Multivariate probability calibration is the problem of predicting class membership probabilities from classification scores of multiple classifiers. To achieve better performance, the calibrating function is often required to be coordinate-wise non-decreasing; that is, for every classifier, the higher the score, the higher the probability of the class labeling being positive. To this end, we propose a multivariate regression method based on shape-restricted Bernstein polynomials. This method is universally flexible: it can approximate any continuous calibrating function with any specified error, as the polynomial degree increases to infinite. Moreover, it is universally consistent: the estimated calibrating function converges to any continuous calibrating function, as the training size increases to infinity. Our empirical study shows that the proposed method achieves better calibrating performance than benchmark methods.

In many real-world classification problems, accurate prediction of membership probabilities is critical for further decision making. The probability calibration problem studies how to map scores obtained from one classification algorithm to membership probabilities. The requirement of non-decreasingness for this mapping involves an infinite number of inequality constraints, which makes its estimation computationally intractable. For the sake of this difficulty, existing methods failed to achieve four desiderata of probability calibration: universal flexibility, non-decreasingness, continuousness and computational tractability. This paper proposes a method with shape-restricted polynomial regression, which satisfies all four desiderata. In the method, the calibrating function is approximated with monotone polynomials, and the continuously-constrained requirement of monotonicity is equivalent to some semidefinite constraints. Thus, the calibration problem can be solved with tractable semidefinite programs. This estimator is both strongly and weakly universally consistent under a trivial condition. Experimental results on both artificial and real data sets clearly show that the method can greatly improve calibrating performance in terms of reliability-curve related measures.

Novelty detection models aim to find the minimum volume set covering a given probability mass. This paper proposes a robust single-class support vector machine (SSVM) for novelty detection, which is mainly based on the worst case conditional value-at-risk minimization. By assuming that every input is subject to an uncertainty with a specified symmetric support, this robust formulation results in a maximization term that is similar to the regularization term in the classical SSVM. When the uncertainty set is l1 -norm, l∞ -norm or box, its training can be reformulated to a linear program; while the uncertainty set is l2 -norm or ellipsoidal, its training is a tractable second-order cone program. The proposed method has a nice consistent statistical property. As the training size goes to infinity, the estimated normal region converges to the true provided that the magnitude of the uncertainty set decreases in a systematic way. The experimental results on three data sets clearly demonstrate its superiority over three benchmark models.

Copula has become the standard tool in financial dependence modeling. This paper proposes a novel nonparametric bivariate copula estimation method which smooths empirical copula with shape-restricted least squares support vector regression. This meth

Due to recent financial crises and regulatory concerns, financial intermediaries' credit risk assessment is an area of renewed interest in both the academic world and the business community. In this paper, we propose a new fuzzy support vector machine to discriminate good creditors from bad ones. Because in credit scoring areas we usually cannot label one customer as absolutely good who is sure to repay in time, or absolutely bad who will default certainly, our new fuzzy support vector machine treats every sample as both positive and negative classes, but with different memberships. By this way we expect the new fuzzy support vector machine to have more generalization ability, while preserving the merit of insensitive to outliers, as the fuzzy support vector machine (SVM) proposed in previous papers. We reformulate this kind of two-group classification problem into a quadratic programming problem. Empirical tests on three public datasets show that it can have better discriminatory power than the standard support vector machine and the fuzzy support vector machine if appropriate kernel and membership generation method are chosen.