Harold Hotelling

The resolution of a set of n tests o r other var ia tes into components 7~, each of which accounts for the greates t possible port ion 71, ~,~, . , of the total variance of the tests unaccounted for by the previous components, has been dealt with by the author in a previous paper (2). Such "factors ," on account of their analogy with the principal axes of a quadric, have been called principal components. The present paper describes a modification of the i terat ive scheme of calculating principal components there presented, in a fashion tha t materially accelerates convergence. The application of the i terative process is not confined to statistics, bu t may be used to obtain the magnitudes and orientations of the principal axes of a quadric or hyperquadric in a manner which will ordinari ly be f a r less laborious than those given in books on geometry. This is t rue whether the quadrics are ellipsoids or hyperboloids; the proof o f convergence given in an earl ier paper is applicable to all kinds of central quadrics. Fo r hyperboloids some of the roots k~ of the characteris t ic equation would be negative, while for ellipsoids all are positive. I f in a statistical problem some of the roots should come out negative, this would indicate ei ther an error in calculation, or that, if correlations corrected for a t tenuat ion had been used, the same type of inconsistency had crept in tha t sometimes causes such correlations to exceed unity. Another method of calculating principal components has been discovered by Professor Truman L. Kelley, which involves less labor than the original i terat ive method, a t least in the examples to which he has applied it (5). How it would compare with the present accelerated method is not clear, except tha t some experience at Columbia Univers i ty has suggested that the method here set for th is the more efficient. I t is possible tha t Kelley's method is more suitable when all the characterist ic roots are desired, but not the corresponding correlations of the variates with the components. The present method seems to the computers who have tr ied both to be superior when the components themselves, as well as their contr ibutions to the total variance, are to be specified. The advantage of the present method is enhanced when, as will of ten be the case in dealing with numerous variates, not all the characterist ic roots but only a few of the largest are required.