In this paper, we propose novel strategies for neutral vector variable decorrelation. Two fundamental invertible transformations, namely, serial nonlinear transformation and parallel nonlinear transformation, are proposed to carry out the decorrelation. For a neutral vector variable, which is not multivariate-Gaussian distributed, the conventional principal component analysis cannot yield mutually independent scalar variables. With the two proposed transformations, a highly negatively correlated neutral vector can be transformed to a set of mutually independent scalar variables with the same degrees of freedom. We also evaluate the decorrelation performances for the vectors generated from a single Dirichlet distribution and a mixture of Dirichlet distributions. The mutual independence is verified with the distance correlation measurement. The advantages of the proposed decorrelation strategies are intensively studied and demonstrated with synthesized data and practical application evaluations.

This paper addresses modelling data using the Watson distribution. The Watson distribution is one of the simplest distributions for analyzing axially symmetric data. This distribution has gained some attention in recent years due to its modeling capability. However, its Bayesian inference is fairly understudied due to difficulty in handling the normalization factor. Recent development of Markov chain Monte Carlo (MCMC) sampling methods can be applied for this purpose. However, these methods can be prohibitively slow for practical applications. A deterministic alternative is provided by variational methods that convert inference problems into optimization problems. In this paper, we present a variational inference for Watson mixture models. First, the variational framework is used to side-step the intractability arising from the coupling of latent states and parameters. Second, the variational free energy is further lower bounded in order to avoid intractable moment computation. The proposed approach provides a lower bound on the log marginal likelihood and retains distributional information over all parameters. Moreover, we show that it can regulate its own complexity by pruning unnecessary mixture components while avoiding over-fitting. We discuss potential applications of the modeling with Watson distributions in the problem of blind source separation, and clustering gene expression data sets.

A novel Bayesian matrix factorization method for bounded support data is presented. Each entry in the observation matrix is assumed to be beta distributed. As the beta distribution has two parameters, two parameter matrices can be obtained, which matrices contain only nonnegative values. In order to provide low-rank matrix factorization, the nonnegative matrix factorization (NMF) technique is applied. Furthermore, each entry in the factorized matrices, i.e., the basis and excitation matrices, is assigned with gamma prior. Therefore, we name this method as beta-gamma NMF (BG-NMF). Due to the integral expression of the gamma function, estimation of the posterior distribution in the BG-NMF model can not be presented by an analytically tractable solution. With the variational inference framework and the relative convexity property of the log-inverse-beta function, we propose a new lower-bound to approximate the objective function. With this new lower-bound, we derive an analytically tractable solution to approximately calculate the posterior distributions. Each of the approximated posterior distributions is also gamma distributed, which retains the conjugacy of the Bayesian estimation. In addition, a sparse BG-NMF can be obtained by including a sparseness constraint to the gamma prior. Evaluations with synthetic data and real life data demonstrate the good performance of the proposed method.

This paper addresses the Bayesian estimation of the von-Mises Fisher (vMF) mixture model with variational inference (VI). The learning task in VI consists of optimization of the variational posterior distribution. However, the exact solution by VI does not lead to an analytically tractable solution due to the evaluation of intractable moments involving functional forms of the Bessel function in their arguments. To derive a closed-form solution, we further lower bound the evidence lower bound where the bound is tight at one point in the parameter distribution. While having the value of the bound guaranteed to increase during maximization, we derive an analytically tractable approximation to the posterior distribution which has the same functional form as the assigned prior distribution. The proposed algorithm requires no iterative numerical calculation in the re-estimation procedure, and it can potentially determine the model complexity and avoid the over-fitting problem associated with conventional approaches based on the expectation maximization. Moreover, we derive an analytically tractable approximation to the predictive density of the Bayesian mixture model of vMF distributions. The performance of the proposed approach is verified by experiments with both synthetic and real data.

We address the problem of blind source separation in acoustic applications where there is no prior knowledge about the number of mixing sources. The presented method employs a mixture of complex Watson distributions in its generative model with a sparse Dirichlet distribution over the mixture weights. The problem is formulated in a fully Bayesian inference with assuming prior distributions over all model parameters. The presented model can regulate its own complexity by pruning unnecessary components by which we can possibly relax the assumption of prior knowledge on the number of sources.

