ABC-PMC: Population Monte Carlo Sampler of Hydrograph Signatures in MATLAB and Python

Description

Approximate Bayesian Computation avoids the use of an explicit likelihood function in favor a (number of) summary statistics that measure the distance between the model simulation and the data. This ABC approach is a vehicle for diagnostic model calibration and evaluation for the purpose of learning and model correction. The ABC-PMC toolbox in MATLAB and Python implements the Population Monte Carlo sampler to approximate the posterior distribution of the summary metrics using the distance function $\rho(\cdot)$ and sequence of successively smaller epsilon values, $\epsilon_{1},\ldots,epsilon_{J}$. Samples are accepted if $\rho(\cdot) \leq \epsilon$. The PMC sampler starts out as ABC-REJ during the first iteration, $j = 1$, but using a much larger initial value for $\epsilon$, the acceptance threshold. During each successive next step (iteration), $j = (2,\ldots,J)$, the value of $\epsilon$ is decreased and the proposal distribution, $q_{j}(\theta_{k}^{j−1},\cdot) = N_{d}(\theta_{k}^{j−1},\Sigma^{j})$ adapted using $\Sigma^{j} = \text{Cov}(\theta_{1}^{j−1},\ldots,\theta_{N}^{j−1})$ with $\theta_{k}$ drawn from a multinomial distribution, $F(\theta_{1:N}^{j−1} \vert w_{1:N}^{j-1})$ where $w_{1:N}^{j-1}$ are the posterior weights $(w_{l}^{j-1} \geq 0; \sum_{l=1}^{N} w_{l}^{j-1} = 1)$. Through a sequence of successive (multi)normal proposal distributions, the prior sample is thus iteratively refined until a sample of the posterior distribution is obtained. This approach, similar in spirit to the adaptive Metropolis sampler of Haario et al. (1999, 2001), receives a much higher sampling efficiency than ABC-REJ, particularly for cases where the prior sampling distribution $p(\theta)$ is a poor approximation of the posterior distribution. Details of the ABC-PMC procedure are given in Appendix B of Sadegh and Vrugt (2013). Another significant improvement in efficiency is obtained from MCMC simulation using the DREAM$_{(ABC)}$ algorithm of Sadegh and Vrugt (2014). This code is part of DREAM-Suite.

Getting Started

Installing: MATLAB

• Download and unzip the zip file 'MATLAB_code_ABC_PMC_V1.0.zip' in a directory 'ABC_PMC'
• Add the toolbox to your MATLAB search path by running the script 'install_ABC_PMC.m' available in the root directory
• You are ready to run the built-in examples

Executing program

• After intalling, you can simply direct to each example folder and execute the local 'example_X.m' file
• Please make sure you read carefully the instructions (i.e., green comments) in 'install_ABC_PMC.m'

Installing: Python

• Download and unzip the zip file 'Python_code_ABC_PMC_V1.0.zip' to a directory called 'ABC_PMC'

Executing program

• Go to Command Prompt and directory of example_X in the root of ABC_PMC
• Now you can execute this example by typing "python example_X.py".
• Instructions can be found in the file 'ABC_PMC.py'

Authors

• Vrugt, Jasper A. (jasper@uci.edu)

Literature

1. Turner, B.M, and T. van Zandt (2012), A tutorial on approximate Bayesian computation, Journal of Mathematical Psychology, 56, pp. 69-85
2. Sadegh, M., and J.A. Vrugt (2014), Approximate Bayesian computation using Markov chain Monte Carlo simulation: DREAM_(ABC), Water Resources Research, https://doi.org/10.1002/2014WR015386
3. Vrugt, J.A., and M. Sadegh (2013), Toward diagnostic model calibration and evaluation: Approximate Bayesian computation, Water Resources Research, 49, pp. 4335–4345, https://doi.org/10.1002/wrcr.20354
4. Sadegh, M., and J.A. Vrugt (2013), Bridging the gap between GLUE and formal statistical approaches: approximate Bayesian computation, Hydrology and Earth System Sciences, 17, pp. 4831–4850
5. Sisson, S.A., Y. Fan, and M.M. Tanaka (2007), Sequential Monte Carlo without likelihoods, Proceedings of the National Academy of Sciences of the United States of America, 104(6), pp. 1760-1765, https://doi.org/10.1073/pnas.0607208104

Version History

• 1.0
  • Initial Release
  • Built-in Case Studies
  • Basic Postprocessing
  • Python Implementation

Built-in Case Studies

1. Example 1: Toy example from Sisson et al. (2007)
2. Example 2: Linear regression example from Vrugt and Sadegh (2013)
3. Example 3: Hydrologic modeling using hydrograph functionals

Acknowledgments