The PRINCOMP Procedure

Getting Started

The following example uses the PRINCOMP procedure to analyze mean daily temperatures in selected cities in January and July. Both the raw data and the principal components are plotted to illustrate how principal components are orthogonal rotations of the original variables.

The following statements create the Temperature data set:

   data Temperature;
      title 'Mean Temperature in January and July for 
             Selected Cities';
      input City $1-15 January July;
      datalines;
   Mobile          51.2 81.6  
   Phoenix         51.2 91.2
   Little Rock     39.5 81.4  
   Sacramento      45.1 75.2
   Denver          29.9 73.0  
   Hartford        24.8 72.7
   Wilmington      32.0 75.8  
   Washington DC   35.6 78.7
   Jacksonville    54.6 81.0  
   Miami           67.2 82.3
   Atlanta         42.4 78.0  
   Boise           29.0 74.5
   Chicago         22.9 71.9  
   Peoria          23.8 75.1
   Indianapolis    27.9 75.0  
   Des Moines      19.4 75.1
   Wichita         31.3 80.7  
   Louisville      33.3 76.9
   New Orleans     52.9 81.9  
   Portland, ME    21.5 68.0
   Baltimore       33.4 76.6  
   Boston          29.2 73.3
   Detroit         25.5 73.3  
   Sault Ste Marie 14.2 63.8
   Duluth           8.5 65.6  
   Minneapolis     12.2 71.9
   Jackson         47.1 81.7  
   Kansas City     27.8 78.8
   St Louis        31.3 78.6  
   Great Falls     20.5 69.3
   Omaha           22.6 77.2  
   Reno            31.9 69.3
   Concord         20.6 69.7  
   Atlantic City   32.7 75.1
   Albuquerque     35.2 78.7  
   Albany          21.5 72.0
   Buffalo         23.7 70.1  
   New York        32.2 76.6
   Charlotte       42.1 78.5  
   Raleigh         40.5 77.5
   Bismarck         8.2 70.8  
   Cincinnati      31.1 75.6
   Cleveland       26.9 71.4  
   Columbus        28.4 73.6
   Oklahoma City   36.8 81.5  
   Portland, OR    38.1 67.1
   Philadelphia    32.3 76.8  
   Pittsburgh      28.1 71.9
   Providence      28.4 72.1  
   Columbia        45.4 81.2
   Sioux Falls     14.2 73.3  
   Memphis         40.5 79.6
   Nashville       38.3 79.6  
   Dallas          44.8 84.8
   El Paso         43.6 82.3  
   Houston         52.1 83.3
   Salt Lake City  28.0 76.7  
   Burlington      16.8 69.8
   Norfolk         40.5 78.3  
   Richmond        37.5 77.9
   Spokane         25.4 69.7  
   Charleston, WV  34.5 75.0
   Milwaukee       19.4 69.9  
   Cheyenne        26.6 69.1
   ;

The following statements plot the temperature data set. For information on the %PLOTIT macro, see Appendix B, "Using the %PLOTIT Macro."

   title2 'Plot of Raw Data';
   %plotit(data=Temperature, labelvar=City, 
           plotvars=July January, color=black, colors=blue);
   run;

The results are displayed in Figure 52.1, which shows a scatter diagram of the 64 pairs of data points with July temperatures plotted against January temperatures.

Figure 52.1: Plot of Raw Data

The following statement requests a principal component analysis on the Temperature data set and outputs the scores to the Prin data set (OUT= Prin):

   proc princomp data=Temperature cov out=Prin;
   title2;
   var July January;
   run;

Figure 52.2 displays the PROC PRINCOMP output, beginning with simple statistics. The standard deviation of January (11.712) is higher than the standard deviation of July (5.128). The COV option in the PROC PRINCOMP statement requests the principal components to be computed from the covariance matrix. The total variance is 163.474. The first principal component explains about 94 percent of the total variance, and the second principal component explains only about 6 percent. Note that the eigenvalues sum to the total variance.

From the Eigenvectors matrix, you can represent the first principal component Prin1 as a linear combination of the original variables

${{\hv Prin1}} = 0.3435 x ({{\hv July}} -\overline{{{\hv July}}}) + 0.9391 x ({{\hv January}} -\overline{{{\hv January}}})$

and, similarly, the second principal component Prin2 as

${{\hv Prin2}} = 0.9391 x ({{\hv July}} - \overline{{{\hv July}}}) - 0.3435 x ({{\hv January}} - \overline{{{\hv January}}})$

where $\overline{{{\hv July}}}$ and $\overline{{{\hv January}}}$ are the means of July temperatures and January temperatures, respectively. Note that January receives a higher loading on Prin1 because it has a higher standard deviation than July, and the PRINCOMP procedure calculates the scores using the centered variables rather than the standardized variables.

Mean Temperature in January and July for Selected Cities

The PRINCOMP Procedure

Observations	64
Variables	2

Simple Statistics
	July	January
Mean	75.60781250	32.09531250
StD	5.12761910	11.71243309

Covariance Matrix
	July	January
July	26.2924777	46.8282912
January	46.8282912	137.1810888

Total Variance	163.47356647

Eigenvalues of the Covariance Matrix
	Eigenvalue	Difference	Proportion	Cumulative
1	154.310607	145.147647	0.9439	0.9439
2	9.162960		0.0561	1.0000

Eigenvectors
	Prin1	Prin2
July	0.343532	0.939141
January	0.939141	-.343532

Figure 52.2: Results of Principal Component Analysis

The following statement plots the Prin data set created from the previous PROC PRINCOMP statement:

   title2 'Plot of Principal Components';
   %plotit(data=Prin, labelvar=City, 
          plotvars=Prin2 Prin1, color=black, colors=blue);
   run;

Figure 52.3 displays a plot of the second principal component Prin2 against the first principal component Prin1. It is clear from this plot that the principal components are orthogonal rotations of the original variables and that the first principal component has a larger variance than the second principal component. In fact, Prin1 has a larger variance than either of the original variables July and January.

Figure 52.3: Plot of Principal Components

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