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A Bayesian approach for estimating breath from transdermal alcohol concentration
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A Bayesian approach for estimating breath from transdermal alcohol concentration
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A Bayesian Approach for Estimating Breath from Transdermal Alcohol Concentration by Bowen Zheng A Thesis Presented to the FACULTY OF THE USC DORNSIFE COLLEGE OF LETTERS, ARTS AND SCIENCES UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree MASTER OF SCIENCE (Applied Mathematics) May 2020 Copyright 2020 Bowen Zheng Table of Contents List Of Tables iii List Of Figures iv Abstract v Chapter 1: Introduction 1 1.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Chapter 2: Mathematical Models for Conversion of TAC to BrAC 4 2.1 Physiological Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Bayesian Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Empirical Distribution Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Mixed Model Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Chapter 3: Numerical Experiments 13 3.1 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Data Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2.1 Spline Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2.2 Least Squares Splines Approximation . . . . . . . . . . . . . . . . . . . . . 15 3.2.3 Lagrange Multiplier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2.4 Smoothing Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 An Example of Estimating BrAC by Bayesian Approach . . . . . . . . . . . . . . . 19 3.4 Performance of Bayesian Approach Model . . . . . . . . . . . . . . . . . . . . . . . 23 Chapter 4: Conclusion 26 Reference List 27 Appendix A Additional MATLAB code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ii List Of Tables 3.1 Mean square errors for models with dierent step size . . . . . . . . . . . . . . . . 25 iii List Of Figures 3.1 Results of Data Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 An example of basis functions for convolution kernel based on cubic spline function 20 3.3 An example of basis functions for the convolution of BrAC with k (t) . . . . . . . 21 3.4 An example of estimated convolution kernel . . . . . . . . . . . . . . . . . . . . . . 22 3.5 An example of reproduced TAC from BrAC and estimated convolution kernel . . . 23 3.6 An example of estimation of BrAC compared to raw BrAC . . . . . . . . . . . . . 24 iv Abstract Compared to Breath alcohol concentration (BrAC) which is the most commonly used measure- ment for alcohol, Transdermal alcohol concentration (TAC) obtained from the wearable alcohol biosensors is more reliable as it based on more complex physiological model. However, raw TAC data does not consistently associate with BrAC which is more interpretable across individuals, devices, and environmental conditions. Currently, there is no well-established method for predict- ing reliable quantitative values of BrAC from TAC data. This thesis discusses how to apply the Bayesian approach on the conversion of TAC to BrAC. Key Words: Transdermal Alcohol Concentration, Cubic Spline, Least Square, Bayesian Approach v Chapter 1 Introduction 1.1 Motivations As a part of our society, alcohol is widely used for people to celebrate, relax, socialize, and improve the enjoyment of meals. As shown by data, almost 90 percent of adults in the United States have experience drinking. Alcohol consumption can impact health and well-being, safety, and social behavior. It is necessary to do some related researches which can help people understand why alcohol-related injury or disease happens and prevent it. When people consume alcohol, they can eliminate it in several ways, like through your breath, blood or sweat. There exist several methods to measure alcohol content. The most common ways to measure alcohol content are the blood and breath alcohol tests. The blood alcohol test measures alcohol content through investigating the blood samples, and the breath alcohol test measures how much alcohol is in the air you breathe out. Measurements obtained by these two alcohol tests are called blood alcohol concentration (BAC) and breath alcohol concentration (BrAC) respectively. The two measurements, BAC and BrAC, have been veried to have a direct correlation of each other.[4] Also considering measurement of BrAC using breathalyser has been widely adopted by law-enforcement agencies and alcohol research community, we mainly use BrAC in this research which is equivalent to use BAC. 1 The National Institute on Alcohol Abuse and Alcoholism (NIAAA) 2017-2021 Strategic Plan emphasized the importance to develop the wearable biosensor that can provide accurate and quantiable measurement of alcohol. Not only the scientists would benet greatly from these accurate data to study alcohol-related health outcomes, treatment ecacy, and recovery, but also the individuals could make better health choices by a source of reliable data. When the alcohol biosensor wore by people on their arms or legs, it can provide values of transdermal alcohol concentration (TAC) real-time, which represent the amount of alcohol diusing through the skin. The measurement of TAC has been shown to be relatively easy to measure by electrochemical sensors placed on the skin. However, raw TAC data does not consistently correlate with the values of BrAC because of individuals, devices, and environment conditions. Currently, there is no well-established method for predicting reliable quantitative values of BrAC from TAC data. The objective of this research is to build mathematical models for the conversion of TAC to BrAC, and test the accuracy. 