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Showing papers on "Gaussian process published in 2001"


Journal ArticleDOI
TL;DR: In this article, the authors develop an asymptotic theory of inference for an unrestricted two-regime TAR model with an autoregressive unit root, which is based on a new set of tools that combine unit root and empirical process methods.
Abstract: This paper develops an asymptotic theory of inference for an unrestricted two-regime Ž. threshold autoregressive TAR model with an autoregressive unit root. We find that the asymptotic null distribution of Wald tests for a threshold are nonstandard and different from the stationary case, and suggest basing inference on a bootstrap approximation. We also study the asymptotic null distributions of tests for an autoregressive unit root, and find that they are nonstandard and dependent on the presence of a threshold effect. We propose both asymptotic and bootstrap-based tests. These tests and distribution theory Ž. Ž allow for the joint consideration of nonlinearity thresholds and nonstationary unit . roots . Our limit theory is based on a new set of tools that combine unit root asymptotics with empirical process methods. We work with a particular two-parameter empirical process that converges weakly to a two-parameter Brownian motion. Our limit distributions involve stochastic integrals with respect to this two-parameter process. This theory is entirely new and may find applications in other contexts. We illustrate the methods with an application to the U.S. monthly unemployment rate. We find strong evidence of a threshold effect. The point estimates suggest that the threshold effect is in the short-run dynamics, rather than in the dominate root. While the conventional ADF test for a unit root is insignificant, our TAR unit root tests are arguably significant. The evidence is quite strong that the unemployment rate is not a unit root process, and there is considerable evidence that the series is a stationary TAR process.

719 citations


Journal ArticleDOI
TL;DR: An information-theoretic perspective on optimum transmitter strategies, and the gains obtained by employing them, for systems with transmit antenna arrays and imperfect channel feedback is provided.
Abstract: The use of channel feedback from receiver to transmitter is standard in wireline communications. While knowledge of the channel at the transmitter would produce similar benefits for wireless communications as well, the generation of reliable channel feedback is complicated by the rapid time variations of the channel for mobile applications. The purpose of this paper is to provide an information-theoretic perspective on optimum transmitter strategies, and the gains obtained by employing them, for systems with transmit antenna arrays and imperfect channel feedback. The spatial channel, given the feedback, is modeled as a complex Gaussian random vector. Two extreme cases are considered: mean feedback, in which the channel side information resides in the mean of the distribution, with the covariance modeled as white, and covariance feedback, in which the channel is assumed to be varying too rapidly to track its mean, so that the mean is set to zero, and the information regarding the relative geometry of the propagation paths is captured by a nonwhite covariance matrix. In both cases, the optimum transmission strategies, maximizing the information transfer rate, are determined as a solution to simple numerical optimization problems. For both feedback models, our numerical results indicate that, when there is a moderate disparity between the strengths of different paths from the transmitter to the receiver, it is nearly optimal to employ the simple beamforming strategy of transmitting all available power in the direction which the feedback indicates is the strongest.

703 citations


Journal ArticleDOI
TL;DR: A simple closed-form approximation for the distribution of the peak-to-average power ratio (PAPR) in strictly band-limited orthogonal frequency-division multiplexing (OFDM) signals is developed, based on the level-crossing rate analysis.
Abstract: The distribution of the peak-to-average power ratio (PAPR) in strictly band-limited orthogonal frequency-division multiplexing (OFDM) signals is studied. Assuming that the base-band OFDM signal is characterized as a band-limited complex Gaussian process, we first attempt to derive the exact distribution of the PAPR in the band-limited OFDM signals. Since this distribution cannot be expressed in a closed form, we further develop a simple closed-form approximation, based on the level-crossing rate analysis. Comparisons of the proposed distributions with those obtained by computer simulations show good agreement and convergence with an increase in the number of subcarriers.

