This paper presents the recurrent estimation of distributions (RED) for modeling real-valued data in a semiparametric fashion. RED models make two novel uses of recurrent neural networks (RNNs) for density estimation of general real-valued data. First, RNNs are used to transform input covariates into a latent space to better capture conditional dependencies in inputs. After, an RNN is used to compute the conditional distributions of the latent covariates. The resulting model is efficient to train, compute, and sample from, whilst producing normalized pdfs. The effectiveness of RED is shown via several real-world data experiments. Our results show that RED models achieve a lower held-out negative log-likelihood than other neural network approaches across multiple dataset sizes and dimensionalities. Further context of the efficacy of RED is provided by considering anomaly detection tasks, where we also observe better performance over alternative models.
Nonparametric density estimation has been a well studied problem in statistics and machine learning @cite_0 . Unfortunately, nonparametric approaches like kernel density estimation suffer greatly from the curse of dimensionality and do not perform well when data does not have a small number of dimensions ( @math ). To alleviate this, several semiparametric approaches have been explored. Such approaches include forest density estimation @cite_15 , which assumes that the data has a forest (i.e. a collection of trees) structured graph. This assumption leads to a density which factorizes in a first order Markovian fashion through a tree traversal of the graph. Another common semiparametric approach is to use a nonparanormal type model @cite_21 . This approach uses a Gaussian copula with a rank-based transformation and a sparse precision matrix. While both approaches are well-understood theoretically, their strong assumptions often lead to inflexible models.
@cite_0 @cite_15 @cite_21
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Recent methods for estimating sparse undirected graphs for real-valued data in high dimensional problems rely heavily on the assumption of normality. We show how to use a semiparametric Gaussian copula---or "nonparanormal"---for high dimensional inference. Just as additive models extend linear models by replacing linear functions with a set of one-dimensional smooth functions, the nonparanormal extends the normal by transforming the variables by smooth functions. We derive a method for estimating the nonparanormal, study the method's theoretical properties, and show that it works well in many examples. We study graph estimation and density estimation in high dimensions, using a family of density estimators based on forest structured undirected graphical models. For density estimation, we do not assume the true distribution corresponds to a forest; rather, we form kernel density estimates of the bivariate and univariate marginals, and apply Kruskal's algorithm to estimate the optimal forest on held out data. We prove an oracle inequality on the excess risk of the resulting estimator relative to the risk of the best forest. For graph estimation, we consider the problem of estimating forests with restricted tree sizes. We prove that finding a maximum weight spanning forest with restricted tree size is NP-hard, and develop an approximation algorithm for this problem. Viewing the tree size as a complexity parameter, we then select a forest using data splitting, and prove bounds on excess risk and structure selection consistency of the procedure. Experiments with simulated data and microarray data indicate that the methods are a practical alternative to Gaussian graphical models. This paper develops a nonparametric density estimator with parametric overtones. Suppose f(x, θ) is some family of densities, indexed by a vector of parameters θ. We define a local kernel-smoothed likelihood function which, for each x, can be used to estimate the best local parametric approximant to the true density. This leads to a new density estimator of the form f(x, o(x)), thus inserting the best local parameter estimate for each new value of x. When the bandwidth used is large, this amounts to ordinary full likelihood parametric density estimation, while for moderate and small bandwidths the method is essentially nonparametric, using only local properties of data and the model. Alternative ways more general than via the local likelihood are also described. The methods can be seen as ways of nonparametrically smoothing the parameter within a parametric class. Properties of this new semiparametric estimator are investigated. Our preferred version has approximately the same variance as the ordinary kernel method but potentially a smaller bias. The new method is seen to perform better than the traditional kernel method in a broad nonparametric vicinity of the parametric model employed, while at the same time being capable of not losing much in precision to full likelihood methods when the model is correct. Other versions of the method are approximately equivalent to using particular higher order kernels in a semiparametric framework. The methodology we develop can be seen as the density estimation parallel to local likelihood and local weighted least squares theory in nonparametric regression. We study the density estimation problem with observations generated by certain dynamical systems that admit a unique underlying invariant Lebesgue density. Observations drawn from dynamical systems are not independent and moreover, usual mixing concepts may not be appropriate for measuring the dependence among these observations. By employing the @math -mixing concept to measure the dependence, we conduct statistical analysis on the consistency and convergence of the kernel density estimator. Our main results are as follows: First, we show that with properly chosen bandwidth, the kernel density estimator is