Is there a difference between the Karhunen-Loève expansion and pca?

Question

Jaime De La Mota Sanchis 2021-5-11

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https://ww2.mathworks.cn/matlabcentral/answers/827625-is-there-a-difference-between-the-karhunen-loeve-expansion-and-pca

回答： Ernesto Staffetti 2021-5-11

Hello everyone: I have a vector called listOfUVValues of size 1000*50, it contains 50 realizations of a stochastic process. I need to calculate the Karhunen-Loève expansion of this process and obtain the uncorrelated random variables as explained in wikipedia. For that, I have used the pca function of Matlab as

[coeffUV, score_vectorUV, latentUV, tsquaredUV, explainedUV, muUV]=pca(listOfUVValues, 'Centered',false);

and as far as I understand, coeffUV has to be the vector containing said random variables. However, if I plot a histogram of the first column of coeffUV, I obtain what can be seen in the attached figure. The histogram of said random variables moves between intervals way smaller than the data in listOfUVValues.

Can someone please tell me if I am getting it right and apc can be used as the Karhunen-Loève expansion or if I am missunderstanding the coeff is not the equivalent to the random variable?

Any answer is appreciated.

Regards.

Jaime.