# LjungBox¶

class hyppo.time_series.LjungBox(max_lag=0)

Ljung-Box for Cross Correlation (CorrX) test statistic and p-value.

Parameters

max_lag (int, default: 0) -- The maximum number of lags in the past to check dependence between x and the shifted y. If None, then max_lag=np.ceil(np.log(n)). Also the M hyperparmeter below.

Notes

The statistic can be derived as follows 1:

Let $$x$$ and $$y$$ be $$(n, 1)$$ and $$(n, 1)$$ 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,

$\mathrm{Ljung-Box}_n (x, y) = n(n+2)\sum_{j=1}^M \frac{ \rho^2(x[j:n], y[0:(n-j)])}{n-j}$

where $rho$ is the Pearson correlation coefficient. The p-value returned is calculated either via chi-squared distribution or using a permutation test.

References

1

Ronak Mehta, Jaewon Chung, Cencheng Shen, Ting Xu, and Joshua T. Vogelstein. Independence Testing for Multivariate Time Series. arXiv:1908.06486 [cs, stat], May 2020. arXiv:1908.06486.

Methods Summary

 Helper function that calculates the Ljung-Box cross correlation test statistic. LjungBox.test(x, y[, reps, workers, auto, ...]) Calulates the time-series test test statistic and p-value.

LjungBox.statistic(x, y)

Helper function that calculates the Ljung-Box cross correlation test statistic.

Parameters

x,y (ndarray of float) -- Input data matrices. x and y must have the same number of samples. That is, the shapes must be (n, 1) and (n, 1) where n is the number of samples.

Returns

LjungBox.test(x, y, reps=1000, workers=1, auto=True, random_state=None)

Calulates the time-series test test statistic and p-value.

Parameters
• x,y (ndarray of float) -- 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, 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, default: 1) -- The number of cores to parallelize the p-value computation over. Supply -1 to use all cores available to the Process.

• is_distsim (bool, default: False) -- Whether or not x and y are input matrices.

Returns