multiscale_graphcorr#
- scipy.stats.multiscale_graphcorr(x, y, compute_distance=<function _euclidean_dist>, reps=1000, workers=1, is_twosamp=False, random_state=None)[source]#
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
andy
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
andy
can have shapes(n, n)
if they are distance or similarity matrices, andcompute_distance
must be sent toNone
. Ifx
andy
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
ifx
andy
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 formcompute_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 intoworkers
sections and evaluated in parallel (usesmultiprocessing.Pool <multiprocessing>
). Supply-1
to use all cores available to the Process. Alternatively supply a map-like callable, such asmultiprocessing.Pool.map
for evaluating the p-value in parallel. This evaluation is carried out asworkers(func, iterable)
. Requires that func be pickleable. The default is1
.- is_twosampbool, optional
If True, a two sample test will be run. If
x
andy
have shapes(n, p)
and(m, p)
, this optional will be overridden and set toTrue
. Set toTrue
ifx
andy
both have shapes(n, p)
and a two sample test is desired. The default isFalse
. Note that this will not run if inputs are distance matrices.- random_state{None, int,
numpy.random.Generator
, numpy.random.RandomState
}, optionalIf seed is None (or np.random), the
numpy.random.RandomState
singleton is used. If seed is an int, a newRandomState
instance is used, seeded with seed. If seed is already aGenerator
orRandomState
instance then that instance is used.
- Returns:
- resMGCResult
An object containing attributes:
- statisticfloat
The sample MGC test statistic within [-1, 1].
- pvaluefloat
The p-value obtained via permutation.
- mgc_dictdict
Contains additional useful results:
- mgc_mapndarray
A 2D representation of the latent geometry 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:
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].
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:
\[c^{kl} = \frac{\sum_{ij} A G_k B H_l} {\sqrt{\sum_{ij} A^2 G_k \times \sum_{ij} B^2 H_l}}\]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:
\[X = [U | V] \in \mathcal{R}^{p \times (n + m)} Y = [0_{1 \times n} | 1_{1 \times m}] \in \mathcal{R}^{(n + m)}\]Then, the MGC statistic can be calculated as normal. This methodology can be extended to similar tests such as distance correlation [4].
Added 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
[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
Examples
>>> import numpy as np >>> from scipy.stats import multiscale_graphcorr >>> x = np.arange(100) >>> y = x >>> res = multiscale_graphcorr(x, y) >>> res.statistic, res.pvalue (1.0, 0.001)
To run an unpaired two-sample test,
>>> x = np.arange(100) >>> y = np.arange(79) >>> res = multiscale_graphcorr(x, y) >>> res.statistic, res.pvalue (0.033258146255703246, 0.023)
or, if shape of the inputs are the same,
>>> x = np.arange(100) >>> y = x >>> res = multiscale_graphcorr(x, y, is_twosamp=True) >>> res.statistic, res.pvalue (-0.008021809890200488, 1.0)