices and vectors will be given below for each model separately. The approximating estimation problems are all solved on either MAC or PC laptops using MATLAB. To solve nite dimensional approximating estimation problems stated in (4.9) which turned out to be constrained optimization problems, we use MATLAB routine FMINCON. Before solving the deconvolution problem, we obtain the optimal regularization parameters based on the data and the results of parameter estimation problem which requires solving another optimization problem, (5.19). For this problem, we use FMINSEARCH a built-in routine from MATLAB. Finally, to solve the nite dimensional optimization problems in (5.9) for obtaining the estimated input, we use another MATLAB routine, LSQNONNEG. 7.2 Numerical Results with Simulated Data In this section, we state the numerical results obtained using simulated data. The simulated data is generated by rst sampling the target distribution to obtain 100 samples ofq. A spline based Galerkin approximation to the system (7.6) -(7.11) using a 128 equally spaced point grid on [0; 1] is then solved using eachq-sample. The resulting 100 output signals is then averaged at each time point. Note that we compute gradients with either FMINCON's built-in nite dierencing or the adjoint method, (7.1)-(7.5). Which method is used had only a negligible eect on the results. The input signal used is u(t) =j cos(t)j [0;2] (t), t2 [0; 20], and the sampling interval is = 0:1. 69 In our rst example, we use a diusion equation model with one random parameter, and in the second one, we use another diusion equation model with two random parameters. For the rst example we estimate the distribution of the random parameter based on the data simulated under the assumption that the random parameter has (i) a truncated uniform distribution, (ii) a truncated exponential distribution, and (iii) a truncated normal distribution, whereas for the second example, we assume the random parameters have a truncated bivariate normal distribution, and simulated data accordingly. 7.2.1 One-Parameter Model In this series of examples we consider the system (7.6),(7.7),(7.9)-(7.11) withq 1 random andq 2 = 1 which models the phenomenon in Figure 7.1. In this case we have q =q 1 2Q = [a;b], W =f'2 H 2 (0; 1); D ' = 0g, H =H 1 L (0; 1) =f'2H 1 (0; 1); D ' = 0g, Dom(A(q)) =f'2V : 1 ' = 0g, and (q) = 1 where (q) is as in (4.19). It follows thata(q;'; ) =q R 1 0 ' 0 () 0 ()d for'; 2V andhb(q); i V ;V =hb; i V ;V = (1) = ( 1); 2 V , and R T 0 w(t;s)hc(q);'(s)i V ;V ds = R T 0 w(t;s)'(s; 1=3)ds, where '2 L 2 ([0;T];V ) and w(t;s) is a positive weight function as it is described in Chapter 4, and 0 = 1=3. Standard arguments [5, 6] show that Assumptions 4.1-4.3 are satised. To carry out the nite dimensional discretization, we let n;m be positive integers and set N = (n;m). In this case we have either D = [a;b] (uniform and normal) or D = [0;R] (exponential). In what follows we describe the q discretization for the uniform and normal cases; the exponential is similar. The basis for the approximating subspacesW N are taken to be tensor Figure 7.1: One-parameter model 70 products of the standard linear spline basis elements ' n i corresponding to the uniform mesh f0; 1 n ; 2 n ;:::; n1 n ; 1g on [0; 1], and the characteristic function basis m j for the interval [a;b]. Thej th element corresponds to the j th sub-interval [a + (j 1) ba m ;a +j ba m ), j = 1; 2;:::;m. In this way W N =spanf N i;j g, i = 1; 2;:::;n, j = 1; 2;:::;m where N i;j (;q) =' n i () m j (q), 2 [0; 1], q2 [a;b] with dim(W N )=nm. Using standard estimates [29] it is not dicult to show that Assumption 4.6 holds. Re-numbering N i;j 's so that N i;j = N k where k = (j 1)n +i, the matrix representation for the operatorsA N is given by [A N ] =(M N ) 1 K N with M N s;r = N r ; N s H = Z b a Z 1 0 N r N s f(q;)ddq = Z b a m j1 m j2 f(q;)dq Z 1 0 ' n i1 ' n i2 d; (7.15) and K N s;r =a(q; N r ; N s ) = Z b a q Z 1 0 @ N r @ @ N s @ f(q;)ddq = Z b a q m j1 m j2 f(q;)dq Z 1 0 ' n0 i1 ' n0 i2 d; (7.16) where r = (j 1 1)n +i 1 , s = (j 2 1)n +i 2 , i 1 ;i 2 = 1; 2;:::;n, j 1 ;j 2 = 1; 2;:::;m. We also have B N s = Z b a N s (1;q)f(q;)dq =' n i2 (1) Z b a m j2 f(q;)dq; (7.17) and C N r = Z b a N r (1=3;q)f(q;)dq =' n i1 (1=3) Z b a m j1 (q)f(q;)dq; (7.18) r;s = 1; 2;:::;nm, r = (j 1 1)n +i 1 , s = (j 2 1)n +i 2 , i 1 ;i 2 = 1; 2;:::;n, j 1 ;j 2 = 1; 2;:::;m. 71 N Uniform Exponential Normal n m a b R a b 4 4 1.76 4.27 2e-5 3.61 2.61 5.44 4.05 0.62 8 8 1.91 4.05 4e-5 3.81 2.29 5.42 4.01 0.40 16 16 1.94 4.00 0.20 4.34 2.17 5.42 4.01 0.37 32 32 1.95 3.99 0.30 5.95 2.15 5.42 4.00 0.35 64 64 1.96 3.99 0.30 11.08 2.14 5.42 4.00 0.35 True Values 2 4 1/3 | | | 4 0.25 Table 7.1: Convergence results for Examples 6.1, 6.2 and 6.3; estimation of the parameters in truncated uniform, exponential and normal distributions With the density f =f 0 (;) =f 0 (; (a;b;)) as given in Denition 7.1 above, if we dene f 1 (;;) = Z f(q;)dq and f 2 (;;) = Z qf(q;)dq; it is a straight forward, albeit somewhat tedious, exercise to compute the partial derivatives @fi @ , @fi @ , @fi @ , @fi @a , @fi @b , i = 0; 1; 2. These partial derivatives show up in the matrices that appear in the adjoint equations (7.1)-(7.5). We test our scheme on truncated uniform ( = (a;b)), exponential ( = (0;R;)) and normal ( = (a;b;;)) distributions. Our results are shown in Table 7.1 and Figure 7.2 below. In the top row of Figure 7.2, we have plotted the converged estimated population models together with the data and the 75% credible band for the truncated uniform, exponential and normal densities. The credible bands can be obtained directly from the solution to the population model. Indeed,q is sampled from the estimated distribution and thenC(q)x N j (;q) Figure 7.2: Estimation results for one-parameter model 72 is evaluated at the sampled q's wherex N j is given by (4.40). Now theq dependence of the solution to the population model is only valid almost everywhere and our convergence framework is an L 2 (in q) theory. Consequently, pointwise evaluation is, strictly speaking, undened. However, the results appear to be useful so we have included them. It is interesting to note that the credible band for the exponential distribution is quite wide, almost to the point of making the population model not that useful. This is because the exponential distribution, especially one with a mean and variance of = 1= = 3, has a rather "fat" tail. The lower left and the lower right panels of Figure 7.2 show the converging estimated pdfs for the truncated exponential and normal distributions, respectively. The lower middle panel shows how the output of the population model compares to the data when the resolution of the nite element discretizations of q and and the truncation point of the densities are varied. It appears from the gure that it is the q discretization that determines the rate of convergence, while a rather coarse discretization seems to suce. We believe that this explains the slow convergence of (the exponential parameter) and (the standard deviation of the normal) observed in Table 7.1 and the lower right panel of Figure 7.2. The truncation of the density appears to have only a negligible eect. 7.2.2 Two-Parameter Model In this example we consider the system (7.6), (7.8)-(7.11), by taking the Robin boundary condition (7.8) at = 0. In this case, q = [q 1 ;q 2 ] is the vector of random parameters with q 2 Q = [a 1 ;b 1 ] [a 2 ;b 2 ], H =L 2 (0; 1), V =H 1 (0; 1), W =H 2 (0; 1), and Dom(A(q)) =f'2H 2 (0; 1) : R ' = 0; 1 ' = u(t)g and (q) = 