Computationally Efficient Bayesian Sequential Monitoring for Functional Profiles and Density Estimation

We consider change-point detection and estimation in two different settings. The objective is to halt a process when the process generating observations deviates from a specified in control standard, in which case the process is referred to as out of control. We propose a Bayesian sequential monitoring methodology in two different settings- functional profile monitoring and density estimation. The goal is to halt a process when the process generating these observations- functional profiles or density estimates- deviates from a specified in control standard. In functional sequential process monitoring, a process is characterized by sequences of observations called profiles which are monitored over time for stability. We propose a Bayesian sequential process control (SPC) methodology which uses wavelets to monitor the functional responses and detect out of control profiles. Our contribution is to propose a solution to the growing computational cost by constructing an efficient and accurate approximation to the posterior distribution of the wavelet coefficients, without recourse to Markov chain Monte Carlo. In sequential density estimation monitoring, a process is characterized by data from which a density estimate is developed and updated when new data arrives. We propose a methodology that sequentially estimates the density of univariate data. We utilize Polya trees to again develop a Bayesian SPC methodology. Our proposed methodology monitors the distribution of observed data and detects when the generating density differs from the in-control standard. We also propose an approximation that merges the probability mass of multiple data points being a change point in the proposed methodology to curb computational complexity while maintaining accuracy in detecting a change point.