Preface to First Edition - lib

288 PRINCIPAL COMPONENT ANALYSISTo maximise a function of several variables subject to one or more constraints,the method of Lagrange multipliers is used. In this case this leadsto the solution that a 1 is the eigenvector of the sample covariance matrix,S, corresponding to its largest eigenvalue – full details are given in Morrison(2005).The other components are derived in similar fashion, with a j being theeigenvector of S associated with its jth largest eigenvalue. If the eigenvaluesof S are λ 1 , λ 2 , ...,λ q , then since a ⊤ j a j = 1, the variance of the jth componentis given by λ j .The total variance of the q principal components will equal the total varianceof the original variables so thatq∑λ j = s 2 1 + s 2 2 + · · · + s 2 qj=1where s 2 j is the sample variance of x j. We can write this more concisely asq∑λ j = trace(S).j=1Consequently, the jth principal component accounts for a proportion P j ofthe total variation of the original data, whereP j =λ jtrace(S) .The first m principal components, where m < q, account for a proportionm∑λ jj=1P (m) =trace(S) .When the variables are on very different scales principal component analysis isusally carried out on the correlation matrix rather than the covariance matrix.16.3 Analysis Using RTo begin it will help to score all seven events in the same direction, so that‘large’ values are ‘good’. We will recode the running events to achieve this;R> data("heptathlon", package = "HSAUR2")R> heptathlon$hurdles heptathlon$run200m heptathlon$run800m