svd, pod, pca, etc

POD, SVD, PCA, etc Christian Himpe ([email protected])

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POD, SVD, PCA, etc

Christian Himpe ([email protected])

[email protected]

Overview

1 SVD2 POD3 PCA

SVD - Singular Value Decomposition

GivenX ∈ Rm×n, m ≤ nσi =

√λi (XXT ), σ1 ≥ σ2 · · · ≥ σm

then

X = UDV T

withU ∈ Rm×m, UUT = 1 (Left Singular Vectors)D ∈ Rm×n, Dij = σiδij (Singular Values)V ∈ Rn×n, VV T = 1 (Right Singular Vectors)

POD - Proper Orthogonal Decomposition

Given[x1, . . . , xn] = X ∈ Rm×n

R = E[XXT ] (Autocorrelation Matrix)then

Rwi = λiwiSVD⇒ X = UDV T

withU = [u1, . . . , um] = [w1, . . . ,wm] (POD Modes)

λi (R) =σ2

in , (Empirical Eigenvalues)

notealso called KLT - Karhunen Loeve Transformation

PCA - Principal Component Analysis

Given[x1, . . . , xn] = X ∈ Rm×n, 1

n∑n

i=1 xi = 0C = E[(X − E[X ])(X − E [X ])T ] (Covariance Matrix)

then

Cwi = λiwiSVD⇒ (X − E[X ]) = UDV T

withU = [u1, . . . , um] = [w1, . . . ,wm] (Principal Components)

λi (C ) =σ2

in (Weights)

notealso called EOF - Empirical Orthogonal Functions

tl;dr

POD(X )⇔ SVD(X )

PCA(X )⇔ SVD(X − E[X ])

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