Conditional probability calculations for the nonlinear Schrödinger equation with additive noise

The method for the computation of the conditional probability density function for the nonlinear Schrödinger equation with additive noise is developed. We present in a constructive form the conditional probability density function in the limit of small noise and analytically derive it in a weakly nonlinear case. The general theory results are illustrated using fiber-optic communications as a particular, albeit practically very important, example.

Randomized Push-Out Mechanisms in Priority Queuing and their Probability Characteristics

2002 Mathematics Subject Classification: 60K25.

Special Classes of Orthogonal Polynomials and Corresponding Quadratures of Gaussian Type

MSC 2010: 33C47, 42C05, 41A55, 65D30, 65D32

Numerical modelling of spectral, temporal and statistical properties of Raman fiber lasers

Efficient numerical modelling of the power, spectral and statistical properties of partially coherent quasi-CW Raman fiber laser radiation is presented. XPM between pump wave and generated Stokes wave is not important in the generation spectrum broadening and XPM term can be omitted in propagation equation what sufficiently speeds-up simulations. The time dynamics of Raman fiber laser (RFL) is stochastic exhibiting events several times more intense that the mean value on the ps timescale. However, the RFL has different statistical properties on different time scales. The probability density function of spectral power density is exponential for the generation modes located either in the spectrum centre or spectral wings while the phases are distributed uniformly. The pump wave preserves the initial Gaussian statistics during propagation in the laser cavity. Intense pulses in the pump wave are evolved under the SPM influence and are not disturbed by the dispersion. Contrarily, in the generated wave the dispersion plays a significant role that results in stochastic behavior. © 2012 Elsevier B.V. All rights reserved.

Green's function for paraxial equation

The theory and experimental applications of optical Airy beams are in active development recently. The Airy beams are characterised by very special properties: they are non-diffractive and propagate along parabolic trajectories. Among the striking applications of the optical Airy beams are optical micro-manipulation implemented as the transport of small particles along the parabolic trajectory, Airy-Bessel linear light bullets, electron acceleration by the Airy beams, plasmonic energy routing. The detailed analysis of the mathematical aspects as well as physical interpretation of the electromagnetic Airy beams was done by considering the wave as a function of spatial coordinates only, related by the parabolic dependence between the transverse and the longitudinal coordinates. Their time dependence is assumed to be harmonic. Only a few papers consider a more general temporal dependence where such a relationship exists between the temporal and the spatial variables. This relationship is derived mostly by applying the Fourier transform to the expressions obtained for the harmonic time dependence or by a Fourier synthesis using the specific modulated spectrum near some central frequency. Spatial-temporal Airy pulses in the form of contour integrals is analysed near the caustic and the numerical solution of the nonlinear paraxial equation in time domain shows soliton shedding from the Airy pulse in Kerr medium. In this paper the explicitly time dependent solutions of the electromagnetic problem in the form of time-spatial pulses are derived in paraxial approximation through the Green's function for the paraxial equation. It is shown that a Gaussian and an Airy pulse can be obtained by applying the Green's function to a proper source current. We emphasize that the processes in time domain are directional, which leads to unexpected conclusions especially for the paraxial approximation.

A lower bound on the per soliton capacity of the nonlinear optical fibre channel

A closed-form expression for a lower bound on the per soliton capacity of the nonlinear optical fibre channel in the presence of (optical) amplifier spontaneous emission (ASE) noise is derived. This bound is based on a non-Gaussian conditional probability density function for the soliton amplitude jitter induced by the ASE noise and is proven to grow logarithmically as the signal-to-noise ratio increases.

Projected sequential Gaussian processes:a C++ tool for interpolation of large datasets with heterogeneous noise

Heterogeneous datasets arise naturally in most applications due to the use of a variety of sensors and measuring platforms. Such datasets can be heterogeneous in terms of the error characteristics and sensor models. Treating such data is most naturally accomplished using a Bayesian or model-based geostatistical approach; however, such methods generally scale rather badly with the size of dataset, and require computationally expensive Monte Carlo based inference. Recently within the machine learning and spatial statistics communities many papers have explored the potential of reduced rank representations of the covariance matrix, often referred to as projected or fixed rank approaches. In such methods the covariance function of the posterior process is represented by a reduced rank approximation which is chosen such that there is minimal information loss. In this paper a sequential Bayesian framework for inference in such projected processes is presented. The observations are considered one at a time which avoids the need for high dimensional integrals typically required in a Bayesian approach. A C++ library, gptk, which is part of the INTAMAP web service, is introduced which implements projected, sequential estimation and adds several novel features. In particular the library includes the ability to use a generic observation operator, or sensor model, to permit data fusion. It is also possible to cope with a range of observation error characteristics, including non-Gaussian observation errors. Inference for the covariance parameters is explored, including the impact of the projected process approximation on likelihood profiles. We illustrate the projected sequential method in application to synthetic and real datasets. Limitations and extensions are discussed. © 2010 Elsevier Ltd.

