An Application of Filtered Renewal Processes in Hydrology

Filtered renewal processes are used to forecast daily river flows. For these processes, contrary to filtered Poisson processes, the time between consecutive events is not necessarily exponentially distributed, which is more realistic. The model is applied to obtain one- and two-day-ahead forecasts of the flows of the Delaware and Hudson Rivers, both located in the United States. Better results are obtained than with filtered Poisson processes, which are often used to model river flows. 1. Introduction Let be a Poisson process with rate . A filtered Poisson (sometimes called shot noise) process is a continuous-time stochastic process defined by in which the random variables denote the arrival times of the events of the Poisson process and are assumed to be independent and identically distributed random variables that are also independent of . The function is called the response function. In many applications, the response function is chosen of the form where is a parameter that must be estimated. It then gives the value at time of an event of magnitude of the Poisson process that occurred at time . Moreover, the random variables are generally assumed to be exponentially distributed with parameter . With the above response function, the filtered Poisson process behaves as in Figure 1. Actually, this behavior depends on the form of the response function, but not on the distribution of the time between the events. Therefore, the same behavior would be observed in the case when is a renewal process. Remember that a Poisson process is a particular renewal process. Figure 1: Example of a trajectory of a filtered Poisson process defined by ( 1) and ( 2). To model the daily flows of rivers given their most recent observed values, conceptual and physical models based on the filtered Poisson process have been widely used successfully for many decades; see, for example, Weiss [1], Kelman [2], Koch [3], and Konecny [4]. Filtered Poisson processes are still being used to model various phenomena in civil engineering; see Yin et al. [5] and Miyamoto et al. [6]. Now, especially in hydrological applications, the form of the response function in (2) is taken for granted rather than being justified by the observations of the variable of interest. Actually, it is generally simply an assumption made to obtain a mathematically tractable model. Similarly, the actual distribution of the ’s is not investigated. However, if one is only interested in forecasting the value of , based on the values of the process up to time , this does not really cause a problem because the forecast