1 where (q) is as in 4.19. As we show in Chapter 6, the sesquilinear form onVV is given bya(q;'; ) =q 1 R 1 0 ' 0 0 d +'(0) (0) with<b(q); > V ;V = q 2 (1) = q 2 ( 1), 2 V , and R T 0 w(t;s) < c(q);'(s) > V ;V ds = R T 0 w(t;s)'(s; 0)ds, where '2L 2 ([0;T];V ) and w(t;s) is a positive weight function as in Chapter 4 and we set 0 = 0. In this case N = (n;m 1 ;m 2 ), where n is again the level of discretization of the space variable and 73 m i is the level of discretization of q i , i = 1; 2. Once again the approximating subspaces were constructed using tensor products as we dened in Chapter 6. We re-number N i;j;k (;q 1 ;q 2 ) so that i;j;k = N l where l = [(j 1)m 2 +k 1](n + 1) +i with i = 1;:::;n + 1, j = 1;:::;m 1 , and k = 1;:::;m 2 and we obtain the matrices given in (7.14) as follows. M N s;r = N r ; N s H = Z b1 a1 Z b2 a2 Z 1 0 N r N s f(q;)ddq 2 dq 1 = Z b1 a1 Z b2 a2 m1 1;j1 m2 2;k1 m1 1;j2 m2 2;k2 f(q;)dq 2 dq 1 Z 1 0 ' n i1 ' n i2 d; (7.19) K N s;r =a(q; N r ; N s ) + Z b1 a1 Z b2 a2 N r (0;q) N s (0;q)f(q;)dq 2 dq 1 = Z b1 a1 Z b2 a2 Z 1 0 q 1 @ N r @ @ N s @ ddq 2 dq 1 +' n i1 (0)' n i2 (0) Z b1 a1 Z b2 a2 m1 1;j1 m2 2;k1 m1 1;j2 m2 2;k2 f(q;)dq 2 dq 1 = Z b1 a1 Z b2 a2 q 1 m1 1;j1 m2 2;k1 m1 1;j2 m2 2;k2 f(q;)dq 2 dq 1 Z 1 0 (' n i1 ) 0 (' n i2 ) 0 d +' n i1 (0)' n i2 (0) Z b1 a1 Z b2 a2 m1 1;j1 m2 2;k1 m1 1;j2 m2 2;k2 f(q;)dq 2 dq 1 (7.20) B N s = Z b1 a1 Z b2 a2 q 2 N s (1;q)f(q;)dq 2 dq 1 =' n i2 (1) Z b1 a1 Z b2 a2 q 2 m1 1;j2 m2 2;k2 f(q;)dq 2 dq 1 C N r = Z b1 a1 Z b2 a2 N r (0;q)f(q;)dq =' n i1 (0) Z b1 a1 Z b2 a2 m1 1;j1 m2 2;k1 f(q;)dq 2 dq 1 where r;s = 1;:::; (n + 1)m 1 m 2 with r = [(j 1 1)m 2 +k 1 1](n + 1) +i 1 , s = [(j 2 1)m 2 +k 2 1](n + 1) +i 2 for i 1 ;i 2 = 1;:::;n + 1, j 1 ;j 2 = 1;:::;m 1 , and k 1 ;k 2 = 1;:::;m 2 . 74 In this example the truncated exponential family is based on the bivariate normal. Once again, it is possible to compute all the partial derivatives (although of course their evaluation requires the numerical evaluation of single and double integrals) that are required to form the matrices that appear in the state and adjoint equations (7.1)-(7.5). We obtained simulated data by generating samples for q from a N( ; ) distribution with = 2 6 6 4 12 10 3 7 7 5 and = 2 6 6 4 9 3 3 5 3 7 7 5 . n m 1 m 2 a b c d 4 8 8 5.88 18.15 4.85 14.63 11:72 9:88 12:13 5:76 5:76 7:35 8 8 8 5.67 18.35 5.17 14.46 11:68 9:87 10:15 4:04 4:04 5:97 16 8 8 5.79 18.17 5.06 14.66 11:67 9:86 9:29 3:03 3:03 5:21 Table 7.2: Convergence results for Example 6.4; estimation of the parameters in truncated bivariate normal distribution Our results are shown in Table 7.2 and Figure 7.3, where it can be seen that we obtain reasonably good approximations to the actual parameters that we use to simulate the data. We parameterize the covariance matrix as = L T L, where the 2 2 matrix L is upper triangular with L 11 and L 22 both positive so as to guarantee that at each step in the optimization, is positive denite symmetric. The plot of the optimal joint density in the left panel of Figure 7.3 corresponds to n = 16 and m 1 =m 2 = 8. In the right panel of Figure 7.3 we have the output of the t population model and the 75% credible band. Figure 7.3: Estimated probability density function and 75% credible band for forward model 75 7.3 Numerical Results with Alcohol Biosensor Data In this section, we present both our parameter estimation and input estimation results by using actual data. We use the same model that we used in Section 7.2.2. For the parameter estimation problem, all the matrices and vectors in (7.14) are the same as they are described in Section 7.2.2. However, for the deconvolution problem, we consider two dierent approaches for some of the examples to show dierent possibilities. One in which we allow the estimated BrAC signal to depend on the random parameters,q 1 andq 2 , in addition to depending on time, and the other where we sought estimated BrAC as a function of time only. The rst method has the benet of allowing for the ecient computation of credible bands by simply sampling the distribution of the random parameters and then directly substituting the samples into an expression for the estimated BrAC as a function of q 1 and q 2 . In the second method credible bands had to be computed via sampling and simulation. With the rst method, training takes longer because of the time involved in solving the optimization problem (5.20) associated with the computation of the optimal regularization parameters. Then the deconvolution of the estimated BrAC signal and computation of the associated credible bands is very quick. On the other hand, for the second approach, the training is quicker than the rst, but because of the need to simulate the deterministic (4.14) model with each of the random samples ofq 1 andq 2 , the deconvolution and credible band determination step is considerably slower than the rst approach. Since the training is o-line and the deconvolution and credible band determination is generally on-line, the rst approach is desirable. In comparing the results for the two methods, we observe generally not remarkable dierences. When u is a function of both t and q, by the denition in (5.3), B L in (7.14) will become an (n + 1)m 1 m 2 mm 1 m 2 dimensional matrix as follows B L s;p = Z b1 a1 Z b2 a2 q 2 N s (1;q) M p (t;q)f(q;)dq 2 dq 1 : (7.21) 76 However, when u is a function of only t, B L wll be an (n + 1)m 1 m 2 m dimensional matrix as B L s;p = Z b1 a1 Z b2 a2 q 2 N s (1;q) M p (t)f(q;)dq 2 dq 1 ; (7.22) where M p = m i and setS M =spanf M P g M P . 7.3.1 Dataset 1 - Single Subject with Multiple Drinking Episodes A WrisTAS TM 7 alcohol biosensor was worn for 18 days by a single individual. In addition, contemporaneous breath analyzer BrAC measurements were collected. The WrisTAS TM 7 (Figure 7.4; Giner, Inc, Waltham, MA) was set to measure the local ethanol vapor concentration over the skin surface at 5-minute intervals. The rst drinking episode was conducted in the laboratory with BrAC measured and recorded every 15 minutes from the start of the drinking session until BrAC returned to 0.000. Then the TAC device was worn in the eld and consumed alcohol ad libitum for the following 17 days. For each drinking episode, BrAC readings were taken every 30 minutes until the BrAC returned to 0.000. Figure 7.5 shows the entire 18 day TAC signal along with the contemporaneous BrAC measurements. The 11 individual drinking episodes are marked. The TAC measurements provided by the sensor are in units of milligrams per deciliter (mg/dl), and the BrAC measurements are in units of percent alcohol. We t the population consisting of all eleven drinking episodes, but we also visually stratied the population into two groups by using Figure Figure 7.4: Alcohol Biosensor Device: The WrisTAS TM 7 77 Figure 7.5: BrAC and TAC measurements for dataset 1 7.5. The rst group contains the data belonging to drinking episodes 1, 2, 4, 6, 7, 8, 11. In each of these drinking episodes, the peak BrAC value is higher than the bench calibrated peak TAC value. The second group contains the remaining drinking episodes, 3, 5, 9, and 10 for which the peak TAC value is higher than the peak BrAC value. Before performing our estimation procedure, we rst picked randomly two datasets, episodes 1 and 6, from the rst group and one dataset, episode 9, from the second group. We excluded the drinking episodes that were chosen for cross validation and obtained estimation results by using the remaining eight drinking episodes for training. As described previously in Section 7.1, we take two