A first-order autoregressive model for the joint distribution of a time dependent random variable and its derivative

作者：Mole Nils^*; Chan Lawrence

来源：Environmetrics, 2011, 22(7): 826-846.

DOI：10.1002/env.1103

• first-order autoregressive process
• joint probability density function
• exponential, gamma and uniform distributions
• turbulent dispersion

A first-order autoregressive (AR(1)) model is analysed to investigate the relationship between a time dependent random variable Gamma(t) and its derivative Y(t) = partial derivative Gamma(t)/partial derivative(t). This is motivated by the problem of the turbulent dispersion of a gas cloud or plume in the atmosphere, where the square of the derivative, D(t) = Y(t)(2), is also of interest. It is shown that in general Gamma(t) and D(t) are correlated, with their correlation proportional to the skewness of Gamma(t). The probability distributions of the innovations and of Y(t) and D(t), and the joint distributions of Gamma(t) and Y(t), and of Gamma(t) and D(t), are derived for a number of assumed distributions for Gamma(t). The latter include the normal, exponential, gamma (of integer order), uniform, and sum of uniform distributions. The lack of time reversibility of the AR(1) model is apparent in the asymmetric distributions of the derivative when Gamma(t) has the exponential or gamma distribution. Comparisons are made with some experimental observations.

• 出版日期2011-11