An Investigation into Water Consumption Data Using Parametric Probability Density Functions

Abstract

Many datasets can be expressed through the usage of frequency-based histograms that provide compact visualizations of large volumes of data. A motivating hypothesis of this work is that the relative frequency of measurements represents important information for characterizing causal relationships. The basis of this work is transforming discrete histograms into continuous probability density functions (PDFs) using conservation of probability. In essence, this thesis will investigate whether conservation of probability can be applied as a governing law that characterizes how both physical and abstract measurement histograms will evolve through time. The main application within this thesis involves transforming bimonthly residential water consumption histograms into parametric PDFs for 60 sequential billing periods. Consistent parameterization for each billing period allows for regression analysis to infer a causal relationship between the PDF statistics and ambient conditions such as price and weather. This method generates partial differential equations (PDEs) for each statistic that combine to reproduce measurement data PDFs for varying ambient conditions through time using an “advection-dispersion” like relationship. The significance of this methodology is that parametric PDEs can describe the historical relationship between the measurement PDF and ambient conditions. This relationship may be exploited in future work to generate parametric PDEs that forecast the evolution of measurement PDFs through their location, scale, and shape with respect to influential ambient conditions. This thesis also demonstrates a relationship between measurement PDFs and the governing PDEs for the physical process of molecular diffusion. This outcome provides compelling evidence that conservation of probability applies to both abstract and physical systems, which suggests conservation of information is a unifying concept for modeling systemic response. Ultimately, conservation of probability provides a mechanism for reconciliation that ensures no information is either created or destroyed, while generating PDEs to reproduce measurement data as spatially- and temporally-continuous PDFs.