1.2 Literature Review In the past decade, several methods were come up for estimating BrAC from raw TAC data. I.G.Rosen developed a mathematical model which is a parabolic partial dierential equation with input BrAC and output TAC on the boundary. The inverse problem of estimating the BrAC is formulated as a blind decovolution problem.[3] In S.E.Luczak's research, a two-step mathematical modeling process was used for the inversion of TAC to BrAC. The rst step is the calibration phase that t rst principles physics-based models (i.e., the forward model) to capture the propagation of alcohol through the skin, and its measurement by the sensor. Then the second step used the individualized parameters determined in the calibration phase to convert TAC to BrAC.[7] These methods showed relatively good performance on predicting BrAC from TAC data, and some drinking summary scores (e.g., peak estimated BrAC, time of peak estimated BrAC) are 2 highly agreement with the raw BrAC for testing data. However, the participant and researcher burden for obtaining the simultaneously collected TAC and breath analyzer BrAC data in the laboratory reduced the feasibility for non-researchers to use these approaches. 3 Chapter 2 Mathematical Models for Conversion of TAC to BrAC 2.1 Physiological Models TAC is obtained by a wearable biosensor which can measure the alcohol content derived from alcohol excretion through the skin. So we can consider to use the physiological models for our research. Diusion equation is a highly simplied physiological model of this transport which has the following form: @ t (t;x) =D@ 2 x (t;x);t> 0;x2 (0; 1) (2.1) where (t;x) represents the alcohol content in depth x of skin with normalized thickness of 1 at time t. The parameter D represents the diusion rate through the skin.[2] To be more specic, we can assume the BrAC is measured at boundary x = 0 and the TAC is measured at x = 1. Therefore we get the boundary conditions @ x (t; 0) =D(t; 0) = BrAC (t) @ x (t; 1) =D(t; 1) = TAC (t) (2.2) where BrAC (t) and TAC (t) represent the breath and transdermal alcohol contents respectively. Although the physiological models will be more sophisticated in reality, the model dened by 4 (2.1) and (2.2) can show us some relationships between BrAC and TAC. The mapping between BrAC (t) and TAC (t) has the form TAC (t) = Z t 1 K(ts;D) BrAC (s)ds (2.3) where the convolution kernel K(t;D) has the form K(t;D) =He A(D)t B (2.4) where e A(D)t represents a semi-group of linear operators from a Hilbert space V to V generated by an unbounded linear operator A(D). B andH are often referred to as the input and observa- tion operators, respectively in system theoretic framework. To be specic, B represents a linear operator from R 1 to V , and H represents a linear functional on V .[8][9] Even though the system of partial dierential equations (2.1) and (2.2) provide a functional form of the convolution kernel K, and it always has support on innite time interval, we have to mod- ify the convolution kernels considering the presence of alcohol in sweat varnishes in nite time following removal of alcohol in the blood stream. In fact, instead of the physiologically motivated parameterization of the convolution kernel provided by the partial dierential equation models like (2.1) and (2.2), other alternative parameterization can also be considered. 2.2 Bayesian Approach Now we consider a probabilistic model based on 2.3. We use random functions BrAC(t;!) and K(t;!) to replaceu BrAC (t) andK(t;D). These functions take values from functional spaceV BrAC and V K . Hence, the measured TAC is given by TAC(t;!) = Z t 1 K(ts;!)BrAC(s;!)ds +(t;!);t> 0 (2.5) 5 where the function (t;!) represents the measurement noise. It is reasonable to make some assumptions on (t;!). First, (t;!) is independent of BrAC and K. What's more, we can assume(t;!) has the normal distribution with mean 0. That is the joint probability distribution of (t 1 ;!), (t 2 ;!), ::: , (t m ;!) has the form f (v) = exp( 1 2 v T P 1 m (t 1 ;:::;t m )v) p (2) m j P m (t 1 ;:::;t m )j ;v2R m (2.6) where the mm matrix P 1 m (t 1 ;:::;t m ) is the covariance matrix. In practice, we may consider the spacesV BrAC andV K to be nite-dimensional. Hence the random functions BrAC(t;!) and K(t;!) can be represented by their coordinates X(!) 2 R n B and Y (!)2 R n K . Denote the joint probability distribution of these random vectors as f X;Y . Based on the Baye's theorem,[6] the conditional probability density function of BrAC and convolution kernels f X;YjTAC has the form f X;YjTAC (x;yjTAC(t 1 );:::;TAC(t m )) = f (v(x;y))f X;Y (x;y) v 0 (2.7) where v2R m ; v i (x;y) =TAC(t i ) Z ti 1 K(t i s;y)BrAC(s;x)ds;i = 1;:::;m and v 0 = Z R n BR n K f (v(x;y))f X;Y (x;y)dxdy The prior distributions f X;Y andf can be obtained by analysis of collected data. After that, we can predict the likelihood for BrAC and convolution kernel by the following methods. 6 - Maximum likelihood predictor. Find BrAC(t; ^ x) and K(t; ^ y) where (^ x; ^ y) = arg max x;y f X;YjTAC (x;yjTAC(t 1 );:::;TAC(t m )) - Least square predictor. Find BrAC(t; ^ x) and K(t; ^ y) where ^ x =E(XjTAC(t 1 );:::;TAC(t m )) = Z R n BR n K xf (v(x;y))f X;Y (x;y)dxdy; ^ y =E(YjTAC(t 1 );:::;TAC(t m )) = Z R n BR n K yf (v(x;y))f X;Y (x;y)dxdy - Minimum variance predictor. Find BrAC(t; ^ x) and K(t; ^ y) where ^ x = arg min x E((XE(X)) 2 jTAC(t 1 );:::;TAC(t m )) = Z R n BR n K jjxE(x)jj 2 2 f (v(x;y))f X;Y (x;y)dxdy; ^ y = arg min y E((YE(Y )) 2 jTAC(t 1 );:::;TAC(t m )) = Z R n BR n K jjYE(Y )jj 2 2 f (v(x;y))f X;Y (x;y)dxdy Note that when the dimensions of space V BrAC andV K are very large, the cost of calculation for the estimators BrAC(t; ^ x) and K(t; ^ x) can be signicantly huge. In the next sections, we will discuss two possible ways for computing these estimators. 