658 citations


Journal ArticleDOI
TL;DR: In this paper, a stochastic calculus with respect to a Gaussian process of the form B_t = \int^t_0 K(t, s), dW_s, where W$ is a Wiener process and K(T, s) is a square integrable kernel was developed.
Abstract: In this paper we develop a stochastic calculus with respect to a Gaussian process of the form $B_t = \int^t_0 K(t, s)\, dW_s$, where $W$ is a Wiener process and $K(t, s)$ is a square integrable kernel, using the techniques of the stochastic calculus of variations. We deduce change-of-variable formulas for the indefinite integrals and we study the approximation by Riemann sums.The particular case of the fractional Brownian motion is discussed.

542 citations


Proceedings Article
03 Jan 2001
TL;DR: An extension to the Mixture of Experts model, where the individual experts are Gaussian Process (GP) regression models, using an input-dependent adaptation of the Dirichlet Process to implement a gating network for an infinite number of Experts.
Abstract: We present an extension to the Mixture of Experts (ME) model, where the individual experts are Gaussian Process (GP) regression models. Using an input-dependent adaptation of the Dirichlet Process, we implement a gating network for an infinite number of Experts. Inference in this model may be done efficiently using a Markov Chain relying on Gibbs sampling. The model allows the effective covariance function to vary with the inputs, and may handle large datasets – thus potentially overcoming two of the biggest hurdles with GP models. Simulations show the viability of this approach.

507 citations


Book ChapterDOI
TL;DR: In this paper, the authors focus on the inequalities, small-ball probabilities, and application of Gaussian processes, and find that the small ball probability is a key step in studying the lower limits of the Gaussian process.
Abstract: Publisher Summary This chapter focuses on the inequalities, small ball probabilities, and application of Gaussian processes. It is well-known that the large deviation result plays a fundamental role in studying the upper limits of Gaussian processes, such as the Strassen type law of the iterated logarithm. However, the complexity of the small ball estimate is well-known, and there are only a few Gaussian measures for which the small ball probability can be determined completely. The small ball probability is a key step in studying the lower limits of the Gaussian process. It has been found that the small ball estimate has close connections with various approximation quantities of compact sets and operators, and has a variety of applications in studies of Hausdorff dimensions, rate of convergence in Strassen's law of the iterated logarithm, and empirical processes.

442 citations


Journal ArticleDOI
TL;DR: In this paper, an asymptotic theory for nonlinear regression with integrated processes is developed, and sufficient conditions for weak consistency are given and a limit distribution theory is provided, which is mixed normal with mixing variates that depend on the sojourn time of the limiting Brownian motion of the integrated process.
Abstract: An asymptotic theory is developed for nonlinear regression with integrated processes. The models allow for nonlinear effects from unit root time series and therefore deal with the case of parametric nonlinear cointegration. The theory covers integrable and asymptotically homogeneous functions. Sufficient conditions for weak consistency are given and a limit distribution theory is provided. The rates of convergence depend on the properties of the nonlinear regression function, and are shown to be as slow as n 1/4 for integrable functions, and to be generally polynomial in n 1/2 for homogeneous functions. For regressions with integrable functions, the limiting distribution theory is mixed normal with mixing variates that depend on the sojourn time of the limiting Brownian motion of the integrated process.

435 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that in contrast to continuous processes, the variance of the estimators cannot be reduced by smoothing beyond a scale set by the number of point events in the interval.
Abstract: The spectrum and coherency are useful quantities for characterizing the temporal correlations and functional relations within and between point processes. This article begins with a review of these quantities, their interpretation, and how they may be estimated. A discussion of how to assess the statistical significance of features in these measures is included. In addition, new work is presented that builds on the framework established in the review section. This work investigates how the estimates and their error bars are modified by finite sample sizes. Finite sample corrections are derived based on a doubly stochastic inhomogeneous Poisson process model in which the rate functions are drawn from a low-variance gaussian process. It is found that in contrast to continuous processes, the variance of the estimators cannot be reduced by smoothing beyond a scale set by the number of point events in the interval. Alternatively, the degrees of freedom of the estimators can be thought of as bounded from above by the expected number of point events in the interval. Further new work describing and illustrating a method for detecting the presence of a line in a point process spectrum is also presented, corresponding to the detection of a periodic modulation of the underlying rate. This work demonstrates that a known statistical test, applicable to continuous processes, applies with little modification to point process spectra and is of utility in studying a point process driven by a continuous stimulus. Although the material discussed is of general applicability to point processes, attention will be confined to sequences of neuronal action potentials (spike trains), the motivation for this work.

