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Obtaining breath alcohol concentration from transdermal alcohol concentration using Bayesian approaches
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Obtaining breath alcohol concentration from transdermal alcohol concentration using Bayesian approaches
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ObtainingBreathAlcoholConcentrationfromTransdermalAlcohol ConcentrationUsingBayesianApproaches by KeenanJ.Hawekotte ADissertationPresentedtothe FACULTYOFTHEGRADUATESCHOOL UNIVERSITYOFSOUTHERNCALIFORNIA InPartialFulfillmentofthe RequirementsfortheDegree DOCTOROFPHILOSOPHY (AppliedMathematics) August2021 Copyright 2021 KeenanJ.Hawekotte Dedication Toyou. ii Acknowledgements I want to thank my advisor, Dr. I. Gary Rosen for the many hours spent on the phone at all hours of the day and night. Also, I would like to thank all those on my committee, namely, Dr. Susan E. Luczak and Dr. Chunming Wang. Further, I want to thank Dr. Susan E. Luczak for letting meusedatacollectedfromherlabintheDepartmentofPsychologyattheUniversityofSouthern CaliforniaaspartofIRB-approvedhumansubjectexperiments. I would also like to thank all of my friends, both in and out of math, for the many, many conversations. Austin Pollok taught me more about my own research than I care to admit. Ju- lian Aronowitz let me talk at him for hours to help clear my mind and get my thoughts straight. Stephanie and Charlie Viana helped me keep a cool head through the most challenging years of graduate school. Josh Batra was always willing to lend an ear no matter the topic at hand. Kelly andAveryFink,andJ.BrandonWelchspentyearslisteningtometalkaboutmyresearchprogres- sionandconstantlyencouragedmetopushforward. AlexWolpaandTevinGriffinencouragedme topursuemypassionslongbeforeknowingwhattheywereandsupportedmethroughthebestand worstoftimes. HankRobinsonwasalwaystheretoactasaguideformeandhelpedmemaintain focusonmygoals. IwanttodirectlythankthemembersoftheGonz´ alezfamily,Jos´ e,Laura,Julia,Charlie,Pick- les,andQuixote. Theyopenedtheirhometomeandshowedunconditionalsupport. Ofcourse,Iwouldliketothankmyfamily. Ifnotforthem,Iwouldn’tbehere. Last but not least, I would love to thank my partner, Olivia Gonz´ alez, and her wonderful cat, George. Withoutthem,IwouldnotbewhoIamtoday,andthisthesisverywellmightceasetobe. Thankyou. iii TableofContents Dedication ii Acknowledgements iii ListofTables vi ListofFigures viii Abstract ix Chapter1: Introduction 1 1.1 LiteratureReview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Chapter2: Preliminaries 9 2.1 FunctionalAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Analysis,andMisc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Chapter3: ADistributedParameterModelfortheTransdermalTransportofAlcohol 27 Chapter4: BayesianEstimationofDynamicalSystemParameters 44 4.1 ConvergenceinDistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.3 DeconvolutionofBrACfromTAC . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.4 NumericalResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.4.1 ConvergenceinDistribution . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.4.2 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.4.3 Deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Chapter5: BayesianEstimationofDynamicalSystemInput 77 5.1 Infinite-DimensionalFormulation . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2 Finite-DimensionalFormulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.3 NumericalResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.3.1 ConvergenceinDistributionofthePredictivePosterior . . . . . . . . . . . 85 5.3.2 PredictiveDeconvolutionofBrACfromTAC . . . . . . . . . . . . . . . . 85 iv 5.3.3 NaiveStratificationofPopulationData . . . . . . . . . . . . . . . . . . . 86 Chapter6: DiscussionandConcludingRemarks 94 6.1 BayesianEstimationofModelParameters . . . . . . . . . . . . . . . . . . . . . . 94 6.1.1 DeconvolutionofBrACfromTAC . . . . . . . . . . . . . . . . . . . . . . 95 6.1.2 ConcludingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.2 BayesianEstimationofDynamicalSystemInput . . . . . . . . . . . . . . . . . . 98 6.2.1 PredictiveDeconvolutionofBrACfromTAC . . . . . . . . . . . . . . . . 98 6.2.2 NaiveStratificationofPopulationData . . . . . . . . . . . . . . . . . . . 99 6.2.3 ConcludingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 References 102 v ListofTables 4.1 90%crediblecirclesforMetropolis-HastingsMCMCsampledposteriorswithnoise distributionN(0;0:005 2 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2 90%crediblecirclesforMCMCsampledposteriorsfromidealizedTACdatawith noise distributedN(0;0:025 2 ) and prior distribution equivalent to the one in Sec- tion6of[76],namelyatMVNwith = (0.63181.0295),Σ = (0.0259 0.00770.0077 0.1232), andq bounds[0:01;2:2877]and[0:01;2:1410]. . . . . . . . . . . . . . . . . . . . 71 4.3 90%crediblecirclesforMetropolis-HastingsMCMCsampledposteriorswithnoise distributedN(0;0:025 2 ) and prior distribution equivalent to the one in Section 6 of[76],namelyatMVNwith = (0.63181.0295),Σ = (0.0259 0.00770.0077 0.1232),and q bounds[0:01;2:2877]and[0:01;2:1410]. . . . . . . . . . . . . . . . . . . . . . . 72 4.4 DataassociatedwithposteriordeterminationanddeconvolutionusedinFigure4.4 forparametervaluesina70%crediblecircle. . . . . . . . . . . . . . . . . . . . . 74 4.5 DataassociatedwithposteriordeterminationanddeconvolutionusedinFigure4.4 forparametervaluesina90%crediblecircle. . . . . . . . . . . . . . . . . . . . . 74 4.6 DataassociatedwithposteriordeterminationanddeconvolutionusedinFigure4.5 forparametervaluesina70%crediblecircle. . . . . . . . . . . . . . . . . . . . . 75 4.7 DataassociatedwithposteriordeterminationanddeconvolutionusedinFigure4.5 forparametervaluesina90%crediblecircle. . . . . . . . . . . . . . . . . . . . . 76 5.1 Calculatedmeansandstandarddeviationsforvaryingvaluesofnfortestdrinking episodesubjectBT323LeftArmwithdrinkingpatterndual. . . . . . . . . . . . . 86 5.2 Means for all metrics of every utilized configuration. The mean ground truth BrACdescendingrateis0:015834andthemeangroundtruthBrACascend- ingrateis0:065255. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.3 Standard deviations for all metrics of every utilized configuration. The standard deviation ground truth BrAC descending rate is 0:004001 and the standard deviationgroundtruthBrACascendingrateis0:028918. . . . . . . . . . . . . 91 vi 5.4 Mediansforallmetricsofeveryutilizedconfiguration. Themediangroundtruth BrAC descending rate is 0:015887 and the median ground truth BrAC as- cendingrateis0:061444. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 vii ListofFigures 4.1 (Left) WristTAS TM 7 Continuous Alcohol Monitoring device, (Right) SCRAM SystemsContinuousAlcoholMonitoringdevice. . . . . . . . . . . . . . . . . . . 68 4.2 Posterior distribution surf plots for varying finite dimensional approximations of thekernelfrom(4.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3 Posterior distribution surface plots for varying amounts of collected data, m. All images use noise distributedN(0,0.025 2 ) and prior distribution equivalent to that of Section 6 of [76], namely a tMVN with µ = (0.63181.0295), Σ = (0.0259 0.00770.0077 0.1232), andq bounds[0.01,2.2877]and[0.01,2.1410]. . . . . . . . . . . . . . . . . . . . . . . . . 72 4.4 BrAC deconvolution given TAC and predicted q values for varying participants’ right arm data with gray error regions. Across sub-figures, all training data re- mained constant with R = 25. Prior used was tMVN with bounds [0:01;10] [0:01;10], = (55), and Σ = (0.7 0.10.1 0.55). Associated data is contained in Tables4.4and4.5,for70%and90%parametercredibleregions,respectively. . . . 74 4.5 BrAC deconvolution given TAC and predicted q values derived using varying amount of training data,R, with gray error regions. All sub-figures use the same test TAC data from a single left arm session from BT333. Prior used was tMVN with bounds [0:01;10] [0:01;10], = (55), and Σ = (0.7 0.10.1 0.55). Associ- ateddataarecontainedinTables4.6and4.7,for70%and90%parametercredible regions,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.1 Plots displaying the given testing TAC graph, predicted mean BrAC graphs, and credible bands for3 standard deviations for the predicted mean BrAC function usingallavailabletrainingdata. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 viii Abstract We study the problem of determining blood or breath alcohol (BAC or BrAC) concentration from transdermal alcohol concentration (TAC). In this way, we make use of two different solution ap- proaches. In the initial method, we estimate the posterior distribution of random parameters in a distributed parameter, diffusion equation-based population model for the transdermal transport of alcohol. WethensequentiallyestimatethedistributionoftheinputBrACtothemodel. Theoutput of the model is TAC, which via linear semigroup theory can be expressed as the convolution of BrAC with a filter that depends on the individual participant or subject, the biosensor hardware itself, and environmental conditions, all of which can be considered to be random under the pre- sentedframework. WeutilizeaBayesianapproachtoestimatetheposteriordistributionofthepa- rametersand/orthedeconvolvedBrACconditionedonanindividualsmeasuredTAC(andBrAC). Priors for the models are obtained from temporal population observations of BrAC and TAC via deterministicorstatisticalmethods. Therequisitecomputationsrequirefinite-dimensionalapprox- imation of the underlying state equation, which is achieved through standard finite element (i.e., Galerkin) techniques. We establish consistency of our Bayesian estimators, and we demonstrate theconvergenceofourposteriordistributionscomputedbasedonthefiniteelementmodeltothose based on the underlying infinite-dimensional model. We also present some of the results of our human subject data-based numerical studies. The second method we employ involves the estima- tion of the posterior distribution of a random signal describing BrAC, the previously mentioned distributed parameter, diffusion equation-based model for the transdermal transport of ethanol. We develop a multivariate normal-based Bayesian approach to estimate the posterior distribution ix of the deconvolved BrAC signal conditioned on an individuals measured TAC (and the popula- tion’smeasuredBrACandTAC).Priorsforthemodelsareobtainedbyfittingmultivariatenormal distributions using temporal population observations of BrAC and TAC via deterministic or sta- tistical methods. Again, computations require finite-dimensional approximation of the underlying state equation and the associated semigroup of operators that determines the convolution kernel or filter, and this is achieved through standard finite element Galerkin techniques. We establish the convergence of our approximating posterior distributions to the posterior distributions corre- spondingtotheoriginalinfinite-dimensionalmodelusingapproximationresultsfromthetheoryof linearsemigroupsofoperators. Wepresentasampleofresultsfromsomeofournumericalstudies and validate our convergence result by demonstrating the convergence of our finite-dimensional approximating posterior distributions to the posterior distribution determined by the underlying infinite-dimensional models. In both Bayesian approaches previously mentioned, the posteriors providecredibleregionsandyieldpopulationmodelsthateliminatetheneedtocalibratethemodel foreveryindividual,differentbiosensordevices,andvaryingenvironmentalconditions. x Chapter1 Introduction What follows are the ideas, results, and possible future paths for a problem revolving around the calibration of transdermal alcohol biosensors and their usage in determining blood (BAC) or breathalcoholconcentration(BrAC).Outsideofacademia,peoplewhohavepreviouslycommitted offensessuchasdrivingundertheinfluence(DUI)ofalcoholmayhavetheiralcoholconsumption monitored. Currently, the most prevalent system that is built-in to a user’s car does not allow for continuous monitoring of a person’s BAC but instead requires them to take BrAC readings when theywishtostarttheirvehicles. Duetothis,thesysteminplaceplacesheavyemphasisonpunitive actions, whereas with a less discrete knowledge of BAC, a shift towards a rehabilitative structure canoccur. Inacademia,continuousknowledgeofBACisalsobeneficial. Ifpossible,ratherthanrequiring studies to be done in laboratories under sterile conditions, researchers could then allow for more naturalistic drinking to occur while capturing near-continuous alcohol readings. Unfortunately, the current laboratory methods of blood draws and urinalysis are quite prohibitive to furthering continuous knowledge in such areas of research. BrAC measurements and drinker’s self-reports arealsonotentirelyreliable,Usingabreathalcoholanalyzercorrectlyrequiresspecializedtraining andcanproduceerroneousmeasurementsduetomouthalcoholand/orreadingsbasedonashallow breath by the subject. Self-reports often lead to misrepresentation as subjects may deviate from natural actions due to the reporting requirement feeling unnatural, and alcohol directly impairs subjects ability to be active participants. Due to these disadvantages and caveats, for our work, 1 we seek other methods that allow us to approach real-time BAC knowledge while quantifying uncertainty. Anapproachwithtechnologicalsupport(see,[81,82,83])thatcouldcombatthedisadvantages outlinedaboverevolvesaroundreadingandmeasuringtransdermalalcoholcontent(TAC),i.e.,the amount of alcohol that diffuses through the skin (see, [17, 37, 55, 60]). The benefits derived from the creation of such a device include the availability of near-continuous measurements and the ability to collect them passively (i.e., without the active participation of the subject). This gives researchersand cliniciansthe potentialto continuouslyobservenaturalisticdrinking behaviorand patterns. Alongwiththis,thereisthepossibilityofmakingthesedevicesavailableontheconsumer market (e.g., wearable body system monitoring technology like Fitbits, Apple watches, etc.). The ideas we discuss here may also apply to the monitoring of other substances once the appropriate sensorhardwarehasbeendeveloped. The challenge in using transdermal alcohol sensors is that they provide TAC, whereas alcohol researchers and clinicians have always based their studies and treatments on measurements of BrACandoccasionally,whenavailable,onBAC.Thus,ameanstoreliablyandaccuratelyconvert TAC to BAC or BrAC would be desirable. Although BrAC correlates well with BAC [44, 49] (at least up to higher levels above 0.08, see [47, 68]), the correlation between TAC and BrAC/BAC varies due to several confounding factors. These factors include, but are not limited to, stable features of the skin like its thickness, tortuosity, and porosity, particularly as they apply to the epidermallayeroftheskin,whichdoesnothaveanactivebloodsupply. Inaddition,environmental factors such as ambient temperature and humidity can affect both perspiration and vasodilation, and thus may alter skin conductance, blood flow, the amount of alcohol passing below the skin in the blood, and the amount and rate of alcohol diffusing through the skin. Manufacturing and operationalvariationsamongdifferentsensorsarealsootherexpectedreal-worldhurdles. To convert TAC measurements to BAC/BrAC, earlier attempts used deterministic models [24, 28, 29, 30, 31, 63, 64]. Regression-based models were utilized by some [28], whereas others utilized first principles physics-based models that, on occasion, included modeling the transport 2 of alcohol from ingestion to excretion through the skin [88, 89]. Other initial efforts modeled the transport of alcohol from the blood in the dermal layer through the epidermal layer and its eventual measurement by the sensor using a one-dimensional diffusion equation [24, 63]. The parameters in the diffusion equation model then had to be fit or tuned (i.e., calibrated) to each individual subject, the environmental conditions, and the device through the use of simultaneous BrAC/TAC training data collected in the laboratory or clinic through a procedure known as an alcohol challenge. Once the model was fit, it could then be used to deconvolve BrAC from TAC. This two-pass approach and the related studies were relatively successful [24, 31, 63, 64, 76, 77]. However, this calibration procedure is quite burdensome to researchers, clinicians, subjects, and patients; andbecausethemodelswerefittoasingleuni-modaldrinkingepisode, unaccountedfor variation and uncertainty in the relationship between BrAC and TAC frequently arose, making it difficulttoaccuratelyconvertTACcollectedinthefieldtoBrAC[41,45]. Thus, more recent approaches have had the stated goal of eliminating the need for individual calibration and reframed the problem by seeking to fit a population model to BrAC/TAC training data from across a cohort of subjects, devices, and environmental conditions. To achieve this, re- searchers considered random diffusion equations wherein the parameters in the transport model are defined to be random variables. Then it is the distribution of these parameters that are esti- matedfromdataratherthanthevaluesoftheparametersthemselves. Then,usingthesepopulation models, BrAC is estimated based on TAC, and the uncertainty in the estimates is quantified using the distributions of the random parameters to produce conservative error bands that surround the estimatedBrACsignal[76,78]. Combiningthisknowledgewiththeproblemdepictionfromabovetherearisesanopportunity. In particular, the next step would be to investigate the inclusion of randomness into the first prin- ciples physics-first models already shown to be viable deterministically. As in [76, 77, 78], the main approach for including randomness revolved around putting distributions over the param- eters of the models and thus letting them be random variables; for which the distributions will beparameterizedanddeterminedfrompopulationdata. Thesolutionapproachstillretainsitstwo 3 parts,parameterestimationoftheforwardprocessanddeconvolutionofthebackwardstep;though nowtheparametersbeingestimatedarethoseofthedistributiondescribingthemodelparameters. Similar to before, once the parameters have been estimated, the backward deconvolution can be formulated as a quadratic programming or linear-quadratic tracking problem. The key to this new formulationliesinthepowerofapplyingdistributionsovertheparameterswhereinitallowsdirect computation of credible intervals/bands of BrAC given field TAC data. This not only provides userswithmoreinformationbutalsoexpandstherangeofdatathatcanbeusedwithinthemodel. The current model is based on diffusion, or Fick’s Law, and takes the form of an abstract parabolic or hyperbolic system that has been extended to contain random parameters. We now insteadturnourfocustomoreBayesian-basedapproaches. Withinthese,thereisanassumedprior (“known” before) distribution that is then used alongside the distribution of our newly acquired data to provide an updated distribution (the posterior distribution) that should better reflect the population from which the data was collected. There are two main ways in which this view can be applied to the problem. The first is to apply Bayesian methods to the parameters of the model, whereinthesolutionwillcontainthreesteps. Theinitialstepischoosingapriordistribution,which isfullycontrolledbythemodelersandisaimedatcapturingaperceiveddistribution. Thenextstep involvesusingdataandtheposteriordistributiontodeterminecrediblebandsonmodelparameters. Then in the last step, deconvolution, the posterior is used to estimate a BrAC curve. The second methodthatintegratesBayesianapproachesappliesadistributiondirectlyovertheBrACfunction itself, or more specifically, over a finite vector of times within which we seek to estimate the associated BrAC values. Thus, assuming model parameters are known, we instead need only two steps prior selection and credible band determination. In both of our Bayesian approaches, we must consider the necessary finite dimensionalization and subsequent desired convergences. For the former method involving parameters, our general approach is built on ideas that seek to produce convergences of the posterior distribution itself (see, [19, 35, 72, 87]). For the latter, BrAC-based Bayesian approach, we predominately utilize classical probabilistic results focusing 4 on convergences with respect to the expectation operator. Utilizing these convergences, we may formcomputationalschemesthatutilizemoderncomputationalpowertobedeployedinthefield. Inshort,weimmediatelyfollowwithabriefliteraturereviewoftopicsutilizedintheremainder ofthethesis;Chapter3coversthesetupoftheunderlyingmodel;Chapter4coverstheproblemand numericalresultsofusingBayesiantechniquestoestimatemodelparameters;Chapter5coversthe problem and numerical results of using Bayesian methods to estimate BrAC directly from TAC; andChapter6coversdiscussionofallnumericalresultsandconcludingremarks. 