Multivariate Probability Density Function Research Articles

Estimation of Seismic Wavelets Based on the Multivariate Scale Mixture of Gaussians Model

This paper proposes a new method for estimating seismic wavelets. Suppose a seismic wavelet can be modeled by a formula with three free parameters (scale, frequency and phase). We can transform the estimation of the wavelet into determining these three parameters. The phase of the wavelet is estimated by constant-phase rotation to the seismic signal, while the other two parameters are obtained by the Higher-order Statistics (HOS) (fourth-order cumulant) matching method. In order to derive the estimator of the Higher-order Statistics (HOS), the multivariate scale mixture of Gaussians (MSMG) model is applied to formulating the multivariate joint probability density function (PDF) of the seismic signal. By this way, we can represent HOS as a polynomial function of second-order statistics to improve the anti-noise performance and accuracy. In addition, the proposed method can work well for short time series.

Probabilistic Modeling of Scene Dynamics for Applications in Visual Surveillance

We propose a novel method to model and learn the scene activity, observed by a static camera. The proposed model is very general and can be applied for solution of a variety of problems. The motion patterns of objects in the scene are modeled in the form of a multivariate nonparametric probability density function of spatiotemporal variables (object locations and transition times between them). Kernel Density Estimation is used to learn this model in a completely unsupervised fashion. Learning is accomplished by observing the trajectories of objects by a static camera over extended periods of time. It encodes the probabilistic nature of the behavior of moving objects in the scene and is useful for activity analysis applications, such as persistent tracking and anomalous motion detection. In addition, the model also captures salient scene features, such as the areas of occlusion and most likely paths. Once the model is learned, we use a unified Markov Chain Monte Carlo (MCMC)-based framework for generating the most likely paths in the scene, improving foreground detection, persistent labeling of objects during tracking, and deciding whether a given trajectory represents an anomaly to the observed motion patterns. Experiments with real-world videos are reported which validate the proposed approach.

Spatial variability of satellite visible radiances in dust and dust‐cloud mixed conditions: Implications for dust detection

Using data from MODIS for the 2000–2004 time period, we performed a statistical analysis of visible radiances observed in the presence of mineral dust in cloud‐free and cloudy conditions over oceans. Spatial variability of the 555 nm radiances was examined by introducing two different measures: standard deviation (STD) and a local inhomogeneity parameter (LIP). We demonstrate that introducing the probability density function (PDF) of STD offers a new framework for probabilistic dust detection. We show that the probabilistic approach gives more accurate discrimination of dust from clouds compared with the MODIS fixed STD threshold method. Furthermore, the probabilistic approach enables one to determine the confidence level and skill of dust detection. In addition, we examined the capability of the probabilistic detection of dust‐cloud mixed pixels currently not considered by operational algorithms. A low classification skill of dust‐cloud pixels was found. Introducing multivariate PDFs by including multispectral data might help to overcome this problem.

3D Model Retrieval Using Probability Density-Based Shape Descriptors

We address content-based retrieval of complete 3D object models by a probabilistic generative description of local shape properties. The proposed shape description framework characterizes a 3D object with sampled multivariate probability density functions of its local surface features. This density-based descriptor can be efficiently computed via kernel density estimation (KDE) coupled with fast Gauss transform. The non-parametric KDE technique allows reliable characterization of a diverse set of shapes and yields descriptors which remain relatively insensitive to small shape perturbations and mesh resolution. Density-based characterization also induces a permutation property which can be used to guarantee invariance at the shape matching stage. As proven by extensive retrieval experiments on several 3D databases, our framework provides state-of-the-art discrimination over a broad and heterogeneous set of shape categories.