375 citations


Journal ArticleDOI
24 Jun 2001
TL;DR: It is shown that several previously proposed low-complexity algorithms based on interference cancellation can be derived in a simple direct way by approximating the extrinsic PMF output by the SISO decoders either as a single mass-point PMF or as a Gaussian PDF with the same mean and variance.
Abstract: The synchronous chip-rate discrete-time CDMA channel with channel coding is analysed and the corresponding factor graph is presented. Iterative joint decoding can be derived in a simple and direct way by applying the sum-product algorithm to the factor graph. Since variables have degree 2, no computation takes place at the variable nodes. Computation at the code nodes is just soft-in soft-out (SISO) decoding, whose output is the extrinsic PMF of the coded symbols. Computation at the channel transition nodes is equivalent to MAP symbol-by-symbol multiuser detection, whose complexity is generally exponential in K. We show that several previously proposed low-complexity algorithms based on interference cancellation (IC) can be derived in a simple direct way by approximating the extrinsic PMF output by the SISO decoders either as a single mass-point PMF (hard decision) or as a Gaussian PDF with the same mean and variance (moment matching). Differently from all previously presented methods (derived from heuristic reasoning), we see clearly that extrinsic rather than a posteriori PMF should be fed back. This yields important advantages in terms of limiting achievable throughput.

331 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the process D has the law of the process of the largest eigenvalues of the main minors of an infinite random matrix drawn from Gaussian Unitary Ensemble.
Abstract: Consider the process D k , k = 1,2,…, given by B i being independent standard Brownian motions. This process describes the limiting behavior “near the edge” in queues in series, totally asymmetric exclusion processes or oriented percolation. The problem of finding the distribution of D. was posed in [GW]. The main result of this paper is that the process D. has the law of the process of the largest eigenvalues of the main minors of an infinite random matrix drawn from Gaussian Unitary Ensemble.

276 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered a variant of the competing risks problem in which a terminal event censors a non-terminal event, but not vice versa, and formulated the joint distribution of the events via a gamma frailty model in the upper wedge where data are observable, with the marginal distributions unspecified.
Abstract: SUMMARY We consider a variation of the competing risks problem in which a terminal event censors a non-terminal event, but not vice versa. The joint distribution of the events is formulated via a gamma frailty model in the upper wedge where data are observable (Day et al., 1997), with the marginal distributions unspecified. An estimator for the association parameter is obtained from a concordance estimating function. A novel plug-in estimator for the marginal distribution of the non-terminal event is shown to be uniformly consistent and to converge weakly to a Gaussian process. The assumptions on the joint distribution outside the upper wedge are weaker than those usually made in competing risks analyses. Simulations demonstrate that the methods work well with practical sample sizes. The proposals are illustrated with data on morbidity and mortality in leukaemia patients.

Journal ArticleDOI
TL;DR: A flexible class of Cox processes whose stochastic intensity is a space–time Ornstein–Uhlenbeck process is described, and moment‐based methods of parameter estimation are developed.
Abstract: Space–time point pattern data have become more widely available as a result of technological developments in areas such as geographic information systems. We describe a flexible class of space–time point processes. Our models are Cox processes whose stochastic intensity is a space–time Ornstein–Uhlenbeck process. We develop moment-based methods of parameter estimation, show how to predict the underlying intensity by using a Markov chain Monte Carlo approach and illustrate the performance of our methods on a synthetic data set.

Journal ArticleDOI
TL;DR: In this article, a scaling limit of the height function on the domino tiling model (dimer model) on simply connected regions in $\mathbf{Z}^2$ and show that it is the "massless free field", a Gaussian process with independent coefficients when expanded in the eigenbasis of the Laplacian.
Abstract: We define a scaling limit of the height function on the domino tiling model (dimer model) on simply connected regions in $\mathbf{Z}^2$ and show that it is the “massless free field,” a Gaussian process with independent coefficients when expanded in the eigenbasis of the Laplacian.

