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function [MX,PE,arg3] = lattice(Y,lc,Mode);
% Estimates AR(p) model parameter with lattice algorithm (Burg 1968)
% for multiple channels.
% If you have the NaN-tools, LATTICE.M can handle missing values (NaN),
%
% [...] = lattice(y [,Pmax [,Mode]]);
%
% [AR,RC,PE] = lattice(...);
% [MX,PE] = lattice(...);
%
% INPUT:
% y signal (one per row), can contain missing values (encoded as NaN)
% Pmax max. model order (default size(y,2)-1))
% Mode 'BURG' (default) Burg algorithm
% 'GEOL' geometric lattice
%
% OUTPUT
% AR autoregressive model parameter
% RC reflection coefficients (= -PARCOR coefficients)
% PE remaining error variance
% MX transformation matrix between ARP and RC (Attention: needs O(p^2) memory)
% AR(:,K) = MX(:, K*(K-1)/2+(1:K)); = MX(:,sum(1:K-1)+(1:K));
% RC(:,K) = MX(:,cumsum(1:K)); = MX(:,(1:K).*(2:K+1)/2);
%
% All input and output parameters are organized in rows, one row
% corresponds to the parameters of one channel
%
% see also ACOVF ACORF AR2RC RC2AR DURLEV SUMSKIPNAN
%
% REFERENCE(S):
% J.P. Burg, "Maximum Entropy Spectral Analysis" Proc. 37th Meeting of the Society of Exp. Geophysiscists, Oklahoma City, OK 1967
% J.P. Burg, "Maximum Entropy Spectral Analysis" PhD-thesis, Dept. of Geophysics, Stanford University, Stanford, CA. 1975.
% P.J. Brockwell and R. A. Davis "Time Series: Theory and Methods", 2nd ed. Springer, 1991.
% S. Haykin "Adaptive Filter Theory" 3rd ed. Prentice Hall, 1996.
% M.B. Priestley "Spectral Analysis and Time Series" Academic Press, 1981.
% W.S. Wei "Time Series Analysis" Addison Wesley, 1990.
% Version 2.90
% last revision 06.04.2002
% Copyright (C) 1996-2002 by Alois Schloegl <a.schloegl@ieee.org>
%
% .changelog TSA-toolbox
% 06.04.02 LATTICE.M V2.90
% 27.02.02 LATTICE.M minor bug fix
% 08.02.02 LATTICE.M bootstrap shows that V2.83 is preferable
% 08.02.02 LATTICE.M V2.83 saved as lattice283
% 08.02.02 LATTICE.M V2.82 saved as lattice282
% 04.02.02 LATTICE.M V2.83
% normalization changed from 1 (mean) to (k-1)/k (sum)
% 08.11.01 LATTICE.M V2.75
% help improved
% 11.04.01 LATTICE.M V2.73
% 1) sum (and sumskipnan's) were replaced by mean, this has the effect of
% normalizing with actual number of elements. This seem to improve the estimates
% 2) residual tested, seem to be smaller than for estimates with AR.M
% 3) handling of NaN (i.e. Missing values) is hidden in NaN/mean
% in other words, if NaN/mean is used this algorithm can be used for data with missing values, too.
% This library is free software; you can redistribute it and/or
% modify it under the terms of the GNU Library General Public
% License as published by the Free Software Foundation; either
% Version 2 of the License, or (at your option) any later version.
%
% This library is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
% Library General Public License for more details.
%
% You should have received a copy of the GNU Library General Public
% License along with this library; if not, write to the
% Free Software Foundation, Inc., 59 Temple Place - Suite 330,
% Boston, MA 02111-1307, USA.
if nargin<3, Mode='BURG';
else Mode=upper(Mode(1:4));end;
BURG=~strcmp(Mode,'GEOL');
% Inititialization
[lr,N]=size(Y);
if nargin<2, lc=N-1; end;
F=Y;
B=Y;
[DEN,nn] = sumskipnan((Y.*Y),2);
PE = [DEN./nn,zeros(lr,lc)];
if nargout<3 % needs O(p^2) memory
MX = zeros(lr,lc*(lc+1)/2);
idx= 0;
% Durbin-Levinson Algorithm
for K=1:lc,
[TMP,nn] = sumskipnan(F(:,K+1:N).*B(:,1:N-K),2);
MX(:,idx+K) = TMP./DEN; %Burg
if K>1, %for compatibility with OCTAVE 2.0.13
MX(:,idx+(1:K-1))=MX(:,(K-2)*(K-1)/2+(1:K-1))-MX(:,(idx+K)*ones(K-1,1)).*MX(:,(K-2)*(K-1)/2+(K-1:-1:1));
end;
tmp = F(:,K+1:N) - MX(:,(idx+K)*ones(1,N-K)).*B(:,1:N-K);
B(:,1:N-K) = B(:,1:N-K) - MX(:,(idx+K)*ones(1,N-K)).*F(:,K+1:N);
F(:,K+1:N) = tmp;
[PE(:,K+1),nn] = sumskipnan([F(:,K+1:N).^2,B(:,1:N-K).^2],2);
if ~BURG,
[f,nf] = sumskipnan(F(:,K+1:N).^2,2);
[b,nb] = sumskipnan(B(:,1:N-K).^2,2);
DEN = sqrt(b.*f);
else
DEN = PE(:,K+1);
end;
idx=idx+K;
PE(:,K+1) = PE(:,K+1)./nn; % estimate of covariance
end;
else % needs O(p) memory
arp=zeros(lr,lc-1);
rc=zeros(lr,lc-1);
% Durbin-Levinson Algorithm
for K=1:lc,
[TMP,nn] = sumskipnan(F(:,K+1:N).*B(:,1:N-K),2);
arp(:,K) = TMP./DEN; %Burg
rc(:,K) = arp(:,K);
if K>1, % for compatibility with OCTAVE 2.0.13
arp(:,1:K-1) = arp(:,1:K-1) - arp(:,K*ones(K-1,1)).*arp(:,K-1:-1:1);
end;
tmp = F(:,K+1:N) - rc(:,K*ones(1,N-K)).*B(:,1:N-K);
B(:,1:N-K) = B(:,1:N-K) - rc(:,K*ones(1,N-K)).*F(:,K+1:N);
F(:,K+1:N) = tmp;
[PE(:,K+1),nn] = sumskipnan([F(:,K+1:N).^2,B(:,1:N-K).^2],2);
if ~BURG,
[f,nf] = sumskipnan(F(:,K+1:N).^2,2);
[b,nb] = sumskipnan(B(:,1:N-K).^2,2);
DEN = sqrt(b.*f);
else
DEN = PE(:,K+1);
end;
PE(:,K+1) = PE(:,K+1)./nn; % estimate of covariance
end;
% assign output arguments
arg3=PE;
PE=rc;
MX=arp;
end; %if