Half-Moments

Half-moment measures for data analysis.

This module provides functions for: 1. Half-moment based measures: hm, ham, asymmetry, extremity, cextremity 2. Mean Root Error (MRE)

References

Svetunkov I., Kourentzes N., Svetunkov S. “Half Central Moment for Data Analysis”. Working Paper of Department of Management Science, Lancaster University, 2023:3, 1-21.
Kourentzes N. (2014). The Bias Coefficient: a new metric for forecast bias.

greybox.hm.asymmetry(x: ndarray, center: float | None = None) → float[source]

Asymmetry coefficient.

The asymmetry coefficient measures the asymmetry of a distribution around a center point based on the half moment.

Formula:

asymmetry = 1 - Arg(hm(x, C)) / (pi/4)

Values:

1: Maximum negative asymmetry (all values below center)
0: Symmetric distribution
-1: Maximum positive asymmetry (all values above center)

Parameters:

x (np.ndarray) – Data vector.
center (float, optional) – Centering parameter. If None, uses mean(x).

Returns:

Asymmetry coefficient.

Return type:

float

Examples

>>> x = np.array([1, 2, 3, 4, 5])
>>> asymmetry(x)
0.0
>>> x_skewed = np.array([1, 1, 1, 4, 5])
>>> asymmetry(x_skewed)
-0.20...

References

Svetunkov I., Kourentzes N., Svetunkov S. “Half Central Moment for Data Analysis”.

greybox.hm.cextremity(x: ndarray, center: float | None = None) → complex[source]

Complex Extremity coefficient.

The complex extremity coefficient extends the extremity coefficient to capture both magnitude and phase information.

Formula:: cextremity = 2 * (CH * 2)^(log(0.5)/log(2*3^(-0.75))) - 1

where CH = hm(x, C) / (std(x)^0.25)

Parameters:

x (np.ndarray) – Data vector.
center (float, optional) – Centering parameter. If None, uses mean(x).

Returns:

Complex extremity coefficient.

Return type:

complex

Examples

>>> x = np.array([1, 2, 3, 4, 5])
>>> cextremity(x)
(-0.04...+0...j)

References

Svetunkov I., Kourentzes N., Svetunkov S. “Half Central Moment for Data Analysis”.

greybox.hm.extremity(x: ndarray, center: float | None = None) → float[source]

Extremity coefficient.

The extremity coefficient measures how extreme the values in a distribution are relative to the center.

Formula:: extremity = 2 * (ham(x, C) / (std(x)^0.5))^(log(0.5)/log(2*3^(-0.75))) - 1

Parameters:

x (np.ndarray) – Data vector.
center (float, optional) – Centering parameter. If None, uses mean(x).

Returns:

Extremity coefficient.

Return type:

float

Examples

>>> x = np.array([1, 2, 3, 4, 5])
>>> extremity(x)
-0.04...

References

Svetunkov I., Kourentzes N., Svetunkov S. “Half Central Moment for Data Analysis”.

greybox.hm.ham(x: ndarray, center: float | None = None) → float[source]

Half Absolute Moment (HAM).

The half absolute moment is the absolute value of the half moment. It captures the magnitude of deviations without direction.

Formula:: ham = mean(sqrt(abs(x - C)))

Parameters:

x (np.ndarray) – Data vector.
center (float, optional) – Centering parameter. If None, uses mean(x).

Returns:

Half absolute moment.

Return type:

float

Examples

>>> x = np.array([1, 2, 3, 4, 5])
>>> ham(x)
1.0

References

Svetunkov I., Kourentzes N., Svetunkov S. “Half Central Moment for Data Analysis”.

greybox.hm.hm(x: ndarray, center: float | None = None) → complex[source]

Half Moment (HM).

The half moment is a measure that captures the asymmetry of a distribution by considering only the positive or negative deviations from a center point.

Formula:: hm = mean(sqrt(x - C))

where C is the centering parameter (default is mean of x).

Parameters:

x (np.ndarray) – Data vector.
center (float, optional) – Centering parameter. If None, uses mean(x).

Returns:

Half moment (complex number when mean of sqrt is taken).

Return type:

complex

Examples

>>> x = np.array([1, 2, 3, 4, 5])
>>> hm(x)
(0.79...+0.79...j)

References

Svetunkov I., Kourentzes N., Svetunkov S. “Half Central Moment for Data Analysis”.

greybox.hm.mre(actual: ndarray, forecast: ndarray, na_rm: bool = True) → float[source]

Mean Root Error (MRE).

The MRE measures the average signed root error. It is used as a bias coefficient (Kourentzes, 2014) and can detect systematic over or under-forecasting.

Formula:: MRE = mean(sqrt(actual - forecast))

Note: When actual < forecast, the result is complex. The function returns the real part of the mean.

Parameters:

actual (np.ndarray) – Actual (observed) values.
forecast (np.ndarray) – Forecasted values.
na_rm (bool, default=True) – Remove NA values.

Returns:

Mean Root Error.

Return type:

float

Examples

>>> actual = np.array([1, 2, 3, 4, 5])
>>> forecast = np.array([1.1, 2.0, 3.2, 3.9, 5.1])
>>> mre(actual, forecast)

References

Kourentzes N. (2014). The Bias Coefficient: a new metric for forecast bias. https://kourentzes.com/forecasting/2014/12/17/the-bias-coefficient-a-new-metric-for-forecast-bias/