RollingOU

Description

RollingOU applies a rolling discrete-time Ornstein-Uhlenbeck (OU) model to a data series within a specified window. The OU model, commonly used to model mean-reverting processes, estimates the rate of reversion to the mean and the variance of the noise in each window. As a rolling estimator, RollingOU recalculates model parameters at each step based on a sliding window, making it suitable for non-stationary time series where parameters may evolve over time.

The discretized form of the OU model is:

\[ x[i] = \mu - (x[i-1] - \mu) \cdot \text{mrr} + \sigma \cdot \epsilon \]

where \( x[i] \) is the current value, \( x[i-1] \) is the previous value, \( \mu \) is the mean, \( \text{mrr} \) is the mean reversion rate, \( \sigma \) is the standard deviation of the noise, and \( \epsilon \sim \mathcal{N}(0,1) \) is a standard normal random variable. The parameter mrr controls the rate and nature of mean reversion:

  • 0 < mrr < 1: Values tend to revert to the mean at a rate determined by mrr, with a higher mrr closer to 1 indicating faster mean reversion.

  • mrr < 0: Values display mean repulsion, where deviations from the mean amplify rather than attenuate.

  • mrr > 1: Oscillatory behavior or instability arises, as the model overshoots the mean.

Parameters

window_size (int): The size of the rolling window used to fit the OU model. A larger window provides a smoother estimate but may be less responsive to rapid changes.

output (int): Specifies the output type:

  • 0: Returns the mean reversion rate (mrr).

  • 1: Returns the estimated mean (\(\mu\)).

  • 2: Returns the relative mean (\(\mu^\prime = \mu - x[i]\)), the mean relative to the last value.

  • 3: Returns the estimated standard deviation of the noise (\(\sigma\)).

  • start_policy: Defines how the function handles the initial phase when fewer than window_size data points are available. This parameter accepts one of the following three values:

    • "strict": Returns NaN for all calculations until window_size elements have been processed.

    • "expanding": Adapts the computation by dynamically reducing the window size to include all available data, starting from a single point and growing until window_size is reached.

    • "zero": Simulates a full initial window of zeros, effectively pre-filling the data stream with window_size zeros before processing the actual input.

Usage Example and Plot

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

# Generate synthetic data with cumulative noise
np.random.seed(0)
data = np.cumsum(np.random.normal(0, 1, 500))

# Define a rolling OU model with window size of 50
rolling_ou_mrr = RollingOU(window_size=50, output=0)
rolling_ou_mean = RollingOU(window_size=50, output=1)
rolling_ou_std = RollingOU(window_size=50, output=3)

# Apply RollingOU on the data for each output type
mrr_estimates = [rolling_ou_mrr(d) for d in data]
mean_estimates = [rolling_ou_mean(d) for d in data]
std_estimates = [rolling_ou_std(d) for d in data]

# Create subplots with original and estimated parameters
fig = make_subplots(rows=4, cols=1, shared_xaxes=True,
                    row_heights=[1/4, 1/4, 1/4, 1/4], vertical_spacing=0.05)

fig.add_trace(go.Scatter(y=data, mode='lines', name='Original Data'), row=1, col=1)
fig.add_trace(go.Scatter(y=mrr_estimates, mode='lines', name='Mean Reversion Rate (mrr)'), row=2, col=1)
fig.add_trace(go.Scatter(y=mean_estimates, mode='lines', name='Estimated Mean (μ)', line=dict(color='green')), row=3, col=1)
fig.add_trace(go.Scatter(y=std_estimates, mode='lines', name='Noise Std Dev (σ)', line=dict(color='purple')), row=4, col=1)

fig.update_layout(
    title="Rolling Discrete-Time Ornstein-Uhlenbeck Model Parameters",
    xaxis_title="Index",
    yaxis=dict(title="Original Data"),
    yaxis2=dict(title="mrr"),
    yaxis3=dict(title="mean"),
    yaxis4=dict(title="noise"),
    margin=dict(l=20, r=20, t=120, b=20),
    legend=dict(orientation="h", yanchor="bottom", y=1.02, xanchor="right", x=1)
)

fig.show()

Formula Details

The RollingOU function estimates parameters for the OU model in a rolling window. For a window of data \( \{x[i - W], \dots, x[i]\} \), it calculates:

  1. Mean Reversion Rate (mrr): The rate at which the model reverts to the mean is estimated from the correlation between current and prior values. For \( \text{mrr} > 0 \), the process exhibits mean-reverting behavior, with values tending to return to the mean. Negative values imply mean repulsion, while values greater than 1 indicate oscillatory or potentially unstable behavior.

  2. Realtive mean (\(\mu\)): The model estimates the long-term mean by extrapolating the model, giving the steady-state value toward which the process reverts in the absence of noise.

  3. Realtive mean (\(\mu^\prime\)): The relative mean is expressed relative to the last observation \(\mu^\prime = \mu - x[i]\). A positive relative means indicated that the mean is above the current signal value.

  4. Standard Deviation (\(\sigma\)): This parameter estimates the noise level by calculating the standard deviation of residuals within the window, capturing the typical magnitude of deviations around the mean.

where \( \epsilon \) is drawn from a standard normal distribution. This iterative update leverages a rolling window for dynamic re-estimation of the parameters, allowing adaptation to time-varying behavior in the data.