Damage detection using the improved Kullback-Leibler divergence

The structural characteristic deflection shapes (CDS’s) such as mode shapes and operational deflection shapes are highly sensitive to structural damage in beam- or plate-type structures. Nevertheless, they are vulnerable to measurement noise and could result in unacceptable identification errors. In order to increase the accuracy and noise robustness of damage identification based on CDS’s using vibration responses of random excitation, joint approximate diagonalization (JAD) technique and gapped smoothing method (GSM) are combined to form a sensitive and robust damage index (DI), which can simultaneously detect the existence of damage and localize its position. In addition, it is possible to apply this approach to damage identification of structures under ambient excitation. First, JAD method which is an essential technique of blind source separation is investigated to simultaneously diagonalize a set of power spectral density matrices corresponding to frequencies near a certain natural frequency to estimate a joint unitary diagonalizer. The columns of this joint diagonalizer contain dominant CDS’s. With the identified dominant CDS’s around different natural frequencies, GSM is used to extract damage features and a robust damage identification index is then proposed. Numerical and experimental examples of beams with cracks are used to verify the validity and noise robustness of JAD based CDS estimation and the proposed DI. Furthermore, damage identification using dominant CDS’s estimated by JAD method is demonstrated to be more accurate and noise robust than by the commonly used singular value decomposition method.

The strength/intensity of the stimulus in the random dot motion task (RDMT) [1] is determined by the percentage of dots in the kinematogram moving towards a saccadic target, a. Due to the uncertainty in the stimuli, neurons in sensory systems have evolved to transform environmental information, comprising evidence upon which a decision can be made (e.g. saccading to a). The neurons in the middle-temporal area (MT) appear to produce such evidence during the RDMT, given their tuning to a 'preferred' direction of visual motion. If the dots move predominantly in the preferred direction of an MT neuron, it generates inter-spike intervals (ISI) supporting a saccade to a. These ISIs seem randomly sampled from a distribution, fa, with mean, μa. Otherwise, the ISIs follow another distribution, fb, with mean, μb, where μb is larger than μa and this difference increases with stimulus strength. The accuracy vs motion-strength function of an ideal observer provided with empirical distributions like fa and fb, from a single MT neuron, approximates the subject's psychometric function (at the behavioral level) [1]. The distributions fa and fb are non-negative, positively skewed and have a mode larger than 0 (figure ​(figure1A),1A), as is typical for neural events recorded in many brain areas. Here we investigate why this is advantageous for decision-making. As theoretical decision-making units, we produced 5 new instantiations of the multi-hypothesis sequential probability ratio test (MSPRT) [2]. Each unit assumes its stream of input evidence to follow 1 of 5 probability density functions (PDF) whose compatibility with the empirical distribution of ISIs varies (figure ​(figure1A).1A). These include the Inverse Gaussian, Lognormal, Gamma, Inverse-Gamma and Exponential PDFs (the latter is the distribution of the inter-event times in the oft-used Poisson process). Under equal and appropriate conditions, we then compared their mean decision sample with that of an MSPRT instantiation that assumes Gaussian inputs proposed in [3] and discussed in general in [4], as exemplified in figure ​figure1B.1B. The mean decision sample is the mean number of observations required by a unit to identify which of N parallel information sources supports saccading to a, with a given accuracy. This decision sample is a model of the 'neural decision time'; the psychophysical reaction time also includes sensory and motor delays. The pattern of our results is explicable using a measure of the discrimination information between fa and fb, i.e. the Kullback-Leibler divergence (KLD). We found that, the mean decision sample decreases with increasing fa to fb KLD and, crucially, this follows a power law (figure ​(figure1C).1C). At the behavioral level, Pieron [5] reported the mean reaction time to the presentation of a stimulus (go/no-go decision-making) being shorter for more intense stimuli, and that a power law relates these measures. The universality of Pieron's law indicates that it can inform us of something fundamental about sensorimotor decision-making. Our results suggest that its explanation could lie in the power law relationship between the mean neural decision time and the discrimination information (KLD) among the distributions of sensory evidence. Figure 1 (A) PDFs of interest fitted to ISIs (grey bars) recorded in [1]from the MT during the RDMT. (B) mean decision sample for each MSPRT realisation (accuracy 95% over 1000 trials). (C) the values in (B) vs the fa to fb KLD. The dashed line is a fitted power ...

