非中心高斯过程的最优和次优二次型
Optimal and sub-optimal quadratic forms for non-centered Gaussian processes
摘要：
随机过程的单个随机轨迹常采用二次型分析，如时间平均（TA）均方位移（MSD）或速度自关联函数（VACF）。适当的二次型预期具有窄的概率分布，以减少单个测量的统计不确定性。我们考虑的问题是找到最佳的二次型，最小化选定的累积矩（例如，方差）的概率分布，在固定的平均值的约束下。对于离散非中心高斯过程，我们利用累积矩的谱表示构造了最优二次型。此外，我们得到了一个简单的明确的公式，可作为其他二次型的质量基准的最小可实现的累积矩。我们说明了最优性问题，通过比较最优方差与TA MSD和TA VACF的分数布朗运动叠加有恒定漂移和独立的高斯噪声的方差。
英文摘要：
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 non-centered Gaussian processes, we construct the optimal quadratic form by using the spectral representation of the 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.
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