Errors and residuals | EPFL Graph Search

In statistics and optimization, errors and residuals are two closely related and easily confused measures of the deviation of an observed value of an element of a statistical sample from its "true value" (not necessarily observable). The error of an observation is the deviation of the observed value from the true value of a quantity of interest (for example, a population mean). The residual is the difference between the observed value and the estimated value of the quantity of interest (for example, a sample mean). The distinction is most important in regression analysis, where the concepts are sometimes called the regression errors and regression residuals and where they lead to the concept of studentized residuals. In econometrics, "errors" are also called disturbances. Introduction Suppose there is a series of observations from a univariate distribution and we want to estimate the mean of that distribution (the so-called location model). In this case, the errors are the

On Variable-Scale Piecewise Stationary Spectral Analysis of Speech Signals for ASR

Hervé Bourlard, Vivek Tyagi

It is often acknowledged that speech signals contain short-term and long-term temporal properties that are difficult to capture and model by using the usual fixed scale (typically 20ms) short time spectral analysis used in hidden Markov models (HMMs), based on piecewise stationarity and state conditional independence assumptions of acoustic vectors. For example, vowels are typically quasi-stationary over 40-80ms segments, while plosive typically require analysis below 20ms segments. Thus, fixed scale analysis is clearly sub-optimal for ``optimal'' time-frequency resolution and modeling of different stationary phones found in the speech signal. In the present paper, we investigate the potential advantages of using variable size analysis windows towards improving state-of-the-art speech recognition systems. Based on the usual assumption that the speech signal can be modeled by a time-varying autoregressive (AR) Gaussian process, we estimate the largest piecewise quasi-stationary speech segments, based on the likelihood that a segment was generated by the same AR process. This likelihood is estimated from the Linear Prediction (LP) residual error. Each of these quasi-stationary segments is then used as an analysis window from which spectral features are extracted. Such an approach thus results in a variable scale time spectral analysis, adaptively estimating the largest possible analysis window size such that the signal remains quasi-stationary, thus the best temporal/frequency resolution tradeoff. The speech recognition experiments on the OGI Numbers95 database, show that the proposed variable-scale piecewise stationary spectral analysis based features indeed yield improved recognition accuracy in clean conditions, compared to features based on minimum cross entropy spectrum as well as those based on fixed scale spectral analysis.

Elsevier2006

On Variable-Scale Piecewise Stationary Spectral Analysis of Speech Signals for ASR

Hervé Bourlard, Vivek Tyagi

It is often acknowledged that speech signals contain short-term and long-term temporal properties that are difficult to capture and model by using the usual fixed scale (typically 20ms) short time spectral analysis used in hidden Markov models (HMMs), based on piecewise stationarity and state conditional independence assumptions of acoustic vectors. For example, vowels are typically quasi-stationary over 40-80ms segments, while plosives typically require analysis below 20ms segments. Thus, fixed scale analysis is clearly sub-optimal for ``optimal'' time-frequency resolution and modeling of different stationary phones found in the speech signal. In the present paper, we investigate the potential advantages of using variable size analysis windows towards improving state-of-the-art speech recognition systems. Based on the usual assumption that the speech signal can be modeled by a varying autoregressive (AR) Gaussian process, we estimate the largest piecewise quasi-stationary speech segments, based on the likelihood that a segment was generated by the same AR process. This likelihood is estimated from the Linear Prediction (LP) residual error. Each of these quasi-stationary segments is then used as an analysis window from which spectral features are extracted. Such an approach thus results in variable scale time spectral analysis, adaptively estimating the largest possible analysis window size such that the signal remains quasi-stationary, thus the best temporal/frequency resolution tradeoff. Speech recognition experiments on the OGI Numbers95 database show that the proposed multi-scale piecewise stationary spectral analysis based features indeed yield improved recognition accuracy in clean conditions, compared to features based on minimum cross entropy spectrum as well as those based on fixed scale spectral analysis.

IDIAP2005

A posteriori error estimation for partial differential equations with random input data

Diane Sylvie Guignard

This thesis is devoted to the derivation of error estimates for partial differential equations with random input data, with a focus on a posteriori error estimates which are the basis for adaptive strategies. Such procedures aim at obtaining an approximation of the solution with a given precision while minimizing the computational costs. If several sources of error come into play, it is then necessary to balance them to avoid unnecessary work. We are first interested in problems that contain small uncertainties approximated by finite elements. The use of perturbation techniques is appropriate in this setting since only few terms in the power series expansion of the exact random solution with respect to a parameter characterizing the amount of randomness in the problem are required to obtain an accurate approximation. The goal is then to perform an error analysis for the finite element approximation of the expansion up to a certain order. First, an elliptic model problem with random diffusion coefficient with affine dependence on a vector of independent random variables is studied. We give both a priori and a posteriori error estimates for the first term in the expansion for various norms of the error. The results are then extended to higher order approximations and to other sources of uncertainty, such as boundary conditions or forcing term. Next, the analysis of nonlinear problems in random domains is proposed, considering the one-dimensional viscous Burgers' equation and the more involved incompressible steady-state Navier-Stokes equations. The domain mapping method is used to transform the equations in random domains into equations in a fixed reference domain with random coefficients. We give conditions on the mapping and the input data under which we can prove the well-posedness of the problems and give a posteriori error estimates for the finite element approximation of the first term in the expansion. Finally, we consider the heat equation with random Robin boundary conditions. For this parabolic problem, the time discretization brings an additional source of error that is accounted for in the error analysis. The second part of this work consists in the analysis of a random elliptic diffusion problem that is approximated in the physical space by the finite element method and in the stochastic space by the stochastic collocation method on a sparse grid. Considering a random diffusion coefficient with affine dependence on a vector of independent random variables, we derive a residual-based a posteriori error estimate that controls the two sources of error. The stochastic error estimator is then used to drive an adaptive sparse grid algorithm which aims at alleviating the so-called curse of dimensionality inherent to tensor grids. Several numerical examples are given to illustrate the performance of the adaptive procedure.

EPFL2016

A posteriori error estimation for partial differential equations with random input data

Diane Sylvie Guignard

EPFL2016