dierent approaches in estimating the input: (1) we assume that u was a function of t only and (2) we assume u was a function of both t and q. Note that these two approaches dier from each other when the deconvolution problem is solved, but the parameter estimation problems are formulated in the same way for both of them. Therefore, the optimal parameters are the same. We found the optimal support to Figure 7.6: Optimal pdfs obtained by total and stratied populations for dataset 1 78 be [a 1 ;b 1 ] [a 2 ;b 2 ] = [0; 2:4063] [0; 2:6376] and the optimal parameters to be = 2 6 6 4 0:5738 1:2387 3 7 7 5 and = 2 6 6 4 0:0665 0:0019 0:0019 0:1943 3 7 7 5 . However, the optimal regularization parameters turned out to be dierent in the two cases. They were r 1 = 1:5051 and r 2 = 0:5658 for the rst approach and r 1 = 0:8238 and r 2 = 0:4203 for the second approach. The optimal pdf can be seen on the left panel of Figure 7.6. The estimation results, under the assumption that u is a function of only t, for the training drinking episodes which are 2, 3, 4, 5, 7, 8, 10, and 11 can be seen in Figure 7.7. Those results that are obtained under the assumption that u is a function of both t and q for the training drinking episodes can be seen in the Figure 7.8 and the estimation results for these two approaches for the cross validation datasets can be seen in the rst and second row of Figure 7.10, respectively. Note that in each plot, solid black line represents our estimation (eBrAC) for actual BrAC and dashed black line represents (sTAC) which is the output obtained by using eBrAC and our population model. Figure 7.7: Estimation results for the training drinking episodes by assuming u is a function of only t using total population for dataset 1 We also obtain estimation results after stratifying the dataset into two groups as described above under the assumption that u is a function of both t andq. For the rst group, we nd optimal support to be [a 1 ;b 1 ][a 2 ;b 2 ] = [0; 1:2529][0; 2:2773] and the optimal mean and covariance matrix 79 Figure 7.8: Estimation results for the training drinking episodes by assuming u is a function of both t andq using total population for dataset 1 to be = 2 6 6 4 0:3944 1:0860 3 7 7 5 and = 2 6 6 4 0:0179 0:0066 0:0066 0:1727 3 7 7 5 , and the optimal regularization parameters to ber 1 = 0:8469 andr 2 = 0:2342. On the other hand, the optimal support for the second group was [0; 3:4097] [0; 3:2392] and the optimal parameters were = 2 6 6 4 2:0657 1:5094 3 7 7 5 , = 2 6 6 4 0:4797 0:0091 0:0091 0:2883 3 7 7 5 , r 1 = 0:0083, and r 2 = 0:7266. The optimal pdfs of the rst and second groups can be seen in the middle and right panels of Figure 7.6, respectively. The estimation results for the training drinking episodes in both groups are shown in Figure 7.9, and the results for the cross validation drinking episodes are shown on the third row of Figure 7.10. Figure 7.9: Estimation results for the training drinking episodes by assuming u is a function of both t andq after stratication for dataset 1 80 Figure 7.10: Estimation results for the cross validation drinking episodes obtained by assuming u is a function of only t using total population, and assuming u is a function of both t andq using total and stratied population for dataset 1 In the upper left panel of Figure 7.11 we have the mean or expected value of the approximating impulse response function or convolution kernel for the population model, given by h N l ( ) =C( ) ^ A N ( ) l1 A N ( ) 1 ( ^ A N ( )I)B N ( ) =C( ) ^ A N ( ) l1 A N ( ) 1 ( ^ A N ( )I) P N H B( ); l = 1; 2;:::;K, together with the 75% credible band. Recall that for each l = 1; 2;:::;K, h N l ( ), or at least its representer, is an element in L 2 ( ) (Q ) and thus is a function ofq, and consequently, the mean and credible bands can be obtained by simply substituting in samples ofq. Note that as a result of the fact that the bases for our approximating subspaces are tensor products, the impulse response function for the scheme in which we simply sought an estimate for BrAC that was a function of t only (and not of q 1 and q 2 ), turn out to be the mean of h N l ( ), l = 1; 2;:::;K, E ( ) [h N l ( )], and as such are plotted in this gure as well. In the upper right panel of Figure 7.11 we plot the surface, h N l ( ) as a function of q 1 and q 2 at the time at which E ( ) [h N l ( )] 81 Figure 7.11: Convolution kernel, Estimated BrAC, and pdf of estimated BrAC obtained for dataset 1 is at its peak. In the lower left panel of Figure 7.11 we have the estimated BrAC for drinking episode 7, which is chosen randomly, as a function of q 1 and q 2 at the time at which its mean is at its peak, and nally in the lower right panel of Figure 7.11 we plot the pdf for the estimated BrAC at the time it is at its peak. Once again we note that since the estimated BrAC at each time t is an element L 2 ( ) (Q ) and thus is a function ofq, this pdf can be obtained by simply generating samples ofq from the bivariate normal distribution determined by the parameters and then simply substituting them into the obtained expression for the estimated BrAC, (7.13). In addition to the estimated continuous BrAC signals shown in the plots in the previous section, clinicians and alcohol researchers are interested in a number of statistics associated with the BrAC for a drinking episode. Specically, for each drinking episode, they look at the maximum or peak value of the BrAC, the time (since the start of the episode) at which the peak value of the BrAC is attained, the area underneath the episode's BrAC curve, the BrAC absorption rate, and the BrAC elimination rate. The BrAC absorption rate is dened to be the peak BrAC value divided by the amount of time that elapses from the last zero BrAC measurement (more precisely the time at which the BrAC level rst rises above the predened threshold) until the time at which the peak BrAC value was attained, and the BrAC elimination rate is dened to be the peak BrAC 82 value divided by the the amount of time that elapses from the time at which the peak BrAC value was attained until the rst zero BrAC measurement (more precisely the time at which the BrAC level rst sinks below a predened threshold). In Tables 7.3-7.7 we provide these statistics and their credible intervals for the drinking episodes in the single subject data. In these tables, we use our estimated BrAC surfaces and samples from the bivariate normal distribution corresponding to the parameters to compute 75% credible intervals for each of the statistics. Note that in all of these tables, the episodes and subjects without an asterisk (*) are used in training the two peak BrAC Total Population Stratied Population Ep. actual estimated credible band actual estimated credible band 1* 0.0520 0.0452 [0.0347,0.0634] 0.0520 0.0546 [0.0366,0.0829] 2 0.0230 0.0113 [0.0074,0.0184] 0.0230 0.0131 [0.0053,0.0290] 3 0.0360 0.0468 [0.0362,0.0650] 0.0360 0.0333 [0.0122,0.0591] 4 0.0570 0.0495 [0.0344,0.0767] 0.0570 0.0563 [0.0273,0.1079] 5 0.0520 0.0713 [0.0564,0.0974] 0.0520 0.0529 [0.0196,0.0936] 6* 0.0230 0.0160 [0.0088,0.0290] 0.0230 0.0183 [0.0045,0.0486] 7 0.0180 0.0104 [0.0048,0.0204] 0.0180 0.0117 [0.0014,0.0349] 8 0.0170 0.0155 [0.0115,0.0227] 0.0170 0.0186 [0.0113,0.0314] 9* 0.0260 0.0302 [0.0187,0.0506] 0.0260 0.0241 [0.0081,0.0438] 10 0.0130 0.0195 [0.0119,0.0333] 0.0130 0.0148 [0.0051,0.0272] 11 0.0480 0.0441 [0.0296,0.0702] 0.0480 0.0513 [0.0226,0.1058] Table 7.3: Peak BrAC for each drinking episode in dataset 1 obtained for total population and stratied population time of peak