A toy data set holds configurations of two ferromagnetically coupled spins,

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Linear algebra and vector spaces

Scientific Computing for Physicists, Ch Learning from physical data: regression, classification and principal components

Problem

A toy data set holds

N=16

configurations

\mathbf{x}=(s_1,s_2)

of two ferromagnetically coupled spins,

s_j=\pm1

, sampled at a temperature where aligned pairs are the more likely:

5

copies each of

(+1,+1)

and

(-1,-1)

, and

3

copies each of

(+1,-1)

and

(-1,+1)

. No configuration carries a label. ((a)) Find the mean configuration and the covariance matrix

C=X^{\mathsf{T}}X/(N-1)

of the centred data

X

. ((b)) Find the eigenvalues and unit eigenvectors of

C

, and the fraction of the total variance carried by the first principal component. ((c)) Write the first principal component

\mathbf{x}\cdot\mathbf{v}_1

in terms of the magnetisation per spin

m=(s_1+s_2)/2

, and give its value for each of the four configurations. ((d)) A second data set, drawn at a temperature so high that the coupling is irrelevant, holds

4

copies of each of the four configurations. Find its covariance matrix, and say what principal component analysis can report about its direction of greatest variance.

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