Bayesian Inference in Large-scale Problems

Many modern applications fall into the category of "large-scale" statistical problems, in which both the number of observations n and the number of features or parameters p may be large. Many existing methods focus on point estimation, despite the continued relevance of uncertainty quantification in the sciences, where the number of parameters to estimate often exceeds the sample size, despite huge increases in the value of n typically seen in many fields. Thus, the tendency in some areas of industry to dispense with traditional statistical analysis on the basis that "n=all" is of little relevance outside of certain narrow applications. The main result of the Big Data revolution in most fields has instead been to make computation much harder without reducing the importance of uncertainty quantification. Bayesian methods excel at uncertainty quantification, but often scale poorly relative to alternatives. This conflict between the statistical advantages of Bayesian procedures and their substantial computational disadvantages is perhaps the greatest challenge facing modern Bayesian statistics, and is the primary motivation for the work presented here.

Two general strategies for scaling Bayesian inference are considered. The first is the development of methods that lend themselves to faster computation, and the second is design and characterization of computational algorithms that scale better in n or p. In the first instance, the focus is on joint inference outside of the standard problem of multivariate continuous data that has been a major focus of previous theoretical work in this area. In the second area, we pursue strategies for improving the speed of Markov chain Monte Carlo algorithms, and characterizing their performance in large-scale settings. Throughout, the focus is on rigorous theoretical evaluation combined with empirical demonstrations of performance and concordance with the theory.

One topic we consider is modeling the joint distribution of multivariate categorical data, often summarized in a contingency table. Contingency table analysis routinely relies on log-linear models, with latent structure analysis providing a common alternative. Latent structure models lead to a reduced rank tensor factorization of the probability mass function for multivariate categorical data, while log-linear models achieve dimensionality reduction through sparsity. Little is known about the relationship between these notions of dimensionality reduction in the two paradigms. In Chapter 2, we derive several results relating the support of a log-linear model to nonnegative ranks of the associated probability tensor. Motivated by these findings, we propose a new collapsed Tucker class of tensor decompositions, which bridge existing PARAFAC and Tucker decompositions, providing a more flexible framework for parsimoniously characterizing multivariate categorical data. Taking a Bayesian approach to inference, we illustrate empirical advantages of the new decompositions.

Latent class models for the joint distribution of multivariate categorical, such as the PARAFAC decomposition, data play an important role in the analysis of population structure. In this context, the number of latent classes is interpreted as the number of genetically distinct subpopulations of an organism, an important factor in the analysis of evolutionary processes and conservation status. Existing methods focus on point estimates of the number of subpopulations, and lack robust uncertainty quantification. Moreover, whether the number of latent classes in these models is even an identified parameter is an open question. In Chapter 3, we show that when the model is properly specified, the correct number of subpopulations can be recovered almost surely. We then propose an alternative method for estimating the number of latent subpopulations that provides good quantification of uncertainty, and provide a simple procedure for verifying that the proposed method is consistent for the number of subpopulations. The performance of the model in estimating the number of subpopulations and other common population structure inference problems is assessed in simulations and a real data application.

In contingency table analysis, sparse data is frequently encountered for even modest numbers of variables, resulting in non-existence of maximum likelihood estimates. A common solution is to obtain regularized estimates of the parameters of a log-linear model. Bayesian methods provide a coherent approach to regularization, but are often computationally intensive. Conjugate priors ease computational demands, but the conjugate Diaconis--Ylvisaker priors for the parameters of log-linear models do not give rise to closed form credible regions, complicating posterior inference. In Chapter 4 we derive the optimal Gaussian approximation to the posterior for log-linear models with Diaconis--Ylvisaker priors, and provide convergence rate and finite-sample bounds for the Kullback-Leibler divergence between the exact posterior and the optimal Gaussian approximation. We demonstrate empirically in simulations and a real data application that the approximation is highly accurate, even in relatively small samples. The proposed approximation provides a computationally scalable and principled approach to regularized estimation and approximate Bayesian inference for log-linear models.