BrAC Total Population Stratied Population Ep. actual estimated credible band actual estimated credible band 1* 0.7500 0.5500 [0.2167,0.7167] 0.7500 0.4333 [0.2667,0.6000] 2 0.5000 0.9500 [0.5667,1.0500] 0.5000 0.7667 [0.5833,0.9667] 3 1.5000 0.8500 [0.4833,1.0000] 1.5000 1.1000 [1.0667,1.1333] 4 2.0000 2.8667 [2.3833,3.0500] 2.0000 2.7500 [2.4667,3.0167] 5 3.0000 2.2167 [1.8833,2.3167] 3.000 2.3833 [2.3500,2.4000] 6* 0.5833 1.2833 [0,1.3833] 0.5833 1.0833 [0.8833,1.2833] 7 1.5833 1.6167 [0,1.7333] 1.5833 1.3333 [1.0500,1.5833] 8 0.8333 0.7833 [0.4167,0.9167] 0.8333 0.6667 [0.4667,0.8333] 9* 2.0000 2.4167 [1.8333,2.6000] 2.0000 2.6333 [2.6000,2.6667] 10 1.0833 0.7500 [0.3667,0.8833] 1.0833 0.9333 [0.8833,0.9667] 11 0.9167 1.6333 [1.1000,1.8667] 0.9167 1.4833 [1.2000,1.7833] Table 7.4: Time of peak BrAC for each drinking episode in dataset 1 obtained for total population and stratied population 83 area under the BrAC curve Total Population Stratied Population Ep. actual estimated credible band actual estimated credible band 1* 0.1019 0.0958 [0.0562,0.1367] 0.1019 0.1082 [0.0724,0.1609] 2 0.0163 0.0122 [0.0027,0.0218] 0.0163 0.0145 [0.0048,0.0335] 3 0.0750 0.1086 [0.0638,0.1540] 0.0750 0.0941 [0.0329,0.1693] 4 0.1666 0.1283 [0.0451,0.2137] 0.1666 0.1477 [0.0682,0.2923] 5 0.1988 0.2272 [0.1562,0.2992] 0.1988 0.1964 [0.0701,0.3490] 6* 0.0267 0.0173 [0,0.0394] 0.0267 0.0203 [0.0034,0.0601] 7 0.0265 0.0146 [0,0.0356] 0.0265 0.0170 [0.0012,0.0555] 8 0.0154 0.0224 [0.0101,0.0362] 0.0154 0.0255 [0.0148,0.0447] 9* 0.0554 0.0606 [0.0124,0.1108] 0.0554 0.0518 [0.0168,0.0948] 10 0.0175 0.0220 [0.0018,0.0458] 0.0175 0.0184 [0.0062,0.0335] 11 0.1181 0.0727 [0.0150,0.1342] 0.1181 0.0848 [0.0294,0.1919] Table 7.5: Area under the BrAC curve for each drinking episode in dataset 1 obtained for total population and stratied population population models, while those marked with an asterisk are held back for cross validation. All of these results are computed using the rst approach (BrAC a function of both time andq 1 and q 2 ). We compute these statistics and their credible intervals using the rst approach. absorption rate Total Population Stratied Population Ep. actual estimated credible band actual estimated credible band 1* 0.0693 0.0287 [0.0216,0.0614] 0.0693 0.0398 [0.0290,0.0655] 2 0.0460 0.0098 [0.0035,0.0201] 0.0460 0.0112 [0.0057,0.0223] 3 0.0240 0.0241 [0.0178,0.0377] 0.0240 0.0191 [0.0072,0.0337] 4 0.0285 0.0121 [0.0048,0.0262] 0.0285 0.0151 [0.0079,0.0361] 5 0.0260 0.0218 [0.0154,0.0374] 0.0260 0.0178 [0.0068,0.0315] 6* 0.0394 0.0141 [0,0.0276] 0.0394 0.0153 [0.0057,0.0320] 7 0.0114 0.0060 [0,0.0126] 0.0114 0.0072 [0.0016,0.0169] 8 0.0204 0.0087 [0.0057,0.0225] 0.0204 0.0117 [0.0084,0.0269] 9* 0.0130 0.0086 [0.0020,0.0170] 0.0130 0.0072 [0.0027,0.0129] 10 0.0120 0.0176 [0.0041,0.0309] 0.0120 0.0141 [0.0052,0.0250] 11 0.0524 0.0271 [0.0114,0.0469] 0.0524 0.0342 [0.0186,0.0621] Table 7.6: Absorption rate for each drinking episode in dataset 1 obtained for total population and stratied population 7.3.2 Dataset 2 - Multiple Subjects each with Single Drinking Episode This dataset was collected from multiple subjects wearing WrisTAS TM 7 using the same protocol as in the rst episode in dataset 1. Thirty two participants (47% female) were administered 84 elimination rate Total Population Stratied Population Ep. actual estimated credible band actual estimated credible band 1* 0.0173 0.0065 [0.0059,0.0118] 0.0173 0.0139 [0.0105,0.0217] 2 0.0230 0.0130 [0.0040,0.0183] 0.0230 0.0105 [0.0070,0.0281] 3 0.0180 0.0177 [0.0105,0.0269] 0.0180 0.0031 [0.0012,0.0054] 4 0.0258 0.0166 [0.0127,0.0299] 0.0258 0.0245 [0.0178,0.0466] 5 0.0130 0.0065 [0.0057,0.0100] 0.0130 0.0049 [0.0019,0.0088] 6* 0.0162 0.0142 [0,0.0293] 0.0162 0.0177 [0.0057,0.0463] 7 0.0196 0.0094 [0,0.0252] 0.0196 0.0113 [0.0023,0.0322] 8 0.0204 0.0063 [0.0047,0.0100] 0.0204 0.0091 [0.0074,0.0145] 9* 0.0120 0.0227 [0.0066,0.0524] 0.0120 0.0201 [0.0076,0.0365] 10 0.0130 0.0140 [0.0042,0.0273] 0.0130 0.0103 [0.0035,0.0196] 11 0.0137 0.0194 [0.0100,0.0350] 0.0137 0.0270 [0.0194,0.0562] Table 7.7: Elimination rate for each drinking episode in dataset 1 obtained for total population and stratied population a moderate, gender- and weight-adjusted dose of alcohol designed to reach a peak BrAC of approximately 0.05%. All participants were monitored in the laboratory with breathalyzer readings taken by study sta at 15-minute intervals until BrAC returned to 0.000. The study was approved by the University of California, San Diego Human Research Protection Program and written informed consent for participation was obtained. Figure 7.12: Optimal pdfs and scatter plots for q samples obtained by total and stratied populations for dataset 2 85 For dataset 2, we rst used the raw BrAC and TAC from each drinking episode to estimate the two parameters q 1 and q 2 deterministically for the 32 subjects separately, which provided general information about the range and distribution of these parameters and could be done for any data screened to be included as part of our population model approach. Seven of the episodes produced parameter values that were outliers (i.e., orders of magnitude out of range). To avoid over-tting of the models or increasing computational burden and to be consistent with standard procedure in optimization problems, these episodes were excluded from being considered for the population sample. In addition, due to sensor failures (i.e., peak TAC < 0:02, which is the threshold typically required to indicate a drinking episode, n = 4) and glitches/artifacts in the TAC recordings (i.e., spikes that could be attributed to exposure of the device to external alcohol sources, n = 3, and dips due to lack of direct contact with the skin, n = 4), we removed an additional 11 episodes from these analyses (see [24] for additional details on sensor failure rates). This resulted in a nal sample of 15 subjects (47% female). Figure 7.13: Estimation results for the training drinking episodes using total population for dataset 2 86 We then plotted the TAC and BrAC data of the 15 episodes. Similar to dataset 1, we saw that in 11 episodes peak BrAC was higher than peak TAC, and in 4 episodes peak TAC was higher than peak BrAC. We randomly selected two of the 11 episodes with higher BrAC (episodes 13 and 21) and one of the four episodes with higher TAC (episode 14) to use as the three test episodes, which were held out of the data used for the population model estimates. The remaining 12 episodes were used as the training data for establishing the population parameters. Figure 7.14: Estimation results for the training drinking episodes using stratied population for dataset 2 We have plots and tables similar to those for dataset 1 showing the results for this dataset. For the total population, we obtained the optimal support as [a 1 ;b 1 ] [a 2 ;b 2 ] = [0; 2:6619] [0; 2:5524], and the optimal mean and covariance function as = 2 6 6 4 0:8515 1:3286 3 7 7 5 , = 2 6 6 4 0:0807 0:0019 0:0019 0:1805 3 7 7 5 . The corresponding optimal pdf can be seen on the upper left panel of Figure 7.12. In addition to the optimal pdfs, we provide scatter plots on the bottom row of Figure 7.12. These scatter plots show the q values generated by the optimal distribution for each case. Among 87 Figure 7.15: Estimation results for the testing drinking episodes using total and stratied population for dataset 2 all these values, the red ones are the samples that lie in a circular region centered at the mean with 75% probability. Figure 7.13 shows the estimation results for dataset 2 