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#

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.

NaN handling#

Policy: ignore. A NaN in any input at index t causes the function to skip that step: output at t is NaN and internal state is unchanged. Subsequent finite samples are processed as if step t had not occurred.

Examples#

Description#

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()