Journal ArticleDOI
TL;DR: A ridge theorem of the Gaussian chirp dictionary is proved, from which an estimate of the locally optimal scale and chirP is built and the efficiency and speed of the method is demonstrated on a sound signal.
Abstract: We introduce a modified matching pursuit algorithm, called fast ridge pursuit, to approximate N-dimensional signals with M Gaussian chirps at a computational cost O(MN) instead of the expected O(MN/sup 2/logN). At each iteration of the pursuit, the best Gabor atom is first selected, and then, its scale and chirp rate are locally optimized so as to get a "good" chirp atom, i.e., one for which the correlation with the residual is locally maximized. A ridge theorem of the Gaussian chirp dictionary is proved, from which an estimate of the locally optimal scale and chirp is built. The procedure is restricted to a sub-dictionary of local maxima of the Gaussian Gabor dictionary to accelerate the pursuit further. The efficiency and speed of the method is demonstrated on a sound signal.

Journal ArticleDOI
TL;DR: In this article, the moments of Wiener integrals of fractional Brownian motion with respect to the Lp-norm of the integrand were studied and it was shown that when the self-similarity index H> 1 2, we can have only an upper inequality.

Journal ArticleDOI
24 Jun 2001
TL;DR: Aspects of the duality between the information-embedding problem and the Wyner-Ziv (1976) problem of source coding with side information at the decoder are developed and used to establish a spectrum new results on these and related problems, with implications for a number of important applications.
Abstract: Aspects of the duality between the information-embedding problem and the Wyner-Ziv (1976) problem of source coding with side information at the decoder are developed and used to establish a spectrum new results on these and related problems, with implications for a number of important applications. The single-letter characterization of the information-embedding problem is developed and related to the corresponding characterization of the Wyner-Ziv problem, both of which correspond to optimization of a common mutual information difference. Dual variables and dual Markov conditions are identified, along with the dual role of noise and distortion in the two problems. For a Gaussian context with quadratic distortion metric, a geometric interpretation of the duality is developed. From such insights, we develop a capacity-achieving information-embedding system based on nested lattices. We show the resulting encoder-decoder has precisely the same decoder-encoder structure as the corresponding Wyner-Ziv system based on nested lattices that achieves the rate-distortion limit. For a binary context with Hamming distortion metric, the information-embedding capacity is developed, along with its relationship to the corresponding Wyner-Ziv rate-distortion function. In turn, an information-embedding system for this case based on nested linear codes is constructed having an encoder-decoder that is identical to the decoder-encoder structure for the corresponding system that achieves the Wyner-Ziv rate-distortion limit. Finally, based on these results, a simple layered joint source-channel coding system is developed with a perfectly symmetric encoder-decoder structure. Its application and performance is discussed in a broadcast setting in which there is a need to control the fidelity experienced by different receivers. Among other results, we show that such systems and their multilayer extensions retain attractive optimality properties in the Gaussian-quadratic case, but not in the binary-Hamming case.

Proceedings ArticleDOI
07 Oct 2001
TL;DR: The EM ofGM can be regarded as a special EM of HMM and the EM algorithm of GMM based on symbols is faster in implementation than the EM algorithms based on samples (or on observation) traditionally.
Abstract: The HMM (hidden Markov model) is a probabilistic model of the joint probability of a collection of random variables with both observations and states. The GMM (Gaussian mixture model) is a finite mixture probability distribution model. Although the two models have a close relationship, they are always discussed independently and separately. The EM (expectation-maximum) algorithm is a general method to improve the descent algorithm for finding the maximum likelihood estimation. The EM of HMM and the EM of GMM have similar formulae. Two points are proposed in this paper. One is that the EM of GMM can be regarded as a special EM of HMM. The other is that the EM algorithm of GMM based on symbols is faster in implementation than the EM algorithm of GMM based on samples (or on observation) traditionally.

