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Biplots are two-dimensional plots that help to visualize relationships in high dimensional data. A previous article discusses how to interpret biplots for continuous variables. The biplot projects observations and variables onto the span of the first two principal components. The observations are plotted as markers; the variables are plotted as vectors. The observations and/or vectors are not usually on the same scale, so they need to be rescaled so that they fit on the same plot. There are four common scalings (GH, COV, JK, and SYM), which are discussed in the previous article.

This article shows how to create biplots in SAS. In particular, the goal is to create the biplots by using modern ODS statistical graphics. You can obtain biplots that use the traditional SAS/GRAPH system by using the %BIPLOT macro by Michael Friendly. The %BIPLOT macro is very powerful and flexible; it is discussed later in this article.

There are four ways to create biplots in SAS by using ODS statistical graphics:

• You can use PROC PRINQUAL in SAS/STAT software to create the COV biplot.
• If you have a license for SAS/GRAPH software (and SAS/IML software), you can use Friendly's %BIPLOT macro and use the OUT= option in the macro to save the coordinates of the markers and vectors. You can then use PROC SGPLOT to create a modern version of Friendly's biplot.
• You can use the matrix computations in SAS/IML to "manually" compute the coordinates of the markers and vectors. (These same computations are performed internally by the %BIPLOT macro.) You can use the Biplot module to create a biplot, or you can use the WriteBiplot module to create a SAS data set that contains the biplot coordinates. You can then use PROC SGPLOT to create the biplot.

For consistency with the previous article, all methods in this article standardize the input variables to have mean zero and unit variance (use the SCALE=STD option in the %BIPLOT macro). All biplots show projections of the same four-dimensional Fisher's iris data. The following DATA step assigns a blank label. If you do not supply an ID variable, some biplots display observations numbers.

data iris;
   set Sashelp.iris;
   id = " ";        /* create an empty label variable */
run;

Use PROC PRINQUAL to compute the COV biplot

The PRINQUAL procedure can perform a multidimensional preference analysis, which is visualized by using a MDPREF plot. The MDPREF plot is closely related to biplot (Jackson (1991), A User’s Guide to Principal Components, p. 204). You can get PROC PRINQUAL to produce a COV biplot by doing the following:

• Use the N=2 option to specify you want to compute two principal components.
• Use the MDPREF=1 option to specify that the procedure should not rescale the vectors in the biplot. By default, MDPREF=2.5 and the vectors appear 2.5 larger than they should be. (More on scaling vectors later.)
• Use the IDENTITY transformation so that the variables are not transformed in a nonlinear manner.

The following PROC PRINQUAL statements produce a COV biplot (click to enlarge):

proc prinqual data=iris plots=(MDPref) 
              n=2       /* project onto Prin1 and Prin2 */
              mdpref=1; /* use COV scaling */
   transform identity(SepalLength SepalWidth PetalLength PetalWidth);  /* identity transform */
   id ID;
   ods select MDPrefPlot;
run;

Use Friendly's %BIPLOT macro

Friendly's books [SAS System for Statistical Graphics (1991) and Visualizing Categorical Data (2000)] introduced many SAS data analysts to the power of using visualization to accompany statistical analysis, and especially the analysis of multivariate data. His macros use traditional SAS/GRAPH graphics from the 1990s. In the mid-2000s, SAS introduced ODS statistical graphics, which were released with SAS 9.2. Although the %BIPLOT macro does not use ODS statistical graphics directly, the macro supports the OUT= option, which enables you to create an output data set that contains all the coordinates for creating a biplot.

The following example assumes that you have downloaded the %BIPLOT macro and the %EQUATE macro from Michael Friendly's web site.

/* A. Use %BIPLOT macro, which uses SAS/IML to compute the biplot coordinates. 
      Use the OUT= option to get the coordinates for the markers and vectors.
   B. Transpose the data from long to wide form.
   C. Use PROC SGPLOT to create the biplot
*/
%let FACTYPE = SYM;   /* options are GH, COV, JK, SYM */
title "Biplot: &FACTYPE, STD";
%biplot(data=iris, 
        var=SepalLength SepalWidth PetalLength PetalWidth, 
        id=id,
        factype=&FACTYPE,  /* GH, COV, JK, SYM */
        std=std,           /* NONE, MEAN, STD  */
        scale=1,           /* if you do not specify SCALE=1, vectors are auto-scaled */
        out=biplotFriendly,/* write SAS data set with results */ 
        symbols=circle dot, inc=1);
 