A note on the convergence analysis of a sparse grid multivariate probability density estimator

With the recent growth in volume and complexity of available data has come a renewed interest in the problem of estimating multivariate probability density functions. However, traditional methods encounter the curse of dimensionality (complexity grows exponentially with dimension). Here we provide an outline of a convergence analysis of a sparse grid based probability density estimation, which supports the use of the method for moderately complex (up to 15 dimensions) data sets, as has already been demonstrated for sparse grid quadrature and interpolation. References Hans-Joachim Bungartz and Michael Griebel. A note on the complexity of solving {P}oisson's equation for spaces of bounded mixed derivatives. J. Complexity , 15(2):167--199, 1999. http://dx.doi.org/10.1006/jcom.1999.0499 Hans-Joachim Bungartz and Michael Griebel. Sparse grids. Acta Numerica , 13:147--269, 2004. http://dx.doi.org/10.1017/S0962492904000182 J. Garcke. Sparse grid tutorial. http://www.math.tu-berlin.de/ garcke/paper/sparseGridTutorial.pdf [29 January, 2009], 2008. J. Garcke, M. Griebel, and M. Thess. Data mining with sparse grids. Computing , 67:225--253, 2001. http://dx.doi.org/10.1007/s006070170007 D. Scott. Multivariate Density Estimation: Theory, Practice, and Visualization . Wiley, New York, 1992. C. Zenger. Sparse grids. In W. Hackbusch, editor, Parallel Algorithms for Partial Differential Equations: Proceedings of the Sixth GAMM-Seminar, Kiel, 1990 , volume 31 of Notes on Num. Fluid Mech. , pages 241--251. Vieweg, 1991.

A note on the convergence analysis of a sparse grid multivariate probability density estimator

With the recent growth in volume and complexity of available data has come a renewed interest in the problem of estimating multivariate probability density functions. However, traditional methods encounter the curse of dimensionality (complexity grows exponentially with dimension). Here we provide an outline of a convergence analysis of a sparse grid based probability density estimation, which supports the use of the method for moderately complex (up to 15 dimensions) data sets, as has already been demonstrated for sparse grid quadrature and interpolation. References Hans-Joachim Bungartz and Michael Griebel. A note on the complexity of solving {P}oisson's equation for spaces of bounded mixed derivatives. J. Complexity, 15(2):167--199, 1999. http://dx.doi.org/10.1006/jcom.1999.0499 Hans-Joachim Bungartz and Michael Griebel. Sparse grids. Acta Numerica, 13:147--269, 2004. http://dx.doi.org/10.1017/S0962492904000182 J. Garcke. Sparse grid tutorial. http://www.math.tu-berlin.de/ garcke/paper/sparseGridTutorial.pdf [29 January, 2009], 2008. J. Garcke, M. Griebel, and M. Thess. Data mining with sparse grids. Computing, 67:225--253, 2001. http://dx.doi.org/10.1007/s006070170007 D. Scott. Multivariate Density Estimation: Theory, Practice, and Visualization. Wiley, New York, 1992. C. Zenger. Sparse grids. In W. Hackbusch, editor, Parallel Algorithms for Partial Differential Equations: Proceedings of the Sixth GAMM-Seminar, Kiel, 1990, volume 31 of Notes on Num. Fluid Mech., pages 241--251. Vieweg, 1991.

The Estimation of Radial Exponential Random Vectors in Additive White Gaussian Noise

Image signals are always disturbed by noise during their transmission, such as in mobile or network communication. The received image quality is significantly influenced by noise. Thus, image signal denoising is an indispensable step during image processing. As we all know, most commonly used methods of image denoising is Bayesian wavelet transform estimators. The Performance of various estimators, such as maximum a posteriori (MAP), or minimum mean square error (MMSE) is strongly dependent on correctness of the proposed model for original data distribution. Therefore, the selection of a proper model for distribution of wavelet coefficients is important in wavelet-based image denoising. This paper presents a new image denoising algorithm based on the modeling of wavelet coefficients in each subband with multivariate Radial Exponential probability density function (PDF) with local variances. Generally these multivariate extensions do not result in a closed form expression, and the solution requires numerical solutions. However, we drive a closed form MMSE shrinkage functions for a Radial Exponential random vectors in additive white Gaussian noise (AWGN). The estimator is motivated and tested on the problem of wavelet-based image denoising. In the last, proposed, the same idea is applied to the dual-tree complex wavelet transform (DT-CWT), This Transform is an over-complete wavelet transform.

