Time Series¶

Cross Multiscale Graph Correlation (MGCX)¶

class hyppo.time_series.MGCX(compute_distance='euclidean', max_lag=0, **kwargs)[source]¶

Class for running the MGCX test for independence of time series.

MGCX is an independence test between two (paired) time series of not necessarily equal dimensions. The population parameter is 0 if and only if the time series are independent. It is based upon energy distance between distributions.

Parameters:

compute_distance : callable(), optional (default: euclidean): A function that computes the distance among the samples within each data matrix. Set to None if x and y are already distance matrices. To call 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.
max_lag : int, optional (default: 0): The maximum number of lags in the past to check dependence between x and the shifted y. Also the \(M\) hyperparmeter below.

See also

MGC: Multiscale graph correlation test statistic and p-value.
DcorrX: Cross distance correlation test statistic and p-value.

Notes

The statistic can be derived as follows:

Let \(x\) and \(y\) be \((n, p)\) and \((n, q)\) series respectively, which each contain \(y\) observations of the series \((X_t)\) and \((Y_t)\). Similarly, let \(x[j:n]\) be the \((n-j, p)\) last \(n-j\) observations of \(x\). Let \(y[0:(n-j)]\) be the \((n-j, p)\) first \(n-j\) observations of \(y\). Let \(M\) be the maximum lag hyperparameter. The cross distance correlation is,

\[\mathrm{MGCX}_n (x, y) = \sum_{j=0}^M frac{n-j}{n} \mathrm{MGC}_n (x[j:n], y[0:(n-j)])\]

References

[1]	Mehta, R., Chung, J., Shen C., Xu T., Vogelstein, J. T. (2019). A Consistent Independence Test for Multivariate Time-Series. ArXiv

test(self, x, y, reps=1000, workers=1)[source]¶

Calculates the MGCX test statistic and p-value.

Parameters:

x, y : ndarray: Input data matrices. x and y must have the same number of samples. That is, the shapes must be (n, p) and (n, q) where n is the number of samples and p and q are the number of dimensions. Alternatively, x and y can be distance matrices, where the shapes must both be (n, n).
reps : int, optional (default: 1000): The number of replications used to estimate the null distribution when using the permutation test used to calculate the p-value.
workers : int, optional (default: 1): The number of cores to parallelize the p-value computation over. Supply -1 to use all cores available to the Process.
auto : bool (default: True): Automatically uses fast approximation when sample size and size of array is greater than 20. If True, and sample size is greater than 20, a fast chi2 approximation will be run. Parameters reps and workers are irrelevant in this case.

Returns:

stat : float

The computed MGCX statistic.

pvalue : float

The computed MGCX p-value.

mgcx_dict : dict

Contains additional useful returns containing the following keys:

opt_lag : int

The optimal lag that maximizes the strength of the relationship with respect to lag.

opt_scale : tuple

The optimal scale that maximizes the strength of the relationship with respect to scale.

Examples

The optimal scale should be global [n, n] for cases of linear correlation.

>>> import numpy as np
>>> from hyppo.time_series import MGCX
>>> np.random.seed(456)
>>> x = np.arange(7)
>>> y = x
>>> stat, pvalue, mgcx_dict = MGCX().test(x, y, reps = 100)
>>> '%.1f, %.2f, [%d, %d]' % (stat, pvalue, mgcx_dict['opt_scale'][0],
... mgcx_dict['opt_scale'][1])
'1.0, 0.02, [7, 7]'

The increasing the max_lag can increase the ability to identify dependence.

>>> import numpy as np
>>> from hyppo.time_series import MGCX
>>> np.random.seed(1234)
>>> x = np.random.permutation(10)
>>> y = np.roll(x, -1)
>>> stat, pvalue, mgcx_dict = MGCX(max_lag=1).test(x, y, reps=1000)
>>> '%.1f, %.2f, %d' % (stat, pvalue, mgcx_dict['opt_lag'])
'1.1, 0.01, 1'

In addition, the inputs can be distance matrices. Using this is the, same as before, except the compute_distance parameter must be set to None.

