Some models for rainfall based on stochastic point processes

1987 ◽  
Vol 410 (1839) ◽  
pp. 269-288 ◽  
Keyword(s):  
Poisson Process ◽  
Stochastic Models ◽  
Point Processes ◽  
Rainfall Intensity ◽  
Storm Events ◽  
Total Rainfall ◽  
Random Intensity ◽  
Hourly Rainfall ◽  
Complex Models ◽  
Stochastic Point Processes

Stochastic models are discussed for the variation of rainfall intensity at a fixed point in space. First, models are analysed in which storm events arise in a Poisson process, each such event being associated with a period of rainfall of random duration and constant but random intensity. Total rainfall intensity is formed by adding the contributions from all storm events. Then similar but more complex models are studied in which storms arise in a Poisson process, each storm giving rise to a cluster of rain cells and each cell being associated with a random period of rain. The main properties of these models are determined analytically. Analysis of some hourly rainfall data from Denver, Colorado shows the clustered models to be much the more satisfactory.

The coincidence approach to stochastic point processes

1975 ◽  
Vol 7 (1) ◽  
pp. 83-122 ◽  
Author(s):  
Odile Macchi
Keyword(s):  
Poisson Process ◽  
Probability Space ◽  
Point Processes ◽  
Counting Process ◽  
Regular Point ◽  
General Point ◽  
Completely Regular ◽  
Photon Process ◽  
Stochastic Point Processes

The structure of the probability space associated with a general point process, when regarded as a counting process, is reviewed using the coincidence formalism. The rest of the paper is devoted to the class of regular point processes for which all coincidence probabilities admit densities. It is shown that their distribution is completely specified by the system of coincidence densities. The specification formalism is stressed for ‘completely’ regular point processes. A construction theorem gives a characterization of the system of coincidence densities of such a process. It permits the study of most models of point processes. New results on the photon process, a particular type of conditioned Poisson process, are derived. New examples are exhibited, including the Gauss-Poisson process and the ‘fermion’ process that is suitable whenever the points are repulsive.

scholarly journals A fluid limit for a cache algorithm with general request processes

2010 ◽  
Vol 42 (03) ◽  
pp. 816-833 ◽  
Author(s):  
Takayuki Osogami
Keyword(s):  
Stochastic Models ◽  
Point Processes ◽  
Original System ◽  
Poisson Processes ◽  
Fluid Limit ◽  
Average Probability ◽  
Inhomogeneous Poisson Processes ◽  
Formal Limit ◽  
Stochastic Point Processes ◽  
General Stochastic

We introduce a formal limit, which we refer to as a fluid limit, of scaled stochastic models for a cache managed with the least-recently-used algorithm when requests are issued according to general stochastic point processes. We define our fluid limit as a superposition of dependent replications of the original system with smaller item sizes when the number of replications approaches ∞. We derive the average probability that a requested item is not in a cache (average miss probability) in the fluid limit. We show that, when requests follow inhomogeneous Poisson processes, the average miss probability in the fluid limit closely approximates that in the original system. Also, we compare the asymptotic characteristics, as the cache size approaches ∞, of the average miss probability in the fluid limit to those in the original system.

A marked Poisson process model of summer rainfall in southern Ontario

1985 ◽  
Vol 12 (4) ◽  
pp. 886-898 ◽  
Author(s):  
J. D. Bonser ◽  
T. E. Unny ◽  
K. Singhal
Keyword(s):  
Poisson Process ◽  
Process Model ◽  
Point Processes ◽  
Summer Rainfall ◽  
Model Parameters ◽  
Southern Ontario ◽  
Hourly Rainfall ◽  
Storm Duration ◽  
Rainfall Volume ◽  
Rainfall Occurrences

A mathematical description of summer rainfall occurrences in southern Ontario is developed in this paper. The theory of Poisson point processes with specific application to rainfall modelling is presented with a critical review of previous literature on Poisson rainfall models. A marked Poisson process model of summer storms is formulated, using the marks to represent the random duration and intensity of the events. Model parameters are estimated for four locations in southern Ontario using a total of 48 seasons of hourly rainfall data. The model is applied to calculate the seasonal return period of extreme storms and the probability distribution of total seasonal rainfall volume. These two examples demonstrate the accuracy and usefulness of the model. Key words: rainfall, storm duration, storm intensity, temporal storm pattern, probabilistic model, Poisson process, exponential distribution, gamma distribution, Weibull distribution.