Nonnegative matrix factorization is an appealing technique for many audio applications. However, in it's basic form it does not use temporal structure, which is an important source of information in speech processing. In this paper, we propose NMF-based filtering and smoothing algorithms that are related to Kalman filtering and smoothing. While our prediction step is similar to that of Kalman filtering, we develop a multiplicative update step which is more convenient for nonnegative data analysis and in line with existing NMF literature. The proposed smoothing approach introduces an unavoidable processing delay, but the filtering algorithm does not and can be readily used for on-line applications. Our experiments using the proposed algorithms show a significant improvement over the baseline NMF approaches. In the case of speech denoising with factory noise at 0 dB input SNR, the smoothing algorithm outperforms NMF with 3.2 dB in SDR and around 0.5 MOS in PESQ, likewise source separation experiments result in improved performance due to taking advantage of the temporal regularities in speech.

In this paper a model-based dual-channel noise reduction approach is presented which is an alternative to conventional noise reduction algorithms essentially due to its independence of the noise power spectral density estimation and of any prior knowledge about the spatial noise field characteristics. We use a mixture of circular-symmetric complex-Gaussian distributions projected on the unit hypersphere for modeling the complex discrete Fourier transform coefficients of noisy speech signals in the frequency domain. According to the derived mixture model, clustering of the noise and the target speech components is performed depending on their direction of arrival. A soft masking strategy is proposed for speech enhancement based on responsibilities assigned to the target speech class in each time-frequency bin. Our experimental results show that the proposed approach is more robust than conventional dual-channel noise reduction systems based on the single- and dual-channel noise power spectral density estimators.

This paper addresses text-independent speaker identification (SI) based on line spectral frequencies (LSFs). The LSFs are transformed to differential LSFs (MLSF) in order to exploit their boundary and ordering properties. We show that the square root of MLSF has interesting directional characteristics implying that their distribution can be modeled by a mixture of von-Mises Fisher (vMF) distributions. We analytically estimate the mixture model parameters in a fully Bayesian treatment by using variational inference. In the Bayesian inference, we can potentially determine the model complexity and avoid overfitting problem associated with conventional approaches based on the expectation maximization. The experimental results confirm the effectiveness of the proposed SI system.

The derivation of MMSE estimators for the DFT coefficients of speech signals, given an observed noisy signal and super-Gaussian prior distributions, has received a lot of interest recently. In this letter, we look at the distribution of the periodogram coefficients of different phonemes, and show that they have a gamma distribution with shape parameters less than one. This verifies that the DFT coefficients for not only the whole speech signal but also for individual phonemes have super-Gaussian distributions. We develop a spectral domain speech enhancement algorithm, and derive hidden Markov model (HMM) based MMSE estimators for speech periodogram coefficients under this gamma assumption in both a high uniform resolution and a reduced-resolution Mel domain. The simulations show that the performance is improved using a gamma distribution compared to the exponential case. Moreover, we show that, even though beneficial in some aspects, the Mel-domain processing does not lead to better results than the algorithms in the high-resolution domain.

In this letter the focus is on linear filtering of speech before degradation due to additive background noise. The goal is to design the filter such that the speech intelligibility index (SII) is maximized when the speech is played back in a known noisy environment. Moreover, a power constraint is taken into account to prevent uncomfortable playback levels and deal with loudspeaker constraints. Previous methods use linear approximations of the SII in order to find a closed-form solution. However, as we show, these linear approximations introduce errors in low SNR regions and are therefore suboptimal. In this work we propose a nonlinear approximation of the SII which is accurate for all SNRs. Experiments show large intelligibility improvements with the proposed method over the unprocessed noisy speech and better performance than one state-of-the art method.

