EwKurt

Description

EwKurt computes the exponentially weighted moving kurtosis, indicating the “tailedness” of the data distribution while emphasizing recent values.

Parameters

One of the following decay parameters is required to calculate alpha, where a higher alpha value gives recent points more influence:

  • com: Center of mass. alpha = 1 / (1 + com)

  • span: Span. alpha = 2 / (span + 1)

  • halflife: Half-life. alpha = 1 - exp(-log(2) / halflife)

  • alpha: Directly specifies the smoothing factor, where 0 < alpha < 1

NaN handling: NaN values are ignored in the mean calculation.

Usage Example and Plot

import numpy as np
import plotly.graph_objects as go
from plotly.subplots import make_subplots

from screamer import EwKurt

data = np.cumsum(np.random.normal(size=300))
ewkurt_data = EwKurt(span=20)(data)

fig = make_subplots(rows=2, cols=1, shared_xaxes=True, row_heights=[1/2, 1/2], vertical_spacing=0.1)

fig.add_trace(go.Scatter(y=data, mode='lines', name='Original Data'), row=1, col=1)
fig.add_trace(go.Scatter(y=ewkurt_data, mode='lines', name='EwKurt', line=dict(color='red')), row=2, col=1)

fig.update_layout(
    title="Exponentially Weighted Moving Kurtosis",
    xaxis_title="Index",
    yaxis=dict(title="Original Data"),
    yaxis2=dict(title="EwKurt", range=[-1, 2]),
    margin=dict(l=20, r=20, t=40, b=20),
    legend=dict(orientation="h", yanchor="bottom", y=1.02, xanchor="right", x=1)
)

fig.show()

Formula Details

EwKurt computes the exponentially weighted excess kurtosis recursively, with a bias correction that uses an effective sample size, \(N_{eff}\), derived from the sum of weights. This approach ensures an unbiased estimate even as older observations’ influence diminishes over time, aligning with Pandas’ convention for excess kurtosis.

Let:

  • alpha be the smoothing factor, calculated from com, span, halflife, or specified directly, where 0 < alpha < 1.

For each new data point \(x_t\), EwKurt updates five cumulative sums, \(S_x\), \(S_{xx}\), \(S_{xxx}\), \(S_{xxxx}\), \(S_w\), and \(S_{ww}\), as follows:

  1. Adjust \(S_x\), the weighted sum of values, by retaining a fraction \((1 - \alpha)\) of the previous weighted sum and adding the new value:

\[ S_x = S_x \times (1 - \alpha) + x_t \]

  1. Adjust \(S_{xx}\), the weighted sum of squared values, by retaining a fraction of the previous sum and adding the square of the new value:

\[ S_{xx} = S_{xx} \times (1 - \alpha) + x_t^2 \]

  1. Adjust \(S_{xxx}\), the weighted sum of cubed values, by retaining a fraction of the previous sum and adding the cube of the new value:

\[ S_{xxx} = S_{xxx} \times (1 - \alpha) + x_t^3 \]

  1. Adjust \(S_{xxxx}\), the weighted sum of fourth powers, by retaining a fraction of the previous sum and adding the fourth power of the new value:

\[ S_{xxxx} = S_{xxxx} \times (1 - \alpha) + x_t^4 \]

  1. Adjust \(S_w\), the cumulative weight, by retaining a fraction of the previous weight and adding a weight of \(1\):

\[ S_w = S_w \times (1 - \alpha) + 1 \]

  1. Adjust \(S_{ww}\), the cumulative squared weight, by retaining a fraction \((1 - \alpha)^2\) of the previous squared weight and adding \(1\):

\[ S_{ww} = S_{ww} \times (1 - \alpha)^2 + 1 \]

  1. Calculate the effective sample size, \(N_{eff}\), as:

\[ N_{eff} = \frac{S_w^2}{S_{ww}} \]

  1. Compute the exponentially weighted mean, \(\text{EwMean}\), as:

\[ \text{EwMean} = \frac{S_x}{S_w} \]

  1. Compute the exponentially weighted variance, \(\text{EwVar}\), with bias correction:

\[ \text{EwVar} = \left( \frac{S_{xx}}{S_w} - \left( \frac{S_x}{S_w} \right)^2 \right) \times \frac{N_{eff}}{N_{eff} - 1} \]

  1. Calculate the standard deviation, \(\text{EwStd}\), as:

\[ \text{EwStd} = \sqrt{\text{EwVar}} \]

  1. Compute the fourth central moment, \(m_4\), as:

\[ m_4 = \frac{S_{xxxx}}{S_w} - 4 \cdot \frac{S_x}{S_w} \cdot \frac{S_{xxx}}{S_w} + 6 \cdot \left( \frac{S_x}{S_w} \right)^2 \cdot \frac{S_{xx}}{S_w} - 3 \cdot \left( \frac{S_x}{S_w} \right)^4 \]

  1. Calculate the raw kurtosis, \(g_2\), as:

\[ g_2 = \frac{m_4}{\text{EwVar}^2} \]

  1. Finally, compute the exponentially weighted excess kurtosis with bias correction:

\[ \text{EwKurt} = \frac{N_{eff} \cdot (N_{eff} + 1) \cdot g_2 - 3 \cdot (N_{eff} - 1)^2}{(N_{eff} - 1) \cdot (N_{eff} - 2) \cdot (N_{eff} - 3)} \]

This calculation uses \(N_{eff}\) to correct for the effective sample size, ensuring that the excess kurtosis measure remains unbiased by scaling based on the decreasing influence of older values.