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The Common Principal Component (CPC) Approach to Functional time Series (FTS) Models

Received: 11 January 2015     Accepted: 1 February 2015     Published: 9 February 2015
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Abstract

The functional time series (FTS) models are used for analyzing, modeling and forecasting age-specific mortality rates. However, the application of these models in presence of two or more groups within similar populations needs some modification. In these cases, it is desirable for the disaggregated forecasts to be coherent with the overall forecast. The 'coherent' forecasts are the non-divergent forecasts of sub-groups within a population. Reference [1] first proposed a coherent functional model based on product and ratios of mortality rates. In this paper, we relate some of the functional time series models to the common principal components (CPC) and partial common principal components (PCPC) models introduced by [2] and provide the methods to estimate these models. We call them common functional principal component (CFPC) models and use them for coherent mortality forecasting. Here, we propose a sequential procedure based on Johansen methodology to estimate the model parameters. We use vector approach and make use of error correction models to forecast the specific time series coefficient for each sub-group.

Published in Science Journal of Applied Mathematics and Statistics (Volume 3, Issue 1)
DOI 10.11648/j.sjams.20150301.14
Page(s) 22-26
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Mortality, Forecast, Coherent Forecasts, Functional Data, Life Expectancy, Sex-Ratio, Cointegration

References
[1] Hyndman, R.J., Booth, H. and Yasmeen, F. (2013). Coherent mortality forecasting: the product-ratio method with functional time series models. Demography 50(1), 261-283.
[2] Flury, B. (1988). Common Principal Components & Related Multivariate Models, Wiley Series in Probability and Mathematical Statistics.
[3] Hyndman, R.J. and Ullah, M.S. (2007). Robust forecasting of mortality and fertility rates: a functional data approach. Computational Statistics and Data Analysis, 51, 4942-4956.
[4] Yasmeen, F., Hyndman, R.J. and Erbas, B. (2010). Forecasting age-related changes in breast cancer mortality among white and black US women: A functional data approach. Cancer Epidemiology, 34(5), 542-549.
[5] Flury, B. D., Nel, D. G. & Pienaar, I. (1995), ‘Simultaneous detection of shift in means andvariances’, Journal of the American Statistical Association 90(432), 1474–1481.
[6] Fengler, M., H¨ardle, W. & Mammen, E. (2005), ‘A dynamic semiparametric factor modelfor implied volatility string dynamics’. SFB 649 Discussion Paper No. 2005-20, SFB 649Humboldt-Univestitat zu Berlin.
[7] Fengler, M. R., H¨ardle, W. K. & Villa, C. (2003), ‘The dynamics of implied volatilities: Acommon principal component approach’, Review of Derivatives Research 6, 179–202.
[8] Trendafilov (2010). Stepwise estimation of common principal components. Computational Statistics & Data Analysis, 54(12), 3446-3457. doi:10.1016/j.csda.2010.03.010
[9] NeuenschwanderB. E and Flury, B.D (2000). Common Principal Components for Dependent Random Vectors, Journal of Multivariate Analysis, 75(2),163–183
[10] Li, N. & Lee, R. (2005), Coherent mortality forecasts for a group of populations: Anextension of the Lee-Carter method, Demography 42(3), 575–594.
[11] Hyndman, R. J., Koehler, A. B., Ord, J. K. & Snyder, R. D. (2008), Forecasting with exponentialsmoothing: the state space approach,Springer-Verlag, Berlin.URL: www.exponentialsmoothing.net
[12] Hyndman, R. & Khandakar, Y. (2008), ‘Automatic time series forecasting: the forecastpackage for R’, Journal of Statistical software 27(3) URL: http://www.jstatsoft.org/v27/i03
[13] Engle, R. and Granger, C. (1987). Co-integration and error correction: representation, estimation and testing. Econometrica, 55, 251-276.
[14] Hamilton, J.D. (1994). Time Series Analysis, Princeton University Press, Princeton, NJ.
[15] Johansen, S. (1988). Statistical analysis of cointegration vectors. Journal of Economics Dynamics and Control, 12, 231-254.
[16] Hyndman, R.J. (2008). addb: Australian Demographic Data Bank. R package version 3.222. URL: robjhyndman.com/software/addb
Cite This Article
  • APA Style