The Construction and Properties of Ellipsoidal Probability Density Functions

[enter Abstract Body]A recipe for the construction of a multivariate probability density function from a normalized symmetric univariate function using a distance metric methodology is developed. A method to perform multivariate integration in a polar coordinate system in an arbitary number of dimensions is described. This method is then used to compute a constant that normalizes the multivariate p.d.f. constructed using the recipe specified here. The use of the Kolmogorov test, as applied to the univariate distribution of the square of the metric distance, for distributional identification is described. A summary of the general properties of the constructed p.d.f. is given including the computation of: the principal moments; Mardia's multivariate kurtosis measure; and, the characteristic and moment generatic functions. Some elements of maximum likelihood estimation are explored.

On a multivariate gamma distribution

A multivariate probability model possessing a dependence structure that is reflected in its variance–covariance structure and gamma distributed univariate margins is introduced and studied. In particular, the higher order moments and cumulants, Chebyshev-type inequalities and multivariate probability density functions are derived. The model suggested herein is believed to be capable of describing dependent insurance losses.

An Information Theoretic Exploratory Method for Learning Patterns of Conditional Gene Coexpression from Microarray Data

In this article, we introduce an exploratory framework for learning patterns of conditional co-expression in gene expression data. The main idea behind the proposed approach consists of estimating how the information content shared by a set of M nodes in a network (where each node is associated to an expression profile) varies upon conditioning on a set of L conditioning variables (in the simplest case represented by a separate set of expression profiles). The method is non-parametric and it is based on the concept of statistical co-information, which, unlike conventional correlation based techniques, is not restricted in scope to linear conditional dependency patterns. Moreover, such conditional co-expression relationships can potentially indicate regulatory interactions that do not manifest themselves when only pair-wise relationships are considered. A moment based approximation of the co-information measure is derived that efficiently gets around the problem of estimating high-dimensional multi-variate probability density functions from the data, a task usually not viable due to the intrinsic sample size limitations that characterize expression level measurements. By applying the proposed exploratory method, we analyzed a whole genome microarray assay of the eukaryote Saccharomices cerevisiae and were able to learn statistically significant patterns of conditional co-expression. A selection of such interactions that carry a meaningful biological interpretation are discussed.

Elucidating Model Inadequacies in a Cloud Parameterization by Use of an Ensemble-Based Calibration Framework

AbstractEvery cloud parameterization contains structural model errors. The source of these errors is difficult to pinpoint because cloud parameterizations contain nonlinearities and feedbacks. To elucidate these model inadequacies, this paper uses a general-purpose ensemble parameter estimation technique. In principle, the technique is applicable to any parameterization that contains a number of adjustable coefficients. It optimizes or calibrates parameter values by attempting to match predicted fields to reference datasets. Rather than striving to find the single best set of parameter values, the output is instead an ensemble of parameter sets. This ensemble provides a wealth of information. In particular, it can help uncover model deficiencies and structural errors that might not otherwise be easily revealed. The calibration technique is applied to an existing single-column model (SCM) that parameterizes boundary layer clouds. The SCM is a higher-order turbulence closure model. It is closed using a multivariate probability density function (PDF) that represents subgrid-scale variability. Reference datasets are provided by large-eddy simulations (LES) of a variety of cloudy boundary layers. The calibration technique locates some model errors in the SCM. As a result, empirical modifications are suggested. These modifications are evaluated with independent datasets and found to lead to an overall improvement in the SCM’s performance.

Object Trajectory-Based Activity Classification and Recognition Using Hidden Markov Models

Motion trajectories provide rich spatiotemporal information about an object's activity. This paper presents novel classification algorithms for recognizing object activity using object motion trajectory. In the proposed classification system, trajectories are segmented at points of change in curvature, and the subtrajectories are represented by their principal component analysis (PCA) coefficients. We first present a framework to robustly estimate the multivariate probability density function based on PCA coefficients of the subtrajectories using Gaussian mixture models (GMMs). We show that GMM-based modeling alone cannot capture the temporal relations and ordering between underlying entities. To address this issue, we use hidden Markov models (HMMs) with a data-driven design in terms of number of states and topology (e.g., left-right versus ergodic). Experiments using a database of over 5700 complex trajectories (obtained from UCI-KDD data archives and Columbia University Multimedia Group) subdivided into 85 different classes demonstrate the superiority of our proposed HMM-based scheme using PCA coefficients of subtrajectories in comparison with other techniques in the literature.