2.3 Empirical Distribution Approach After we obtain the data from laboratory experiments, we can regroup them into two datasets. The rst dataset consists of a collection of matched BrAC and TAC measurements f(BrAC(t;! k );TAC(t;! k ));k = 1;:::;N 1 g. Note we can obtain the continuous values for BrAC and TAC after the process of data smoothing, and the time stamps for the data are set relative 7 to the rst non-zero TAC measurement. As a result, the functions BrAC(t;! k ) and TAC(t;! k ) are assumed to be dened over the interval (1;1). The second dataset consists of a collection of matched TAC measurementsf(TAC 1 (t; ^ ! k );TAC 2 (t; ^ ! k ));k = 1;:::;N 2 g for identical under- lying BrAC. The dierence between TAC 1 and TAC 2 are due to measurement errors. That is TAC(t;!) = TAC 1 (t; ^ !)TAC 2 (t; ^ !) is a random sample of the measurement error. Note the time stamp for each pair of TAC 1 and TAC 2 are set relative to the rst non-zero TAC mea- surement. One of the benets of this approach is that the sample mean of measurement error is always near zero. To calculate the estimators, we need to get the covariance matrix P m (t 1 ;:::;t m ). An unbiased estimator of P m (t 1 ;:::;t m ) is the sample covariance matrix ^ X m (t 1 ;:::;t m ) = 1 N 2 1 N2 X i=1 (v i v k )(v i v k ) T (2.8) where v k = 0 B B B B B B @ TAC(t 1 ;! k ) . . . TAC(t m ;! k ) 1 C C C C C C A The sample mean of measurement error is always near zero, that is v k 0. So the covariance matrix P m (t 1 ;:::;t m ) can be estimated by ^ X m (t 1 ;:::;t m ) = 1 N 2 1 N2 X i=1 v i v T i (2.9) Therefore, we can estimate BrAC from the new TAC data by the following estimators. - Maximum likelihood predictor. Find BrAC(t;! k ) where k = arg max k v T k X m (t 1 ;:::;t m )v k 8 - Least square predictor. The estimation of BrAC(t;!) can be computed by d BrAC(t;!) = P N1 k=1 BrAC(t;! k )e v T k P m (t1;:::;tm)v k P N1 k=1 e v T k P m (t1;:::;tm)v k - Minimum variance predictor. Find BrAC(t;! k ) where k = arg min k P N1 k=1 jjBrAC(t;! k ) BrAC(t)jj 2 2 e v T k P m (t1;:::;tm)v k P N1 k=1 e v T k P m (t1;:::;tm)v k where BrAC(t) = 1 N 1 N1 X k=1 BrAC(t;! k ) The premise of this approach is that the amount of data must be sucient since the estimated BrAC are only given as a combination of the training data. When the training dataset is not rich enough, the estimations of BrAC may not reliable. Unfortunately, most training data collected from laboratory experiments are limited. For instance, drinking in laboratory may not last that long, and the consumption of alcohol may take less than one hour, which is not like drinking in the reality. In the next section, we will discuss a mixed model which may help retrieval of BrAC in more natural drinking events. 2.4 Mixed Model Approach In this mixed model approach, we still use the equation TAC(t;!) = Z t 1 K(ts;!)BrAC(s;!)ds +(t;!);t> 0 (2.10) as our basic model. Here BrAC(t) is an unknown parameter to be estimated. To estimate BrAC(t) given known values of TAC, we need to know the distribution of K(t). In fact, the 9 distribution of K(t;!) can be estimated by our rst dataset of matched BrAC and TAC pairs (BrAC(t;! k );TAC(t;! k ));k = 1;:::;N 1 . For each ! k , we use the least square to determine K(t;! k )2V k , that is to nd K(t;! k ) such that J(K) = Z tmax tmin [TAC(t;! k ) Z t 1 K(ts;! k )BrAC(s;! k )ds] 2 dt (2.11) is minimized. Also, we need to constraint the convolution kernel K(t) by the following principles - The support of K(t) is bounded. In fact, the support should be less than the length of the time interval between the time of the last non-zero BrAC and the the time of the last non-zero TAC measurement. - The function K(t) must be non-negative. Here we choose the cubic spline as the function form ofK(t) since cubic spline can well re ect the relationship between BrAC and TAC. For each basis cubic spline function k , we can compute the convolution of BrAC with k to obtain function k dened by k (t) = Z t 1 k (ts)BrAC(s)ds: (2.12) The function K(t) has the form K(t) = m X k=1 k k (t): (2.13) To estimate K(t) we have to nd coecients k such that the result of convolution of BrAC and K(t) is close to the measured TAC. We can optimize either L 1 or L 2 norms of the dierence between measured TAC and the convolution ofBrAC andK(t), that is to nd2R m such that J 1 () = Z t1 t0 jTAC(t) m X k=1 k k (t)jdt; (2.14) 10 or J 2 () = Z t1 t0 jTAC(t) m X k=1 k k (t)j 2 dt; (2.15) is minimized. In order to guarantee K(t) to be non-negative, we can constraint either the coe- cients k to be non-negative or the values ofK(t) to be non-negative at all numerical integration points. After we obtain the estimation of the convolution kernel K(t), the next step is to estimate BrAC for a given set of measured TAC, which is known as deconvolution of the TAC measurement. We can assume BrAC can be represented as a linear combination of basis functionsb k (t);k = 1;:::;n. Therefore we can estimate BrAC by minimizing the following least square functional Z tm t0 jTAC(t;!) Z t 1 K(ts) n X k=1 k b k (s)dsj 2 dt (2.16) In fact, the process of deconvolution is quite similar to that of estimation of convolution kernel. We can also assume that the unknown function BrAC(t) is a cubic spline function dened over an interval from the time of the rst non-zero TAC measurement minus the length of support of the convolution kernel to the latest TAC measurement. Hence, given the basis spline functions b k (t);k = 1;:::;n, we dene the covolution of b k (t) with the estimated convolution kernel as k (t) = Z t 1 K(ts)b k (s)ds: (2.17) The deconvolution problem consists of nding optimal estimation for BrAC of the form d BrAC(t) = n X k=1 k b k (t) (2.18) 11 such that the following function is minimized J() = Z t b t0 jTAC(t) n X k=1 k k (t)j 2 dt (2.19) where t 0 denotes the time of rst TAC reading, and t b denotes the time of most recent TAC reading. We can also constraint either the coecients k to be non-negative or the values of d BrAC(t) to be non-negative at all numerical integration points in order to guarantee the value of d BrAC(t) to be non-negative. 