>>> import numpy as np
>>> from hyppo.time_series import MGCX
>>> np.random.seed(789)
>>> x = np.ones((10, 10)) - np.identity(10)
>>> y = 2 * x
>>> mgcx = MGCX(compute_distance=None)
>>> stat, pvalue, mgcx_dict = mgcx.test(x, y)
>>> '%.1f, %.2f' % (stat, pvalue)
'0.0, 1.00'

Cross Distance Correlation (DcorrX)¶

class hyppo.time_series.DcorrX(compute_distance='euclidean', max_lag=0, **kwargs)[source]¶

Class for running the DcorrX test for independence of time series.

DcorrX is an independence test between two (paired) time series of not necessarily equal dimensions. The population parameter is 0 if and only if the time series are independent. It is based upon energy distance between distributions.

Parameters:

compute_distance : callable(), optional (default: euclidean): A function that computes the distance among the samples within each data matrix. Set to None if x and y are already distance matrices. To call 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.
max_lag : int, optional (default: 0): The maximum number of lags in the past to check dependence between x and the shifted y. Also the \(M\) hyperparmeter below.

See also

Dcorr: Distance correlation test statistic and p-value.
MGCX: Cross multiscale graph correlation test statistic and p-value.

Notes

The statistic can be derived as follows:

Let \(x\) and \(y\) be \((n, p)\) and \((n, q)\) series respectively, which each contain \(y\) observations of the series \((X_t)\) and \((Y_t)\). Similarly, let \(x[j:n]\) be the \((n-j, p)\) last \(n-j\) observations of \(x\). Let \(y[0:(n-j)]\) be the \((n-j, p)\) first \(n-j\) observations of \(y\). Let \(M\) be the maximum lag hyperparameter. The cross distance correlation is,

\[\mathrm{DcorrX}_n (x, y) = \sum_{j=0}^M frac{n-j}{n} \mathrm{Dcorr}_n (x[j:n], y[0:(n-j)])\]

References

[2]	Mehta, R., Chung, J., Shen C., Xu T., Vogelstein, J. T. (2019). A Consistent Independence Test for Multivariate Time-Series. ArXiv

test(self, x, y, reps=1000, workers=1)[source]¶

Calculates the DcorrX test statistic and p-value.

Parameters:

x, y : ndarray: Input data matrices. x and y must have the same number of samples. That is, the shapes must be (n, p) and (n, q) where n is the number of samples and p and q are the number of dimensions. Alternatively, x and y can be distance matrices, where the shapes must both be (n, n).
reps : int, optional (default: 1000): The number of replications used to estimate the null distribution when using the permutation test used to calculate the p-value.
workers : int, optional (default: 1): The number of cores to parallelize the p-value computation over. Supply -1 to use all cores available to the Process.
auto : bool (default: True): Automatically uses fast approximation when sample size and size of array is greater than 20. If True, and sample size is greater than 20, a fast chi2 approximation will be run. Parameters reps and workers are irrelevant in this case.

Returns:

stat : float

The computed DcorrX statistic.

pvalue : float

The computed DcorrX p-value.

dcorrx_dict : dict

Contains additional useful returns containing the following keys:

opt_lag : int

The optimal lag that maximizes the strength of the relationship.

Examples

>>> import numpy as np
>>> from hyppo.time_series import DcorrX
>>> np.random.seed(456)
>>> x = np.arange(7)
>>> y = x
>>> stat, pvalue, dcorrx_dict = DcorrX().test(x, y, reps = 100)
>>> '%.1f, %.2f, %d' % (stat, pvalue, dcorrx_dict['opt_lag'])
'1.0, 0.01, 0'

The increasing the max_lag can increase the ability to identify dependence.

>>> import numpy as np
>>> from hyppo.time_series import DcorrX
>>> np.random.seed(1234)
>>> x = np.random.permutation(10)
>>> y = np.roll(x, -1)
>>> stat, pvalue, dcorrx_dict = DcorrX(max_lag=1).test(x, y, reps=1000)
>>> '%.1f, %.2f, %d' % (stat, pvalue, dcorrx_dict['opt_lag'])
'1.1, 0.01, 1'

In addition, the inputs can be distance matrices. Using this is the, same as before, except the compute_distance parameter must be set to None.

>>> import numpy as np
>>> from hyppo.time_series import DcorrX
>>> np.random.seed(789)
>>> x = np.ones((10, 10)) - np.identity(10)
>>> y = 2 * x
>>> dcorrx = DcorrX(compute_distance=None)
>>> stat, pvalue, dcorrx_dict = dcorrx.test(x, y)
>>> '%.1f, %.2f' % (stat, pvalue)
'0.0, 1.00'