title: Time-dependent ARMA modeling of genomic sequences.
abstract: BACKGROUND
  Over the past decade, many investigators have used sophisticated time series tools for the analysis of genomic sequences. Specifically, the correlation of the nucleotide chain has been studied by examining the properties of the power spectrum. The main limitation of the power spectrum is that it is restricted to stationary time series. However, it has been observed over the past decade that genomic sequences exhibit non-stationary statistical behavior. Standard statistical tests have been used to verify that the genomic sequences are indeed not stationary. More recent analysis of genomic data has relied on time-varying power spectral methods to capture the statistical characteristics of genomic sequences. Techniques such as the evolutionary spectrum and evolutionary periodogram have been successful in extracting the time-varying correlation structure. The main difficulty in using time-varying spectral methods is that they are extremely unstable. Large deviations in the correlation structure results from very minor perturbations in the genomic data and experimental procedure. A fundamental new approach is needed in order to provide a stable platform for the non-stationary statistical analysis of genomic sequences.
  
  
  RESULTS
  In this paper, we propose to model non-stationary genomic sequences by a time-dependent autoregressive moving average (TD-ARMA) process. The model is based on a classical ARMA process whose coefficients are allowed to vary with time. A series expansion of the time-varying coefficients is used to form a generalized Yule-Walker-type system of equations. A recursive least-squares algorithm is subsequently used to estimate the time-dependent coefficients of the model. The non-stationary parameters estimated are used as a basis for statistical inference and biophysical interpretation of genomic data. In particular, we rely on the TD-ARMA model of genomic sequences to investigate the statistical properties and differentiate between coding and non-coding regions in the nucleotide chain. Specifically, we define a quantitative measure of randomness to assess how far a process deviates from white noise. Our simulation results on various gene sequences show that both the coding and non-coding regions are non-random. However, coding sequences are "whiter" than non-coding sequences as attested by a higher index of randomness.
  
  
  CONCLUSIONS
  We demonstrate that the proposed TD-ARMA model can be used to provide a stable time series tool for the analysis of non-stationary genomic sequences. The estimated time-varying coefficients are used to define an index of randomness, in order to assess the statistical correlations in coding and non-coding DNA sequences. It turns out that the statistical differences between coding and non-coding sequences are more subtle than previously thought using stationary analysis tools: Both coding and non-coding sequences exhibit statistical correlations, with the coding regions being "whiter" than the non-coding regions. These results corroborate the evolutionary periodogram analysis of genomic sequences and revoke the stationary analysis' conclusion that coding DNA behaves like random sequences.
authors.affiliation: Department of Systems Engineering, University of Arkansas at Little Rock, Little Rock, AR, USA. jszielinski@ualr.edu
authors.forename: Jerzy S
authors.initials: JS
authors.lastname: Zielinski
authors.forename: Nidhal
authors.initials: N
authors.lastname: Bouaynaya
authors.forename: Dan
authors.initials: D
authors.lastname: Schonfeld
authors.forename: William
authors.initials: W
authors.lastname: O'Neill
mesh_terms: D000465:Algorithms; D001483:Base Sequence; D002874:Chromosome Mapping / Q000379:methods*; D003198:Computer Simulation; D008957:Models, Genetic; D015233:Models, Statistical; D008969:Molecular Sequence Data; D012044:Regression Analysis; D017422:Sequence Analysis, DNA / Q000379:methods*
references.citation: Nature. 1992 Oct 29;359(6398):782
references.pmid: 1436053
references.citation: Phys Rev A. 1992 Jun 15;45(12):8902-8913
references.pmid: 9906993
references.citation: Phys Rev Lett. 1992 Jun 22;68(25):3805-3808
references.pmid: 10045801
references.citation: Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 1995 May;51(5):5084-91
references.pmid: 9963221
references.citation: Phys Rev A. 1991 May 15;43(10):5240-5260
references.pmid: 9904836
references.citation: Nature. 1993 Jan 21;361(6409):212-3
references.pmid: 8423849
references.citation: Phys Rev E Stat Nonlin Soft Matter Phys. 2005 Apr;71(4 Pt 1):041910
references.pmid: 15903704
references.citation: Nature. 1992 Mar 12;356(6365):168-70
references.pmid: 1301010
references.citation: Phys Rev E Stat Nonlin Soft Matter Phys. 2007 Mar;75(3 Pt 1):032903
references.pmid: 17500745
references.citation: J Theor Biol. 1996 Dec 7;183(3):341-3
references.pmid: 9015453
references.citation: Proc Natl Acad Sci U S A. 1994 Dec 20;91(26):12972-5
references.pmid: 7809157
references.citation: Physica A. 1999;273(1-2):1-18
references.pmid: 11542924