Gaussian Classes Research Articles

Elementary transformations of systems of equations over quasigroups and generalized identities

Abstract The paper is devoted to the study of equations with the left-hand side having the form of a composition of operations which belong to given sets S1, …, Sn, … of quasigroup operations. Elementary transformations are described which allow reducing systems of this kind to the form where all equations except one do not depend essentially on the variable xn. A class of systems is said to be Gaussian if every system obtained via such transformations also belongs to this class. It is evident that for Gaussian classes of systems of equations there is an efficient solving algorithm. This motivates the problem of finding conditions under which the class is Gaussian. In this work it is shown that for a class of systems to be Gaussian the operations in the sets Si should satisfy the generalized distributivity law. Sets of operations obeying this condition are to be investigated in the future.

Selective atrophy of the connected deepest cortical layers following small subcortical infarct.

To explore whether in patients with chronic small subcortical infarct the cortical layers of the connected cortex are differentially affected and whether these differences correlate with clinical symptomatology. Twenty patients with a history of chronic small subcortical infarct affecting the corticospinal tracts and 15 healthy controls were included. Connected primary motor cortex was identified with tractography starting from infarct. T1-component probability maps were calculated from T1 relaxation 3T MRI, dividing the cortex into 5 laminar gaussian classes. Focal cortical thinning was observed in the connected cortex and specifically only in its deepest laminar class compared to the nonaffected mirrored cortex (p < 0.001). There was loss of microstructural integrity of the affected corticospinal tract with increased mean diffusivity and decreased fractional anisotropy compared to the contralateral nonaffected tract (p ≤ 0.002). Clinical scores were correlated with microstructural damage of the corticospinal tracts and with thinning of the cortex and specifically only its deepest laminar class (p < 0.001). No differences were found in the laminar thickness pattern of the bilateral primary motor cortices or in the microstructural integrity of the bilateral corticospinal tracts in the healthy controls. Our results support the concept of secondary neurodegeneration of connected primary motor cortex after a small subcortical infarct affecting the corticospinal tract, with observations that the main cortical thinning occurs in the deepest cortex and that the clinical symptomatology is correlated with this cortical atrophy pattern. Our findings may contribute to a better understanding of structural reorganization and functional outcomes after stroke.

Pseudo-Hermitian ensemble of random Gaussian matrices.

It is shown how pseudo-Hermiticity, a necessary condition satisfied by operators of PT symmetric systems can be introduced in the three Gaussian classes of random matrix theory. The model describes transitions from real eigenvalues to a situation in which, apart from a residual number, the eigenvalues are complex conjugate.

A Note on Linear SVM in Gaussian Classes

The linear support vector machine(SVM) is motivated by the maximal margin separating hyperplane and is a popular tool for binary classification tasks. Many studies exist on the consistency properties of SVM; however, it is unknown whether the linear SVM is consistent for estimating the optimal classification boundary even in the simple case of two Gaussian classes with a common covariance, where the optimal classification boundary is linear. In this paper we show that the linear SVM can be inconsistent in the univariate Gaussian classification problem with a common variance, even when the best tuning parameter is used.

SOME REMARKS ON THE DOZIER–SILVERSTEIN THEOREM FOR RANDOM MATRICES WITH DEPENDENT ENTRIES

The Dozier–Silverstein theorem asserts the almost sure convergence of the empirical spectral distribution of information plus noise matrices, i.e. perturbations of deterministic matrices whose spectral distribution converges (information matrices) by random matrices with i.i.d. entries (noise matrices). We show that a modification of the original proof given by Dozier and Silverstein allows to extend this result to more general noise matrices, in particular matrices with independent columns satisfying a natural concentration inequality for quadratic forms, matrices with independent entries, satisfying a Lindeberg-type condition (recovering a recent result by Xie), matrices with heavy-tailed entries in the domain of attraction of the Gaussian distribution and certain classes of matrices with dependencies among columns (generalizing those investigated recently by O'Rourke and containing matrices with exchangeable entries). As a corollary we obtain the circular law for random matrices with independent log-concave isotropic rows.

