function?varargout?=?pca（X?N?method）
%PCA?PRINCIPLE?COMPONENTS?ANALYSIS
%
%??Performing?principal?components?analysis?on?the?N1-by-N2?real-valued
%??data?matrix?X?where?N1?and?N2?are?the?number?of?features?（N1?variables）
%??and?observations?（N2?samples）?respectively.
%
%??P?=?pca（XNmethod）?returns?the?N-by-N1?matrix?P?of?basis?vectors?one
%??basis?vector?per?row?in?order?of?decreasing?corresponding?eigenvalues
%??i.e.?P*X?retains?the?first?N?principle?components.?If?N?is?not
%??specified?all?components?（N=N1）?are?kept.
%
%??Two?methods?are?available:?‘eig‘?（default）?and?‘svd‘?which?solve?the
%??problem?by?eigenvalue?decomposition?and?singular?value?decomposition
%??respectively.?‘svd‘?is?running?in?‘economy‘?mode.?If?there?are?more?
%??variables?than?samples?（N1>N2）?‘svd‘?is?recommended?for?this?code.
%
%??[P?D]?=?pca（XNmethod）?also?returns?all?eigenvalues?（normalized）?in?D
%??in?descending?order.
%
%??[P?D?Y]?=?pca（XNmethod）?further?returns?the?N-by-N2?matrix?Y?=?P*X
%??each?column?of?whom?is?the?projection?of?the?corresponding?observation
%??from?X?onto?the?basis?vectors?contained?in?P.
%
%?Siqing?Wu?<6sw21@queensu.ca>
%?Version:?1.1?Date:?2008-07-30

error（nargchk（13nargin））?%?check?the?number?of?arguments
error（nargoutchk（03nargout））

[nr?nc]?=?size（X）;

if?nargin<3
????method?=?‘eig‘;?%?default?method
end
if?nargin<2
????N?=?nr;?%?keep?all?components
elseif?N??nr?||?round（N）~=N
????fprintf（‘Input?N=%g?is?not?valid;?all?components?will?be?retained.\n‘?N）
????N?=?nr;
end

X?=?X-repmat（mean（X2）[1?nc]）;?%?center?data

switch?method
????case?‘eig‘
????????C?=?X*X.‘;
????????%?should?be?C/（nc-1）?for?unbiased?estimate?of?the?covariance?matix
????????%?but?won‘t?affect?P.
????????[ED]?=?eig（C）;?%?D?lists?eigenvalues?from?small?to?large
????????%?rearrange?D?and?E
????????D?=?diag（D）;
????????D?=?D（end:-1:1）/max（D）;
????????E?=?E（:end:-1:1）‘;
????????P?=?E（1:N:）;
????????
????case?‘svd‘
????????%?Instead?of?solving?X*X‘?=?E*D*E‘?solve?X?=?U*Sigma*V‘
????????%?X*X‘?=?U*Sigma*V‘*V*Sigma*U‘?=?U*Sigma^2*U‘?->?P?=?U‘;
????????[Usigma]?=?svd（X‘econ‘）;?%?“economy?size“?decomposition
????????if?N?>?nc
????????????fprintf（‘SVD?->?N=%d?is?used.\n‘?nc）
????????????N?=?nc;
????????end
????????D?=?diag（sigma）.^2;
????????D?=?D/max（D）;
????????P?=?U（:1:N）‘;
????????
????otherwise
????????error（‘Undefined?method!‘）
end
fprintf（‘Top?%d?components?are?retained;?cumulative?eigenvalue?contribution?is?%1.2f\n‘?N?sum（D（1:N））/sum（D））

switch?nargout
????case?{01}
????????varargout?=?{P};
????case?{2}
????????varargout?=?{P?D};
????case?{3}
????????Y?=?P*X;
????????varargout?=?{P?D?Y};
end

?屬性????????????大小?????日期????時間???名稱
-----------?---------??----------?-----??----
?????文件????????2748??2012-02-21?11:12??pca.m
?????文件????????1400??2012-02-21?10:57??pca2D.m
?????文件????????5211??2012-02-21?10:38??PCAexample.m
?????文件???????12236??2009-06-11?16:03??PCA_Nicolas\PCA_Nicolas.m
?????文件?????????218??2009-06-09?22:24??PCA_Nicolas\dim3_data.mat
?????文件?????????240??2009-06-09?14:05??PCA_Nicolas\dim_data.mat