Convert a covariance matrix to a correlation matrix in SAS


I have previously blogged about how to convert a covariance matrix into a correlation matrix in SAS (and the other way around). However, I still get questions about it, perhaps because my previous post demonstrated more than one way to accomplish each transformation.

To eliminate all confusion, the following SAS/IML program implements a function that converts a covariance matrix into the associated correlation matrix. For an explanation of why this works, see my previous article.

proc iml;
/* convert a covariance matrix, S, to a correlation matrix */
start Cov2Corr(S);
   D = sqrt(vecdiag(S));
   return( S / D` / D ); /* divide columns, then divide rows */
S = {1.0  1.0  8.1,  /* test it */
     1.0 16.0 18.0,
     8.1 18.0 81.0 };
Corr = Cov2Corr(S);
print Corr;

There are many covariance matrices that produce the same correlation matrix, therefore the inverse operation (which produces a covariance matrix from a correlation matrix) is not uniquely defined. However, if you specify the standard deviations for each variable, you can write a SAS/IML function that converts a correlation matrix to a covariance matrix:

/* R = correlation matrix
   sd = (vector of) standard deviations for each variable 
   Return covariance matrix with sd##2 on the diagonal */
start Corr2Cov(R, sd);
   std = colvec(sd);
   return( std` # R #std );
/* convert correlation matrix to covariance matrix */
sd = {1 4 9}; /* std devs = sqrt(vecdiag(S)) */
Cov = Corr2Cov(Corr, sd);
print Cov;

New in SAS/IML 12.3 software

Update: As of SAS/IML 12.3, these functions are now built into SAS/IML software. See the documentation for the CORR2COV function and the COV2CORR function.

Converting a covariance matrix to a correlation matrix with SAS/STAT software

You can, of course, use the DATA step to convert a covariance matrix to a correlation matrix (and the other way around), but here's a neat trick that you can do in SAS/STAT software: the FACTOR procedure can read a covariance matrix (technically, a TYPE=COV data set) and write the corresponding correlation matrix to a SAS data set, as shown in the following example.

/* define covariance matrix as TYPE=COV data set (lower triangular) */
data cov(type=cov);
input x1-x3;
1    .   .
1   16   .
8.1 18  81
/* convert covariance matrix to correlation matrix */
proc factor data=cov outstat=outstat noprint; 
proc print data=outstat noobs; 
where _TYPE_="CORR";

Does anyone know of a slick trick that converts a correlation matrix (and standard deviations) into a covariance matrix without using SAS/IML software? Post your solution in the comments.


About Author

Rick Wicklin

Distinguished Researcher in Computational Statistics

Rick Wicklin, PhD, is a distinguished researcher in computational statistics at SAS and is a principal developer of SAS/IML software. His areas of expertise include computational statistics, simulation, statistical graphics, and modern methods in statistical data analysis. Rick is author of the books Statistical Programming with SAS/IML Software and Simulating Data with SAS.


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  3. Love this blog!

    I have a question about singular covariance matrices! I seem to have run into one...and my understanding is that this should not be the case. I am considering a 189x189 covariance matrix produced by a major ratings agency. I tried to use this covariance matrix to produce a correlation matrix, but the covariance matrix appears to be non-invertible, or singular, or have a determinant = 0.

    Does your IML text or your simulation text address this scenario?



    • Rick Wicklin

      Yes, this can happen. It occurs when a variable (column of a matrix) is a linear combination of the other variables (columns). For example:

      x = shape(1:12, 4);
      cov = cov(x);
      d = det(cov);
      print d;

      The workaround is to identify the collinearities in the data. Alternatively, run PROC PRINCOMP or PROC VARCLUS on the data to obtain a lower dimensional set of variables that nevertheless captures most (or all) of the variance.

  4. Pingback: A simple way to construct a large correlation matrix - The DO Loop

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