# Statistical test for determining hot spot in a heat map

I'm analyzing high through proteomic data and have several thousand 5x5 matrices, where in each matrix the rows and columns refer to different biological conditions, and each cell refers to gene expression in that pair of biological conditions.

I expect there to be three kinds of proteins in this dataset:

In most cases I expect there to be no large variation, but random fluctuations between cells:

  0  0 -1 -2  1
2  1  0 -1 -1
0  0  1  2  0
1 -2  1  3  0
-3  1  2  0  0


In a few cases, some of the samples might be contaminated, and there might be some extreme outliers, e.g. in a case where (2,2) and (3,4) are outliers:

  1  0  0 -2  1
0 84  1 -1 -1
-1  0  0  2  0
0  1  99 3  0
1 -3  2  0  0


However, the cases that I'm interested in, are those in which protein expression changes towards a certain optimal pair of biological conditions, e.g. where proteins change expression reaching an optimal point at the (3,4) cell:

  0  0  5 12  8
1  5 20 36 21
0 10 31 52 40
1 -2 17 23 30
-3  1 12  5  8


Is there a statistical test to identify the optimal cell in the above example, that is also robust to handling outliers and random fluctuation as in the first two cases?