# Symbolic

Features based on a discrete symbolization of real-valued time-series data

`entropy_pairs`

`entropy_pairs`

**Naming info**: This features matches the *hctsa* feature called `SB_MotifThree_quantile_hh`

`entropy_pairs`

is computed as the following:

Converts each value in the time series into one of three symbols ('A', 'B', or 'C') using an equi-probable binning in which the lowest 3rd of values are assigned 'A', the middle 3rd 'B', and the highest 3rd of values are given 'C'.

It then analyzes the probabilities of all two-letter sequences ('AA', 'AB', 'BB', …) and outputs the entropy of this set of probabilities.

This feature is based on the *hctsa* code `SB_MotifThree(x_z,'quantile')`

, returning the `hh`

output.

Time series that have predictable two-letter sequences (i.e., some sequences are far more probable than others) will have low values, and time series that are less predictable (i.e., all two-letter sequences are approximately equi-probable) will have high values.

For example, this Ricker map is highly predictable, with letters (A=red, B=blue, C=green) shown below, converting to "ABCABCABCABC…", such that "AB", "BC", and "CA" become highly probable sequences. It has a very low value for this feature (1.1):

Here is another very predictable series, which has a very long string of C, then B, then A, such that "AA", "BB", and "CC" are highly probable sequences yielding low entropy (1.1):

Time series that have predictable patterns on longer than 2-length windows, or that requiring more granularity than a 3-letter symbolization to resolve, have high values, like this map:

`transition_variance`

`transition_variance`

**Naming info**: This feature matches the *hctsa* feature called `SB_TransitionMatrix_3ac_sumdiagcov`

.

This feature first symbolizes the time series into a 3-letter equiprobable alphabet (by quantile). It then computes $\tau$ as first zero-crossing of the autocorrelation function and computes the $\tau$-step transition probabilities between the three states as a 3x3 transition matrix. This feature then returns the sum of column-wise variances of this matrix.

It is a measure of the specificity of source states given a target state. A minimum value would be given to a noisy series where the transition probabilities are approximately uniform, and a maximum value for a highly ordered series with very specific state transition rules. The subtlety with this feature is that it computes the transitions on a timescale of $\tau$ such that it measures such order on the timescale at which the linear autocorrelation has faded.

## Consecutive stretches

*catch22* contains two features that capture the maximum length of time over which similar consecutive local patterns are observed:

`stretch_high`

measures the longest successive period of above-average values.The feature called

`SB_BinaryStats_mean_longstretch1`

in*hctsa*(the`longstretch1`

output from running`SB_BinaryStats(x_z,'mean')`

in*hctsa*);

`stretch_decreasing`

measures the longest successive period of successive decreases.The feature called

`SB_BinaryStats_diff_longstretch0`

in*hctsa*(the`longstretch0`

output from running`SB_BinaryStats(x_z,'diff')`

in*hctsa*).

`stretch_high`

`stretch_high`

### What it does

`stretch_high`

computes the longest sequence of successive values in the time series that are greater than the mean. Algorithmically, this is achieved in two steps:

Transform the time series into a binary sequence: time-series values that are greater than the mean are set to

`1`

and time-series values that are less than or equal to the mean are set to`0`

.Return the longest sequence of successive values that are

`1`

.

### What it measures

Low values are given to time series that tend not to linger too much on either side of the mean, like this moving average process, which has a maximum duration of

`stretch_high = 8`

samples (red) above the mean (zero: dashed line):

High values are given to time series that have at least one long duration of time spent above the mean, like this stochastic sine map (

`stretch_high = 107`

successive time points above the mean):

`stretch_decreasing`

`stretch_decreasing`

### What it does

`stretch_decreasing`

is similar to the above, but it calculates the longest sequence of successive steps in the time series that *decrease*. Algorithmically, this is achieved in two steps:

Transform the time series into a binary sequence: each time-series value is converted to a

`1`

if it is higher than the previous time point, and`0`

if it is lower than the previous time point (starting from the second point in the time series, and thus yielding a sequence of length`N-1`

, where`N`

is the length of the original time series).Return the longest sequence of successive values that are

`0`

.

### What it measures

Here is a time series of a time series from the complex butterfly map, with the longest period (of

`stretch_decreasing = 30`

) successive decreases highlighted in red.

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