12 Chapter 3 Numerical Experiments 3.1 Data Collection Our research was mainly based on the data collected by the department of psychology, University of Southern California. Data were from 40 participants, and most of them completed 4 drinking sessions. The BrAC data were measured by the Breathalyzer, and the TAC data were obtained by the wearable biosensor. Note participants wore the transdermal alcohol sensors on both left and right arms for each drinking session. Therefore, a collection of matched two TAC measurements for identical BrAC data was recorded. In summary, there are totally 150 valid datasets, and each dataset consists of 2 sets of TAC data, and 1 set of BrAC data for each drinking session. We should note that the values of BrAC and TAC were obtained at discrete time during the experiment. However, the continuous values of BrAC and TAC are required for our conversion approach. As a result, we have to smooth the raw data to access the continuous values. The interpolation method will be discussed in the next section. 13 3.2 Data Smoothing In order to obtain the continuous values of BrAC and TAC, we used the cubic spline functions combined with least square approximation and Lagrange multiplier. The results showed a good performance of this approach. 3.2.1 Spline Functions A spline function of order m is a piecewise polynomial function of degree m 1 in a variable x. The values of x where the pieces of polynomial meet are known as knot points. The spline function is continuous and has continuous derivatives of orders 1;:::;m 2 when the knots are distinct.[1] A spline function s(t) of order m 0, on a grid =a =t 0 <t 1 <<t n =b of distinct knots is a real function s with the following properties: 1. For t2 [t i ;t i+1 ];i = 0; 1;:::;n 1, s(t) is a polynomial of degree <m. 2. s(t)2C m2 [a;b] The space of all spine functions of order m on is denoted by S ;m . Clearly S ;m is a linear vector space. Then we dene the basis for S ;m k (x) = [t k+m+1 ;:::;t k ](xt) m + ;k =m; _;n 1 (3.1) which is also called the B-Spline basis. So for the cubic spline, s(t) can be presented to the following equation: s(t) = n1 X k=3 k k (t) (3.2) 14 Back to our data, we can rst obtain f(t) from the linear interpolation. Then the interpolation problem can be written n1 X k=3 k k (t) =f(t) (3.3) There are several ways to compute the coecients k ;k =3;2;:::;n 1. Here we will use the least square to determine k . 3.2.2 Least Squares Splines Approximation Now we consider the linear least squares spline approximation problem, that is we want to mini- mize J() =min Z tn t0 (s(t)f(t)) 2 dt =min Z tn t0 ( n1 X k=3 k k (t)f(t)) 2 dt =min m X j=0 ( n1 X k=3 k k (s j )f(s j )) 2 (s j+1 sj) (3.4) where m is a very large number, and f(t) is the linear interpolation of the data.[10] The minimum of the sum of squares is found by setting the gradient to zero. @J() @ ^ k = 2 m X j=0 ( n1 X k=3 k k (s j )f(s j ))(s j+1 s j ) ^ k (t j ) = 2(s j+1 s j )[ m X j=0 n1 X k=3 k k (s j ) ^ k (s j ) m X j=0 f(s j ) ^ k (s j )] = 0 (3.5) which is equivalent to m X j=0 n1 X k=3 k k (s j ) ^ k st j ) = m X j=0 f(s j ) ^ k (s j ) (3.6) The above equation can be written as the matrix form ~ A~ a = ~ b (3.7) 15 where A i;j = m X k=0 j (s k ) i (s k );i;j =3;:::;n 1 b i = m X k=0 f(s k ) i (s k );i =3;:::;n 1 According to (2:7) we can obtain ~ = (A T A) 1 A T b (3.8) However, the matrix ~ A might be singular matrix. We can add a regularization term to guarantee ~ A be non-singular. The new J() can be represented to J() = min Z tn t0 (s(t)f(t)) 2 dt + n1 X i=3 2 i (3.9) After we add the regularization term, we can obtain ~ = (A T A) +L 1 A T b (3.10) where L = 0 B B B B B B B B B B B B B B @ 1 0 0 0 0 1 0 0 . . . . . . . . . . . . . . . 0 0 1 0 0 0 0 1 1 C C C C C C C C C C C C C C A After we get the coecients ~ , we can construct the spline function s(t) = P n1 k=3 k k (t) 16 3.2.3 Lagrange Multiplier In order to obtain more accurate result, we can add a constraint which guarantees the two areas under s(t) and f(t) are the same, that is Z tn t0 n1 X k=3 k k (t)dt = Z tn t0 f(t)dt N X j=0 n1 X k=3 k k (s j )(s j+1 s j ) = N X j=0 f(s j )(s j+1 s j ) (3.11) Now we can use the method of Lagrange multiplier and obtain the Lagrange function L() =J() + ( N X j=0 n1 X k=3 k k (s j )(s j+1 s j ) N X j=0 f(s j )(s j+1 s j ) (3.12) where is called the Lagrange multiplier.[5] The minimum of L() is found by setting the gradient to zero. @L() @ ^ k = 2 N X j=0 ( n1 X k=3 k k (s j )f(s j )) ^ k (s j )(s j+1 s j ) + N X j=0 ^ k (s j )(s j+1 s j ) = 0 (3.13) which is equivalent to 2 N X j=0 n1 X k=3 k k (s j ) ^ k (s j ) = 2 N X j=0 f(s j ) ^ k (s j ) N X j=0 ^ k (s j ) (3.14) The above equation can be written as the matrix form 2A~ = 2 ~ b~ p (3.15) 17 where A i;j = N X k=0 j (s k ) i (s k );i;j =3;:::;n 1 b i = N X k=0 f(s k ) i (s k );i =3;:::;n 1 p i = N X k=0 i (s k ) From (2:15) we can obtain ~ = (A) 1 ( ~ b 2 ~ p) (3.16) Let c = N X j=0 f(s j ) Then the constraint condition can be written to the form ~ p T ~ =c ~ p T (A) 1 ( ~ b 2 ~ p) =c By solving the above equation we can get = 2 ~ p T (A) 1 ~ bc ~ p T (A) 1 ~ p By (2:16), ~ can be presented to the form ~ = (A) 1 ( ~ b ~ p T (A) 1 ~ bc ~ p T (A) 1 ~ p ~ p) (3.17) 18 Figure 3.1: Results of Data Smoothing 3.2.4 Smoothing Results After using the above interpolation techniques, the cubic spline can be constructed. The result of data smoothing for one subject is shown in Figure 3.1. We can see from the above gure that the smoothing BrAC and TAC eliminate the eects of outliers. 