universally consistent under @math -norm; Second, we establish convergence rates for the estimator with respect to several classes of dynamical systems under @math -norm. In the analysis, the density function @math is only assumed to be H " o lder continuous which is a weak assumption in the literature of nonparametric density estimation and also more realistic in the dynamical system context. Last but not least, we prove that the same convergence rates of the estimator under @math -norm and @math -norm can be achieved when the density function is H " o lder continuous, compactly supported and bounded. The bandwidth selection problem of the kernel density estimator for dynamical system is also discussed in our study via numerical simulations. Kernel density estimation is the most widely-used practical method for accurate nonparametric density estimation. However, long-standing worst-case theoretical results showing that its performance worsens exponentially with the dimension of the data have quashed its application to modern high-dimensional datasets for decades. In practice, it has been recognized that often such data have a much lower-dimensional intrinsic structure. We propose a small modification to kernel density estimation for estimating probability density functions on Riemannian submanifolds of Euclidean space. Using ideas from Riemannian geometry, we prove the consistency of this modified estimator and show that the convergence rate is determined by the intrinsic dimension of the submanifold. We conclude with empirical results demonstrating the behavior predicted by our theory. Elementary approaches to classic strong laws of large numbers use a monotonicity argument or a Tauberian argument of summability theory. Together with results on variance of sums of dependent random variables they allow to establish various strong laws of large numbers in case of dependence, especially under mixing conditions. Strong consistency of nonparametric regression estimates of local averaging type (kernel and nearest neighbor estimates), pointwise as well as in L 2, can be considered as a generalization of strong laws of large numbers. Both approaches can be used to establish strong universal consistency in the case of independence and, mostly by sharpened integrability assumptions, consistency under ρ-mixing or α-mixing. In a similar way Rosenblatt-Parzen kernel density estimates are treated. We present simple and computationally efficient nonparametric estimators of R 'enyi entropy and mutual information based on an i.i.d. sample drawn from an unknown, absolutely continuous distribution over @math . The estimators are calculated as the sum of @math -th powers of the Euclidean lengths of the edges of the generalized nearest-neighbor' graph of the sample and the empirical copula of the sample respectively. For the first time, we prove the almost sure consistency of these estimators and upper bounds on their rates of convergence, the latter of which under the assumption that the density underlying the sample is Lipschitz continuous. Experiments demonstrate their usefulness in independent subspace analysis. A mixture of factor analyzer is a semiparametric density estimator that performs clustering and dimensionality reduction in each cluster (component) simultaneously. It performs nonlinear dimensionality reduction by modeling the density as a mixture of local linear models. The approach can be used for classification by modeling each class-conditional density using a mixture model and the complete data is then a mixture of mixtures. We propose an incremental mixture of factor analysis algorithm where the number of components (local models) in the mixture and the number of factors in each component (local dimensionality) are determined adaptively. Our results on different pattern classification tasks prove the utility of our approach and indicate that our algorithms find a good trade-off between model complexity and accuracy. We introduce the Gaussian Process Convolution Model (GPCM), a two-stage non-parametric generative procedure to model stationary signals as the convolution between a continuous-time white-noise process and a continuous-time linear filter drawn from Gaussian process. The GPCM is a continuous-time nonparametric-window moving average process and, conditionally, is itself a Gaussian process with a nonparametric kernel defined in a probabilistic fashion. The generative model can be equivalently considered in the frequency domain, where the power spectral density of the signal is specified using a Gaussian process. One of the main contributions of the paper is to develop a novel variational free-energy approach based on inter-domain inducing variables that efficiently learns the continuous-time linear filter and infers the driving white-noise process. In turn, this scheme provides closed-form probabilistic estimates of the covariance kernel and the noise-free signal both in denoising and prediction scenarios. Additionally, the variational inference procedure provides closed-form expressions for the approximate posterior of the spectral density given the observed data, leading to new Bayesian nonparametric approaches to spectrum estimation. The proposed GPCM is validated using synthetic and real-world signals. We propose and analyse estimators for statistical functionals of one or more distributions under nonparametric assumptions. Our estimators are derived from the von Mises expansion and are based on the theory of influence functions, which appear in the semiparametric statistics literature. We show that estimators based either on data-splitting or a leave-one-out technique enjoy fast rates of convergence and other favorable theoretical properties. We apply this framework to derive estimators for several popular information theoretic quantities, and via empirical evaluation, show the advantage of this approach over existing estimators. This