I am working on a project where I am given a large table of numbers, in which we are hoping to see certain patterns. For example (using R):
set.seed(77)
mat <- matrix(rnorm(100), 10, 10)
colnames(mat) <- letters[1:10]
rownames(mat) <- 1:10
round(mat, 3)
a b c d e f g h i j
1 -0.550 -2.941 -0.254 -2.362 0.003 -1.305 0.399 -1.638 0.751 0.877
2 1.091 -0.243 1.519 -0.551 -0.531 0.887 0.027 -0.332 -0.067 0.835
3 0.640 -0.141 1.781 -0.305 -0.710 2.336 -1.001 0.448 -0.504 -0.048
4 1.043 -0.033 -0.879 -0.750 -0.291 0.503 0.009 0.272 -0.160 -3.410
5 0.170 0.280 -1.529 0.144 0.885 -2.268 -0.164 -0.254 -0.093 -1.513
6 1.138 0.590 0.136 -0.549 -0.154 -2.032 0.423 2.348 0.474 0.252
7 -0.971 1.024 -0.709 0.160 -0.954 -0.138 -0.424 -0.213 0.131 -0.473
8 -0.132 2.107 -1.410 -0.088 0.667 -0.953 -0.470 0.051 0.717 0.977
9 0.146 0.155 1.831 0.081 0.388 1.578 0.172 -2.246 -0.003 2.435
10 1.441 0.913 1.290 0.899 0.549 -1.248 1.847 0.920 -2.177 -0.082
My goal is to be able to sort arbitrary sized matrices (not necessarily square) by switching rows and columns to minimize cell-wise differences. For example, all the values on line 1 should stay on line 1, but perhaps it makes more sense distance-wise for line 1 to appear on line 8, and vice versa regarding the columns. This problem reminds me of finding the inverse of a matrix using linear algebra.
Note: I realize there is not a unique solution to this problem, so resampling is OK (ideally, rerunning the function will produce different clusterings). As a starting point, optimally larger values would float to the top while smaller values would tend toward the bottom.
In the above example, perhaps the largest numbers are all in row 9 and so one move might be to bring it to the top. Similarly, perhaps the largest numbers are found in column c and that is moved to the left. Basically, I want to reorder the rows and columns of the table so that the magnitude of the difference between each entry (e.g. a1 vs a2/b1; c2 vs. the four surrounding entries) is minimized - but with the restriction that the operations happen row- and column-wise.
I am primarily interested in a theoretical approach that will show me how to accomplish this, but will eventually be implementing the solution in R, so assistance on that front would also be appreciated.
To make this more concrete: this idea is to be applied to the results from large-scale simulation studies, where the entries might be Type I error or power rates, the rows might pertain to a design condition (like sample size) and the columns might pertain to different multiple comparison procedures or something like that. The goal is to rearrange the entries of this table so that we might be able to glance at the table and see where clusters of "good" procedures are and where the "bad" ones are.
