Discriminating Gaussian processes via quadratic form statistics

Abstract

Gaussian processes are powerful tools for modeling and predicting various numerical data. Hence, checking their quality of fit becomes a vital issue. In this article, we introduce a testing methodology for general Gaussian processes based on a quadratic form statistic. We illustrate the methodology on three statistical tests recently introduced in the literature, which are based on the sample autocovariance function, time average mean-squared displacement, and detrended moving average statistics. We compare the usefulness of the tests by taking into consideration three very important Gaussian processes: the fractional Brownian motion, which is self-similar with stationary increments (SSSIs), scaled Brownian motion, which is self-similar with independent increments (SSIIs), and the Ornstein–Uhlenbeck (OU) process, which is stationary. We show that the considered statistics’ ability to distinguish between these Gaussian processes is high, and we identify the best performing tests for different scenarios. We also find that there is no omnibus quadratic form test; however, the detrended moving average test seems to be the first choice in distinguishing between same processes with different parameters. We also show that the detrended moving average method outperforms the Cholesky method. Based on the previous findings, we introduce a novel procedure of discriminating between Gaussian SSSI, SSII, and stationary processes. Finally, we illustrate the proposed procedure by applying it to real-world data, namely, the daily EURUSD currency exchange rates, and show that the data can be modeled by the OU process.

Boosting the performance of anomalous diffusion classifiers with the proper choice of features

“…X τ i = 1 τ +1 τ j=0 X i− j . As mentioned in reference [82], the DMA-based statistical tests can help in the detection of the SBM. In our model, we used two values of DMA for each trajectory as input features, namely DMA(1) and DMA(2).…”

Anomalous diffusion, aging, and nonergodicity of scaled Brownian motion with fractional Gaussian noise: overview of related experimental observations and models

“…The flurry of new single-particle-tracking (SPT) datasets reporting on and novel theoretical-analysis tools assessing the properties of anomalous diffusion has established an unprecedented need for novel theoretical models of diffusion and transport. Such models should desirably embody certain characteristic features of different ''standard'' anomalous-diffusion processes [28][29][30][31][32][33] such as, i.e., conventional Brownian motion (BM), [34][35][36][37][38][39][40][41] fractional BM (FBM) [42][43][44][45][46][47][48] governed by fractional Gaussian noise, scaled BM (SBM) [49][50][51][52][53][54][55][56][57][58][59][60][61] and ultraslow SBM 62,63 with a power-law timedependent diffusivity D(t) p t aÀ1 , continuous-time random walks (CTRWs), 31,64,65 Le ´vy walks and flights, 66 heterogeneous diffusion processes (HDPs) 52,53,[67][68][69] w...…”

“…Such models should desirably embody certain characteristic features of different “standard” anomalous-diffusion processes 28–33 such as, i.e. , conventional Brownian motion (BM), 34–41 fractional BM (FBM) 42–48 governed by fractional Gaussian noise, scaled BM (SBM) 49–61 and ultraslow SBM 62,63 with a power-law time-dependent diffusivity D ( t ) ∝ t α −1 , continuous-time random walks (CTRWs), 31,64,65 Lévy walks and flights, 66 heterogeneous diffusion processes (HDPs) 52,53,67–69 with a power-law position-dependent diffusivity, D ( x ) ∝ | x | , etc.…”

Boosting the performance of anomalous diffusion classifiers with the proper choice of features

“…As mentioned in Ref. [61], the DMA-based statistical tests can help in the detection of the scaled Brownian motion. In our model, we used two values of DMA for each trajectory as input features, namely DM A(1) and DM A(2).…”