/* transpose from long to wide */
data Biplot;
set biplotFriendly(where=(_TYPE_='OBS') rename=(dim1=Prin1 dim2=Prin2 _Name_=_ID_))
    biplotFriendly(where=(_TYPE_='VAR') rename=(dim1=vx dim2=vy _Name_=_Variable_));
run;
 
proc sgplot data=Biplot aspect=1 noautolegend;
   refline 0 / axis=x; refline 0 / axis=y;
   scatter x=Prin1 y=Prin2 / datalabel=_ID_;
   vector  x=vx    y=vy    / datalabel=_Variable_
                             lineattrs=GraphData2 datalabelattrs=GraphData2;
   xaxis grid offsetmin=0.1 offsetmax=0.2;
   yaxis grid;
run;

Because you are using PROC SGPLOT to display the biplot, you can easily configure the graph. For example, I added grid lines, which are not part of the output from the %BIPLOT macro. You could easily change attributes such as the size of the fonts or add additional features such as an inset. With a little more work, you can merge the original data and the biplot data and color-code the markers by a grouping variable (such as Species) or by a continuous response variable.

Notice that the %BUPLOT macro supports a SCALE= option. The SCALE= option applies an additional linear scaling to the vectors. You can use this option to increase or decrease the lengths of the vectors in the biplot. For example, in the SYM biplot, shown above, the vectors are long relative to the range of the data. If you want to display vectors that are only 25% as long, you can specify SCALE=0.25. You can specify numbers greater than 1 to increase the vector lengths. For example, SCALE=2 will double the lengths of the vectors. If you omit the SCALE= option or set SCALE=0, then the %BIPLOT macro automatically scales the vectors to the range of the data. If you use the SCALE= option, you should tell the reader that you did so.

SAS/IML modules that compute biplots

The %BIPLOT macro uses SAS/IML software to compute the locations of the markers and vectors for each type of biplot. I wrote three SAS/IML modules that perform the three steps of creating a biplot:

• The CalcBiplot module computes the projections of the observations and scores onto the first few principal components. This module (formerly named CalcPrinCompBiplot) was written in the mid-2000s and has been distributed as part of the SAS/IML Studio application. It returns the scores and vectors as SAS/IML matrices.
• The WriteBiplot module calls the CalcBiplot module and then writes the scores to a SAS data set called _SCORES and the vectors (loadings) to a SAS data set called _VECTORS. It also creates two macro variables, MinAxis and MaxAxis, which you can use if you want to equate the horizontal and vertical scales of the biplot axes.
• The Biplot function calls the WriteBiplot module and then calls PROC SGPLOT to create a biplot. It is the "raw SAS/IML" version of the %BIPLOT macro.

You can use the CalcBiplot module to compute the scores and vectors and return them in IML matrices. You can use the WriteBiplot module if you want that information in SAS data sets so that you can create your own custom biplot. You can use the Biplot module to create standard biplots. The Biplot and WriteBiplot modules are demonstrated in the next sections.

Use the Biplot module in SAS/IML

The syntax of the Biplot module is similar to the %BIPLOT macro for most arguments. The input arguments are as follows:

• X: The numerical data matrix
• ID: A character vector of values used to label rows of X. If you pass in an empty matrix, observation numbers are used to label the markers. This argument is ignored if labelPoints=0.
• varNames: A character vector that contains the names of the columns of X.
• FacType: The type of biplot: 'GH', 'COV', 'JK', or 'SYM'.
• StdMethod: How the original variables are scaled: 'None', 'Mean', or 'Std'.
• Scale: A numerical scalar that specifies additional scaling applied to vectors. By default, SCALE=1, which means the vectors are not scaled. To shrink the vectors, specify a value less than 1. To lengthen the vectors, specify a value greater than 1. (Note: The %BIPLOT macro uses SCALE=0 as its default.)
• labelPoints: A binary 0/1 value. If 0 (the default) points are not labeled. If 1, points are labeled by the ID values. (Note: The %BIPLOT macro always labels points.)

The last two arguments are optional. You can specify them as keyword-value pairs outside of the parentheses. The following examples show how you can call the Biplot module in a SAS/IML program to create a biplot:

ods graphics / width=480px height=480px;
proc iml;
/* assumes the modules have been previously stored */
load module=(CalcBiplot WriteBiplot Biplot);
use sashelp.iris;
read all var _NUM_ into X[rowname=Species colname=varNames];
close;
 
title "COV Biplot with Scaled Vectors and Labels";
run Biplot(X, Species, varNames, "COV", "Std") labelPoints=1;   /* label obs */
 
title "JK Biplot: Relationships between Observations";
run Biplot(X, NULL, varNames, "JK", "Std");
 
title "JK Biplot: Automatic Scaling of Vectors";
run Biplot(X, NULL, varNames, "JK", "Std") scale=0;            /* auto scale; empty ID var */
 
title "SYM Biplot: Vectors Scaled by 0.25";
run Biplot(X, NULL, varNames, "SYM", "Std") scale=0.25;        /* scale vectors by 0.25 */

The program creates four biplots, but only the last one is shown. The last plot uses the SCALE=0.25 option to rescale the vectors of the SYM biplot. You can compare this biplot to the SYM biplot in the previous section, which did not rescale the length of the vectors.

