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# Incremental differences

Properties of the 1-point incremental differences of the time series

### MD_hrv_classic_pnn40

MD_hrv_classic_pnn40 computes the proportion of difference magnitudes that are greater than 4% of the standard deviation of the time series.
This feature will give low values to series that have periods in which the series stays approximately constant (within
$0.04\sigma$
), and high values to series that do (e.g., they 'jump around alot' from point to point).
This is a common statistic to measure about heart rate time series, cf. "The pNNx files: re-examining a widely used heart rate variability measure", J.E. Mietus et al., Heart 88(4) 378 (2002).
For example, the Rossler attractor time series below has many near-constant stretches (Just 14.5% of all step increments are larger than
$0.04\sigma$
), yielding a low value for this statistic:
These log returns of opening prices of a stock, on the other hand, fluctuate alot more: 95% of successive increments exceed the
$0.04\sigma$
threshold, yielding a high value for this statistic:

## FC_LocalSimple_mean1_tauresrat

This feature is the tauresrat output of running the code FC_LocalSimple(x_z,'mean',1) in hctsa. It computes:
1. 1.
The set of incremental differences between successive pairs of time-series values (imagining this as the set of residuals from a naive 1-point forecast).
2. 2.
Computes the first zero-crossing of the autocorrelation function for the residuals, tau_resid, and for the original time series, tau.
3. 3.
Returns the ratio of these two values, tau_resid/tau.
Here's an example of a time series, where the autocorrelation function (black, lower plot) decays slowly due to slower trends in the time series (black, upper plot), whereas the incremental differences (blue in the upper plot) are much noisier, leading to a rapid drop of the autocorrelation function, and reduction in the first zero-crossing from 15 -> 1.
By contrast, in this AR(2) time series, the increments are still highly autocorrelated, and we end up with a value of the feature near 1.