EwStd

Description

EwStd calculates the exponentially weighted moving standard deviation, providing insight into the spread of recent values, with an emphasis on more recent observations.

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 EwStd

# Generate example data
data = np.cumsum(np.random.normal(size=300))

# Compute exponentially weighted standard deviation with a span of 20
ewstd_data = EwStd(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=ewstd_data, mode='lines', name='EwStd', line=dict(color='red')), row=2, col=1)

fig.update_layout(
    title="Exponentially Weighted Moving Standard Deviation",
    xaxis_title="Index",
    yaxis=dict(title="Original Data"),
    yaxis2=dict(title="EwStd", range=[0, 8]),
    margin=dict(l=20, r=20, t=80, b=20),
    legend=dict(orientation="h", yanchor="bottom", y=1.02, xanchor="right", x=1)
)

fig.show()

Formula Details

EwStd computes the exponentially weighted moving standard deviation recursively, with a bias correction that uses an effective sample size, \(N_{eff}\), calculated from the sum of weights. This ensures an unbiased estimate even as the influence of older observations diminishes over time.

Let:

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

For each new data point \(x_t\), EwStd updates four cumulative sums, \(S_x\), \(S_{xx}\), \(S_w\), 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 \(S_w\), 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. Finally, calculate the exponentially weighted moving standard deviation as:

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

The term \(N_{eff}\) adjusts for the effective sample size, ensuring that the standard deviation calculation remains unbiased by accounting for the diminishing weight of older values.