%%?Principal?Component?Analysis?and?Partial?Least?Squares
%?Principal?Component?Analysis?（PCA）?and?Partial?Least?Squares?（PLS）?are
%?widely?used?tools.?This?code?is?to?show?their?relationship?through?the
%?Nonlinear?Iterative?PArtial?Least?Squares?（NIPALS）?algorithm.

%%?The?Eigenvalue?and?Power?Method
%?The?NIPALS?algorithm?can?be?derived?from?the?Power?method?to?solve?the
%?eigenvalue?problem.?Let?x?be?the?eigenvector?of?a?square?matrix?A
%?corresponding?to?the?eignvalue?s:
%
%?$$Ax=sx$$
%
%?Modifying?both?sides?by?A?iteratively?leads?to
%
%?$$A^kx=s^kx$$
%
%?Now?consider?another?vectro?y?which?can?be?represented?as?a?linear
%?combination?of?all?eigenvectors:
%
%?$$y=\sum_i^n?b_ix_i=Xb$$
%
%?where
%
%?$$X=\left[x_1\\\?\cdots\\\?x_n?\right]$$
%
%?and
%
%?$$b?=?\left[b_1\\\?\cdots\\\?b_n?\right]^T$$
%
%?Modifying?y?by?A?gives
%
%?$$Ay=AXb=XSb$$
%
%?Where?S?is?a?diagnal?matrix?consisting?all?eigenvalues.?Therefore?for
%?a?large?enough?k
%
%?$$A^ky=XS^kb\approx?\alpha?x_1$$
%
%?That?is?the?iteration?will?converge?to?the?direction?of?x_1?which?is?the
%?eigenvector?corresponding?to?the?eigenvalue?with?the?maximum?module.
%?This?leads?to?the?following?Power?method?to?solve?the?eigenvalue?problem.

A=randn（105）;
%?sysmetric?matrix?to?ensure?real?eigenvalues
B=A‘*A;
%find?the?column?which?has?the?maximum?norm
[dumidx]=max（sum（A.*A））;
x=A（:idx）;
%storage?to?judge?convergence
x0=x-x;
%convergence?tolerant
tol=1e-6;
%iteration?if?not?converged
while?norm（x-x0）>tol
????%iteration?to?approach?the?eigenvector?direction
????y=A‘*x;
????%normalize?the?vector
????y=y/norm（y）;
????%save?previous?x
????x0=x;
????%x?is?a?product?of?eigenvalue?and?eigenvector
????x=A*y;
end
%?the?largest?eigen?value?corresponding?eigenvector?is?y
s=x‘*x;
%?compare?it?with?those?obtained?with?eig
[VD]=eig（B）;
[didx]=max（diag（D））;
v=V（:idx）;
disp（d-s）
%?v?and?y?may?be?different?in?signs
disp（min（norm（v-y）norm（v+y）））

%%?The?NIPALS?Algorithm?for?PCA
%?The?PCA?is?a?dimension?reduction?technique?which?is?based?on?the
%?following?decomposition:
%
%?$$X=TP^T+E$$
%
%?Where?X?is?the?data?matrix?（m?x?n）?to?be?analysed?T?is?the?so?called
%?score?matrix?（m?x?a）?P?the?loading?matrix?（n?x?a）?and?E?the?residual.
%?For?a?given?tolerance?of?residual?the?number?of?principal?components?a
%?can?be?much?smaller?than?the?orginal?variable?dimension?n.?
%?The?above?power?algorithm?can?be?extended?to?get?T?and?P?by?iteratively
%?subtracting?A?（in?this?case?X）?by?x*y‘?（in?this?case?t*p‘）?until?the
%?given?tolerance?satisfied.?This?is?the?so?called?NIPALS?algorithm.

%?The?data?matrix?with?normalization
A=randn（105）;
meanx=mean（A）;
stdx=std（A）;
X=（A-meanx（ones（101）:））./stdx（ones（101）:）;
B=X‘*X;
%?allocate?T?and?P
T=zeros（105）;
P=zeros（5）;
%?tol?for?convergence
tol=1e-6;
%?tol?for?PC?of?95?percent
tol2=（1-0.95）*5*（10-1）;
for?k=1:5
????%find?the?column?which?has?the?maxim

?屬性????????????大小?????日期????時間???名稱
-----------?---------??----------?-----??----
?????文件????????8947??2014-07-11?16:56??pls\learningpcapls.m
?????文件????????1327??2014-07-11?16:56??pls\license.txt
?????文件????????4642??2014-08-01?17:13??pls\NIPLS_for_PLS.m
?????文件????????3732??2014-07-24?16:28??pls\pls.m
?????文件????????3543??2014-07-11?16:56??pls\plsnc.m
?????文件???????23709??2014-07-11?16:56??pls\html\learningpcapls.html
?????文件????????2766??2014-07-11?16:56??pls\html\learningpcapls_01.png
?????文件????????2544??2014-07-11?16:56??pls\html\learningpcapls_eq14726.png
?????文件????????1651??2014-07-11?16:56??pls\html\learningpcapls_eq1475.png
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?????文件????????2086??2014-07-11?16:56??pls\html\learningpcapls_eq2092.png
?????文件????????4390??2014-07-11?16:56??pls\html\learningpcapls_eq29314.png
?????文件????????1293??2014-07-11?16:56??pls\html\learningpcapls_eq38356.png
?????文件????????1368??2014-07-11?16:56??pls\html\learningpcapls_eq48172.png
?????文件????????2334??2014-07-11?16:56??pls\html\learningpcapls_eq7260.png
?????文件?????????850??2014-07-11?16:56??pls\html\learningpcapls_eq955.png