Inferring visuomotor priors for sensorimotor learning.

Sensorimotor learning has been shown to depend on both prior expectations and sensory evidence in a way that is consistent with Bayesian integration. Thus, prior beliefs play a key role during the learning process, especially when only ambiguous sensory information is available. Here we develop a novel technique to estimate the covariance structure of the prior over visuomotor transformations--the mapping between actual and visual location of the hand--during a learning task. Subjects performed reaching movements under multiple visuomotor transformations in which they received visual feedback of their hand position only at the end of the movement. After experiencing a particular transformation for one reach, subjects have insufficient information to determine the exact transformation, and so their second reach reflects a combination of their prior over visuomotor transformations and the sensory evidence from the first reach. We developed a Bayesian observer model in order to infer the covariance structure of the subjects' prior, which was found to give high probability to parameter settings consistent with visuomotor rotations. Therefore, although the set of visuomotor transformations experienced had little structure, the subjects had a strong tendency to interpret ambiguous sensory evidence as arising from rotation-like transformations. We then exposed the same subjects to a highly-structured set of visuomotor transformations, designed to be very different from the set of visuomotor rotations. During this exposure the prior was found to have changed significantly to have a covariance structure that no longer favored rotation-like transformations. In summary, we have developed a technique which can estimate the full covariance structure of a prior in a sensorimotor task and have shown that the prior over visuomotor transformations favor a rotation-like structure. Moreover, through experience of a novel task structure, participants can appropriately alter the covariance structure of their prior.

Copula Processes

We define a copula process which describes the dependencies between arbitrarily many random variables independently of their marginal distributions. As an example, we develop a stochastic volatility model, Gaussian Copula Process Volatility (GCPV), to predict the latent standard deviations of a sequence of random variables. To make predictions we use Bayesian inference, with the Laplace approximation, and with Markov chain Monte Carlo as an alternative. We find both methods comparable. We also find our model can outperform GARCH on simulated and financial data. And unlike GARCH, GCPV can easily handle missing data, incorporate covariates other than time, and model a rich class of covariance structures.

Generalised Wishart Processes

We introduce a stochastic process with Wishart marginals: the generalised Wishart process (GWP). It is a collection of positive semi-definite random matrices indexed by any arbitrary dependent variable. We use it to model dynamic (e.g. time varying) covariance matrices. Unlike existing models, it can capture a diverse class of covariance structures, it can easily handle missing data, the dependent variable can readily include covariates other than time, and it scales well with dimension; there is no need for free parameters, and optional parameters are easy to interpret. We describe how to construct the GWP, introduce general procedures for inference and predictions, and show that it outperforms its main competitor, multivariate GARCH, even on financial data that especially suits GARCH. We also show how to predict the mean of a multivariate process while accounting for dynamic correlations.

MMI training for continuous phoneme recognition on the TIMIT database

This paper reports our experiences with a phoneme recognition system for the TIMIT database which uses multiple mixture continuous density monophone HMMs trained using MMI. A comprehensive set of results are presented comparing the ML and MMI training criteria for both diagonal and full covariance models. These results using simple monophone HMMs show clear performance gains achieved by MMI training, and are comparable to the best reported by others including those which use context-dependent models. In addition, the paper discusses a number of performance and implementation issues which are crucial to successful MMI training.

Extended VTS for noise-robust speech recognition

Model compensation is a standard way of improving the robustness of speech recognition systems to noise. A number of popular schemes are based on vector Taylor series (VTS) compensation, which uses a linear approximation to represent the influence of noise on the clean speech. To compensate the dynamic parameters, the continuous time approximation is often used. This approximation uses a point estimate of the gradient, which fails to take into account that dynamic coefficients are a function of a number of consecutive static coefficients. In this paper, the accuracy of dynamic parameter compensation is improved by representing the dynamic features as a linear transformation of a window of static features. A modified version of VTS compensation is applied to the distribution of the window of static features and, importantly, their correlations. These compensated distributions are then transformed to distributions over standard static and dynamic features. With this improved approximation, it is also possible to obtain full-covariance corrupted speech distributions. This addresses the correlation changes that occur in noise. The proposed scheme outperformed the standard VTS scheme by 10% to 20% relative on a range of tasks. © 2006 IEEE.

Structured precision modelling with Cholesky basis superposition for speech recognition

Structured precision modelling is an important approach to improve the intra-frame correlation modelling of the standard HMM, where Gaussian mixture model with diagonal covariance are used. Previous work has all been focused on direct structured representation of the precision matrices. In this paper, a new framework is proposed, where the structure of the Cholesky square root of the precision matrix is investigated, referred to as Cholesky Basis Superposition (CBS). Each Cholesky matrix associated with a particular Gaussian distribution is represented as a linear combination of a set of Gaussian independent basis upper-triangular matrices. Efficient optimization methods are derived for both combination weights and basis matrices. Experiments on a Chinese dictation task showed that the proposed approach can significantly outperformed the direct structured precision modelling with similar number of parameters as well as full covariance modelling. © 2011 IEEE.