Another challenging and somewhat non-standard joint modeling problem is inference on tail dependence in stochastic processes. In applications where extreme dependence is of interest, data are almost always time-indexed. Existing methods for inference and modeling in this setting often cluster extreme events or choose window sizes with the goal of preserving temporal information. In Chapter 5, we propose an alternative paradigm for inference on tail dependence in stochastic processes with arbitrary temporal dependence structure in the extremes, based on the idea that the information on strength of tail dependence and the temporal structure in this dependence are both encoded in waiting times between exceedances of high thresholds. We construct a class of time-indexed stochastic processes with tail dependence obtained by endowing the support points in de Haan's spectral representation of max-stable processes with velocities and lifetimes. We extend Smith's model to these max-stable velocity processes and obtain the distribution of waiting times between extreme events at multiple locations. Motivated by this result, a new definition of tail dependence is proposed that is a function of the distribution of waiting times between threshold exceedances, and an inferential framework is constructed for estimating the strength of extremal dependence and quantifying uncertainty in this paradigm. The method is applied to climatological, financial, and electrophysiology data.

The remainder of this thesis focuses on posterior computation by Markov chain Monte Carlo. The Markov Chain Monte Carlo method is the dominant paradigm for posterior computation in Bayesian analysis. It has long been common to control computation time by making approximations to the Markov transition kernel. Comparatively little attention has been paid to convergence and estimation error in these approximating Markov Chains. In Chapter 6, we propose a framework for assessing when to use approximations in MCMC algorithms, and how much error in the transition kernel should be tolerated to obtain optimal estimation performance with respect to a specified loss function and computational budget. The results require only ergodicity of the exact kernel and control of the kernel approximation accuracy. The theoretical framework is applied to approximations based on random subsets of data, low-rank approximations of Gaussian processes, and a novel approximating Markov chain for discrete mixture models.

Data augmentation Gibbs samplers are arguably the most popular class of algorithm for approximately sampling from the posterior distribution for the parameters of generalized linear models. The truncated Normal and Polya-Gamma data augmentation samplers are standard examples for probit and logit links, respectively. Motivated by an important problem in quantitative advertising, in Chapter 7 we consider the application of these algorithms to modeling rare events. We show that when the sample size is large but the observed number of successes is small, these data augmentation samplers mix very slowly, with a spectral gap that converges to zero at a rate at least proportional to the reciprocal of the square root of the sample size up to a log factor. In simulation studies, moderate sample sizes result in high autocorrelations and small effective sample sizes. Similar empirical results are observed for related data augmentation samplers for multinomial logit and probit models. When applied to a real quantitative advertising dataset, the data augmentation samplers mix very poorly. Conversely, Hamiltonian Monte Carlo and a type of independence chain Metropolis algorithm show good mixing on the same dataset.

Learning from Geometry

Subspaces and manifolds are two powerful models for high dimensional signals. Subspaces model linear correlation and are a good fit to signals generated by physical systems, such as frontal images of human faces and multiple sources impinging at an antenna array. Manifolds model sources that are not linearly correlated, but where signals are determined by a small number of parameters. Examples are images of human faces under different poses or expressions, and handwritten digits with varying styles. However, there will always be some degree of model mismatch between the subspace or manifold model and the true statistics of the source. This dissertation exploits subspace and manifold models as prior information in various signal processing and machine learning tasks.

A near-low-rank Gaussian mixture model measures proximity to a union of linear or affine subspaces. This simple model can effectively capture the signal distribution when each class is near a subspace. This dissertation studies how the pairwise geometry between these subspaces affects classification performance. When model mismatch is vanishingly small, the probability of misclassification is determined by the product of the sines of the principal angles between subspaces. When the model mismatch is more significant, the probability of misclassification is determined by the sum of the squares of the sines of the principal angles. Reliability of classification is derived in terms of the distribution of signal energy across principal vectors. Larger principal angles lead to smaller classification error, motivating a linear transform that optimizes principal angles. This linear transformation, termed TRAIT, also preserves some specific features in each class, being complementary to a recently developed Low Rank Transform (LRT). Moreover, when the model mismatch is more significant, TRAIT shows superior performance compared to LRT.

The manifold model enforces a constraint on the freedom of data variation. Learning features that are robust to data variation is very important, especially when the size of the training set is small. A learning machine with large numbers of parameters, e.g., deep neural network, can well describe a very complicated data distribution. However, it is also more likely to be sensitive to small perturbations of the data, and to suffer from suffer from degraded performance when generalizing to unseen (test) data.

