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蒙托卡罗模拟奇异谱分析

%
% MCSSA - Tools for implementing Monte Carlo testing
%         of Singular Spectrum Analysis results.
%
%-------------------------------------------------------------
% Written by Eric Breitenberger
%            Geophysical Institute
%            University of Alaska                   
%  
% Please send comments and suggestions to [e-mail]
%-------------------------------------------------------------
% Functions included in the toolkit:
%
% MCSSA1   - Tests the null hypothesis that a signal is composed
%            of pure AR(1) noise.
%
% MCSSA2   - Tests the null hypothesis that a signal is composed
%            of pure AR(1) noise + signal.
%
% MCSSADEM - A script which demonstrates the uses of the toolkit by
%            recreating many of the figures in Allen and Smith 1996.
%
% AR1NV    - Estimates the AR(1) parameters for a time series in a 
%            fairly naive way, for the AR(1) noise null hypothesis.
%
% AR1      - Estimates the AR(1) parameters for a time series in a 
%            more correct way, for the AR(1) noise null hypothesis. 
%
% AR1SIGNV - Estimates the AR(1) parameters for a time series in a fairly
%            naive way, for the signal + AR(1) noise null hypothesis.
%
% AR1SIG   - Estimates the AR(1) parameters for a time series in a more
%            sophisticated way, for the signal + AR(1) noise null hypothesis.
%
% GAMMEST  - Used by AR1.
%
% GAMMEST2 - Used by AR1SIG.
%
% AR1EOF   - Computes the theoretical EOFs of an AR(1) process. 
% 
% AR1EOF2  - Computes the theoretical EOFs for a signal contaminated
%            with AR(1) noise. 
%
% AR1NOISE - Generates multiple AR(1) noise realizations (surrogates)
%            given the AR(1) parameters.
%
% AR1PSD   - returns the theoretical spectral density of an AR(1) process.
%
% CONFLIM  - Computes the confidence limits for an ensemble of surrogates.  
%
% EOFCORR  - examine correlations between two sets of eigenvectors.
%
% EOFFREQ  - Calculates the dominant frequency associated with each
%            of the EOFs.
%
% EOFPROJ  - Projects a covariance matrix on an eigenbasis.
%
% EOFSYM   - Determines whether EOFs are symmetric, anti-symmetric, 
%            or neither.
%
% CHI2CONF - calculate approximate confidence limits for an eigenspectrum
%            using the chi-squared distribution with 3N/M d. o. f.
%
% WILHIL   - Wilson-Hilferty approximation to chi-squared. Used by CHI2CONF.
%
% MCBINOM  - cumulative binomial distribution, in a form for MCSSA.
%
% PLOTEIG  - Makes a fancy plot of an eigenspectrum with confidence limits.
%
% Also, a file called MRATEST.DAT is included, which contains the test
% data used in Allen and Smith 1995. Typing "load mratest.dat" in Matlab
% will load the data, a 200 by 3 matrix where the first column is the 
% data (signal + noise), the second column is the noise, and the third
% column is the signal.
%----------------------------------------------------------------------
% INTRODUCTION:
%
% This toolkit contains Matlab routines to enable Monte Carlo testing
% of results from Singular Spectrum Analysis (e.g. the SSA and MSSA
% toolkits. The code is designed in a modular fashion for maximum 
% flexibility and future extensibility.
% 
% The white noise hypothesis often used to establish confidence limits
% in SSA is clearly inadequate for many purposes. These routines offer
% an alternative, that of testing against randomly generated AR(1) 
% noise. The basic idea is to form a null hypothesis about the form
% of the data, generate many surrogate data which conform to the null
% hypothesis, project these surrogates on the EOFs to get surrogate
% eigenspectra, and then use these to assign confidence limits to the
% data eigenspectrum. If significant EOFs are found, a composite null-
% hypothesis (that the data consist of some specified signal with 
% superimposed AR(1) noise) may be formed and the procedure repeated 
% until the null-hypothesis cannot be rejected.
%
% Suppose, for example, that we form the null-hypothesis that a series
% is composed of AR(1) noise. After appropriately generating a few 
% thousand series of AR(1) noise and projecting them on the EOFs, we
% may find that the first EOF (possibly the trend) is significant at 
% the 95% level. We now form a new null-hypothesis: that the data 
% consist of AR(1) noise + trend. We again generate appropriate 
% surrogates, project them on the eigenbasis, and determine confidence
% limits. We might now find that the eigenvalues corresponding to the
% annual cycle in the data are significant, so we form a third 
% null-hypothesis that the data consist of noise + trend + annual cycle.
% We regenerate surrogates, etc... This procedure continues until we
% have a null hypothesis that we cannot reject.
%
% References:
%
% Allen, M. R., 1992, Interactions between the atmosphere and oceans on time-
%   scales of weeks to years, Ph. D. Thesis, University of Oxford, 202 pp.
% Allen, M. R., and L. A. Smith, 1994, Investigating the origins and 
%   significance of low-frequency modes of climate variability, Geoph. Res.
%   Letters 21(10), 883-886.
% Allen, M. R., and L. A. Smith, 1996, Monte Carlo SSA: detecting irregular
%   oscillations in the presence of coloured noise, J. Climate, in press. 
% Allen, M. R., and A. W. Robertson, 1996, Distinguishing modulated 
%   oscillations from coloured noise in multivariate datasets, submitted to
%   Climate Dynamics. 
%   
% I wish to thank Myles Allen for many patient answers to my questions
% concerning MCSSA. His input has been extremely helpful to me in developing
% this toolkit. Thanks, Myles!
%
% Version history: 
%   1/24/96 : First released.
%   2/6/96  : Minor bug fixes in MCSSA2 and AR1SIG.
%   2/26/96 : New files: CHI2CONF, WILHIL, AR1EOF2, and MCSSADEM.
%             Eigenspectrum shape testing added to MCSSA1 and MCSSA2.
%             Many minor improvements in documentation and functionality.
%   6/27/97 : Bug fixes in MCSSA2 and AR1EOF2, which occasionally gave 
%             incorrect results.
%             New files: AR1PSD, MCBINOM, PLOTEIG, EOFCORR.