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Functional maximum-likelihood estimation of ARH(p) models

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Abstract

In this paper the problem of functional filtering of an autoregressive Hilbertian (ARH) process, affected by additive Hilbertian noise, is addressed when the functional parameters defining the ARH(p) equation are unknown. The maximum-likelihood estimation of such parameters is obtained from the implementation of an expectation-maximization algorithm. Specifically, a finite-dimensional matrix approximation of the state equation is considered from its diagonalization in terms of the spectral decomposition of the functional parameters involved (Principal-Oscillation-Pattern-based diagonalization). The Expectation step and maximization step are then computed from the forward Kalman filtering followed by a backward Kalman smoothing recursion in terms of the Fourier coefficients associated with such a decomposition.

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References

  • Alter O (2000) Singular value decomposition for genome-wide expression data processing and modelling. Proc Natl Acad Sci 97:10101–10106 (Online)

    Article  CAS  Google Scholar 

  • Alter O, Golub GH (2003) Generalized singular value decomposition for comparative analysis of genome-scale expression data sets of two different organisms. Proc Natl Acad Sci 100:3351–3356 (Online)

    Article  CAS  Google Scholar 

  • Antoniadis A, Sapatinas T (2003) Wavelet methods for continuous-time prediction using Hilbert-valued autoregressive processes. J Multivar Anal 87:133–158

    Article  Google Scholar 

  • Bar-Joseph Z, Gerber G, Jaakkola T, Gifford D, Simon I (2003) Continuous representation of time-series gene expression data. J Comput Biol 10:341–356

    Article  CAS  Google Scholar 

  • Bar-Joseph Z (2004) Analyzing time series gene expression data. Bioinformatics 20:2493–503

    Article  CAS  Google Scholar 

  • Besse P, Cardot H, Stephenson DB (2000) Autoregressive forecasting of some functional climatic variations. Scand J Stat 27:673–687

    Article  Google Scholar 

  • Bosq D (1991) Nonparametric statistics for stochastic processes, estimation and prediction. Lectures Notes in Statistics, vol 110. Springer, New York

  • Bosq D (2000) Linear processes in function spaces. Springer, New York

    Google Scholar 

  • Bosq D (2008) A note on asymptotic parametric prediction. J Stat Plann Infer. doi:10.1016/j.jspi.2008.07.018

  • Bosq D, Blanke D (2007) Inference and prediction in large dimensions, Wiley Series in Probability and Statistics. Wiley, New York

  • Cardot H (1997) Contribution á l’estimation et á la prévision statistique de doneées fonctionnelles. Ph.D. thesis, University of Toulouse 3, France

  • Cavanaugh JE (1997) Unifying the derivation for the Akaike and corrected Akaike criteria. Stat Prob Lett 33:202–208

    Article  Google Scholar 

  • Christakos G (1985) Recursive parameter estimation with applications in Earth sciences. Math Geol 17:489–515

    Article  Google Scholar 

  • Christakos G (1988) On-line estimation of nonlinear physical systems. Math Geol 20:111–133

    Article  Google Scholar 

  • Christakos G (2005) Random field models in earth sciences. Academic Press, San Diego

    Google Scholar 

  • Christakos G, Hristopulos DT (1998) Spatiotemporal environmental health modelling. Kluwer, Boston

    Google Scholar 

  • Damon J, Guillas S (2005) Estimation and simulation of autoregressive hilbertian processes with exogenous variables. Stat Infer Stoch Proc 8:185–204

    Article  Google Scholar 

  • Dautray R, Lions JL (1992) Mathematical analysis and numerical methods for science and technology 3. Spectral theory and applications. Springer, Berlin

    Google Scholar 

  • Dunford N, Schwartz JT (1971) Linear operators, part iii, spectral operators. Wiley, New York Interscience

    Google Scholar 

  • Ferraty F, Vieu P (2006) Nonparameric functional data analysis, Springer series in statistics. Springer, New York

    Google Scholar 

  • Germain F, Doisy A, Ronot X, Tracqui P (1999) Characterization of cell deformation and migration using a parametric estimation of image motion. IEEE Trans Biomed Eng 46:584–600

    Article  CAS  Google Scholar 

  • Goia A (2003) Selection model in functional linear regression models for scalar response. In: Ferligoj A, Mrvar A (eds) Developments in Applied Statistics, Metodoloski zvezki, vol 19, FDV, Ljubljana

  • Guillas S (2001) Rates of convergence of autocorrelation estimates for autoregressive Hilbertian processes. Stat Prob Lett 55:281–291

    Article  Google Scholar 

  • Hall P, Poskitt DS, Presnell B (2001) A functional data-analytic approach to signal discrimination. Technometrics 43:1–9

    Article  Google Scholar 

  • Haoudi A, Bensmail H (2006) Bioinformatics and data mining in proteomics. Expert Rev Proteomics 3:333–343

