RollingFracDiff

Description

RollingFracDiff computes the rolling fractional differentiation of a time series, retaining long-term memory while applying a fractional degree of differencing. This method is useful for transforming a time series to achieve stationarity while preserving as much of the memory as possible. Fractional differentiation allows for non-integer degrees of differencing.

Formulas

For each new data point \(x_t\), RollingFracDiff computes weights \(w_k\) based on the desired degree of differentiation \(d\) and convolves the input time series with these weights over a rolling window of size window_size.

  1. Weight Calculation: Calculate weights \(w_k\) recursively based on \(d\):

\[ w_k = (-1)^k \cdot \binom{d}{k}, \]

which allows us to compute the weights in the following way:

\[ w_{0} = 1, \]
\[ w_k = - \frac{d - k + 1}{k} w_{k-1} \]

  1. Thresholding: Any weight \(w_k\) that falls below the specified threshold is ignored, effectively truncating the weight sequence.

  2. Rolling Fractional Differentiation: For each point \(t\), apply the computed weights over the rolling window, yielding the fractionally differentiated value \(y_t\):

\[ y_t = \sum_{k=0}^{\min(t, \mathrm{window_{size}} - 1)} w_k \cdot x_{t - k} \]

Parameters

  • window_size: Integer specifying the size of the rolling window. This is the number of past values used in the fractional differentiation calculation.

  • frac_order (d): Float representing the degree of fractional differentiation. d = 0 corresponds to an identity transformation of the data, while d=1 results in standard difference. The higher the value of d, the lower the amount of memory preserved.

  • threshold: Float specifying the minimum weight threshold. Any computed weight \(w_k\) smaller than this threshold is set to zero. This optimizes computation by ignoring insignificant contributions.

Usage Example and Plot

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

# Generate example data
data = np.cumsum(np.random.normal(size=1000))

# Apply rolling fractional differentiation with window size 30, frac_order 0.5, threshold 0.001
frac_diff_data = RollingFracDiff(window_size=30, frac_order=0.5, threshold=0.001)(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=frac_diff_data, mode='lines', name='Fractional Differentiation', line=dict(color='blue')), row=2, col=1)

fig.update_layout(
    title="Rolling Fractional Differentiation",
    xaxis_title="Index",
    yaxis=dict(title="Original Data"),
    yaxis2=dict(title="FracDiff Data"),
    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()