for all 12 episodes which are used for training to estimate the distribution of the random parameters. Note that for this dataset we peak BrAC Total Population Stratied Population Ep. actual estimated credible band actual estimated credible band 1 0.0550 0.0318 [0.0182,0.0455] 0.0550 0.0420 [0.0227,0.0632] 2 0.0520 0.0791 [0.0532,0.1301] 0.0520 0.0543 [0.0370,0.0638] 5 0.0640 0.0296 [0.0194,0.0393] 0.0640 0.0396 [0.0261,0.0552] 9 0.0600 0.0458 [0.0305,0.0599] 0.0600 0.0607 [0.0399,0.0841] 12 0.0770 0.0675 [0.0416,0.0931] 0.0770 0.0900 [0.0551,0.1290] 13* 0.0670 0.0238 [0.0145,0.0331] 0.0670 0.0316 [0.0188,0.0460] 14* 0.0390 0.0516 [0.0295,0.0748] 0.0390 0.0363 [0.0200,0.0489] 15 0.0610 0.0976 [0.0656,0.1607] 0.0610 0.0667 [0.0445,0.0843] 16 0.0610 0.0276 [0.0182,0.0456] 0.0610 0.0368 [0.0251,0.0502] 17 0.0460 0.0369 [0.0176,0.0582] 0.0460 0.0485 [0.0208,0.0778] 20 0.0500 0.0228 [0.0139,0.0320] 0.0500 0.0305 [0.0189,0.0436] 21* 0.0610 0.0354 [0.0222,0.0482] 0.0610 0.0467 [0.0282,0.0673] 22 0.0500 0.0677 [0.0442,0.0896] 0.0500 0.0468 [0.0294,0.0608] 24 0.0510 0.0278 [0.0188,0.0457] 0.0510 0.0370 [0.0248,0.0509] 26 0.0460 0.0230 [0.0157,0.0297] 0.0460 0.0306 [0.0218,0.0409] Table 7.8: Peak BrAC for each drinking episode in dataset 2 obtained for total and stratied population 88 time of peak BrAC Total Population Stratied Population Ep. actual estimated credible band actual estimated credible band 1 0.5833 1.7833 [1.5833,1.9500] 0.5833 1.7000 [1.4833,1.8000] 2 0.8333 1.1667 [1.0333,1.2833] 0.8333 1.2500 [1.1333,1.3000] 5 0.5000 0.8667 [0.7500,0.9667] 0.5000 0.8000 [0.6667,0.8667] 9 0.7500 1.7333 [1.5833,1.8500] 0.7500 1.6500 [1.4833,1.7333] 12 0.3333 0.8333 [0.6667,0.9667] 0.3333 0.7500 [0.5667,0.8333] 13* 0.3333 1.3167 [1.1667,1.4333] 0.3333 1.2500 [1.0833,1.3167] 14* 1.1667 2.1167 [1.9833,2.2333] 1.1667 2.2167 [2.0833,2.2667] 15 0.7500 1.1500 [1.0167,1.2500] 0.7500 1.2167 [1.1000,1.2667] 16 0.6667 0.8667 [0.7500,0.9500] 0.6667 0.7833 [0.6667,0.8333] 17 0.5000 2.5333 [2.3000,2.7167] 0.5000 2.4500 [2.2167,2.5333] 20 0.3333 1.6667 [1.5167,1.7833] 0.3333 1.6000 [1.4500,1.6667] 21* 0.7500 1.3167 [1.1333,1.4500] 0.7500 1.2333 [1.0333,1.3167] 22 0.6333 1.2333 [1.1000,1.3333] 0.6333 1.3000 [1.1667,1.3500] 24 0.9167 1.4333 [1.3000,1.5500] 0.9167 1.3667 [1.2000,1.4333] 26 1.2500 1.4833 [1.3333,1.6000] 1.2500 1.4000 [1.2667,1.4667] Table 7.9: Time of peak BrAC for each drinking episode in dataset 2 obtained for total and stratied population obtain all the results under the assumption that u is a function of botht andq. The sTAC curves consistently mappes onto TAC, indicating that the inversion process also workes well for this dataset. As can be observed on the rst row of Figure 7.13, however, there is greater variability in the eBrAC curves, with some of the eBrAC curves close to the BrAC curves (e.g., episode 15) and others not so close (e.g., episode 20). Once again, the respective 75% credible bands capture a good portion of the actual BrAC curves (see Tables 7.8-7.12). Even in episodes where eBrAC is not that close to the BrAC, the general shape of the eBrAC curve is relatively representative of the true BrAC curve. The testing drinking episode results for the three episodes held out of the training data are shown in the top row of Figure 7.15 and in Tables 7.8-7.12. Again, we observe that sTAC mappes onto the raw TAC data well, indicating the inversion process workes well when episodes outside of the population data are run in the model. The eBrAC curve results are more variable, with relatively good t for episode 21, but relatively poor t for episodes 13 and 14. Since we are solving this estimation problem based on the population data and forming a single model for 89 area under the BrAC curve Total Population Stratied Population Ep. actual estimated credible band actual estimated credible band 1 0.1095 0.0791 [0.0416,0.1371] 0.1095 0.1010 [0.0479,0.1558] 2 0.1310 0.1968 [0.1322,0.2552] 0.1310 0.1332 [0.0898,0.1681] 5 0.1226 0.0789 [0.0457,0.1128] 0.1226 0.1001 [0.0524,0.1500] 9 0.1307 0.1123 [0.0742,0.1851] 0.1307 0.1431 [0.0910,0.1995] 12 0.1162 0.1509 [0.0851,0.2577] 0.1162 0.1917 [0.1032,0.2842] 13* 0.1131 0.0529 [0.0295,0.0905] 0.1131 0.0674 [0.0342,0.1020] 14* 0.0767 0.1091 [0.0599,0.1613] 0.0767 0.0730 [0.0365,0.1008] 15 0.1640 0.2435 [0.1567,0.4036] 0.1640 0.1659 [0.1034,0.2148] 16 0.1375 0.0706 [0.0452,0.0944] 0.1375 0.0894 [0.0569,0.1245] 17 0.0983 0.0828 [0.0345,0.1492] 0.0983 0.1056 [0.0376,0.1740] 20 0.0959 0.0472 [0.0243,0.0735] 0.0959 0.0600 [0.0313,0.0898] 21* 0.1220 0.0929 [0.0553,0.1297] 0.1220 0.1186 [0.0656,0.1744] 22 0.1223 0.1535 [0.0917,0.2589] 0.1223 0.1047 [0.0550,0.1457] 24 0.1173 0.0654 [0.0440,0.0847] 0.1173 0.0837 [0.0541,0.1159] 26 0.0857 0.0565 [0.0383,0.0736] 0.0857 0.0716 [0.0502,0.0955] Table 7.10: Area under the BrAC curve for each drinking episode in dataset 2 obtained for total and stratied population everyone in the population, this second dataset, which is collected from multiple subjects and contains more variability than dataset 1 collected on a single subject, yielded less favorable results than our results from dataset 1. absorption rate Total Population Stratied Population Ep. actual estimated credible band actual estimated credible band 1 0.0943 0.0150 [0.0095,0.0208] 0.0943 0.0207 [0.0125,0.0296] 2 0.0624 0.0388 [0.0265,0.0661] 0.0624 0.0290 [0.0208,0.3500] 5 0.1280 0.0177 [0.0114,0.0245] 0.1280 0.0239 [0.0160,0.0340] 9 0.0800 0.0185 [0.0122,0.0315] 0.0800 0.0252 [0.0171,0.0355] 12 0.2310 0.0371 [0.0251,0.0608] 0.2310 0.0517 [0.0354,0.0709] 13* 0.2010 0.0135 [0.0083,0.0187] 0.2010 0.0174 [0.0114,0.0255] 14* 0.0334 0.0209 [0.0128,0.0291] 0.0334 0.0148 [0.0100,0.0190] 15 0.0813 0.0496 [0.0330,0.0844] 0.0813 0.0354 [0.0256,0.0477] 16 0.0915 0.0148 [0.0104, 0.0242] 0.0915 0.0206 [0.0150,0.0274] 17 0.0920 0.0162 [0.0092,0.0277] 0.0920 0.0212 [0.0116,0.0319] 20 0.1500 0.0134 [0.0080,0.0230] 0.1500 0.0180 [0.0113,0.0261] 21* 0.0813 0.0154 [0.0104,0.0198] 0.0813 0.0210 [0.0139,0.0292] 22 0.0789 0.0349 [0.0236,0.0493] 0.0789 0.0284 [0.0192,0.0361] 24 0.0556 0.0122 [0.0083, 0.0204] 0.0556 0.0164 [0.0115,0.0231] 26 0.0368 0.0104 [0.0070,0.0137] 0.0368 0.0139 [0.0099,0.0194] Table 7.11: Absorption rate for each drinking episode in dataset 2 obtained for total and stratied population 90 elimination rate Total Population Stratied Population Ep. actual estimated credible band actual estimated credible band 1 0.0150 0.0124 [0.0075,0.0218] 0.0150 0.0159 [0.0104, 0.0255] 2 0.0160 0.0226 [0.0161,0.0375] 0.0160 0.0156 [0.0105,0.0188] 5 0.0160 0.0066 [0.0046,0.0084] 0.0160 0.0087 [0.0061,0.0124] 9 0.0171 0.0175 [0.0126,0.0295] 0.0171 0.0244 [0.0183,0.0356] 12 0.0197 0.0237 [0.0156,0.0329] 0.0197 0.0307 [0.0226,0.0455] 13* 0.0212 0.0086 [0.0058,0.0114] 0.0212 0.0114 [0.0081,0.0171] 14* 0.0187 0.0250 [0.0175,0.0446] 0.0187 0.0170 [0.0104,0.0237] 15 0.0152 0.0232 [0.0165,0.0312] 0.0152 0.0161 [0.0105,0.0201] 16 0.0152 0.0050 [0.0035,0.0067] 0.0152 0.0070 [0.0050,0.0097] 17 0.0153 0.0172 [0.0099,0.0315] 0.0153 0.0229 [0.0117,0.0396] 20 0.0136 0.0068 [0.0057,0.0117] 0.0136 0.00089 [0.0086,0.0153] 21* 0.0203 0.0130 [0.0088,0.0224] 0.0203 0.0168 [0.0122,0.0259] 22 0.0121 0.0164 [0.0124,0.0224] 0.0121 0.0114 [0.0091, 0.0174] 24 0.0170 0.0085 [0.0069,0.0144] 0.0170 0.0140 [0.0100,0.0198] 26 0.0221 0.0042 [0.0033,0.0072] 0.0221 0.0094 [0.0071,0.0128] Table 7.12: Elimination rate for each drinking episode in dataset 2 obtained for total and stratied population We stratied the dataset into two groups according to their peak raw BrAC and peak raw TAC values. For the