1.1 LiteratureReview Parameter estimation problems permeate the natural phenomena modeling sphere as investigated by [15]. More specifically, parameter estimation for distributed systems in physiologically-based problemsgarneredattentionfromtheworkofbiologistsstudyinginterstitialfluidwithinthebrain [23,48,65]. InitialapproachesprimarilyutilizedFick’sLawtodevelopdiffusion-basedpartialdif- ferentialequations(PDE)thatmodelthedesiredparticlemovementandthenappliedleast-squares approximation techniques to determine optimal parameters given a set of data [14, 79]. Regres- sionanalysistechniques,suchastheanalysisofvariance(ANOVA),werethenutilizedtoperform model evaluation and comparison [9, 10]. Further work wasthen performed that looked into non- linearstatisticalmodelsusingweightedleast-squares(see,[36])andgeneralizednonlinearmodels [73]. In addition, confidence and asymptotic distributions have been investigated for such param- etersin[8],and[26]. Asinitiallyoutlinedin[15],alloftheseapproachesleadtothedevelopment of rigorous theory surrounding approximation and convergence of estimates of least-squares esti- mators. Though, almost all focused solely on utilizing least-squares approaches and maintained a strictdeterministicviewoftheunderlyingdynamicalsystem. Following the initial deterministic approaches, stochastic approaches then followed, leading to the desire to fit random parameters of dynamical systems. To do this, measurement error in- herent to any physical system was introduced between the underlying theoretical PDE model and 5 the observed, and recorded data measurements [12]. Further [12], laid the framework for esti- mation of probability measures in random abstract evolution equations and the convergence of finite-dimensional approximations in the Prohorov metric. Taking the novel approach of treating randomvariablesasanotherindependent,space-likevariable,[40]developedatheorysurrounding randomabstractparabolicpartialdifferentialequationswithdynamicsdefinedintermsofbounded and coercive sesquilinear forms. Combining these two previous frameworks with approximation results from linear semigroup theory (e.g., the Trotter-Kato theorem, [13]), [76, 78] develop a least-squares framework that appropriately describes the distributions of random parameters in distributed systems. Further, approximation and convergence results are displayed, and computa- tional approaches to the problem are outlined. Another approach, [75], that was attempted made useofnonlinearmixed-effectsmodelsasin[26]anddeterminedamaximumlikelihoodestimator for the random parameters. Returning to work surrounding least squares estimators, [11] utilized the Prohorov metric to establish general (ordinary, generalized, and other) least squares estimates and estimators for measure estimation problems surrounding specific PDEs. Building from this, [6] seeks to develop a Prohorov metric-based nonparametric approach to estimating the distribu- tion of random parameters in dynamical systems. As such, they also demonstrate approximation and convergence results of the determined estimator. Continuing the nonparametric approach, [5] seekstoincorporatethemixed-effectsmodeltodetermineamaximumlikelihoodestimatorforthe random parameters and showcasesstatistical consistencyof the determined point estimator. More recentapproaches from more members of thesame research group haveutilized theworkof [80], wherein a framework is built upon the idea that the stochastic error should be introduced directly into the underlying PDE model. Utilizing this framework along with control theory ideals, [91] develop a theory of real-time control and estimation schemes for a specific set of PDEs based on intravenous-infusedalcoholstudies. Further,in[92],thesamegroupdeterminedalinear-quadratic- Gaussian tracking compensator (see, [7]) and established approximation and convergence results 6 of the compensator. The most recent work from the same group has elected to forego any least- squares optimization and has instead opted to investigate the usage of hidden Markov models and physics-informedgenerativeadversarialneuralnetworks,[57]and[56],respectively. Another possible approach to the problem of estimating the distribution of random parame- ters in dynamical systems is based on the centuries-old ideals of Bayes’ Theorem. As outlined in [52] and dating back to the 1740s, Reverend Thomas Bayes developed a framework that allows people to update prior beliefs given newly emerged data. In the late 18th century and early 19th century, Pierre-Simon Laplace independently discovered the underlying concept of Bayes’ Theo- rem,developedthecentrallimittheorem,andrealizedwhatwouldbecomethegeneraltheoremfor Bayes’methods. Afteroveracenturyofdecline,HaroldJeffreysmadeBayes’Theoremusefulfor scientists in his 1939 book, Theory of Probability [42]. While a contentious topic through much of the 20th century, Bayes’ methods experienced a revolution during the 1990s as an increase in computing power allowed the fundamental computational complexity issue with Bayesian meth- odstobeovercome. CombinedwiththeadventofMarkovChainMonteCarlomethods,Bayesian statisticians were then able to expand the amount of work they were able to complete. With this growth in computational power, Bayesian methods have been adopted into nearly every field pos- sible. One primary field of interest is machine learning (see, [62]), with such things as Bayesian neural networks (see, [34, 51]) and Gaussian process regression (see, [62, 85]) being used across a wide range of problems. On the topic of asymptotic theories in Bayesian frameworks, initial investigations began in the 1960s with [35, 72], where convergence properties of posteriors were investigated. Recentworkin[87]demonstratedgeneralhypothesesthatcanbeutilizedwhenhan- dlingtheconsistencyofBayesianposteriordistributions. A key component of the problem we will study is the alcohol biosensor device. Dating back to the 1930s, alcohol (ethanol) consumed in the form of alcoholic beverages has been known to be excreted from the human body through sweat glands, and perspiration [17, 37, 55, 60]. This is because water and ethanol are highly miscible [86] and the alcohol rapidly finds its way into all of the water in the body. While such insights were heavily used for modeling purposes, in recent 7 decades, these observations have paved the way for the development of a device to measure the amount of alcohol excreted transdermally through the skin [81, 82, 83]. The two largest modern devices being the SCRAM Continuous Alcohol Monitoring (CAM) device manufactured by Al- cohol Monitoring Systems (AMS) in Littleton, Colorado, and the WrisTAS device manufactured by Giner, Inc. in Newton, Massachusetts (see, [3]). Though, there may be more wearable device integrations on the market in the near future due to the National Institute on Alcohol Abuse and Alcoholism (NIAAA) having held two wearable alcohol biosensor challenges in 2016 and 2017 (see,[1,54]). In recent years, other approaches to determining deterministic and random parameters in dy- namicalsystemsfortheproblemsurroundingestimatingBrACfromTAChavebeenresearchedby other project groups. Specifically, in [33] the team took a machine learning approach that utilized features extraction methods (see, [22]) to subsequently apply random forests (see, [16, 38]) in or- der to determine BrAC directly from TAC. In this approach, no underlying model parameters are directlyestimated,butinstead,themachinelearningmethodsusedaretrainedsothattheycontain latent information regarding relationships within the training data. This effectively accomplishes the same task that the parameters are used for. In the group behind [41], the problem was shifted to be focused on determining the number of consumed alcoholic beverages from measured TAC values. To accomplish this, statistical significance calculations were used to determine covariate (e.g., identified sex, age, etc.) inputs to a multiple regression model. In [30], a model was devel- opedtodeterminepeakBrACfromTACandsoughttoaccountfordifferencesinidentifiedsexof thestudyparticipants. Toaccountforpossibledifferencesinidentifiedsex,thegroupuseusedthe fixed effect components of a mixed-effect model to determine coefficients for a linear regression peakBrACestimator. 8 Chapter2 Preliminaries 2.1 FunctionalAnalysis We begin this section with definitions that are utilized to formulate a solution to the underlying problem in Chapter 3. We then follow with further results and their usage in an approach that commonly arises when seeking to take a functional analytic approach to solving linear partial differential equations (PDEs). We finish the section with further results that are of use to us when handlingapproximationsofoperators. Definition2.1. LetV beacomplexBanachspaceandfT(t) : 0t<1gbeafamilyofoperators inL(V), the bounded linear operators fromV toV. The familyfT(t) : 0 t <1g is called a stronglycontinuousorC 0 -semigroupinV if 1. T(0) =I, 2. T(t+s) =T(t)T(s)foreveryt;s 0,and 3. foreveryx2V,themappingt!T(t)xiscontinuouson[0;1). Ifonly(1)and(2)abovearesatisfied,wecallfT(t) : 0t<1gasemigroup. Definition2.2. LetV be a complex Banach space andT(t) be aC 0 -semigroup. The infinitesimal generatorAofT(t)isdefinedby 9 D(A) =fx2V : lim t→0 (T(t)xx)=t existsg and Ax = lim t→0 (T(t)xx)=t for x2D(A): Remark 1. For aC 0 semigroup,T(t) with generatorA, we necessarily have thatA is closed and denselydefined. Further,wehavethatitsatisfiesthefollowingresolventboundforRe()> 0, (IA) −1 M Re() ; whereM 1isafixedconstant. Definition 2.3. Let V be a complex Banach space and fT(t) : 0 t < 1g be a family of operators inL(X), the bounded linear operators from V to V. The familyfT(t)g is called an analyticsemigroupinV ifitisaC 0 -semigroupand,foreachx2V,t!T(t)xisrealanalyticon (0;1). Definition 2.4. A linear operatorA on a complex Banach spaceV with dualV ∗ , is called dissi- pative if for everyx 2 D(A), there exists anx ∗ 2 n v :v2V ∗ and hv;xi V ∗ ,V =kxk 2 =kvk 2 o suchthatRe(hx ∗ ;Axi V ∗ ,V ) 0. Definition2.5. AlinearoperatorAonacomplexBanachspaceV iscalled maximally dissipative ifitsonlydissipativeextensionisAitself. Theorem 2.1 (Theorem 2.1.6 of [84]). Let A be a closed linear operator on complex Banach spaceV with its domainD(A) dense. BothA andA ∗ are dissipative if and only if the half-plane f : Re> 0giscontainedin(A)andk(A) −1 k (Re) −1 holdsinthehalf-plane. 10 Theorem2.2. LetA be a closed, dissipative linear operator on complex Banach spaceV with its domainD(A) dense. A is maximally dissipative if and only ifA ∗ is dissipative. In this case,A ∗ is maximallydissipative. Theorem 2.3 (Hille-Yosida Theorem). A linear (possibly unbounded) operator A on a complex BanachspaceV istheinfinitesimalgeneratorofaC 0 -semigroupofcontractionsT(t);t 0ifand onlyif 1. Aisclosedanddenselydefined,and 2. Theresolventset(A)ofAcontainsR + andforeveryRe()> 0, (IA) −1 1 Re() : Theorem2.4(Lumer-PhillipsTheorem). LetAbealinearoperatordefinedonalinearsubspace D(A)oftheBanachspaceV. ThenAgeneratesacontractionsemigroupifandonly 1. D(A)isdenseinV, 2. Aisclosed, 3. Aisdissipative,and 4. A 0 I issurjectiveforsome 0 > 0. Anoperatorsatisfyingthelasttwoconditionsismaximallydissipative. Definition2.6(Definition2.3.1of[84]). LetAbeaclosedoperatordenselydefinedonacomplex BanachspaceV. TheoperatorAissaidtobeof type (!;M)ifthereexist 0 !g andk(A) −1 k M for < 0, and if there exists a numberM ε suchthatk(A) −1 kM ε holdsinjargj>! +"forall"> 0. Theorem2.5(Theorem3.3.1of[84]). LetV beacomplexBanachspace. Supposethereexistreal numbers;M and an angle! 2 [0;=2] such thatA+ is of the type (!;M), then operator 11 A defined on V is the generator of a semigroupfT(t)g:T(t) can be continued holomorphically with respect tot into the sectorft : jargtj⩽ =2!g, where (2) holds. Let be an arbitrary angle satisfying 0 < < =2!, then e −βt T(t) is uniformly bounded in the closed subsector ftjargtj g and, whent approaches 0 inside this subsector,T(t) converges strongly toI, For anynaturalnumbernwehave limsup t→+0 t n d dt n T(t) = limsup t→+0 t n kA n T(t)k<1: Theorem2.6(Trotter-KatoTheorem Theorem2.1of[13]). LetA N andAbeinfinitesimalgenera- torsofC 0 -semigroupsS N (t)andS(t)onHilbertspacesH andH N ,withH N H fororthogonal projection operatorP N :H !H N with the property thatP N x!x for allx2X. IfA N andA satisfy 1. Thereexistconstants!;M suchthatjS N (t)jMe ωt foreachN 2Z + ,and 2. There exists2(A)\ ∞ N=1 (A N ) with Re()>! suchthatR λ (A N )P N x!R λ (A)x for eachx2H, thenforeachx2H,S N (T)P N x!S(t)xuniformlyintonanycompactinterval[0;t 1 ]. Remark 2. There is an alternate version of Theorem 2.6 commonly used in parameter estimation problemsthatreplaces(2)withthecondition: • There exists a setD dense in H such that for some , (I A)D is dense in H and A N P N x!Axforallx2D. Remark 3. Another version of Theorem 2.6 commonly used in parameter estimation problems reducesH toonlybeingaBanachspace. Inthissetting,theprojectionoperatorP N isreplacedby thecondition: • Foreveryx2H,thereexistsx N 2H N suchthatx N !xinH asN !1. 12 Some of the conditions for the previous results can be difficult to verify. As such, we look at the usage of semigroup theory in a more structured setting, specifically within Hilbert space. Let V andH be Hilbert spaces withV ,!H, that is,V is densely, and continuously embedded inH. Therefore, identifyingH with its dual,H ∗ , we find that we also get thatH ,! V ∗ . This leads to what is called a Gelfand Triple ofV ,! H ,! V ∗ . Under this setting, we then seek an operator thatwillultimatelygenerateaholomorphicsemigroupasinTheorem2.5. Definition2.7. LetV beacomplexvectorspace. Themapa :V V !Ciscalled sesquilinear if 1. a(u 1 +u 2 ;v 1 +v 2 ) =a(u 1 ;v 1 )+a(u 1 ;v 2 )+a(u 2 ;v 1 )+a(u 2 ;v 2 ), 2. a(u 1 ;v 1 ) = ¯ a(u 1 v 1 ), forallu 1 ;u 2 ;v 1 ;v 2 2V andall;2C. Further,themapais 1. bounded ifja(u;v)jmjjujjjjvjj,and 2. coerciveifRe(a(u;u))jjujj 2 , foru;v2V,andm;> 0. Theorem2.7 (Lax-Milgram Theorem). LetH be a Hilbert space, whose inner product and norm will be denoted by (;) andkk, respectively. Assume thata(u;v) isa sesquilinear form defined onHH andthatthereexistpositiveconstantsC andcsuchthat ja(u;v)jCkukkvk ja(u;u)jckuk 2 forallu;v2H. Undertheseconditions,ifF 2H ∗ ,i.e.,ifF isacontinuousantilinearfunctional onX,thereexistsanelementusuchthatF(v) =a(u;v)forallv2H:Furthermore,uisuniquely determinedbyF. 13 Definition 2.8 (Definition 2.2.1 of [84]). Given Hilbert spacesV ,! X, that is,V is densely and continuously embedded inX, and an operatorA defined such that for the bounded and coercive sesquilinearforma(;) :V V !Cifthereexistsanelementf inX suchthata( ;') = (f;') forall 2V,then 2D(A)andA =f,thenAiscalledregularlydissipative. Theorem2.8(Theorem2.1of[90]). LetV;H beHilbertspaceswithV ,!H,a(;)beabounded and coercive sesquilinear form onV, andA be a linear operator onH which is determined from theform. Then,Asatisfies (A) Σ ω =f2C;jargj 0 and all'2V, and defines the operatorA. So for Re()> 0, there exists aboundedinverseofAwhichhasvariousboundsforevery'2H orV ∗ : (A) −1 f H M 1 jj −1 jfj 2 (A) −1 f H M 2 jj −β/2 kfk V ∗ (A) −1 f V M 2 jj −1/2 jfj; (A+) −1 f V −1 kfk 0 (A) −1 f V ∗ M 1 jj −1 kfk V ∗ whereM 1 = 1+M= andM 2 = [(1+M)=] 1/2 . Theorem2.10(Theorem2.3of[13]). LetV;H beHilbertspaceswithV ,!H andapproximating subspaces H N for N 2 Z + with projection operator P N , and Q be a parameter space with distancemetricd andparameterdependentsesquilinearform(q;;) :V V !C,q2Q. If satisfies, • Forq;˜ q2Q,andforall'; 2V, j(q;'; )(˜ q;'; )jd(q;˜ q)jj'jj V jj jj V ; • Thereexistc 1 > 0andsomereal` 0 suchthatforq2Q,'2V wehave (q;';')+` 0 jj'jj 2 H c 1 jj'jj 2 V ; 15 • Thereexistc 2 > 0suchthatforq2Q,'; 2V wehave j(q;'; )jc 2 jj'jj V jj jj V ; andifthefollowingaresatisfied, • Foreachz2V,thereexists ˆ Z N 2H N suchthat z ˆ z N V ! 0asN !1,and • Thereexistfq N gQsuchthatq N !q asN !1, then for parameter-dependent semigroups T(q;) generated by and T N (q;) generated by restricted toH N H N , we have that for eachz 2 H,T N (q N ;t)P N z ! T(q;t)z in theV norm fort> 0,uniformlyintoncompactsubintervals. Now, under the Gelfand Triple setup with regularly dissipative operator A as defined previ- ously, we may employ finite-dimensional approximation approaches and seek guaranteed conver- gence given appropriately chosen approximations. Commonly, these approximations are deter- mined by picking a basis (e.g., splines, etc.), and then for finite dimension N, defining A N by restrictingthesesquilinearforma(;)tothesubspaceH N asdeterminedbyanorthogonalprojec- tion operatorP N . Under this scheme, Theorems 2.6 and 2.10 may be directly applied to achieve convergence. The final result we list is not typically utilized when handling the Gelfand Triple setup, but can be applied when seeking to investigate statistical convergence results with distributions that involvesemigroups. Theorem 2.11 (Theorem 5.3 of [74]). For parametersq 2 Q a subset of the positive orthant of R n , and q-dependent semigroupfT(;q) : t > 0g with infinitesimal generator A(q) defined in terms of aq-dependent, bounded and coercive sesquilinear forma(q) :V V !C on a Hilbert spaceV,assume 16 1. (Affine)Themapq7!a(q)isaffine,inthesensethatforanyu;v2V, a(q;u;v) =a 0 (u;v)+a 1 (q;u;v) wherea 0 isindependentofq andthemapq7!a 1 (q;;)islinear,and 2. (Continuous)Foranyq; ¯ q2Qwithmetricd Q (;)wehave ja(q;u;v)a(¯ q;u;v)jd Q (q; ¯ q)kuk V kvk V 8u;v2V: Then, the semigroupT(;q) is (Fr´ echet) differentiable inq in the interior ofQ, where fort > 0, ¯ q2Q,andactingonq2Qthederivativeisgivenby T q (t; ¯ q)q = 1 2i Z ∂Σγ e λt R λ (A(¯ q))A(q)R λ (A(¯ q))d (2.1) for A(q) the generator of T(;q), R λ (A(q)) = (A(q)I) −1 the resolvent of A(q), and the obtusesectorΣ γ =f2C : arg( 0 ) π 2 +arctan((1+ 0 0 )(1 ))gwith 2 (0;1), 0 asdeterminedbytheboundednessofa(q),and 0 , 0 asdeterminedbythecoercivityofa(q). 2.2 Probability Inthissection,wecompiletheoremsandresultsthatwillbeofusetousthroughoutthethesis. Theorem2.12(Bayes’Theorem). FortwoeventsAandB wehave P(AjB) = P(BjA)P(A) P(B) = P(BjA)P(A) P(BjA)P(A)+P(BjA ′ )P(A ′ ) : 17 Theorem 2.13 (Jensen’s Inequality). Let (Ω;Σ;) be a probability space. If g is a real-valued, -integrablefunction,andif'isaconvexfunctionontherealline,then ' Z gd Z '(g)d: Theorem 2.14 (Strong Law of Large Numbers). Let X 1 ;X 2 ;::: be independent identically dis- tributedrandomvariableswithE[X i ] =andVar(X i ) = 2 <1,anddefine ¯ X n = 1 n P n i=1 X i . Thenforevery"> 0, P lim n→∞ j ¯ X n j<" = 1; thatis, ¯ X n convergesalmostsurelyto. Theorem 2.15 (Strong Law of Large Numbers (i.n.i.d) [58]). LetX 1 ;X 2 ;::: be independent non- identicallydistributedrandomvariableswithE[X i ] =and ∞ X i=1 Var(X i ) i 2 <1: Define ¯ X n = 1 n P n i=1 X i . Thenforevery"> 0, P lim n→∞ j ¯ X n j<" = 1; thatis, ¯ X n convergesalmostsurelyto. 