Quantization of the linear predictive coding parameters is an important part in speech coding. Probability density function (PDF)-optimized vector quantization (VQ) has been previously shown to be more efficient than VQ based only on training data. For data with bounded support, some well-defined bounded-support distributions (e.g., the Dirichlet distribution) have been proven to outperform the conventional Gaussian mixture model (GMM), with the same number of free parameters required to describe the model. When exploiting both the boundary and the order properties of the line spectral frequency (LSF) parameters, the distribution of LSF differences LSF can be modelled with a Dirichlet mixture model (DMM). We propose a corresponding DMM based VQ. The elements in a Dirichlet vector variable are highly mutually correlated. Motivated by the Dirichlet vector variable's neutrality property, a practical non-linear transformation scheme for the Dirichlet vector variable can be obtained. Similar to the Karhunen-Loève transform for Gaussian variables, this non-linear transformation decomposes the Dirichlet vector variable into a set of independent beta-distributed variables. Using high rate quantization theory and by the entropy constraint, the optimal inter- and intra-component bit allocation strategies are proposed. In the implementation of scalar quantizers, we use the constrained-resolution coding to approximate the derived constrained-entropy coding. A practical coding scheme for DVQ is designed for the purpose of reducing the quantization error accumulation. The theoretical and practical quantization performance of DVQ is evaluated. Compared to the state-of-the-art GMM-based VQ and recently proposed beta mixture model (BMM) based VQ, DVQ performs better, with even fewer free parameters and lower computational cost

Deriving a good model for multitalker babble noise can facilitate different speech processing algorithms, e.g., noise reduction, to reduce the so-called cocktail party difficulty. In the available systems, the fact that the babble waveform is generated as a sum of N different speech waveforms is not exploited explicitly. In this paper, first we develop a gamma hidden Markov model for power spectra of the speech signal, and then formulate it as a sparse nonnegative matrix factorization (NMF). Second, the sparse NMF is extended by relaxing the sparsity constraint, and a novel model for babble noise (gamma nonnegative HMM) is proposed in which the babble basis matrix is the same as the speech basis matrix, and only the activation factors (weights) of the basis vectors are different for the two signals over time. Finally, a noise reduction algorithm is proposed using the derived speech and babble models. All of the stationary model parameters are estimated using the expectation-maximization (EM) algorithm, whereas the time-varying parameters, i.e., the gain parameters of speech and babble signals, are estimated using a recursive EM algorithm. The objective and subjective listening evaluations show that the proposed babble model and the final noise reduction algorithm significantly outperform the conventional methods.

Reducing the interference noise in a monaural noisy speech signal has been a challenging task for many years. Compared to traditional unsupervised speech enhancement methods, e.g., Wiener filtering, supervised approaches, such as algorithms based on hidden Markov models (HMM), lead to higher-quality enhanced speech signals. However, the main practical difficulty of these approaches is that for each noise type a model is required to be trained a priori. In this paper, we investigate a new class of supervised speech denoising algorithms using nonnegative matrix factorization (NMF). We propose a novel speech enhancement method that is based on a Bayesian formulation of NMF (BNMF). To circumvent the mismatch problem between the training and testing stages, we propose two solutions. First, we use an HMM in combination with BNMF (BNMF-HMM) to derive a minimum mean square error (MMSE) estimator for the speech signal with no information about the underlying noise type. Second, we suggest a scheme to learn the required noise BNMF model online, which is then used to develop an unsupervised speech enhancement system. Extensive experiments are carried out to investigate the performance of the proposed methods under different conditions. Moreover, we compare the performance of the developed algorithms with state-of-the-art speech enhancement schemes using various objective measures. Our simulations show that the proposed BNMF-based methods outperform the competing algorithms substantially.

We present a speech enhancement algorithm which is based on a Bayesian Nonnegative Matrix Factorization (NMF). Both Minimum Mean Square Error (MMSE) and Maximum a-Posteriori (MAP) estimates of the magnitude of the clean speech DFT coefficients are derived. To exploit the temporal continuity of the speech and noise signals, a proper prior distribution is introduced by widening the posterior distribution of the NMF coefficients at the previous time frames. To do so, a recursive temporal update scheme is proposed to obtain the mean value of the prior distribution; also, the uncertainty of the prior information is governed by the shape parameter of the distribution which is learnt automatically based on the nonstationarity of the signals. Simulations show a considerable improvement compared to the maximum likelihood NMF based speech enhancement algorithm for different input SNRs.

