Nonlinear autocorrelation

These features capture nonlinear autocorrelation properties of a time series.

catch22 contains 2 features which each capture some aspect of the nonlinear autocorrelation properties of a time series. Select one of the cards below to discover more information:

trev

ami2

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1. trev

What it does

computes computes the average across the time series of the cube of successive time-series differences. It will be close to zero for time series for which the distribution of successive decreases in the time series matches the distribution of successive increases, but will be positive if increases tend to be larger in magnitude and negative if decreases tend to be larger in magnitude.

It is computed as:

$$\langle (x_{i+1} - x_i)^3\rangle_t\,,$$

for time-series values, x, averaged across all possible time points, t (i.e., from index 1 to N-1, for a time series of length N).

It is based on a statistic used in nonlinear time-series analysis, cf. Surrogate time series, Schreiber and Schmitz, Physica D, 142: 346 (2000).

The original hctsa implementation is the code CO_trev(x_z,1) , for a z-scored time series, x_z.

To give an intuition, below we plot some examples of the outputs of this feature for different scenarios:

Example 1: Chua Map

This Chua map time series has increments that are approximately symmetric (blue histogram), as are the cubic increments (orange histogram), and so this statistic has a value (the mean of the orange distribution) of approximately zero:

Click to enlarge image.

Feature output: trev =-0.021

Example 2: Simple Piecewise linear chaotic flow

This simple flow has a multi-modal distribution of successive increments (blue), with an asymmetry towards larger increases compared to smaller decreases. This is accentuated after taking a cube (orange), yielding a positive value of this statistic:

Click to enlarge image.

Feature output: trev =1.083

Example 3: Lozi Map

This Lozi map is an example of the opposite behaviour: time-series decreases can be sudden and large in magnitude, whereas increases are relatively gradual. This leads to an asymmetry towards large negative values, leading to a negative value for this statistic:

Click to enlarge image.

Feature output: trev =-3.497

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2. ami2

What it does

is a nonlinear version of the autocorrelation function: using a nonlinear correlation metric (mutual information) instead of a conventional linear correlation metric, evaluated using a histogram with 5 bins and at a time delay τ = 2 (from the hctsa code CO_HistogramAMI(x_z,2,'even',5)).

Explore the tabs below to see examples of the typical outputs of this feature for various time series:

Example 1: Chaotic Web Map

This feature gives high values to time series like this Chaotic Web map:

which has clear dependence structure of the time-series value at the current point, $x_t$, and the value two time points ahead, $x_{t+2}$ , yielding a high value for this feature of 1.25:

Feature output: ami2 =1.250

Example 2: Tent Map

This Tent map also has a clear (nonlinear) dependence of present time-series value with that two-steps into the future (and a moderate value of this feature):

as seen in its embedding:

Feature output: ami2 =0.473

Example 3: Heart Rhythm

This feature gives a moderate value for this heart rhythm ECG time series:

and its embedding:

Feature output: ami2 =0.25

Example 4: Earthquake

We obtain a low value for this earthquake time series:

and embedding:

Feature output: ami2 =0.019

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