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# scipy.stats.multiscale_graphcorr

scipy.stats.multiscale_graphcorr(x, y, compute_distance=<function _euclidean_dist>, 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, y ndarray

      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_distance callable, 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.

    reps int, optional

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

    workers int 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_twosamp bool, 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 overridden 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`. Note that this will not run if inputs are distance matrices.

    random_state{None, int, `numpy.random.Generator`,

      > `numpy.random.RandomState`}, optional

      If seed is None (or np.random), the `numpy.random.RandomState` singleton is used. If seed is an int, a new `RandomState` instance is used, seeded with seed. If seed is already a `Generator` or `RandomState` instance then that instance is used.

  Returns:
    res MGCResult

      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:

  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:

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

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

  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

  [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)