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1	Amos-type bounds for modified Bessel function ratios. Hornik, Kurt, Grün, Bettina January 2013 (has links) (PDF) (please take a look at the pdf)
2	On Maximum Likelihood Estimation of the Concentration Parameter of von Mises-Fisher Distributions Hornik, Kurt, Grün, Bettina 10 1900 (has links) (PDF) Maximum likelihood estimation of the concentration parameter of von Mises-Fisher distributions involves inverting the ratio R_nu = I_{nu+1} / I_nu of modified Bessel functions. Computational issues when using approximative or iterative methods were discussed in Tanabe et al. (Comput Stat 22(1):145-157, 2007) and Sra (Comput Stat 27(1):177-190, 2012). In this paper we use Amos-type bounds for R_nu to deduce sharper bounds for the inverse function, determine the approximation error of these bounds, and use these to propose a new approximation for which the error tends to zero when the inverse of R is evaluated at values tending to 1 (from the left). We show that previously introduced rational bounds for R_nu which are invertible using quadratic equations cannot be used to improve these bounds. / Series: Research Report Series / Department of Statistics and Mathematics
3	movMF: An R Package for Fitting Mixtures of von Mises-Fisher Distributions Hornik, Kurt, Grün, Bettina 07 1900 (has links) (PDF) Finite mixtures of von Mises-Fisher distributions allow to apply model-based clustering methods to data which is of standardized length, i.e., all data points lie on the unit sphere. The R package movMF contains functionality to draw samples from finite mixtures of von Mises-Fisher distributions and to fit these models using the expectation-maximization algorithm for maximum likelihood estimation. Special features are the possibility to use sparse matrix representations for the input data, different variants of the expectationmaximization algorithm, different methods for determining the concentration parameters in the M-step and to impose constraints on the concentration parameters over the components. In this paper we describe the main fitting function of the package and illustrate its application. In addition we compare the clustering performance of finite mixtures of von Mises-Fisher distributions to spherical k-means. We also discuss the resolution of several numerical issues which occur for estimating the concentration parameters and for determining the normalizing constant of the von Mises-Fisher distribution. (authors' abstract)
4	Highway Development Decision-Making Under Uncertainty: Analysis, Critique and Advancement El-Khatib, Mayar January 2010 (has links) While decision-making under uncertainty is a major universal problem, its implications in the field of transportation systems are especially enormous; where the benefits of right decisions are tremendous, the consequences of wrong ones are potentially disastrous. In the realm of highway systems, decisions related to the highway configuration (number of lanes, right of way, etc.) need to incorporate both the traffic demand and land price uncertainties. In the literature, these uncertainties have generally been modeled using the Geometric Brownian Motion (GBM) process, which has been used extensively in modeling many other real life phenomena. But few scholars, including those who used the GBM in highway configuration decisions, have offered any rigorous justification for the use of this model. This thesis attempts to offer a detailed analysis of various aspects of transportation systems in relation to decision-making. It reveals some general insights as well as a new concept that extends the notion of opportunity cost to situations where wrong decisions could be made. Claiming deficiency of the GBM model, it also introduces a new formulation that utilizes a large and flexible parametric family of jump models (i.e., Lévy processes). To validate this claim, data related to traffic demand and land prices were collected and analyzed to reveal that their distributions, heavy-tailed and asymmetric, do not match well with the GBM model. As a remedy, this research used the Merton, Kou, and negative inverse Gaussian Lévy processes as possible alternatives. Though the results show indifference in relation to final decisions among the models, mathematically, they improve the precision of uncertainty models and the decision-making process. This furthers the quest for optimality in highway projects and beyond. highway development decision making under uncertainty real options jump processes infinitely divisible distributions Lévy processes measure theory characteristic function characteristic exponent Lévy jump measure probability measure cádlág stochastic process Wiener process Poisson process compound Poisson process finite activity models jump diffusion models Merton model Gaussian jump process Kou model double exponential jump process leptokurtic volatility smile infinite activity models pure jumps process generalized hyperbolic model modified Bessel function negative inverse Gaussian model geometric Brownian motion model log-normal distribution model