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GraphPeelOpt.m
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GraphPeelOpt.m
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function [thetaOpt,outOpt] = GraphPeelOpt(x,y,varargin)
% Perfom graph peeling and optimisation to determine number of sum of
% exponentials in data. y = sum aexp(bx). Number of data points should be
% more than 4.
%
% Input arguments:
% x: exponent
% y: function value
%
% Optional input arguments. Create a structure variable with the following field:
% nExp: Set the maximum number of exponenetials to fit up to. Default
% nExp = [1:3]
% plotGP: Set plotGP to 'y' to plot graph-peeling curve. Default 'n'
%
% Outputs:
% thetaOpt: Overall optimum parameters based on the lowest cost-function.
% theta = [a1;..;an, b1;...;bn] size nExp x 2
% outOpt: A structured varaible with the following fields as optional extras
% theta: Optimised set of parameters for each nExp
% cF: Value of cost-function for each nExp
% modelOrder: Optimum number of exponentials (model
% order)
% slowestTimeConst: Slowest time constant for the optimum model
% order
%
% Copyright (C) W. D. Widanage - WMG, University of Warwick, U.K. 16/10/2016 (Calm)
% All Rights Reserved
% Software may be used freely for non-comercial purposes only
p = inputParser; % Create an input parse object to handle positional and property-value arguments
% Create variable names and assign default values after checking the value
addRequired(p,'x', @isnumeric);
addRequired(p,'y', @isnumeric);
% Optional parameters
addParameter(p,'plotGP','n')
addParameter(p,'nExp',[1:3]);
% Re-parse parObj
parse(p, x, y, varargin{:})
x = p.Results.x(:);
y = p.Results.y(:);
nData = length(x);
nExp = p.Results.nExp; % Maximum number of exponentials
loopCnt = 0; % Initialise
for nn = nExp % Loop over maximum possible number of exp segments
loopCnt = loopCnt + 1; % Loop counter for each nExp
z = y; % Make a copy of the data for graph peeling
idxZero = find(z<=0);
z(idxZero) = 1E-3;
nPts = floor(nData/nn);
for pp = nn:-1:1 % Loop over each exp segment
lnz = log(z); % Log of raw data
sIdx = nPts*(pp-1) + 1;
if pp == nn
eIdx = nData;
else
eIdx = sIdx + nPts - 1;
end
lnzSeg = lnz(sIdx:eIdx);
xSeg = x(sIdx:eIdx);
K = [xSeg,ones(length(lnzSeg),1)];
pLin = Lls(K,lnzSeg); % Slope and intercept
b(pp,1) = pLin(1); % Exponent (slope of line)
a(pp,1) = exp(pLin(2)); % Coefficient
% Peel off exponential term
z = z - a(pp)*exp(b(pp)*x);
idxZero = find(z<=0);
z(idxZero) = 1E-3;
end
% Peform parameter optimastion for given graph-peel values
fh = @SumOfExp;
optionsExp.Jacobian = 'on';
optionsExp.termMsg = 'n';
optionsExp.iterMax = 100;
[thetaOptTmp,infoTheta] = LMAlgorithm_varIdx(fh, [a;b], x, y, optionsExp);
aOpt = thetaOptTmp(1:nn);
bOpt = thetaOptTmp(nn+1:end);
% options = optimset('TolX',0.1);
% fh = @(lambda)sumExpErr(lambda,x,y);
% bOpt = fminsearch(fh,b,options);
% [sse,aOpt] = sumExpErr(bOpt,x,y);
% Store optimised and intial values
theta(loopCnt,1) = {[aOpt, bOpt]};
theta0(loopCnt,1) = {[a,b]};
cF(loopCnt,1) = infoTheta.cF_iter(end);
thetaFracErr(loopCnt,1) = {infoTheta.fracErr};
% Corrected AIC
dof = 2*nn+1;
AIC(loopCnt,1) = nData*log(infoTheta.cF_iter(end)/nData) + 2*dof + 2*dof*(dof+1)/(nData-dof-1);
% Note any poitive exponents to avoid and flag warning
if any(bOpt>0)
outOpt.msg = sprintf('Positive exponent for nExp = %d', nn);
