# Namespace Statistics¶

namespace statistics

Functions

af::array covariance(const af::array &tss, bool unbiased = true)

Returns the covariance matrix of the time series contained in tss.

Return
af::array The covariance matrix of the time series.
Parameters
• tss: Expects an input array whose dimension zero is the length of the time series (all the same) and dimension one indicates the number of time series.
• unbiased: Determines whether it divides by n - 1 (if false) or n (if true).

af::array kurtosis(const af::array &tss)

Returns the kurtosis of tss (calculated with the adjusted Fisher-Pearson standardized moment coefficient G2).

Return
af::array The kurtosis of tss.
Parameters
• tss: Expects an input array whose dimension zero is the length of the time series (all the same) and dimension one indicates the number of time series.

af::array moment(const af::array &tss, int k)

Returns the kth moment of the given time series.

Return
af::array The kth moment of the given time series.
Parameters
• tss: Expects an input array whose dimension zero is the length of the time series (all the same) and dimension one indicates the number of time series.
• k: The specific moment to be calculated.

af::array ljungBox(const af::array &tss, long lags)

The Ljung–Box test checks that data within the time series are independently distributed (i.e. the correlations in the population from which the sample is taken are 0, so that any observed correlations in the data result from randomness of the sampling process). Data are no independently distributed, if they exhibit serial correlation.

The test statistic is:

$Q = n\left(n+2\right)\sum_{k=1}^h\frac{\hat{\rho}^2_k}{n-k}$

where ‘’n’’ is the sample size, $$\hat{\rho}k$$ is the sample autocorrelation at lag ‘’k’‘, and ‘’h’’ is the number of lags being tested. Under $$H_0$$ the statistic Q follows a $$\chi^2{(h)}$$. For significance level $$\alpha$$, the $$critical region$$ for rejection of the hypothesis of randomness is:

$Q > \chi_{1-\alpha,h}^2$

where $$\chi_{1-\alpha,h}^2$$ is the $$\alpha$$-quantile of the chi-squared distribution with ‘’h’’ degrees of freedom.

 G. M. Ljung G. E. P. Box (1978). On a measure of lack of fit in time series models. Biometrika, Volume 65, Issue 2, 1 August 1978, Pages 297–303.

Return
af::array Ljung-Box statistic test.
Parameters
• tss: Expects an input array whose dimension zero is the length of the time series (all the same) and dimension one indicates the number of time series.
• lags: Number of lags being tested.

af::array quantile(const af::array &tss, const af::array &q, float precision = 100000000)

Returns values at the given quantile.

Return
af::array Values at the given quantile.
Parameters
• tss: Expects an input array whose dimension zero is the length of the time series (all the same) and dimension one indicates the number of time series. NOTE: the time series should be sorted.
• q: Percentile(s) at which to extract score(s). One or many.
• precision: Number of decimals expected.

af::array quantilesCut(const af::array &tss, float quantiles, float precision = 0.00000001)

Discretizes the time series into equal-sized buckets based on sample quantiles.

Return
af::array Matrix with the categories, one category per row, the start of the category in the first column and the end in the second category.
Parameters
• tss: Expects an input array whose dimension zero is the length of the time series (all the same) and dimension one indicates the number of time series. NOTE: the time series should be sorted.
• quantiles: Number of quantiles to extract. From 0 to 1, step 1/quantiles.
• precision: Number of decimals expected.

af::array sampleStdev(const af::array &tss)

Estimates standard deviation based on a sample. The standard deviation is calculated using the “n-1” method.

Return
af::array The sample standard deviation.
Parameters
• tss: Expects an input array whose dimension zero is the length of the time series (all the same) and dimension one indicates the number of time series.

af::array skewness(const af::array &tss)

Calculates the sample skewness of tss (calculated with the adjusted Fisher-Pearson standardized moment coefficient G1).

Return
af::array Array containing the skewness of each time series in tss.
Parameters
• tss: Expects an input array whose dimension zero is the length of the time series (all the same) and dimension one indicates the number of time series.