18 Lemma 2.16. Let f and g be probability densities. Letkf gk 1 = R jf gj stand for the L 1 -distance and let K + (f;g) = R f log + (f=g) and K − (f;g) = R f log − (f=g), where log − x = max(logx;0). Then K − (f;g) 1 2 kfgk p K(f;g)=2 and K + (f;g) 1 2 kfgk+K(f;g)K(f;g)+ p K(f;g)=2: 2.3 Statistics Inthefollowingsection,wesuccinctlypresentsomeofthemainworksofFreedman[35],Schwartz [72],andWalker[87],specificallythroughthelensofChoiandRamamoorthi[19]. AssumethatΘ isaparameterspace,ff θ :2 Θgisafamilyofdensitiesallwithrespecttosome-finitemeasure on a measurable spaceX. Further, we assume that Θ andX are complete separable metric spaces, as well that the mapping 7! f θ is one-to-one and (;x) 7! f θ (x) is measurable. We denote the prior (on Θ) by 0 , data byfX 1 ;:::;X n g and posterior by(jX 1 ;:::;X n ). Further, let P ∞ θ 0 stand for the join distribution offX i g ∞ i=1 when 0 2 Θ is the true parameter value of . Note, we will assume all data points are independent and identically distributed (i.i.d.) unless otherwisestated. Definition2.9. For 0 aprior overparameter andf(jX 1 ;X 2 ;:::;X n )g asequence of posterior distributions,thesequenceofposteriorsissaidtobeconsistentat 0 iff(UjX 1 ;X 2 ;:::;X n )g! 1 a.s.P ∞ θ 0 forallneighborhoodsU ofΘ,thefeasiblesetofparameters,where 0 2U. Definition2.10. Wedenotetheaffinitybetweentwoprobabilitymeasuresf,g onΘas Aff(f;g) = Z Θ p fgd 19 Definition2.11. TheKullback-Leibler(KL)divergenceisdefinedas K( 0 ;) =E θ 0 log f θ 0 f θ andaKLneighborhood ofsome 0 isK ε ( 0 ) =f2 ΘjK( 0 ;)<"gfor"> 0. Definition 2.12. We say a point 0 is in the KL support of 0 if for every" > 0, 0 (K ε ( 0 )) = 0 (fjK( 0 ;)<"g)> 0. Definition 2.13. Given a measurable set G ⋐ Θ and 0 2 Θ as the true value of , let P ∞ θ 0 denote the join distribution of fX i g ∞ i=1 . Then the posterior (GjX 1 ;:::;X n ) converges to zero exponentiallywithP ∞ θ 0 probability1ifthereexistsa > 0suchthat P ∞ θ 0 (f(GjX 1 ;:::;X n )>e −nβ inf. ofteng) = 0: Proposition2.17([39],Lemma4.4.1). If 0 isintheKLsupportofprior 0 thenforall > 0, lim n→∞ e nβ P(J(X 1 ;:::;X n )) =1 a:s: P ∞ θ 0 wherewithf (n) θ beginthejointdensityoffX i g n i=1 distributedwithdensityf θ , J(X 1 ;:::;X n ) = Z Θ f θ f θ 0 (X 1 ;:::;X n ) 0 (d): Definition2.14. Foraprobabilitymeasure onΘ,letv n ν bethemarginaldensityoffX i g n i=1 , v n ν (X 1 ;:::;X n ) = Z Θ f (n) θ (X 1 ;:::;X n )(d) withf (n) θ (X 1 ;:::;X n ) the joint probability of the data under, that is, the likelihood of the given data. 20 Definition 2.15. LetG Θ and > 0. The setG and 0 are strongly separated if for every probabilitymeasure onG, Aff(f (1) θ 0 ;v 1 ν )< where f (1) θ 0 is the data distribution of X 1 under 0 , andv 1 ν is the marginal density of X 1 as in Definition 2.14. We will say thatG and 0 are strongly separated if they are strongly separated forsome> 0. Lemma 2.18 (Lemma 3.6 of [19]). LetA Θ. If 0 andG are strongly separated then for all probabilitymeasures onG,foralln, Aff(f θ 0 ;v (1) ν )<e nβ 0 where 0 =log(): Theorem2.19(Schwartz,[72]). LetG Θ. If 1. 0 isintheKLsupportofprior 0 , 2. forsomek,Gand 0 arestronglyseparatedfortheparametrization7!f k θ . Then(GjX 1 ;:::;X n )goestozeroexponentiallya.s.P ∞ θ 0 . Theorem2.20(Walker,[87]). LetG Θ. If 1. 0 isintheKLsupportofprior 0 , 2. G = S i≥1 G i suchthat: (a) Forsome> 0alloftheG i ’sarestrong separatedfrom 0 and (b) P i≥1 p 0 (G i )<1. Then(GjX 1 ;:::;X n )goestozeroexponentiallya.s.P ∞ θ 0 . Theorem2.21. Assume 21 1. 0 isintheKLsupportof 0 , 2. For all > 0, there exists sets G 1 ;G 2 ;::: such that the diameter of G i , diam(G i ) < , S G i = Θ,and P p 0 (G i )<1. Then for anyL 1 neighborhoodU of 0 , the posterior probability ofU c goes to zero exponentially a.s.P ∞ θ 0 . Note1. Explicitly,theL 1 neighborhoodU forany"> 0is R X jf θ (x)f θ 0 (x)jdx<" . The previous results set the groundwork and methods to be used when we wish to investigate posterior convergence with respect to independent, identically distributed (i.i.d.) data. That is, whengivenprior 0 overparameterspaceΘ,i.i.d. datafX i g n i=1 ,andasetG Θ,wefocusonthe posteriordistribution, (GjX 1 ;:::;X n ) = R G f θ (X 1 ;:::;X n ) 0 (d) R Θ f θ (X 1 ;:::;X n ) 0 (d) = R G f θ (X 1 ;:::;X n )=f θ 0 (X 1 ;:::;X n ) 0 (d) R Θ f θ (X 1 ;:::;X n )=f θ 0 (X 1 ;:::;X n ) 0 (d) : (2.2) We then wish to apply one of Theorems 2.19, 2.20, or 2.21—which make use of the posterior in (2.2),Proposition2.17,andLemma2.18—butallofthemareheavilydependentuponourpriorand data distributions. Commonly, we will have that Θ is a compact subset of the positive orthant of R m forsomem2Z + ,andboththeprioranddatadistributionshavecontinuousdensityfunctions with support Θ. One primary example being truncated multivariate normal distributions on an interval of the real line. As such, it is a straightforward calculation to determine that will be in the KL support of 0 (and if not, recall that the prior is a modeling choice and only attempts to best capture current knowledge). Then by the compactness of Θ and full-support of 0 , we have that condition (2) of Theorem 2.21 is satisfied, and hence we have posterior consistency. While this setup works for i.i.d. data, a regular occurrence when using time-series-based data is to have instead independent, non-identically distributed (i.n.i.d.) data. The previous results do not cover thiscase,buttheycanbeextendedtodoso. 22 Theorem 2.22 (Theorem 1 of Appendix A.2 in [20]). Let 0 be a prior over parameter space Θ, fX i g ∞ i=1 be independent but not identically distributed data with distribution generated byf i,θ for 2 Θ,and 0 2 Θthetrueparametervalue. For2 Θdefine Λ i ( 0 ;) = log f i,θ 0 (X i ) f i,θ (X i ) ; κ i ( 0 ;) =E θ 0 [Λ i ( 0 ;)]; S i ( 0 ;) = Var θ 0 [Λ i ( 0 ;)]: IfthereexistsasetB Θwith 0 (B)> 0suchthat 1. P i≥1 S i ( 0 ;) i 2 <1 82B,and 2. Forevery"> 0, 0 (B\f : κ i ( 0 ;)<" 8ig)> 0. Then8 > 0, e mβ J(fX i g m i=1 )!1 a.s.P ∞ θ 0 asm!1forJ(fX i g) = R Θ Q m i=1 [f i,θ (X i )=f i,θ 0 (X i )] 0 (d). Theorem 2.23. Let 0 be a prior over parameter space Θ,fX i g ∞ i=1 be independent but not iden- tically distributed data with distribution generated byf i,θ for 2 Θ, 0 2 Θ the true parameter value,andG Θwith 0 = 2G. IfG = S i≥1 G i suchthat 1. Forsome> 0,allG i ’sarestrongly separatedfrom 0 forthemodel7!f i,θ ,and 2. P i≥1 p 0 (G i )<1, thenforsome 0 > 0, e mβ 0 J G (fX i g m i=1 )! 0 a.s.P ∞ θ 0 asm!1forJ G (fX i g m i=1 ) = R G Q m i=1 [f i,θ (X i )=f i,θ 0 (X i )] 0 (d). 23 So for i.n.i.d. data under the same framework as previously discussed, we note that (2.2) may still be utilized. Our goal of showing posterior consistency is now reduced to utilizing Theorems 2.22 and 2.23. While these theorems are more heavily dependent on underlying statistical struc- tures, we note that the requisite hypotheses are typically able to be checked analytically when utilizingparametricdistributionswithacontinuousdensityfunction. Definition2.16. Fordistribution overΘand2 [0;1],the-credibleregionisaregionC Θ, thesetoffeasible,suchthat Z C ()d = 1: Note,wewilltypicallywritethisasthe100(1)%-credibleregionandhavestatedimplicitly. 2.4 Analysis,andMisc. As before with probability, this section is meant for disparate results that will be of use to use throughoutthethesis. Theorem 2.24 (Young’s Inequality). If a and b are nonnegative real numbers and p and q are postiverealnumberssuchthat1=p+1=q = 1,then ab a p p + b q q : Theorem 2.25 (Fatou’s Lemma). Letff n g be a sequence of non-negative measurable functions, then Z liminf n→∞ f n d liminf n→∞ Z f n d: 24 Theorem2.26(Cauchy-SchwarzInequality). Forallelementsf andg ofaninner-productspace, itholdsthat jhf;gij 2 hf;fihg;gi whereh;iistheinnerproduct. Theorem 2.27 (Morrey’s Inequality). Assume n < p 1. Then there exists a constant C, dependingonlyonnandp,suchthat kuk C 0,γ (R n ) Ckuk W 1,p (R n ) forallu2C 1 (R n ),where = 1n=p. Theorem2.28(H¨ older’sInequality). Letp;q2 (1;1) with 1=p+1=q = 1. Then for all measur- ablefunctionsf andg, kfgk 1 kfk p kgk q withequalitywhenjgj =cjfj p−1 forsomec. Theorem2.29(MeanValueTheoremonNormedLinearSpacesfromTheorem4ofSection3.2in [18]). Letf be a map from an open setD in one normed linear space into another normed linear space. Ifthelinesegment S =fta+(1t)b : 0t 1g liesinD andiff ′ (x)existsateachpointofS,then kf(b)f(a)kkbaksup x∈S kf ′ (x)k: 25 Remark4. AsaBanachspaceisacomplete(allCauchysequencesconverge),normedlinearspace, Theorem2.29directlyapplies. 26 Chapter3 ADistributedParameterModelfortheTransdermalTransport ofAlcohol WemodelthealcoholbiosensorproblemdescribedinChapter1usingaone-dimensionaldiffusion equation to describe alcohol transport through the epidermal layer of the skin. As in [63] and [76], and making use of an idea recently introduced in [91], we consider a bounded domain one- dimensional equation coupled with a well-mixed compartment representing the vapor collection chamberofthebiosensor. Thecouplingisachievedthroughafluxtermatthe = 0boundaryofthe domain. Inthisway,weareabletoavoidhavingtodealwithanunboundedoutputorobservation operator in the form of a point-wise measurement in L 2 . The resulting hybrid ordinary-partial differentialequationinput/outputsystemtakestheform @x @t (t;) =q 1 @ 2 x @ 2 (t;); 0<< 1; t> 0; dw dt (t) =q 3 @x @ (t;0)q 4 w(t); t> 0; x(t;0) =w(t); t> 0 q 1 @x @ (t;1) =q 2 u(t); t> 0; w(0) =w 0 ; x(0;) =' 0 (); 0<< 1; y(t) =w(t); t 0; (3.1) 27 wherex(t;) is the concentration of alcohol at depth in the epidermal layer of the skin at time t > 0, w(t) is the concentration of alcohol in the biosensor’s vapor collection chamber at time t > 0,u(t) is the BrAC or BAC at timet > 0,y(t) is the TAC at timet > 0, the initial dataw 0 is inR + and' 0 is inL 2 (0;1) and q = [q 1 ;q 2 ] are unknown, un-measurable, dimensionless and, in general, subject-dependent physiological parameters. The parametersq 3 andq 4 are device (i.e hardware)dependentparametersthatwillbeassumedtobebench-measurable,althoughtheycould alsobeestimatedalongwithq 1 andq 2 usingthetechniqueswewilldevelopbelow. Forclarity,we willfocusourattentionhereonthedevelopmentofapopulationmodelforacohortofsubjectsby estimatingthedistributionofthephysiologicalparametersq 1 andq 2 . Weconsiderthissystemonafinite-timehorizon,T > 0,andweassumezero-orderholdinput, u(t) =u k ,t2 [k;(k +1)),k = 0;1;2;:::, where denotesthesamplingtimeofthebiosensor. We setx k =x k () =x(k;),w k =w(k), andy k =y(k),k = 0;1;:::;K, where we assume T =K. Fork = 0;1;2;::: we consider the system (3.1) on the interval [k;(k +1)] and make thechangeofvariable: v(t;) =x(t;)()u k where() = q 2 q 1 . Itistheneasilyverifiedthat w andv satisfythefollowinghybridsystem @v @t (t;) =q 1 @ 2 v @ 2 (t;); 0<< 1; k <t< (k +1); dw dt (t) =q 3 @v @ (t;0)q 4 w(t)+ q 3 q 2 q 1 u k ; k <t< (k +1); v(t;0) =w(t); k <t< (k +1); q 1 @v @ (t;1) = 0; k <t< (k +1); (3.2) withinitialconditions v(k;) =x(k;)()u k =x k u k ; on [0;1]; w(k) =w k : (3.3) We then use linear semigroup theory to rewrite the system (3.2), (3.3) in state space form in an appropriately chosen Hilbert space and subsequently obtain a discrete time evolution system 28 for (w k ;x k ),k = 0;1;2;::::K which is equivalent to (3.1). LetQ denote a compact subset of the positiveorthantofR 4 ,andforq = [q 1 ;q 2 ;q 3 ;q 4 ] T 2QwedefinetheHilbertspaces H q =RL 2 (0;1) (3.4) V =f(; )2H q : 2H 1 (0;1); = (0)g (3.5) withrespectiveinnerproducts h( 1 ; 1 );( 2 ; 2 )i q = (q 1 =q 3 ) 1 2 + Z 1 0 1 () 2 ()d h( 1 (0); 1 );( 2 (0); 2 )i V = 1 (0) 2 (0)+ Z 1 0 ′ 1 () ′ 2 ()d (3.6) and respective corresponding normsjj q , andkk V . Note that the Sobolev Embedding Theorem (see, [2]) yields that the norm induced by the V inner product is equivalent to the standard H 1 norm onV. To show thatV andH are Hilbert, andV is densely and continuously embedded in H q ,wefirstnotethatR,L 2 (0;1),andH 1 (0;1)areallHilbertSpaces. SoforanyCauchysequence f( n ;' n )g n≥1 H q wefind j n m j 2 j( n ;' n )( m ;' m )j 2 q ! 0 asm;n!1. ThusbythecompletenessofRwehavethatthereexistsa2Rsuchthat n ! asn!1:Further,wehave k' n ' m k 2 L 2 (0,1) j( n ;' n )( m ;' m )j 2 q ! 0 asm;n!1. Thus by the completeness ofL 2 (0;1) we have that there exists a'2R such that ' n ! ' asn ! 1. Thus ( n ;' n ) ! (;') asn ! 1 and we have thatH q is complete and Hilbert. WenotethatV completenessfollowsdirectlyfromthecompletenessofH 1 (0;1)andthus V isalsoHilbert. 29 Now, sinceQ R 4 is compact in the positive orthant, there exist constants ˜ M;M > 0 such thatM 0,x2 [0;1] ' n (x) = n' 1 n n x1 [0,1/n] (x)+'(x)1 (1/n,1] (x): Soforeveryn,' n (0) = and(' n (0);' n )2V,and' n !'a.e.H 1 (0;1). ThusV ,!H q . Then based on the weak formulation of the system (3.2), for eachq 2 Q define bilinear form a(q;;) :V V !Rby a(q; ˆ '; ˆ ) = (q 1 q 4 =q 3 )'(0) (0)+q 1 Z 1 0 ' ′ () ′ ()d (3.9) for ˆ '; ˆ 2 V, where ˆ ' = ('(0);'), ˆ = ( (0); ), andq = [q 1 ;q 2 ;q 3 ;q 4 ] T 2 Q. We now apply standardargumentstoarguethattheforma(q;;)satisfiesthefollowingthreeproperties. 30 1. Boundedness There exists a constant 0 > 0 such thatja(q; ˆ 1 ; ˆ 2 )j 0 jj ˆ 1 jj V jj ˆ 2 jj V ; ˆ 1 ; ˆ 2 2V, 2. Coercivity There exists constants 0 2 R and 0 > 0 such that a(q; ˆ ; ˆ ) + 0 j ˆ j 2 q 0 jj ˆ jj 2 V ; ˆ 2V. 3. Continuity For all ˆ 1 ; ˆ 2 2 V, we have that q 7! a(q; ˆ 1 ; ˆ 2 ) is a continuous mapping fromQintoR. First,forboundednesswehavethatfor ˆ '; ˆ 2V andq2Q, ja(q; ˆ '; ˆ )j q 1 q 4 q 3 j'(0) (0)j+q 1 1 Z 0 ' ′ () ′ ()d q 1 q 4 q 3 j'(0) (0)j+q 1 0 @ 1 Z 0 ' ′ () 2 d 1 A 1 2 0 @ 1 Z 0 ′ () 2 d 1 A 1 2 q 1 q 4 q 3 j'(0) (0)j+q 1 kˆ 'k V k ˆ k V where the second inequality follows from the Cauchy-Schwarz inequality (2.26), and the third inequalityfollowsfromthedefinitionoftheinnerproductonV,(3.6). Since' and are continuous on [0;1], and by Morrey’s Inequality (2.27) withn = 1,p = 2, and = 1n=p = 1=2, thereexists ¯ (n;p)> 0 suchthatj'(0) (0)j ¯ kˆ 'k v k ˆ k v . Duetothe compactnessofQwemaychoose 0 independentofq2Qsuchthat ja(q; ˆ '; ˆ )j 0 kˆ 'k V k ˆ k V : 31 Second,forcoercivitywefindthatgiven ˆ '2V and 0 > 0 a(q; ˆ '; ˆ ')+ 0 jˆ 'j 2 q = q 1 q 4 q 3 '(0) 2 +q 1 1 Z 0 ' ′ () 2 d + 0 2 4 q 1 q 3 '(0) 2 +q 1 1 Z 0 '() 2 d 3 5 = q 1 q 4 + 0 q 1 q 3 '(0) 2 +q 1 0 @ 1 Z 0 ' ′ () 2 d + 0 1 Z 0 '() 2 d 1 A : SinceQR 4 iscompact,thereexists 0 > 0independentofqinQsuchthat a(q; ˆ '; ˆ ')+ 0 jˆ 'j 2 q 0 0 @ '(0) 2 + 1 Z 0 ' ′ () 2 d + 1 Z 0 '() 2 d 1 A 0 kˆ 'k V : Third,forcontinuitywefindthatgiven ˆ ; ˆ '2V andq; ¯ q2Q ja(q; ˆ '; ˆ )a(¯ q; ˆ '; ˆ )j = q 1 q 4 q 3 '(0) (0)+q 1 1 Z 0 ' ′ () ′ ()d ¯ q 1 ¯ q 4 ¯ q 3 '(0) (0) ¯ q 1 1 Z 0 ' ′ () ′ ()d = q 1 q 4 ¯ q 3 ¯ q 1 ¯ q 4 q 3 q 3 ¯ q 3 '(0) (0)+(q 1 ¯ q 1 ) 1 Z 0 ' ′ () ′ ()d : Noticethat jq 1 q 4 ¯ q 3 ¯ q 1 ¯ q 4 q 3 j =jq 1 q 4 (¯ q 3 q 3 )q 3 (¯ q 1 ¯ q 4 q 1 q 4 )j =jq 1 q 4 (¯ q 3 q 3 )q 3 [¯ q 1 (¯ q 4 q) 4 )q 4 (q 1 ¯ q 1 )]j (q 1 q 4 +q 3 q 1 +q 3 q 4 )kq ¯ qk 1 : 32 SinceQiscompactthereexists ˜ M > 0,independentofallq2Qsuchthat jq 1 q 4 ¯ q 3 ¯ q 1 ¯ q 4 q 3 j ˜ Mkq ¯ qk 1 : In addition, there existsM > 0, such thatM < q 1 ;q 2 ;q 3 ;q 4 for all [q 1 ;q 2 ;q 3 ;q 4 ] T 2 Q. Thus we find ja(q; ˆ '; ˆ )a(¯ q; ˆ '; ˆ )j ˜ M M 2 j'(0) (0)jkq ¯ qk 1 +kq ¯ qk 1 1 Z 0 ' ′ () ′ ()d ˜ Ckq ¯ qk 1 kˆ 'k V k ˆ k V where ˜ C is independent of q, ˆ ', and ˆ , and follows from Morrey’s Inequality (2.27). Note that in (1)-(3) above, Q compact allowed that the constants 0 , 0 , and 0 all be chosen independent ofq 2 Q. Furthermore, properties (1)-(3) immediately yield that the forma(q;;) definesabounded,coerciveoperatorA(q)2L(V;V ∗ )givenby hA(q)ˆ '; ˆ i =hA(q)('(0);');( (0); )i =a(q;'; ) for ˆ ' = ('(0);'); ˆ = ( (0); ) 2 V. If we define the setD = fˆ ' 2 V :A(q)ˆ ' 2 H q g = f('(0);') 2 V : ' 2 H 2 (0;1);' ′ (1) = 0g which is independent ofq forq 2 Q, we obtain the closed,denselydefinedlinearoperatorA(q) :DH q !H q givenby A(q)ˆ ' =A(q)('(0);') = (q 3 ' ′ (0)q 4 '(0);q 1 ' ′′ ); ˆ ' = ('(0);')2D: (3.10) The operatorA(q) :DH q !H q is regularly dissipative (see, Definition 2.8) and (see, for ex- ample,[84])istheinfinitesimalgeneratorofaholomorphicsemigroupofboundedlinearoperators fe A(q)t : t 0g onH q andV ∗ . We note that this follows directly from the work of Section 2.1, wherein we have a bounded and coercive sesquilinear form a(q;;) (see, Definition 2.7) that is 33 usedtodefineaclosedanddenseoperatorontheHilbertpivotspacewithinaGelfandtriple. Thus, we satisfy Definition 2.6 and may apply Theorem 2.5 to obtain our result. Moreover, the system (3.2),(3.3)canthenbewritteninstatespaceformas dˆ v dt (t) =A(q)ˆ v(t)+( q 3 q 2 q 1 ;0)u k ; k <t< (k +1); ˆ v(k) = (w k ;x k u k ): Thenfortimestep > 0andk = 0;1;:::;K,letting ˆ x k = (w k ;x k ),itfollowsthat ˆ x k+1 = (w k+1 ;x k+1 ) = (w((k +1));x((k +1);)) = ˆ v((k +1))+(0;)u k =e A(q)τ (w k ;x k u k )+ τ Z 0 e A(q)s ( q 3 q 2 q 1 ;0)dsu k +(0;)u k = ˆ A(q)ˆ x k +(I ˆ A(q))(0;)u k +A(q) −1 ( ˆ A(q)I)( q 3 q 2 q 1 ;0)u k = ˆ A(q)ˆ x k + ˆ B(q)u k (3.11) where ˆ A(q) =e A(q)τ 2L(H q ;H q ) and ˆ B(q) = (I ˆ A(q)) (0;)A(q) −1 ( q 3 q 2 q 1 ;0) 2L(R;H q ); (3.12) where in (3.12) we have used the fact that the operator A(q) −1 commutes with the semigroup generated byA(q). Note that the operator ˆ B(q) is in fact an element inH q and that (0;) 2 V, but that ( q 3 q 2 q 1 ;0) is only an element inH q . It follows from (3.11) that our discrete time model is giveninstatespaceformas ˆ x k+1 = ˆ A(q)ˆ x k + ˆ B(q)u k ; k = 0;1;2;:::; ˆ x 0 = (w 0 ;' 0 ); y k = ˆ Cˆ x k ; k = 0;1;2;:::; (3.13) 34 where ˆ x k = (w k ;x k ),k = 0;1;2;::: and the operator ˆ C 2L(H q ;R) is given by ˆ C(; ) = , where(; )2H q . From (3.13) it is immediately clear that if we assume zero initial conditions, ˆ x = (w 0 ;' 0 ) = (0;0),theoutputy canbewrittenasadiscretetimeconvolutionoftheinput,u,withafilter,h(q), as y k = k−1 X j=0 ˆ C ˆ A(q) k−j−1 ˆ B(q)u j (3.14) = k−1 X j=0 h k−j−1 (q)u j ; k = 0;1;2;:::; (3.15) whereforq2Q,h i (q) = ˆ C ˆ A(q) i ˆ B(q),i = 0;1;2;:::. Buildingoffoftheworkwithin[13]anditsapplicationsoftheTrotter-Katosemigroupapprox- imationtheorem(see,Theorem2.6,andforexample,[61]),thefollowingresultcanbeshown. Lemma 3.1. For Q a compact subset of the positive orthant orR 4 , T = K for constant time step > 0, andh i as defined in (3.15), we have that the mapping q 7! h i (q) fromQ intoR is continuous,uniformlyinq andi,forq2Qandi2f0;1;2;:::;Kg. Proof. A direct consequence of Theorem 2.10 (see, [13]) is that the mapping q 7! e A(q)t is continuous in q for q 2 Q and uniformly continuous for t 2 [0;T]. Further, the compact- ness of Q raises the continuity in q to be uniform. Now, by (3.11) and (3.12) we have ˆ B(q) = (I ˆ A(q))(0;)+ R τ 0 e A(q)s q 3 q 2 q 1 ;0 dsandthuswefindthatq7! ˆ B(q)isalsouniformlycontin- uousinq sinceitisasumandproductoffunctionsuniformlycontinuousinq forq2Q. As ˆ C as in (3.14) does not depend onq, we have thatq 7!h i (q) = ˆ C ˆ A(q) i ˆ B(q) is uniformly continuous inq andi forq 2Q andi2 0;1;:::;K since if ˆ A(q) has continuity constant ¯ M, then ˆ A(q) i has continuityconstantboundedbymaxf ¯ M; ¯ M(K +1)g,whichisindependentofi. 35 Remark 5. In the proof above, a useful bound to see the uniform continuity is that forq; ¯ q 2 Q, wehave q 3 q 2 q 1 ¯ q 3 ¯ q 2 ¯ q 1 ˜ M 3 M jjq ¯ qjj 1 forsomeM; ˜ M > 0bythecompactnessofQ. Nowalthoughtheinput/outputmodelgivenin(3.14)isastraightforwardconvolutioninR,the filter,fh k (q)ginvolvesthesemigroupfe A(q)t :t 0gwhichisdefinedontheinfinite-dimensional HilbertspaceH q . Consequently,computationswillrequirefinite-dimensionalapproximation. For n = 1;2;:::letf' n i ()g n i=0 denotethestandardlinearB-splinesontheinterval [0;1]definedwith respect to the uniform meshf0; 1 n ; 2 n ;:::; n−1 n ;1g. That is,' n i () = (ni+1)1 [ i−1 n , i n ] +(1 n +i)1 [ i n , i+1 n ] . Thenset V n = spanfˆ ' n i g = spanf(' n i (0);' n i )g n i (3.16) and letP n q : H q ! V n denote the orthogonal projection ofH q on toV n along (V n ) ⊥ . Standard argumentsfromthetheoryofsplines(see,forexample,[70])canbeusedtoarguethatjP n q (; ) (; )j q ! 0, asn ! 1, for all (; ) 2 H q , and thatkP n q ˆ ' ˆ 'k V ! 0, asn ! 1, for all ˆ '2V withtheconvergenceuniforminq forq2Q. Forn = 1;2;:::andk = 0;1;2;:::weset ˆ x n k () = n X i=0 X n,k i ˆ ' n i (); andweapproximatethe operatorA(q) usingaGalerkinapproach. Thatis, wedefinethe operator A n (q)2L(V n ;V n )byrestrictingtheforma(q;;)toV n V n . Wethenset ˆ A n (q) =e A n (q)τ ; and ˆ B n (q) = (I n ˆ A n (q)) (0;)A n (q) −1 P n q ( q 3 q 2 q 1 ;0) ; (3.17) 36 where ˆ B n (q) 2L(R;V n ) = V n . The matrix representations for these operators with respect to the basisfˆ ' n i g n i=0 are then given by [A n (q)] = [M n (q)] −1 K n (q), [ ˆ A n (q)] = e −[M n (q)] −1 K n (q)τ , and[ ˆ B n (q)] = (I[ ˆ A n (q)]) Ξ n [A n (q)] −1 [M n (q)] −1 [q 2 ;0;0;:::;0] T = (I[ ˆ A n (q)]) Ξ n + [K n (q)] −1 [q 2 ;0;0;:::;0] T ,where[M] n i,j=0 =h(' n i (0);' n i );(' n j (0);' n j )i q ,[K] n i,j=0 = a(q;' n i (0);' n i );(' n j (0);' n j )), and Ξ n = q 2 q 1 [0; 1 n ; 2 n ;:::; n−1 n ;1] T . Letting [ ˆ C n ] = [1;0;0;:::;0]2 R 1×(n+1) ,weconsiderthediscretetimedynamicalsysteminV n givenby ˆ x n k+1 = ˆ A n (q)ˆ x n k + ˆ B n (q)u k ; k = 0;1;2;:::;K1 y n k = ˆ Cˆ x n k ; k = 0;1;2;:::;K; ˆ x n 0 = (0;0)2V n (3.18) orequivalentlyinR n+1 givenby X n,k+1 = [ ˆ A n (q)]X n,k +[ ˆ B n (q)]u k ; k = 0;1;2;:::;K1; y n k = [ ˆ C n ]X n,k ; k = 0;1;2;:::;K; X n,0 = [0;0;:::;0] T 2R n+1 weobtainthat y n k = k−1 X j=0 ˆ C n ˆ A n (q) k−j−1 ˆ B n (q)u j ; (3.19) = k−1 X j=0 [ ˆ C n ][ ˆ A n (q)] k−j−1 [ ˆ B n (q)]u j ; = k−1 X j=0 h n k−j−1 (q)u j ; k = 0;1;2;:::;K; (3.20) whereh n i (q) = [ ˆ C n ][ ˆ A n (q)] i [ ˆ B n (q)]2R,i = 0;1;2;:::;K1. 