Applications of the Quadratic Covariation Differentiation Theory: Variants of the Clark-Ocone and Stroock's Formulas

In a 2006 article [1], Allouba gave his quadratic covariation differentiation theory for Itô's integral calculus. In it, he defined the derivative of a semimartingale with respect to a Brownian motion as the time derivative of their quadratic covariation and a generalization thereof. He then obtained a systematic pathwise stochastic differentiation theory that comes complete with a fundamental theorem of stochastic calculus relating this derivative to Itô's integral, a differential stochastic chain rule, a differential stochastic mean value theorem, and other differentiation rules. In this current article, we use this differentiation theory in [1] to obtain variants of the celebrated Clark-Ocone and Stroock representation formulas, with and without change of measure. We prove our variants of the Clark-Ocone formula under L 2-type conditions on the random variable but with no L p conditions on the derivative. We do not use Malliavin calculus, weak distributional or Radon-Nikodym type derivatives, or the significant extra machinery of the Hida-Malliavin calculus. Moreover, unlike with Malliavin or Hida-Malliavin calculi, the form of our variant of the Clark-Ocone formula under change of measure is as simple as it is under no change of measure, and without requiring any further differentiability conditions on the Girsanov transform integrand beyond the standard Novikov condition. This is a consequence of the invariance under change of measure of the first author's derivative in [1]. The formulations and proofs are simple and natural applications of the differentiation theory in [1] and standard Itô integral calculus. Iterating our variants of the Clark-Ocone formula, we obtain variants of Stroock's formula. We illustrate the applicability of these formulas and the theory in [1] by easily, and without Hida-Malliavin methods, obtaining the representation of the Brownian indicator F = 𝕀[K, ∞)(W T ), which is not standard Malliavin differentiable, and by applying them to digital options in finance. We then identify the chaos expansion of the Brownian indicator. The first author further extends and applies his differentiation theory in forthcoming articles and obtains a general stochastic calculus for a large class of processes with different orders and types of variations, including many that fall outside the classical Gaussian, Markovian, or semimartingale classes.

Comparing alternative Lévy base correlation models for pricing and hedging CDO tranches

In this paper we investigate alternative Lévy base correlation models that arise from the Gamma, Inverse Gaussian and CMY distribution classes. We compare these models with the basic (exponential) Lévy base correlation model and the classical Gaussian base correlation model. For all investigated models, the Lévy base correlation curve is significantly flatter than the corresponding Gaussian curve, which indicates better correspondence of the Lévy models with reality. Furthermore, we present the results of pricing bespoke tranchlets and comparing deltas of both standard and custom-made tranches under all the considered models. We focus on deltas with respect to the CDS index and individual CDSs, and the hedge ratio for hedging the equity tranche with the junior mezzanine.

On the window size for classification in changing environments

Classification in changing environments (commonly known as concept drift) requires adaptation of the classifier to accommodate the changes. One approach is to keep a moving window on the streaming data and constantly update the classifier on it. Here we consider an abrupt change scenario where one set of probability distributions of the classes is instantly replaced with another. For a fixed 'transition period' around the change, we derive a generic relationship between the size of the moving window and the classification error rate. We derive expressions for the error in the transition period and for the optimal window size for the case of two Gaussian classes where the concept change is a geometrical displacement of the whole class configuration in the space. A simple window resize strategy based on the derived relationship is proposed and compared with fixed-size windows on a real benchmark data set data set (Electricity Market).

Local complementation rule for continuous-variable four-mode unweighted graph states

The local complementation rule is applied for continuous-variable (CV) graph states in the paper, which is an elementary graph transformation rule and successive application of which generates the orbit of any graph states. The corresponding local Gaussian transformations of local complementation for four-mode unweighted graph states were found, which do not mirror the form of the local Clifford unitary of qubit exactly. This work is an important step to characterize the local Gaussian equivalence classes of CV graph states.

A case-study on naïve labelling for the nearest mean and the linear discriminant classifiers

The abundance of unlabelled data alongside limited labelled data has provoked significant interest in semi-supervised learning methods. “Naïve labelling” refers to the following simple strategy for using unlabelled data in on-line classification. A new data point is first labelled by the current classifier and then added to the training set together with the assigned label. The classifier is updated before seeing the subsequent data point. Although the danger of a run-away classifier is obvious, versions of naïve labelling pervade in on-line adaptive learning. We study the asymptotic behaviour of naïve labelling in the case of two Gaussian classes and one variable. The analysis shows that if the classifier model assumes correctly the underlying distribution of the problem, naïve labelling will drive the parameters of the classifier towards their optimal values. However, if the model is not guessed correctly, the benefits are outweighed by the instability of the labelling strategy (run-away behaviour of the classifier). The results are based on exact calculations of the point of convergence, simulations, and experiments with 25 real data sets. The findings in our study are consistent with concerns about general use of unlabelled data, flagged up in the recent literature.