In this paper, we propose a variational Bayes approach to the underdetermined blind source separation and show how a variational treatment can open up the possibility of determining the actual number of sources. The procedure is performed in a frequency bin-wise manner. In every frequency bin, we model the time-frequency mixture by a variational mixture of Gaussians with a circular-symmetric complex-Gaussian density function. In the Bayesian inference, we consider appropriate conjugate prior distributions for modeling the parameters of this distribution. The learning task consists of estimating the hyper-parameters characterizing the parameter distributions for the optimization of the variational posterior distribution. The proposed approach requires no prior knowledge on the number of sources in a mixture.

A new text-independent speaker identification (SI) system is proposed. This system utilizes the line spectral frequencies (LSFs) as alternative feature set for capturing the speaker characteristics ...

Bayesian estimation of the parameters in beta mixture models (BMM) is analytically intractable. The numerical solutions to simulate the posterior distribution are available, but incur high computational cost. In this paper, we introduce an approximation to the prior/posterior distribution of the parameters in the beta distribution and propose an analytically tractable (closed form) Bayesian approach to the parameter estimation. The approach is based on the variational inference (VI) framework. Following the principles of the VI framework and utilizing the relative convexity bound, the extended factorized approximation method is applied to approximate the distribution of the parameters in BMM. In a fully Bayesian model where all of the parameters of the BMM are considered as variables and assigned proper distributions, our approach can asymptotically find the optimal estimate of the parameters posterior distribution. Also, the model complexity can be determined based on the data. The closed-form solution is proposed so that no iterative numerical calculation is required. Meanwhile, our approach avoids the drawback of overfitting in the conventional expectation maximization algorithm. The good performance of this approach is verified by experiments with both synthetic and real data.

Parameter estimation for the beta distribution is analytically intractable due to the integration expression in the normalization constant. For maximum likelihood estimation, numerical methods can be used to calculate the parameters. For Bayesian estimation, we can utilize different approximations to the posterior parameter distribution. A method based on the variational inference (VI) framework reported the posterior mean of the parameters analytically but the approximating distribution violated the correlation between the parameters. We now propose a method via the expectation propagation (EP) framework to approximate the posterior distribution analytically and capture the correlation between the parameters. Compared to the method based on VI, the EP based algorithm performs better with small amounts of data and is more stable.

Statistical modeling plays an important role in various research areas. It provides away to connect the data with the statistics. Based on the statistical properties of theobserved data, an appropriate model can be chosen that leads to a promising practicalperformance. The Gaussian distribution is the most popular and dominant probabilitydistribution used in statistics, since it has an analytically tractable Probability DensityFunction (PDF) and analysis based on it can be derived in an explicit form. However,various data in real applications have bounded support or semi-bounded support. As the support of the Gaussian distribution is unbounded, such type of data is obviously notGaussian distributed. Thus we can apply some non-Gaussian distributions, e.g., the betadistribution, the Dirichlet distribution, to model the distribution of this type of data.The choice of a suitable distribution is favorable for modeling efficiency. Furthermore,the practical performance based on the statistical model can also be improved by a bettermodeling. An essential part in statistical modeling is to estimate the values of the parametersin the distribution or to estimate the distribution of the parameters, if we consider themas random variables. Unlike the Gaussian distribution or the corresponding GaussianMixture Model (GMM), a non-Gaussian distribution or a mixture of non-Gaussian dis-tributions does not have an analytically tractable solution, in general. In this dissertation,we study several estimation methods for the non-Gaussian distributions. For the Maxi-mum Likelihood (ML) estimation, a numerical method is utilized to search for the optimalsolution in the estimation of Dirichlet Mixture Model (DMM). For the Bayesian analysis,we utilize some approximations to derive an analytically tractable solution to approxi-mate the distribution of the parameters. The Variational Inference (VI) framework basedmethod has been shown to be efficient for approximating the parameter distribution byseveral researchers. Under this framework, we adapt the conventional Factorized Approx-imation (FA) method to the Extended Factorized Approximation (EFA) method and useit to approximate the parameter distribution in the beta distribution. Also, the LocalVariational Inference (LVI) method is applied to approximate the predictive distributionof the beta distribution. Finally, by assigning a beta distribution to each element in thematrix, we proposed a variational Bayesian Nonnegative Matrix Factorization (NMF) forbounded support data. The performances of the proposed non-Gaussian model based methods are evaluatedby several experiments. The beta distribution and the Dirichlet distribution are appliedto model the Line Spectral Frequency (LSF) representation of the Linear Prediction (LP)model for statistical model based speech coding. For some image processing applications,the beta distribution is also applied. The proposed beta distribution based variationalBayesian NMF is applied for image restoration and collaborative filtering. Comparedto some conventional statistical model based methods, the non-Gaussian model basedmethods show a promising improvement.