Journal ArticleDOI
TL;DR: This study is motivated by the case of a high-speed network where a large number of sources are expected to be multiplexed and by appealing to Central Limit Theorem type of arguments, the input process is model as a general Gaussian process.
Abstract: In this paper, we propose an approximation for the loss probability, PL (x), in a finite buffer system with buffer size x. Our study is motivated by the case of a high-speed network where a large number of sources are expected to be multiplexed. Hence, by appealing to Central Limit Theorem type of arguments, we model the input process as a general Gaussian process. Our result is obtained by making a simple mapping from the tail probability in an infinite buffer system to the loss probability in a finite buffer system. We also provide a strong asymptotic relationship between our approximation and the actual loss probability for a fairly large class of Gaussian input processes. We derive some interesting asymptotic properties of our approximation and illustrate its effectiveness via a detailed numerical investigation.

Proceedings Article
03 Jan 2001
TL;DR: Kernel Meta-Training, which is a method of learning a Gaussian Process kernel from a distribution of functions that generates the learned function, is introduced.
Abstract: This paper presents AutoDJ: a system for automatically generating music playlists based on one or more seed songs selected by a user. AutoDJ uses Gaussian Process Regression to learn a user preference function over songs. This function takes music metadata as inputs. This paper further introduces Kernel Meta-Training, which is a method of learning a Gaussian Process kernel from a distribution of functions that generates the learned function. For playlist generation, AutoDJ learns a kernel from a large set of albums. This learned kernel is shown to be more effective at predicting users' playlists than a reasonable hand-designed kernel.

Journal ArticleDOI
TL;DR: In this article, the Sampson and Guttorp approach is used to model the non-stationary correlation function r(x, x′) of a Gaussian spatial process through a bijective space deformation, f, so that in the deformed space the spatial correlation function can be considered isotropic.
Abstract: We use the Sampson and Guttorp approach to model the non-stationary correlation function r(x, x′) of a Gaussian spatial process through a bijective space deformation, f, so that in the deformed space the spatial correlation function can be considered isotropic, namely r(x, x′) = ρ(∣ f(x)−f(x′)∣), where ρ belongs to a known parametric family. Given the locations in the deformed space of a number of geographic sites at which data are available, we smoothly extrapolate the deformation to the whole region of interest. Using a Bayesian framework, we estimate jointly these locations, as well as the parameters of the correlation function and the variance parameters. The advantage of our Bayesian approach is that it allows us to obtain measures of uncertainty of all these parameters. As the parameter space is of a very high dimension, we implement an MCMC method for obtaining samples from the posterior distributions of interest. We demonstrate our method through a simulation study, and show an application to a real data set. Copyright © 2001 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this paper, the sensitivity of DFT molecular properties to the radial and angular numerical integration grid meshes, as well as to the partitioning scheme, is discussed for a number of molecules using the Gaussian 98 program system.

Journal ArticleDOI
TL;DR: An image segmentation method based on texture analysis that determines a novel set of texture features derived from a Gaussian-Markov random fields model and a recently proposed method for integrating relaxation oscillator networks is proposed.
Abstract: We propose an image segmentation method based on texture analysis. Our method is composed of two parts. The first part determines a novel set of texture features derived from a Gaussian-Markov random fields (GMRF) model. Unlike a GMRF-based approach, our method does not employ model parameters as features or require the extraction of features for a fixed set of texture types a priori. The second part is a 2D array of locally excitatory globally inhibitory oscillator networks (LEGION). After being filtered for noise suppression, features are used to determine the local couplings in the network. When LEGION runs, the oscillators corresponding to the same texture tend to synchronize, whereas different texture regions tend to correspond to distinct phases. In simulations, a large system of differential equations is solved for the first time using a recently proposed method for integrating relaxation oscillator networks. We provide results on real texture images to demonstrate the performance of our method.

Journal ArticleDOI
TL;DR: In this paper, a valid asymptotic expansion for the covariance of functions of multivariate normal vectors is applied to approximate autocovariances of time series generated by nonlinear transformation of Gaussian latent variates, and nonlinear functions of these, with special reference to long memory stochastic volatility models, serve to identify the roles played by the underlying Gaussian processes.

Journal ArticleDOI
TL;DR: In this article, the authors proposed the use of the subsampling method for inferential purposes and applied it to Manski's maximum score estimator and its small sample performance was highlighted via a simulation study.

