# Time Series¶

## Cross Multiscale Graph Correlation (MGCX)¶

class hyppo.time_series.MGCX(compute_distance=None, max_lag=0)[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.

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(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. 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)
'1.0, 0.00'


## Cross Distance Correlation (DcorrX)¶

class hyppo.time_series.DcorrX(compute_distance=None, max_lag=0)[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.

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(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. 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)
'1.0, 0.00'