The line spectral frequencies (LSF) are known to be the mostefficient representation of the linear predictive coding (LPC) parametersfrom both the distortion and perceptual point of view.By considering the bounded property of the LSF parameters,we apply beta mixture models (BMM) to model the distributionof the LSF parameters. Meanwhile, by following the principlesof probability density function (PDF) optimized vector quantization(VQ), we derive the bit allocation strategy for the BMM.The LSF parameters are obtained from the TIMIT database anda practical VQ is designed. By taking the Bayesian informationcriterion (BIC), the square error (SE) and the spectral distortion(SD) as the criteria, the BMM based VQ outperforms theGaussian mixture model based VQ with uncorrelated Gaussiancomponent (UGMVQ) by about 1-2 bits/vector.

Statistical pattern recognition is one of the most studied and applied approaches in the area of pattern recognition. Mixture modelling of densities is an efficient statistical pattern recognition method for continuous data. We propose a classifier based on the beta mixture models for strictly bounded and asymmetrically distributed data. Due to the property of the mixture modelling, the statistical dependence in a multi-dimensional variable is captured, even with the conditional independence assumption in each mixture component. A synthetic example and the USPS handwriting digit data was used to verify the effectiveness of this approach. Compared to the conventional Gaussian mixture models (GMM), the beta mixture models has a better performance on data which has strictly bounded value and asymmetric distribution. The performance of beta mixture models is about equivalent to that of GMM applied to data transformed via a strictly increasing link function.

With A-V recordings. ten normal hearing people took recognition tests at different signal-to-noise ratios (SNR). The AV recognition results are predicted by the fuzzy logical model of perception (FLMP) and the post-labelling integration model (POSTL). We also applied hidden Markov models (HMMs) and multi-stream HMMs (MSHMMs) for the recognition. As expected, all the models agree qualitatively with the results that the benefit gained from the visual signal is larger at lower acoustic SNRs. However, the FLMP severely overestimates the AV integration result, while the POSTL model underestimates it. Our automatic speech recognizers integrated the audio and visual stream efficiently. The visual automatic speech recognizer could be adjusted to correspond to human visual performance. The MSHMMs combine the audio and visual streams efficiently, but the audio automatic speech recognizer must be further improved to allow precise quantitative comparisons with human audio-visual performance.

A computational method to predict speech intelligibility in noisy environments has been developed. By modeling speech and noise as stochastic signals, the information transmission through a given a ...

A novel test method was designed to measure the spectro-temporal discrimination ability of cochlear implant (CI) users. The test signals are bandpass filtered, speech weighted noise, with the long term spectrum of speech. The goal of the test is to measure the amplitude difference between spectral bands in two presented signals which is required for the listener just to discriminate between the two sounds. Twenty CI users were tested with the spectro-temporal discrimination test and a conventional speech recognition test. For test stimuli differing in only two spectral bands, CI and normal hearing users show nearly equally good results on the spectro-temporal test. For spectra with four and more bands, the spread in the CI users' results is great.