3.3 AnExampleofEstimatingBrACbyBayesianApproach After the above data processing, we have already obtained the continuous values of BrAC and TAC. Then we can estimate BrAC by using Bayesian approach. First, we need to estimate the convolution kernel K(t) by given matched BrAC and TAC. We assume K(t) is a cubic spline function which has the form K(t) = m X k=1 k k (t); (3.18) where coecients k is unknown parameter we need to determine, and k (t) is the basis functions for convolution kernel which are shown in Figure 3.2. To reproduce TAC by BrAC and convolution 19 Figure 3.2: An example of basis functions for convolution kernel based on cubic spline function kernel, we dene the basis functions of the convolution of BrAC with k (t) as k (t) = Z t 1 k (ts)BrAC(s)ds; (3.19) which are shown in Figure 3.3. Therefore, we can determine the coecients k (t)2 R m by optimizing either L 1 or L 2 norms of the dierence between measured TAC and the reproducing TAC, that is to minimize J 1 () = Z t1 t0 jTAC(t) m X k=1 k k (t)jdt; (3.20) 20 Figure 3.3: An example of basis functions for the convolution of BrAC with k (t) or J 2 () = Z t1 t0 jTAC(t) m X k=1 k k (t)j 2 dt; (3.21) At the same time, we need to guarantee K(t) to be non-negative. We can constraint either the coecients k to be non-negative or the values of K(t) to be non-negative at all numerical integration points. The estimated convolution kernel are shown in Figure 3.4. From the Figure 3.4 we can see that the estimated kernel obtained by using either L 1 orL 2 norms with positivity constraints on either coecients or kernel itself produce similar results. 21 Figure 3.4: An example of estimated convolution kernel After obtaining the estimated convolution kernel, we can then reproduce TAC. The Figure 3.5 shows the result of reproducing TAC. The nal step is to estimate BrAC when new TAC data are available. We can similarly dene BrAC function as the cubic spline function of the form d BrAC(t) = n X k=1 k b k (t): (3.22) The objective of the deconvolution problem is to nd optimal k such that one of the following functions is minimized: J 1 () = Z t b t0 TAC(t) n X k=1 k k (t)dt; (3.23) 22 Figure 3.5: An example of reproduced TAC from BrAC and estimated convolution kernel J 2 () = Z t b t0 jTAC(t) n X k=1 k k (t)j 2 dt (3.24) where k (t) = Z t 1 K(ts)b k (s)ds: (3.25) is the convolution of estimated kernel with k . Here we use the raw TAC data to estimate BrAC with the correspond estimated convolution kernel. Figure 3.6 shows the estimation of BrAC compared to raw BrAC data. From the Figure 3.6 we can see that the estimation results are signicantly dierent between four dierent techniques. It seems the L 2 estimations are more erratic and L 1 estimations tend to under estimate the BrAC. 3.4 Performance of Bayesian Approach Model In our research, we used 10-fold cross validation method to set training and testing data. One important parameter in the Bayesian Approach Model is the step size between the grid points when we construct the cubic spline function. In order to evaluate models with dierent step size, 23 Figure 3.6: An example of estimation of BrAC compared to raw BrAC we need to compute the mean square errors for each model. First, we need to calculate the mean square error j ;j = 1;:::;k for each data in the testing dataset which has the size k. j = 1 m m X i=1 (y i ^ y i ) 2 (3.26) 24 where y i represents the raw BrAC values, and ^ y i represents the corresponding estimated BrAC values. m represents the number of raw BrAC points. Then, we compute the mean square error e for the whole testing dataset. e = 1 k k X j=1 j (3.27) Finally, we need to repeat the above two steps 10 times, record e n ;n = 1;:::; 10 for each time, and compute the mean square error E for the model. E = 1 10 10 X n=1 e n (3.28) We choose 6 dierent step size (10, 20, 30, 40, 50, 60/min) and estimate BrAC with 4 dierent techniques (L 1 , L 1alt , L 2 , L 2alt ). The results are shown in Table 3.1. Step size 10 20 30 40 50 60 E L 1 0.00093 0.00085 0.00167 0.00122 0.00199 0.00203 L 1-alt 0.00056 0.00094 0.00110 0.00102 0.00148 0.00223 L 2 0.00018 0.00048 0.00062 0.00032 0.00066 0.00093 L 2-alt 0.00035 0.00051 0.00097 0.00072 0.00077 0.00112 Table 3.1: Mean square errors for models with dierent step size 25 Chapter 4 Conclusion This study discusses how to apply the Bayesian approach on the problem of estimating BrAC from TAC. The mathematical model is based on the diusion equation combined with Baye's theorem. We can think of TAC as the convolution function of BrAC. Therefore, we have to estimate convolution kernel rst from the raw BrAC and TAC data. Then we deconvolve the new TAC data with estimated convolution kernel to predict BrAC. Since the data collected is mostly discrete in reality, we need to smooth data to obtain continuous values before using the model. In this research, we mainly used the cubic spline least square approximation with Lagrange multiplier to smooth data. In future work, machine learning approaches can be used to address remaining limitations. For example, we cannot estimate BrAC only from TAC data when we consider more components like temperature, skin hydration, and vasodilation. 26 Reference List [1] Germund Dahlquist and ke Bjrck. Numerical Methods in Scientic Computing: Volume 1. Society for Industrial and Applied Mathematics, USA, 2008. [2] Adolf Fick. Ueber diusion. Annalen der Physik, 170(1):59{86, 1855. [3] W. W. Hu I. G. Rosen, S. E. Luczak and M. Hankin. Discrete-time blind deconvolution for distributed parameter systems with dirichlet boundary input and unbounded output with application to a transdermal alcohol biosensor. 