Discriminating Gaussian processes via quadratic form statistics | JoVE Visualize

Area of Science:

Statistics
Time Series Analysis
Machine Learning

Background:

Gaussian processes are essential for numerical data modeling and prediction.
Assessing the quality of fit for Gaussian processes is crucial for reliable analysis.
Existing statistical tests have limitations in distinguishing between various Gaussian process types.

Purpose of the Study:

To introduce a novel testing methodology for general Gaussian processes using quadratic form statistics.
To compare the effectiveness of different statistical tests in discriminating between key Gaussian processes.
To propose a new procedure for distinguishing between Gaussian self-similar with stationary increments (SSSI), self-similar with independent increments (SSII), and stationary processes.

Main Methods:

Utilizing a quadratic form statistic for hypothesis testing.
Illustrating the methodology with three statistical tests: autocovariance function, mean-squared displacement, and detrended moving average.
Comparing test performance on fractional Brownian motion (SSSI), scaled Brownian motion (SSII), and Ornstein-Uhlenbeck (OU) processes.
Evaluating the detrended moving average method against the Cholesky method.

Main Results:

The considered statistics demonstrate high ability in distinguishing between different Gaussian processes.
The detrended moving average test is identified as the best performing for distinguishing processes with different parameters.
The detrended moving average method shows superior performance compared to the Cholesky method.
No single omnibus quadratic form test was found to be universally optimal.

Conclusions:

A new procedure for discriminating between Gaussian SSSI, SSII, and stationary processes is introduced.
The detrended moving average test is a highly effective tool for Gaussian process model selection.
The proposed methodology and procedure are validated using real-world financial exchange rate data (EURUSD).
The daily EURUSD currency exchange rates are shown to be well-modeled by the Ornstein-Uhlenbeck process.