Abstract
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.
Highlights
The Poisson process is a basic stochastic process for events that occur at random in various natural and social phenomena, such as the number of decay of radioactive atoms, the occurrence of a failure of elements in devices, and daily sales amount of commodities [1]
We revealed the advantage of our method considering the term of Taylor’s fluctuation scaling to the conventional
Our model distinguishes non-stationarity of the parameter from the large fluctuation caused by this fluctuation scaling
Summary
The Poisson process is a basic stochastic process for events that occur at random in various natural and social phenomena, such as the number of decay of radioactive atoms, the occurrence of a failure of elements in devices, and daily sales amount of commodities [1]. The basic process follows a Poisson process with a constant λ, but there exists fluctuation in the population of the observing objects, that gives the standard deviation of fluctuation proportional to the mean value, λ [1,10,11,12] This fluctuation scaling is unavoidable in many cases since the total number of the observing objects is practically uncontrollable in the real system. The State Space Model (SSM) [16] is applicable for unpredictable non-stationarity with abrupt and indifferentiable changes, in this paper we generalize this method to handle non-stationary Poisson time series with Taylor’s fluctuation law.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
