Optimal and suboptimal quadratic forms for noncentered Gaussian processes

Individual random trajectories of stochastic processes are often analyzed by using quadratic forms such as time averaged (TA) mean square displacement (MSD) or velocity auto-correlation function (VACF). The appropriate quadratic form is expected to have a narrow probability distribution in order to reduce statistical uncertainty of a single measurement. We consider the problem of finding the optimal quadratic form that minimizes a chosen cumulant moment (e.g., the variance) of the probability distribution, under the constraint of fixed mean value. For discrete noncentered Gaussian processes, we construct the optimal quadratic form by using the spectral representation of cumulant moments. Moreover, we obtain a simple explicit formula for the smallest achievable cumulant moment that may serve as a quality benchmark for other quadratic forms. We illustrate the optimality issues by comparing the optimal variance with the variances of the TA MSD and TA VACF of fractional Brownian motion superimposed with a constant drift and independent Gaussian noise.

Figure 1

The ratio ${\kappa }_2/\left(2{\kappa }_1^2\right)$ for the TA MSD of the discrete fBm versus the Hurst exponent $H$ (with $N=100$, $\sigma =1$, $\mu =\varepsilon =0$), with different lag times: $n=1$ (blue solid line), $n=10$ (green dashed line), and $n=50$ (red dash-dotted line). This ratio is normalized by optimal value $1/N$.

Figure 2

The ratio ${\overline{\kappa }}_2/{\kappa }_2^{\mathrm{opt}}$ for the TA MSD of the discrete fBm versus the Hurst exponent $H$ (with $N=100$, $\sigma =1$): (a) for three drifts $\mu$ (without noise, $\varepsilon =0$); (b) for four noise levels $\varepsilon$ (without drift, $\mu =0$). Here ${\overline{\kappa }}_2$ is the variance at the lag time at which ${\kappa }_2/\left(2{\kappa }_1^2\right)$ is minimal.

Figure 3

The ratio ${\overline{\kappa }}_2/{\kappa }_2^{\mathrm{opt}}$ for the TA VACF of the discrete fBm of the Hurst exponent $H$ (with $N=100$, $\sigma =1$). (a) For three drifts $\mu$ (without noise, $\varepsilon =0$); (b) for four noise levels $\varepsilon$ (without drift, $\mu =0$). Here ${\overline{\kappa }}_2$ is the variance at the lag time at which ${\kappa }_2/\left(2{\kappa }_1^2\right)$ is minimal.

Figure 4

The ratio ${\overline{\kappa }}_2/{\kappa }_2^{\mathrm{opt}}$ for the TA MSD of the discrete Ornstein-Uhlenbeck process, versus the inverse time scale $\theta$ (with $N=100$, $\sigma =1$): (a) no drift ($\mu =0$) and four values of noise variance ${\varepsilon }^2$ (shown by four curves); (b) no noise ($\varepsilon =0$) and four values of drift $\mu$ (shown by four curves). Here ${\overline{\kappa }}_2$ is the variance at the lag time at which ${\kappa }_2/\left(2{\kappa }_1^2\right)$ is minimal.