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

Author

Listed:
  • Gen Sakoda

    (Department of Mathematical and Computing Sciences, School of Computing, Tokyo Institute of Technology, 4259 Nagatsuta-cho, Midori-ku, Yokohama 226-8502, Japan
    These authors contributed equally to this work.)

  • Hideki Takayasu

    (Sony Computer Science Laboratories, 3-14-13 Higashi-Gotanda, Shinagawa-ku, Tokyo 141-0022, Japan
    Institute of Innovative Research, Tokyo Institute of Technology, 4259 Nagatsuta-cho, Midori-ku, Yokohama 226-8502, Japan
    These authors contributed equally to this work.)

  • Misako Takayasu

    (Department of Mathematical and Computing Sciences, School of Computing, Tokyo Institute of Technology, 4259 Nagatsuta-cho, Midori-ku, Yokohama 226-8502, Japan
    Institute of Innovative Research, Tokyo Institute of Technology, 4259 Nagatsuta-cho, Midori-ku, Yokohama 226-8502, Japan
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Gen Sakoda & Hideki Takayasu & Misako Takayasu, 2019. "Tracking Poisson Parameter for Non-Stationary Discontinuous Time Series with Taylor’s Abnormal Fluctuation Scaling," Stats, MDPI, vol. 2(1), pages 1-15, January.
    • Handle: RePEc:gam:jstats:v:2:y:2019:i:1:p:5-69:d:200321
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    References listed on IDEAS

    as
    1. Gaku Fukunaga & Hideki Takayasu & Misako Takayasu, 2016. "Property of Fluctuations of Sales Quantities by Product Category in Convenience Stores," PLOS ONE, Public Library of Science, vol. 11(6), pages 1-19, June.
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    4. Cerqueti, Roy & Foschi, Rachele & Spizzichino, Fabio, 2009. "A spatial mixed Poisson framework for combination of excess-of-loss and proportional reinsurance contracts," Insurance: Mathematics and Economics, Elsevier, vol. 45(1), pages 59-64, August.
    5. Harvey, Andrew, 2006. "Forecasting with Unobserved Components Time Series Models," Handbook of Economic Forecasting, in: G. Elliott & C. Granger & A. Timmermann (ed.), Handbook of Economic Forecasting, edition 1, volume 1, chapter 7, pages 327-412, Elsevier.
    6. Cerqueti Roy & Ventura Marco, 2015. "Patent Valuation under Spatial Point Processes with Delayed and Decreasing Jump Intensity," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 15(2), pages 433-456, July.
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