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function [a,e,REV,TOC,CPUTIME,ESU] = aar(y, Mode, arg3, arg4, arg5, arg6, arg7, arg8, arg9);
% Calculates adaptive autoregressive (AAR) and adaptive autoregressive moving average estimates (AARMA)
% of real-valued data series using Kalman filter algorithm.
% [a,e,REV] = aar(y, mode, MOP, UC, a0, A, W, V);
%
% The AAR process is described as following
% y(k) - a(k,1)*y(t-1) -...- a(k,p)*y(t-p) = e(k);
% The AARMA process is described as following
% y(k) - a(k,1)*y(t-1) -...- a(k,p)*y(t-p) = e(k) + b(k,1)*e(t-1) + ... + b(k,q)*e(t-q);
%
% Input:
% y Signal (AR-Process)
% Mode is a two-element vector [aMode, vMode],
% aMode determines 1 (out of 12) methods for updating the co-variance matrix (see also [1])
% vMode determines 1 (out of 7) methods for estimating the innovation variance (see also [1])
% aMode=1, vmode=2 is the RLS algorithm as used in [2]
% aMode=-1, LMS algorithm (signal normalized)
% aMode=-2, LMS algorithm with adaptive normalization
%
% MOP model order, default [10,0]
% MOP=[p] AAR(p) model. p AR parameters
% MOP=[p,q] AARMA(p,q) model, p AR parameters and q MA coefficients
% UC Update Coefficient, default 0
% a0 Initial AAR parameters [a(0,1), a(0,2), ..., a(0,p),b(0,1),b(0,2), ..., b(0,q)]
% (row vector with p+q elements, default zeros(1,p) )
% A Initial Covariance matrix (positive definite pxp-matrix, default eye(p))
% W system noise (required for aMode==0)
% V observation noise (required for vMode==0)
%
% Output:
% a AAR(MA) estimates [a(k,1), a(k,2), ..., a(k,p),b(k,1),b(k,2), ..., b(k,q]
% e error process (Adaptively filtered process)
% REV relative error variance MSE/MSY
%
%
% Hint:
% The mean square (prediction) error of different variants is useful for determining the free parameters (Mode, MOP, UC)
%
% REFERENCE(S):
% [1] A. Schloegl (2000), The electroencephalogram and the adaptive autoregressive model: theory and applications.
% ISBN 3-8265-7640-3 Shaker Verlag, Aachen, Germany.
%
% More references can be found at
% http://www.dpmi.tu-graz.ac.at/~schloegl/publications/
%
% $Revision: 1.10 $
% $Id: aar.m,v 1.10 2005/05/25 02:52:38 pkienzle Exp $
% Copyright (C) 1998-2003 by Alois Schloegl <a.schloegl@ieee.org>
%
%
% 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.
% for compilation with the Matlab compiler mcc -V1.2 -ir
%#realonly
%#inbounds
[nc,nr]=size(y);
%if nc<nr y=y'; end; tmp=nr;nc=nr; nr=tmp;end;
if nargin<2 Mode=0; end;
% check Mode (argument2)
if prod(size(Mode))==2
aMode=Mode(1);
vMode=Mode(2);
end;
if any(aMode==(0:14)) & any(vMode==(0:7)),
fprintf(1,['a' int2str(aMode) 'e' int2str(vMode) ' ']);
else
fprintf(2,'Error AAR.M: invalid Mode argument\n');
return;
end;
% check model order (argument3)
if nargin<3 MOP=[10,0]; else MOP= arg3; end;
if length(MOP)==0 p=10; q=0; MOP=p;
elseif length(MOP)==1 p=MOP(1); q=0; MOP=p;
elseif length(MOP)>=2 p=MOP(1); q=MOP(2); MOP=p+q;
end;
if nargin<4 UC=0; else UC= arg4; end;
a0=zeros(1,MOP);
A0=eye(MOP);
if nargin>4,
if all(size(arg5)==([1,1]*(MOP+1))); % extended covariance matrix of AAR parameters
a0 = arg5(1,2:size(arg5,2));
A0 = arg5(2:size(arg5,1),2:size(arg5,2)) - a0'*a0;
else
a0 = arg5;
if nargin>5
A0 = arg6;
end;
end;
end;
if nargin<7, W = []; else W = arg7; end;
if all(size(W)==MOP),
if aMode ~= 0,
fprintf(1,'aMode should be 0, because W is given.\n');
end;
elseif isempty(W),
if aMode == 0,
fprintf(1,'aMode must be non-zero, because W is not given.\n');
end;
elseif any(size(W)~=MOP),
fprintf(1,'size of W does not fit. It must be %i x %i.\n',MOP,MOP);
return;
end;
if nargin<8, V0 = []; else V0 = arg8; end;