ABSTRACTAs a probabilistic distance between two probability density functions, Kullback–Leibler divergence is widely used in many applications, such as image retrieval and change detection. Unfortunately, for some models, e.g., Gaussian Mixture Models (GMMs), Kullback–Leibler divergence is not analytically tractable. One has to resort to approximation methods. A number of methods have been proposed to address this issue. In this article, we compare seven methods, namely Monte Carlo method, matched bound approximation, product of Gaussians, variational method, unscented transformation, Gaussian approximation and min-Gaussian approximation, for approximating the Kullback–Leibler divergence between two Gaussian mixture models for satellite image retrieval. Two experiments using two public data sets have been performed. The comparison is carried out in terms of retrieval accuracy and computational time.

This article addresses the robust estimation of the output layer linear parameters in a radial basis function network (RBFN). A prominent method used to estimate the output layer parameters in an RBFN with the predetermined hidden layer parameters is the least-squares estimation, which is the maximum-likelihood (ML) solution in the specific case of the Gaussian noise. We highlight the connection between the ML estimation and minimizing the Kullback-Leibler (KL) divergence between the actual noise distribution and the assumed Gaussian noise. Based on this connection, a method is proposed using a variant of a generalized KL divergence, which is known to be more robust to outliers in the pattern recognition and machine-learning problems. The proposed approach produces a surrogate-likelihood function, which is robust in the sense that it is adaptive to a broader class of noise distributions. Several signal processing experiments are conducted using artificially generated and real-world data. It is shown that in all cases, the proposed adaptive learning algorithm outperforms the standard approaches in terms of mean-squared error (MSE). Using the relative increase in the MSE for different noise conditions, we compare the robustness of our proposed algorithm with the existing methods for robust RBFN training and show that our method results in overall improvement in terms of absolute MSE values and consistency.

The eigenfrequencies of a bridge are influenced by the environment and by structural integrity. The task of damage identification is to conclude structural integrity from measured eigenfrequencies. Environmental frequency changes are a negative effect that hinders reliable damage identification. In this paper, it is experimentally shown how temperature and traffic load affect the eigenfrequencies, and reduction of this effect using a frequency prediction model is presented. Random variations are described using a probability density function, and a damage indicator based on this function is proposed. The paper addresses the issue of reliability of damage detection considering random and systematic errors that are present in the eigenfrequencies extracted from measurements. Various damage scenarios are presented, both with identifiable and non-identifiable damages.

In this paper, we propose a new objective evaluation method for hidden Markov model (HMM)-based speech synthesis using Kullback-Leibler divergence (KLD). The KLD is used to measure the difference between the probability density functions (PDFs) of the acoustic feature vectors extracted from natural training and synthetic speech data. For the evaluation, Gaussian mixture model (GMM) is used to model the distribution of acoustic feature vectors, including the fundamental frequency (F0). Continuous F0, obtained with linear interpolation, is used in the evaluation. In essence, the KLD is the expectation of the logarithmic difference between the likelihoods calculated on training and synthetic speech. This likelihood difference is appropriate to characterize the quality of a HMM-based speech synthesis system in generating synthetic speech using a maximum likelihood criterion. The objective evaluation is tested with 3 different HMM-based speech synthesis systems which use multi-space distribution (MSD) to model discontinuous F0. These systems are trained on a common speech corpus in French. We propose an index to evaluate HMM-based speech synthesis system which takes into account the relative variation of the KLDs on test sets of synthetic and natural speech. This index correlates inversely with the result of the MOS (mean opinion score) perceptual test.