Multivariate distribution of returns in financial time series

Multivariate probability density functions of returns are constructed in order to model the empirical behavior of returns in a financial time series. They describe the well-established deviations from the Gaussian random walk, such as an approximate scaling and heavy tails of the return distributions, long-ranged volatility-volatility correlations (volatility clustering) and return-volatility correlations (leverage effect). Free parameters of the model are fixed over the long term by fitting 100+ years of daily prices of the Dow Jones 30 Industrial Average. The multivariate probability density functions which we have constructed can be used for pricing derivative securities and risk management.

Color image histogram equalization by absolute discounting back-off

A novel color image histogram equalization approach is proposed that exploits the correlation between color components and it is enhanced by a multi-level smoothing technique borrowed from statistical language engineering. Multi-level smoothing aims at dealing efficiently with the problem of unseen color values, either considered independently or in combination with others. It is applied here to the HSI color space for the probability of intensity and the probability of saturation given the intensity, while the hue is left unchanged. Moreover, the proposed approach is extended by an empirical technique, which is based on a hue preserving non-linear transformation, in order to eliminate the gamut problem. This is the second method proposed in the paper. The equalized images by the two methods are compared to those produced by other well-known methods. The better quality of the images equalized by the proposed methods is judged in terms of their visual appeal and objective figures of merit, such as the entropy and the Kullback–Leibler divergence estimates between the resulting color histogram and the multivariate uniform probability density function.

E233 乱流拡散火炎の確率的モデリング手法の比較検討(乱流燃焼)

A comparative study of stochastic modeling of turbulent combustion has been conducted in the configuration of jet diffusion flame. Flamelet model, multivariate presumed probability density function (PDF) model, and PDF model are evaluated in terms of numerical accuracy. Fuel is a mixture of hydrogen and nitrogen. A fast chemistry is assumed in the flamelet model, a one-step irreversible reaction in the multivariate presumed PDF, and a detailed chemistry in the PDF model. The flamelet model overestimates the reaction than the others but shows an agreement with experiment results. This is attributable to turbulent models employed here. Therefore, the accuracy of combustion modeling strongly depends on the turbulent model of the flow field.

Approximation of joint PDFs by discrete distributions generated with Monte Carlo methods

A new method for the approximation of multivariate scalar probability density functions (PDFs) in turbulent reacting flow by means of a joint presumed discrete distribution (jPDD) is presented. The jPDDs can be generated with specified mean values and variances as well as covariances. Correlations between variables – e.g. fluctuating mixture fractions and/or reaction progress – can thereby be taken into account. In this way the new approach overcomes an important limitation of ordinary presumed PDF methods, where statistical independence between the variables is often assumed. Different methods are presented to generate discrete distributions, based either on biased random number generators or on mixing models familiar from PDF transport models. The new approach is extensively validated on a turbulent flow configuration with simultaneous mixing and reaction. Large eddy simulation data as well as results from a transported PDF model are used for the validation of the jPDD approach. The comparison shows that in particular distributions generated with mixing models are able to predict mean reaction rates accurately. For the configuration considered, the neglect of correlations results in significant underestimation of reaction rates. Moreover it is found that higher statistical moments (e.g. the skewness) can influence reaction rates. The consequences for the generation of jPDDs are discussed. In summary, the new jPDD model has the potential to be significantly more accurate than established presumed PDF methods, because correlations between fluctuating variables can be taken into account. At the same time, the new approach is nearly as efficient as standard presumed PDF formulations, since mean rates are computed in a pre-processing step and stored in look-up tables as a function of the first and second moments of the relevant variables.