paper presents a patch-based approach for crowd density estimation in public scenes. We formulate the problem of estimating density in a structured learning framework applied to random decision forests. Our approach learns the mapping between patch features and relative locations of all objects inside each patch, which contribute to generate the patch density map through Gaussian kernel density estimation. We build the forest in a coarse-to-fine manner with two split node layers, and further propose a crowdedness prior and an effective forest reduction method to improve the estimation accuracy and speed. Moreover, we introduce a semi-automatic training method to learn the estimator for a specific scene. We achieved state-of-the-art results on the public Mall dataset and UCSD dataset, and also proposed two potential applications in traffic counts and scene understanding with promising results. Visual features are commonly modeled with probability density functions in computer vision problems, but current methods such as a mixture of Gaussians and kernel density estimation suffer from either the lack of flexibility by fixing or limiting the number of Gaussian components in the mixture or large memory requirement by maintaining a nonparametric representation of the density. These problems are aggravated in real-time computer vision applications since density functions are required to be updated as new data becomes available. We present a novel kernel density approximation technique based on the mean-shift mode finding algorithm and describe an efficient method to sequentially propagate the density modes over time. Although the proposed density representation is memory efficient, which is typical for mixture densities, it inherits the flexibility of nonparametric methods by allowing the number of components to be variable. The accuracy and compactness of the sequential kernel density approximation technique is illustrated by both simulations and experiments. Sequential kernel density approximation is applied to online target appearance modeling for visual tracking, and its performance is demonstrated on a variety of videos. This review presents a unified, efficient model of random decision forests which can be applied to a number of machine learning, computer vision, and medical image analysis tasks. Our model extends existing forest-based techniques as it unifies classification, regression, density estimation, manifold learning, semi-supervised learning, and active learning under the same decision forest framework. This gives us the opportunity to write and optimize the core implementation only once, with application to many diverse tasks. The proposed model may be used both in a discriminative or generative way and may be applied to discrete or continuous, labeled or unlabeled data. The main contributions of this review are: (1) Proposing a unified, probabilistic and efficient model for a variety of learning tasks; (2) Demonstrating margin-maximizing properties of classification forests; (3) Discussing probabilistic regression forests in comparison with other nonlinear regression algorithms; (4) Introducing density forests for estimating probability density functions; (5) Proposing an efficient algorithm for sampling from a density forest; (6) Introducing manifold forests for nonlinear dimensionality reduction; (7) Proposing new algorithms for transductive learning and active learning. Finally, we discuss how alternatives such as random ferns and extremely randomized trees stem from our more general forest model. This document is directed at both students who wish to learn the basics of decision forests, as well as researchers interested in the new contributions. It presents both fundamental and novel concepts in a structured way, with many illustrative examples and real-world applications. Thorough comparisons with state-of-the-art algorithms such as support vector machines, boosting and Gaussian processes are presented and relative advantages and disadvantages discussed. The many synthetic examples and existing commercial applications demonstrate the validity of the proposed model and its flexibility. In this paper, we present a novel way of analyzing and summarizing a collection of curves, based on piecewise constant density estimation. The curves are partitioned into clusters, and the dimensions of the curves points are discretized into intervals. The cross-product of these univariate partitions forms a data grid of cells, which represents a nonparametric estimator of the joint density of the curves and point dimensions. The best model is selected using a Bayesian model selection approach and retrieved using combinatorial optimization algorithms. The proposed method requires no parameter setting and makes no assumption regarding the curves; beyond functional data, it can be applied to distributional data. The practical interest of the approach for functional data and distributional data exploratory analysis is presented on two real world datasets. Two central limit theorems for sample Fr 'echet means are derived, both significant for nonparametric inference on non-Euclidean spaces. The first one, Theorem 2.2, encompasses and improves upon most earlier CLTs on Fr 'echet means and broadens the scope of the methodology beyond manifolds to diverse new non-Euclidean data including those on certain stratified spaces which are important in the study of phylogenetic trees. It does not require that the underlying distribution @math have a density, and applies to both intrinsic and extrinsic analysis. The second theorem, Theorem 3.3, focuses on intrinsic means on Riemannian manifolds of dimensions @math and breaks new ground by providing a broad CLT without any of the earlier restrictive support assumptions. It makes the statistically reasonable assumption of a somewhat smooth density of @math . The excluded case of dimension @math proves to be an