scipy.stats.multiscale_graphcorr

scipy.stats.multiscale_graphcorr(x, y, compute_distance=<function _euclidean_dist at 0x7f0b666fd510>, reps=1000, workers=1, is_twosamp=False, random_state=None)

Computes the Multiscale Graph Correlation (MGC) test statistic.

Specifically, for each point, MGC finds the \(k\)-nearest neighbors for one property (e.g. cloud density), and the \(l\)-nearest neighbors for the other property (e.g. grass wetness) [1]. This pair \((k, l)\) is called the “scale”. A priori, however, it is not know which scales will be most informative. So, MGC computes all distance pairs, and then efficiently computes the distance correlations for all scales. The local correlations illustrate which scales are relatively informative about the relationship. The key, therefore, to successfully discover and decipher relationships between disparate data modalities is to adaptively determine which scales are the most informative, and the geometric implication for the most informative scales. Doing so not only provides an estimate of whether the modalities are related, but also provides insight into how the determination was made. This is especially important in high-dimensional data, where simple visualizations do not reveal relationships to the unaided human eye. Characterizations of this implementation in particular have been derived from and benchmarked within in [2].

Parameters

x, yndarray

If x and y have shapes (n, p) and (n, q) where n is the number of samples and p and q are the number of dimensions, then the MGC independence test will be run. Alternatively, x and y can have shapes (n, n) if they are distance or similarity matrices, and compute_distance must be sent to None. If x and y have shapes (n, p) and (m, p), an unpaired two-sample MGC test will be run.

compute_distancecallable, optional

A function that computes the distance or similarity among the samples within each data matrix. Set to None if x and y are already distance matrices. The default uses the euclidean norm metric. If you are calling a custom function, either create the distance matrix before-hand or create a function of the form compute_distance(x) where x is the data matrix for which pairwise distances are calculated.

repsint, optional

The number of replications used to estimate the null when using the permutation test. The default is 1000.

workersint or map-like callable, optional

If workers is an int the population is subdivided into workers sections and evaluated in parallel (uses multiprocessing.Pool <multiprocessing>). Supply -1 to use all cores available to the Process. Alternatively supply a map-like callable, such as multiprocessing.Pool.map for evaluating the p-value in parallel. This evaluation is carried out as workers(func, iterable). Requires that func be pickleable. The default is 1.

is_twosampbool, optional

If True, a two sample test will be run. If x and y have shapes (n, p) and (m, p), this optional will be overriden and set to True. Set to True if x and y both have shapes (n, p) and a two sample test is desired. The default is False.

random_stateint or np.random.RandomState instance, optional

If already a RandomState instance, use it. If seed is an int, return a new RandomState instance seeded with seed. If None, use np.random.RandomState. Default is None.

Returns

statfloat

The sample MGC test statistic within [-1, 1].

pvaluefloat

The p-value obtained via permutation.

mgc_dictdict

Contains additional useful additional returns containing the following keys:

• mgc_mapndarray

  A 2D representation of the latent geometry of the relationship. of the relationship.

• opt_scale(int, int)

  The estimated optimal scale as a (x, y) pair.

• null_distlist

  The null distribution derived from the permuted matrices

See also

pearsonr

Pearson correlation coefficient and p-value for testing non-correlation.

kendalltau

Calculates Kendall’s tau.

spearmanr

Calculates a Spearman rank-order correlation coefficient.

Notes

A description of the process of MGC and applications on neuroscience data can be found in [1]. It is performed using the following steps:

1. Two distance matrices \(D^X\) and \(D^Y\) are computed and modified to be mean zero columnwise. This results in two \(n \times n\) distance matrices \(A\) and \(B\) (the centering and unbiased modification) [3].

