12.07.2015 Views

Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)

Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)

Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)

SHOW MORE
SHOW LESS
  • No tags were found...

Create successful ePaper yourself

Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.

76 4. Interpreting <strong>Principal</strong> <strong>Component</strong>s: Examplesπ and MR. Morgan (1981) also reports PCAs for a number of other similardata sets, in several of which the PCs provide useful interpretations.4.5 Stock Market PricesThe data in this example are the only set in this chapter that previouslyappeared in a textbook (Press, 1972, Section 9.5.2). Both the data, andthe PCs have interesting structures. The data, which were originally analysedby Feeney and Hester (1967), consist of 50 quarterly measurementsbetween 1951 and 1963 of US stock market prices for the 30 industrialstocks making up the Dow-Jones index at the end of 1961. Table 4.7 gives,in the simplified form described for Table 4.2, the coefficients of the firsttwo PCs, together with the percentage of variation accounted for by eachPC, for both covariance and correlation matrices.Looking at the PCs for the correlation matrix, the first is a ‘size’ component,similar to those discussed in Section 4.1. It reflects the fact thatall stock prices rose fairly steadily during the period 1951–63, with the exceptionof Chrysler. It accounts for roughly two-thirds of the variation inthe 30 variables. The second PC can be interpreted as a contrast between‘consumer’ and ‘producer’ stocks. ‘Consumer’ companies are those thatmainly supply goods or services directly to the consumer, such as AT&T,American Tobacco, General Foods, Proctor and Gamble, Sears, and Woolworth,whereas ‘producer’ companies sell their goods or services mainly toother companies, and include Alcoa, American Can, Anaconda, Bethlehem,Union Carbide, and United Aircraft.The PCs for the covariance matrix can be similarly interpreted, albeitwith a change of sign for the second component, but the interpretationis slightly confounded, especially for the first PC, by the different-sizedvariances for each variable.Feeney and Hester (1967) also performed a number of other PCAs usingthese and related data. In one analysis, they removed a linear trend from thestock prices before calculating PCs, and found that they had eliminated thesize (trend) PC, and that the first PC was now very similar in form to thesecond PC in the original analyses. They also calculated PCs based on ‘rateof-return’rather than price, for each stock, and again found interpretablePCs. Finally, PCs were calculated for subperiods of 12 years of data inorder to investigate the stability of the PCs, a topic that is discussed moregenerally in Section 10.3.To conclude this example, note that it is of a special type, as each variableis a time series, in which consecutive observations are not independent.Further discussion of PCA for time series data is given in Chapter 12. Apossible technique for finding PCs that are free of the trend in a vector oftime series, which is more general than the technique noted above for thepresent example, is described in Section 14.3.

Hooray! Your file is uploaded and ready to be published.

Saved successfully!

Ooh no, something went wrong!