    Article  CAS  Google Scholar 

  • Kato T (1995) Perturbation theory of linear operators. Springer, New York

    Google Scholar 

  • Klevezc RR, Murray DB (2001) Genome wide oscillations in expression: wavelet analysis of time series data from yest expression arrays uncovers the dynamic architecture of phenotype. Mol Biol Rep 28:73–82

    Article  Google Scholar 

  • Leng X, Müller H-G (2006) Classification using functional data analysis for temporal gene expression data. Bioinformatics 22:68–76

    Article  CAS  Google Scholar 

  • Mas A (1999) Normalité asymptotique de l’estimateur empirique de l’opérateur d’autocorrélation d’un processus ARH(1). C R Acad Sci Paris, 329 Série I, pp 899–902

  • Mas A (2007) Weak convergence in the functional autoregressive model. J Multivar Anal 98:1231–1261

    Article  Google Scholar 

  • Matheron G, Traité de géoestatistique apliquée, tome I, mémoires du Bureau de Recherches Géologiques et Miniéres, Editions Bureau de Recherches Geologiques et Miniéres, Paris, 24, 1962

  • Merlévede F (1996) Processus linéaires Hilbertiens: inversibilité, theorémes limites, estimation et prévision, Ph.D. Thesis, University of París 6, France

  • Monk NAM (2003) Unravelling nature’s networks. Biochem Soc Trans 31:1457–1561

    Article  CAS  Google Scholar 

  • Mourid T (1995) Contribution á la statistique des processus autiregréssifs á temps continu. Ph.D. thesis, University of Paris 6, France

  • Mourid T, Bensmain N (2006) Sieves estimator of the operator of a functional autoregressive process. Stat Prob Lett 76:93–108

    Article  Google Scholar 

  • Müller HG (2005) Functional modelling and classification of longitudinal data. Scand J Statist 32:223–240

    Article  Google Scholar 

  • Müller HG, Stadmüller U (2005) Generalized functional models. Ann Stat 33:774–885

    Article  Google Scholar 

  • Pumo B (1992) Estimation et prévision de processus autirégressifs fonctionnels. Application aux processus á temps continu. Ph.D. thesis, University of París 6, France

  • Ramsay JO, Silverman BW (2005) Functional data analysis, Springer series in statistics. Springer, New York

  • Raychadhuri S, Stuart JM, Altman RB (2000) Principal component analysis to summarize microarray experiments: application to sporulation time series. Pacif Sympos Bicomp 5:452–63

    Google Scholar 

  • Ruiz-Medina MD, Salmerón R, Angulo JM (2007) Kalman filtering from POP-based diagonalization of ARH(1). Comput Stat Data Anal 51:4994–5008

    Article  Google Scholar 

  • Salmerón R, Ruiz-Medina MD (2009a) Multispectral decomposition of FAR(p) models. Stoch Env Res Risk Assess 23:289–297

    Article  Google Scholar 

  • Salmerón R, Ruiz-Medina MD (2009b) Functional SEM algorithm in ARH(1) models. XXXI Congreso Nacional de Estadística e Investigación Operativa, Spain

    Google Scholar 

  • Schwartz G (1978) Estimating the dimension of a model. Ann Stat 6:461–464

    Article  Google Scholar 

  • Shibata R (1981) An optimal selection of regression variables. Biometrika 68:45–54

    Article  Google Scholar 

  • Song JJ, Lee H-J, Morris JS, Kangd S (2007) Clustering of time-course gene expression data using fuctional data analysis. Comput Biol Chem 31:265–274

    Article  CAS  Google Scholar 

  • Vieu P (1995) Order choice in nonlinear autoregressive models. Statistics 26:307–328

    Article  Google Scholar 

  • Yao F, Müller HG, Clifford AJ, Dueker SR, Lin Follett J, Buchholz BAY, Vogel JS (2003) Shrinkage estimation for functional principal component scores with application to the population kinetics of plasma folate. Biometrics 59:676–685

    Article  Google Scholar 

  • Yao F, Müller HG, Wang JL (2005) Functional data analysis for sparse longitudinal data. J Am Stat Assoc100:577–590

    Article  CAS  Google Scholar 

Download references

Acknowledgments

This work has been supported in part by projects MTM2008-03903, MTM2005-08597 of the DGI, MEC, and P05-FQM-00990, P06-FQM-02271 of the Andalousian CICE, Spain.

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Authors and Affiliations

  1. Departamento de Estadística e Investigación Operativa, Universidad de Granada, Campus de Fuente Nueva s/n, 18071, Granada, Spain

    M. D. Ruiz-Medina

  2. Departamento de Métodos Cuantitativos para la Economía y la Empresa, Universidad de Granada, Campus Universitario de Cartuja s/n, 18011, Granada, Spain

    R. Salmerón

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  1. M. D. Ruiz-Medina
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Correspondence to M. D. Ruiz-Medina.

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Ruiz-Medina, M.D., Salmerón, R. Functional maximum-likelihood estimation of ARH(p) models. Stoch Environ Res Risk Assess 24, 131–146 (2010). https://doi.org/10.1007/s00477-009-0306-2

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