scholarly journals A fluid limit for a cache algorithm with general request processes

2010 ◽  
Vol 42 (3) ◽  
pp. 816-833 ◽  
Author(s):  
Takayuki Osogami
Keyword(s):  
Stochastic Models ◽  
Point Processes ◽  
Original System ◽  
Poisson Processes ◽  
Fluid Limit ◽  
Average Probability ◽  
Inhomogeneous Poisson Processes ◽  
Formal Limit ◽  
Stochastic Point Processes ◽  
General Stochastic

We introduce a formal limit, which we refer to as a fluid limit, of scaled stochastic models for a cache managed with the least-recently-used algorithm when requests are issued according to general stochastic point processes. We define our fluid limit as a superposition of dependent replications of the original system with smaller item sizes when the number of replications approaches ∞. We derive the average probability that a requested item is not in a cache (average miss probability) in the fluid limit. We show that, when requests follow inhomogeneous Poisson processes, the average miss probability in the fluid limit closely approximates that in the original system. Also, we compare the asymptotic characteristics, as the cache size approaches ∞, of the average miss probability in the fluid limit to those in the original system.

Topics in stochastic point processes, by applications in hydrology

1984 ◽  
Vol 16 (1) ◽  
pp. 20-20
Author(s):  
P. Todorovic
Keyword(s):  
Porous Media ◽  
Stochastic Models ◽  
Random Function ◽  
Point Processes ◽  
Marked Point ◽  
Renewal Theory ◽  
Marked Point Processes ◽  
Martingale Approach ◽  
Stochastic Point Processes ◽  
Hydrological Applications

The strong impetus for the research in the theory of point processes comes from applications, real or potential, to a multitude of engineering, industrial and biological problems. Here we discuss some particular topics in this area and their applications in hydrology. Specifically, we consider stochastic models of a variety of geophysical phenomena, such as the rainfall, floods, sediment transport, dispersion in porous media and some others. The first part of this presentation is concerned with point processses on R+. Our discussion includes the stochastic intensity, some remarks on the martingale approach and a brief expose of the elements of renewal theory. In addition, we discuss in some detail the case when the counting random function represents a Markov process. In the second part we give an introduction to the theory of point processes on an abstract topological space. Elements of marked point processes, which are of particular interest in hydrological investigations, are also included. The rest of the paper is concerned with some hydrological applications.

scholarly journals Typical Rainfall Distribution Pattern of Flood Event Caused by Tropical Cyclone at Bima City, West Nusa Tenggara, Indonesia

2019 ◽  
Vol 5 (1) ◽  
pp. 1
Author(s):  
Rachna Sok
Keyword(s):  
Tropical Cyclone ◽  
Distribution Pattern ◽  
Tropical Cyclones ◽  
Rainfall Intensity ◽  
Heavy Precipitation ◽  
Rainfall Event ◽  
Total Rainfall ◽  
Rainfall Distribution ◽  
Hourly Rainfall ◽  
The Impact

Tropical cyclones are the most serious meteorological phenomena that hit Bima city in December 2016. The strong winds and heavy precipitation associated with a typhoon significantly affect the weather in this city. The impact of a tropical cyclone on precipitation variability in Bima is studied using rainfall data for analyzing hourly rainfall distribution pattern during the event. Depend on the geographic situation and climate characteristic, the hourly rainfall distribution pattern of one area is different to others area. The research aims to analyze hourly rainfall distribution pattern in the form of the rainfall intensity distribution. This research is conducted using one automatic rainfall gauge in Bima city, West Nusa Tenggara province that obtained from Regional Disaster Management Agency (BPBD). The results showed that two events of rainfall were recorded. The first rainfall event was on 20th to 21st December 2016 with a total rainfall 191.4 mm. The second rainfall event occurred on 22nd to 23rd December 2016 with a total rainfall 126.2 mm. The rainfall distribution pattern has rainfall intensity peak at 45% of duration with cumulative rainfall reached 70%. It was found there is no common pattern of temporal rainfall distribution for rainfall induced by tropical cyclones.

