Tracking Poisson Parameter for Non-Stationary Discontinuous Time Series with Taylor's Abnormal Fluctuation Scaling

We propose a parameter estimation method for non-stationary Poisson time series with the abnormal fluctuation scaling, known as Taylor's law. By introducing the effect of Taylor's fluctuation scaling into the State Space Model with the Particle Filter, the underlying Poisson parameter's time evolution is estimated correctly from given non-stationary time series data with abnormally large fluctuations. We also developed a discontinuity detection method which enables tracking the Poisson parameter even for time series including sudden discontinuous jumps. As an example of application of this new general method, we analyzed Point-of-Sales data in convenience stores to estimate change of probability of purchase of commodities under fluctuating number of potential customers. The effectiveness of our method for Poisson time series with non-stationarity, large discontinuities and Taylor's fluctuation scaling is verified by artificial and actual time series.