In PCA, based on the eigen values of the covariance matrix of the dataset, it is decided as to which column/variable is of importance. I want to know what is the real and practical use of finding the eigen values in general and how is it related in finding the principal component in PCA? Thanks to all in advance
The typical practical use is to find the direction which the data set has maximum variance. The higher is the eigenvalue, the higher will be the variance along an covariance matrix's eigenvector direction (principal component).
Dear Gustavo,
Thanks for your reply.
Here, I did not understand the importance of the direction which you are emphasizing on. Would you please elaborate on this?
The object of PCA is to reduce the number of variables to explain a scientific phenomenon. You can easily demostrate, that under certain conditions the variance covariance matrix can be decomposed into the sum of the product of the eigenvalues time the eigenvectors.
All next to zero eigenvalues contribute only a little to the explanation of the variability of the phenomenon under research; In this way you have to consider, which are the most important eigenvalues. And the software you are using can give you many ideas. For example, if the first 3 eigenvuels they explain 98% of the variability, what is the reason to include 4 or 5 out of 5, if the latest can contribute only little to explain the variability .... ?. Simply the PCA, is a technique to reduce the problem, as stated at the beginning.
Dear Rafael,
Your answer is very clear and convincing.based on to the accuracy and the reduction in the number of variables. As you mentioned, finding the eigen values help in achieving so. My doubt is why only eigen values? And if it is only this way we can do this, what property of eigen value convinces us in accepting that what ever PCA does and says is correct. Again, Mr. Gustavo was saying something about direction....what does this direction signifies?
Dear Paul,
Unfortunately, neither do I have access to these articles nor I am able to arrange it. I would be glad if you please help me in getting these articles.
I wrote small articles in Wikipedia on PCA (go to "talk" tab and see "PCA in simple explanation" or go to the following address
http://en.wikipedia.org/wiki/Talk:Principal_component_analysis
PCA is about transforming a data set to a coordinate system where the axes are orthonormal and have maximum/minimum length. They are called principal axes, thus forming an ellipsoid with major and minor axes (using a 2-D analogy). The smaller principal components (have smaller length) and thus do not contribute much to the total energy (variance) in the data. These can be removed as they don't contribute much, thus obtaining a reduced set of coordinates.
Good Jose ... That is a nive anwer from a geometrical point of view......!!! That is what this thing is about
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