Abstract:
The beta process has recently been widely used as a nonparametric prior for different models in machine learning, including latent feature models. In this paper, we prove the asymptotic consistency of the finite dimensional approximation of the beta process due to Paisley \& Carin (2009). In addition, we derive an almost sure approximation of the beta process. This approximation provides a direct method to efficiently simulate the beta process. A simulated example, illustrating the work of the method and comparing its performance to several existing algorithms, is also included.

Abstract:
In this paper, we develop simple, yet efficient, procedures for sampling approximations of the two-Parameter Poisson-Dirichlet Process and the normalized inverse-Gaussian process. We compare the efficiency of the new approximations to the corresponding stick-breaking approximations, in which we demonstrate a substantial improvement.

Abstract:
Ferguson's Dirichlet process plays an important role in nonparametric Bayesian inference. Let $P_a$ be the Dirichlet process in $\mathbb{R}$ with a base probability measure $H$ and a concentration parameter $a>0.$ In this paper, we show that $\sqrt {a} \big(P_a((-\infty,t]) -H((-\infty,t])\big)$ converges to a certain Brownian bridge as $a \to \infty.$ We also derive a certain Glivenko-Cantelli theorem for the Dirichlet process. Using the functional delta method, the weak convergence of the quantile process is also obtained. A large concentration parameter occurs when a statistician puts too much emphasize on his/her prior guess. This scenario also happens when the sample size is large and the posterior is used to make inference.

Abstract:
We describe a simple and efficient procedure for approximating the L\'evy measure of a $\text{Gamma}(\alpha,1)$ random variable. We use this approximation to derive a finite sum-representation that converges almost surely to Ferguson's representation of the Dirichlet process based on arrivals of a homogeneous Poisson process. We compare the efficiency of our approximation to several other well known approximations of the Dirichlet process and demonstrate a substantial improvement.

Abstract:
In this paper, we present some asymptotic properties of the normalized inverse-Gaussian process. In particular, when the concentration parameter is large, we establish an analogue of the empirical functional central limit theorem, the strong law of large numbers and the Glivenko-Cantelli theorem for the normalized inverse-Gaussian process and its corresponding quantile process. We also derive a finite sum-representation that converges almost surely to the Ferguson and Klass representation of the normalized inverse-Gaussian process. This almost sure approximation can be used to simulate efficiently the normalized inverse-Gaussian process.

Abstract:
The robustness to the prior of Bayesian inference procedures based on a measure of statistical evidence are considered. These inferences are shown to have optimal properties with respect to robustness. Furthermore, a connection between robustness and prior-data conflict is established. In particular, the inferences are shown to be effectively robust when the choice of prior does not lead to prior-data conflict. When there is prior-data conflict, however, robustness may fail to hold.

Abstract:
In recent years, Bayesian nonparametric statistics has gathered extraordinary attention. Nonetheless, a relatively little amount of work has been expended on Bayesian nonparametric hypothesis testing. In this paper, a novel Bayesian nonparametric approach to the two-sample problem is established. Precisely, given two samples $\mathbf{X}=X_1,\ldots,X_{m_1}$ $\overset {i.i.d.} \sim F$ and $\mathbf{Y}=Y_1,\ldots,Y_{m_2} \overset {i.i.d.} \sim G$, with $F$ and $G$ being unknown continuous cumulative distribution functions, we wish to test the null hypothesis $\mathcal{H}_0:~F=G$. The method is based on the Kolmogorov distance and approximate samples from the Dirichlet process centered at the standard normal distribution and a concentration parameter 1. It is demonstrated that the proposed test is robust with respect to any prior specification of the Dirichlet process. A power comparison with several well-known tests is incorporated. In particular, the proposed test dominates the standard Kolmogorov-Smirnov test in all the cases examined in the paper.

Abstract:
The extreme temperature trends are analyzed for a meteorological data collection station in Jeddah, Saudi Arabia over approximately last four decades stretching between years 1970 and 2006. The long-term change in temperature has been assessed by Mann-Kendell rank statistics and linear trend analysis. The study also includes the estimation of hot and cold days and nights frequencies and finally the temperature anomalies on yearly basis. The ratio between the seasonal mean temperatures (Tmmean) of the daily mean of hottest (July) and coldest (January) months was 1.032. Similarly the ratios between the seasonal mean temperature of daily maximum (Tmmax) of hottest and coldest months was 1.033 while for seasonal mean temperature of daily minimum (Tmmin) was 1.030. Significant increase was observed in hot days per year and relatively smaller decrease in hot nights. Significant increase in summer time temperatures was confirmed by both linear regression analysis and M-K rank statistics. The monthly and annual mean maximum temperatures have increased more than the mean and mean minimum temperatures.

Abstract:
The temperature variability was studied using linear regression models, Man-Kendall (M-K) rank statistics, mean monthly and annual temperature anomalies, number of hot and cold days and nights per annum. The temperature time series (mean, minimum and maximum) has shown a warming trend of the local air during last two decades. The maximum and minimum of the monthly mean temperatures of 23.35°C and 13.25°C were observed in June and January, respectively. Similarly, the maximum and minimum temperature ranges of 13.4°C and 8.2°C were found in December and June. The ratios between the hottest and coldest month’s monthly mean maximum and monthly mean minimum temperatures were 1.039 and 1.033, respectively. The best fit linear line showed an increase of 0.048°C per annum in the annual mean temperature with a coefficient of determination of 52%. An overall increase of 1.01°C was found between 1985. The Mann-Kendall rank statistics test confirmed the warming trends of the local atmospheric environment of Abha city.

Abstract:
This study investigated the use of neural networks in function approximation, data fitting and prediction. Due to its superior performance, the counterpropagation network was considered and an attempt was made to enhance its performance. As a result of this research, we proposed a new neural network architecture named Single Layer Linear Counterpropagation (SLLIC) network. The SLLIC neural net has the following additional features: weight Initialization, automatic structure determination and higher order neural network concepts. The SLLIC network was tested and results show that the performance of the system in terms of good approximation or prediction is comparable to and some times better than other neural nets architecture s and traditional techniques.