Mean and covariance of the partial derivatives (gradient, velocity) of a stochastic process

Antonio Sala, UPV

Difficulty: ***** ,       Relevance: ,      Duration: 17:54

*Enlace a Spanish version

Summary:

Certain stochastic processes (“second order” or higher, whatever that means, we’ll skip it for the moment being) are differentiable... for example, the speed of a moving mass subjected to random accelerations can be characterised: that speed is a stochastic process. Position is the integral of velocity, but velocity is the derivative of position.

Formally defining the derivative of a stochastic process requires some rigour (convergence in probability or in mean square) but, well, if we make a “leap of faith” and believe that we understand what the speed mentioned above is, then, from the statistical characteristics of the position in mean and variance (covariance function $k(position1,position2)$ ) the speed characteristics in terms of mean and variance, and speed-position covariance can be calculated.

In a multivariable stochastic process $f(x)$ (not a time series $f(t)$ as seemed to be implicitly assumed before), the idea is generalized to estimating partial derivatives of said process $f:{ℝ}^n\to ℝ$ (if they exist), from covariance function $k(x,x^{\prime })$.

This video proves that the gradient of the covariance $k(x,x^{\prime })$ is the covariance between the process and its derivative, and that the Hessian (second derivatives) of the covariance gives us the covariance between partial derivatives at different points.

Of course, in time series, operations could also be done with associated stochastic differential equations (for example, Kalman filtering) or with the power spectral density. These issues are not considered here.

*Link to my [ whole collection] of videos in English. Link to larger [ Colección completa] in Spanish.