2013 Proceedings of the Conference on Control and its Applications, pages pp. 125{130, 2013. [4] Victoria C. Spencer Heike Wollersen Reinhard Dettmeyer Marcel A. Verho Immanuel Roiu, Christoph G. Birngruber. A comparison of breath- and blood-alcohol test results from real- life policing situations: A one-year study of data from the central hessian police district in germany. Forensic Science International, 232:125{130, 2013. [5] Leon S Lasdon. Optimization theory for large systems. Dover Books on Mathematics. Dover, Mineola, NY, 2002. [6] Peter M. Lee. Bayesian Statistics: An Introduction. Wiley Publishing, 4th edition, 2012. [7] Susan E. Luczak and I. Gary Rosen. Estimating brac from transdermal alcohol concentration data using the brac estimator software program. Alcohol Clin Exp Res, pages pp. 2243{2252, 2014. [8] Steven W. Smith. The Scientist and Engineer's Guide to Digital Signal Processing. California Technical Publishing, USA, 1997. [9] Ioan I. Vrabie. Chapter 2 - semigroups of linear operators. In C0-Semigroups and Application, volume 191 of North-Holland Mathematics Studies, pages 35 { 50. North-Holland, 2003. [10] Jerey H Williams. Quantifying Measurement. 2053-2571. Morgan & Claypool Publishers, 2016. 27 Appendix A Additional MATLAB code 1 function [ v]= BSplineBasis0 (m, x ) 2 % 3 % Description : Compute one b a s i s function f o r Bs p l i n e with uniformly 4 % spaced grid with spacing 1 . 5 % 6 v=ones (m+2 ,1) reshape (x , 1 , numel ( x ) )[0:m+1] ' ones (1 , length ( x ) ) ; 7 v (v<0)=0; 8 v=v . ^m; 9 f o r k=1:m +1 10 f o r j =1:m +2k 11 v ( j , : ) =(v ( j +1 ,:)v ( j , : ) ) /k ; 12 end 13 end 14 v=v ( 1 , : ) ; 15 return 16 end 1 function [ v]= BSplineBasis (m, x0 , dx , n , x ) 2 % 3 % Description : Compute b a s i s f u n c t i o n s f o r Bs p l i n e with uniformly 4 % spaced grid with spacing dx . 5 % 6 v=zeros (n m, length ( x ) ) ; 7 % 8 % Scaling x . 9 % 10 x=(xx0 ) /dx ; 11 f o r k=1:n m 12 [ v (k , : ) ]= BSplineBasis0 (m, xk+1) ; 13 end 14 return 15 end 1 function [ ACFit]= LSQFit constr (DateNum ,AC, DateNum0 , DeltaT ) 2 % 3 % Description : Create a l e a s t square f i t of an a l c o h o l content time h i s t o r y 28 4 % by cubic s p l i n e s on grid t0+kDeltaT where k i s s e l e c t e d so that the 5 % i n t e r v a l over which AC i s nonzero i s covered by the support of cubic 6 % polynomial b a s i s f u n c t i o n s . The cubic s p l i n e approximation i s 7 % constrained to have the same i n t e g r a l as the AC. 8 % 9 ACFit=s t r u c t ( 'RawData ' , [ ] ) ; 10 ACFit . RawData=s t r u c t ( 'DateNum ' ,DateNum , 'AC' ,AC, 'DateNum0 ' ,DateNum0 , ' DeltaT ' , DeltaT ) ; 11 % 12 % Defining the cubic s p l i n e grid . 13 % 14 eps =10^(4) ; 15 ind=f i n d (AC>=eps ) ; 16 DateNum range=[min(DateNum( ind ) )4DeltaT , max(DateNum( ind ) )+4DeltaT ] ; 17 indBasis min=f l o o r ( ( DateNum range (1)DateNum0) /DeltaT ) ; 18 indBasis min=min ( [ indBasis min , 0 ] ) ; 19 indBasis max=c e i l ( ( DateNum range (2)DateNum0) /DeltaT ) ; 20 indBasis max=max ( [ indBasis max , 0 ] ) ; 21 ACFit .LSQ=s t r u c t ( ' Grid ' , [ indBasis min : indBasis max ] DeltaT+DateNum0 , ' nGridPts ' , indBasis maxindBasis min , ' Step ' , DeltaT ) ; 22 % 23 % Compute the i n t e g r a l s used in the l e a s t square approximation . 24 % 25 nIntegrationPts =360010; 26 tGrid=DateNum0+indBasis minDeltaT+(indBasis maxindBasis min )DeltaT [ 0 : nIntegrationPts ] / nIntegrationPts ; 27 [ vBasis ]= BSplineBasis (3 , ACFit .LSQ. Grid (1) , ACFit .LSQ. Step , ACFit .LSQ. nGridPts , tGrid ) ; 28 ind=f i n d (~ isnan (AC) ) ; 29 DateNum=DateNum( ind ) ; 30 AC = AC( ind ) ; 31 ACInterp=interp1 (DateNum ,AC, tGrid , ' l i n e a r ' ,0) ; 32 Mnormal=vBasis vBasis ' ; 33 RHS =vBasis reshape ( ACInterp , numel ( ACInterp ) ,1) ; 34 % 35 % I f condition number of normal matrix i s too large , add a r e g u l a r i z a t i o n 36 % term . 37 % 38 CondLimit =10^5; 39 minRegularization=1/CondLimit ; 40 i f cond (Mnormal)>CondLimit 41 ACFit .LSQ. Regularization=minRegularization ; 42 Mnormal=Mnormal+minRegularization eye ( s i z e (Mnormal) ) ; 43 e l s e 44 ACFit .LSQ. Regularization =0; 45 end 46 % 47 % Compute the i n t e g r a l of AC and s l i n e b a s i s elements . 29 48 % 49 sumAC=sum( ACInterp ) ; 50 sumBasis=sum( vBasis , 2 ) ; 51 % 52 % Compute the value of Lagrange m u l t i p l i e r . 53 % 54 lambda=(sumBasis ' ( MnormalnRHS)sumAC) /( sumBasis ' ( MnormalnsumBasis ) ) ; 55 ACFit .LSQ. Coef=Mnormaln(RHSlambdasumBasis ) ; 56 % 57 % Compute the r e s i d u a l of l e a s t square approximation . 58 % 59 ACappr=ACFit .LSQ. Coef ' vBasis ; 60 ACappr(ACappr<0)=0; 61 ACFit .LSQ. Residual=sum ( ( tGrid (2)tGrid (1) )abs (ACapprACInterp ) . ^ 2 ) ; 62 ACFit .LSQ. R e s i d u a l r e l=ACFit .LSQ. Residual /sum ( ( tGrid (2)tGrid (1) )abs ( ACInterp ) . ^ 2 ) ; 63 return ; 64 end 1 function [ ACappr]=getLSQFitValue (LSQ, tGrid , varargin ) 2 % 3 % Description : Get approximated and smoothed AC values . 4 % 5 [ vBasis ]= BSplineBasis (3 ,LSQ. Grid (1) ,LSQ. Step ,LSQ. nGridPts , tGrid ) ; 6 Coef Label=' Coef ' ; 7 i f nargin>3 8 Coef Label=vararginf2g; 9 end 10 Coef=g e t f i e l d (LSQ, Coef Label ) ; 11 ACappr=Coef ' vBasis ; 12 TruncationFlag =1; 13 i f nargin>2 14 i f ~ isempty ( vararginf1g) 15 TruncationFlag=vararginf1g; 16 end 17 end 18 i f TruncationFlag==1 19 ACappr(ACappr<0)=0; 20 end 21 return 22 end 1 function [ MatchedBrACTAC]= estimateKernel (MatchedBrACTAC , varargin ) 2 % 3 % Description : Using a matched BrAC and TAC pair of measurements to 4 % estimate the convolution kernel between them . 5 % 6 StartEnd Consistent =1; 7 ACFit=' Fit 1 ' ; 8 i f nargin>1 9 i f ~ isempty ( vararginf1g) 30 10 i f i s f i e l d (MatchedBrACTAC .TAC, vararginf1g) 11 ACFit=vararginf1g; 12 end 13 end 14 end 15 TAC =g e t f i e l d (MatchedBrACTAC .TAC, ACFit ) ; 16 BrAC=g e t f i e l d (MatchedBrACTAC .BrAC, ACFit ) ; 17 % 18 % Determine the length of support of the convolution kernel . 