Use the WriteBiplot module in SAS/IML

If you prefer to write an output data set and then create the biplot yourself, use the WriteBiplot module. After loading the modules and the data (see the previous section), you can write the biplot coordinates to the _Scores and _Vectors data sets, as follows. A simple DATA step appends the two data sets into a form that is easy to graph:

run WriteBiplot(X, NULL, varNames, "JK", "Std") scale=0;   /* auto scale vectors */
QUIT;
 
data Biplot;
   set _Scores _Vectors;    /* append the two data sets created by the WriteBiplot module */
run;
 
title "JK Biplot: Automatic Scaling of Vectors";
title2 "FacType=JK; Std=Std";
proc sgplot data=Biplot aspect=1 noautolegend;
   refline 0 / axis=x; refline 0 / axis=y;
   scatter x=Prin1 y=Prin2 / ; 
   vector  x=vx    y=vy    / datalabel=_Variable_
                             lineattrs=GraphData2 datalabelattrs=GraphData2;
   xaxis grid offsetmin=0.1 offsetmax=0.1 min=&minAxis max=&maxAxis;
   yaxis grid min=&minAxis max=&maxAxis;
run;

In the program that accompanies this article, there is an additional example in which the biplot data is merged with the original data so that you can color-code the observations by using the Species variable.

Summary

This article shows four ways to use modern ODS statistical graphics to create a biplot in SAS. You can create a COV biplot by using the PRINQUAL procedure. If you have a license for SAS/IML and SAS/GRAPH, you can use Friendly's %BIPLOT macro to write the biplot coordinates to a SAS data set, then use PROC SGPLOT to create the biplot. This article also presents SAS/IML modules that compute the same biplots as the %BIPLOT macro. The WriteBiplot module writes the data to two SAS data sets (_Score and _Vector), which can be appended and used to plot a biplot. This gives you complete control over the attributes of the biplot. Or, if you prefer, you can use the Biplot module in SAS/IML to automatically create biplots that are similar to Friendly's but are displayed by using ODS statistical graphics.

You can download the complete SAS program that is used in this article. For convenience, I have also created a separate file that defines the SAS/IML modules that create biplots.

The post Create biplots in SAS appeared first on The DO Loop.

Principal component analysis (PCA) is an important tool for understanding relationships in multivariate data. When the first two principal components (PCs) explain a significant portion of the variance in the data, you can visualize the data by projecting the observations onto the span of the first two PCs. In a PCA, this plot is known as a score plot. You can also project the variable vectors onto the span of the PCs, which is known as a loadings plot. See the article "How to interpret graphs in a principal component analysis" for a discussion of the score plot and the loadings plot.

A biplot overlays a score plot and a loadings plot in a single graph. An example is shown at the right. Points are the projected observations; vectors are the projected variables. If the data are well-approximated by the first two principal components, a biplot enables you to visualize high-dimensional data by using a two-dimensional graph.

In general, the score plot and the loadings plot will have different scales. Consequently, you need to rescale the vectors or observations (or both) when you overlay the score and loadings plots. There are four common choices of scaling. Each scaling emphasizes certain geometric relationships between pairs of observations (such as distances), between pairs of variables (such as angles), or between observations and variables. This article discusses the geometry behind two-dimensional biplots and shows how biplots enable you to understand relationships in multivariate data.

Some material in this blog post is based on documentation that I wrote in 2004 when I was working on the SAS/IML Studio product and writing the SAS/IML Studio User's Guide. The documentation is available online and includes references to the literature.

The Fisher iris data

A previous article shows the score plot and loadings plot for a PCA of Fisher's iris data. For these data, the first two principal components explain 96% of the variance in the four-dimensional data. Therefore, these data are well-approximated by a two-dimensional set of principal components. For convenience, the score plot (scatter plot) and the loadings plot (vector plot) are shown below for the iris data. Notice that the loadings plot has a much smaller scale than the score plot. If you overlay these plots, the vectors would appear relatively small unless you rescale one or both plots.

The mathematics of the biplot

You can perform a PCA by using a singular value decomposition of a data matrix that has N rows (observations) and p columns (variables). The first step in constructing a biplot is to center and (optionally) scale the data matrix. When variables are measured in different units and have different scales, it is usually helpful to standardize the data so that each column has zero mean and unit variance. The examples in this article use standardized data.

The heart of the biplot is the singular value decomposition (SVD). If X is the centered and scaled data matrix, then the SVD of X is
X = U L V`
where U is an N x N orthogonal matrix, L is a diagonal N x p matrix, and V is an orthogonal p x p matrix. It turns out that the principal components (PCs) of X`X are the columns of V and the PC scores are the columns of U. If the first two principal components explain most of the variance, you can choose to keep only the first two columns of U and V and the first 2 x 2 submatrix of L. This is the closest rank-two approximation to X. In a slight abuse of notation,
X ≈ U L V`
where now U, L, and V all have only two columns.

