Principal Component Analysis Action Set

This section contains PROC CAS code.

Note: Input data must be accessible in your CAS session, either as a CAS table or as a transient-scope table. A CAS table has a two-level name: the first level is your CAS engine libref, and the second level is the table name. You refer to this table in the CAS procedure by specifying only the second level. For more information about two-level names, see Chapter 2, Shared Concepts. A transient-scope table is called directly from the action and exists in memory for the duration of the action. For more information about accessing data, see SAS Viya: System Programming Guide. For more information about PROC CAS and programming in CASL, see SAS Cloud Analytic Services: CASL Programmer’s Guide and SAS Cloud Analytic Services: CASL Reference.

The following DATA step creates the mycas.Crime data table—which provides crime rates per 100,000 people in seven categories for each of the 50 US states in 1977—in your CAS session. This DATA step assumes that your CAS engine libref is named mycas, but you can substitute any appropriately defined CAS engine libref.

data mycas.Crime;
   input State $1-15 Murder Rape Robbery Assault
         Burglary Larceny Auto_Theft;
   datalines;
Alabama        14.2 25.2  96.8 278.3 1135.5 1881.9 280.7
Alaska         10.8 51.6  96.8 284.0 1331.7 3369.8 753.3
Arizona         9.5 34.2 138.2 312.3 2346.1 4467.4 439.5
Arkansas        8.8 27.6  83.2 203.4  972.6 1862.1 183.4
California     11.5 49.4 287.0 358.0 2139.4 3499.8 663.5
Colorado        6.3 42.0 170.7 292.9 1935.2 3903.2 477.1
Connecticut     4.2 16.8 129.5 131.8 1346.0 2620.7 593.2
Delaware        6.0 24.9 157.0 194.2 1682.6 3678.4 467.0
Florida        10.2 39.6 187.9 449.1 1859.9 3840.5 351.4
Georgia        11.7 31.1 140.5 256.5 1351.1 2170.2 297.9
Hawaii          7.2 25.5 128.0  64.1 1911.5 3920.4 489.4
Idaho           5.5 19.4  39.6 172.5 1050.8 2599.6 237.6
Illinois        9.9 21.8 211.3 209.0 1085.0 2828.5 528.6
Indiana         7.4 26.5 123.2 153.5 1086.2 2498.7 377.4
Iowa            2.3 10.6  41.2  89.8  812.5 2685.1 219.9
Kansas          6.6 22.0 100.7 180.5 1270.4 2739.3 244.3
Kentucky       10.1 19.1  81.1 123.3  872.2 1662.1 245.4
Louisiana      15.5 30.9 142.9 335.5 1165.5 2469.9 337.7
Maine           2.4 13.5  38.7 170.0 1253.1 2350.7 246.9
Maryland        8.0 34.8 292.1 358.9 1400.0 3177.7 428.5
Massachusetts   3.1 20.8 169.1 231.6 1532.2 2311.3 1140.1
Michigan        9.3 38.9 261.9 274.6 1522.7 3159.0 545.5
Minnesota       2.7 19.5  85.9  85.8 1134.7 2559.3 343.1
Mississippi    14.3 19.6  65.7 189.1  915.6 1239.9 144.4
Missouri        9.6 28.3 189.0 233.5 1318.3 2424.2 378.4
Montana         5.4 16.7  39.2 156.8  804.9 2773.2 309.2
Nebraska        3.9 18.1  64.7 112.7  760.0 2316.1 249.1
Nevada         15.8 49.1 323.1 355.0 2453.1 4212.6 559.2
New Hampshire   3.2 10.7  23.2  76.0 1041.7 2343.9 293.4
New Jersey      5.6 21.0 180.4 185.1 1435.8 2774.5 511.5
New Mexico      8.8 39.1 109.6 343.4 1418.7 3008.6 259.5
New York       10.7 29.4 472.6 319.1 1728.0 2782.0 745.8
North Carolina 10.6 17.0  61.3 318.3 1154.1 2037.8 192.1
North Dakota    0.9  9.0  13.3  43.8  446.1 1843.0 144.7
Ohio            7.8 27.3 190.5 181.1 1216.0 2696.8 400.4
Oklahoma        8.6 29.2  73.8 205.0 1288.2 2228.1 326.8
Oregon          4.9 39.9 124.1 286.9 1636.4 3506.1 388.9
Pennsylvania    5.6 19.0 130.3 128.0  877.5 1624.1 333.2
Rhode Island    3.6 10.5  86.5 201.0 1489.5 2844.1 791.4
South Carolina 11.9 33.0 105.9 485.3 1613.6 2342.4 245.1
South Dakota    2.0 13.5  17.9 155.7  570.5 1704.4 147.5
Tennessee      10.1 29.7 145.8 203.9 1259.7 1776.5 314.0
Texas          13.3 33.8 152.4 208.2 1603.1 2988.7 397.6
Utah            3.5 20.3  68.8 147.3 1171.6 3004.6 334.5
Vermont         1.4 15.9  30.8 101.2 1348.2 2201.0 265.2
Virginia        9.0 23.3  92.1 165.7  986.2 2521.2 226.7
Washington      4.3 39.6 106.2 224.8 1605.6 3386.9 360.3
West Virginia   6.0 13.2  42.2    .  597.4 1341.7 163.3
Wisconsin       2.8 12.9  52.2  63.7  846.9 2614.2 220.7
Wyoming          .  21.9  39.7 173.9  811.6 2772.2 282.0
;

