A self-similar Christmas tree

Self-similar Christmas tree created in SAS

Happy holidays to all my readers! My greeting-card to you is an image of a self-similar Christmas tree. The image (click to enlarge) was created in SAS by using two features that I blog about regularly: matrix computations and ODS statistical graphics.

Self-similarity in Kronecker products

I have previously shown that the repeated Kronecker product of a binary matrix leads to self-similar structures.

Specifically, if M is a binary matrix then the Kronecker product M@M is a block matrix in which each 1 in the original matrix is replaced with a copy of M and each 0 is replaced by a zero block. (Here '@' is the Kronecker product (or direct product) operator, which is implemented in SAS/IML software.) If you repeat this process three or more times and create a heat map of the product matrix, the self-similar structure is apparent.

For example, the following 5 x 7 binary matrix has a 1 in positions that approximate the shape of a Christmas tree. The direct product M@M@M is a matrix that has 53 = 125 rows and 73 = 343 columns. If you use the HEATMAPDISC subroutine in SAS/IML to create a heat map of the direct product, you obtain the following image in which a green cell represents a 1:

proc iml;
M = {0 0 0 1 0 0 0,
     0 0 1 1 1 0 0,
     0 1 1 1 1 1 0,
     1 1 1 1 1 1 1,
     0 0 0 1 0 0 0};
Tree = M @ M @ M;     /* M is 5 x 7, so Tree is 125 x 343 */
ods graphics / width=300px height=450px  ANTIALIASMAX=50000;
call heatmapdisc(Tree) title="Happy Holidays to All my Readers!"
     colorramp={White Green} displayoutlines=0 ShowLegend=0;

That's not a bad result for three SAS/IML statements! You can see that the "tree" has a self-similar structure: the large tree is composed of 17 smaller trees, each of which is composed of 17 mini trees.

Adding a star

My readers are worth extra effort, so I want to put a star on the top of the tree. One way to do this is to use a scatter plot and plot a special observation by using a star symbol. To plot the "tree" by using a scatter plot, it is necessary to "unpack" the 125 x 343 matrix into a list of (x, y) values (this is a wide-to-long conversion of the matrix). You can use the NDX2SUB function to extract the row and column of each 1 in the matrix, as follows:

/* extract (row, column) pairs for matrix elements that are '1' */
idx = loc(Tree = 1);                /* indices for the 1s */
s = ndx2sub(dimension(Tree), idx);  /* convert indices to subscripts */
create Tree from s[c={"RNames" "CNames"}];
append from s;
call symputx("XPos", range(s[,2])/2); /* midrange = horiz position of star */

The previous statements create a SAS data set called TREE that contains the (x, y) positions of the 1s in the direct product matrix. The statements also saved the midrange value to a macro variable. The following SAS procedures create the location of the star, concatenate the two data sets, and create a scatter plot of the result, which is shown at the top of this article.

data star;
x = &XPos; y = 0; 
data Tree2;
set Tree star;
ods graphics / width=600px height=800px  ANTIALIASMAX=50000;
title "Happy Holidays";
proc sgplot data=Tree2 noborder noautolegend;
scatter x=CNames y=RNames / markerattrs=(color=ForestGreen symbol=SquareFilled size=3);
scatter x=x y=y / markerattrs=(symbol=StarFilled size=20)             /* yellow star */
                  filledoutlinedmarkers markerfillattrs=(color=yellow);
yaxis reverse display=none;
xaxis display=none;

Previous SAS-created Christmas cards

All year long I blog about how to use SAS for serious pursuits like statistical modeling, data analysis, optimization, and simulation. Consequently, I enjoy my occasional forays into a fun and frivolous topic such as how to create a greeting card with SAS. If you like seeing geeky SAS-generated images, here is a list of my efforts from previous years:

Wherever you are and whatever holiday you celebrate, may you enjoy peace and happiness this season and in the New Year.


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 PROC IML and SAS/IML Studio. 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.


  1. My favourite tree this season... Your annual geeky SAS images are always a delight. I especially LOVE your extra effort (and coding) to create the star!

    Wishing you a jolly and peaceful festive season too Rick.

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