37 Usinglinearsemigrouptheory(see,forexample,[13,15,63])andinparticulartheTrotter-Kato semigroup approximation theorem (see, Theorem 2.6, and for example, [46, 61]) the following resultscanbeestablished. Theorem3.2. ForQ a compact subset of the positive orthant ofR 4 ,n = 1;2;:::,f' n i ()g n i=0 the standard linear B-splines on the interval [0;1] defined with respect to the uniform meshf0; 1 n ; 2 n ; :::; n−1 n ;1g, V n = spanf(' n i (0);' n i )g n i ,P n q : H q ! V n the orthogonal projection of H q on toV n along (V n ) ⊥ , and ˆ A n (q) and ˆ B n (q) defined as in (3.17), we have thatj ˆ A n (q)P n q (; ) ˆ A(q)(; )j q ! 0,asn!1,forall(; )2H q ,thatk ˆ A n (q)P n q ˆ ' ˆ A(q)ˆ 'k V ! 0,asn!1, for all ˆ ' 2 V, and thatk ˆ B n (q) ˆ B(q)k V ! 0, asn ! 1, with the convergence in all cases uniforminq forq2Q. Proof. First, we show that there exists 2 (A(q))\ ∞ n=1 (A n (q)) such that R λ (A n )P n q z ! R λ (A)z for all z 2 H q with R · () the resolvent operator. Note that since ˆ A(q) results from a(q;;) being restricted to V n V n , we can choose = 0 as in Assumption 2 so that 2 (A(q))\ ∞ n=1 (A n (q)) with > ! for! as inje ˆ A(q)t j Me ωt (from the fact that our operators generateaholormophicsemigroup,see[84])foralln. Letz 2H q and definew =w(q) :=R λ (A(q))z andw n =w n (q) :=R λ (A n (q)P n q z). Since w2D = Dom(A(q))V we have by standard arguments from the theory of splines (see, [70]) that P n q ˆ ' ˆ ' V ! 0 asn ! 1 for ˆ ' 2 V, uniformly inq. Now, we may define a sequence fˆ w n gV n satisfyingjjˆ w n wjj V ! 0asn!1. Letz n :=w n ˆ w n sothatz n 2V n . Wefind that, a(q;w;z n ) =h(A(q))w;z n ihw;z n i =hz;z n ihw;z n i 38 and a(q;w n ;z n ) =h(A n (q))w n ;z n ihw n ;z n i =hP n q z;z n ihw n ;z n i =hz;z n ihw n ;z n i: Hence, a(q;w;z n ) =a(q;w n ;z n )+hw n w;z n i: ByAssumption2wehave 0 jjz n jj 2 V a(q;z n ;z n )+jz n j 2 q =a(q;w n ;z n )a(q; ˆ w n ;z n )+jz n j 2 q =a(q;w;z n )+hww n ;z n ia(q; ˆ w n ;z n )+jz n j 2 q =a(q;w ˆ w n ;z n )+ hww n ;z n i+jz n j 2 q : UsingAssumption1wehave 0 jjz n jj 2 V 0 jjw ˆ w n jj V jjz n jj V + hww n ;z n i+jz n j 2 q : Nownotice, hww n ;z n i =hw ˆ w n ;z n i+hˆ w n w n ;z n i =hw ˆ w n ;z n ihz n ;z n i: 39 Sincejj q ˜ Cjjjj V forsome ˜ C > 0wehavethat 0 jjz n jj 2 V 0 jjw ˆ w n jj V jjz n jj V +hw ˆ w n ;z n i 0 jjw ˆ w n jj V jjz n jj V +jjjw ˆ w n j q jz n j q ( 0 +jj ˜ C 2 )jjw ˆ w n jj V jjz n jj V ! 0 (3.21) asn!1bydefinitionoffˆ w n g. Thus,z n =w n ˆ w n ! 0asn!1intheV norm. Thus,bythe triangle inequality,jjww n jj V ! 0 asn!1 and furtherw n !w in theH q norm asn!1. ThuswithanapplicationoftheTrotter-Katosemigroupapproximationtheorem(Theorem2.6)we findj ˆ A n (q)P n q ˆ ' ˆ A(q)ˆ 'j q ! 0asn!1. Now,letP n V :V !V n betheorthogonalprojection ofV ontoV n along(V n ) ⊥ (usingtheV innerproduct). UsingLemma2.9,ourfirstboundisfor ˆ '2V, jjR λ (A n (q))ˆ 'jj V ¯ C jj 1 2 jˆ 'j q (3.22) where ¯ C is independent ofn. Using this with ˆ ' =P n V zP n q z forz 2V, along with the result following (3.21), we obtainR λ (A n (q))P n V z ! R λ (A(q))z in theV norm for allz 2 V. This is onehalfofwhatisneededfortheTrotter-Katosemigroupapproximationtheorem. Fortheotherhalf,weusethelastboundofLemma2.9. Inthatway,wemayreadilyarguethat for ˆ '2V jjR λ (A n (q))ˆ 'jj V M 1 jj jjˆ 'jj V whereM 1 isindependentofn. Usingthisandtheidentity e A n (q)t = 1 2i Z Γ e λt R λ (A n (q))d (3.23) 40 forΓacontourabout 0 (similartothatgivenin[61]). Wenowhavethatforz2V n , e A n (q)t z V M 2 e λ 0 t jjzjj V withM 2 independentofn. SonowwemayapplytheTrotter-Katosemigroupap- proximationtheoremtoobtain ˆ A n (q)P n V ˆ ' ˆ A(q)ˆ ' V ! 0asn!1forall ˆ '2V uniformly inq. This implies, ˆ A n (q)P n q ˆ ' ˆ A(q)ˆ ' q ! 0 asn ! 1 for all ˆ ' 2 V uniformly inq since V H q . Now,usingbound(3.22)in(3.23)wehavethat(similarto[61]) e A n (q)t z V M 3 e λ 0 t t − 1 2 jzj q (3.24) fort> 0andz2V n ,whereM 3 isindependentofn. SinceV isdenseinH q ,givenz2H q and"> 0,wemaychoosez V 2V suchthatjz V zj q < ". NotingthatA(q)z2V fort> 0,wefind A n (q)P n q zA(q)z V A n (q) P n q zP n V z V V +jjA n (q)P n V z V A(q)z V jj V +jjA(q)(z V z)jj V : By the bound in (3.24), the first term on the right-hand side of the above equation is bounded by M 3 e λ 0 t t − 1 2 jP n q zP n V z V j q ! M 3 e λ 0 t t − 1 2 jzz V j q asn!1. The third term on the right-hand side is handled similarly. The second term approaches zero as n ! 1 by previous arguments. Thus, ˆ A n (q)P n q ˆ '! ˆ A(q)ˆ 'intheV normfor ˆ '2H q uniformlyinq. Lastly,notice ˆ B n (q) ˆ B(q) V jj(I n I)(0;)jj V + ˆ A(q) −1 q 3 q 2 q 1 ;0 ˆ A n (q) −1 P n q q 3 q 2 q 1 ;0 V + ˆ A(q)(0;) ˆ A n (q)(0;) V + P n q q 3 q 2 q 1 ;0 q 3 q 2 q 1 ;0 V ! 0 41 as n ! 1 by the results above and by the fact that ˆ A n (q) and ˆ A n (q) −1 are bounded (so all previousresultsapply). Theorem3.3. Under the same hypotheses as Theorem 3.2, we have thatkˆ x n k (q) ˆ x k (q)k V ! 0, thatjˆ x n k (q)ˆ x k (q)j q ! 0,thatjy n k y k j! 0,andthatjh n k (q)h k (q)j! 0asn!1uniformly ink fork2f0;1;2;:::;Kganduniformlyinq forq2Q. Proof. Recall that fork 2 f0;1;:::;Kg,n 2Z + , andq 2 Q, as per Systems (3.13) and (3.18) we have ˆ x k = K−1 P j=0 ˆ A(q) k−j−1 ˆ B(q)u j and ˆ x n k = K−1 P j=0 ˆ A n (q) k−j−1 ˆ B n (q)u j , respectively. Thus by notingthatBrAC(u)isalwaysboundedbyone,wehave jjˆ x n k ˆ x k jj V k−1 X j=0 ˆ A n (q) k−j−1 ˆ B n (q) ˆ A(q) k−j−1 ˆ B(q) V K−1 X j=0 ˆ A n (q) k−j−1 ˆ B n (q) ˆ A(q) k−j−1 ˆ B(q) V : (3.25) Recall from Theorem (3.2) that for ˆ '2H q , ˆ A n (q)ˆ '! ˆ A(q)ˆ ' and ˆ B n (q)! ˆ B(q) uniformly in q intheV normasn!1. Thus,everyterminthesumfrom(3.25)convergestozeroasn!1 uniformlyinq. Hence, ˆ x n k ! ˆ x k asn!1uniformlyinbothq andk intheV normas(3.25)has nodependenceonk. Next, notice that by the continuous embedding of V in H q we have that there exists some ˜ C > 0suchthatjj q ˜ Cjjjj V . Sowehavethat jˆ x n k ˆ x k j q ˜ Cjjˆ x n k ˆ x k jj V ! 0 asn!1uniformlyinq andk. 42 Again from Systems (3.13) and (3.18), recall that y k = ˆ Cˆ x k and y n k = ˆ C n ˆ x n k , respectively. Further,notethat ˆ C n isequivalentto ˆ C. Thus, jy n k y k j =j ˆ C(ˆ x n k ˆ x k )j ˜ C 2 ˆ C V jjˆ x n k ˆ x k jj V ! 0 (3.26) asn!1uniformlyinbothq andk. Lastly,forh k (q)andh n k (q)asin(3.15)and(3.20),respectively,wehave jh n k (q)h k (q)j = ˆ C ˆ A n (q) k ˆ B n (q) ˆ A(q) k ˆ B(q) ˜ C 3 ˆ C V K−1 X j=0 ˆ A n (q) j ˆ B n (q) ˆ A(q) j ˆ B(q) V (3.27) ! 0 asn!1uniformlyinq andk byTheorem3.2. Remark 6. In (3.26) and (3.27) we also made use of the fact that for ˆ C onV its norm is trivially boundedbyone. Thatis,j ˆ Cˆ 'j 2 ='(0) 2 '(0) 2 + R R ' 2 (s)ds =jjˆ 'jj 2 V forall ˆ '2V. 43 Chapter4 BayesianEstimationofDynamicalSystemParameters In this chapter, we develop a Bayesian framework to estimate the unknown parameters q = [q 1 ;q 2 ] T in the system (3.1). To illustrate our approach, for simplicity but without loss of gen- erality, we have assumed that the sensor parametersq 3 andq 4 have been bench-measured and are therefore known and concentrate our effort on estimating the physiological subject-dependent pa- rametersq 1 andq 2 . Inaddition,wewillassumethatwehavetrainingdata, fu i k g K k=0 ;fy i k g K k=0 R i=1 , fromR participants or subjects where without loss of generality (i.e., by padding with zeros) we haveassumedthatalltraininginput/outputdatasetshavethesamenumber,K,ofobservations. In thiscase,fori = 1;:::;Rasin(3.15)and(3.19)wehave, y i k = k−1 X j=0 h k−j−1 (q)u i j ; k = 0;1;2;:::;K; (4.1) and y n,i k = k−1 X j=0 h n k−j−1 (q)u i j ; k = 0;1;2;:::;K; (4.2) whereh j (q) = ˆ C ˆ A(q) j ˆ B(q) 2 R andh n j (q) = [ ˆ C n ][ ˆ A n (q)] j [ ˆ B n (q)] 2 R, forj = 0;1;2;:::; K1. This formulation is taking the position of finding population parametersq. If we wish to find the parametersq for a specific person the methods outlined in Chapter 4 can still be applied by takingi = 1;:::;R to be differing BrAC/TAC measurement events each withk = 0;:::;K measurementtimesforthedesiredperson. 44 Allofwhatfollowsbelowcaneasilybeextendedtoestimatingallfouroftheparametersinthe model. Ourunderlyingstatisticalmodelincorporatingnoiseisbasedontheobservationoffy i j gas in(4.1)andisgivenby V i j =y i j +" i j = j−1 X ℓ=0 h j−ℓ−1 u i ℓ +" i j (4.3) whereV i j are our measured TAC values, and" i j are the i.i.d. noise terms corresponding to person i at timej with > 0; > 0. Commonly, as we will assume in Section 4.2 and beyond," i j N(0; 2 ). Inordertobeabletocarryouttherequisitecomputations,weconsidertheapproximating statisticalmodelbasedon(4.2) V i j =y n,i j +" i j = j−1 X ℓ=0 h n j−ℓ−1 u i ℓ +" i j ; (4.4) where once again the V i j ’s are assumed to be the measured TAC values. We considerq to be a random vector on some probability spacefΩ;Σ;Pg with support Q and assume that the prior distributionofqisgivenbythepushforwardmeasure 0 . ThatisforGQ P(fq2Gg) =P(q −1 (G)) = Z q −1 (G) dP = Z G d 0 (q): Weassumeindependenceacrossbothi(individuals)andj(samplingtimesforeachindividual), foreachiandjwehaveV i j y i j =" i j (commonlydistributedN(0; 2 ))andsimilarly,V i j y n,i j =" i j (again commonly distributed N(0; 2 )). Letting ' denote the density of " i j ’s, for q 2 Q the likelihoodandtheapproximatinglikelihoodfunctionsaregivenby(see,forexample,[25,80]) L(qjfV i j g) = R Y i=1 K Y j=1 '(V i j y i j ) and L n (qjfV i j g) = R Y i=1 K Y j=1 '(V i j y n,i j ); respectively. 45 AnapplicationofBayes’Theorem(see,Theorem2.12,andforexample,Theorem1.31in[69]) yields that the posterior distribution ofq or the conditional distribution ofq conditioned on the data,fV i j g,isapushforwardmeasure =(jfV i j g)thatisabsolutelycontinuouswithrespectto 0 andwhoseRadon-Nikodymderivative,ordensity,forq2Qisgivenby d d 0 (qjfV i j g) = L(qjfV i j g) R Q L(qjfV i j g)d 0 (q) = 1 Z R Y i=1 K Y j=1 '(V i j y i j (q)); (4.5) where Z = Z Q L(qjfV i j g)d 0 (q) = Z Q R Y i=1 K Y j=1 '(V i j y i j (q))d 0 (q): (4.6) Inthisway,forGQ,wehave P(fq2GgjfV i j g) = Z G d(q) = Z G d(qjfV i j g): (4.7) If, in addition, we have 0 with density d 0 d =f 0 where denotes Lebesgue measure onQ, thenwithconditionaldensityf givenby f(q) =f(qjfV i j g) = d d (qjfV i j g) (4.8) = L(qjfV i j g)f 0 (q) R Q L(qjfV i j g)f 0 (q)d(q) = 1 Z R Y i=1 K Y j=1 '(V i j y i j (q))f 0 (q); with Z = Z Q L(qjfV i j g)f 0 (q)d(q) = Z Q R Y i=1 K Y j=1 '(V i j y i j (q))f 0 (q)d(q); (4.9) and P(fq2GgjfV i j g) = Z G f(q)d(q) = Z G f(qjfV i j g)d(q): (4.10) 46 Analogously, in the case of the approximating likelihood (4.5), (4.6), (4.7), (4.8), (4.9), and (4.10)become d n d 0 (qjfV i j g) = L n (qjfV i j g) R Q L n (qjfV i j g)d 0 (q) = 1 Z R Y i=1 K Y j=1 '(V i j y n,i j (q)); (4.11) Z n = Z Q L n (qjfV i j g)d 0 (q) = Z Q R Y i=1 K Y j=1 '(V i j y n,i j (q))d 0 (q); (4.12) P(fq2GgjfV i j g) = Z G d n (q) = Z G d n (qjfV i j g); f n (q) =f n (qjfV i j g) = d n d (qjfV i j g) (4.13) = L n (qjfV i j g)f 0 (q) R Q L n (qjfV i j g)f 0 (q)d(q) = 1 Z n R Y i=1 K Y j=1 '(V i j y n,i j (q))f 0 (q); Z n = Z Q L n (qjfV i j g)f 0 (q)d(q) = Z Q R Y i=1 K Y j=1 '(V i j y n,i j (q))f 0 (q)d(q); and P(q2GjfV i j g) = Z G f n (q)d(q) = Z G f n (qjfV i j g)d(q); respectively. 4.1 ConvergenceinDistribution Consider the random variableq with posterior distribution described by the measure given in (4.5) and (4.6), and letq n denote the random variableq but with posterior distribution n given 47 by (4.11) and (4.12). In this section, we establish thatq n Dist !q as n ! 1; that is thatq n converges in distribution toq. Recall that due to the the physical constraints based on our model for the alcohol biosensor problem, (3.1), we require that the parametersq lie in the interior of the positiveorthantofR 2 . Theorem 4.1. For a continuous prior 0 with compact support Q in the interior of the positive orthantofR 2 andanoisedistributionwithboundeddensityfunction'andsupportonR,q n ,the randomvariablewithposteriordistribution n givenby(4.11)and(4.12)convergesindistribution tothetherandomvariableqwithposteriordistribution givenby(4.5)and(4.6). Proof. ForS asubsetQ,thetriangleinequalityyields jP(fq n 2SgjfV i j g)P(fq2SgjfV i j g)j (4.14) = Z S d n (q) Z S d(q) = 1 Z n Z S L n (qjfV i j g)d 0 (q) 1 Z Z S L(qjfV i j g)d 0 (q) = 1 Z n Z S R Y i=1 K Y j=1 '(V i j y n,i j (q))d 0 (q) 1 Z Z S R Y i=1 K Y j=1 '(V i j y i j (q))d 0 (q) 1 Z 1 Z n 0 @ Z S R Y i=1 K Y j=1 '(V i j y i j (q))d 0 (q) 1 A + 1 Z n Z S R Y i=1 K Y j=1 '(V i j y i j (q)) R Y i=1 K Y j=1 '(V i j y n,i j (q)) d 0 (q); where' is the normal density describing the distributionof the noise term in (4.3), andZ andZ n areasin(4.6)and(4.12),respectively. Focusing first on the limit ofj1=Z1=Z n j asn!1, by Lemma 3.1 we have that they j i (q) are continuous inq forq2Q,i2f0;1;:::;Rg, andj 2f0;1;:::;Kg. SinceQ is compact, the 48 fy i j (q)g are bounded and thus 0<Z <1. By Theorem 3.3, sincey n,i j (q)!y i j (q) uniformly in q forq 2Q asn!1, 0<Z n <1 forn large enough. Again by Theorem 3.3 it follows from (4.6), (4.12), and the Bounded Convergence Theorem thatZ n !Z asn!1 and therefore that j1=Z1=Z n j! 0asn!1. Then,essentiallythesameargumentsyieldthat Z S jL(qjfV i j g)L n (qjfV i j g)jd 0 (q)! 0; fromwhichitthenimmediatelyfollowsthat 1 Z n Z S jL(qjfV i j g)L n (qjfV i j g)jd 0 (q)! 0; andthereforefrom(4.14)thatq n Dist !qasn!1andthetheoremisproven. Forthepushforwardmeasures and n from(4.5)and(4.11),respectively,wearecommonly interested in their respective expected values and the convergences there within. SinceQ is com- pact,adirectinvocationofthePortmanteautheorem(see,[69])establishesthefollowingcorollary whichguaranteestheconvergenceofallmomentsdescribedby n tothoseof. Corollary4.2. UnderthehypothesesofTheorem4.1,andforanycontinuousfunctiong :Q!R wehavethatasn!1 E π n[g(q)] = Z Q g(q)d n (q)! Z Q g(q)d(q) =E π [g(q)]: 4.2 Consistency Inthissection,wedemonstratethestrongconsistencyoftheposteriordistribution(asinDefinition 2.9)withrespecttotheparameters,q,byimposingstrongerassumptionsonthedistributionofthe noiseterms" i j in(4.3)andontheprior, 0 ,byrestrictingQtoarectangleinthepositiveorthantof R 2 ,andbyapplyingtheframeworksummarizedin[19]. 49 As in [19], we show consistency of the posterior distribution as in (4.8), rather than consis- tency of a point estimator based on the posterior distribution (see, Definition 2.9). As such, for prior 0 over Q, posterior (jfV i j g) from (4.5), and i.i.d. noise " i j N(0; 2 ) for > 0, we also consider that for q 2 Q assumed known we have random variablesV i j N(y i j (q); 2 ) as determined by (4.3) for i = 1;2;:::;R and j = 1;2;:::;K. Further, we have thatfV i j g i,j are independentiniandj,butarenon-identicallydistributed(i.n.i.d.). For clarity and brevity we consider the random vectorV i = (V i 0 ;V i 1 ;:::;V i K ) T with values inR K+1 andindependententriesderivedfromthematrixequivalentto(4.3)and(4.4),namely V i =y i +ε i =Hu i +ε i ; and V i =y n,i +ε i =H n u i +ε i with noise vectorsε i = (" i 0 ;" i 1 ;:::;" i K ) T , observed TAC vectorsV i = (V i 0 ;V i 1 ;:::;V i K ) T , theo- retical TAC vectorsy i = (y i 0 ;j i 1 ;:::;y i K ) T and analogousy n,i , BrAC data vectorsu i = (u i 0 ;u i 1 ; :::;u i K ) T and analogousu n,i , and kernel matricesH andH n with entries [H] i,j = h i−j 1 j≤i and [H n ] i,j =h n i−j 1 j≤i ,respectively. Inthisway,foreveryi2f1;2;:::;Rg,byindependenceinj we havethefamilyofjointdistributions ( f i,q (·) = K Y j=0 '( j y i j (q)) :q2Q ) representing the possible densities ofV i for' the noise density andy i j as in (4.1). We are inter- estedinthescenariowherethenumberofsubjectsR!1. Withourreframing,forGQbyindependenceiniwemayrewrite(4.5)as (GjfV i g) = R G L(qjfV i g)d 0 (q) R Q L(qjfV i g)d 0 (q) = R G R Q i=1 f i,q (V i )d 0 (q) R Q R Q i=1 f i,q (V i )d 0 (q) ; (4.15) 50 where for alli we have data vectorsV i . For our purposes we will be interested in the following equivalentform (GjfV i g) = R G L(qjfV i g)=L(q 0 jfV i g)d 0 (q) R Q L(qjfV i g)=L(q 0 jfV i g)d 0 (q) = J G (fV i g) J(fV i g) ; (4.16) whereq 0 2Qisthetruevalueofourparameters[q 1 ;q 2 ] T . WefirstformalizetheresultsdiscussedinSection7of[19]thathandlethei.n.i.d. caseofposte- riorconsistency. WeemphasizeDefinition2.9withadirectrepetition,andassuchwesaythatour posteriordistributionsf(jfV i g)gasin(4.15)are(strongly)consistentatq 0 iff(UjfV i g)g! 1 a.sP ∞ q 0 foreveryneighborhoodU ofq 0 asR!1,whereP ∞ q 0 = Q ∞ i=1 P i,q 0 withP i,q 0 theproba- bilitydistributiongeneratedbyf i,q 0 withdatasamplesfV i g. For this we show that for sets G Q with q 0 = 2 G, J G (fV k g) ! 0 and J(fV i g) ! 1 as R ! 1 in some appropriate manner to be made precise below. ForJ G (fV i g) ! 0 we take the same approach as expressed in [19] and thus will make use of Definitions 2.10, 2.14, and 2.15 utilizingthei.n.i.d. motivationfromSection2.3. FromthesedefinitionswecontinueusingtheideasfromSection2.3andmakeuseofTheorem 2.23 . For proof see Sections 3 and 7 of [19] (or, for examples, [87]). To showJ(fV i g)!1 as R!1weapplyTheorem2.22,whichutilizestheapproachasoutlinedintheproofofTheorem 1ofAppendixA.2in[20](specificallytheproofofeq. (8)inAppendixA.2). Weincludetheproof hereforcompleteness. Forasimilarapproachsee[21]. 51 Theorem4.3. Let 0 be a prior over parameter spaceQ,fV i g ∞ i=1 be independent but not identi- cally distributed data with distribution generated byf i,q forq 2 Q, andq 0 2 Q the true value of theparameters[q 1 ;q 2 ] T . Forq2Qdefine Λ i (q 0 ;q) = log f i,q 0 (V i ) f i,q (V i ) ; κ i (q 0 ;q) =E q 0 [Λ i (q 0 ;q)]; S i (q 0 ;q) = Var q 0 [Λ i (q 0 ;q)]: IfthereexistsasetBQwith 0 (B)> 0suchthat 1. P i≥1 S i (q 0 ;q) i 2 <1 8q2B,and 2. Forevery"> 0, 0 (B\fq : κ i (q 0 ;q)<" 8ig)> 0. Then8 > 0, e Rβ J(fV i g R i=1 )!1 a.s.P ∞ q 0 asR!1forJ(fV i g)asin(4.16). Proof. Forq2Q,wedefinethefollowing, log + (x) = maxf0;log(x)g log − (x) = minf0;log(x)g W i = log + f i,q (V i ) f i,q 0 (V i ) κ + i (q 0 ;q) =E q 0 log + f i,q 0 (V i ) f i,q (V i ) κ − i (q 0 ;q) =E q 0 log − f i,q 0 (V i ) f i,q (V i ) : 52 Now,notice Var q 0 (W i ) =E q 0 (W 2 i ) κ + i 2 E q 0 (W 2 i ) κ + i (q 0 ;q)K − i (q 0 ;q) 2 =E(W 2 i )κ i (q 0 ;q) 2 E q 0 " log + f i,q 0 (V i ) f i,q (V i ) 2 # +E q 0 " log − f i,q 0 (V i ) f i,q (V i ) 2 # κ i (q 0 ;q) 2 =E q 0 " log + f i,q 0 (V i ) f i,q (V i ) log − f i,q 0 (V i ) f i,q (V i ) 2 # κ i (q 0 ;q) 2 (4.17) = Var i (q 0 ;q) where (4.17) follows from the fact that log + (x)log − (x) = 0 for all x > 0. For set B as in the theoremstatement,itfollowsthat P i≥1 Varq 0 (W i ) i 2 <1forallq2B. By Kolmogorov’s Strong Law of Large Numbers for independent, non-identically distributed randomvariables(Theorem2.15), 1 R R X i=1 W i κ + i (q 0 ;q) ! 