normality assumption heavy-tailed distribution asymmetric distribution quantile-quantile plot Rydberg algorithm parameter calibration moments cummulants sum of squares Monte Carlo simulation least-squares regression backward dynamic programming discrete-time Markov chain land price traffic demand highway service quality index data collection traffic volume Annual Average Daily Traffic Ontario Ministry of Transportation Land Registry Office Freedom of Information Act seasonality value of transportation types of transportation systems importance of highway systems cost of wrong decisions monetary cost private sector cost human cost land expropriation environmental cost irreversibility cost factor time cost factor political cost factor opportunity cost opportunity cost of wrong decisions economic cost of wrong decisions economic cost of mis-estimation economic profit of wrong decisions cost of inaction price of making right decisions scope of decisions rehabilitation decision land acquisition decision expansion decision uncertainty modeling land acquisition cost expropriation cost construction cost material cost Montréal-Mirabel International Airport Pickering Airport 407 Express Toll Route Highway 407 Statistics
5	Highway Development Decision-Making Under Uncertainty: Analysis, Critique and Advancement El-Khatib, Mayar January 2010 (has links) While decision-making under uncertainty is a major universal problem, its implications in the field of transportation systems are especially enormous; where the benefits of right decisions are tremendous, the consequences of wrong ones are potentially disastrous. In the realm of highway systems, decisions related to the highway configuration (number of lanes, right of way, etc.) need to incorporate both the traffic demand and land price uncertainties. In the literature, these uncertainties have generally been modeled using the Geometric Brownian Motion (GBM) process, which has been used extensively in modeling many other real life phenomena. But few scholars, including those who used the GBM in highway configuration decisions, have offered any rigorous justification for the use of this model. This thesis attempts to offer a detailed analysis of various aspects of transportation systems in relation to decision-making. It reveals some general insights as well as a new concept that extends the notion of opportunity cost to situations where wrong decisions could be made. Claiming deficiency of the GBM model, it also introduces a new formulation that utilizes a large and flexible parametric family of jump models (i.e., Lévy processes). To validate this claim, data related to traffic demand and land prices were collected and analyzed to reveal that their distributions, heavy-tailed and asymmetric, do not match well with the GBM model. As a remedy, this research used the Merton, Kou, and negative inverse Gaussian Lévy processes as possible alternatives. Though the results show indifference in relation to final decisions among the models, mathematically, they improve the precision of uncertainty models and the decision-making process. This furthers the quest for optimality in highway projects and beyond. highway development decision making under uncertainty real options jump processes infinitely divisible distributions Lévy processes measure theory characteristic function characteristic exponent Lévy jump measure probability measure cádlág stochastic process Wiener process Poisson process compound Poisson process finite activity models jump diffusion models Merton model Gaussian jump process Kou model double exponential jump process leptokurtic volatility smile infinite activity models pure jumps process generalized hyperbolic model modified Bessel function negative inverse Gaussian model geometric Brownian motion model log-normal distribution model normality assumption heavy-tailed distribution asymmetric distribution quantile-quantile plot Rydberg algorithm parameter calibration moments cummulants sum of squares Monte Carlo simulation least-squares regression backward dynamic programming discrete-time Markov chain land price traffic demand highway service quality index data collection traffic volume Annual Average Daily Traffic Ontario Ministry of Transportation Land Registry Office Freedom of Information Act seasonality value of transportation types of transportation systems importance of highway systems cost of wrong decisions monetary cost private sector cost human cost land expropriation environmental cost irreversibility cost factor time cost factor political cost factor opportunity cost opportunity cost of wrong decisions economic cost of wrong decisions economic cost of mis-estimation economic profit of wrong decisions cost of inaction price of making right decisions scope of decisions rehabilitation decision land acquisition decision expansion decision uncertainty modeling land acquisition cost expropriation cost construction cost material cost Montréal-Mirabel International Airport Pickering Airport 407 Express Toll Route Highway 407 Statistics

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