Statistical Analysis of Minimum Classification Error Learning for Gaussian and Hidden Markov Model Classifiers

Minimum classification error learning realized via generalized probabilistic descent, usually referred to as (MCE/GPD), is a very popular and powerful framework for building classifiers. This paper first presents a theoretical analysis of MCE/GPD. The focus is on a simple classification problem for estimating the means of two Gaussian classes. For this simple algorithm, we derive difference equations for the class means and decision threshold during learning, and develop closed form expressions for the evolution of both the smoothed and true error. In addition, we show that the decision threshold converges to its optimal value, and provide an estimate of the number of iterations needed to approach convergence. After convergence the class means drift towards increasing their distance to infinity without contributing to the decrease of the classification error. This behavior, referred to as mean drift, is then related to the increase of the variance of the classifier. The theoretical results perfectly agree with simulations carried out for a two-class Gaussian classification problem. In addition to the obtained theoretical results we experimentally verify, in speech recognition experiments, that MCE/GPD learning of Gaussian mixture hidden Markov models qualitatively follows the pattern suggested by the theoretical analysis. We also discuss links between MCE/GPD learning and both batch gradient descent and extended Baum-Welch re-estimation. The latter two approaches are known to be popular in large scale implementations of discriminative training. Hence, the proposed analysis can be used, at least as a rough guideline, for better understanding of the properties of discriminative training algorithms for speech recognition.

Comparison and validation of tissue modelization and statistical classification methods in T1-weighted MR brain images

This paper presents a validation study on statistical nonsupervised brain tissue classification techniques in magnetic resonance (MR) images. Several image models assuming different hypotheses regarding the intensity distribution model, the spatial model and the number of classes are assessed. The methods are tested on simulated data for which the classification ground truth is known. Different noise and intensity nonuniformities are added to simulate real imaging conditions. No enhancement of the image quality is considered either before or during the classification process. This way, the accuracy of the methods and their robustness against image artifacts are tested. Classification is also performed on real data where a quantitative validation compares the methods' results with an estimated ground truth from manual segmentations by experts. Validity of the various classification methods in the labeling of the image as well as in the tissue volume is estimated with different local and global measures. Results demonstrate that methods relying on both intensity and spatial information are more robust to noise and field inhomogeneities. We also demonstrate that partial volume is not perfectly modeled, even though methods that account for mixture classes outperform methods that only consider pure Gaussian classes. Finally, we show that simulated data results can also be extended to real data.

The Optimal Classification Using a Linear Discriminant for Two Point Classes Having Known Mean and Covariance

The current study provides a simple algorithm for finding the optimal ROC curve for a linear discriminant between two point distributions, given only information about the classes' means and covariances. The method makes no assumptions concerning the exact type of distribution and is shown to provide the best possible discrimination for any physically reasonable measure of the classification error. This very general solution is shown to specialise to results obtained in other papers which assumed multi-dimensional Gaussian distributed classes, or minimised the maximum classification error. Some numerical examples are provided which show the improvement in classification of this method over previously used methods.

Generalization Error of Randomized Linear Zero Empirical Error Classifier: Simple Asymptotics for Centered Data Case

An estimation of the generalization performance of classifier is one of most important problems in pattern clasification and neural network training theory. In this paper we estimate the generalization error (mean expected probability of classification) for randomized linear zero empirical error (RLZEE) classifier which was considered by Raudys, Diciūnas and Basalykas. Instead of “non-explicit” asymptotics of a generalization error of RLZEE classifier for centered multivariate spherically Gaussian classes proposed by Basalykas et al. (1996) we obtain an “explicit” and more simple asymptotics. We also present the numerical simulations illustrating our theoretical results and comparing them with each other and previously obtained results.

Misclassification Probability Bounds for Multivariate Gaussian Classes

Classification systems based on linear discriminant analysis are employed in a variety of communications applications, in which the classes are most commonly characterized by known Gaussian PDFs. The performance of these classifiers is analyzed in this paper in terms of the conditional probability of misclassification. Easily computed lower and upper bounds on this error probability are presented and shown to provide corresponding bounds on the number of Monte Carlo trials required to obtain a desired level of accuracy. The error probability bounds yield an exact and easily computed expression for the error probability in the case where there are only two classes and a single hyperplane. In the special case where misclassification into a nominated class is independent of all other misclassifications, successively tighter upper and lower bounds can be computed at the expense of successively higher-order products of the individual misclassification probabilities. Finally, bounds are provided on the number of Monte Carlo trials required to improve, with suitably high confidence level, on the confidence interval formed by the error probability bounds.

On feature extraction in multi-class pattern recognition†

The problem of feature selection in multi-class pattern recognition is viewed as that of a mapping of vector samples from n-dimensional space to that in m-dimensional space (m<n) by a linear transformation. The transformation matrix is obtained for Gaussian distributed pattern classes by maximizing the expected divergence between any pair of pattern classes. Furthermore, for zero mean Gaussian distributed pattern classes an upperbound on the probability of error is obtained in terms of the expected divergence