19 % 20 [maxBrAC, indBrAC]=max(BrAC. RawData .AC) ; 21 [maxTAC, indTAC]=max(TAC. RawData .AC) ; 22 StepSize Default =10; 23 % 24 % Find the l a s t zero BrAC and TAC reading before reaching t h e i r peak . 25 % 26 indBrAC0=f i n d (BrAC. RawData .AC( 1 : indBrAC)>0) ; 27 i f indBrAC0 (1)>1 28 indBrAC0=indBrAC0 (1)1; 29 e l s e 30 indBrAC0=1; 31 end 32 indTAC0=f i n d (TAC. RawData .AC( 1 : indTAC)>0) ; 33 i f indTAC0 (1)>1 34 indTAC0=indTAC0 (1)1; 35 e l s e 36 indTAC0=1; 37 end 38 Support LowerBound= TAC. RawData . DateNum(indTAC0)BrAC. RawData . DateNum( indBrAC0 ) ; 39 i f Support LowerBound<0 40 disp ( ' estimateKernel : TAC has nonzero value before BrAC. ' ) ; 41 StartEnd Consistent =0; 42 Support LowerBound=0; 43 end 44 % 45 % Find the upper bound of support . F i r s t f i n d the f i r s t zero BrAC reading 46 % a f t e r reaching peak value . 47 % 48 ind=f i n d (BrAC. RawData .AC(indBrAC : end )==0) ; 49 i f isempty ( ind ) 50 indBrAC=length (BrAC. RawData .AC) ; 51 e l s e 52 indBrAC=indBrAC+ind (1)1; 53 end 54 indTAC=f i n d (TAC. RawData . DateNum>BrAC. RawData . DateNum(indBrAC) ) ; 55 % 56 % Find the end of nonzero TAC. 57 % 58 indTAC=indTAC(1) ; 31 59 ind=f i n d (TAC. RawData .AC(indTAC : end )>0) ; 60 i f isempty ( ind ) 61 % 62 % I f there i s no nonzero TAC reading f o l l o w i n g BrAC reaching zero . 63 % 64 KernelSupport=6 StepSize Default /(2460) ; 65 Support UppererBound=Support LowerBound+6 StepSize Default /(2460) ; 66 StartEnd Consistent =0; 67 disp ( ' estimateKernel : Beginning and ending of TAC and BrAC are i n c o n s i s t e n t . ' ) ; 68 i f nargin>2 69 i f ~ isempty ( vararginf2g) 70 KernelSupport=vararginf2g; 71 end 72 end 73 e l s e 74 indTAC=min ( [ indTAC+ind ( end ) , length (TAC. RawData . DateNum) ] ) ; 75 Support UppererBound=(TAC. RawData . DateNum(indTAC)BrAC. RawData . DateNum(indBrAC) ) ; 76 i f Support UppererBound<Support LowerBound 77 disp ( ' estimateKernel : Beginning and ending of TAC and BrAC are i n c o n s i s t e n t . ' ) ; 78 StartEnd Consistent =0; 79 KernelSupport=6 StepSize Default /(2460) ; 80 i f nargin>2 81 i f ~ isempty ( vararginf2g) 82 KernelSupport=vararginf2g; 83 end 84 end 85 e l s e 86 KernelSupport=Support UppererBoundSupport LowerBound ; 87 end 88 end 89 KernelSupport=max ( [ KernelSupport ,6 StepSize Default /(2460) ] ) ; 90 % 91 % Set s p l i n e grid . 92 % 93 StepSize=KernelSupport /8; 94 nGrid=max ( [ 8 , c e i l ( KernelSupport / StepSize ) ] ) ; 95 MatchedBrACTAC . Kernel=s t r u c t ( ' Grid ' , Support LowerBound +[0: nGrid ] StepSize , . . . 96 ' nGridPts ' , nGrid , ' Step ' , StepSize , ' Fit ' ,ACFit , ' StartEnd Consistency ' , StartEnd Consistent ) ; 97 % 98 % BSpline b a s i s grid points are s h i f t e d by 1 . 99 % 100 nKernelGrid=c e i l (max ( [ Support UppererBound , Support LowerBound , MatchedBrACTAC . Kernel . Grid ( end ) ] ) / StepSize ) ; 101 tKernelGrid =[0: nKernelGrid100] StepSize /100; 32 102 [ vBasis ]= BSplineBasis (3 ,MatchedBrACTAC . Kernel . Grid (1) , . . . 103 MatchedBrACTAC . Kernel . Step , MatchedBrACTAC . Kernel . nGridPts , tKernelGrid ) ; 104 [ nKernelBasis , nKernelPts ]= s i z e ( vBasis ) ; 105 minDateNum= TAC.LSQ. Grid (1) ; 106 maxDateNum= TAC.LSQ. Grid ( end ) ; 107 minDateNum=min ( [ minDateNum ,BrAC.LSQ. Grid (1) ] ) ; 108 maxDateNum=max ( [ maxDateNum ,BrAC.LSQ. Grid ( end ) ] ) ; 109 nGridPts=100 c e i l ( (maxDateNumminDateNum+1/24)/ StepSize ) ; 110 tGrid=minDateNum0.5/24+[0: nGridPts ] StepSize /100; 111 [ TAC Approx]=getLSQFitValue (TAC.LSQ, tGrid ) ; 112 [ BrAC Approx]=getLSQFitValue (BrAC.LSQ, tGrid ) ; 113 yBasis=zeros ( nKernelBasis , nGridPts+1) ; 114 % 115 % Evalute the convolution of BrAC with b a s i s of kernel 116 % 117 f o r iKernelPts =1: nKernelPts 118 yBasis ( : , iKernelPts : end )=yBasis ( : , iKernelPts : end )+vBasis ( : , iKernelPts )BrAC Approx ( 1 : endiKernelPts +1) ; 119 end 120 % 121 % Find optimal kernel 122 % 123 MatchedBrACTAC . Kernel . Coef=L1PositiveRegression ( yBasis , TAC Approx) ; 124 MatchedBrACTAC . Kernel . C o e f a l t=L 1 P o s i t i v e R e g r e s s i o n a l t ( vBasis , yBasis , TAC Approx) ; 125 MatchedBrACTAC . Kernel . Coef2=L2PositiveRegression ( yBasis , TAC Approx) ; 126 MatchedBrACTAC . Kernel . C o e f 2 a l t=L 2 P o s i t i v e R e g r e s s i o n a l t ( vBasis , yBasis , TAC Approx) ; 127 MatchedBrACTAC . Kernel . Residual=sum ( ( tGrid (2)tGrid (1) )abs (TAC Approx MatchedBrACTAC . Kernel . Coef ' yBasis ) ) ; 128 MatchedBrACTAC . Kernel . R e s i d u a l r e l=MatchedBrACTAC . Kernel . Residual / . . . 129 sum ( ( tGrid (2)tGrid (1) )abs (TAC Approx) ) ; 1 function [ MatchedBrACTAC]= estimateKernel (MatchedBrACTAC , varargin ) 2 % 3 % Description : Using a matched BrAC and TAC pair of measurements to 4 % estimate the convolution kernel between them . 5 % 6 StartEnd Consistent =1; 7 ACFit=' Fit 1 ' ; 8 i f nargin>1 9 i f ~ isempty ( vararginf1g) 10 i f i s f i e l d (MatchedBrACTAC .TAC, vararginf1g) 11 ACFit=vararginf1g; 12 end 13 end 14 end 15 TAC =g e t f i e l d (MatchedBrACTAC .TAC, ACFit ) ; 16 BrAC=g e t f i e l d (MatchedBrACTAC .BrAC, ACFit ) ; 17 % 33 18 % Determine the length of support of the convolution kernel . 19 % 20 [maxBrAC, indBrAC]=max(BrAC. RawData .AC) ; 21 [maxTAC, indTAC]=max(TAC. RawData .AC) ; 22 StepSize Default =10; 23 % 24 % Find the l a s t zero BrAC and TAC reading before reaching t h e i r peak . 25 % 26 indBrAC0=f i n d (BrAC. RawData .AC( 1 : indBrAC)>0) ; 27 i f indBrAC0 (1)>1 28 indBrAC0=indBrAC0 (1)1; 29 e l s e 30 indBrAC0=1; 31 end 32 indTAC0=f i n d (TAC. RawData .AC( 1 : indTAC)>0) ; 33 i f indTAC0 (1)>1 34 indTAC0=indTAC0 (1)1; 35 e l s e 36 indTAC0=1; 37 end 38 Support LowerBound= TAC. RawData . DateNum(indTAC0)BrAC. RawData . DateNum( indBrAC0 ) ; 39 i f Support LowerBound<0 40 disp ( ' estimateKernel : TAC has nonzero value before BrAC. ' ) ; 41 StartEnd Consistent =0; 42 Support LowerBound=0; 43 end 44 % 45 % Find the upper bound of support . F i r s t f i n d the f i r s t zero BrAC reading 46 % a f t e r reaching peak value . 