fprintf([outOpt.msg,'\n'])
avoidExp(loopCnt,1) = true;
else
avoidExp(loopCnt,1) = false;
end
end
% Only select valid optimised values
thetaValid = theta(~avoidExp,1);
thetaFracErrValid = thetaFracErr(~avoidExp,1);
theta0Valid = theta0(~avoidExp,1);
nExpValid = nExp(~avoidExp);
cFValid = cF(~avoidExp);
AICValid = AIC(~avoidExp);
% Find optimum number of exponentials based on cost function or AIC
[~,idx] = min(AICValid);
optModelOrder = nExpValid(idx);
thetaOpt = thetaValid{idx,1};
thetaOptFracErr = thetaFracErrValid{idx,1};
thetaOpt0 = theta0Valid{idx,1};
% Largest time constant and standard deviation in optimum solution
[minExponent,idxMin] = min(-thetaOpt(:,2));
timeConstMax = 1./minExponent;
fracErrMinExponent = thetaOptFracErr(optModelOrder + idxMin);
stdMaxTimeConst = timeConstMax*fracErrMinExponent;
% stdMaxTimeCont = (-1./(minExponent+stdMinExponent)) - timeConstMax;
timeSegments = linspace(0, x(end), optModelOrder+1);
timePoints = timeSegments(2:end);
outOpt.theta = theta; % Save all optimised values for each of the nExp
outOpt.cF = cF; % Save the cost-function calue for each of the nExp
outOpt.optModelOrder = optModelOrder; % Save optimum model order
outOpt.slowestTimeConst = timeConstMax; % Save the slowest time constant
outOpt.slowestTimeConstStd = stdMaxTimeConst; % Save the slowest time constant std
outOpt.volGP = SumOfExp(thetaOpt(:),x);
outOpt.volGPOpt = SumOfExp(thetaOpt0(:),x);
outOpt.timeConstants = sort(-1./thetaOpt(:,2));
outOpt.timePoints = timePoints;
% Plot cost-fucntion value for each nExp and optimum fit for smallest cF
if strcmpi(p.Results.plotGP,'y')
figure
subplot(2,1,1)
plot(nExpValid,AICValid,'-o')
xlabel('Number of exponentials'); ylabel('AIC value')
subplot(2,1,2)
yOpt = outOpt.volGP;
yhat = outOpt.volGPOpt;
plot(x,y,'-o',x,yhat,'- x',x,yOpt,'. -');
xlabel('x');
ylabel('y');
legend('Measured','Graph peel','Optimised');
title(sprintf('Fit for nExp = %g',nExpValid(idx)))
end
end
function [y,J] = SumOfExp(theta,x)
% Sum of exponentials y = sum_i^M aexp(bx). i = 1...M.
%
% Input arguments:
% theta: [a1,...,aM,b1,...,bM], size M+1 x 1 with M the number of
% exponentials
% x: Exponent values, size N x 1
%
% Output
% y: function value size N x 1
% J: Jacobian matrix size N x M
%
% Copyright (C) W. D. Widanage - WMG, University of Warwick, U.K. 22/10/2016 (Black!)
% All Rights Reserved
% Software may be used freely for non-comercial purposes only
p = inputParser; % Create an input parse object to handle positional and property-value arguments
% Create variable names and assign default values after checking the value
addRequired(p,'theta', @isnumeric);
addRequired(p,'x', @isnumeric);
% Re-parse parObj
parse(p, theta, x)
x = p.Results.x(:);
theta = p.Results.theta(:);
nData = length(x);
nExp = length(theta)/2;
a = theta(1:nExp);
b = theta(nExp+1:end);
y = zeros(nData,1);
for ii = 1:nExp
yTmp = a(ii)*exp(b(ii)*x);
y = y + yTmp;
J1(:,ii) = exp(b(ii)*x);
J2(:,ii) = a(ii)*x.*exp(b(ii)*x);
end
J = [J1,J2];
end
% % Cost-function
% function [sse,a] = sumExpErr(b,x,y)
%
% A = zeros(length(x),length(b));
% for ii = 1:length(b)
% A(:,ii) = exp(b(ii)*x);
% end
% a = A\y;
% z = A*a;
% sse = norm(z-y)^2; % Sum of squared errors
% end
%
%
% % Sum of exp
% function yhat = sumExp(a,b,x)
% yhat = zeros(size(x));
%
% for ii = 1:length(a)
% yhatTmp = a(ii)*exp(b(ii)*x);
% yhat = yhat +yhatTmp;
% end
%
% end