Overview

This sample shows one way of computing Mahalanobis distance in each of the following scenarios:

• from each observation to the mean
• from each observation to a specific observation
• from each observation to all other observations (all possible pairs)

1) To compute the Mahalanobis distance from each observation to the mean, first run PROC PRINCOMP with the STD option to produce principal component scores in the OUT= data set having an identity covariance matrix. The Mahalanobis distance and Euclidean distances are equivalent for these scores. Then use a DATA step with a statement such as:

   mahalanobis_distance_to_mean = sqrt(uss(of prin:));

to complete the required distance.

2) To compute the Mahalanobis distance from each observation to a specific point, compute the principal component score for that point using the original scoring coefficients. Then compute the Euclidean distance from each observation to the reference point. One easy way to do this is to use PROC FASTCLUS treating the reference point as the SEED.

3) To compute Mahalanobis distances between all possible pairs, run PROC DISTANCE on the OUT= data set as created by PRINCOMP in the steps above. PROC DISTANCE will automatically calculate all possible pairs.

Additional Documentation

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These sample files and code examples are provided by SAS Institute Inc. "as is" without warranty of any kind, either express or implied, including but not limited to the implied warranties of merchantability and fitness for a particular purpose. Recipients acknowledge and agree that SAS Institute shall not be liable for any damages whatsoever arising out of their use of this material. In addition, SAS Institute will provide no support for the materials contained herein.

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title 'Mahalanobis Distance';
options pagno=1  dtreset nodate;

/* SAS Institute Inc.

updated:  8/29/07

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/*
    To get the Mahalanobis distance of each observation
    to the mean, first run PRINCOMP with the STD option to produce
    principal component scores in the OUT=data set with
    an identity covariance matrix.  (Hence the Mahalanobis distance and
    Euclidean distance are the same for these scores.)  Then use a DATA step
    with a statement such as :  Mah_dist=sqrt (uss(of PRIN1-PRINn)); to
    complete the required distance.

    To get the Mahalanobis distance of each observation to a specific point,
    compute the principal component score for that point
    using the original scoring coefficients.  Then compute the
    Euclidean distance from each observation to the reference point.
    One easy way to do this is to use PROC FASTCLUS treating the reference
    point as the SEED.

	To get Mahalanobis distances between all possible pairs, run PROC DISTANCE
    on the OUT= data set as created by PRINCOMP in the steps above.
*/


data points;
     drop i;
     do i=1 to 10;
    id=put(i,$9.);
    x=rannor(34343);
y=rannor(12345);
z=rannor(54321);
output;
  end;
 run;


title2 'find Mahalanobis distance from each point to the mean';
proc princomp data=points std out=out outstat=outstat noprint;
     var x y z;
 run;
data mahalanobis_to_mean;
     set out;
 mahalanobis_distance_to_mean = sqrt(uss(of prin:));
 run;
proc print data=mahalanobis_to_mean uniform noobs;
     id id;
     run;

title2 'find Mahalanobis distance from each point to a reference point';
data reference_point;
     x=-1; y=-2; z=-2.5;
 id='reference';
 run;
proc score data=reference_point score=outstat out=reference_point;
     var x y z;
 run;
proc append data=out base=reference_point;
     run;
proc fastclus data=reference_point maxc=1 replace=none maxiter=0 noprint
     out=mahalanobis_to_point(rename=(distance=mahalanobis_distance_to_point)
         drop=cluster);
     var prin:;
 run;
proc print data=mahalanobis_to_point uniform noobs;
     id id;
     run;

title2 'find Mahalanobis distances between all possible pairs of points';
proc distance data=out out=distance_matrix;
	var interval(prin:);
	id id;
	run;
proc print data=distance_matrix uniform noobs;
	run;

---

These sample files and code examples are provided by SAS Institute Inc. "as is" without warranty of any kind, either express or implied, including but not limited to the implied warranties of merchantability and fitness for a particular purpose. Recipients acknowledge and agree that SAS Institute shall not be liable for any damages whatsoever arising out of their use of this material. In addition, SAS Institute will provide no support for the materials contained herein.

Type:

Sample

Date Modified:	2008-06-16 12:59:08
Date Created:	2007-11-30 13:34:37

Operating System and Release Information

Product Family	Product	Host	SAS Release
			Starting	Ending
SAS System	SAS/STAT	z/OS	9.1 TS1M0
		Microsoft® Windows® for 64-Bit Itanium-based Systems	9.1 TS1M0
		Microsoft Windows Server 2003 Datacenter 64-bit Edition	9.1 TS1M0
		Microsoft Windows Server 2003 Enterprise 64-bit Edition	9.1 TS1M0
		Microsoft Windows 2000 Advanced Server	9.1 TS1M0
		Microsoft Windows 2000 Datacenter Server	9.1 TS1M0
		Microsoft Windows 2000 Server	9.1 TS1M0
		Microsoft Windows 2000 Professional	9.1 TS1M0
		Microsoft Windows NT Workstation	9.1 TS1M0
		Microsoft Windows Server 2003 Datacenter Edition	9.1 TS1M0
		Microsoft Windows Server 2003 Enterprise Edition	9.1 TS1M0
		Microsoft Windows Server 2003 Standard Edition	9.1 TS1M0
		Microsoft Windows XP Professional	9.1 TS1M0
		64-bit Enabled AIX	9.1 TS1M0
		64-bit Enabled HP-UX	9.1 TS1M0
		64-bit Enabled Solaris	9.1 TS1M0
		HP-UX IPF	9.1 TS1M0
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