Using filtered Poisson processes to model a river flow

Let X(t) be the flow of a river at time t  . Models of the form X(t)=∑n=1N(t)Yn(t-τn)ke-(t-τn)/c are considered, where the τn’s are the arrival times of the events of the Poisson process {N(t), t ⩾ 0} with rate λ, and the Yn’s are independent exponentially distributed random variables with parameter μ. The parameters c, λ and μ must be estimated. An application to the Delaware River is presented. We find that the basic model, namely that for which k = 0, can be significantly improved. We also use the model to forecast the river flow at time t + 1, based on the history of the process in the interval [0, t].