Since L is a diagonal matrix, you can write L = L^c L^1-c for any number c in the interval [0, 1]. You can then write
X ≈ (U L^c)(L^1-c V`)
   = A B

This the factorization that is used to create a biplot. The most common choices for c are 0, 1, and 1/2.

The four types of biplots

The choice of the scaling parameter, c, will linearly scale the observations and vectors separately. In addition, you can write X ≈ (β A) (B / β) for any constant β. Each choice for c corresponds to a type of biplot:

• When c=0, the vectors are represented faithfully. This corresponds to the GH biplot. If you also choose β = sqrt(N-1), you get the COV biplot.
• When c=1, the observations are represented faithfully. This corresponds to the JK biplot.
• When c=1/2, the observations and vectors are treated symmetrically. This corresponds to the SYM biplot.

The GH biplot for variables

If you choose c = 0, then A = U and B = L V`. The literature calls this biplot the GH biplot. I call it the "variable preserving" biplot because it provides the most faithful two-dimensional representation of the relationship between vectors. In particular:

• The length of each vector (a row of B) is proportional to the variance of the corresponding variable.
• The Euclidean distance between the i_th and j_th rows of A is proportional to the Mahalanobis distance between the i_th and j_th observations in the data.

In preserving the lengths of the vectors, this biplot distorts the Euclidean distance between points. However, the distortion is not arbitrary: it represents the Mahalanobis distance between points.

The GH biplot is shown to the right, but it is not very useful for these data. In choosing to preserve the variable relationships, the observations are projected onto a tiny region near the origin. The next section discusses an alternative scaling that is more useful for the iris data.

The COV biplot

If you choose c = 0 and β = sqrt(N-1), then A = sqrt(N-1) U and B = L V` / sqrt(N-1). The literature calls this biplot the COV biplot. This biplot is shown at the top of this article. It has two useful properties:

• The length of each vector is equal to the variance of the corresponding variable.
• The Euclidean distance between the i_th and j_th rows of A is equal to the Mahalanobis distance between the i_th and j_th observations in the data.

In my opinion, the COV biplot is usually superior to the GH biplot.

The JK biplot

If you choose c = 1, you get the JK biplot, which preserves the Euclidean distance between observations. Specifically, the Euclidean distance between the i_th and j_th rows of A is equal to the Euclidean distance between the i_th and j_th observations in the data.

In faithfully representing the observations, the angles between vectors are distorted by the scaling.

The SYM biplot

If you choose c = 1/2, you get the SYM biplot (also called the SQ biplot), which attempts to treat observations and variables in a symmetric manner. Although neither the observations nor the vectors are faithfully represented, often neither representation is very distorted. Consequently, some people prefer the SYM biplot as a compromise between the COV and JK biplots. The SYM biplot is shown in the next section.

How to interpret a biplot

As discussed in the SAS/IML Studio User's Guide, you can interpret a biplot in the following ways:

• The cosine of the angle between a vector and an axis indicates the importance of the contribution of the corresponding variable to the principal component.
• The cosine of the angle between pairs of vectors indicates correlation between the corresponding variables. Highly correlated variables point in similar directions; uncorrelated variables are nearly perpendicular to each other.
• Points that are close to each other in the biplot represent observations with similar values.
• You can approximate the relative coordinates of an observation by projecting the point onto the variable vectors within the biplot. However, you cannot use these biplots to estimate the exact coordinates because the vectors have been centered and scaled. You could extend the vectors to become lines and add tick marks, but that becomes messy if you have more than a few variables.

If you want to faithfully interpret the angles between vectors, you should equate the horizontal and vertical axes of the biplot, as I have done with the plots on this page.

If you apply these facts to the standardized iris data, you can make the following interpretations:

• The PetalLength and PetalWidth variables are the most important contributors to the first PC. The SepalWidth variable is the most important contributor to the second PC.
• The PetalLength and PetalWidth variables are highly correlated. The SepalWidth variable is almost uncorrelated with the other variables.
• Although I have suppressed labels on the points, you could label the points by an ID variable or by the observation number and use the relative locations to determine which flowers had measurements that were most similar to each other.

Summary

This article presents an overview of biplots. A biplot is an overlay of a score plot and a loadings plot, which are two common plots in a principal component analysis. These two plots are on different scales, but you can rescale the two plots and overlay them on a single plot. Depending upon the choice of scaling, the biplot can provide faithful information about the relationship between variables (lengths and angles) or between observations (distances). It can also provide approximates relationships between variables and observations.

In a subsequent post, I will show how to use SAS to create the biplots in this article.

The post What are biplots? appeared first on The DO Loop.