0 a.sP ∞ q 0 : (4.18) Foreveryq2B withP ∞ q 0 probability1,wehave liminf R→∞ 1 R R X i=1 log f i,q (V i ) f i,q 0 (V i ) limsup R→∞ 1 R R X i=1 log + f i,q (V i ) f i,q 0 (V i ) =limsup R→∞ 1 R R X i=1 κ + i (q 0 ;q) (4.19) limsup R→∞ 1 R R X i=1 κ i (q 0 ;q)+ 1 R R X i=1 r κ i (q 0 ;q) 2 ! (4.20) limsup R→∞ 0 @ 1 R R X i=1 κ i (q 0 ;q)+ v u u t 1 R R X i=1 κ i (q 0 ;q) 2 1 A (4.21) 53 where (4.19) follows from (4.18), (4.20) follows from Lemma 2.16, and (4.21) follows from Jensen’sinequality(Theorem2.13). Now,let > 0,andchoose"sothat"+ p "=2=2. Also,let ¯ B =B\fq2Q : κ i (q 0 ;q)< " 8ig:Forq2 ¯ B, 1 R P R i=1 κ i (q 0 ;q)<"sothatforeveryq2 ¯ B liminf R→∞ 1 R R X i=1 log f i,q (V i ) f i,q 0 (V i ) ("+ p "=2)=2: Now, J(fVg R i=1 ) Z C L(qjfVg R i=1 ) L(q 0 jfVg R i=1 d 0 (q) wherebyFatou’slemma(2.25)itthenfollowsthatforevery > 0 e Rβ J(fVg R i=1 )!1 a.s.P ∞ q 0 : Hence,thetheoremhasbeenproven. Before moving on to our main theorem we apply Theorem 2.11 and subsequent Corollary 4.4 to prove a lemma that will be of use to us later. The following corollary is an immediate consequence of the work in [86]. Specifically, that for R λ (A(q)) the resolvent of the generator ofT(;q), the mapq 7! R λ (A(q)) is analytic as a map fromQ toL(V ∗ ;V), and from (2.1) the map ¯ q 7! T q (t; ¯ q) depends continuously onR λ (A(¯ q)). Note that Σ γ is independent of q as the constants 0 and 0 , 0 fromConditions1and2,respectively,ofChapter3areindependentofq. Corollary 4.4. Under the same hypotheses as Theorem 2.11, we have that the map ¯ q 7! T q (t; ¯ q) iscontinuousintheoperatornormonL(Q;L(V ∗ ;V))for ¯ q intheinteriorofQ. Lemma4.5. ForQ a rectangle in the positive orthant ofR 2 , Hilbert spacesH q andV as in (3.4) and (3.5), respectively, bilinear forma(q;;) : V V ! R as in (3.9) withq 3 andq 4 assumed known, and generated infinitesimal operator A(q) as in (3.10), then the generated holomorphic 54 semigroup of bounded linear operatorsfe A(q)t : t 0g onH q andV ∗ is (Fr´ echet) differentiable andLipschitzinqintheinteriorofQ. Further,fori = 1;:::;Randj = 1;:::;K,y i j andy n,i j asin (4.1)and(4.2)are(Fr´ echet)differentiableandLipschitzinq withLipschitzconstantsindependent ofiandj. Proof. First, by Condition 3 of Chapter 3 we have that forq 2 Q,q 7! a(q;;) is continuous in q. Second,noticefor ˆ ; ˆ '2V with ˆ = ( (0); )and ˆ ' = ('(0);'), a(q; ˆ ; ˆ ') = q 1 q 4 q 3 (0)'(0)+q 1 1 Z 0 ′ (x)' ′ (x)dx =a 0 (q; ;')+a 1 (q; ;') whereq 7!a 0 (q; ;') andq 7!a 1 (q; ;') are clearly linear inq forq 2Q. Hence the bilinear forma(q;;) is affine and continuous in q Thus, by Theorem 2.11, we have that the semigroup generated bya(q;;),fe A(q)t : t 0g = fT(t;q) : t 0g, is (Fr´ echet) differentiable inq for q2Q. Denotethederivativeinq and ¯ q actingonq2QbyT q (t; ¯ q)q. Moreoverfort2 (0;T]wehaveforq 1 ;q 2 2QandlinesegmentS =fsq 1 +(1s)q 2 : 0 s 1g, kT(t;q 1 )T(t;q 2 )k L(V ∗ ,V) kq 1 q 2 k 1 sup x∈S kT q (t;x)k L(Q,L(V ∗ ,V)) kq 1 q 2 k 1 max x∈Q kT q (t;x)k L(Q,L(V ∗ ,V)) ˜ Ckq 1 q 2 k 1 (4.22) withkk L(V ∗ ,V) theoperatornorm,wherethefirstinequalityfollowsfromtheMeanValueTheorem onBanachspaces(seeTheorem2.29),andthesecondandthirdinequalitiesfollowfromthecom- pactnessofQ,Corollary4.4,andthecontinuityofthemapT q (t; ¯ q))7!kT q (t; ¯ q))k L(Q,L(V ∗ ,V)) . Further,underzero-orderholdthedifferentiabilityandLipschitzpropertiesof ˆ A(q) =e A(q)τ = T(;q) still remain. Now, considering ˆ B(q) from (3.12), that is ˆ B(q) = (I ˆ A(q)) (0;) 55 A(q) −1 ( q 3 q 2 q 1 ;0) , we find that it is a sum and product ofq-differentiable andq-Lipschitz terms and thus is differentiable and Lipschitz inq. Sinceh andy as in (4.1) are a composition and sum ofq-differentiabletermstheyremaindifferentiable. Using(4.22)wehavethefollowingLipschitz boundforallj2f0;1;:::;Kgandq; ¯ q intheinteriorofQ, jh j (q)h j (¯ q)j =j ˆ C ˆ A(q) j ˆ B(¯ q) ˆ C ˆ A(q) j ˆ B(¯ q)j k ˆ Ck L(Hq,R) h k ˆ A(q) j ˆ A(¯ q) j k L(Hq) k ˆ B(q)k L(R,Hq) + k ˆ A(¯ q) j k L(Hq) k ˆ B(q) ˆ B(¯ q)k L(R,Hq) i C 1 h k ˆ A(q) j ˆ A(¯ q) j k L(Hq) +k ˆ B(q) ˆ B(¯ q)k L(R,Hq) i C 1 ( ˜ C ˆ A + ˜ C ˆ B )kq 1 q 2 k 1 (4.23) forC 1 the max of the operator norms for ˆ A(q), ˆ B(q), ˆ C overQ, and ˜ C ˆ A , ˜ B ˆ A the max Lipschitz constants of ˆ A(q) and ˆ B(q) over all k and Q. The final inequality above follows from (4.22) by noticing that for all ˆ ' 2 H q , k ˆ A(q)ˆ ' ˆ A(¯ q)ˆ 'k Hq C V k ˆ A(q)ˆ ' ˆ A(¯ q)ˆ 'k V and that by identificationthesupremumoverV ∗ islargerthanthatoverH q . For ˆ A n (q)and ˆ B n (q)asin(3.17) byarepetitionoftheaboveargumentswemaintaindifferentiabilityinq,andthush n andy n asin (4.2)aredifferentiableandLipschitzinq. Lastly,foralli = 1;:::;R,j = 1;:::;K wehavethatforq; ¯ q2Q jy i j (q)y i j (¯ q)j j−1 X ℓ=0 jh j−ℓ−1 (q)h j−ℓ−1 (¯ q)ju i j = j−1 X ℓ=0 j ˆ C ˆ A(q) j−ℓ−1 ˆ B(q) ˆ C ˆ A(¯ q) j−ℓ−1 ˆ B(¯ q)ju i j ˜ M (K+1) (K +1)kq ¯ qk 1 (4.24) wherefu i j gareBrACvaluesboundedbydefinitiontobein[0;1], ˜ M istheLipschitzconstantfrom (4.23),andK +1isthefixedupperboundonthenumberoftemporalobservations,j. Hence,the 56 Lipschitzconstantfory i j isindependentof(i;j). Bynoticingthatthepreviousstatementholdsfor y i,n j witharepetitionoftheworkleadingto(4.24),ourlemmaisproven. AdirectconsequenceofLemma4.5isthatforalli = 1;2;:::;Rwithκ i (¯ q;q)andΛ i (¯ q;q)as inthestatementofTheorem4.3,wehave jκ i (¯ q;q)j =jE ¯ q [Λ i (¯ q;q)]j K X j=0 1 2 2 y i j (¯ q) 2 y i j (q)+2y i j (¯ q)(y i j (q)y i j (¯ q)) = 1 2 2 K X j=0 (y i j (¯ q)y i j (q)) 2 K ¯ M 2 2 2 k¯ qqk 1 = ˜ `k¯ qqk 1 (4.25) where > 0 is the standard deviation of the N(0; 2 ) noise density, and ¯ M is the Lipschitz constant from (4.24) that is independent ofi (andj). Thus, for any ∗ > 0,i 2 f1;:::;Rg, and ¯ q;q2fq2Q :kq 0 qk 1 < ∗ g,wehavethat kf i,¯ q f i,q k L 1 (2jκ i (¯ q;q)j) 1 2 (2 ˜ `k¯ qqk 1 ) 1 2 < 2( ˜ ` ∗ ) 1 2 (4.26) by the relationship between total variation and Kullback-Leibler distances, for ˜ ` as in (4.25). A keyusageofthisworkistouchedoninthefollowingexample. Example4.1. Let ∗ beasin(4.26). Ifweletq ∗ 2Qbesuchthat kf i,q 0 f i,q ∗k L 1 > ∗ (4.27) and consider the setG =fq 2Q :kq ∗ qk 1 < ( ∗ ) 2 =(16 ˜ `)g, thenG is strongly separated from q 0 (see,Definition2.15). 57 To see this, first notice that for all ¯ q;q 2 G, we find kf i,¯ q f i,q k L 1 2 ∗ =4 = ∗ =2. Now, let be an density onG. As per Definition 2.14, the marginal density of V 1 ,v 1 ν (V 1 ) = R Q L(qjV 1 )d(q),satisfies kf i,q ∗v 1 ν (V 1 )k L 1 = Z R K+1 jf i,q ∗(x) Z G f i,q (x)d(q)jdx Z G Z R K+1 jf i,q ∗(x)f i,q (x)jdxd(q) = Z G kf i,q ∗f i,q k L 1 d(q) ∗ 2 (G) = ∗ 2 : BytherelationshipbetweentheHellingerdistanceandAffinity,wehavethat 8(1Aff(f i,q 0 ;v 1 ν (V 1 )))kf i,q 0 v 1 ν (V 1 )k 2 L 1 : Thus, Aff(f i,q 0 ;v 1 ν (V 1 )) 1 1 8 kf i,q 0 v 1 ν (V 1 )k 2 L 1 1 1 8 kf i,q 0 f i,q ∗k 2 L 1 kv 1 ν (V 1 )k 2 L 1 f i,q ∗ 2 (4.28) 1 1 8 ∗ ∗ 2 2 = 1 1 8 ( ∗ ) 2 4 = 1 ( ∗ ) 2 32 2 (0;1) (4.29) where(4.28))followsfromthereversetriangleinequality. Hence,byDefinition2.15wehavethatG isstronglyseparatedfromq 0 (for asin(4.29). WiththisexampleinmindwenotethatConditions1and2ofTheorem2.23aresatisfiedifthe followingspecialconditionismet: 1. Forevery ∗ > 0,thereexistsetsG 1 ;G 2 ;:::withL 1 diameterlessthan ∗ , diam(G i )< ∗ , S i≥1 G i =Q,and P i≥1 p 0 (G i )<1forthemappingsq7!f i,q 58 where 0 is the prior overQ. This follows from the fact that if special Condition 1 holds then we may take an" ∗ -neighborhood ofq 0 ,U = fq 2 Q : kf i,q 0 f i,q k L 1 < " ∗ 8ig. Since (4.26) is independentofi,U isnonemptyandcontainsthesetfq2Q :kq 0 qk 1 < (" ∗ ) 2 =(4 ˜ `)g. Nowset ∗ = (" ∗ ) 2 =(16 ˜ `),andbycompactnesscoverQwithafinitenumberofdisjointsetsA i determined by the ballsfq 2 Q : k¯ q i qk 1 < ∗ g with model q 7! f i,q , wheref¯ q i g γ i=1 represents a finite set of points inQ chosen so that S i≥1 G i = Q. From theseG i ’s we have that the finite subset that intersect withU c must coverU c . This finite subset ofG i ’s subsequently satisfies the assumptions of Theorem 2.23. Specifically, the strong separation condition is satisfied as per the discussion surrounding (4.27) by noticing that on eachG i we havekf i,q 0 f i,¯ q i k 1 > " ∗ , and the convergent sum condition is satisfied by the fact that the G i ’s can be considered (made) mutually exclusive withunioncontainedinQ. Wenowstateandproveourmaintheorem. Theorem 4.6. For a continuous prior 0 with compact supportQ, a rectangle in the interior of the positive orthant ofR 2 , i.i.d noise distributed asN(0; 2 ) for > 0, datafV i g R i=1 drawn from independent but not identically distributed distributions generated by f i,q as in (4.15), Hilbert spacesH q andV as in (3.4) and (3.5), respectively, bilinear forma(q;;) : V V ! R as in (3.9) withq 3 andq 4 assumed known, generated infinitesimal operatorA(q) as in (3.10), and true parameterq 0 2Q,wehavethatourposterior(jfV i g)asin(4.15)isconsistent(asinDefinition 2.9)forq 0 asR!1. Proof. ForanysetGQwithq 0 = 2Awewillusetheformof(GjfV i g)asin(4.16)andhandle thenumerator,J G anddenominator,J separately. First, as Q is compact, for any > 0 we may cover Q by a finite number of sets G i , i = 1;2;:::; where each G i is a subset of an L 1 ball in Q. That is, for every i and q; ¯ q 2 A i we have thatkq ¯ qk 1 < . ForR large enough, if on eachG i we consider the modelq 7! f i,q for i2f1;:::;Rgandf i,q thedensityoftherandomvariableV i withqassumedknown,thenspecial Condition 1 is satisfied for prior 0 . Hence, by Theorem 2.23 we have that for some 0 > 0, e Rβ 0 J G (fV i g)! 0a.s.P ∞ q 0 asR!1. 59 NowforΛ i ,κ i ,andS i asinthestatementofTheorem4.3,wehavethatfori = 1;2;:::;Rand q2Q jκ i (q 0 ;q)j ˜ `kq 0 qk 1 ; and (4.30) S i (q 0 ;q) = K X j=0 Var q 0 1 2 2 y i j (q 0 ) 2 y i j (q) 2 +2V k (y i j (q 0 )y i j (q)) = K X j=0 1 4 4 Var q 0 2V i j (y i j (q 0 )y i j (q)) = K X j=0 4 2 4 4 (y i j (q)y i j (q 0 )) 2 ˜ ` 2 4 kq 0 qk 2 1 for(4.30)and ˜ `asdeterminedby(4.25). Thus,forq 0 ;q2Qwefindthat X i≥1 S i (q 0 ;q) i 2 ˜ ` 2 4 kq 0 qk 2 1 ! X i≥1 1 i 2 <1: Further, by (4.30) we have that for every" > 0 andi,fq :jκ i (q 0 ;q)j < "g is non-empty and our choice in such q does not depend on i. Hence the setfq : jκ i (q 0 ;q)j < " 8ig is non-empty. Thus, for B = Q we satisfy the assumptions of Theorem 4.3 and therefore find that 8 > 0, e Rβ J(fV i g)!1a.s.P ∞ q 0 ask!1. So for any set G Q with q 0 = 2 A, from (4.16) we have that (GjfV i g) ! 0 a.s. P ∞ q 0 as R!1andthusthetheoremhadbeenproved. FromLemma4.5wefindthatwemaintainthedifferentiabilityandLipschitzpropertiesofthe finite-dimensional semigroup as in (3.17) and respective kernel as in (4.2). Thus, with a straight- forward rewriting of (4.15) and (4.16) in terms of the the finite-dimensional posterior (4.11), and repetitionoftheworkfollowingLemma4.5throughtheproofofTheorem4.6wehavethefollow- ingcorollary. 60 Corollary 4.7. Under the same hypotheses as Theorem 4.6, for fixed positive integern we have that our finite-dimensional posterior n (jfV i g) as in(4.11) is consistent (as in Definition 2.9) at q 0 asR!1. Remark 7. In both Theorem 4.6 and Corollary 4.7, we wish to reemphasize the definition of con- sistency,thatisDefinition2.9. Akeypartofthedefinitionisthatthesequenceofposteriorproba- bilities for any neighborhoodU containingq 0 converges almost surely with respect to the infinite product measure over the data, P ∞ q 0 . This means that on null sets of P ∞ q 0 , which contain infinite sequences of data,fV i g ∞ i=1 , we may find that the sequence of posterior probabilities for any setU containing q 0 does not go to one. Further, a key component of any Bayesian setup that remains important for us is the prior overQ, 0 . The prior can often be viewed as a regularization term, and thus for our model, consistency may be affected—most likely having a slower convergence rate—if our prior is not as informed. In addition, if we wish to understand what sequences or sets of sequences of observation data maintain consistency, then we must consider the interplay between the non-null sets ofP ∞ q 0 and the information level of the prior. In understanding the non- null sets of P ∞ q 0 , we may be able to identify characteristics within the observation data sets that have meaning in terms of the underlying problem. Though, the determination of P ∞ q 0 as well as theinterpretationofcharacteristicsaboutobservationdatasetsmustbedoneonamodel-by-model basis. In understanding the information-level of the prior, we may identify the decay constants in Theorems 2.22 and 2.23, and thus may be able to identify information density levels within oberservation data that aid in achieving consistency for specific sequences of data. Combining thiswithinformationregardingP ∞ q 0 meansthatidentifiabilityproblemswithrespecttoobservation data may be considered with the added possible benefit of rate determination. If we wish to take this further by attempting to determine characteristics about the input BrAC data that guarantee consistency, we still have to consider all of the previous remarks regarding the determination of observation (TAC) data characteristics. In addition, we would have to take the underlying PDE model (3.1) into account and would have to use some method (e.g., deconvolution as in Section 4.3, direct simulation of BrAC to TAC, etc.) to determine the sets of input data that generate the 61 foundP ∞ q 0 non-nullsetsofobservationdata. Fromhere,wemaybeabletoidentifycharacteristics withinthesetsofinputdataandmaybeabletodetermineconstraintsthatguaranteeconsistency. 4.3 DeconvolutionofBrACfromTAC In this section, we consider the problem of using the biosensor measured TAC signal to estimate BrAC. We do this by deconvolving it; to wit we invert the convolution given in (4.2) subject to a positivity constraint and regularization to mitigate the inherent ill-posedness of the inversion. Recallthattheconvolutiongivenin(4.2)wasfoundbysolvingthefinite-dimensionaldiscretetime system(3.18)derivedfrom(3.1). Weemploythemethodoriginallydescribedin[77],whereinthe problemisformulatedasaconstrained,regularized,optimizationproblem(see,forexample,[50]). We first briefly summarize the treatment in [77] and then follow by showing how our work is able to make direct use of this theory. Let ˜ V and ˜ H be Hilbert spaces forming a Gelfand Triple, ˜ V ,! ˜ H ,! ˜ V ∗ . For an admissible setQ, a compact subset of the positive orthant ofR 2 , withq 2 Q, letA(q) be an abstract parabolic operator defined by a sesquilinear forma(q;;) : V V ! R (i.e., one that satisfies conditions 1, 2 and 3 in Chapter 3) that when restricted to f'2 ˜ V :A(q)'2 ˜ Hg generates a holomorphic semigroup on ˜ H,fe A(q)t :t 0g. For bounded operatorsB(q)2L(R; ˜ H)andC(q)2L( ˜ H;R)considertheinput/outputsystem ˙ x(t) =A(q)x(t)+B(q)u(t) x(0) =x 0 2 ˜ H; y(t) =C(q)x(t) (4.31) whereontheinterval[0;T],u2L 2 (0;T)istheinput,y theoutput,andxisthestatevariable. For sampling interval > 0 and zero-order hold input u(t) = u j , t 2 [j;(j + 1)), j = 0;1;2;:::thecorrespondingsampled-timesystemisgivenby x j+1 = ˆ A(q)x j + ˆ B(q)u j x 0 2 ˜ H; y j = ˆ C(q)x j (4.32) 62 where ˆ A(q) = e A(q)τ 2L( ˜ H; ˜ H), ˆ B(q) = R τ 0 e A(q)s B(q)ds 2L(R; ˜ H), and ˆ C(q) = C(q) 2 L( ˜ H;R),x j =x(j)2 ˜ H,y j =y(j)2Rj = 0;1;2;:::,andforallj,fu j gRarezero-order holdinputvalues. Now letq be a random variable with support the parameter space Q. For ˜ the probability measure ofq, define the Bochner spacesV =L 2 ˜ π (Q; ˜ V),H =L 2 ˜ π (Q; ˜ H), andU =L 2 ˜ π (Q;R). It is easily shown that the spacesV andH form a Gelfand tripleV ,! H ,! V ∗ . Definea(;) : V V ! R bya('; ) = E ˜ π [a(q;'(q); (q))] = R Q a(q;'(q); (q))d˜ (q) for'; 2 V. Then, as in Chapter 3, the form a(;) satisfies conditions 1, 2 and 3 and therefore defines a linear mapA that when restricted tof'2V :A'2Hg, generates an analytic semigroup onH, fe At :t 0g. Assumefurtherthatthemapq7!B(q)isinL ∞ (Q;L(R; ˜ H))andthatthemapq7!C(q)is inL 2 (Q;L( ˜ H;R)) with respect to the measure ˜ . (Note that it then follows thatB 2L ∞ (Q; ˜ H) and by the Riesz Representation Theorem, that effectively C 2 L 2 (Q; ˜ H) = H). Then define bounded linear operators B 2 L(U;H) and C 2 L(H;R) by hBu; i H = E ˜ π [hB(q)u(q); (q)i ˜ H ] = R Q hB(q)u(q); (q)i ˜ H d˜ (q) andC = E ˜ π [C(q) (q)] = R Q C(q) (q)d˜ (q), re- spectively, foru 2 U and 2 H. It can then be shown [40, 71] that foru 2 L 2 ([0;T];U) the solutionto(4.31)withu(t) =u(t;q) =u(t)agreeswiththesolutionto ˙ x(t) =Ax(t)+Bu(t) x(0) =x 0 2H; y(t) =Cx(t); (4.33) for ˜ almosteveryq2Q. Then with sampling interval > 0 as in (4.32), and zero-order hold inputu(t) = u j , t 2 [j;(j +1)),u j 2U,j = 0;1;2;:::,(4.33)becomes x j+1 = ˆ Ax j + ˆ Bu j x 0 2H y j = ˆ Cx j (4.34) 63 forj = 0;1;2;:::where ˆ A =e Aτ 2L(H;H), ˆ B = R τ 0 e As Bds2L(U;H),and ˆ C 2L(H;R). Now, with (4.34), note thatfu j g U is obtained by zero-order hold sampling a continuous time signal. That is, the input to (4.34) isu(j) =u j 2 U withu at least continuous on [0;T]. We seek an estimate for the input based on this model, wherein the input estimateu is a function of both time and the random parametersq. For optimization purposes (more precisely, to be able to include regularization) we require additional regularity. Given the time interval [0;T], let u2S(0;T) =H 1 (0;T;U)andletU beacompactsubsetofS(0;T). Theinputestimationordeconvolutionproblemisthengivenby u ∗ = argmin U J(u) = argmin U K X k=1 jy k (u) ˆ y k j 2 +kuk 2 S(0,T) (4.35) wherekk 2 S(0,T) is a norm onS(0;T) that will be defined below (see, (4.41)),fˆ y k g are measured outputvalues,thetermkuk 2 S(0,T) servesasregularization,and y k (u) = k−1 X j=0 hh k−j−1 ;u j i U ; k = 1;2;:::;K withu j =u(j)forj = 1;2;:::;K thezero-orderholdinputtothediscretetimesystem(4.34), and convolution filterh ℓ = ˆ C ˆ A ℓ−1 ˆ B 2L(U;R) =U ∗ (which is equal toU, by the Riesz Repre- sentationTheorem)where ˆ C 2L(H;R), ˆ A2L(H;H),and ˆ B = R Q e As Bds2L(U;H). Solving(4.35)requiresfinitedimensionalapproximations. ForindexM,letU M defineanap- proximatingfamilyofclosedsubsetsofU,whereeachsubsetiscontainedwithinacorresponding finite dimensional subspace,S M ofS(0;T). Further we require that for eachu2U there exists a sequencefu M g withu M 2 U M such thatu M ! u in S(0;T) as M ! 1. For index N, letV N be an element of an approximating family of finite-dimensional subspaces ofV, and let P N H :H!V N betheorthogonalprojectionofHontoV N . WealsorequireofthespacesV N that foreachv2V,P N H v!v inV asN !1. 64 We next specify finite-dimensional operators ˆ A N 2 L(V N ;V N ), ˆ B N 2 L(U;V N ), and ˆ C N 2L(V N ;R) that define the finite-dimensional system analogous to (4.34). That is, letA N : V N ! V N be given byhA N ' N ; N i H = a(' N ; N ) for' N ; N 2 V N , ˆ A N = e A N τ ,B N = P N H B, ˆ B N = R τ 0 e A N s B N ds, and ˆ C N = C. In this way, we obtain a doubly-indexed sequence of approximatingfinite-dimensionaloptimizationordeconvolutionproblemsgivenby u ∗ L = argmin U M J L (u) = argmin U M K X k=1 jy N k (u) ˆ y k j 2 +kuk 2 S(0,T) (4.36) where y N k (u) = k−1 X j=0 hh N k−j−1 ;u j i U ; k = 1;2;:::;K withL = (M;N)andh N ℓ = ˆ C N ( ˆ A N ) ℓ ˆ B N 2U. Using the approximation properties of the subspacesV N andU M (that is, that for eachu 2 U M there exists a sequence fu M g withu M 2 U M and ku M uk S(0,T) ! 0 as M ! 1, and that for eachv 2V,kP N H vvk V ! 