Scalar and its dissipation in the near field of turbulent lifted jet flame

In this study various scalar dissipation rates and their modelling in the context of partially premixed flame are investigated. A DNS dataset of the near field of a turbulent hydrogen lifted jet flame is processed to analyse the mixture fraction and progress variable dissipation rates and their cross dissipation rate at several axial positions. It is found that the classical model for the passive scalar dissipation rate ε{lunate}̃ZZ gives good agreement with the DNS, while models developed based on premixed flames for the reactive scalar dissipation rate ε{lunate}̃cc only qualitatively capture the correct trend. The cross dissipation rate ε{lunate}̃cZ is mostly negative and can be reasonably approximated at downstream positions once ε{lunate}̃ZZ and ε{lunate}̃cc are known, although the sign cannot be determined. This approach gives better results than one employing a constant ratio of turbulent timescale and the scalar covariance c'Z'̃. The statistics of scalar gradients are further examined and lognormal distributions are shown to be very good approximations for the passive scalar and acceptable for the reactive scalar. The correlation between the two gradients increases downstream as the partially premixed flame in the near field evolves ultimately to a diffusion flame in the far field. A bivariate lognormal distribution is tested and found to be a reasonable approximation for the joint PDF of the two scalar gradients. © 2011 The Combustion Institute.

Predicting surface vibration from underground railways through inhomogeneous soil

Noise and vibration from underground railways is a major source of disturbance to inhabitants near subways. To help designers meet noise and vibration limits, numerical models are used to understand vibration propagation from these underground railways. However, the models commonly assume the ground is homogeneous and neglect to include local variability in the soil properties. Such simplifying assumptions add a level of uncertainty to the predictions which is not well understood. The goal of the current paper is to quantify the effect of soil inhomogeneity on surface vibration. The thin-layer method (TLM) is suggested as an efficient and accurate means of simulating vibration from underground railways in arbitrarily layered half-spaces. Stochastic variability of the soils elastic modulus is introduced using a KL expansion; the modulus is assumed to have a log-normal distribution and a modified exponential covariance kernel. The effect of horizontal soil variability is investigated by comparing the stochastic results for soils varied only in the vertical direction to soils with 2D variability. Results suggest that local soil inhomogeneity can significantly affect surface velocity predictions; 90 percent confidence intervals showing 8 dB averages and peak values up to 12 dB are computed. This is a significant source of uncertainty and should be considered when using predictions from models assuming homogeneous soil properties. Furthermore, the effect of horizontal variability of the elastic modulus on the confidence interval appears to be negligible. This suggests that only vertical variation needs to be taken into account when modelling ground vibration from underground railways. © 2012 Elsevier Ltd. All rights reserved.

Sensor graphs for guaranteed cooperative localization performance

A group of mobile robots can localize cooperatively, using relative position and absolute orientation measurements, fused through an extended Kalman filter (ekf). The topology of the graph of relative measurements is known to affect the steady-state value of the position error covariance matrix. Classes of sensor graphs are identified, for which tight bounds for the trace of the covariance matrix can be obtained based on the algebraic properties of the underlying relative measurement graph. The string and the star graph topologies are considered, and the explicit form of the eigenvalues of error covariance matrix is given. More general sensor graph topologies are considered as combinations of the string and star topologies, when additional edges are added. It is demonstrated how the addition of edges increases the trace of the steady-state value of the position error covariance matrix, and the theoretical predictions are verified through simulation analysis.

Factor analysis based VTS discriminative adaptive training

Vector Taylor Series (VTS) model based compensation is a powerful approach for noise robust speech recognition. An important extension to this approach is VTS adaptive training (VAT), which allows canonical models to be estimated on diverse noise-degraded training data. These canonical model can be estimated using EM-based approaches, allowing simple extensions to discriminative VAT (DVAT). However to ensure a diagonal corrupted speech covariance matrix the Jacobian (loading matrix) relating the noise and clean speech is diagonalised. In this work an approach for yielding optimal diagonal loading matrices based on minimising the expected KL-divergence between the diagonal loading matrix and "correct" distributions is proposed. The performance of DVAT using the standard and optimal diagonalisation was evaluated on both in-car collected data and the Aurora4 task. © 2012 IEEE.

