Probability density approximations of linear system responses to intensity-modulated Gaussian processes

Probability density function (PDF) estimation is a fundamentally important problem for statistical pattern recognition. Independent component analysis (ICA) can be applied to the feature vectors so that the PDF estimation of a high dimensional vector can be converted to the PDF estimation of several 1-dimensional variables. However in practice we find that this PDF is in poor generalization ability for pattern classification because of the implied noise. Hence, this paper proposes an improvement of ICA based PDF estimation method. A latent variable model is built to separate the noise from the feature vector so that the pattern information and the noise can be dealt with respectively. Based on the latent variable model, a modified ICA based PDF is deduced. The validity of our proposed method is demonstrated by the experiments of off-line handwritten numeral recognition.

The non-line-of-sight (NLOS) propagation caused by intermittently blocking of the direct path between the base station and the mobile station is the main difficulty for mobile user location in cellular wireless communication system. So it is indispensable to investigate the techniques about the NLOS error identification and correction. In this paper, we suggest a robust and accurate approach using probability density function (PDF) estimator to identify the NLOS situation, and then an improved biased Kalman filter which depends on PDF estimator as the nonlinearity weighting coefficient in recursive operation is used to mitigate the NLOS error for time of arrival (TOA) measurements. As the PDF estimator is calculated from actual TOA samples, it is adaptive for the complex practical NLOS environments. The simulation results also indicate that with the less prior information about communication environments, the approach we present has a good location performance even in severe NLOS situations.

In this paper, we propose a fast and versatile blind source separation including closed-form estimation of sources' probability density functions (PDFs), where the ICA's activation function is automatically adapted to various noise conditions. In the proposed method, closed-form second-order ICA and closed-form PDF estimation are introduced as a computational-cost-efficient preprocessing to extract sources' PDFs. Compared with various type of conventional ICAs, e.g., fixed activation-function type and ML-based type, our proposed algorithm can give a faster and higher convergence. Experimental assessment reveals that the proposed method is versatile for handling non-speech sound sources.

The joint probability density function (PDF) of different structural responses is a very important topic in the stochastic response analysis of nonlinear structures. In this paper, the probability density evolution method, which is successfully developed to capture the instantaneous PDF of an arbitrary single response of interest, is extended to evaluate the joint PDF of any two responses. A two-dimensional partial differential equation in terms of the joint PDF is established. The strategy of selecting representative points via the number theoretical method and sieved by a hyper-ellipsoid is outlined. A two-dimensional difference scheme is developed. The free vibration of an SDOF system is examined to verify the proposed method, and a frame structure exhibiting hysteresis subjected to stochastic ground motion is investigated. It is pointed out that the correlation of different responses results from the fact that randomness of different responses comes from the same set of basic random parameters involved. In other words, the essence of the probabilistic correlation is a physical correlation.

A generalized method for the stationary probabilistic response of nonlinear dynamical system

The error estimate of Borgonovo's moment-independent index δi is considered, and it shows that the possible computational complexity of δi is mainly due to the probability density function (PDF) estimate because the PDF estimate is an ill-posed problem and its convergence rate is quite slow. So it reminds us to compute Borgonovo's index using other methods. To avoid the PDF estimate, δi, which is based on the PDF, is first approximatively represented by the cumulative distribution function (CDF). The CDF estimate is well posed and its convergence rate is always faster than that of the PDF estimate. From the representation, a stable approach is proposed to compute δi with an adaptive procedure. Since the small probability multidimensional integral needs to be computed in this procedure, a computational strategy named asymptotic space integration is introduced to reduce a high-dimensional integral to a one-dimensional integral. Then we can compute the small probability multidimensional integral by adaptive numerical integration in one dimension with an improved convergence rate. From the comparison of numerical error analysis of some examples, it can be shown that the proposed method is an effective approach to uncertainty importance measure computation.

Constructing transient response probability density of non-linear system through complex fractional moments

: The structure of the asymptotic log-likelihood ratio decision procedure for the discrimination of alternative stationary zero mean multivariate Gaussian processes is developed. The log-likelihood ratio statistic is shown to be asymptotically normally distributed. New time and frequency domain formulas for the conditional mean (The Kullback-Liebler information measure) and variance of the log-likelihood ratio statistic under the alternative hypotheses are given and the probability of misclassification is shown to be bounded exponentially with n, the number of observations.

Direct probability integral method for stochastic response analysis of static and dynamic structural systems

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.