0 asN !1), and the corresponding operators ˆ A N 2 L(V N ;V N ), ˆ B N 2L(U;V N ),and ˆ C N 2L(V N ;R),itcanbeshownthat1)foreachmulti-index L, (4.36) admits a solutionu ∗ L , and 2) there exists a subsequence offu ∗ L g,fu ∗ L k g fu ∗ L g with u ∗ L !u ∗ stronglyask!1withu ∗ asolutionof(4.35). Further,ifinadditionU isassumedto beaclosedandconvexsubsetofS(0;T),foreachM,U M isaclosedandconvexsubsetofU,and theoptimizationproblemgivenin(4.35)admitsaunique(withrespecttosampling)solution,then the sequence of solutions to (4.36),fu ∗ L g converges strongly, or inS(0;T) to the unique solution of(4.35),u ∗ . FortheproofsoftheseresultsseeSection5of[77]. Tonumericallycarryouttherequisitecomputationstoactuallydetermineu ∗ L forgivenvalues ofM, N andL = (M;N), we continue to apply the results in [77] while also connecting them to our treatment in Chapters 3-4 above. We assume that the feasible parameter set Q is a com- pact rectangle in the positive orthant ofR 2 , we set ˜ H = H q and ˜ V = V as in (3.4) and (3.5), respectively,andweidentifytheoperatorsin(4.32)withthosein(3.13). Ourdistributionoverq, 65 ˜ , is the finite-dimensional posterior n (jfV i j g) for fixedn as in (4.11) and we proceed with the BochnerspacesV =L 2 π n (·|{V i j }) (Q;V)andH =L 2 π n (·|{V i j }) (Q;H q )toachieve(4.34). For the state variablesx j (;q) we have that 2 [0;1] and q 2 Q = [a 1 ;b 1 ] [a 2 ;b 2 ] for 0 < a i < b i , i = 1;2. Further, for the inputsu(t;q) we have that t 2 [0;T] and q 2 Q. Let n be as in (3.16) and m a positive integer, and we discretize [0;1] and [0;T] using the sets of linearB-splines,f n j g n j=0 andf m j g m j=0 ,respectively,ontheuniformmeshes,f j n g n j=0 andf iT m g m j=0 , respectively. Further,forpositiveintegersm 1 andm 2 ,wediscretizeQwiththe0 th -orderB-splines f m i i,j g m i j=1 ,i = 1;2definedwithrespecttotheuniformgridsfa i (b i −a i )j m i g m i j=0 ,on[a i ;b i ],i = 1;2. Then for multi-indicesN = (n;m 1 ;m 2 ) andM = (m;m 1 ;m 2 ) we define the approximating subspacesV N andS M asfollowsusingtensorproducts. Welet V N =spanf( n j (0) m 1 1,j 1 m 2 2,j 2 ; n j m 1 1,j 1 m 2 2,j 2 )g n,m 1 ,m 2 j=0,j 1 =1,j 2 =1 = spanf ˆ N i g N i=1 ; and (4.37) S M =spanf( m j m 1 1,j 1 m 2 2,j 2 g m,m 1 ,m 2 j=0,j 1 =1,j 2 =1 = spanf ˆ M i g M i=1 ; whereN = (n+1)m 1 m 2 andM = (m+1)m 1 m 2 . Standardapproximationtheoreticarguments (see,forexample,[70])canbeusedtoarguethatthesubspacesdefinedin(4.37)satisfytherequired approximationassumptionsonV N andS M . Thenx N 2V N andu M 2S M ,canbewrittenas x N (;q) = n,m 1 ,m 2 X i=0,i 1 =1,i 2 =1 x N i,i 1 ,i 2 n i () m 1 1,i 1 (q 1 ) m 2 2,i 2 (q 2 ) and (4.38) u M (t;q) = m,m 1 ,m 2 X i=0,i 1 =1,i 2 =1 u M i,i 1 ,i 2 m i (t) m 1 1,i 1 (q 1 ) m 2 2,i 2 (q 2 ); respectively. Then with the bases forV N andS M as chosen above, it is an elementary exercise to determine the matrix representations for the operatorsA N ,B N ,C N , ˆ A N , ˆ B N , and ˆ C N . It then followsthat(4.34)takestheform M N X N k+1 =K n X N k +B L U N k ; k = 1;2;:::;K; y L k =C N X N k ; k = 0;1;:::;K; (4.39) 66 whereX N k 2R N arethecoefficientsofthebasiselementsf ˆ N i g,U N k 2R M arethecoefficientsof the basis elementsf ˆ M i g as in (4.38),M N 2R N×N is a matrix with entries [M N ] i,j =h ˆ i ; ˆ j i H , K N 2 R N×N is a matrix with entries [K N ] i,j = hA N ˆ N i ; ˆ N j i H ,B L 2 R N×M is a matrix with entrieshB N ˆ M i ; ˆ N j i H , andC N 2R 1×M is given by [1;0;:::;0]. From here the matrix represen- tation ofh L k (withL = (M;N) in place ofN due to the joint dependence on the multi-indicesM andN)canbefoundusing(4.39). Wenotethattheoptimizationproblem(4.36)isaconstrainedproblem,inthatU N k of(4.39)are tobenon-negative. Withaproperplacementoffh L k gintotheblockmatrixH L ,theapproximating deconvolutionproblem(4.36)isnowgivenby u ∗ L = argmin U M J L (u) = argmin U M 2 6 4 H L r ∗ 1 Q M 1 +r ∗ 2 Q M 2 1 2 3 7 5 U M Y M 2 R K+KM (4.40) whereU M is theKM dimensional column vector of the coefficients ofu 2U M , andY M is the K +KM column vector of measured output valuesfˆ y k g followed byKM zeros. Further,Q M i fori = 1;2 are matrices with entries given by theU inner products of the basis elements for the subspacesS M as determined by the regularization termkuk 2 S(0,T) as given in (4.41) below. Note thattheregularizationtermkuk 2 S(0,T) isderivedfromaweightedinnerproductonS[0;T]andthus correspondstoasquarednormonS(0;T),kk 2 S(0,T) , kuk 2 S(0,T) =r 1 Z T 0 ku(t)k 2 U dt+r 2 Z T 0 k ˙ u(t)k 2 U dt: (4.41) The values of the weights used,r ∗ i > 0 fori = 1;2 are chosen optimally. Indeed, in order to find (r ∗ 1 ;r ∗ 2 )BrAC-TACinput-outputtrainingdatapairs,f(u i j ;V i j )g P,K i=0,j=0 areusedtooptimize(r 1 ;r 2 ) accordingtothefollowingscheme (r ∗ 1 ;r ∗ 2 ) = argmin (r 1 ,r 2 )∈R + ×R + R X i=1 K X j=1 ¯ u i,∗ L;j−1 u i j−1 2 + ˜ y i,∗ L;j V i j 2 (4.42) 67 Figure 4.1: (Left) WristTAS TM 7 Continuous Alcohol Monitoring device, (Right) SCRAM Sys- temsContinuousAlcoholMonitoringdevice. where ¯ u i,∗ L;j = E ˜ π(θ ∗ ) [˜ u i,∗ L;j ], and ¯ u i,∗ L;j ;˜ y i,∗ L;j are the predicted input and output values usingr 1 ;r 2 in (4.40). 4.4 NumericalResults All of the data used in the studies detailed below were collected in IRB-approved human subject experiments designed and run by researchers in the laboratory of Dr. Susan E. Luczak of the Department of Psychology at the University of Southern California as part of a National Institute on Alcohol Abuse and Alcoholism (NIAAA) grant, number R01 AA026368 (see, [67]). These experiments were carried out in controlled environments wherein 40 participants completed one to four drinking episodes, with data from a total of 146 drinking episodes recorded. All data were collected using an Alco-sensor IV from Intoximeters, Inc, St. Louis, MO for BrAC and two SCRAM System Continuous Alcohol Monitoring devices manufactured by Alcohol Monitoring Systems(AMS)inLittleton, Colorado(seeFigure4.1)simultaneouslyplacedontheparticipants’ left and right arms for TAC. Participants started their readings with a TAC and BrAC of 0.000, consumedalcohol(equivalentacrossallsessions)inoneofthreedifferentdrinkingpatterns(single: over 15 minutes; dual: over two 15-min periods spaced 30-minutes apart; or steady: over 60 minutes), and then ended their session when their TAC and BrAC had returned to 0.000. We note thattheplacementofthetwosensorschallengestheindependenceassumptionfromChapter4,but for experimental purposes we will include all of the data with this caveat in mind. In addition, 68 we did not focus on any specific drinking pattern as including all possible patterns is in line with real-world,varyingdrinkingpatternsandmayimprovethegeneralizabilityofourmodel. In the calculations of Sections 4.4.1-4.4.2, as in (3.11), time is discretized by a constant sam- pling time of 5 minutes and is subject to our zero-order hold assumption. While this challenges theimplicationsofourzeroorderholdassumption,namelythat = 5impliesthatsubjects’BAC isconstantfor5minutes,thisrestrictionisneededascomputationalcomplexitybecomesunstable as decreases. In order to achieve this sampling time, we first linearly interpolate all of the data (both BrAC and TAC), and then re-sample at our desired rate of = 5. For Section 4.4.3, a will be be discussed. Further, in all sections we assume a truncated multivariate normal (tMVN) prior 0 (as in (4.11)) onq with mean and covariance matrix Σ which varies from example to example. For ease of computations, withq 3 andq 4 as in (3.3) we assumed thatq 3 = q 4 = 1. All computational work was done in Python 3.7.2 and includes ported Matlab code from the work of [24, 63, 76, 77], in particular with respect to the creation of the finite-dimensional, discrete-time kernelasin(4.2). Portedcodewasverifiedagainsttheoriginalcodethroughtheuseofunittests. 4.4.1 ConvergenceinDistribution WeusedsurfaceplotsaswellasMetropolisHastings(MH)MarkovChainMonteCarlo(MCMC) methodstovalidateourconvergenceindistributionresults. Throughouttheresultsdescribedhere we have that from (4.4) for all sample times the i.i.d. noise" is distributed asN(0;0:005 2 ), and prior 0 as in (4.11) is distributed as the optimal distribution found in Section 6 of [77]. Specifi- cally, the prior is a tMVN random variable with mean, = ( 0.6318 1.0295 ) and covariance matrix, Σ = ( 0.0259 0.0077 0.0077 0.1232 ) with the feasible parameter set,Q, taken to beQ = [0:01;2:2877] [0:01;2:1410] forq = [q 1 ;q 2 ] T . For computational reasons, we limit ourselves to measurements from a random subgroup ofR = 3 subject drinking episode measurements. Figure 4.2 contains the resulting sur- face plots forn values of 1;3, and 25. Further, Table 4.1 contains the means and credible regions forn values of 1;2;3; and 25 as determined by respective 1000 sample (1100 draws with a 100 drawburn-inperiod)MHMCMCsamplingruns. 69 (a)n = 1,R = 3,σ noise = 0.005 (b)n = 3,R = 3,σ noise = 0.005 (c)n = 25,R = 3,σ noise = 0.005 Figure 4.2: Posterior distribution surf plots for varying finite dimensional approximations of the kernelfrom(4.2). nDimension Mean(q 1 ;q 2 ) CredibleCircleRadius 1 (0.7185,0.8512) 0.1173 2 (0.6829,0.8651) 0.1097 3 (0.6776,0.8686) 0.1029 25 (0.6719,0.8716) 0.1289 Table 4.1: 90% credible circles for Metropolis-Hastings MCMC sampled posteriors with noise distributionN(0;0:005 2 ). 4.4.2 Consistency We again used surface plots as well as MH MCMC sampling methods to verify our consistency results. For these studies we have assumed that the noise " is now distributed as N(0;0:025 2 ) while our prior 0 from (4.11) is still a tMVN random variable with mean and covariance that differbetweenexamples. To test Theorem 4.6, we generated 276 TAC values using subject-measured BrAC values via (4.4) with a predetermined q 0 value of [1;1], n = 24, and noise variance of 0:025 2 . We utilized theoptimaldistributionfoundinSection6of[77]astheprior 0 from(4.11). Specifically, 0 isa tMVNwith = ( 0.6318 1.0295 )andcovariancematrix,Σ = ( 0.0259 0.0077 0.0077 0.1232 )withbounds[0:01;2:2877]and 70 [0:01;2:1410]forq 1 andq 2 ,respectively. Table4.2displaysthecalculatedmeansand90%credible circle radii for the posterior distribution (4.13) for increasing amounts of idealized (BrAC, TAC) data pairs (R) all generated using the “true”q 0 value previously stated. To calculate these values, MHMCMCsamplesweredrawnwithasampleofsize1400(1500datapointswitha100sample burn-inphase). R Mean(q 1 ;q 2 ) CredibleCircleRadius 1 (0.683,1.023) 0.2854 11 (0.734,1.031) 0.1963 26 (0.776,1.025) 0.1677 101 (0.877,1.011) 0.1590 276 (0.942,1.003) 0.0787 Table4.2: 90%crediblecirclesforMCMCsampledposteriorsfromidealizedTACdatawithnoise distributedN(0;0:025 2 )andpriordistributionequivalenttotheoneinSection6of[76],namelya tMVNwith = ( 0.6318 1.0295 ),Σ = ( 0.0259 0.0077 0.0077 0.1232 ),andq bounds[0:01;2:2877]and[0:01;2:1410]. We now investigate the results of Section 4.2 with respect to the field-measured (BrAC, TAC) data pairs. Note that we no longer are able to know the true value of the parameters,q 0 . Surface plotsforincreasingamountsofsubjectdrinkingepisodemeasurements,R = 1;26;76,and101are contained within Figure 4.3. In these computations for the prior 0 from (4.11), we again utilized the optimal distribution found in Section 6 of [77]. That is, 0 is a tMVN with = ( 0.6318 1.0295 ) and covariance matrix, Σ = ( 0.0259 0.0077 0.0077 0.1232 ) with bounds [0:01;2:2877] and [0:01;2:1410] forq 1 andq 2 , respectively. Table 4.3 displays the calculated means and 90% credible circle radii for increasing amounts of subjects, and thus data (corresponding toR as in Section 4.2) included in determination of the prior. Inordertocalculatethesevalues,MHMCMCsampleswereusedwhereforeachRasample of size 1400 (1500 data points with a 100 sample burn-in phase) was determined using the same tMVNpriorasthesurfaceplots. 71 (a)R = 1,n = 5 (b)R = 11,n = 5 (c)R = 26,n = 5 (d)R = 101,n = 5 Figure 4.3: Posterior distribution surface plots for varying amounts of collected data, m. All images use noisedistributedN(0,0.025 2 )andpriordistributionequivalenttothatofSection6of[76],namelyatMVN withµ = ( 0.6318 1.0295 ),Σ = ( 0.0259 0.0077 0.0077 0.1232 ),andq bounds[0.01,2.2877]and[0.01,2.1410]. R Mean(q 1 ;q 2 ) CredibleCircleRadius 1 (0.913,1.251) 0.2824 11 (1.426,1.629) 0.1497 26 (1.900,1.551) 0.1560 101 (2.183,1.231) 0.1254 Table 4.3: 90% credible circles for Metropolis-Hastings MCMC sampled posteriors with noise distributedN(0;0:025 2 )andpriordistributionequivalenttotheoneinSection6of[76],namelya tMVNwith = ( 0.6318 1.0295 ),Σ = ( 0.0259 0.0077 0.0077 0.1232 ),andq bounds[0:01;2:2877]and[0:01;2:1410]. 4.4.3 Deconvolution As in Section 4.3, we rely on the treatment in [77] for deconvolving BrAC from TAC using a distributionoverQ,q. Thechosendistributionwastheposterior(4.13)withn = 3. Todetermine theposterior,weplacedapriorofatMVNrandomvariablewithbounds[0:01;10][0:01;10],and parameters = ( 5 5 ) and Σ = ( 0.7 0.1 0.1 0.55 ). The noise used was distributed asN(0;0:025 2 ). Further, in determining (4.37) and (4.38), we set our space discretization to ben = 3, time discretization asm = 1300,anddiscretizedQwithm 1 =m 2 = 20. Thetestdatasetusedconsistedoffivedrinkingepisodesfromfourdifferentparticipantschosen heuristically. BylinearlyinterpolatingtheBrACandTACdataforeachsubjectinthetestandtrain 72 datasets, we are able to re-sample our data to receive = 45 seconds, and the time discretization m = 1300 previously mentioned. The associated participant IDs, TAC device placement (left vs. right arm), type of drinking pattern used (single, dual, or steady), and number of subjects used in posterior distribution determination (R) are labeled in Figures 4.4 and 4.5. Participants in the test dataset were not included in any size of the training dataset used for the posterior whenever possible. ThemainexceptionsbeingFigures4.5cand4.5f,whereinweallowedalldatathatwasn’t thecurrenttestdatapointtobeincludedinthetrainingsets. Asin(4.42),weutilizedallavailable non-test subject drinking episode measurements (R = 136) to determine population parameters (r ∗ 1 ;r ∗ 2 )tobe(4:7733;1:7020). Figure 4.4, shows varying deconvolution attempts for three participants, whereas Figure 4.5 shows deconvolution attempts for the same participant reading (BT333), with varying amounts of subject data within the posterior (4.13), R = 25;75;145. In both Figures 4.4 and 4.5, gray bands represent error regions, at both a 70% and 90% level, that are determined by sampling respective parameter posterior distributions and utilizing these samples with (4.40) to determine estimated BrAC values. Thus, these error regions contain the 70% and 90% credible regions for therespectiveunderlyingparameterposteriordistributions. 73 (a)BT322-Rightarm-Dual- 70%errorband (b)BT335-Rightarm-Single- 70%errorband (c)BT319-Rightarm-Dual- 70%errorband (d)BT322-Rightarm-Dual- 90%errorband (e)BT335-Rightarm-Single- 90%errorband (f)BT319-Rightarm-Dual- 90%errorband Figure 4.4: BrAC deconvolution given TAC and predictedq values for varying participants’ right arm data with gray error regions. Across sub-figures, all training data remained constant with R = 25. Prior used was tMVN with bounds [0:01;10][0:01;10], = ( 5 5 ), and Σ = ( 0.7 0.1 0.1 0.55 ). Associated data is contained in Tables 4.4 and 4.5, for 70% and 90% parameter credible regions, respectively. Arm R q mean 70.0%Credible q 1 range q 2 range CircleRadius Right 25 [4.512,1.346] 1.104 [3.408,5.616] [0.242,2.450] Table 4.4: Data associated with posterior determination and deconvolution used in Figure 4.4 for parametervaluesina70%crediblecircle. Arm R q mean 90.0%Credible q 1 range q 2 range CircleRadius Right 25 [4.512,1.346] 1.431 [3.082,5.943] [0.01,2.777] Table 4.5: Data associated with posterior determination and deconvolution used in Figure 4.4 for parametervaluesina90%crediblecircle. 74 (a)BT333-Leftarm-Single- R = 25-70%errorband (b)BT333-Leftarm-Single- R = 75-70%errorband (c)BT333-Leftarm-Single- R = 145-70%errorband (d)BT333-Leftarm-Single- R = 25-90%errorband (e)BT333-Leftarm-Single- R = 75-90%errorband (f)BT333-Leftarm-Single- R = 145-90%errorband Figure 4.5: BrAC deconvolution given TAC and predictedq values derived using varying amount of training data, R, with gray error regions. All sub-figures use the same test TAC data from a single left arm session from BT333. Prior used was tMVN with bounds [0:01;10] [0:01;10], = ( 5 5 ), and Σ = ( 0.7 0.1 0.1 0.55 ). Associated data are contained in Tables 4.6 and 4.7, for 70% and 90%parametercredibleregions,respectively. Arm R q mean 70.0%Credible q 1 range q 2 range CircleRadius Left 25 [4.512,1.346] 1.104 [3.408,5.616] [0.242,2.450] Left 75 [3.450,1.215] 1.149 [2.301,4.5991] [0.066,2.364] Left 145 [2.824,1.000] 0.832 [1.992,3.656] [0.168,1.832] Table 4.6: Data associated with posterior determination and deconvolution used in Figure 4.5 for parametervaluesina70%crediblecircle. 75 Arm R q mean 90.0%Credible q 1 range q 2 range CircleRadius Left 25 [4.512,1.346] 1.431 [3.082,5.943] [0.01,2.777] Left 75 [3.450,1.215] 1.490 [2.006,4.986] [0.01,2.705] Left 145 [2.824,1.000] 1.066 [1.758,3.890] [0.01,2.066] Table 4.7: Data associated with posterior determination and deconvolution used in Figure 4.5 for parametervaluesina90%crediblecircle. 76 Chapter5 BayesianEstimationofDynamicalSystemInput In this chapter, we develop a Bayesian framework to estimate unknown BrAC values u in the system(3.1)givenavectorofknownTACvaluesatspecifiedmeasurementtimes,t = (t 0 ;t 1 ;:::; t K ) ⊤ . More specifically, this is accomplished by utilizing the infinite-dimensional state-space problem(3.13)andthesubsequentfinite-dimensionalformulation(3.18). Wemaketheassumption thatthe subject-dependent physiologicalparametersq 1 andq 2 , as well as the sensor parametersq 3 andq 4 areassumedtobeeitherknownortohavebeendeterminedfromacohortofsubjectsinthe formofapopulationmodel(see,forexample,[63,76,77])andbenchmeasurement,respectively. Inaddition,weassumethatwehaveBrAC-TACtrainingdatapairs, fu i k g K k=0 ;fy i k g K k=0 R i=1 ,from R subjectswhereforeachparticipanti,wehaveK +1BrAC-TACobservations. Wenotethatall ofthemotivationandworkabovewasderivedforanarbitrarysamplingtime,whichisassumed fixed and constant across subjects, i = 1;:::;R. Depending on the training data chosen for the work that follows, subsequent formulations can either be used to determine a population-level predicted BrAC function or a sub-population level predicted BrAC function. The former may be implementedifothersystemparameters,suchasq2Qfrom(3.1),aredeterminedtohaveanon- presentornegligibleeffectontheBrACpredictions. Whereasthelatterisabletobeimplemented in the opposite case. Typically such an implementation is paired with a data filtering or selection processthatensuresthepropersub-populationisidentified. Inbothcases,themethodsoutlinedin the following chapter can still be applied. Lastly, note that with the assumption of a fixedq, weh forh(q)andh n forh n (q),asin(3.15)and(3.19),respectively. 77 5.1 Infinite-DimensionalFormulation Buildingfrom(3.13),givenafixedtimestep > 0andassociatedtimevectort = (t 0 ;t 1 ;:::;t K ) ⊤ we have BrAC and TAC vectorsu = (u 0 ;u 1 ;:::;u K ) ⊤ andy = (y 0 ;y 1 ;:::;y K ) ⊤ , respectively. Thus, we may rewrite (3.15) as y = Hu where H 2 R K+1×K+1 with [H] i,j = h i−j 1 j≤i = ˆ C ˆ A i−j ˆ B 1 j≤i fori;j = 1;2;:::;K + 1. To account for measurement error, we let our statistical modelbegivenby V =Hu+ε (5.1) whereV is a random vector inR K+1 of TAC values at times t,u is a random vector inR K+1 of BrAC values at times t, and ε is a noise vector that is assumed to have a multivariate normal distribution,thatis,εMVN( ~ 0; 2 I K+1 )forsome> 0. Consequently,givenobservedBrAC- TAC training data vector pairs, f(u i ;V i )g R i=1 we consider V i = Hu i +ε i with independence acrossbothsubject,i = 1;:::;R,andmeasurementtimes,j = 0;:::;K. We place a multivariatenormal (MVN) prior on the BrAC vectoru with mean and covari- anceΣ. Todeterminetheoptimal(insquared-error)priorparameters,wefollowtheworkofEvans (see,[32]),whereinweplaceanaivejointprioroverandΣwithdensityoftheform P(;Σ −1 )/jΣj − 1 2 exp 1 2 tr Σ −1 2 I K+1 +(ζ)(ζ) ⊤ (5.2) for some> 0 andζ 2R K+1 . This then implies,jΣMVN(ζ;Σ) and Σ −1 W R ( −2 I K+1 ; K + 1), for Wishart distribution W. As such, Evans (see, [32]) further showed that for BrAC trainingvectorsfu i g R i=1 ,theoptimal(insquared-error)parametersforandΣaregivenby f µ = R¯ µ+ζ R+1 2R K+1 ; and (5.3) f Σ = 1 R1 S + 2 I K+1 + R R+1 (¯ µζ)(¯ µζ) ⊤ 2R K+1×K+1 ; (5.4) 78 respectively,for ¯ µ =R −1 P R i=1 u i andS = P R i=1 (u i ¯ µ)(u i ¯ µ) ⊤ . WiththeprioroverunowknowntobeMVN(f µ ;f Σ ),by(5.1)wehavethatV =Hu+ε MVN(Hf µ ;Hf Σ H ⊤ + 2 I K+1 ). Todetermine> 0weutilizethetrainingTACvectorsfV i g R i=1 andsolvethefollowingminimizationproblem = min s>0 Hf Σ H ⊤ +s 2 I K+1 S V 2 F ; (5.5) wherejjjj F is the Frobenius norm, andS V = (R 1) −1 P R i=1 (V i ¯ V )(V i ¯ V ) ⊤ with ¯ V = R −1 P R i=1 V i . We note that another common method of selecting makes use of (5.7) below. Namely, is chosen so as to minimize the mean squared error between the predicted BrAC given TACandtrueBrACacrosstheentiretyofthetrainingset. GivenanewvectorofTACvaluesattimest,V ∗ ,wemaynowutilizethetrainedMVN prior overutodetermineanestimatefortheassociatedBrACvector,u ∗ . Noticethat 0 B @ u V 1 C A MVN 0 B @ 0 B @ f µ Hf µ 1 C A ; 0 B @ f Σ f Σ H ⊤ Hf Σ Hf Σ H ⊤ + 2 I K+1 1 C A 1 C A d =MVN(µ ∗ ;Σ ∗ ); (5.6) andsobystandardpropertiesofmultivariatenormaldistributionswefind ujV =V ∗ MVN f ∗ µ ;f ∗ Σ (5.7) where we definef ∗ µ = f µ +f Σ H ⊤ (Hf Σ H ⊤ + 2 I K+1 ) −1 (V ∗ Hf µ ) 2R K+1 andf ∗ Σ = f Σ f Σ H ⊤ (Hf Σ H ⊤ + 2 I K+1 ) −1 Hf Σ 2R K+1×K+1 . 5.2 Finite-DimensionalFormulation In order to compute the entries in the filter matrix, H, from (5.1), we must deal with fact that the semigroup,fe At : t 0g, is defined on an infinite dimensional Hilbert space. Consequently, 79 we cannot directly carry out the requisite calculations in (5.7). Thus, we make use of the finite- dimensionalapproximationwhichledto(3.18),inparticular,thesequenceofapproximatingfinite dimensional operators H n as in (3.19). In practice, the requisite computations are carried using H n for some fixed value for n 2 Z + . Given the same time step > 0, associated time vector, t = (t 0 ;t 1 ;:::;t K ) ⊤ , and BrAC-TAC data vectors,f(u i ;V i )g R i=1 as in Section 5.1, we now have theapproximatingstatisticalmodel V n =H n u+ε (5.8) whereV n is a random vector inR K+1 of TAC values at timest,u is a random vector inR K+1 of BrAC values at times t, ε MVN( ~ 0; 2 I K+1 ) for some > 0, and H n 2 R K+1×K+1 has entries [H n ] i,j = h n i−j 1 j≤i = ˆ C n ( ˆ A n ) i−j ˆ B n 1 j≤i for i;j = 1;2;:::;K + 1. Note that the data vectors for this model are equivalent to those of (5.1) and thus fori = 1;:::;R we useV n,i and V i interchangeablywhenreferringtothedata. As was the case with (5.1)-(5.7) we are able to estimate an input BrAC given measured TAC, V n,∗ ,usingthedistribution ujV n =V n,∗ MVN f n,∗ µ ;f n,∗ Σ (5.9) whereu has a MVN(f µ ;f Σ ) prior,V n = H n u +ε, f n,∗ µ = f µ +f Σ (H n ) ⊤ (H n f Σ (H n ) ⊤ + 2 I K+1 ) −1 (V ∗ H n f µ )2R K+1 andf n,∗ Σ =f Σ f Σ (H n ) ⊤ (H n f Σ (H n ) ⊤ + 2 I K+1 ) −1 H n f Σ 2 R K+1×K+1 . Forn> 0,considertherandomvectorsX ⊤ := (u ⊤ ;V ⊤ ) ⊤ asin(5.6),and (X n ) ⊤ := (u ⊤ ;(V n ) ⊤ ) ⊤ (5.10) MVN 0 B @ 0 B @ f µ H n f µ 1 C A ; 0 B @ f Σ f Σ (H n ) ⊤ H n f Σ H n f Σ (H n ) ⊤ + 2 I K+1 1 C A 1 C A d =MVN(µ n,∗ ;Σ n,∗ ): 80 WenowestablishthatX n L 2 !X asn!1;thatisthatX n convergesinmeansquaretoX. We firstprovealemmathatwillbeofuseinsubsequentproofsandthenfollowwiththetheorem. Lemma 5.1. For K 2 Z + associated with time step > 0 fixed, H as in (5.1), and H n as in (5.8), we have thatH n ! H in Frobenius norm asn ! 