pyhctsa.operations.correlation.crinkle_statistic¶
-
pyhctsa.operations.correlation.crinkle_statistic(y, tau=
1)¶ Crinkle statistic: normalized fourth moment of the second difference.
With \(\tilde{x} = x - \bar{x}\),
\[C = \frac{\langle (\tilde{x}_{t-1} - 2\tilde{x}_t + \tilde{x}_{t+1})^4 \rangle} {\langle \tilde{x}_t^2 \rangle^{2}},\]where the numerator averages over the \(N - 2\) interior points and the denominator over all \(N\) samples. Statistic as used in D. Kaplan’s code (hctsa
DK_crinkle), where it is credited to J. Theiler.- Parameters:¶
- y : array-like¶
The input time series.
- tau : int or str, optional¶
The step of the second difference, in samples: an integer, or a rule for
get_tau()(e.g.'ac1e'). The numerator then averages \(( ilde{x}_{t- au} - 2 ilde{x}_t + ilde{x}_{t+ au})^4\) over every interior point, which pools thetauphases of the series sampled everytausamples. Default 1. NaN when a rule cannot set the delay.
- Returns:¶
The crinkle statistic (NaN if the series is constant or has fewer than
2 tau + 1samples, where it is undefined).- Return type:¶
float
Notes
With \(\Delta^2 \tilde{x}_t = \tilde{x}_{t-1} - 2\tilde{x}_t + \tilde{x}_{t+1}\), C factors exactly as the kurtosis of \(\Delta^2 \tilde{x}\) times \((\langle (\Delta^2 \tilde{x})^2 \rangle / \langle \tilde{x}^2 \rangle)^2\). For a stationary process the ratio tends to \(6 - 8\rho_1 + 2\rho_2\), with \(\rho_k\) the lag-k autocorrelation, so C responds to the lag-1 and lag-2 autocorrelation and to the value distribution, not only to spikes in the curvature.
crinkle_statistic_kurtosis()is a variant without these dependences.No published definition of the statistic is known. Theiler et al. [1] define the companion skewed-difference statistic
theiler_q()and name higher moments as a class of discriminating statistic for surrogate-data tests, but not this one. A plausible origin is as a check for the spurious high-frequency “crinkliness” of Fourier surrogates of data whose end points do not match [2], which a fourth moment of the second difference detects.References