Example 8

Case-Study VIII: Approximate Bayesian Computation of mixture distribution

This case study illustrates the ability of DREAM to perform statistical inference involving the use of summary metrics. This functionality, also known as Approximate Bayesian Computation (ABC) has been implemented in DREAM and the underlying theory and concepts have been published by Sadegh and Vrugt (2014). The ABC approach is particularly useful for diagnostic model evaluation as it relaxes the need for an explicit likelihood function in favor of one or multiple different summary statistics rooted in the relevant theory that together have a clearer and more compelling diagnostic power than some average (likelihood)-measure of the size of the error residuals. Vrugt and Sadegh (2013) have shown that this diagnostics approach enhances considerably our chances to diagnose and detect model structural errors. Indeed, the use of behavioral signatures helps to analyze and pinpoint which part of the model is malfunctioning and in need of further improvement.

This eight case study benchmarks the implementation of the ABC method against a simple mixture of two univariate normal distributions,

where and denote the mean of the first and second normal pdf of the mixture distribution, and and define their corresponding variances, respectively. The probability density function of in Equation (2.01) is given by the following weighted sum of two normal densities

This Equation is easy to derive if you follow the steps in case study III. Indeed, the determinant of and is simply equivalent to 1 and 1/10 respectively, and and 00.

We can easily generate samples from this target distribution with DREAM if we evaluate directly the density of Equation (2.02). Instead, we follow a different procedure herein and use the ABC procedure in DREAM. This requires the user to define signatures of system behavior, which in the present context is equivalent to a summary metric of Equation (2.02). This summary metric, , is defined as follows

where is a vector with hundred elements and |·| denotes the modulus operator (absolute value). The values of the vector Y are drawn randomly from the univariate normal distribution with zero mean and unit standard deviation, thus   , . The tilde symbol ‟” means ‟distributed according to".

The following distance function, is used in DREAM to help determine whether we should accept the value of the parameter, x or not

The variable in Equation (8.04) denotes the observed value of the summary metric. This value is equal to zero in the present study. If the value of the distance function is smaller than some small nominal value then is accepted, and otherwise the proposed parameter value is rejected. The initial population is drawn from the univariate uniform distribution with and using Latin hypercube sampling. A total of N = 8 chains are created with DREAM using standard settings for the algorithmic variables.

Implementation of plugin functions

The complete source code can be found in DREAM SDK - Examples\D3\Drm_Example08\Plugin\Src_Cpp