1 forH andH n as in (5.1) and (5.8), respectively. Proof. Forh i andh n i asin(3.15)and(3.19)withtimestep,wefind jjH n Hjj 2 F = K+1 X i=1 i X j=1 jh n i−j h i−j j! 0 asn!1byTheorem3.3. Hence,H n !H inFrobeniusnormasn!1. Theorem 5.2. Let R 2 Z + , i = 1;2;:::;R, and K 2 Z + with associated time step > 0 be fixed. ForX ⊤ = (u ⊤ ;V ⊤ ) ⊤ asin(5.6),and(X n ) ⊤ := (u ⊤ ;(V n ) ⊤ ) ⊤ asin(5.10),wehavethat X n L 2 !X asn!1. Proof. Notice,X n X MVN(˜ µ; ˜ Σ)where ˜ µ =µ n,∗ µ ∗ ; and ˜ Σ = Σ n,∗ +Σ ∗ Σ XX nΣ X n X where Σ XX n = Cov(X;X n ), and Σ X n X = Cov(X n ;X). The covariance matrices are readily foundusingthedefinitionsofV andV n from(5.1)and(5.8),respectively. Wefind, E jjX n Xjj 2 2 = 2(K+1) X i=1 E (X n i X i ) 2 = 2(K+1) X i=1 Var(X n i X i )+E(X n i X i ) 2 = tr( ˜ Σ)+jj˜ jj 2 2 (5.11) 81 wheretrdenotesthematrixtraceoperator. Wedealwitheachtermin(5.11)separatelyandinturn. First,fortr( ˜ Σ)notethat tr( ˜ Σ) = tr(Σ n,∗ )+tr(Σ ∗ )tr(Σ XX n)tr(Σ X n X ) = tr Hf Σ (H n ) ⊤ tr Hf Σ H ⊤ + tr H n f Σ H ⊤ tr H n f Σ (H n ) ⊤ Hf Σ (H n ) ⊤ Hf Σ H ⊤ 2 F + H n f Σ H ⊤ H n f Σ (H n ) ⊤ 2 F jjHjj 2 F jjf Σ jj 2 F jjH n Hjj 2 F +jjH n jj 2 F jjf Σ jj 2 F jjH n Hjj 2 F ! 0 asn!1byLemma5.1. Nowfor ˜ µ,wehave jj˜ µjj 2 2 =jjµ n,∗ µ ∗ jj 2 2 =jjH n f µ Hf µ jj 2 2 jjf µ jj 2 2 jjH n Hjj 2 F ! 0 asn!1,againbyLemma5.1. HencewefindthatE[jjX n Xjj 2 2 ]! 0asn!1andthusthetheoremisproven. The functiong(x;y) = y for (x;y) 2 R s 1 R s 2 withs 1 ;s 2 2 Z + is clearly continuous and triviallyLipschitzwithconstant1. Thus,wehave E jjg(x n ;y n )g(x;y)jj 2 E jj(x n ;y n ;)(x;y)jj 2 for(x n ;y n );(x;y)2R s 1 R s 2 . Hencewehavetheimmediatecorollary. Corollary5.3. Under the same assumptions of Theorem 5.2, we have thatV n L 2 !V asn!1 forV andV n asin(5.1)and(5.8),respectively. Now, recall the random variablesujV = V ∗ MVN f ∗ µ ;f ∗ Σ andujV n = V n,∗ MVN f n,∗ µ ;f n,∗ Σ as in (5.7) and (5.9), respectively. We establish thatujV n = V n,∗ Dist ! ujV =V ∗ ;thatisthatujV n =V n,∗ convergesindistributiontoujV =V ∗ . 82 Theorem 5.4. LetK 2Z + with time step > 0. Letu ∗ ,V ∗ , andV n,∗ be known realizations of therandomvariablesu,V,andV n ,respectively. ForujV =V ∗ asin(5.7)andujV n =V n,∗ asin(5.9),wehavethatujV n =V n,∗ Dist !ujV =V ∗ asn!1. Proof. Notethatbyconstructionofthestatisticalassumptionsunderlying(5.7)and(5.9),theTAC data vectors are equivalent. That is, V ∗ = V n,∗ . To avoid cumbersome notation when handling densities, we omit subscripts when the associated random variable is clear from the context. For example,f u,V (u;V ) =f(u;V ). Then,foru ∗ 2R K+1 wehave f u|V n (u ∗ jV n,∗ )f u|V (u ∗ jV ∗ ) = f u,V n (u ∗ ;V n,∗ ) f V n (V n,∗ ) f u,V (u ∗ ;V ∗ ) f V (V ∗ ) jf(u ∗ ;V n,∗ )f(u ∗ ;V ∗ )jf(V ∗ ) f(V n,∗ )f(V ∗ ) + jf(V n,∗ )f(V ∗ )jf(u ∗ ;V ∗ ) f(V n,∗ )f(V ∗ ) := I 1 f(V ∗ )+I 2 f(u ∗ ;V ∗ ) f(V n,∗ )f(V ∗ ) : RecallingtheequivalenceofthedatavectorsV ∗ =V n,∗ ,wefindthat I 1 =jf u,V n (u ∗ ;V n,∗ )f u,V (u ∗ ;V ∗ )j! 0 asn!1 byTheorem5.2,andthat I 2 =jf V n (V n,∗ )f V (V ∗ )j! 0 asn!1 byCorollary5.3. Further,wehavethat f u,V n (u ∗ ;V n,∗ )!f u,V (u ∗ ;V ∗ ) asn!1 83 byTheorem5.2,andthat f V n (V n,∗ )!f V (V ∗ ) asn!1 byCorollary5.3. Hence,itfollowsthat f u|V n (u ∗ jV n,∗ )f u|V (u ∗ jV ∗ ) ! 0 asn!1forallu ∗ 2R K+1 . Thus,ujV n =V n,∗ Dist !ujV =V ∗ asn!1andthetheorem isproven. 5.3 NumericalResults All of the data used in the studies presented below was collected in IRB-approved human subject experiments designed and carried out in the laboratory of Dr. Susan E. Luczak of the Department of Psychology at the University of Southern California as part of a National Institute on Alcohol Abuse and Alcoholism (NIAAA) funded investigation, grant number R01 AA026368 (see, [67]). Allexperimentswereconductedincontrolledenvironmentsundercloseobservationbylaboratory personnel. Figure4.1showstwotransdermalalcoholmonitoringdevices. In these experiments, 40 participants participated in one to four drinking episodes wherein each episode participants started with a BrAC of 0.000, consumed alcohol (equivalent across all sessions)inoneofthreedifferentdrinkingpatterns(single: over15minutes;dual: overtwo15-min periods spaced 30-minutes apart; or steady: over 60 minutes), and then ended their session when BrAC had returned to 0.000. To measure BrAC/BAC, all data was collected using a breathalyzer, and for TAC all data was collected using two SCRAM System Continuous Alcohol Monitoring devices manufactured by Alcohol Monitoring Systems (AMS) in Littleton, Colorado (shown on therightinFigure4.1simultaneouslyplacedontheparticipants’leftandrightarms. Wenotethat thesimultaneousplacementoftheSCRAMsensorschallengestheindependenceassumptionfrom 84 Chapter 5, but for experimental purposes, we will allow all data to be included with this caveat in mind. Whenever possible, in cross-validating our results, we ensured that no two simultaneous arm readings were split between training and testing sets. In what follows, all codes were written inPython3.7.2andallcomputationswerecarriedoutonastandardlaptopcomputer. 5.3.1 ConvergenceinDistributionofthePredictivePosterior First, we corroborate the theoretical work of Section 5.2. Specifically, we verify the results of Theorem 5.4 as it makes direct use of Theorem 5.2 and is the main focus of further examples below. To focus on the results of increasing the approximating dimension n, we select one test drinking episode at random from the 146 available and limit our training data to two randomly selected drinking episodes from the remaining data (excluding other measurements of the same subject). Weuseanexaggeratedsamplingtimeof = 0:9471hourswithK = 25,andfrom(5.2) we set = 0:005 and ζ = (0:2;0:2;:::;0:2) ⊤ 2 R K+1 . We then make use of (5.3), (5.4), and (5.5)todeterminef µ ;f Σ ;and,respectively. From[76],wesetq = (0:6318;1:0295;0:6318;1) ⊤ andthenlettheapproximatingdimensionforthefilter(3.20)vary. Table5.1displaysthepredicted means and standard deviations across an array of times for varyingn values. All predicted mean vectorsandcovariancematriceswerefoundusing(5.9). 5.3.2 PredictiveDeconvolutionofBrACfromTAC Next,weapplyandseektofurtherverifythetheorypresentedinSection5toestimateBrACfrom TACgivenatrainingcorpusofBrAC-TACtrainingdatapairs. Todothisweselectonetestdrinking episode at random from the available 146, and use the remaining episodes (not including other measurementsofthesamesubject)astrainingepisodes. Thesamplingtimeusedinthisstudywas = 0:2786hourswithK = 85. Asin(5.2)wenaivelyset = 0:5andζ = (0:2;0:2;:::;0:2) ⊤ 2 R K+1 , and determine f µ ;f Σ ; and as in (5.3), (5.4), and (5.5), respectively. In addition, as per theabovenumericalwork,theapproximatingdimensionforthefilter(3.20)ischosenheuristically to be of size n = 7 and uses the fixed vector, q = (0:6318;1:0295;0:6318;1) ⊤ as determined 85 Time(hours) 0 0.9471 1.8942 4.7355 9.471 mean n 1 0.0066341 0.0536181 0.0405103 0.0066526 0.0065613 2 0.0066293 0.0536095 0.0405027 0.00665 0.0065563 3 0.0066287 0.053608 0.0405014 0.0066499 0.0065556 7 0.0066282 0.053607 0.0405006 0.0066498 0.0065552 25 0.0066282 0.0536068 0.0405004 0.0066498 0.0065551 175 0.0066282 0.0536068 0.0405004 0.0066498 0.0065551 standard deviation n 1 0.0200178 0.0209157 0.0202906 0.0199361 0.019926 2 0.0200312 0.0209306 0.0203053 0.0199496 0.019939 3 0.0200332 0.0209329 0.0203076 0.0199516 0.019941 7 0.0200344 0.0209343 0.020309 0.0199529 0.019942 25 0.0200346 0.0209346 0.0203093 0.0199531 0.019942 175 0.0200346 0.0209346 0.0203093 0.0199531 0.019942 Table 5.1: Calculated means and standard deviations for varying values of n for test drinking episodesubjectBT323LeftArmwithdrinkingpatterndual. in [76]. Figure 5.1 displays the resulting predicted BrAC function values, ground truth BrAC function values, calculated BrAC credible bands, and test TAC values for three separately chosen testepisodes. 5.3.3 NaiveStratificationofPopulationData Now,weseektoinvestigatetheapplicabilityofthetheorypresentedinChapter5whenappliedto atrainingcorpusofBrAC-TACtrainingdatapairsdeterminedbynaivecovariatefilters. Todothis weselectedarandomsampleof15drinkingepisodesfromthe146available(ensuringnooverlap between drinking episodes) to act as a test set. In addition, we generated a list of 47 covariate filters that can be used to filter the overall remaining training data drinking episodes. These 47 covariatefilterswerebuiltfromabaseofninesubject-levelcovariates,namely: age,height,weight, identifiedgender(femaleormale,codedas0or1,respectively),totalbodywater(TBW),alcohol dose administered in the drinking session (dose), standard (std) drink size in ounces, mean non- zerorecordedTACvalue(MNTAC),andmean(ambient)temperature. Then,foreachtestdrinking episodeweusedeachcovariateconfigurationtodetermineatrainingdatasetofsizeR = 11closest (in standard Euclidean distance) drinking sessions that were not produced by the subject being 86 (a)BT323-Leftarm-Dual (b)BT322-Leftarm-Dual (c)BT328-Rightarm-Single Figure 5.1: Plots displaying the given testing TAC graph, predicted mean BrACgraphs, and cred- ible bands for3 standard deviations for the predicted mean BrAC function using all available trainingdata. tested (so as to not bias the training data). All metrics for each covariate configuration were then averagedovertheresultsforeachtestdrinkingepisodefromthe15testepisodes. Weelectedtouse the following metrics comparing estimated to true BrAC: mean square error (MSE), peak BrAC valuedifference,peakBrACtimedifference,truevs. estimatedBrACdescendingrate(peakBrAC to zero), and true vs. estimated BrAC ascending rate (zero to peak BrAC). As a set of controls, for the case when no covariates are used, we employed a random selection of R = 11 training episodes(random)aswellasusingallavailabletrainingepisodes(all). The sampling time used in this investigation was = 0:2786 hours with K = 85. As in (5.2) we again set = 0:5 and ζ = (0:2;0:2;:::;0:2) ⊤ 2 R K+1 , and determine f µ ;f Σ ; and as in (5.3), (5.4), and (5.5), respectively. In addition, as per our previous numerical work, the approximating dimension for the filter (3.20) is set to be of size n = 7 and uses the fixed vector,q = (0:6318;1:0295;0:6318;1) ⊤ as determined in [76]. Tables 5.2-5.4, display the results means, standard deviations, and medians, respectively, for all utilized metrics across all covariate configurations. Wenotethatduetolayoutrestrictions,eachtablecaptioncontainsthemeanground truth values for BrAC descending and ascending rates. Further, covariate configurations with the bestscoresareinboldforeachmetricwithinthetables. 87 mean runtime mean MSE mean peakdiff. meanpeak timediff. meanest. BrACdesc. rate meanest. BrACasc rate age 0.857191 0.001966 0.009884 0.33825 -0.010203 0.061661 weight 0.839689 0.001878 0.010261 0.238765 -0.009721 0.046154 height 0.852853 0.001701 0.00996 0.318353 -0.010495 0.05668 gender 0.835248 0.001844 0.009878 0.33825 -0.010391 0.075703 totalbodywater (TBW) 0.861275 0.001922 0.0107 0.258662 -0.00981 0.044792 dose 0.877112 0.001926 0.010949 0.258662 -0.009856 0.045012 stddrink 0.880418 0.001897 0.010439 0.258662 -0.009921 0.045471 all 0.846865 0.001544 0.009313 0.218868 -0.010795 0.050403 averagenon-zero TAC(ANTAC) 0.878605 0.001529 0.009034 0.238765 -0.009503 0.051946 ageandANTAC 0.867979 0.001608 0.009192 0.238765 -0.010356 0.048008 weightand ANTAC 0.851895 0.001845 0.010783 0.218868 -0.009645 0.045575 heightandANTAC 0.877049 0.001656 0.009672 0.298456 -0.010428 0.053212 genderand ANTAC 0.875454 0.001552 0.010232 0.238765 -0.010611 0.048703 TBWandANTAC 0.858425 0.001975 0.011162 0.258662 -0.009639 0.046753 doseandANTAC 0.868232 0.00196 0.01112 0.258662 -0.009685 0.046968 stddrinkand ANTAC 0.852244 0.001941 0.010921 0.238765 -0.009649 0.047942 ageandweight 0.855451 0.001799 0.010273 0.238765 -0.009559 0.046215 ageandheight 0.894325 0.001719 0.009586 0.358147 -0.010732 0.058106 ageandgender 0.878065 0.001768 0.009861 0.218868 -0.010463 0.04797 ageandTBW 0.848861 0.001836 0.01102 0.258662 -0.009968 0.046506 ageanddose 0.92262 0.001867 0.011055 0.238765 -0.010057 0.045735 ageandstddrink 0.913597 0.001889 0.010079 0.258662 -0.010103 0.047359 weightandheight 0.875662 0.001759 0.010472 0.358147 -0.010082 0.046772 weightandgender 0.836092 0.001916 0.010259 0.238765 -0.009729 0.046226 weightandTBW 0.894153 0.001866 0.010365 0.33825 -0.010059 0.047902 weightanddose 0.866253 0.00189 0.010087 0.278559 -0.010188 0.045188 weightandstd drink 0.851028 0.001916 0.010259 0.238765 -0.009729 0.046226 heightandgender 0.879132 0.001645 0.009593 0.298456 -0.01053 0.05403 heightandTBW 0.874102 0.001932 0.0118 0.258662 -0.01049 0.052373 heightanddose 0.846581 0.001769 0.010552 0.258662 -0.009992 0.052136 heightandstd drink 0.866634 0.001701 0.00996 0.318353 -0.010495 0.05668 genderandTBW 0.869186 0.002004 0.011268 0.278559 -0.009827 0.044141 genderanddose 0.854978 0.001983 0.010974 0.238765 -0.009706 0.045232 genderandstd drink 0.846304 0.001995 0.010375 0.278559 -0.010127 0.044993 TBWanddose 0.877953 0.001926 0.010949 0.258662 -0.009856 0.045012 TBWandstddrink 0.856947 0.001922 0.0107 0.258662 -0.00981 0.044792 88 doseandstddrink 0.865442 0.001926 0.010949 0.258662 -0.009856 0.045012 random 0.851676 0.001493 0.010263 0.238765 -0.010088 0.04883 avgtemp 0.839095 0.001537 0.010261 0.208919 -0.00928 0.045334 avgtempandage 0.886209 0.001688 0.011793 0.232132 -0.009792 0.047008 avgtempand weight 0.857501 0.001882 0.011857 0.208919 -0.009558 0.04539 avgtempand height 0.838081 0.001505 0.010524 0.232132 -0.010895 0.048239 avgtempand gender 0.83633 0.001599 0.011446 0.232132 -0.010274 0.046487 avgtempandTBW 0.845611 0.001809 0.01185 0.255346 -0.010058 0.050671 avgtempanddose 0.841658 0.001894 0.012119 0.255346 -0.009748 0.049789 avgtempandstd drink 0.830685 0.00147 0.010621 0.208919 -0.009865 0.046074 avgtempand ANTAC 0.845538 0.001537 0.010261 0.208919 -0.00928 0.045334 Table 5.2: Means for all metrics of every utilized configuration. The mean ground truth BrAC descendingrateis0:015834andthemeangroundtruthBrACascendingrateis0:065255. 89 stddev runtime stddev MSE stddev peakdiff. stddev peaktime diff. stddev est. BrAC desc. rate stddev est. BrAC asc. rate age 0.018703 0.000922 0.006099 0.318974 0.002666 0.022453 weight 0.054534 0.000743 0.007671 0.206776 0.002372 0.006262 height 0.063155 0.000929 0.00731 0.295121 0.002648 0.015961 gender 0.063304 0.001028 0.008483 0.318974 0.002896 0.027517 totalbodywater (TBW) 0.035797 0.001118 0.009446 0.195963 0.00302 0.006194 dose 0.032613 0.001116 0.009327 0.195963 0.003017 0.006218 stddrink 0.021837 0.001143 0.009162 0.195963 0.003011 0.00599 all 0.047419 0.000873 0.006356 0.187708 0.002742 0.002005 averagenon-zero TAC(ANTAC) 0.047986 0.000863 0.0048 0.177965 0.002182 0.007452 ageandANTAC 0.061024 0.000817 0.006666 0.177965 0.002859 0.003882 weightand ANTAC 0.056256 0.000598 0.007042 0.187708 0.002273 0.004836 heightandANTAC 0.074649 0.001009 0.007306 0.287648 0.002946 0.014974 genderand ANTAC 0.072814 0.000808 0.006344 0.177965 0.002993 0.005868 TBWandANTAC 0.046729 0.001034 0.009041 0.165277 0.002888 0.007127 doseandANTAC 0.056008 0.001038 0.009126 0.165277 0.002897 0.007056 stddrinkand ANTAC 0.050906 0.001055 0.009102 0.177965 0.002898 0.007019 ageandweight 0.064407 0.000699 0.007996 0.206776 0.00223 0.005856 ageandheight 0.051011 0.000853 0.008045 0.286959 0.002997 0.017657 ageandgender 0.04582 0.000884 0.00673 0.187708 0.002488 0.00302 ageandTBW 0.049877 0.001087 0.009118 0.222456 0.002888 0.006327 ageanddose 0.074975 0.001085 0.00912 0.206776 0.003039 0.005075 ageandstddrink 0.041024 0.000952 0.008986 0.222456 0.002838 0.005924 weightandheight 0.061875 0.000778 0.00823 0.164075 0.00301 0.010217 weightandgender 0.044127 0.000754 0.007667 0.206776 0.002358 0.006316 weightandTBW 0.057241 0.000968 0.007553 0.155401 0.002885 0.010283 weightanddose 0.01902 0.000991 0.007021 0.210571 0.002825 0.008137 weightandstd drink 0.072382 0.000754 0.007667 0.206776 0.002358 0.006316 heightandgender 0.054941 0.000949 0.007598 0.267688 0.002676 0.012428 heightandTBW 0.07165 0.001 0.008799 0.246113 0.002908 0.018245 heightanddose 0.066819 0.000935 0.009198 0.246113 0.002878 0.018138 heightandstd drink 0.031825 0.000929 0.00731 0.295121 0.002648 0.015961 genderandTBW 0.032991 0.001072 0.008516 0.210571 0.002836 0.007647 genderanddose 0.046153 0.001071 0.008434 0.206776 0.002744 0.005635 genderandstd drink 0.039035 0.001076 0.00728 0.210571 0.002749 0.008379 TBWanddose 0.040091 0.001116 0.009327 0.195963 0.003017 0.006218 TBWandstddrink 0.036669 0.001118 0.009446 0.195963 0.00302 0.006194 90 doseandstddrink 0.039765 0.001116 0.009327 0.195963 0.003017 0.006218 random 0.047779 0.000835 0.006754 0.232038 0.002556 0.006017 avgtemp 0.06076 0.000717 0.008311 0.201033 0.002088 0.003687 avgtempandage 0.078335 0.000975 0.007492 0.191421 0.002533 0.005436 avgtempand weight 0.063342 0.00073 0.007245 0.201033 0.00226 0.005178 avgtempand height 0.039438 0.001054 0.008103 0.222654 0.003194 0.004502 avgtempand gender 0.065617 0.000929 0.008353 0.191421 0.002678 0.004092 avgtempandTBW 0.051103 0.000908 0.00927 0.137331 0.003298 0.008083 avgtempanddose 0.056339 0.001034 0.009798 0.137331 0.003092 0.007722 avgtempandstd drink 0.042956 0.000844 0.008074 0.201033 0.00264 0.002684 avgtempand ANTAC 0.058417 0.000717 0.008311 0.201033 0.002088 0.003687 Table5.3: Standarddeviationsforallmetricsofeveryutilizedconfiguration. Thestandarddevia- tiongroundtruthBrACdescendingrateis0:004001andthestandarddeviationgroundtruth BrACascendingrateis0:028918. 91 median runtime median MSE median peakdiff. medianpeak timediff. medianest. BrACdesc. rate medianest. BrACasc. rate age 0.859414 0.001801 0.008133 0.278559 -0.010448 0.05035 weight 0.843452 0.001945 0.007854 0.278559 -0.009707 0.046771 height 0.862685 0.00165 0.008216 0.278559 -0.01163 0.050799 gender 0.856723 0.001772 0.008466 0.278559 -0.010912 0.093453 TBW 0.865146 0.001542 0.007384 0.278559 -0.009237 0.043989 dose 0.870345 0.001542 0.008893 0.278559 -0.009488 0.04407 stddrink 0.876493 0.001542 0.008893 0.278559 -0.009862 0.044806 all 0.875577 0.001587 0.009933 0.278559 -0.011918 0.049666 averagenon-zero TAC(ANTAC) 0.87288 0.001451 0.008678 0.278559 -0.00968 0.048715 ageandANTAC 0.869251 0.001591 0.007069 0.278559 -0.010795 0.047356 weightand ANTAC 0.86332 0.001995 0.008584 0.278559 -0.009741 0.047063 heightandANTAC 0.874167 0.001399 0.008066 0.278559 -0.010487 0.047836 genderand ANTAC 0.87172 0.00136 0.010342 0.278559 -0.011121 0.047068 TBWandANTAC 0.872172 0.001667 0.008304 0.278559 -0.009571 0.044662 doseandANTAC 0.874833 0.001576 0.008892 0.278559 -0.009571 0.044662 stddrinkand ANTAC 0.859012 0.001576 0.008304 0.278559 -0.00945 0.046028 ageandweight 0.85528 0.002115 0.007798 0.278559 -0.009707 0.046681 ageandheight 0.889361 0.001663 0.007425 0.278559 -0.01163 0.050014 ageandgender 0.880083 0.001687 0.008962 0.278559 -0.010685 0.047991 ageandTBW 0.864163 0.001581 0.010617 0.278559 -0.010125 0.044123 ageanddose 0.916505 0.001773 0.010859 0.278559 -0.010125 0.044123 ageandstddrink 0.915176 0.001773 0.007602 0.278559 -0.010448 0.04489 weightandheight 0.876654 0.001686 0.007442 0.278559 -0.009707 0.046372 weightandgender 0.854648 0.002128 0.007854 0.278559 -0.009707 0.046771 weightandTBW 0.892885 0.001686 0.00826 0.278559 -0.009707 0.047813 weightanddose 0.872001 0.001707 0.00826 0.278559 -0.009707 0.044926 weightandstd drink 0.846184 0.002128 0.007854 0.278559 -0.009707 0.046771 heightandgender 0.87512 0.00152 0.008216 0.278559 -0.01163 0.050799 heightandTBW 0.904809 0.001663 0.009393 0.278559 -0.010889 0.047357 heightanddose 0.848817 0.001528 0.007749 0.278559 -0.009862 0.046486 heightandstd drink 0.86956 0.00165 0.008216 0.278559 -0.01163 0.050799 genderandTBW 0.866989 0.001982 0.008935 0.278559 -0.009406 0.044926 genderanddose 0.852383 0.001957 0.008333 0.278559 -0.009237 0.044121 genderandstd drink 0.853568 0.001982 0.00882 0.278559 -0.009707 0.044926 TBWanddose 0.865951 0.001542 0.008893 0.278559 -0.009488 0.04407 TBWandstddrink 0.854511 0.001542 0.007384 0.278559 -0.009237 0.043989 doseandstddrink 0.857306 0.001542 0.008893 0.278559 -0.009488 0.04407 92 random 0.854029 0.001442 0.009309 0.278559 -0.010426 0.049123 avgtemp 0.852332 0.001353 0.009739 0.278559 -0.009732 0.044567 avgtempandage 0.89062 0.001413 0.008965 0.278559 -0.009864 0.047079 avgtempand weight 0.866662 0.001979 0.010671 0.278559 -0.009908 0.045751 avgtempand height 0.850971 0.001282 0.007224 0.278559 -0.012032 0.046911 avgtempand gender 0.842317 0.001276 0.01043 0.278559 -0.010639 0.046493 avgtempandTBW 0.851146 0.001651 0.011235 0.278559 -0.010548 0.050219 avgtempanddose 0.84332 0.001569 0.009996 0.278559 -0.010268 0.049275 avgtempandstd drink 0.847508 0.001307 0.008929 0.278559 -0.01093 0.045624 avgtempand ANTAC 0.850343 0.001353 0.009739 0.278559 -0.009732 0.044567 Table 5.4: Medians for all metrics of every utilized configuration. The median ground truth BrAC descending rate is 0:015887 and the median ground truth BrAC ascending rate is 0:061444. 