Kurtosis Function¶

The kurtosis function computes the kurtosis of a window. When combined with the SlidingWindow abstraction, the kurtosis function can be used to compute the kurtosis feature of a time series. The kurtosis is defined as:

\[ \kappa = \begin{cases} \frac{m_4}{m_2^2} - 3 & \text{Fisher} \\ \frac{m_4}{m_2^2} & \text{Pearson} \end{cases} \]

where \(m_2\) and \(m_4\) are the second and fourth central moments, respectively. They are defined as:

\[ m_2 = \frac{1}{N} \sum_{i=1}^N (x_i - \bar{x})^2 \]

\[ m_4 = \frac{1}{N} \sum_{i=1}^N (x_i - \bar{x})^4 \]

where \(N\) is the number of samples in the window and \(\bar{x}\) is the mean of the window.

Compute the krutosis of the values in x where where is True.

The krutosis is a measure of the "tailedness" of a distribution. It is defined as the fourth standardized moment of a distribution, and is calculated as:

Parameters:

Name	Type	Description	Default
`x`	`ndarray`	The array to compute the krutosis of.	required
`fisher`	`Union[bool, bool_]`	Whether to use Fisher's definition of kurtosis i.e. subtract 3 from the result. Default is `True`. If `False`, the result is the Pearson's definition of kurtosis.	`True`
`where`	`Callable[[Union[int, float, int_, float_]], Union[bool, bool_]]`	A function that takes a value and returns `True` or `False`. Default is `lambda x: not np.isnan(x)` i.e. a measurement is valid if it is not a `NaN` value.	`lambda : not numpy.isnan(x)`

Returns:

Type	Description
`Union[float, float_]`	The krutosis of the values in `x` where `where` is `True`.

Examples¶

Fisher Kurtosis¶

import numpy as np
import autonfeat as aft
import autonfeat.functional as F

# Random data
n_samples = 100
x = np.random.rand(n_samples)

# Create sliding window
ws = 10
ss = 10
window = aft.SlidingWindow(window_size=ws, step_size=ss)

# Get featurizer
featurizer = window.use(F.kurtosis_tf)

# Get features
features = featurizer(x)

# Print features
print(features)

Pearson Kurtosis¶

import numpy as np
import autonfeat as aft
import autonfeat.functional as F

# Random data
n_samples = 100
x = np.random.rand(n_samples)

# Create sliding window
ws = 10
ss = 10
window = aft.SlidingWindow(window_size=ws, step_size=ss)

# Get featurizer
featurizer = window.use(F.kurtosis_tf)

# Get features
features = featurizer(x, fisher=False)

# Print features
print(features)

If you enjoy using AutonFeat, please consider starring the repository ⭐️.