group with peak BrAC higher than peak TAC we obtained the optimal support to be [a 1 ;b 1 ] [a 2 ;b 2 ] = [0; 2:2877] [0; 2:1410], and optimal mean and covariance function to be = 2 6 6 4 0:6397 1:0424 3 7 7 5 , = 2 6 6 4 0:0596 0:0013 0:0013 0:1287 3 7 7 5 . On the other hand, for the group with peak TAC higher than the peak BrAC, those values turned out to be [a 1 ;b 1 ] [a 2 ;b 2 ] = [0; 3:7651] [0; 3:7855], = 2 6 6 4 1:4571 1:9290 3 7 7 5 , = 2 6 6 4 0:1636 0:0044 0:0044 0:4017 3 7 7 5 . The optimal pdfs for both of these subgroups can be seen on the top middle and top right panel of Figure 7.12, respectively. When we stratied the training episodes based on whether raw TAC or raw BrAC was higher, sTAC continued to t TAC well, and now the eBrAC curves mapped more closely on to the BrAC curves in both the training episodes (see Figure 7.14) and the test episodes (see the bottom row of Figure 7.15 compared with the top row). The results in Tables 7.8-7.12 show the statistics when the dataset is stratied compared with the left columns when the dataset is run as a whole. Considering both the gures and tables, we can conclude that on average, stratication improved the results. 91 7.3.3 Dataset 3 - Multiple Subjects each with Single Drinking Episode In this example, we consider training data collected from multiple subjects at the University of Illinois at Urbana-Champaign (see [18]) using the Secure Continuous Alcohol Monitoring System (SCRAM) manufactured by Alcohol Monitoring Systems in Littleton, Colorado (shown in Figure 7.16). Sixty participants were administered a moderate, gender and weight adjusted dose of alcohol (0.82 g/kg for men, 0.74 g/kg for women) intended to produce a peak BrAC of approximately .08%. The alcoholic beverage was administered over the course of 36 minutes. All participants were then monitored in the laboratory while breathalyzer readings were taken at approximately 30 minute intervals and the SCRAM assessed transdermal readings. Participants were allowed to leave the laboratory once their BrAC dropped below .03%. However, this dataset was not collected specically for what we used it for. Because of this reason, in some cases, the trial was halted before the TAC started to decrease from its peak. Consequently, we had to exclude data belonging to 37 participants. We then selected 18 subjects from the group of 23 of these subjects whose both BrAC and TAC data had returned to zero or close to zero. Note that the data for the remaining 5 subjects was corrupted or unusable for one reason or another unrelated to the focus of this investigation. A scatter plot of their deterministically t values is shown in Figure 7.18. The study was approved by the University of Illinois at Urbana - Champaign Institutional Review Board and all participants provided written informed consent. Figure 7.16: Alcohol Biosensor Device: The SCRAM 92 Figure 7.17: Optimal pdf obtained for dataset 3 We used our deterministic model to estimate the values for q = (q 1 ;q 2 ) for each of the 18 datasets separately. The scatter plot of theseq values can be seen in Figure 7.18. According to this plot, we again visually stratied the datasets by selecting the data of eight of the subjects, subjects 1, 8, 9, 10, 20, 21, 22, and 23, whoseq values clustered around the origin. We then chose subjects 8 and 10 randomly for cross validation and used the remaining six subjects, numbers 1, 9, 20, 21, 22, and 23, for training. The optimal t population truncated bivariate normal pdf is plotted in Figure 7.17 and the converged values for the parameters were [a 1 ;b 1 ] [a 2 ;b 2 ] = [0; 1:1080] [0; 0:7826], Figure 7.18: Scatter plot of deterministically obtained parameter estimates for the 18 datasets considered in the dataset 3 93 Figure 7.19: Estimation results for the training drinking episodes by assuming u is a function of both t andq for dataset 3 = 2 6 6 4 0:3462 0:5618 3 7 7 5 , = 2 6 6 4 0:0140 0:0067 0:0067 0:0361 3 7 7 5 . The optimal regularization parameters are r 1 = 0:0000 and r 2 = 2:7480. The input or BrAC estimates for the subjects 1, 9, 20, 21, 22, and 23, which are the ones used for training, can be seen in Figures 7.19 and 7.20. Our results for the caseu depends on both t andq are in Figure 7.19, and our results for the case u depends on t only are shown in Figure Figure 7.20: Estimation results for the training drinking episodes by assuming u is a function of only t for dataset 3 94 Figure 7.21: Estimation results for the cross validation drinking episodes obtained by assuming u is a function of only t, and assuming u is a function of both t andq for dataset 3 7.20. Analogous results for the cross validation subjects, 8 and 10, can be found in Figure 7.21. Figure 7.22 has the plots for this dataset that are analogous to the plots in 7.11 for dataset 1. The statisrics for peak BrAC, time of peak BrAC, area under the BrAC curve, absorption rate and elimination rate can be found in Tables 7.13-7.17 for all of the drinking episodes in dataset 3. Figure 7.22: Convolution kernel, Estimated BrAC, and pdf of estimated BrAC For dataset 3 95 peak BrAC Ep. actual estimated credible band 1 0.0820 0.0974 [0.0368,0.1719] 8* 0.0740 0.0458 [0.0149,0.0894] 9 0.0640 0.0408 [0.0129,0.0793] 10* 0.0750 0.0862 [0.0285,0.1646] 20 0.1020 0.0696 [0.0275,0.1215] 21 0.0820 0.0706 [0.0255,0.1051] 22 0.0810 0.0762 [0.0265,0.1418] 23 0.0390 0.0422 [0.0143,0.0804] Table 7.13: Peak BrAC for each drinking episode in dataset 3 time of peak BrAC Ep. actual estimated credible band 1 2.3333 4.3333 [3.7833,4.7667] 8* 2.0500 1.8667 [1.5167,2.1167] 9 3.0000 3.0167 [2.6000,3.3000] 10* 1.3833 2.8833 [2.5000,3.1500] 20 1.6833 2.9667 [2.5500,3.2667] 21 2.0000 2.3333 [1.9000,2.6500] 22 2.1000 2.7833 [2.2667,3.1667] 23 2.6500 2.6500 [2.3000,2.9000] Table 7.14: Time of peak BrAC for each drinking episode in dataset 3 area under the BrAC curve Ep. actual estimated credible band 1 0.3822 0.4543 [0.1595,0.8149] 8* 0.2392 0.1296 [0.0377,0.1983] 9 0.2568 0.1312 [0.0364,0.2634] 10* 0.2590 0.2688 [0.0805,0.4094] 20 0.3820 0.2578 [0.0929,0.3809] 21 0.2884 0.2536 [0.0827,0.4853] 22 0.2905 0.2868 [0.0891,0.5521] 23 0.1444 0.1218 [0.0372,0.1848] Table 7.15: Area under the BrAC curve for each drinking episode in dataset 3 absorption rate Ep. actual estimated credible band 1 0.0351 0.0185 [0.0079,0.0312] 8* 0.0361 0.0160 [0.0059,0.0237] 9 0.0213 0.0112 [0.0044,0.0165] 10* 0.0542 0.0229 [0.0092,0.0336] 20 0.0606 0.0180 [0.0079,0.0313] 21 0.0410 0.0212 [0.0088,0.0308] 22 0.0386 0.0203 [0.0082,0.0295] 23 0.0147 0.0120 [0.0048,0.0175] Table 7.16: Absorption rate for each drinking episode in dataset 3 96 elimination rate Ep. actual estimated credible band 1 0.0132 0.0220 [0.0095,0.0406] 8* 0.0180 0.0175 [0.0069,0.0350] 9 0.0130 0.0135 [0.0048,0.0261] 10* 0.0109 0.0304 [0.0114,0.0578] 20 0.0203 0.0151 [0.0076,0,0276] 21 0.0178 0.0186 [0.0082,0.0352] 22 0.0143 0.0211 [0.0082,0.0405] 23 0.0089 0.0154 [0.0060,0.0291] Table 7.17: Elimination rate for each drinking episode in dataset 3 97 Chapter 8 Discussion and Concluding Remarks The results presented in this thesis suggest a number of open mathematical questions that deserve further consideration. In the treatment here, although we evaluate them at specic values of q 1 and q 2 , the surfaces we obtain for the estimated impulse response function and input and the associated convergence theory are in fact only L 2 making point-wise evaluation undened. We are currently looking at the introduction of some form of parabolic regularization [23] into the q-dependence of the population model. In this