pyhctsa.operations.correlation.quantilogram¶

pyhctsa.operations.correlation.quantilogram(y, lag=1)¶

Serial dependence of the series being below its own quantiles (the quantilogram).

Port of hctsa’s CO_Quantilogram. Computes the quantilogram of Linton and Whang [1]: the autocorrelation, at a given lag, of the ‘hit’ process that marks when the series is below its alpha-quantile, for a range of quantile levels alpha from the lower tail through the center to the upper tail. At level alpha, the hit process is h_t = 1(y_t <= q_alpha), where q_alpha is the sample alpha-quantile, and the quantilogram is the sample autocorrelation of h_t at the lag: sum_t (h_t - hbar)(h_{t+lag} - hbar) / sum_t (h_t - hbar)^2. It is positive when excursions below the alpha-quantile cluster in time and negative when they alternate with excursions above it. For an independent series it is zero at every level, with a standard error of about 1/sqrt(N). Looking at the lower and upper quantile levels separately captures dependence in the tails (volatility clustering, or bursts of extreme values that follow one another), at the center (directional persistence), and any asymmetry between them. Because the series enters only through whether each value lies below a quantile, the result at a fixed lag is unchanged by any increasing monotonic rescaling of the series, and is not affected by outliers (a lag set by the timescale of the series is found from the series itself, so it can change with the rescaling).

References

Parameters:¶

y : array-like¶

The input time series (non-finite values are dropped).

lag : int or {'ac', 'ac1e', 'mi'}, optional¶

The time lag (default 1). A positive integer, or a lag set by the timescale of the series (see get_tau()): 'ac' (the first zero-crossing of the autocorrelation function of the series), 'ac1e' (the floor of its first 1/e crossing), or 'mi' (the smaller of the first minimum of the Kraskov automutual information and the 'ac1e' delay).

Returns:¶

The quantilogram at the lag for each quantile level, in the fields q05, q10, q25, q50, q75, q90 and q95, for alpha = 0.05, 0.10, 0.25, 0.50, 0.75, 0.90 and 0.95, respectively. A level for which the hit process is constant (a constant series), or for which the lag is not smaller than N/2 (or cannot be set), is NaN.

Return type:¶

dict

Notes

The sample quantile is the round(alpha*N)-th smallest value, and the hit process is centered at its sample mean rather than at alpha, which is the same for a continuous-valued series (they differ by at most 1/N) and stays well-defined for series with tied values.

The central quantile levels measure the same persistence as the quantile-state transition probabilities and the up-down motif frequencies (transition_matrix, motif_two), so hctsa registers only the tail levels (q05, q10, q90, q95 at the 'mi' lag; q05 and q95 at lag 1). All seven levels are computed.