From the perspective of complexity of function classes, such a learning machine has a huge capacity (complexity), which tends to overfit. The manifold model provides us with a way of regularizing the learning machine, so as to reduce the generalization error, therefore mitigate overfiting. Two different overfiting-preventing approaches are proposed, one from the perspective of data variation, the other from capacity/complexity control. In the first approach, the learning machine is encouraged to make decisions that vary smoothly for data points in local neighborhoods on the manifold. In the second approach, a graph adjacency matrix is derived for the manifold, and the learned features are encouraged to be aligned with the principal components of this adjacency matrix. Experimental results on benchmark datasets are demonstrated, showing an obvious advantage of the proposed approaches when the training set is small.

Stochastic optimization makes it possible to track a slowly varying subspace underlying streaming data. By approximating local neighborhoods using affine subspaces, a slowly varying manifold can be efficiently tracked as well, even with corrupted and noisy data. The more the local neighborhoods, the better the approximation, but the higher the computational complexity. A multiscale approximation scheme is proposed, where the local approximating subspaces are organized in a tree structure. Splitting and merging of the tree nodes then allows efficient control of the number of neighbourhoods. Deviation (of each datum) from the learned model is estimated, yielding a series of statistics for anomaly detection. This framework extends the classical {\em changepoint detection} technique, which only works for one dimensional signals. Simulations and experiments highlight the robustness and efficacy of the proposed approach in detecting an abrupt change in an otherwise slowly varying low-dimensional manifold.

Achievable rates and outage probability of cognitive radio with dynamic frequency hopping under imperfect spectrum sensing

In this study, the authors propose simple methods to evaluate the achievable rates and outage probability of a cognitive radio (CR) link that takes into account the imperfectness of spectrum sensing. In the considered system, the CR transmitter and receiver correlatively sense and dynamically exploit the spectrum pool via dynamic frequency hopping. Under imperfect spectrum sensing, false-alarm and miss-detection occur which cause impulsive interference emerged from collisions due to the simultaneous spectrum access of primary and cognitive users. That makes it very challenging to evaluate the achievable rates. By first examining the static link where the channel is assumed to be constant over time, they show that the achievable rate using a Gaussian input can be calculated accurately through a simple series representation. In the second part of this study, they extend the calculation of the achievable rate to wireless fading environments. To take into account the effect of fading, they introduce a piece-wise linear curve fitting-based method to approximate the instantaneous achievable rate curve as a combination of linear segments. It is then demonstrated that the ergodic achievable rate in fast fading and the outage probability in slow fading can be calculated to achieve any given accuracy level.

Spectral Frame Analysis and Learning through Graph Structure

This dissertation investigates the connection between spectral analysis and frame theory. When considering the spectral properties of a frame, we present a few novel results relating to the spectral decomposition. We first show that scalable frames have the property that the inner product of the scaling coefficients and the eigenvectors must equal the inverse eigenvalues. From this, we prove a similar result when an approximate scaling is obtained. We then focus on the optimization problems inherent to the scalable frames by first showing that there is an equivalence between scaling a frame and optimization problems with a non-restrictive objective function. Various objective functions are considered, and an analysis of the solution type is presented. For linear objectives, we can encourage sparse scalings, and with barrier objective functions, we force dense solutions. We further consider frames in high dimensions, and derive various solution techniques. From here, we restrict ourselves to various frame classes, to add more specificity to the results. Using frames generated from distributions allows for the placement of probabilistic bounds on scalability. For discrete distributions (Bernoulli and Rademacher), we bound the probability of encountering an ONB, and for continuous symmetric distributions (Uniform and Gaussian), we show that symmetry is retained in the transformed domain. We also prove several hyperplane-separation results. With the theory developed, we discuss graph applications of the scalability framework. We make a connection with graph conditioning, and show the in-feasibility of the problem in the general case. After a modification, we show that any complete graph can be conditioned. We then present a modification of standard PCA (robust PCA) developed by Cand\`es, and give some background into Electron Energy-Loss Spectroscopy (EELS). We design a novel scheme for the processing of EELS through robust PCA and least-squares regression, and test this scheme on biological samples. Finally, we take the idea of robust PCA and apply the technique of kernel PCA to perform robust manifold learning. We derive the problem and present an algorithm for its solution. There is also discussion of the differences with RPCA that make theoretical guarantees difficult.

