Statistical test for determining hot spot in a heat map

Question

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?

Thanks in advance.

Answer 1

It looks like you are interested in some measure of local "spatial autocorrelation" --- i.e., the correlation of a cell value with the cell values that are nearby in the spacial matrix. There are many possible measures of this kind, depending on how you look at the spacial properties of your matrix. You can find some information on "local" measures of spatial association in Anselin (1995). This will give you a starting point for determining what exactly you want to look at.

Local measures of spatial association will be able to differentiate between points which are a local peak with ascending values nearby (like your point $(3,4)$ in your last matrix) versus points that have an "outlying" value on their own (like your points $(2,2)$ and $(4,3)$ in your second matrix). I would suggest you start by choosing an appropriate measure of local spatial association for your matrix data, and then compute this for those matrices to see how it differentiates these kinds of points. In constructing your measure, you will need to decide on what constitutes a "close" point (e.g., are diagonals considered the same distance as adjacent squares) and this will affect your chosen spatial measure. Nevertheless, any reasonable measure of this kind should be able to differentiate the two cases of concern to you.

Answer 2

I'm just going to throw out some thoughts:

1. There is no rigorous method to detect outliers for nonparametric data. If you can't say what the distribution really is, then you can't say what is an outlier. Therefore, you need to remove those by hand in a justifiable way.

2. Mathematically, I think by "optimal" you mean "maximum." Assuming you have removed putative outliers, your problem is now how to determine a maximum value that you are confindent in with respect to noise. This is a signal processing problem, not a statistics problem. I would suggest taking a look at image processing algorithms which also deal with noisy 2D data sets.

Best I can offer.