Bayesian Regression Analysis

In this section, Bayesian analysis is performed to identify the uncertain coefficients of the quadratic function. The effective temperature T is assumed to be different from the measured values since there is measurement noise in the data acquisition process and the temperature in different parts of the building could also be non-uniform. The difference is assumed Gaussian with zero mean and variance a. In this study, this standard deviation is taken to be aj = 0.5°C since the difference between the indoor and outdoor temperature measurement was around 1 °C and the average value was used as the measured temperature T for the wth day. On the other hand, the squared fundamental frequency is identified by the Bayesian spectral density approach to be presented in Chapter 3. Therefore, the uncertain parameters include the coefficients of thequadratic function and the effective temperatures 0 = [l o, b, b2, T, Ti,7 ], where iV is the number of data points. The data include the measurements of the temperature and the identified squared fundamental frequencies X = [li,. .., j, 2  

In this study, a non-informative prior PDF for the uncertain parameters is used and the prior PDF is absorbed into the normalizing constant. Then, the updated PDF is proportional to the likelihood function  

To obtain the optimal parameters, the objective function is defined as the negative logarithm of the likelihood function without taking the terms that do not depend on the uncertain parameters  

The optimal values can be obtained by minimizing the objective function but this optimization problem is nonlinear. However, the solution can be obtained by solving a series of linear 

On the other hand, the conditional optimal value for 7), can be found by solving dJ(0)/dT = 0, n = 1,2, N, and it gives the following cubic equation  

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In the second approach, a Bayesian regression analysis (e.g., Straub and Der Kiureghian 2008 loannou and Rossetto 2013) is adopted in order to take into accotmt prior information regarding the model s parameters, 0, especially when the available number of observations is small. Prior information is obtained from existing fragility functions or independent post-earthquake data of similar groups of assets. In addition, this... 

P. O. Maitre, M. Buhrer, D. Thomson, and D. R. Stanski, A three-step approach combining Bayesian regression and NONMEM population analysis application to midazolam. J Pharmacokinet Biopharm 19 377-384 (1991). 

The Bayesian spectral density approach for parametric identification and model updating regression analysis are applied. During the monitoring period, four typhoons flitted over Macao. The structural behavior under such violent wind excitation is treated as discordance and the measurements obtained under these events are not taken into account for the analysis. By excluding these fifteen days of measurements, there are 168 pairs of identified squared fundamental frequency and measured temperature in the data set. Figure 2.28(a) shows the variation of the identified squared fundamental frequencies with their associated uncertainties represented by a confidence interval that is bounded by the plus or minus three standard derivations from the estimated values. It is noticed that this confidence interval contains 99.7% of the probability. Since the confidence intervals are narrow compared with the variation... 

We omit the details, as the analysis is a standard Bayesian linear regression analysis as found in textbooks for Bayesian statistics, see e.g. Carhn Louis (2009) and Bolstad (2007). 

Referring to the situation in question 2, one might think that an informative prior would outweigh the effect of the increasing sample size. With respect to the Bayesian analysis of the linear regression, analyze the way in which the likelihood and an informative prior will compete for dominance in the posterior mean. 

A Bayesian analysis proceeds by placing prior distributions on the regression coefficient vector (3, error standard deviation a, and subset indicator vector 6. One form of prior distribution is given in detail below and other approaches are then discussed. Techniques for choosing hyperparameters of prior distributions, such as the mean of a prior distribution, are discussed later in Section 4. 

St/pen/7sed Data Mining. Searching large volumes of data for hidden predictive relationships. Supervised analysis requires one or more "dependent" or response variables, to be predicted from a set of "independent" or predictor variables. The techniques used include various classification methods (decision tree, support vector, Bayesian) and various estimation methods (regression, neural nets). 

D 3D AD ADME ADMET ANN ARD BCI BCUT BNN C4.5 CART ClogP CoMFA CV Two dimensional Three dimensional Applicability domain Absorption, distribution metabolism, and excretion Absorption, distribution metabolism, excretion, and toxicity Artificial neural network Automatic relevance determination Bernard chemical information Burden, CAS, University of Texas descriptors Bayesian neural network Decision trees using information entropy Classification and regression tree Calculated partition coefficient between octanol and water Comparative molecular field analysis Cross-validation... 

A later chapter will discuss these methods in more detail. For example, support vector machines and traditional neural networks are analogs of multiple regression or discriminant analysis that provide more flexibility in the form of the relationship between molecular properties and bioactivity.Kohonen neural nets are a more flexible analog to principal component analysis. Various Bayesian approaches are alternatives to the statistical methods described earlier. A freely available program oflcrs many of these capabilities. ... 

DuMouchel W. 2011. Multivariate Bayesian logistic regression for analysis of clinical study safety issues. Statistical Science 27(3) 319-339. 

In the paper we have presented and discussed a case study based on this extended Bayesian analysis. The study relates to the problem of comparing two different driUingjars. A standard Bayesian analysis is conducted using a regression model for the downtime which is the key quantity of interest in the study. This model comprises several parameters and the analysis is quite technical. It is a chaUenge to specify the prior distribution for aU the parameters. Using a conjugate prior structure makes it easy to go from the prior distribution to the posterior distribution. 

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