93 Chapter6 DiscussionandConcludingRemarks The theoretical and numerical results presented in this thesis have explored new avenues upon which we have furthered our knowledge of the problem outlined in Chapter 3. Though, traveling theseavenueshassuggestedseveralnewmathematicalquestionsthatdeservefurtherattentionand consideration. In the following chapter, we highlight the findings from Chapters 4 and 5 and then expoundonfurtherworkthatourresultshavegrantedusaccessto. 6.1 BayesianEstimationofModelParameters Figure 4.2 illustrates rapid convergence in dimensionality of our spatial dimensions as n grows, thusbolsteringtheresultsofTheorem4.1. Withintwosteps(n = 3),wehaveagraphthatvisually differsfromthatofn = 25inwaysbarelyperceptible. PairedwiththecrediblecirclesinTable4.1, these provide a strong demonstration that aftern = 3 the mean and radius of the credible circles forq stay consistent. Such results allow us to determine a computationally efficientn value that minimizesdatalostwhenprojecting(3.13)intofinitedimensions,(3.19). For the consistency results, Table 4.2 exemplifies the theoretical prediction in Theorem 4.6 that as the amount of subject data R grows, the posterior distribution better predicts the true q value by localizing the true parameter q 0 in mean with higher confidence (smaller confidence circle). This increasing confidence is backed by the decreasing variance results shown in Figure 4.3. Notice that although the variance decreases, the mean is allowed to shift as more data are 94 incorporated,asevidentfromcomparingFigure4.3ctoFigure4.3d. Thisshiftingmeanisallowed by the theoretical results and is likely due to the incorporation of 70 extra data points. Table 4.3 displays the shifting of the mean as more data are incorporated while quantitatively displaying a decreasing 90% credible circle radius, as expected. As a final note, recall that TAC data were collectedsimultaneouslyfromboththerightandleftarmsofparticipants. 6.1.1 DeconvolutionofBrACfromTAC For Figures 4.4d through 4.4f, all of the images demonstrate the deconvolution method does at- tempttocorrectforthelagtimebetweenBrACandTACreadings(specificallythepeaktimes)but does not always succeed in adjusting for the correct time. While achieving that correction would beideal,aconsolationisthatthe90%crediblebandsmostlycapturethewantedBrACcurve. The main exception being Figure 4.4e, wherein there is little lag between the TAC and BrAC reading. Further,theTACreadingsarefarbelowthoseofBrACwhichappearstobeamechanicaleffectof some SCRAM TAC devices. These are edge cases where an inaccurate deconvolution should be expected. When deployed, real-world scenarios such as these can be handled by communication betweentheowner/overseerofthedeviceandthewearer. Thus,humaninteractioncanbereduced toonlyimportantedgecases. InFigure4.4d,thedeconvolvedmeanBrACcurvemorecloselyresemblestheoverallcurveof themeasuredTACvaluesratherthanthedesiredBrAC,withitsincreasedvaluestowardsthelatter part of the curve. This is to be expected as the measured TAC plays a role in the Bayesian step, butnoticethattheseverityoftheincreaseinthemeanvaluecurveisattenuatedwhencomparedto that of the TAC curve (red vs. yellow curves at the five-hour mark). A similar phenomenon also appearsinFigure4.5d. For Figures 4.5a-4.5c and 4.5d-4.5f, as the number of subject drinking episodesR increases, we find that the mean curve grows towards the actual BrAC curve, an expected convergence phe- nomenongiventhetheoreticalconsistencyresultsfromSection4.2. 95 Lastly,the90%errorbandsaboutthedeconvolvedBrACcurvesappeartoalwayshavealower bound of essentially zero. For the upper bound, the extreme case is shown in Figure 4.4f. These widerangesinBrACvaluesallowustocapturethetrueBrACvaluewithhighprobabilitybutalso leave us capturing far more area under the curve than needed. Thus, there are times when our two-step method would falsely signal that the TAC device wearer is far more inebriated than they are. This incorrect signaling might be due in part to the quantitative inaccurate readings in Figure 4.4f, wherein the TAC curve is greater than the BrAC curve. If our (training) data are mainly composed of the other cases (TAC following BrAC at an attenuated rate), then the algorithm will learn to “guess up” when turning the TAC back into BrAC. This phenomenon may be due to the use of an un-educated prior as the credible regions in Table 4.3 do not approach zero. Hence, in the future, an educated prior could be given preference. Further, when comparing the 90% error bandstothe70%errorbands,wefindthatmostoftheuncertaintyisclaimedbyBrACvaluesthat arebelowthemean. Thisisanimportantdistinctionaswemustalwaysseektoensurethatweare not over-predicting the true BrAC with our estimations. As expected, but worth noting, is that the tendency for the 90% bands to have a lower bound at zero disappears when considering the 70% bands. Lastly,wealsonotethattheupperboundsontheerrorbandsstayrelativelythesame(with only minor variation), which hints at the fact that there is some mixture ofq 1 andq 2 values away fromtheirrespectiveextremesthatcausesthemaximumestimatedBrACtooccur. 6.1.2 ConcludingRemarks We believe that the independent, non-identically distributed (i.n.i.d.) assumption from Chapter 4 (specifically Section 4.2) may not reflect the realities of the data collection method wherein two sensors are worn simultaneously on participants’ left and right arms. We are currently investi- gating the elimination of this i.n.i.d assumption. However, the results from Section 4.4 are quite reasonableandareextremelyusefulwhenseekingtousethisapproachcomputationallyinpractice. Furtherinvestigationisneededregardingthetravelingmeanexhibitedinthenumericalresultsand howitisrelatedtothenon-inclusionofothercovariatedata(age,height,weight,etc.). 96 WealsobelievethatthepackagingofallerrorsourcesintoasinglerandomvariableinChapter 4 may yield larger uncertainties than formulations where many additive errors are considered. Namely, mixed-effects formulations may be utilized in order to separate errors and might lower overall uncertainty. However, the results from Section 4.4 are again quite reasonable, and the usage of mixed-effects formulations can be left as a design choice when considering the main goalsandimplementationsofthePDEmodelfromChapter3. AsmentionedinRemark7,theproblemofidentifiabilitywithrespecttosequencesofobserved dataandconsistencyoftheposteriorsfromSection4.2maybestudiedforourspecificmodelfrom Chapters 3 and 4. An investigation into the exact definition of the infinite product measure over infinitesequencesofobservedTACdatacanbeusedtopossiblyidentifysetsofinfinitesequences thatguaranteeconsistencyandmightbeabletobeabstractedintocommoncharacteristicsinterms ofourunderlyingmodel. Further,ananalysisofchosenpriorscanbeutilizedtodetermineconver- gence rates for consistency as well as possibly aid in determining the information-level required for sequences of TAC data to guarantee consistency. Then finally, these TAC data sets combined with deconvolution methods may help determine sets of input BrAC data that can guarantee con- sistency, which can then further be categorized according to common characteristics and be used toimposeBrACinputconstraints. Of primary interest is the direct inversion of BrAC,u, given TAC as in (4.2) without the need for a two-stepprocess likethat of the method used in this thesis. We believethat as part of a hier- archical Bayesian framework that utilizes Gaussian Processes, we may reduce the problem down toasinglestage(see,Section6.2.3and[62]). Insuchaframework,inplacealongwithaprioron q,weplaceafunctionspaceprioroveru. Inthisway,weobtainamethodthatstatisticallydecon- volvesBrACfromTACwhileprovidingadistributionfromwhichwemayderiveerrorbarsonthe estimated BrAC values. Continuing with hierarchical models, we are examining the inclusion of ahierarchical Bayesian modelthat incorporates covariatesinboth priors placedoverq andu. We believe that this will improve the accuracy of our predictions by allowing the use of all available environmentdata. 97 6.2 BayesianEstimationofDynamicalSystemInput Table 5.1 shows that as the approximating filter dimensionn increases, the predicted means and standard deviations stabilize across all displayed times as theoretically predicted. We note that 0.9471 hours represents the time when BrAC was nearest to its actual peak, and 9.471 hours represents a time when no alcohol was present in the blood. This shows that, as expected, the convergence of our predictive posterior distribution does not depend upon the amount of alcohol present in the bloodstream. Further, Table 5.1 allows us to identify a satisfactory value ofn that maintains both a high level of accuracy and low computational overhead. This was utilized when choosingn = 7incomputingFigure5.1. 6.2.1 PredictiveDeconvolutionofBrACfromTAC InFigure5.1wefindthatusingallavailabletrainingdatayields,ingeneral,asatisfactoryestimated BrAC signal. Figures (5.1a) and (5.1b) both indicate that the true BrAC is contained within the 3 standard deviation credible band about our predicted BrAC. Figure 5.1c, on the other hand, displays a situation where the true BrAC, to some extent, falls outside of the predicted BrAC credible band. Moreover, in this case, we predict a lower estimated peak BrAC than the actual breathalyzermeasuredpeakBrAC.Tocounterthis,wemayelecttostratifythetrainingpopulation beyond naive covariate methods as in Subsection 5.3.1. This may be accomplished by splitting the training data into strata based on factors relating TAC to BrAC (e.g., the peak TAC is greater than vs. is less than the peak BrAC). Alternatively, we might also use the covariates to obtain a more accurate estimate of the parameters q that are used in determining the filter matrixH n , as in (3.20) (see, [4]). Lastly, Figure 5.1b has a rise in predicted BrAC after it has predicted a return to near-zero. More work is needed to determine if this phenomenon is common or if it is due to computationalartifactssuchasunder-oroverflow. 98 6.2.2 NaiveStratificationofPopulationData ForTables5.2-5.4,wefindthatthebestcovariatestouseindeterminingtrainingdatavarydepend- ing upon the goal of the deconvolution as well as the desired statistical metric. If we seek a fit to the BrAC function directly that on average is the closest in mean square error, then we should employ a training data selection processing using Euclidean distance of subjects’ standard drink (in oz) and ambient temperature. Whereas if we sought to determine the time at which the peak BrAC occurred, then there are numerous approaches we may take (average ambient temperature, average ambient temperature and weight, etc.). Further, a key marker across the tables is that by selecting the proper covariates for the desired task, we may achieve better results than using all availabletrainingdata. Forexample,forfittingtheBrACfunctiondirectly,usingaverageambient temperatureandstandarddrink,onaverage,sawa4.8%decreaseinMSE.Moreworkisneededto determineifmorecomplexcombinationsofthecovariatescouldyieldevengreaterresults. 6.2.3 ConcludingRemarks We believe that the independence assumptions in Chapter 3, specifically the assumption that indi- viduals’ right and left arm measurements are independent, may not reflect the laboratory protocol usedincollectingthedatausedinournumericalstudiespresentedinSection4.4. Wearecurrently investigatingtheeffectofthispossiblelackofindependenceonourresultsanditspossibleelimina- tion. Nevertheless, the results from Section 4.4 remain rather impressive. Further investigation is neededintotheeffectsofincreasingtheamountoftrainingdata,andtheinterplaybetweentraining data selection and the differing drinking methods (single, dual, or steady) as described in Section 4.4. While the work of Chapter 5 utilized methods derived from Gaussian process regression, an- other approach that may be of interest involves the use of the expectation-maximization (EM) algorithm(see, [27]). Assuch, wemayutilizethejointdistributions(5.6)and(5.10)toiteratively approximatethemaximumaposterioriestimatesfor(5.7)and(5.9),respectively. 99 An approach of particular interest to us for future research is that of hierarchical models in- volving distributions over covariates and model parameters (such as q from (3.13)). We believe that the inclusion of these distributions could potentially improve the accuracy of our estimated BrAC signals and could be used in tandem with the proposed covariate filtering step investigated inSection5.3.3. Combiningtheseapproachesmayallowustoforegotheneedtohaveatwo-phase (parameter estimation and then deconvolution) approach as we may simultaneously estimate the parameters and the BrAC values. Though, doing such may add computational complexity that is outside the realm of realistic computational boundaries. Further, when both the parameters and BrAC are unknown, we must be careful when handling the product ofq 2 andu(t) in (3.1), as we wouldwish to estimate each parameter separately and seek to avoid estimating their product. An- other future investigation relies on nothing that multivariate normal distributions inherently allow for predicted BrAC to become negative with nonzero probability, contradicting the natural non- negativity constraint on BrAC. Given this, our results remain valid, but the inclusion of bounded linear constraints (see, [43, 85]) or the use of truncated multivariate normal distributions may re- movethisconcernandimproveresults. Finally,wenotethatwearelookingatextendingourresultstothecaseofanabstractparabolic continuous-timemodel(basedonequation(3.11)inChapter3andtheassociatedsemigroup-based continuous-time convolution by replacing the multivariate normal prior and multivariate normal BayesianframeworkwithaGaussianProcesspriorandaGaussianprocess-basedBayesianframe- work (see, [53, 62, 85]). We are optimistic that results analogous to those presented here for the discrete-time formulation can be obtained in the continuous-time setting. One possible initial ap- proach is to utilize the discrete-time models (as utilized in this thesis), namely (3.13) and (3.18) in Chapter 3, and replace the multivariate normal prior over BrAC in Chapter 5 with a Gaussian process prior. As such we may then include additive Gaussian white noise (see, [66]) in (5.1) and (5.8). DuetothefactthatGaussianprocessesareclosedunderlinearoperators(see,[53,59,62]), we would then find that the TAC distributions in (5.1) and (5.8) are simply Gaussian processes. Then,theresultsin(5.7)and(5.9)remainvalidforanygivenfixedtimevector,andwemayapply 100 theworkofChapter5. Notethatcareneedstobetakenwhenhandlingthekernelmatricesin(5.1) and (5.8), but the work of Chapter 3 easily extends to allowing variable time-step values. Lastly, we then may be able to utilize this middle-ground approach to then extend to the continuous-time case, but care would need to be exercised. As in (5.1) and (5.8) we would no longer have linear operators but would instead have Volterra-like operators that depend on an infinite-dimensional kernel. 101 References [1] ASecondChallengeCompetitionfortheWearableAlcoholBiosensor. [2] R.AdamsandJ.Fournier.SobolevSpaces.2nded.Vol.140.AmsterdamBoston:Academic Press,2003. ISBN:9780120441433. 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Abstract (if available)
Abstract
We study the problem of determining blood or breath alcohol (BAC or BrAC) concentration from transdermal alcohol concentration (TAC). In this way, we make use of two different solution approaches. In the initial method, we estimate the posterior distribution of random parameters in a distributed parameter, diffusion equation-based population model for the transdermal transport of alcohol. We then sequentially estimate the distribution of the input BrAC to the model. The output of the model is TAC, which via linear semigroup theory can be expressed as the convolution of BrAC with a filter that depends on the individual participant or subject, the biosensor hardware itself, and environmental conditions, all of which can be considered to be random under the presented framework. We utilize a Bayesian approach to estimate the posterior distribution of the parameters and/or the deconvolved BrAC conditioned on an individual?s measured TAC (and BrAC). Priors for the models are obtained from temporal population observations of BrAC and TAC via deterministic or statistical methods. The requisite computations require finite-dimensional approximation of the underlying state equation, which is achieved through standard finite element (i.e., Galerkin) techniques. We establish consistency of our Bayesian estimators, and we demonstrate the convergence of our posterior distributions computed based on the finite element model to those based on the underlying infinite-dimensional model. We also present some of the results of our human subject data-based numerical studies. The second method we employ involves the estimation of the posterior distribution of a random signal describing BrAC, the previously mentioned distributed parameter, diffusion equation-based model for the transdermal transport of ethanol. We develop a multivariate normal-based Bayesian approach to estimate the posterior distribution of the deconvolved BrAC signal conditioned on an individual?s measured TAC (and the population's measured BrAC and TAC). Priors for the models are obtained by fitting multivariate normal distributions using temporal population observations of BrAC and TAC via deterministic or statistical methods. Again, computations require finite-dimensional approximation of the underlying state equation and the associated semigroup of operators that determines the convolution kernel or filter, and this is achieved through standard finite element Galerkin techniques. We establish the convergence of our approximating posterior distributions to the posterior distributions corresponding to the original infinite-dimensional model using approximation results from the theory of linear semigroups of operators. We present a sample of results from some of our numerical studies and validate our convergence result by demonstrating the convergence of our finite-dimensional approximating posterior distributions to the posterior distribution determined by the underlying infinite-dimensional models. In both Bayesian approaches previously mentioned, the posteriors provide credible regions and yield population models that eliminate the need to calibrate the model for every individual, different biosensor devices, and varying environmental conditions.
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Obtaining breath alcohol concentration from transdermal alcohol concentration using Bayesian approaches
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