Journal ArticleDOI
TL;DR: In this paper, a family of semiparametric transformation models for point processes with positive jumps of arbitrary sizes is proposed, which offer great flexibilities in formulating the effects of covariates on the mean function of the point process.
Abstract: In this article, we propose a family of semiparametric transformation models for point processes with positive jumps of arbitrary sizes. These models offer great flexibilities in formulating the effects of covariates on the mean function of the point process while leaving the stochastic structure completely unspecified. We develop a class of estimating equations for the baseline mean function and the vector-valued regression parameter based on censored point processes and covariate data. These equations can be solved easily by the standard Newton–Raphson algorithm. The resultant estimator of the regression parameter is consistent and asymptotically normal with a covariance matrix that can be estimated consistently. Furthermore, the estimator of the baseline mean function is uniformly consistent and, upon proper normalization, converges weakly to a zero-mean Gaussian process with an easily estimated covariance function. We demonstrate through extensive simulation studies that the proposed inference procedu...

Journal ArticleDOI
TL;DR: The models are shown to be easy to analyse yet flexible enough for a detailed statistical analysis of a particular agricultural experiment concerning the development of two weed species on an organic barley field.
Abstract: Log Gaussian Cox processes as introduced in M0ller et al. (1998) are extended to space-time models called log Gaussian Cox birth processes. These processes allow modelling of spatial and temporal heterogeneity in time series of increasing point processes consisting of different types of points. The models are shown to be easy to analyse, yet flexible enough for a detailed statistical analysis of a particular agricultural experiment concerning the development of two weed species on an organic barley field. Particularly, the aspects of estimation, model validation and intensity surface prediction are discussed.

Journal ArticleDOI
TL;DR: In this paper, the authors study the asymptotic behavior of the likelihood ratio statistic for testing homogeneity in the finite mixture models of a general parametric distribution family and prove that the limiting distribution of this statistic is the squared supremum of a truncated standard Gaussian process.
Abstract: The authors study the asymptotic behaviour of the likelihood ratio statistic for testing homogeneity in the finite mixture models of a general parametric distribution family. They prove that the limiting distribution of this statistic is the squared supremum of a truncated standard Gaussian process. The autocorrelation function of the Gaussian process is explicitly presented. A re-sampling procedure is recommended to obtain the asymptotic p-value. Three kernel functions, normal, binomial and Poisson, are used in a simulation study which illustrates the procedure.

Proceedings Article
03 Jan 2001
TL;DR: An implementation of the framework of mutual information kernels for learning covariance kernels from unlabeled task data using Bayesian techniques is described which uses variational Bayesian mixtures of factor analyzers in order to attack classification problems in high-dimensional spaces where labeled data is sparse, but unlabeling data is abundant.
Abstract: We propose the framework of mutual information kernels for learning covariance kernels, as used in Support Vector machines and Gaussian process classifiers, from unlabeled task data using Bayesian techniques. We describe an implementation of this framework which uses variational Bayesian mixtures of factor analyzers in order to attack classification problems in high-dimensional spaces where labeled data is sparse, but unlabeled data is abundant.

Journal ArticleDOI
TL;DR: It is found that asymptotically, linear parallel interference cancellation diverges for systems loads of greater than about 17% and optimal or near-optimal relaxation parameters for parallel and serial cancellation methods are derived.
Abstract: We consider the convergence in norm of several iterative implementations of linear multiuser receivers, under the assumption of long random spreading sequences. We find that asymptotically, linear parallel interference cancellation diverges for systems loads of greater than about 17%. Using known results from the theory of iterative solutions for linear systems we derive optimal or near-optimal relaxation parameters for parallel (first- and second-order stationary, Chebyshev) and serial cancellation (successive relaxation) methods. An analytic comparison of the asymptotic convergence factor for the various methods is given. Simulations are used to verify results for finite size systems.

Journal ArticleDOI
TL;DR: In this article, the Wiener chaos expansion for the local time of the fractional Brownian motion with Hurst parameter H and a Tanaka formula in the case H> 1 3 were derived.