Previous article Next article Second-Order Markov and Reciprocal Stationary Gaussian ProcessesR. N. MiroshinR. N. Miroshinhttps://doi.org/10.1137/1124099PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] S. N. Bernshtei˘n, On the dependencies among random variablesCollected Works, Vol. IV, Nauka, Moscow, 1964, 235–254, (In Russian.) Google Scholar[2] J. L. Doob, Stochastic processes, John Wiley & Sons Inc., New York, 1953viii+654 MR0058896 0053.26802 Google Scholar[3] Benton Jamison, Reciprocal processes: The stationary Gaussian case, Ann. Math. Statist., 41 (1970), 1624–1630 MR0267637 0248.60030 CrossrefGoogle Scholar[4] I. I. Gikhman and , A. V. Skorokhod, The Theory of Stochastic Processes, Vol. I, Springer-Verlag, New York, 1974 CrossrefGoogle Scholar[5] E. Kamke, Differentialgleichungen. Lösungsmethoden and Lösungen, Vol. 1, Chelsea, New York, 1974 Google Scholar[6] E. Wong, Some results concerning the zero-crossings of Gaussian noise, SIAM J. Appl. Math., 14 (1966), 1246–1254 10.1137/0114099 MR0207059 0147.15905 LinkGoogle Scholar[7] E. Wong, The distribution of intervals between zeros for a stationary Gaussian process, SIAM J. Appl. Math., 18 (1970), 67–73 10.1137/0118008 MR0256456 0221.60024 LinkGoogle Scholar[8] R. N. Miroshin, Convergence of Rice and Longuet-Higgins series for a Wong process, Theory Prob. Applications, 21 (1976), 863–866 10.1137/1121103 0374.60049 LinkGoogle Scholar[9] R. N. Miroshin, Asymptotics of the second factorial moment of the number of upward crossings of the line $kt+a$ by a Gaussian stationary process, Vestnik Leningrad. Univ., 23 (1968), 109–120 MR0235612 0165.52602 Google Scholar[10] M. Krein, Sur le problème du prolongement des fonctions hermitiennes positives et continues, C. R. (Doklady) Acad. Sci. URSS (N.S.), 26 (1940), 17–22 MR0004333 0022.35302 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Characterization of Multivariate Stationary Gaussian Reciprocal DiffusionsJournal of Multivariate Analysis, Vol. 62, No. 1 Cross Ref Gaussian reciprocal diffusions and positive definite sturm-liouville operators4 April 2007 | Stochastics and Stochastic Reports, Vol. 55, No. 3-4 Cross Ref Gaussian reciprocal processes revisitedStatistics & Probability Letters, Vol. 12, No. 4 Cross Ref Reciprocal covariance solutions of some matrix differential equationsStochastic Processes and their Applications, Vol. 37, No. 1 Cross Ref Characterization and Classification of Gaussian Second Order Reciprocal Processes Cross Ref Gaussian reciprocal processes and self-adjoint stochastic differential equations of second order2 May 2007 | Stochastics and Stochastic Reports, Vol. 34, No. 1-2 Cross Ref Reciprocal diffusions and stochastic differential equations of second order ∗Stochastics, Vol. 24, No. 4 Cross Ref Multivariate reciprocal stationary Gaussian processesJournal of Multivariate Analysis, Vol. 23, No. 1 Cross Ref On Miroshin’s Second-Order Reciprocal ProcessesJulia Abrahams3 August 2006 | SIAM Journal on Applied Mathematics, Vol. 44, No. 1AbstractPDF (316 KB)The Use of Rice SeriesR. N. Miroshin28 July 2006 | Theory of Probability & Its Applications, Vol. 28, No. 4AbstractPDF (1011 KB)The zero-crossing problem for some nonstationary Gaussian processes (Corresp.)IEEE Transactions on Information Theory, Vol. 28, No. 4 Cross Ref Convergence of Longuet-Higgins Series for Stationary Gaussian Markov Processes of First OrderR. N. Miroshin17 July 2006 | Theory of Probability & Its Applications, Vol. 26, No. 1AbstractPDF (1546 KB) Volume 24, Issue 4| 1980Theory of Probability & Its Applications History Submitted:11 July 1977Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1124099Article page range:pp. 845-852ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

An approximate analytical method is developed for calculating the probability density of the response of a Duffing oscillator subjected to narrow band random excitation when the stochastic jump phenomenon occurs. The stochastic jump phenomenon is characterized by the existence of multiple states of amplitude and sudden jumps between the states in the sample function of response. In this paper, the trigger for the stochastic jump is considered, and the probabilities of large and small amplitude states of the response are evaluated. The local probability density function of each state is separately assumed using the equivalent linearization method, and the probability density function of the overall response is derived by sum of these local probability density functions weighted with the probability of each state. In the illustrative example, the calculated probability density functions are compared with the Monte Carlo simulation results. It is shown that the present method well expresses the significant characteristic of the stochastic jump phenomenon.

Generalized Laguerre expansions of multivariate probability densities with moments

Stochastic ship roll motion via path integral method

Probability density of the intensity of two partially correlated speckle fields: the influence of finite size detectors