Point process models for diurnal variation rainfall data : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Statistics at Massey University, Albany (Auckland), New Zealand
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Date
2014
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Massey University
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Abstract
The theoretical basis of the point process rainfall models were developed for midlatitude
rainfall that have different temporal characteristics from the tropical rainfall. The diurnal
cycle, a prominent feature in the tropical rainfall, is not represented in the point process
models. An extension of the point process models were developed to address the diurnal
variation in rainfall. An observed indicator of the rainfall, X is added to the point process
models. Two point process models, Poisson white noise (PWN) and Neyman-Scott white
noise (NSWN) model were used as the main rainfall event, Y . The rainfall is modelled
assuming two cases for the variable X, independent and dependent. Bernoulli trials with
Markov dependence are used for the dependent assumption. To allow the model to display
the diurnal variation and correlation between hours, the model was fitted to monthly rain-
fall data by using the properties of two hour blocks for each month of the year. However,
the main point process models were assumed the same for each of the 12 blocks, thus
having only one set of point process parameters for the models for each month. There
are 12 rainfall occurrence parameters and 12 Markov dependence parameters, one for each
block. A total of six models were fitted to the hourly rainfall data from 1974 to 2008 taken
from a rain site in Empangan Genting Klang, Malaysia.
The PWN and NSWN models with X were first fitted with the assumption that
the rainfall indicators are independent between the hours within the two hour block.
Simulation studies showed the model does not fit the moments properties adequently.
The models were then modified based on a dependence assumption between the hours
within the two hour block. These models are known as the Markov X-PWN and Markov
X-NSWN models. Both models improve the fit of the moment properties. However, having
only one point process model to represent the rainfall events for Malaysia rainfall data
was not sufficient. Since tropical rainfall consists of two types of rain, convective and
stratiform, the PWN and Markov X-NSWN model were superposed to represent the two
types of rainfall. A simple method by assuming non-homogenous PWN process for every
two hour block did not fit well the daily diurnal variation. A comparison between the
six models show that the superposed PWN and Markov X-NSWN model improved the
fitting of mean, variance and autocorrelation. The superposed model was then simplified
to an 8-block model to reduce the number of parameters. This modification to the point
process models succeeded in describing the diurnal variation in the rainfall, but some of
the models were not able to fit other properties that were not included in the parameter
estimation process such as the extreme values.
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Keywords
Rain and rainfall, Mathematical models, Point processes, Research Subject Categories::MATHEMATICS::Applied mathematics::Mathematical statistics