Time series modeling and forecasting has fundamental importance to various practical domains. Thus a lot of active research works is going on in this subject during several years. Many important models have been proposed in literature for improving the accuracy and effectiveness of time series forecasting. The aim of this dissertation work is to present a concise description of some popular time series forecasting models used in practice, with their salient features. In this thesis, we have described three important classes of time series models, viz. the stochastic, neural networks and SVM based models, together with their inherent forecasting strengths and weaknesses. We have also discussed about the basic issues related to time series modeling, such as stationarity, parsimony, overfitting, etc. Our discussion about different time series models is supported by giving the experimental forecast results, performed on six real time series datasets. While fitting a model to a dataset, special care is taken to select the most parsimonious one. To evaluate forecast accuracy as well as to compare among different models fitted to a time series, we have used the five performance measures, viz. MSE, MAD, RMSE, MAPE and Theil's U-statistics. For each of the six datasets, we have shown the obtained forecast diagram which graphically depicts the closeness between the original and forecasted observations. To have authenticity as well as clarity in our discussion about time series modeling and forecasting, we have taken the help of various published research works from reputed journals and some standard books.

Support vector machine (SVM) training is an active research area since the dawn of the method. In recent years there has been increasing interest in specialized solvers for the important case of linear models. The algorithm presented by Hsieh et al., probably best known under the name of the "liblinear" implementation, marks a major breakthrough. The method is analog to established dual decomposition algorithms for training of non-linear SVMs, but with greatly reduced computational complexity per update step. This comes at the cost of not keeping track of the gradient of the objective any more, which excludes the application of highly developed working set selection algorithms. We present an algorithmic improvement to this method. We replace uniform working set selection with an online adaptation of selection frequencies. The adaptation criterion is inspired by modern second order working set selection methods. The same mechanism replaces the shrinking heuristic. This novel technique speeds up training in some cases by more than an order of magnitude.

Stochastic Gradient Descent (SGD) has become popular for solving large scale supervised machine learning optimization problems such as SVM, due to their strong theoretical guarantees. While the closely related Dual Coordinate Ascent (DCA) method has been implemented in various software packages, it has so far lacked good convergence analysis. This paper presents a new analysis of Stochastic Dual Coordinate Ascent (SDCA) showing that this class of methods enjoy strong theoretical guarantees that are comparable or better than SGD. This analysis justifies the effectiveness of SDCA for practical applications.

We describe a method for predicting a classification of an object given classifications of the objects in the training set, assuming that the pairs object/classification are generated by an i.i.d. process from a continuous probability distribution. Our method is a modification of Vapnik's support-vector machine; its main novelty is that it gives not only the prediction itself but also a practicable measure of the evidence found in support of that prediction. We also describe a procedure for assigning degrees of confidence to predictions made by the support vector machine. Some experimental results are presented, and possible extensions of the algorithms are discussed.

It is shown that bootstrap approximations of support vector machines (SVMs) based on a general convex and smooth loss function and on a general kernel are consistent. This result is useful to approximate the unknown finite sample distribution of SVMs by the bootstrap approach.

Incorporating feature selection into a classification or regression method often carries a number of advantages. In this paper we formalize feature selection specifically from a discriminative perspective of improving classification/regression accuracy. The feature selection method is developed as an extension to the recently proposed maximum entropy discrimination (MED) framework. We describe MED as a flexible (Bayesian) regularization approach that subsumes, e.g., support vector classification, regression and exponential family models. For brevity, we restrict ourselves primarily to feature selection in the context of linear classification/regression methods and demonstrate that the proposed approach indeed carries substantial improvements in practice. Moreover, we discuss and develop various extensions of feature selection, including the problem of dealing with example specific but unobserved degrees of freedom -- alignments or invariants.

We define a generalized likelihood function based on uncertainty measures and show that maximizing such a likelihood function for different measures induces different types of classifiers. In the probabilistic framework, we obtain classifiers that optimize the cross-entropy function. In the possibilistic framework, we obtain classifiers that maximize the interclass margin. Furthermore, we show that the support vector machine is a sub-class of these maximum-margin classifiers.

The Support Vector Machine (SVM) of Vapnik (1998) has become widely established as one of the leading approaches to pattern recognition and machine learning. It expresses predictions in terms of a linear combination of kernel functions centred on a subset of the training data, known as support vectors. Despite its widespread success, the SVM suffers from some important limitations, one of the most significant being that it makes point predictions rather than generating predictive distributions. Recently Tipping (1999) has formulated the Relevance Vector Machine (RVM), a probabilistic model whose functional form is equivalent to the SVM. It achieves comparable recognition accuracy to the SVM, yet provides a full predictive distribution, and also requires substantially fewer kernel functions. The original treatment of the RVM relied on the use of type II maximum likelihood (the `evidence framework') to provide point estimates of the hyperparameters which govern model sparsity. In this paper we show how the RVM can be formulated and solved within a completely Bayesian paradigm through the use of variational inference, thereby giving a posterior distribution over both parameters and hyperparameters. We demonstrate the practicality and performance of the variational RVM using both synthetic and real world examples.

We introduce the anti-profile Support Vector Machine (apSVM) as a novel algorithm to address the anomaly classification problem, an extension of anomaly detection where the goal is to distinguish data samples from a number of anomalous and heterogeneous classes based on their pattern of deviation from a normal stable class. We show that under heterogeneity assumptions defined here that the apSVM can be solved as the dual of a standard SVM with an indirect kernel that measures similarity of anomalous samples through similarity to the stable normal class. We characterize this indirect kernel as the inner product in a Reproducing Kernel Hilbert Space between representers that are projected to the subspace spanned by the representers of the normal samples. We show by simulation and application to cancer genomics datasets that the anti-profile SVM produces classifiers that are more accurate and stable than the standard SVM in the anomaly classification setting.

We develop a unified approach for classification and regression support vector machines for data subject to right censoring. We provide finite sample bounds on the generalization error of the algorithm, prove risk consistency for a wide class of probability measures, and study the associated learning rates. We apply the general methodology to estimation of the (truncated) mean, median, quantiles, and for classification problems. We present a simulation study that demonstrates the performance of the proposed approach.