偏最小二乘法PLS（matlab自带代码）

偏最小二乘法（NIPALS经典实现--未简化）
偏最小二乘法基本性质推导
偏最小二乘法(SIMPLS---未简化)

前面讲了许多PLS的性质，这里介绍一下PLS的matlab自带的PLS代码，对PLS有一个全面的认识

matlab自带的plsregress函数实现了SIMPLS算法，主要是对X0和Y0的协方差矩阵不断做正交分解

function [Xloadings,Yloadings,Xscores,Yscores, ...beta,pctVar,mse,stats] = plsregress(X,Y,ncomp,varargin)
%PLSREGRESS Partial least squares regression.....
% Center both predictors and response, and do PLS
%对X，Y进行中心化，在模型中去掉截距项
meanX = mean(X,1);
meanY = mean(Y,1);
X0 = bsxfun(@minus, X, meanX);
Y0 = bsxfun(@minus, Y, meanY);% 根据输出的需要，返回不同的参数，下面重点关注SIMPLS，
%这是Dejone在九几年提出的，matlab用这个算法来实现PLS
，可见这个算法的意义。
if nargout <= 2[Xloadings,Yloadings] = simpls(X0,Y0,ncomp);elseif nargout <= 4[Xloadings,Yloadings,Xscores,Yscores] = simpls(X0,Y0,ncomp);
....
end%------------------------------------------------------------------------------
%SIMPLS Basic SIMPLS.  Performs no error checking.
function [Xloadings,Yloadings,Xscores,Yscores,Weights] = simpls(X0,Y0,ncomp)[n,dx] = size(X0);
dy = size(Y0,2);% An orthonormal basis for the span of the X loadings, to make the successive
% deflation X0'*Y0 simple - each new basis vector can be removed from Cov
% separately.%计算协方差矩阵Cov = X0'*Y0;
for i = 1:ncomp% Find unit length ti=X0*ri and ui=Y0*ci whose covariance, ri'*X0'*Y0*ci, is% jointly maximized, subject to ti'*tj=0 for j=1:(i-1).%计算协方差的主成分方向   [ri,si,ci] = svd(Cov,'econ');ri = ri(:,1); ci = ci(:,1); si = si(1);%计算得分ti = X0*ri;normti = norm(ti); ti = ti ./ normti; % ti'*ti == 1%计算载荷Xloadings(:,i) = X0'*ti;qi = si*ci/normti; % = Y0'*tiYloadings(:,i) = qi;if nargout > 2Xscores(:,i) = ti;Yscores(:,i) = Y0*qi; % = Y0*(Y0'*ti), and proportional to Y0*ciif nargout > 4Weights(:,i) = ri ./ normti; % rescaled to make ri'*X0'*X0*ri == ti'*ti == 1endend% Update the orthonormal basis with modified Gram Schmidt (more stable),% repeated twice (ditto).vi = Xloadings(:,i);for repeat = 1:2for j = 1:i-1vj = V(:,j);vi = vi - (vj'*vi)*vj;endendvi = vi ./ norm(vi);V(:,i) = vi;% Deflate Cov, i.e. project onto the ortho-complement of the X loadings.% First remove projections along the current basis vector, then remove any% component along previous basis vectors that's crept in as noise from% previous deflations.
%更新协方差矩阵Cov = Cov - vi*(vi'*Cov);Vi = V(:,1:i);Cov = Cov - Vi*(Vi'*Cov);
endif nargout > 2% By convention, orthogonalize the Y scores w.r.t. the preceding Xscores,% i.e. XSCORES'*YSCORES will be lower triangular.  This gives, in effect, only% the "new" contribution to the Y scores for each PLS component.  It is also% consistent with the PLS-1/PLS-2 algorithms, where the Y scores are computed% as linear combinations of a successively-deflated Y0.  Use modified% Gram-Schmidt, repeated twice.for i = 1:ncompui = Yscores(:,i);for repeat = 1:2for j = 1:i-1tj = Xscores(:,j);ui = ui - (tj'*ui)*tj;endendYscores(:,i) = ui;end
end%------------------------------------------------------------------------------
%PLSCV Efficient cross-validation for X and Y mean squared error in PLS.
function mse = plscv(X,Y,ncomp,cvp,mcreps,ParOptions)[n,dx] = size(X);% Return error for as many components as asked for; some columns may be NaN
% if ncomp is too large for CV.
mse = NaN(2,ncomp+1);% The CV training sets are smaller than the full data; may not be able to fit as
% many PLS components.  Do the best we can.
if isa(cvp,'cvpartition')cvpType = 'partition';maxncomp = min(min(cvp.TrainSize)-1,dx);nTest = sum(cvp.TestSize);
elsecvpType = 'Kfold';
%    maxncomp = min(min( floor((n*(cvp-1)/cvp)-1), dx));maxncomp = min( floor((n*(cvp-1)/cvp)-1), dx);nTest = n;
end
if ncomp > maxncompwarning(message('stats:plsregress:MaxComponentsCV', maxncomp));ncomp = maxncomp;
end% Cross-validate sum of squared errors for models with 1:ncomp components,
% simultaneously.  Sum the SSEs over CV sets, and compute the mean squared
% error
CVfun = @(Xtr,Ytr,Xtst,Ytst) sseCV(Xtr,Ytr,Xtst,Ytst,ncomp);
sumsqerr = crossval(CVfun,X,Y,cvpType,cvp,'mcreps',mcreps,'options',ParOptions);
mse(:,1:ncomp+1) = reshape(sum(sumsqerr,1)/(nTest*mcreps), [2,ncomp+1]);%------------------------------------------------------------------------------
%SSECV Sum of squared errors for cross-validation
function sumsqerr = sseCV(Xtrain,Ytrain,Xtest,Ytest,ncomp)XmeanTrain = mean(Xtrain);
YmeanTrain = mean(Ytrain);
X0train = bsxfun(@minus, Xtrain, XmeanTrain);
Y0train = bsxfun(@minus, Ytrain, YmeanTrain);% Get and center the test data
X0test = bsxfun(@minus, Xtest, XmeanTrain);
Y0test = bsxfun(@minus, Ytest, YmeanTrain);% Fit the full model, models with 1:(ncomp-1) components are nested within
[Xloadings,Yloadings,~,~,Weights] = simpls(X0train,Y0train,ncomp);
XscoresTest = X0test * Weights;% Return error for as many components as the asked for.
outClass = superiorfloat(Xtrain,Ytrain);
sumsqerr = zeros(2,ncomp+1,outClass); % this will get reshaped to a row by CROSSVAL% Sum of squared errors for the null model
sumsqerr(1,1) = sum(sum(abs(X0test).^2, 2));
sumsqerr(2,1) = sum(sum(abs(Y0test).^2, 2));% Compute sum of squared errors for models with 1:ncomp components
for i = 1:ncompX0reconstructed = XscoresTest(:,1:i) * Xloadings(:,1:i)';sumsqerr(1,i+1) = sum(sum(abs(X0test - X0reconstructed).^2, 2));Y0reconstructed = XscoresTest(:,1:i) * Yloadings(:,1:i)';sumsqerr(2,i+1) = sum(sum(abs(Y0test - Y0reconstructed).^2, 2));
end

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