    Farah Yasmeen. (2015). The Common Principal Component (CPC) Approach to Functional time Series (FTS) Models. Science Journal of Applied Mathematics and Statistics, 3(1), 22-26. https://doi.org/10.11648/j.sjams.20150301.14

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    ACS Style

    Farah Yasmeen. The Common Principal Component (CPC) Approach to Functional time Series (FTS) Models. Sci. J. Appl. Math. Stat. 2015, 3(1), 22-26. doi: 10.11648/j.sjams.20150301.14

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    AMA Style

    Farah Yasmeen. The Common Principal Component (CPC) Approach to Functional time Series (FTS) Models. Sci J Appl Math Stat. 2015;3(1):22-26. doi: 10.11648/j.sjams.20150301.14

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  • @article{10.11648/j.sjams.20150301.14,
      author = {Farah Yasmeen},
      title = {The Common Principal Component (CPC) Approach to Functional time Series (FTS) Models},
      journal = {Science Journal of Applied Mathematics and Statistics},
      volume = {3},
      number = {1},
      pages = {22-26},
      doi = {10.11648/j.sjams.20150301.14},
      url = {https://doi.org/10.11648/j.sjams.20150301.14},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20150301.14},
      abstract = {The functional time series (FTS) models are used for analyzing, modeling and forecasting age-specific mortality rates. However, the application of these models in presence of two or more groups within similar populations needs some modification. In these cases, it is desirable for the disaggregated forecasts to be coherent with the overall forecast. The 'coherent' forecasts are the non-divergent forecasts of sub-groups within a population. Reference [1] first proposed a coherent functional model based on product and ratios of mortality rates. In this paper, we relate some of the functional time series models to the common principal components (CPC) and partial common principal components (PCPC) models introduced by [2] and provide the methods to estimate these models. We call them common functional principal component (CFPC) models and use them for coherent mortality forecasting. Here, we propose a sequential procedure based on Johansen methodology to estimate the model parameters. We use vector approach and make use of error correction models to forecast the specific time series coefficient for each sub-group.},
     year = {2015}
    }
    

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  • TY  - JOUR
    T1  - The Common Principal Component (CPC) Approach to Functional time Series (FTS) Models
    AU  - Farah Yasmeen
    Y1  - 2015/02/09
    PY  - 2015
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    DO  - 10.11648/j.sjams.20150301.14
    T2  - Science Journal of Applied Mathematics and Statistics
    JF  - Science Journal of Applied Mathematics and Statistics
    JO  - Science Journal of Applied Mathematics and Statistics
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    PB  - Science Publishing Group
    SN  - 2376-9513
    UR  - https://doi.org/10.11648/j.sjams.20150301.14
    AB  - The functional time series (FTS) models are used for analyzing, modeling and forecasting age-specific mortality rates. However, the application of these models in presence of two or more groups within similar populations needs some modification. In these cases, it is desirable for the disaggregated forecasts to be coherent with the overall forecast. The 'coherent' forecasts are the non-divergent forecasts of sub-groups within a population. Reference [1] first proposed a coherent functional model based on product and ratios of mortality rates. In this paper, we relate some of the functional time series models to the common principal components (CPC) and partial common principal components (PCPC) models introduced by [2] and provide the methods to estimate these models. We call them common functional principal component (CFPC) models and use them for coherent mortality forecasting. Here, we propose a sequential procedure based on Johansen methodology to estimate the model parameters. We use vector approach and make use of error correction models to forecast the specific time series coefficient for each sub-group.
    VL  - 3
    IS  - 1
    ER  - 

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Author Information
  • Department of Statistics, University of Karachi, Karachi, Pakistan

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