way, it may become possible to obtain coercivity of the sesquilinear form stated in Section 7.2.2 with respect to a stronger norm and to obtain H 1 -like well-posedness and convergence of the approximations in the q dependence of the state, input, and output. We also hope to be able to apply the convergence theory based on the Prohorov metric on a space of measures developed in [4] more directly to the class of problems that we have discussed here. More precisely, we would like to be able to eliminate the assumption that the measures are dened in terms of a density, and estimate the measure directly. We believe that such a theory may be possible by assuming that our approximating subspaces are required to satisfy additional regularity (i.e. smoothness) assumptions; in particular that they are required to be contained in the domain of the operator. Then by making use of a slightly dierent version of the Trotter-Kato semigroup approximation theorem (see, for example, [3]) we believe it may now be possible to verify the hypotheses of the more general convergence theorem established in [4] for the estimation 98 of the probability measures directly. Another extension in this same spirit would be to attempt to directly estimate the shape of the density rather than simply its parameters. It seems that in this case, existing results for the estimation of functional parameters in PDEs may be directly applicable. Also, one might consider a parameterization for the random parameters in the model in terms of their polynomial chaos expansions much as is done in [19]. Improved performance of the population model could potentially be obtained with higher delity models with higher dimensional parameterization; for example these might include the addition of an advection term, non-constant or functional coecients (probably with nite dimensional paramerization), damped second order hyperbolic models (e.g. the telegraph equation) for diusion with nite speed of propagation (see, for example, [25]). It would also be interesting to see if an analogous nonlinear theory could be developed for innite dimensional systems governed by maximal monotone operators (see, for example, [7]). In addition, we are looking at the application of the approach in [19] and [30] to control problems for random parabolic systems and in particular, the computation of the feedback solution to the linear quadratic regulator and compensator problems for random distributed parameter systems. As can be seen in our results in Chapter 7, stratifying the dataset into subgroups the members of which share certain common characteristics yielded improved estimation results. In those studies, we stratied the datasets according to a comparison of peak BrAC and peak TAC values, in practice, which generally not be possible (i.e., because BrAC would not be available). Consequently, we are interested in investigating the eect of stratication based on readily observable person-level covariates, for example, height, weight, sex, heavy/light drinker, etc. In addition, if episode-level variations like skin hydration or ambient temperature improve our estimation results, then it is possible to incorporate weather or location data into models to include these types of environmental variation. Finally, it also is possible to include within-subject variation across episodes, such as skin temperature or hydration (e.g., the WrisTAS TM 7 device has temperature and skin conductance sensors in addition to the TAC sensor). This type of a multilevel stratication pr ocess should 99 enhance our estimation results and narrow the credible bands and intervals, with subgroups of the population who share similar person-level and/or episode-level characteristics providing more ecient estimates than those using a single population model for the all people in the population. Finally, if alcohol biosensors were to be incorporated into wearable health monitoring technology (e.g. the Fitbit and the Apple Watch), our approach would have to be modied to produce estimated BrAC in real time. This would be a challenge in light of the inherent latency of the human body's metabolism and transdermal secretion via perspiration of ethanol and the limitations of the analog hardware in the current state-of-the-art transdermal alcohol sensors. 100 Reference List [1] Robert A. Adams and John J. F. Fournier. Sobolev spaces, Jun 2003. [2] H. T. Banks and J. A. Burns. Hereditary control problems:numerical methods based on averaging approximations. SIAM Journal on Control and Optimization, 16(2):169{208, 1976. [3] H. T. Banks, J. A. Burns, and E. M. Cli. Parameter estimation and identication for systems with delays. SIAM Journal on Control and Optimization, 19(6):791{828, 1981. [4] H. T. Banks and W Clayton Thompson. Least squares estimation of probability measures in the prohorov metric framework. Technical report, DTIC Document, 2012. [5] H. Thomas Banks and Karl Kunisch. Estimation techniques for distributed parameter systems. Birkhauser, Boston-Basel-Berlin, 1989. [6] H.T. Banks and K. Ito. A unied framework for approximation in inverse problems for distributed parameter systems. Control Theory Advanced Technology, 4(1):73{90, 1988. [7] Viorel P. Barbu. Nonlinear Semigroups and Dierential Equations in Banach Spaces. Noord- ho International Publishing, Leyden, The Netherlands, 1976. [8] Stephen L Campbell and Carl D Meyer. Generalized inverses of linear transformations. SIAM, 2009. [9] George Casella and Roger L Berger. Statistical inference, volume 2. Duxbury Pacic Grove, CA, 2002. [10] RF Curtain and D Salamon. Finite-dimensional compensators for innite-dimensional systems with unbounded input operators. SIAM Journal on Control and Optimization, 24(4):797{816, 1986. [11] Zheng Dai, I G Rosen, Chunming Wang, Nancy Barnett, and Susan E Luczak. Using drinking data and pharmacokinetic modeling to calibrate transport model and blind deconvolution based data analysis software for transdermal alcohol biosensors. Mathematical biosciences and engineering: MBE, 13(5):911{934, 2016. [12] D. M. Dougherty, N. E. Charles, and S.and Furr R. M.and Hill-Kapturczak N. Acheson, A.and John. Comparing the detection of transdermal and breath alcohol concentrations during periods of alcohol consumption ranging from moderate drinking to binge drinking. Experimental and Clinical Psychopharmacology, 20:373{381, 2012. [13] D. M. Dougherty, N. Hill-Kapturczak, Y. Liang, T. E. Karns, S. E. Cates, S. L. Lake, and J. D. Roache. Use of continuous transdermal alcohol monitoring during a contingency management procedure to reduce excessive alcohol use. Drug and Alcohol Dependence, 142:301{306, 2014. 