In this paper we propose a new distance metric for probability density functions (PDF). The main advantage of this metric is that unlike the popular Kullback-Liebler (KL) divergence it can be computed in closed form when the PDFs are modeled as Gaussian Mixtures (GM). The application in mind for this metric is histogram based image retrieval. We experimentally show that in an image retrieval scenario the proposed metric provides as good results as the KL divergence at a fraction of the computational cost. This metric is also compared to a Bhattacharyya-based distance metric that can be computed in closed form for GMs and is found to produce better results.

Under external load excitation, damage such as breathing cracks and bolt loosening will cause structural time domain acceleration to have nonlinear features. To solve the problem of time domain nonlinear damage identification, a damage identification method based on the Kullback–Leibler (KL) distance of time domain model residuals is proposed in this paper. First, an autoregressive (AR) model order was selected using the autocorrelation function (ACF) and Akaike information criterion (AIC). Then, an AR model was obtained based on the structural acceleration response time series, and the AR model residual was extracted. Finally, the KL distance was used as a damage indicator to judge the structural damage source location. The effectiveness of the proposed method was verified by using a multi-story, multi-span stand model experiment and a simulated eight-story shear structure. The results show that the proposed structural nonlinear damage identification method can effectively distinguish the structural damage location of multi-degree-of-freedom shear structures and complex stand structures, and it is robust enough to detect environmental noise and small damage.

The major sources of error in the measurements of concrete structures are the gauge sensitivities, calibration accuracies, amplitude linearities, and temperature corrections to the gauge sensitivities, which are given in terms of plus–minus ranges, and the round off errors in the measured responses, which are better represented by interval bounds. An algorithm is proposed adapting the existing modified Metropolis Hastings algorithm for estimating the posterior probability of the damage indices as well as the posterior probability of the bounds of the interval parameters, when the measurements are given in terms of interval bounds. A damage index is defined for each element of the finite element model considering the parameters of each element are intervals. Methods are developed for evaluating response bounds in the finite element software ABAQUS, when the parameters of the finite element model are intervals. The proposed method is validated against reinforced concrete beams with three damage scenarios mainly (1) loss of stiffness, (2) loss of mass, and (3) loss of bond between concrete and reinforcement steel, which have been tested in our laboratory. Comparison of the prediction from the proposed method with those obtained from Bayesian analysis and interval optimization technique show improved accuracy and computational efficiency in addition to better representation of measurement uncertainties through interval bounds.

The objective of the chapter is to offer intuitive understandings for statistical learning from an infomation geometric point of view. We introduce a wide class of information divergence indeces that express quantatitavely the degree of departure between any two probability density functions on a data space. In general the information divergence leads to a statistical method by minimization based on empirical data information. We introduce that the information divergence associates with a Riemannian metric and a pair of congugate linear connections for a family of probability density functions. The most familiar example is the Kullback-Leibler divergence, which leads to the maximum likelihood method and associates with the information metric and the pair of the exponential and mixture connections. In the class of statistical methods led to by minimum divergence we discuss statistical properties focusing on robsutness. As applicatuons to statistical learning we discuss the minimum divergence methods for the principal component analysis and independent component analysis, and we also discuss applications to statistical pattern recognition. U -divergence, U -model, U -loss function, robustness, information geometry.

Damage identification in steel girder bridges using modal strain energy-based damage index method and artificial neural network

This article presents a theoretical analysis of optimally distinguishing among environmental parameters from ocean ambient sound. Recent approaches to this problem either focus on parameter estimation or attempt to classify the environment into one of many known types through machine learning. This classification problem is framed as one of hypothesis testing on the received ambient sound snapshots. The resulting test depends on the Kullback-Leibler divergence (KLD) between the distributions corresponding to different environments or sediment types. Analysis of the KLD shows the dependence on the signal-to-noise ratio, the underlying signal subspace, and the distribution of eigenvalues of the respective covariance matrices. This analysis provides insights into both when and why successful hypothesis testing is possible. Experiments demonstrate that our analysis provides insight as to why certain environmental parameters are more difficult to distinguish than others. Experiments on sediment types from the Naval Oceanographic Office Bottom Sediment type database show that certain types are indistinguishable for a given array configuration. Further, the KLD can be used to provide a quantitative alternative to examining bottom loss curves to predict array processing performance.