47 % 48 ind=f i n d (BrAC. RawData .AC(indBrAC : end )==0) ; 49 i f isempty ( ind ) 50 indBrAC=length (BrAC. RawData .AC) ; 51 e l s e 52 indBrAC=indBrAC+ind (1)1; 53 end 54 indTAC=f i n d (TAC. RawData . DateNum>BrAC. RawData . DateNum(indBrAC) ) ; 55 % 56 % Find the end of nonzero TAC. 57 % 58 indTAC=indTAC(1) ; 59 ind=f i n d (TAC. RawData .AC(indTAC : end )>0) ; 60 i f isempty ( ind ) 61 % 62 % I f there i s no nonzero TAC reading f o l l o w i n g BrAC reaching zero . 63 % 64 KernelSupport=6 StepSize Default /(2460) ; 34 65 Support UppererBound=Support LowerBound+6 StepSize Default /(2460) ; 66 StartEnd Consistent =0; 67 disp ( ' estimateKernel : Beginning and ending of TAC and BrAC are i n c o n s i s t e n t . ' ) ; 68 i f nargin>2 69 i f ~ isempty ( vararginf2g) 70 KernelSupport=vararginf2g; 71 end 72 end 73 e l s e 74 indTAC=min ( [ indTAC+ind ( end ) , length (TAC. RawData . DateNum) ] ) ; 75 Support UppererBound=(TAC. RawData . DateNum(indTAC)BrAC. RawData . DateNum(indBrAC) ) ; 76 i f Support UppererBound<Support LowerBound 77 disp ( ' estimateKernel : Beginning and ending of TAC and BrAC are i n c o n s i s t e n t . ' ) ; 78 StartEnd Consistent =0; 79 KernelSupport=6 StepSize Default /(2460) ; 80 i f nargin>2 81 i f ~ isempty ( vararginf2g) 82 KernelSupport=vararginf2g; 83 end 84 end 85 e l s e 86 KernelSupport=Support UppererBoundSupport LowerBound ; 87 end 88 end 89 KernelSupport=max ( [ KernelSupport ,6 StepSize Default /(2460) ] ) ; 90 % 91 % Set s p l i n e grid . 92 % 93 StepSize=KernelSupport /8; 94 nGrid=max ( [ 8 , c e i l ( KernelSupport / StepSize ) ] ) ; 95 MatchedBrACTAC . Kernel=s t r u c t ( ' Grid ' , Support LowerBound +[0: nGrid ] StepSize , . . . 96 ' nGridPts ' , nGrid , ' Step ' , StepSize , ' Fit ' ,ACFit , ' StartEnd Consistency ' , StartEnd Consistent ) ; 97 % 98 % BSpline b a s i s grid points are s h i f t e d by 1 . 99 % 100 nKernelGrid=c e i l (max ( [ Support UppererBound , Support LowerBound , MatchedBrACTAC . Kernel . Grid ( end ) ] ) / StepSize ) ; 101 tKernelGrid =[0: nKernelGrid100] StepSize /100; 102 [ vBasis ]= BSplineBasis (3 ,MatchedBrACTAC . Kernel . Grid (1) , . . . 103 MatchedBrACTAC . Kernel . Step , MatchedBrACTAC . Kernel . nGridPts , tKernelGrid ) ; 104 [ nKernelBasis , nKernelPts ]= s i z e ( vBasis ) ; 105 minDateNum= TAC.LSQ. Grid (1) ; 106 maxDateNum= TAC.LSQ. Grid ( end ) ; 107 minDateNum=min ( [ minDateNum ,BrAC.LSQ. Grid (1) ] ) ; 35 108 maxDateNum=max ( [ maxDateNum ,BrAC.LSQ. Grid ( end ) ] ) ; 109 nGridPts=100 c e i l ( (maxDateNumminDateNum+1/24)/ StepSize ) ; 110 tGrid=minDateNum0.5/24+[0: nGridPts ] StepSize /100; 111 [ TAC Approx]=getLSQFitValue (TAC.LSQ, tGrid ) ; 112 [ BrAC Approx]=getLSQFitValue (BrAC.LSQ, tGrid ) ; 113 yBasis=zeros ( nKernelBasis , nGridPts+1) ; 114 % 115 % Evalute the convolution of BrAC with b a s i s of kernel 116 % 117 f o r iKernelPts =1: nKernelPts 118 yBasis ( : , iKernelPts : end )=yBasis ( : , iKernelPts : end )+vBasis ( : , iKernelPts )BrAC Approx ( 1 : endiKernelPts +1) ; 119 end 120 % 121 % Find optimal kernel 122 % 123 MatchedBrACTAC . Kernel . Coef=L1PositiveRegression ( yBasis , TAC Approx) ; 124 MatchedBrACTAC . Kernel . C o e f a l t=L 1 P o s i t i v e R e g r e s s i o n a l t ( vBasis , yBasis , TAC Approx) ; 125 MatchedBrACTAC . Kernel . Coef2=L2PositiveRegression ( yBasis , TAC Approx) ; 126 MatchedBrACTAC . Kernel . C o e f 2 a l t=L 2 P o s i t i v e R e g r e s s i o n a l t ( vBasis , yBasis , TAC Approx) ; 127 MatchedBrACTAC . Kernel . Residual=sum ( ( tGrid (2)tGrid (1) )abs (TAC Approx MatchedBrACTAC . Kernel . Coef ' yBasis ) ) ; 128 MatchedBrACTAC . Kernel . R e s i d u a l r e l=MatchedBrACTAC . Kernel . Residual / . . . 129 sum ( ( tGrid (2)tGrid (1) )abs (TAC Approx) ) ; 130 QC Flag=0; 131 i f nargin>3 132 i f ~ isempty ( vararginf3g) 133 QC Flag=vararginf3g; 134 end 135 end 136 i f QC Flag==0 137 return ; 138 end 36
Abstract (if available)
Abstract
Compared to Breath alcohol concentration (BrAC) which is the most commonly used measurement for alcohol, Transdermal alcohol concentration (TAC) obtained from the wearable alcohol biosensors is more reliable as it based on more complex physiological model. However, raw TAC data does not consistently associate with BrAC which is more interpretable across individuals, devices, and environmental conditions. Currently, there is no well-established method for predicting reliable quantitative values of BrAC from TAC data. This thesis discusses how to apply the Bayesian approach on the conversion of TAC to BrAC.
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Asset Metadata
Creator
Zheng, Bowen
(author)
Core Title
A Bayesian approach for estimating breath from transdermal alcohol concentration
School
College of Letters, Arts and Sciences
Degree
Master of Science
Degree Program
Applied Mathematics
Publication Date
04/21/2020
Defense Date
04/20/2020
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
Bayesian approach,cubic spline,least square,OAI-PMH Harvest,transdermal alcohol concentration
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Wang, Chunming (
committee chair
), Fulman, Jason (
committee member
), Zhang, Jianfeng (
committee member
)
Creator Email
bowenzhe@usc.edu,zhengbowen197@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c89-284309
Unique identifier
UC11673478
Identifier
etd-ZhengBowen-8280.pdf (filename),usctheses-c89-284309 (legacy record id)
Legacy Identifier
etd-ZhengBowen-8280.pdf
Dmrecord
284309
Document Type
Thesis
Rights
Zheng, Bowen
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
Bayesian approach
cubic spline
least square
transdermal alcohol concentration