The following PROC CAS statements use the eig action in the pca action set to request a principal component analysis of the data. The ODS TRACE command displays the full paths of the output tables in the SAS log.

ods trace on;
proc cas;
   action pca.eig /
      table='Crime';
run;

The default output from this analysis is presented in Output 17.1.1 through Output 17.1.4.

Output 17.1.1 displays the "Model Information," "Number of Variables," "Number of Observations," and "Simple Statistics" tables.

The "Model Information" table identifies the data source and shows that the eigenvalue decomposition method is used to extract principal components.

The "Number of Variables" table indicates that there are seven variables to be analyzed and seven principal components to be computed. By default, if you omit the inputs parameter, all numeric variables defined in the table parameter, except the ones listed in the groupBy subparameter in the table parameter and other parameters, are used in the analysis.

The "Number of Observations" table shows that of the 50 observations in the input data, only 48 observations are used in the analysis, because some observations have incomplete data.

The "Simple Statistics" table displays the mean and standard deviation of the analysis variables.

Output 17.1.2 displays the "Correlation Matrix" table. By default, the eig action requests that principal components be computed from the correlation matrix, so the total variance is equal to the number of variables, 7.

Output 17.1.3 displays the "Eigenvalues" table. The first principal component accounts for about 57.8% of the total variance, the second principal component accounts for about 18.1%, and the third principal component accounts for about 10.7%. Note that the eigenvalues sum to the total variance.

The eigenvalues indicate that two or three components provide a good summary of the data: two components account for 76% of the total variance, and three components account for 87%. Subsequent components account for less than 5% each.

Output 17.1.4 displays the "Eigenvectors" table. From the eigenvectors matrix, you can represent the first principal component, Prin1, as a linear combination of the original variables:

Similarly, the second principal component, Prin2, is

where the variables are standardized.

The first component is a measure of the overall crime rate, because the first eigenvector shows approximately equal loadings on all variables. The second eigenvector has high negative loadings on the variables Auto_Theft and Larceny and high positive loadings on the variables Murder and Assault. There is also a small negative loading on the variable Burglary and a small positive loading on the variable Rape. This component seems to measure the preponderance of property crime compared to violent crime. The interpretation of the third component is not obvious.

s:loadtable{casLib="casuser", path="Crime.csv"}

s:loadactionset{actionset='pca'}
r=s:eig{
   table='Crime'}

print(r.ModelInfo)
print(r.NVars)
print(r.NObs)
print(r.SimpleStatistics)
print(r.Corr)
print(r.Eigenvalues)
print(r.Eigenvectors)
print(r.Timing)

r

s.upload_file('Crime.csv')

s.loadactionset(actionset='pca')
r=s.eig(
   table='Crime')

print(r.ModelInfo)
print(r.NVars)
print(r.NObs)
print(r.SimpleStatistics)
print(r.Corr)
print(r.Eigenvalues)
print(r.Eigenvectors)
print(r.Timing)

print(r)