A presumed joint pdf model for turbulent combustion with varying equivalence ratio

Modeling of the joint probability density function of the mixture fraction and progress variable with a given covariance value is studied. This modeling is validated using experimental and direct numerical simulation (DNS) data. A very good agreement with experimental data of turbulent stratified flames and DNS data of a lifted hydrogen jet flame is obtained. The effect of using this joint pdf modeling to calculate the mean reaction rate with a flamelet closure in Reynolds averaged Navier-Stokes (RANS) calculation of stratified flames is studied. The covariance effect is observed to be large within the flame brush. The results obtained from RANS calculations using this modeling for stratified jet- and rod-stabilized V-flames are discussed and compared to the measurements as a posteriori validation for the joint probability density function model with the flamelet closure. The agreement between the computed and measured values of flame and turbulence quantities is found to be good. © 2012 Copyright Taylor and Francis Group, LLC.

Bayesian non-parametrics and the probabilistic approach to modelling.

Modelling is fundamental to many fields of science and engineering. A model can be thought of as a representation of possible data one could predict from a system. The probabilistic approach to modelling uses probability theory to express all aspects of uncertainty in the model. The probabilistic approach is synonymous with Bayesian modelling, which simply uses the rules of probability theory in order to make predictions, compare alternative models, and learn model parameters and structure from data. This simple and elegant framework is most powerful when coupled with flexible probabilistic models. Flexibility is achieved through the use of Bayesian non-parametrics. This article provides an overview of probabilistic modelling and an accessible survey of some of the main tools in Bayesian non-parametrics. The survey covers the use of Bayesian non-parametrics for modelling unknown functions, density estimation, clustering, time-series modelling, and representing sparsity, hierarchies, and covariance structure. More specifically, it gives brief non-technical overviews of Gaussian processes, Dirichlet processes, infinite hidden Markov models, Indian buffet processes, Kingman's coalescent, Dirichlet diffusion trees and Wishart processes.

Functional requirements for reward-modulated spike-timing-dependent plasticity.

Recent experiments have shown that spike-timing-dependent plasticity is influenced by neuromodulation. We derive theoretical conditions for successful learning of reward-related behavior for a large class of learning rules where Hebbian synaptic plasticity is conditioned on a global modulatory factor signaling reward. We show that all learning rules in this class can be separated into a term that captures the covariance of neuronal firing and reward and a second term that presents the influence of unsupervised learning. The unsupervised term, which is, in general, detrimental for reward-based learning, can be suppressed if the neuromodulatory signal encodes the difference between the reward and the expected reward-but only if the expected reward is calculated for each task and stimulus separately. If several tasks are to be learned simultaneously, the nervous system needs an internal critic that is able to predict the expected reward for arbitrary stimuli. We show that, with a critic, reward-modulated spike-timing-dependent plasticity is capable of learning motor trajectories with a temporal resolution of tens of milliseconds. The relation to temporal difference learning, the relevance of block-based learning paradigms, and the limitations of learning with a critic are discussed.

Bayesian updating of a unified soil compression model

Some amount of differential settlement occurs even in the most uniform soil deposit, but it is extremely difficult to estimate because of the natural heterogeneity of the soil. The compression response of the soil and its variability must be characterised in order to estimate the probability of the differential settlement exceeding a certain threshold value. The work presented in this paper introduces a probabilistic framework to address this issue in a rigorous manner, while preserving the format of a typical geotechnical settlement analysis. In order to avoid dealing with different approaches for each category of soil, a simplified unified compression model is used to characterise the nonlinear compression behavior of soils of varying gradation through a single constitutive law. The Bayesian updating rule is used to incorporate information from three different laboratory datasets in the computation of the statistics (estimates of the means and covariance matrix) of the compression model parameters, as well as of the uncertainty inherent in the model.

Student-t processes as alternatives to Gaussian processes

We investigate the Student-t process as an alternative to the Gaussian process as a non-parametric prior over functions. We derive closed form expressions for the marginal likelihood and predictive distribution of a Student-t process, by integrating away an inverse Wishart process prior over the co-variance kernel of a Gaussian process model. We show surprising equivalences between different hierarchical Gaussian process models leading to Student-t processes, and derive a new sampling scheme for the inverse Wishart process, which helps elucidate these equivalences. Overall, we show that a Student-t process can retain the attractive properties of a Gaussian process - a nonparamet-ric representation, analytic marginal and predictive distributions, and easy model selection through covariance kernels - but has enhanced flexibility, and predictive covariances that, unlike a Gaussian process, explicitly depend on the values of training observations. We verify empirically that a Student-t process is especially useful in situations where there are changes in covariance structure, or in applications such as Bayesian optimization, where accurate predictive covariances are critical for good performance. These advantages come at no additional computational cost over Gaussian processes.