Performance limits of one-bit compressive classification

More and more optimization based decision-makings are taking parameter uncertainty into account, and chance constrained programming (CCP) is an important framework for decision-making under uncertainty. In particular, the quadratically perturbed CCP is an important subclass of general CCP problems, and finds widespread applications from wireless communication, machine learning to control problems.
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\nDespite its importance, previous solutions for quadratically perturbed CCP problem are safe approximation based, and mainly focus on some special classes of uncertainty, e.g., Gaussian and Uniform distributed uncertainty. For CCP with non-Gaussian uncertainty, the optimization is generally intractable owing to the complicated probability density function (PDF). Using a simple fitted distribution with an additional Kullback-Leibler (KL) divergence constraint is a systematic way to tackle CCP with non-Gaussian uncertainty. However, the essential difficulty of using this methodology is choosing the fitted PDF, which should be close to the true PDF while providing a tractable safe approximation to the resulting CCP problem.
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\nIn the first part of this thesis, we derive a novel safe approximation by proposing the flexible t-distribution to be the fitted PDF. More specifically, after the CCP with non-Gaussian uncertainty is transformed into the CCP with a fitted PDF, the property of the regularized outage probability is analysed. Then, the unimodal distributional property of the quadratic form under t-distributed perturbation is established. Based on the unimodal property and the regularized outage analysis, the analytical condition to make the safe approximation with fitted t-distribution having larger feasible set than the safe approximation with fitted Gaussian is obtained.
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\nOn the other hand, although safe approximation is a leading approach in solving CCP problems, and the fitted t-distribution based safe approximation is general enough to handle non-Gaussian uncertainty, a common drawback of this class of method is that they do not provide optimality guarantee for the solutions. In order to improve safe approximation solutions, in the second part of this thesis, a new optimization methodology is proposed to solve the CCP problem under continuous uncertainty distribution. The generally intractable chance constraints and unknown convexity are overcome by novel analyses of local structure of the feasible set. Then a convergent set squeezing procedure is established with local optimal or tight solution guaranteed under mild conditions. Efficient algorithms are also derived for the set squeezing procedure under the widely used quadratically perturbed constraints.
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\nFinally, we illustrate the performance of the two proposed methods for quadratically perturbed CCP with the mean square error (MSE) constrained transceiver design problem in wireless communications. In particular, with Logistic and Gaussian mixture channel uncertainties as examples, simulation results validate the less conservative property of the transceiver design with fitted t-distribution based safe approximation, compared to that with fitted Gaussian and the classic moment method. Furthermore, with safe approximation solutions as initializations, simulation results under Gaussian, uniform, and Gaussian mixture channel uncertainties show that tight solutions for probabilistic MSE constrained transceiver designs are obtained by the proposed set squeezing procedure. The tight control of the outage probability using the proposed approach enables a flexible balance between the MSE outage probability and transmit power, compared to various safe approximation methods.

In the field of target detection and tracking, the resolvability of multiple closely located targets is a significant criterion for measuring the sensing ability of radar and other sensing systems. This paper considers the problem of adaptive radar waveform design to maximize the practical resolvability of closely located targets. Conventional ambiguity function (AF)-based radar resolution considers only the waveform, regardless of the influence of noise. However, noise grreatly influences the achievable resolution as well as radar detection and tracking performance. As a result of this deficiency in conventional radar resolution, Kullback-Leibler Divergence (KLD) is introduced in this paper in order to quantify the “distance” between two probability density functions (PDFs) of radar measurements, and thus to represent practical radar resolution, which includes not only the effect of the waveform but also the signal-to-noise ratio (SNR) and measurement model. Consequently, a new adaptive radar waveform design criterion is proposed, which aims to maximize the practical radar resolution. Simulation results show the effectiveness of the proposed adaptive waveform design method.