Covariance- v. correlation-matrix based PCA

Mathematics Asked by Lucozade on December 5, 2020

In principal component analysis (PCA), one can choose either the covariance matrix or the correlation matrix to find the components. These give different results because, I suspect, the eigenvectors between both matrices are not equal. (Mathematically) similar matrices have the same eigenvalues, but not necessarily the same eigenvectors. Several questions: (1) Why this difference? (2) Does PCA make sense, if you can get two different answers? (3) Which of the two methods is ‘best’? (4) Since PCA operates on standardized (not) raw data in both cases, i.e., scaled by their standard deviation, does it make sense to use the results to draw conclusions about the dominance of variation for the actual, unstandardized data?

eigenvalues eigenvectors linear algebra statistics

One Answer

The problem with not standardizing, i.e. with not scaling the variables by their standard deviation, is that if, for example, one variable is measured in centimeters and another in dollars, then changing centimeters to meters can actually change the eigenvectors, so an arbitrary choice of units can alter the results. Hence I'd use the correlation matrix.

Answered by Michael Hardy on December 5, 2020

Covariance- v. correlation-matrix based PCA

One Answer

Add your own answers!

Related Questions

Evaluate $int_0^{pi/2} frac{arctan{left(frac{2sin{x}}{2cos{x}-1}right)}sin{left(frac{x}{2}right)}}{sqrt{cos{x}}} , mathrm{d}x$

Problem with split exact sequences and free finitely generated modules

Well posedness of Burgers’ equation

degree of minimal polynomial and degree of field extension

The set of all finite subsets of $mathbb{R}_+$ is countable.

What is the probability that balls left are white?

Dual image map restricts to open sets?

How to compute $sum_{n=1}^infty{frac{n}{(2n+1)!}}$?

Prove there is no rational number r such that $2^r = 3$

Rational Roots (with Lots of Square Roots!)

Proving Graph Theory Question

How to say a variable is invertible in Macaulay2?

Given $x, y in mathbb{R}^+$ and $ x^3 + y^3 = x-y$. Prove $x^2 + 4y^2 < 1$

Is a relation that is purely reflexive also symmetric?

$3^{123} mod 100$

Totally order turing machines by halting

Show that there is a unique morphism $E_{c}: R[x] rightarrow R^{prime}$ with $E_{c}(r)=r$ for all $r in R$ and $E_{c}(x)=c$

Implications of an expected value on almost sure convergence

Density of Tensor Products

2010 USAMO #5:Prove that if $frac{1}{p}-2S_q = frac{m}{n}$ for integers $m$ and $n$, then $m – n$ is divisible by $p$.

Ask a Question