Transfer function-noise modeling and spatial interpolation to evaluate the risk of extreme (shallow) water-table levels in the Brazilian Cerrados

Water regimes in the Brazilian Cerrados are sensitive to climatological disturbances and human intervention. The risk that critical water-table levels are exceeded over long periods of time can be estimated by applying stochastic methods in modeling the dynamic relationship between water levels and driving forces such as precipitation and evapotranspiration. In this study, a transfer function-noise model, the so called PIRFICT-model, is applied to estimate the dynamic relationship between water-table depth and precipitation surplus/deficit in a watershed with a groundwater monitoring scheme in the Brazilian Cerrados. Critical limits were defined for a period in the Cerrados agricultural calendar, the end of the rainy season, when extremely shallow levels (< 0.5-m depth) can pose a risk to plant health and machinery before harvesting. By simulating time-series models, the risk of exceeding critical thresholds during a continuous period of time (e.g. 10 days) is described by probability levels. These simulated probabilities were interpolated spatially using universal kriging, incorporating information related to the drainage basin from a digital elevation model. The resulting map reduced model uncertainty. Three areas were defined as presenting potential risk at the end of the rainy season. These areas deserve attention with respect to water-management and land-use planning.

Analysis of human posterior EEG alpha frequency as a function of implicit learning of probabilistic lateralized stimuli in a visual detection task

Alpha oscillatory activity has long been associated with perceptual and cognitive processes related to attention control. The aim of this study is to explore the task-dependent role of alpha frequency in a lateralized visuo-spatial detection task. Specifically, the thesis focuses on consolidating the scientific literature's knowledge about the role of alpha frequency in perceptual accuracy, and deepening the understanding of what determines trial-by-trial fluctuations of alpha parameters and how these fluctuations influence overall task performance. The hypotheses, confirmed empirically, were that different implicit strategies are put in place based on the task context, in order to maximize performance with optimal resource distribution (namely alpha frequency, associated positively with performance): “Lateralization” of the attentive resources towards one hemifield should be associated with higher alpha frequency difference between contralateral and ipsilateral hemisphere; “Distribution” of the attentive resources across hemifields should be associated with lower alpha frequency difference between hemispheres; These strategies, used by the participants according to their brain capabilities, have proven themselves adaptive or maladaptive depending on the different tasks to which they have been set: "Distribution" of the attentive resources seemed to be the best strategy when the distribution probability between hemifields was balanced: i.e. the neutral condition task. "Lateralization" of the attentive resources seemed to be more effective when the distribution probability between hemifields was biased towards one hemifield: i.e., the biased condition task.

Automatic classification of stellar orbits in axisymmetric potentials

Within the classification of orbits in axisymmetric stellar systems, we present a new algorithm able to automatically classify the orbits according to their nature. The algorithm involves the application of the correlation integral method to the surface of section of the orbit; fitting the cumulative distribution function built with the consequents in the surface of section of the orbit, we can obtain the value of its logarithmic slope m which is directly related to the orbit’s nature: for slopes m ≈ 1 we expect the orbit to be regular, for slopes m ≈ 2 we expect it to be chaotic. With this method we have a fast and reliable way to classify orbits and, furthermore, we provide an analytical expression of the probability that an orbit is regular or chaotic given the logarithmic slope m of its correlation integral. Although this method works statistically well, the underlying algorithm can fail in some cases, misclassifying individual orbits under some peculiar circumstances. The performance of the algorithm benefits from a rich sampling of the traces of the SoS, which can be obtained with long numerical integration of orbits. Finally we note that the algorithm does not differentiate between the subtypes of regular orbits: resonantly trapped and untrapped orbits. Such distinction would be a useful feature, which we leave for future work. Since the result of the analysis is a probability linked to a Gaussian distribution, for the very definition of distribution, some orbits even if they have a certain nature are classified as belonging to the opposite class and create the probabilistic tails of the distribution. So while the method produces fair statistical results, it lacks in absolute classification precision.

Relationship Between Orofacial Function, Dentofacial Morphology, And Bite Force In Young Subjects.

The aim was to evaluate the relationship between orofacial function, dentofacial morphology, and bite force in young subjects. Three hundred and sixteen subjects were divided according to dentition stage (early, intermediate, and late mixed and permanent dentition). Orofacial function was screened using the Nordic Orofacial Test-Screening (NOT-S). Orthodontic treatment need, bite force, lateral and frontal craniofacial dimensions and presence of sleep bruxism were also assessed. The results were submitted to descriptive statistics, normality and correlation tests, analysis of variance, and multiple linear regression to test the relationship between NOT-S scores and the studied independent variables. The variance of NOT-S scores between groups was not significant. The evaluation of the variables that significantly contributed to NOT-S scores variation showed that age and presence of bruxism related to higher NOT-S total scores, while the increase in overbite measurement and presence of closed lip posture related to lower scores. Bite force did not show a significant relationship with scores of orofacial dysfunction. No significant correlations between craniofacial dimensions and NOT-S scores were observed. Age and sleep bruxism were related to higher NOT-S scores, while the increase in overbite measurement and closed lip posture contributed to lower scores of orofacial dysfunction.