Compute correlation between distance matrices using the Mantel test.
The Mantel test compares two distance matrices by computing the correlation between the distances in the lower (or upper) triangular portions of the symmetric distance matrices. Correlation can be computed using Pearson’s productmoment correlation coefficient or Spearman’s rank correlation coefficient.
As defined in [R67], the Mantel test computes a test statistic \(r_M\) given two symmetric distance matrices \(D_X\) and \(D_Y\). \(r_M\) is defined as
where
and \(n\) is the number of rows/columns in each of the distance matrices. \(stand(D_X)\) and \(stand(D_Y)\) are distance matrices with their upper triangles containing standardized distances. Note that since \(D_X\) and \(D_Y\) are symmetric, the lower triangular portions of the matrices could equivalently have been used instead of the upper triangular portions (the current function behaves in this manner).
If method='spearman', the above equation operates on ranked distances instead of the original distances.
Statistical significance is assessed via a permutation test. The rows and columns of the first distance matrix (x) are randomly permuted a number of times (controlled via permutations). A correlation coefficient is computed for each permutation and the pvalue is the proportion of permuted correlation coefficients that are equal to or more extreme than the original (unpermuted) correlation coefficient. Whether a permuted correlation coefficient is “more extreme” than the original correlation coefficient depends on the alternative hypothesis (controlled via alternative).
Parameters:  x, y : array_like or DistanceMatrix
method : {‘pearson’, ‘spearman’}
permutations : int, optional
alternative : {‘twosided’, ‘greater’, ‘less’}


Returns:  tuple of floats

Raises:  ValueError

Notes
The Mantel test was first described in [R68]. The general algorithm and interface are similar to vegan::mantel, available in R’s vegan package [R69].
np.nan will be returned for the pvalue if permutations is zero or if the correlation coefficient is np.nan. The correlation coefficient will be np.nan if one or both of the inputs does not have any variation (i.e. the distances are all constant) and method='spearman'.
References
[R67]  (1, 2) Legendre, P. and Legendre, L. (2012) Numerical Ecology. 3rd English Edition. Elsevier. 
[R68]  (1, 2) Mantel, N. (1967). “The detection of disease clustering and a generalized regression approach”. Cancer Research 27 (2): 209220. PMID 6018555. 
[R69]  (1, 2) http://cran.rproject.org/web/packages/vegan/index.html 
Examples
Define two 3x3 distance matrices:
>>> x = [[0, 1, 2],
... [1, 0, 3],
... [2, 3, 0]]
>>> y = [[0, 2, 7],
... [2, 0, 6],
... [7, 6, 0]]
Compute the Pearson correlation between them and assess significance using a twosided test with 999 permutations:
>>> coeff, p_value = mantel(x, y)
>>> round(coeff, 4)
0.7559
Thus, we see a moderatetostrong positive correlation (\(r_M=0.7559\)) between the two matrices.