enigma, although the first theorem does provide a CLT in this case as well under a support restriction. Theorem 3.3 immediately applies to spheres @math , @math , which are also of considerable importance in applications to axial spaces and to landmarks based image analysis, as these spaces are quotients of spheres under a Lie group @math of isometries of @math . People can understand complex auditory and visual information, often using one to disambiguate the other. Automated analysis, even at a low-level, faces severe challenges, including the lack of accurate statistical models for the signals, and their high-dimensionality and varied sampling rates. Previous approaches [6] assumed simple parametric models for the joint distribution which, while tractable, cannot capture the complex signal relationships. We learn the joint distribution of the visual and auditory signals using a non-parametric approach. First, we project the data into a maximally informative, low-dimensional subspace, suitable for density estimation. We then model the complicated stochastic relationships between the signals using a nonparametric density estimator. These learned densities allow processing across signal modalities. We demonstrate, on synthetic and real signals, localization in video of the face that is speaking in audio, and, conversely, audio enhancement of a particular speaker selected from the video. We introduce and investigate a new model of learning probability distributions from independent draws. Our model is inspired by the popular Probably Approximately Correct (PAC) model for learning boolean functions from labeled examples [24], in the sense that we emphasize efficient and approximate learning, and we study the learnability of restricted classes of target distributions. The dist ribut ion classes we examine are often defined by some simple computational mechanism for transforming a truly random string of input bits (which is not visible to the learning algorithm) into the stochastic observation (output) seen by the learning algorithm. In this paper, we concentrate on discrete distributions over O, I n. The problem of inferring an approximation to an unknown probability distribution on the basis of independent draws has a long and complex history in the pattern recognition and statistics literature. For instance, the problem of estimating the parameters of a Gaussian density in highdimensional space is one of the most studied statistical problems. Distribution learning problems have often been investigated in the context of unsupervised learning, in which a linear mixture of two or more distributions is generating the observations, and the final goal is not to model the distributions themselves, but to predict from which distribution each observation was drawn. Data clustering methods are a common tool here. There is also a large literature on nonpararnetric density estimation, in which no assumptions are made on the unknown target density. Nearest-neighbor approaches to the unsupervised learning problem often arise in the nonparametric setting. While we obviously cannot do justice to these areas here, the books of Duda and Hart [9] and Vapnik [25] provide excellent overviews and introductions to the pattern recognition work, as well as many pointers for further reading. See also Izenman’s recent survey article [16]. Roughly speaking, our work departs from the traditional statistical and pattern recognition approaches in two ways. First, we place explicit emphasis on the comput ationrd complexity of distribution learning. It seems fair to say that while previous research has provided an excellent understanding of the information-theoretic issues involved in dis- Network intrusion detection is the problem of detecting anomalous network connections caused by intrusive activities. Many intrusion detection systems proposed before use both normal and intrusion data to build their classifiers. However, intrusion data are usually scarce and difficult to collect. We propose to solve this problem using a novelty detection approach. In particular, we propose to take a nonparametric density estimation approach based on Parzen-window estimators with Gaussian kernels to build an intrusion detection system using normal data only. To facilitate comparison, we have tested our system on the KDD Cup 1999 dataset. Our system compares favorably with the KDD Cup winner which is based on an ensemble of decision trees with bagged boosting, as our system uses no intrusion data at all and much less normal data for training. Density-based clustering algorithms for multivariate data often have difficulties with high-dimensional data and clusters of very different densities. A new density-based clustering algorithm, called KNNCLUST, is presented in this paper that is able to tackle these situations. It is based on the combination of nonparametric k-nearest-neighbor (KNN) and kernel (KNN-kernel) density estimation. The KNN-kernel density estimation technique makes it possible to model clusters of different densities in high-dimensional data sets. Moreover, the number of clusters is identified automatically by the algorithm. KNNCLUST is tested using simulated data and applied to a multispectral compact airborne spectrographic imager (CASI)_image of a floodplain in the Netherlands to illustrate the characteristics of the method. Estimating divergences in a consistent way is of great importance in many machine learning tasks. Although this is a fundamental problem in nonparametric statistics, to the best of our knowledge there has been no finite sample exponential inequality convergence bound derived for any divergence estimators. The main contribution of our work is to provide such a bound for an estimator of R 'enyi- @math divergence for a smooth H "older class of densities on the @math -dimensional unit cube @math . We also illustrate our theoretical results with a numerical experiment.