Bayesian analysis of savings from retrofit projects

Estimates of savings from retrofit projects depend on statistical models, but because of the complicated analysis required to determine the uncertainty of the estimates, savings uncertainty is not often considered. Numerous simplified methods have been proposed to determine savings uncertainty, but in all but the simplest cases, these methods provide approximate results only. The objective of this paper is to show that Bayesian inference provides a consistent framework for estimating savings and savings uncertainty in retrofit projects. We review the mathematical background of Bayesian inference and Bayesian regression, and present two examples of estimating savings and savings uncertainty in retrofit projects. The first is a simple case where both baseline and post-retrofit monthly natural gas use can be modeled as a linear junction of monthly heating degree days. The Efficiency Valuation Organization (EVO 2007) defines two methods of determining savings in such cases: reporting period savings, which is an estimate of the savings during the post-retrofit period; and normalized savings, which is an estimate of the savings that would be obtained during a typical year at the project site. For reporting period savings, classical statistical analysis provides exact analytic results for both savings and savings uncertainty in this case. We use Bayesian analysis to calculate reporting period savings and savings uncertainty and show that the results are identical to the analytical results. For normalized savings, the literature contains no exact expression for the uncertainty of normalized savings; we use Bayesian inference to calculate this quantity for the first time, and compare it with the result of an approximate formula. The second example concerns a problem where the baseline data exhibit nonlinearity and serial autocorrelation, both f which are common in real-world retrofit projects. No analytical solutions exist to determine savings or savings uncertainty in this situation, but several simplified formulas have been proposed. We model the data using a 5-parameter model with first-order autoregressive errors, and use Bayesian inference to develop distributions for the model parameters and for the reporting period savings, which allows us to determine the savings uncertainty. We find the energy savings to be about 5% lower than the result obtained by ignoring the autocorrelation. In addition, the Bayesian analysis finds the savings uncertainty to be narrower than the approximate uncertainty calculated using the simplified formula. These results show that Bayesian inference can be used to determine savings and savings uncertainty for a wide variety of real-world problems.