The coincidence approach to stochastic point processes

1975 ◽  
Vol 7 (01) ◽  
pp. 83-122 ◽  
Author(s):  
Odile Macchi
Keyword(s):  
Poisson Process ◽  
Probability Space ◽  
Point Processes ◽  
Counting Process ◽  
Regular Point ◽  
General Point ◽  
Completely Regular ◽  
Photon Process ◽  
Stochastic Point Processes

The structure of the probability space associated with a general point process, when regarded as a counting process, is reviewed using the coincidence formalism. The rest of the paper is devoted to the class of regular point processes for which all coincidence probabilities admit densities. It is shown that their distribution is completely specified by the system of coincidence densities. The specification formalism is stressed for ‘completely’ regular point processes. A construction theorem gives a characterization of the system of coincidence densities of such a process. It permits the study of most models of point processes. New results on the photon process, a particular type of conditioned Poisson process, are derived. New examples are exhibited, including the Gauss-Poisson process and the ‘fermion’ process that is suitable whenever the points are repulsive.

An analysis of the Pólya point process

1983 ◽  
Vol 15 (01) ◽  
pp. 39-53 ◽  
Author(s):  
Ed Waymire ◽  
Vijay K. Gupta
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Critical Value ◽  
General Representation ◽  
Infinitely Divisible ◽  
Generating Functional ◽  
Representation Problem ◽  
Polya Process ◽  
Pólya Process ◽  
Stochastic Point Processes

The Pólya process is employed to illustrate certain features of the structure of infinitely divisible stochastic point processes in connection with the representation for the probability generating functional introduced by Milne and Westcott in 1972. The Pólya process is used to provide a counterexample to the result of Ammann and Thall which states that the class of stochastic point processes with the Milne and Westcott representation is the class of regular infinitely divisble point processes. So the general representation problem is still unsolved. By carrying the analysis of the Pólya process further it is possible to see the extent to which the general representation is valid. In fact it is shown in the case of the Pólya process that there is a critical value of a parameter above which the representation breaks down. This leads to a proper version of the representation in the case of regular infinitely divisible point processes.

On Updating Algorithms and Inference for Stochastic Point Processes

1975 ◽  
Vol 12 (S1) ◽  
pp. 239-259 ◽  
Author(s):  
D. Vere-Jones
Keyword(s):  
Stochastic Process ◽  
Maximum Likelihood ◽  
Stochastic Approximation ◽  
Approximation Theory ◽  
Point Processes ◽  
Maximum Likelihood Estimates ◽  
Doubly Stochastic ◽  
Stochastic Point Processes ◽  
Asymptotically Efficient ◽  
Estimation And Filtering

This paper is an attempt to interpret and extend, in a more statistical setting, techniques developed by D. L. Snyder and others for estimation and filtering for doubly stochastic point processes. The approach is similar to the Kalman-Bucy approach in that the updating algorithms can be derived from a Bayesian argument, and lead ultimately to equations which are similar to those occurring in stochastic approximation theory. In this paper the estimates are derived from a general updating formula valid for any point process. It is shown that almost identical formulae arise from updating the maximum likelihood estimates, and on this basis it is suggested that in practical situations the sequence of estimates will be consistent and asymptotically efficient. Specific algorithms are derived for estimating the parameters in a doubly stochastic process in which the rate alternates between two levels.

Convergence to equilibrium in a traffic model with restricted passing

1979 ◽  
Vol 16 (4) ◽  
pp. 881-889 ◽  
Author(s):  
Hans Dieter Unkelbach
Keyword(s):  
Poisson Process ◽  
Limit Theorems ◽  
Point Processes ◽  
Road Traffic ◽  
Poisson Processes ◽  
Traffic Model ◽  
Cluster Point ◽  
Convergence To Equilibrium ◽  
Constant Intensity

A road traffic model with restricted passing, formulated by Newell (1966), is described by conditional cluster point processes and analytically handled by generating functionals of point processes.The traffic distributions in either space or time are in equilibrium, if the fast cars form a Poisson process with constant intensity combined with Poisson-distributed queues behind the slow cars (Brill (1971)). It is shown that this state of equilibrium is stable, which means that this state will be reached asymptotically for general initial traffic distributions. Furthermore the queues behind the slow cars dissolve asymptotically like independent Poisson processes with diminishing rate, also independent of the process of non-queuing cars. To get these results limit theorems for conditional cluster point processes are formulated.

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