EwMean

Description

EwMean computes the exponentially weighted moving mean of a data sequence, applying more weight to recent data points and allowing for a smoother, lagged response. You specify the decay rate through alpha, calculated from com, span, halflife, or alpha itself, aligning with the Pandas ewm interface.

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 EwMean

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

# Compute exponentially weighted mean with a span of 20
ewmean_data = EwMean(span=20)(data)

# Create subplots with original and transformed 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=ewmean_data, mode='lines', name='EwMean', line=dict(color='red')), row=2, col=1)

fig.update_layout(
    title="Exponentially Weighted Moving Mean",
    xaxis_title="Index",
    yaxis=dict(title="Original Data"),
    yaxis2=dict(title="EwMean"),
    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

EwMean computes the exponentially weighted moving mean recursively. This calculation approach provides a fast, iterative update for each new value without recalculating the entire window, giving more weight to recent observations. The weighted mean formula aligns with the Pandas ewm interface by adjusting for the weighted contribution of each new element.

Let:

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

For each new data point \(x_t\), EwMean updates two cumulative sums, \(S_x\) and \(S_w\), as follows:

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

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

  1. Adjust \(S_w\), the cumulative weight, by similarly retaining a fraction \((1 - \alpha)\) of the previous weight sum, then adding a weight of \(1\):

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

  1. Compute the exponentially weighted moving mean as:

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