A multivariate population density model of the dLGN/PGN relay

Using a population density approach we study the dynamics of two interacting collections of integrate-and-fire-or-burst (IFB) neurons representing thalamocortical (TC) cells from the dorsal lateral geniculate nucleus (dLGN) and thalamic reticular (RE) cells from the perigeniculate nucleus (PGN). Each population of neurons is described by a multivariate probability density function that satisfies a conservation equation with appropriately defined probability fluxes and boundary conditions. The state variables of each neuron are the membrane potential and the inactivation gating variable of the low-threshold Ca2+ current I(T). The synaptic coupling of the populations and external excitatory drive are modeled by instantaneous jumps in the membrane potential of postsynaptic neurons. The population density model is validated by comparing its response to time-varying retinal input to Monte Carlo simulations of the corresponding IFB network composed of 100 to 1,000 cells per population. In the absence of retinal input, the population density model exhibits rhythmic bursting similar to the 7 to 14 Hz oscillations associated with slow wave sleep that require feedback inhibition from RE to TC cells. When the TC and RE cell potassium leakage conductances are adjusted to represent cholingergic neuromodulation and arousal of the network, rhythmic bursting of the probability density model may either persists or be eliminated depending on the number of excitatory (TC to RE) or inhibitory (RE to TC) connections made by each presynaptic cell. When the probability density model is stimulated with constant retinal input (10-100 spikes/sec), a wide range of responses are observed depending on cellular parameters and network connectivity. These include asynchronous burst and tonic spikes, sleep spindle-like rhythmic bursting, and oscillations in population firing rate that are distinguishable from sleep spindles due to their amplitude, frequency, or the presence of tonic spikes. In this context of dLGN/PGN network modeling, we find the population density approach using 2,500 mesh points and resolving membrane voltage to 0.7 mV is over 30 times more efficient than 1,000-cell Monte Carlo simulations.

Modelling multivariate moments in European Stock Markets

This research extends the results of Mauleón and Perote, and derives analytically a general framework for the multivariate Edgeworth Sargan (ES) density. Its capability to account for multivariate moments beyond correlation is shown–mainly, co-skewness, co-kurtosis and co-volatility. The multivariate ES is then fitted to the residuals of a VAR model applied to three European stock market series of daily data (FTSE, DAX, CAC40), accounting for univariate as well as multivariate departures from normality. The complete model – with nearly 60 parameters – is set up and estimated jointly by maximum likelihood. Two alternative multivariate probability density functions, student's t and the normal skewed, are also estimated and compared to the ES. The empirical results show: (1) in spite of the high nonlinearity and complexity of the model, it is feasible to fit it to empirical data; (2) statistically significant multivariate effects, other than correlations, are found, and (3) the tail fit of the ES is significantly better.

Control design to shape the stationary probability density function

This paper extends a regulatory controller synthesis technique that was previously introduced for first-order processes to higher-order processes. The target of the design is to find a feedback control law such that the multivariate stationary probability density function (PDF) for the closed-loop process reasonably approximates a preselected target PDF. With the idea already motivated and introduced in previous work, the focus of this paper is on dealing with the various complications that arise when the process dimension is greater than one. As in the previous case, the main idea of multivariate PDF-shaping is to reduce the integral equation that governs the relationship between process dynamics and stationary PDF to a set of algebraic equations. The proposed approach relies on parameterization of the feedback control law to simplify manipulation of the integral equation. Numerical simulations with an example process are used to demonstrate application of the technique.

Directional features in online handwriting recognition

The selection of valuable features is crucial in pattern recognition. In this paper we deal with the issue that part of features originate from directional instead of common linear data. Both for directional and linear data a theory for a statistical modeling exists. However, none of these theories gives an integrated solution to problems, where linear and directional variables are to be combined in a single, multivariate probability density function. We describe a general approach for a unified statistical modeling, given the constraint that variances of the circular variables are small. The method is practically evaluated in the context of our online handwriting recognition system frog on hand and the so-called tangent slope angle feature. Recognition results are compared with two alternative modeling approaches. The proposed solution gives significant improvements in recognition accuracy, computational speed and memory requirements.