2. For all values \(k\) and \(l\) from \(1, ..., n\),

   • The \(k\)-nearest neighbor and \(l\)-nearest neighbor graphs are calculated for each property. Here, \(G_k (i, j)\) indicates the \(k\)-smallest values of the \(i\)-th row of \(A\) and \(H_l (i, j)\) indicates the \(l\) smallested values of the \(i\)-th row of \(B\)

   • Let \(\circ\) denotes the entry-wise matrix product, then local correlations are summed and normalized using the following statistic:

\[c^{kl} = \frac{\sum_{ij} A G_k B H_l} {\sqrt{\sum_{ij} A^2 G_k \times \sum_{ij} B^2 H_l}}\]

1. The MGC test statistic is the smoothed optimal local correlation of \(\{ c^{kl} \}\). Denote the smoothing operation as \(R(\cdot)\) (which essentially set all isolated large correlations) as 0 and connected large correlations the same as before, see [3].) MGC is,

\[MGC_n (x, y) = \max_{(k, l)} R \left(c^{kl} \left( x_n, y_n \right) \right)\]

The test statistic returns a value between \((-1, 1)\) since it is normalized.

The p-value returned is calculated using a permutation test. This process is completed by first randomly permuting \(y\) to estimate the null distribution and then calculating the probability of observing a test statistic, under the null, at least as extreme as the observed test statistic.

MGC requires at least 5 samples to run with reliable results. It can also handle high-dimensional data sets.

In addition, by manipulating the input data matrices, the two-sample testing problem can be reduced to the independence testing problem [4]. Given sample data \(U\) and \(V\) of sizes \(p \times n\) \(p \times m\), data matrix \(X\) and \(Y\) can be created as follows:

\[ \begin{align}\begin{aligned}X = [U | V] \in \mathcal{R}^{p \times (n + m)}\\Y = [0_{1 \times n} | 1_{1 \times m}] \in \mathcal{R}^{(n + m)}\end{aligned}\end{align} \]

Then, the MGC statistic can be calculated as normal. This methodology can be extended to similar tests such as distance correlation [4].

New in version 1.4.0.

References

1(1,2)

Vogelstein, J. T., Bridgeford, E. W., Wang, Q., Priebe, C. E., Maggioni, M., & Shen, C. (2019). Discovering and deciphering relationships across disparate data modalities. ELife.

2

Panda, S., Palaniappan, S., Xiong, J., Swaminathan, A., Ramachandran, S., Bridgeford, E. W., … Vogelstein, J. T. (2019). mgcpy: A Comprehensive High Dimensional Independence Testing Python Package. ArXiv:1907.02088 [Cs, Stat].

3(1,2)

Shen, C., Priebe, C.E., & Vogelstein, J. T. (2019). From distance correlation to multiscale graph correlation. Journal of the American Statistical Association.

4(1,2)

Shen, C. & Vogelstein, J. T. (2018). The Exact Equivalence of Distance and Kernel Methods for Hypothesis Testing. ArXiv:1806.05514 [Cs, Stat].

Examples

>>> from scipy.stats import multiscale_graphcorr
>>> x = np.arange(100)
>>> y = x
>>> stat, pvalue, _ = multiscale_graphcorr(x, y, workers=-1)
>>> '%.1f, %.3f' % (stat, pvalue)
'1.0, 0.001'

Alternatively,

>>> x = np.arange(100)
>>> y = x
>>> mgc = multiscale_graphcorr(x, y)
>>> '%.1f, %.3f' % (mgc.stat, mgc.pvalue)
'1.0, 0.001'

To run an unpaired two-sample test,

>>> x = np.arange(100)
>>> y = np.arange(79)
>>> mgc = multiscale_graphcorr(x, y, random_state=1)
>>> '%.3f, %.2f' % (mgc.stat, mgc.pvalue)
'0.033, 0.02'

or, if shape of the inputs are the same,

>>> x = np.arange(100)
>>> y = x
>>> mgc = multiscale_graphcorr(x, y, is_twosamp=True)
>>> '%.3f, %.1f' % (mgc.stat, mgc.pvalue)
'-0.008, 1.0'