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Item Contributions to applied probability : a thesis presented for the degree of Doctor of Science at Massey University, Albany, New Zealand(Massey University, 2004) Hunter, Jeffrey JosephThis thesis covers a selection of published research papers, manuscripts and book chapters of the contributions that the author has made in the field of applied probability. The overall themes focus on the development of the theory and applications of Markov renewal processes, Markov chains, generalized matrix inverses, queueing models, and two dimensional renewal processes. The presentation highlights some strong interconnections between many of the topics including Markov renewal theory and Markov chains to queueing models; generalized matrix inverses to Markov chains and Markov renewal processes; and correlated bivariate processes as two-dimensional renewal processes and arrival processes to queueing models. Many of the research papers have appeared in the "Advances in Applied Probability" or in "Linear Algebra and its Applications" highlighting the strong interdisciplinary links and contributions that the author has made to the both the fields of Applied Probability and Linear Algebra.Item Short-Term Wind Speed Forecasting Based on Hybrid MODWT-ARIMA-Markov Model(IEEE, 2021-06-08) Yousuf MU; Al-Bahadly I; Avci E; Do TDMarkov chains (MC) are statistical models used to predict very short to short-term wind speed accurately. Such models are generally trained with a single moving window. However, wind speed time series do not possess an equal length of behavior for all horizons. Therefore, a single moving window can provide reasonable estimates but is not an optimal choice. In this study, a forecasting model is proposed that integrates MCs with an adjusting dynamic moving window. The model selects the optimal size of the window based on a similar approach to the leave-one-out method. The traditional model is further optimized by introducing a self-adaptive state categorization algorithm. Instead of synthetically generating time series, the modified model directly predicts one-step ahead wind speed. Initial results indicate that adjusting the moving window MC prediction model improved the forecasting performance of a single moving window approach by 50%. Based on preliminary findings, a novel hybrid model is proposed integrating maximal overlap discrete wavelet transform (MODWT) with auto-regressive integrated moving average (ARIMA) and adjusting moving window MC. It is evident from the literature that MC models are suitable for predicting residual sequences. However, MCs were not considered as a primary forecasting model for the decomposition-based hybrid approach in any wind forecasting studies. The improvement of the novel model is, on average, 55% for single deep learning models and 30% for decomposition-based hybrid models.Item Efficient Markov bases for Z-polytope sampling : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatū, New Zealand(Massey University, 2021) McVeagh, MichaelIn this thesis we study the use of lattice bases for fibre sampling, with particular attention paid to applications in volume network tomography. We use a geometric interpretation of the fibre as a Z-polytope to provide insight into the connectivity properties of lattice bases. Fibre sampling is used when we are interested in fitting a statistical model to a random process that may only be observed indirectly via the underdetermined linear system y = Ax. We consider the observed data y and random variable of interest x to contain count data. The likelihood function for such models requires a summation over the fibre Fy, the set of all non-negative integer vectors x satisfying this equation for some particular y. This can be computationally infeasible when Fy is large. One approach to addressing this problem involves sampling from Fy using a Markov Chain Monte Carlo algorithm, which amounts to taking a random walk through Fy . This is facilitated by a Markov basis: a set of moves that can be used construct such a walk, which is therefore a subset of the kernel of the configuration matrix A. Algebraic algorithms for finding Markov bases based on the theory of Gröbner bases are available, but these can fail when the configuration matrix is large and the calculations become computationally infeasible. Instead, we propose constructing a sampler based on a type of lattice basis we call a column partition lattice basis, defined by a matrix U. Constructing such a basis is computationally much cheaper than constructing a Gröbner basis. It is known that lattice bases are not necessarily Markov bases. We give a condition on the matrix U that guarantees that it is a Markov basis, and show for a certain class of configuration matrices how a U matrix that is a Markov basis can be constructed. Construction of lattice bases that are Markov bases is facilitated when the configuration matrix is unimodular, or has unimodular partitions. We consider configuration matrices from volume network tomography, and give classes of traffic network that have configuration matrices with these desirable properties. If a Markov basis cannot be found, one alternative is to sample from some larger set that includes Fy . We give some larger sets that can be used, subject to certain conditions.Item Application of Markov chain model in streamflow forecasting : a thesis submitted in partial fulfilment of the requirements for the degree of Master of Science in Geography at Massey University(Massey University, 1996) Mpelasoka, Freddie SimonThis thesis presents an approach to streamflow forecasting based on a Markov chain model to estimate conditioned probabilities that a one time-step ahead streamflow forecast will be within a certain streamflow range. In this application a set of "states of flow" defined over streamflow ranges (intervals) forms a finite state space of a Markov chain. Flood forecasting is made by focusing on a preselected state of flow as a flood state. A multi-objective (two criteria) function for the quantification of the model performance is introduced. Specifically designed for a flood forecasting and warning system the two criteria are the probability of issuing a false alarm and the probability of failing to forecast a flood event. The goal is to minimize both criteria simultaneously together with a preference of accepting more false alarms than misses. The model has two options of making a forecast: (1) a Threshold Forecast (ThF) approach in which a forecast is based on the probability of making a one-step transition from any state into the flood state; (2) the Most Probable Event (MPE) forecast approach selects the state of flow where the next streamflow is most likely to occur. Forecasts being probabilistic, there are several options for deciding on when it is appropriate to issue a flood warning in the probabilistic framework. A search for the appropriate probability p0 is made on interval [0,1] through evaluation of the objective function at each p0, using data sets from three North Island catchments ( Akitio River, Makakahi River and Kiwitea Stream). The model applying the option of threshold forecasts performed generally well depending on the relative costs assigned to false alarms and misses. The model performed better on the Akitio River which has strongly fluctuating streamflows than on the Makakahi River and Kiwitea Stream which have relatively modest variations in flows. When the Model applied the option of the most probable event forecasts did not perform well as the probabilities of false alarms were found to be too high for the model to be accepted. The outcome of this study suggests a simple short-term flood forecasting procedure especially for rivers with strongly fluctuating flows.