101 [14] D. M. Dougherty, T. E. Karns, J. Mullen, Y. Liang, S. L. Lake, J. D. Roache, and N. Hill- Kapturczak. Transdermal alcohol concentration data collected during a contingency man- agement program to reduce at-risk drinking. Drug and Alcohol Dependence, 148:77{84, 2015. [15] Miguel A Dumett, IG Rosen, J Sabat, A Shaman, L Tempelman, C Wang, and RM Swift. Deconvolving an estimate of breath measured blood alcohol concentration from biosensor collected transdermal ethanol data. Applied Mathematics and Computation, 196(2):724{743, 2008. [16] Lawrence C. Evans. Partial Dierential Equations, volume 19 of Graduate Studies in Mathe- matics. American Mathematical Society, Providence, Rhode Island, 1998. [17] C. E. Fairbairn, K. Bresin, D. Kang, I. G. Rosen, T. Ariss, S. E. Luczak, N. P. Barnett, and N. S. Eckland. A multimodal investigation of contextual eects on alcohols emotional rewards. Journal of Abnormal Psychology, to appear. [18] C. E. Fairbairn, K. Bresin, D. Kang, I. G. Rosen, T. Ariss, Barnett N. P., S. E. Luczak, and Eckland N. S. A multimodal investigation of contextual eects on alcohol's emotional rewards. Manuscript submitted for publication, 2017. [19] Claude Jerey Gittelson, Roman Andreev, and Christoph Schwab. Optimality of adaptive galerkin methods for random parabolic partial dierential equations. J. Computational Applied Mathematics, 263:189{201, 2014. [20] Tosio Kato. Perturbation theory for linear operators, volume 132. Springer Science & Business Media, 2013. [21] Dominick A Labianca. The chemical basis of the breathalyzer: A critical analysis. J. Chem. Educ, 67(3):259{261, 1990. [22] AFJ Levi and IG Rosen. A novel formulation of the adjoint method in the optimal design of quantum electronic devices. SIAM Journal on Control and Optimization, 48(5):3191{3223, 2010. [23] J.L. Lions. Optimal Control of Systems Governed by Partial Dierential Equations. Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Ber ucksichtigung der Anwendungsgebiete. Springer-Verlag, 1971. [24] Susan E Luczak, I Gary Rosen, and Tamara L Wall. Development of a real-time repeated- measures assessment protocol to capture change over the course of a drinking episode. Alcohol and alcoholism, 50(2):180{187, 2015. [25] Akira Okubo and Simon A. Levin. Diusion and Ecological Problems, Modern Perspectives, Second Edition. Springer, New York, 2001. [26] A. Pazy. Semigroups of Linear Operators and Applications to Partial Dierential Equations. Applied Mathematical Sciences. Springer, 1983. [27] Anthony J Pritchard and Dietmar Salamon. The linear quadratic control problem for innite dimensional systems with unbounded input and output operators. SIAM Journal on Control and Optimization, 25(1):121{144, 1987. [28] I. G. Rosen, Susan E. Luczak, and Jordan Weiss. Blind deconvolution for distributed parameter systems with unbounded input and output and determining blood alcohol concentration from transdermal biosensor data. Applied Mathematics and Computation, 231:357{376, 2014. 102 [29] M. H. Schultz. Spline Analysis. Prentice-Hall, 1973. [30] Christoph Schwab and Claude Jerey Gittelson. Sparse tensor discretizations of high- dimensional parametric and stochastic pdes. Acta Numerica, 20:291467, 2011. [31] Melike Sirlanci and I. G. Rosen. Estimation of the distribution of random parameters in discrete time abstract parabolic systems with unbounded input and output: Approximation and convergence. Manuscript submitted for publication, 2017. [32] Blair K Swartz and Richard S Varga. Error bounds for spline and l-spline interpolation. Journal of Approximation Theory, 1972. [33] H. Tanabe. Equations of evolution. Monographs and studies in mathematics. Pitman, 1979. [34] Gregory D Webster and Hampton C Gabler. Feasibility of transdermal ethanol sensing for the detection of intoxicated drivers. In Annual Proceedings/Association for the Advancement of Automotive Medicine, volume 51, page 449. Association for the Advancement of Automotive Medicine, 2007. 103 
Abstract (if available)
Abstract Being motivated by an alcohol biosensor problem, the input to a random parabolic partial differential equation is estimated given the output based on population data. The problem is parsed into two subproblems: a parameter estimation problem and a deconvolution problem. Because of having various sources of uncertainty, the parameters in the mathematical model are assumed to be random and defined in terms of a joint density. In this case, estimating the distribution is equivalent to estimating the joint distribution function. A finite dimensional abstract approximation and convergence theory is developed for estimation of the distribution of random parameters in infinite dimensional discrete time linear systems with dynamics described by regularly dissipative operators and involving, in general, unbounded input and output operators. By taking expectations, the system is re-cast as an equivalent abstract parabolic system in a Gelfand triple of Bochner spaces wherein the random parameters become new space-like variables. Estimating their distribution is now analogous to estimating a spatially varying coefficient in a standard deterministic parabolic system. The estimation problems are approximated by a sequence of finite dimensional problems. Convergence is established using a state space-varying version of the Trotter-Kato semigroup approximation theorem. The resulting system, which is formulated based on the estimated distribution of the random parameters, is referred to as a population model. Once the forward population model has been identified or trained based on a sample from the population, the resulting distribution can then be used to deconvolve the input signal from the observed output signal formulated as either a quadratic programming or linear quadratic tracking problem. In addition, our approach allows for the direct computation of corresponding credible bands without simulation. When applied to the alcohol biosensor problem, the model takes the form of a diffusion equation with the input, which is on the boundary of the domain, being the blood or breath alcohol concentration (BAC/BrAC), and the output, also on the boundary, being the transdermal alcohol concentration (TAC). All the assumptions used to construct our theoretical framework are shown to be satisfied by the model for the alcohol biosensor problem. To assess the efficiency of the approach, numerical results obtained by working with simulated data for a number of examples involving the estimation of exponential families of densities for random parameters in a diffusion equation with boundary input and output are presented and discussed. In addition, the technique is used to estimate bivariate normal distributions and deconvolve BAC/BrAC from TAC based on data from a population that consists of multiple drinking episodes from a single subject and two populations consisting of single drinking episodes from multiple subjects. 
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Creator Sirlanci Tuysuzoglu, Melike (author) 
Core Title Finite dimensional approximation and convergence in the estimation of the distribution of, and input to, random abstract parabolic systems with application to the deconvolution of blood/breath al... 
School College of Letters, Arts and Sciences 
Degree Doctor of Philosophy 
Degree Program Applied Mathematics 
Publication Date 07/20/2018 
Defense Date 04/18/2018 
Publisher University of Southern California (original), University of Southern California. Libraries (digital) 
Tag deconvolution,distribution estimation,linear semigroups of operators,OAI-PMH Harvest,population model,random abstract parabolic systems,random parameters, distributed parameter systems,regularly dissipative operators,system identification,transdermal alcohol biosensor 
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Language English
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Advisor Rosen, Gary (committee chair), Goldstein, Larry (committee member), Luczak, Susan Elizabeth (committee member), Wang, Chunming (committee member) 
Creator Email melikesirlanci@gmail.com,sirlanci@usc.edu 
Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-21622
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Tags
deconvolution
distribution estimation
linear semigroups of operators
population model
random abstract parabolic systems
random parameters, distributed parameter systems
regularly dissipative operators
system identification
transdermal alcohol biosensor