function?[sing_valuesUV]?=?ssa（ts?L）

%
%??Performs?decomposition?stage?of?the?Caterpillar-SSA?algorithm
%??i.e.?construction?of?trajectory?matrix?from?a?time?series?
%??and?its?Singular?Value?Decomposition
%
%??Parameters:
%????ts?--?a?time?series?（a?column）
%????L?--?window?length?（1%
%??Returns:
%????sing_values?--?array?of?M?singular?values?（square?roots?of?eigen?values）
%???????????????????M=min（rank（ts）L）
%????U?--?a?matrix?of?M?eigenvectors?of?length?L?（R^LxM）?M=min（rank（ts）L）
%????V?--?a?matrix?of?M?factorvectors?of?length?K?（R^KxM）?M=min（rank（ts）L）
%
%
%??If?you?modify?this?source?code?somehow?please?send?me?the?modified?version.
%
%
%??（c）?Theodore?Alexandrov?（autossa@gmail.com）
%??Last?modified:?8?May?2006?
%


N=length（ts）;
K=N-L+1;??%?K?must?be?>=1!
if?K<1
????errordlg（‘Wrong?L.?SSA?is?cancelled.‘?‘Error‘）;
????return
end

%form?trajectory?matrix
X=zeros（LK）;
for?i=1:L
????X（i:）=ts（i:i-1+K）‘;
end

%?prepare?to?finite?rank?time?series
M=min（?rank（X）?L?）;


%SVD
[USV]?=?svd（X）;?%?U_i?is?a?column?in?U?{V_i}?are?the?first?L?columns?of?V


%extract?sing.values?and?cut?them?to?the?rank?if?rank%it‘s?necessary?for?finite?rank?time?series
sing_values=diag（S）;
if?M????sing_values=sing_values（1:M）;
end

%cut?eigen?vectors?to?the?rank?if?rankif?M????U=U（:1:M）;
end

%extract?factor?vectors
V=V（:1:M）;

?屬性????????????大小?????日期????時間???名稱
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?????文件????????1464??2007-10-22?13:38??ssa.m