HurstExponent

class eyefeatures.features.measures.HurstExponent(coordinate, n_iters=10, fill_strategy='last', pk=None, eps=1e-22, return_df=True, ignore_errors=False)[source]

Bases: MeasureTransformer

Approximates Hurst Exponent using R/S analysis.

The Hurst exponent is a measure of the long-term memory of time series. It relates to the autocorrelations of the time series, and the rate at which these decrease as the lag between pairs of values increases. $H in (0.5, 1)$ indicates a persistent behavior (trend). $H in (0, 0.5)$ indicates an anti-persistent behavior (mean-reverting). $H = 0.5$ indicates a completely random series (Geometric Brownian Motion).

Parameters:

  • coordinate (str) – coordinate column name (1D Hurst exponent currently available).

  • n_iters (int) – number of iterations to complete. Note: data must be of length more than \(2^{n\_iters}\).

  • fill_strategy (Literal['mean', 'reduce', 'last']) – how to make vector be length of power of \(2\). If “reduce”, then all values after \(2^k\)-th are removed, where \(n < 2^{(k + 1)}\). Other strategies specify the value to fill the vector up to the closest power of \(2\), “mean” being the mean of vector, “last” being the last value (makes time-series constant at the end).

  • pk (list[str]) – list of column names used to split pd.DataFrame.

  • eps (float) – division epsilon.

  • return_df (bool) – Return pd.Dataframe object else np.ndarray.

  • ignore_errors (bool) – If True, return NaN when feature computation fails; otherwise raise.

Example

Quick start with default parameters:

from eyefeatures.features.measures import HurstExponent

transformer = HurstExponent(coordinate="x")
features = transformer.fit_transform(fixations_df)
get_feature_names_out(input_features=None)[source]

Generate feature name that includes coordinate column and hyperparameters.

Return type:

list[str]

  • Time series is divided into segments (blocks) of equal size.

  • The mean is subtracted from each segment to center the data.

  • Compute the cumulative sum of the mean-adjusted data and determine the range (maximum - minimum) of the cumulative deviation.

  • Calculate the standard deviation of the original segment and the ratio of the range to the standard deviation.

  • The slope of he log of the block size and the log of the R/S ratio estimates the Hurst Exponent.

\[\log\frac{R}{S} = \text{HurstExponent} \cdot \log n + C\]

where \(\frac{R}{S}\) is the rescaled range, \(n\) is the block size, and \(C\) is some constant.

Reference

Algorithm on Wiki.