if all(size(V0)==nr),
if vMode ~= 0,
fprintf(1,'vMode should be 0, because V is given.\n');
end;
elseif isempty(V0),
if aMode == 0,
fprintf(1,'vMode must be non-zero, because V is not given.\n');
end;
else
fprintf(1,'size of V does not fit. It must be 1x1.\n');
return;
end;
% if nargin<7 TH=3; else TH = arg7; end;
% TH=TH*var(y);
% TH=TH*mean(detrend(y,0).^2);
MSY=mean(detrend(y,0).^2);
e=zeros(nc,1);
Q=zeros(nc,1);
V=zeros(nc,1);
T=zeros(nc,1);
%DET=zeros(nc,1);
SPUR=zeros(nc,1);
ESU=zeros(nc,1);
a=a0(ones(nc,1),:);
%a=zeros(nc,MOP);
%b=zeros(nc,q);
mu=1-UC; % Patomaeki 1995
lambda=(1-UC); % Schloegl 1996
arc=poly((1-UC*2)*[1;1]);b0=sum(arc); % Whale forgettting factor for Mode=258,(Bianci et al. 1997)
dW=UC/MOP*eye(MOP); % Schloegl
%------------------------------------------------
% First Iteration
%------------------------------------------------
Y=zeros(MOP,1);
C=zeros(MOP);
%X=zeros(q,1);
at=a0;
A=A0;
E=y(1);
e(1)=E;
if ~isempty(V0)
V(1) = V0;
else
V(1) = (1-UC) + UC*E*E;
end;
ESU(1) = 1; %Y'*A*Y;
A1=zeros(MOP);A2=A1;
tic;CPUTIME=cputime;
%------------------------------------------------
% Update Equations
%------------------------------------------------
T0=2;
for t=T0:nc,
%Y=[y(t-1); Y(1:p-1); E ; Y(p+1:MOP-1)]
if t<=p Y(1:t-1)=y(t-1:-1:1); % Autoregressive
else Y(1:p)=y(t-1:-1:t-p);
end;
if t<=q Y(p+(1:t-1))=e(t-1:-1:1); % Moving Average
else Y(p+1:MOP)=e(t-1:-1:t-q);
end;
% Prediction Error
E = y(t) - a(t-1,:)*Y;
e(t) = E;
E2=E*E;
AY=A*Y;
esu=Y'*AY;
ESU(t)=esu;
if isnan(E),
a(t,:)=a(t-1,:);
else
V(t) = V(t-1)*(1-UC)+UC*E2;
if aMode == -1, % LMS
% V(t) = V(t-1)*(1-UC)+UC*E2;
a(t,:)=a(t-1,:) + (UC/MSY)*E*Y';
elseif aMode == -2, % LMS with adaptive estimation of the variance
a(t,:)=a(t-1,:) + UC/V(t)*E*Y';
else % Kalman filtering (including RLS)
if vMode==0, %eMode==4
Q(t) = (esu + V0);
elseif vMode==1, %eMode==4
Q(t) = (esu + V(t));
elseif vMode==2, %eMode==2
Q(t) = (esu + 1);
elseif vMode==3, %eMode==3
Q(t) = (esu + lambda);
elseif vMode==4, %eMode==1
Q(t) = (esu + V(t-1));
elseif vMode==5, %eMode==6
if E2>esu
V(t)=(1-UC)*V(t-1)+UC*(E2-esu);
else
V(t)=V(t-1);
end;
Q(t) = (esu + V(t));
elseif vMode==6, %eMode==7
if E2>esu
V(t)=(1-UC)*V(t-1)+UC*(E2-esu);
else
V(t)=V(t-1);
end;
Q(t) = (esu + V(t-1));
elseif vMode==7, %eMode==8
Q(t) = esu;
end;
k = AY / Q(t); % Kalman Gain
a(t,:) = a(t-1,:) + k'*E;
if aMode==0, %AMode=0
A = A - k*AY' + W; % Schloegl et al. 2003
elseif aMode==1, %AMode=1
A = (1+UC)*(A - k*AY'); % Schloegl et al. 1997
elseif aMode==2, %AMode=11
A = A - k*AY';
A = A + sum(diag(A))*dW;
elseif aMode==3, %AMode=5
A = A - k*AY' + sum(diag(A))*dW;
elseif aMode==4, %AMode=6
A = A - k*AY' + UC*eye(MOP); % Schloegl 1998
elseif aMode==5, %AMode=2
A = A - k*AY' + UC*UC*eye(MOP);
elseif aMode==6, %AMode=2
T(t)=(1-UC)*T(t-1)+UC*(E2-Q(t))/(Y'*Y);
A=A*V(t-1)/Q(t);
if T(t)>0 A=A+T(t)*eye(MOP); end;
elseif aMode==7, %AMode=6
T(t)=(1-UC)*T(t-1)+UC*(E2-Q(t))/(Y'*Y);
A=A*V(t)/Q(t);
if T(t)>0 A=A+T(t)*eye(MOP); end;
elseif aMode==8, %AMode=5
Q_wo = (Y'*C*Y + V(t-1));
C=A-k*AY';
T(t)=(1-UC)*T(t-1)+UC*(E2-Q_wo)/(Y'*Y);
if T(t)>0 A=C+T(t)*eye(MOP); else A=C; end;
elseif aMode==9, %AMode=3
A = A - (1+UC)*k*AY';
A = A + sum(diag(A))*dW;
elseif aMode==10, %AMode=7
A = A - (1+UC)*k*AY' + sum(diag(A))*dW;
elseif aMode==11, %AMode=8
A = A - (1+UC)*k*AY' + UC*eye(MOP); % Schloegl 1998
elseif aMode==12, %AMode=4
A = A - (1+UC)*k*AY' + UC*UC*eye(MOP);
elseif aMode==13
A = A - k*AY' + UC*diag(diag(A));
elseif aMode==14
A = A - k*AY' + (UC*UC)*diag(diag(A));
end;
end;
end;
end;
%a=a(end,:);
TOC = toc;
CPUTIME = cputime - CPUTIME;
%REV = (e'*e)/(y'*y);
REV = mean(e.*e)./mean(y.*y);