Conditional Probability Density Research Articles

Deep neural networks and maximum likelihood search for approximate nearest neighbor in video-based image recognition

We analyzed the way to increase computational efficiency of video-based image recognition methods with matching of high dimensional feature vectors extracted by deep convolutional neural networks. We proposed an algorithm for approximate nearest neighbor search. At the first step, for a given video frame the algorithm verifies a reference image obtained when recognizing the previous frame. After that the frame is compared with a few number of reference images. Each next examined reference image is chosen so that to maximize conditional probability density of distances to the reference instances tested at previous steps. To decrease the required memory space we beforehand calculate only distances from all the images to small number of instances (pivots). When experimenting with either face photos from Labeled Faces in the Wild and PubFig83 datasets or with video data from YouTube Faces we showed that our algorithm allows accelerating the recognition procedure by 1.4–4 times comparing with known approximate nearest neighbor methods.

New Bivariate Exponentiated Modified Weibull Distribution

The new modified Weibull distribution named exponentiated modified Weibull distribution introduced by many authors. In this study, we define and study a bivariate modified Weibull distribution. Some joint probability functions and conditional probability density function are obtained. Also, joint survival function is expressed in compact form. Several statistical properties of this distribution are discussed. In the area of stress-strength models, we estimate the reliability follow a bivariate exponentiated modified Weibull distribution with dependence between stress and strength variables. Parameters estimators using the maximum likelihood method are obtained. Moreover, a numerical illustration is used to obtain Maximum Likelihood Estimators (MLEs).

An Improved Multi-Sensor Fusion Navigation Algorithm Based on the Factor Graph.

An integrated navigation system coupled with additional sensors can be used in the Micro Unmanned Aerial Vehicle (MUAV) applications because the multi-sensor information is redundant and complementary, which can markedly improve the system accuracy. How to deal with the information gathered from different sensors efficiently is an important problem. The fact that different sensors provide measurements asynchronously may complicate the processing of these measurements. In addition, the output signals of some sensors appear to have a non-linear character. In order to incorporate these measurements and calculate a navigation solution in real time, the multi-sensor fusion algorithm based on factor graph is proposed. The global optimum solution is factorized according to the chain structure of the factor graph, which allows for a more general form of the conditional probability density. It can convert the fusion matter into connecting factors defined by these measurements to the graph without considering the relationship between the sensor update frequency and the fusion period. An experimental MUAV system has been built and some experiments have been performed to prove the effectiveness of the proposed method.

Calculation of mutual information for nonlinear optical fiber communication channel at large SNR within path-integral formalism

Using the path-integral technique we calculate the mutual information for the fiber optical channel modelled by the nonlinear Schrödinger equation with additive Gaussian noise. At large signal-to-noise ratio (SNR) we present the mutual information through the path-integral which is convenient for the perturbative expansion both in nonlinearity and dispersion. In the leading order in 1/SNR we demonstrate that the mutual information is determined through the averaged logarithm of the normalization factor Λ of the conditional probability density function P[Y|X]. In the limit of small noise and small nonlinearity we derive analytically the first nonzero nonlinear correction to the mutual information for the channel. For the arbitrary nonlinearity we restrict the mutual information by the low bound obtained from the Jensen’s inequality and analyze the bound for the case of large dispersion.

Bayesian system identification based on hierarchical sparse Bayesian learning and Gibbs sampling with application to structural damage assessment

Bayesian system identification has attracted substantial interest in recent years for inferring structural models based on measured dynamic response from a structural dynamical system. The focus in this paper is Bayesian system identification based on noisy incomplete modal data where we can impose spatially-sparse stiffness changes when updating a structural model. To this end, based on a similar hierarchical sparse Bayesian learning model from our previous work, we propose two Gibbs sampling algorithms. The algorithms differ in their strategies to deal with the posterior uncertainty of the equation-error precision parameter, but both sample from the conditional posterior probability density functions (PDFs) for the structural stiffness parameters and system modal parameters. The effective dimension for the Gibbs sampling is low because iterative sampling is done from only three conditional posterior PDFs that correspond to three parameter groups, along with sampling of the equation-error precision parameter from another conditional posterior PDF in one of the algorithms where it is not integrated out as a “nuisance” parameter. A nice feature from a computational perspective is that it is not necessary to solve a nonlinear eigenvalue problem of a structural model. The effectiveness and robustness of the proposed algorithms are illustrated by applying them to the IASE-ASCE Phase II simulated and experimental benchmark studies. The goal is to use incomplete modal data identified before and after possible damage to detect and assess spatially-sparse stiffness reductions induced by any damage. Our past and current focus on meeting challenges arising from Bayesian inference of structural stiffness serves to strengthen the capability of vibration-based structural system identification but our methods also have much broader applicability for inverse problems in science and technology where system matrices are to be inferred from noisy partial information about their eigenquantities.

Wind speed generation for dynamic analysis

AbstractA simple model to generate large band wind speed time sequences, especially easy to implement with a very reduced number of parameters, is presented. It is based on the calculation of a low‐frequency and a high‐frequency components. Low‐frequency component with 1 h sample time is obtained from a random process based on a conditional probability density function. Using real data from two different wind farms in two different months of the year, it has been found that Weibull distribution centered in the current hourly mean value seems to represent well the 1 h conditional PDF in all cases, and the standard deviation of this conditional Weibull is more or less in the range 1–1.3 m s−1 independently of the season of the year or the location. Regarding to high‐frequency component, low‐frequency samples are used as initial and final values and, between them, the turbulence component values are inserted. For that, it has been used a stochastic process based on a Beta probability function and a simple rescaling procedure with two non‐linear parameters, calculated in a recursive way. Unlike the usual modelling procedures presented in the literature, spectral power density functions are not used. This simplifies the implementation significantly. Ten second sample‐time real speed wind data from two different wind farms have been used to validate the proposed high‐frequency model, obtaining excellent results. A thorough revision of the main models found in the literature to produce wind speed time sequences for dynamic analysis is performed in the paper. Copyright © 2017 John Wiley & Sons, Ltd.

An estimation approach for linear stochastic systems based on characteristic functions

This paper presents an alternative, characteristic function based approach for the Bayesian design of estimators for dynamic linear systems and linear detection problems. For a measurement update, the a posteriori characteristic function of the unnormalized conditional probability density function (ucpdf) of the state given the measurement history is obtained as a convolution of the a priori characteristic function of the ucpdf with the characteristic function of the measurement noise. It is shown that this convolution holds for a general measurement structure. Time propagation involves the product of the updated characteristic function of the ucpdf and the characteristic function of the process noise. Some estimation problems are found to be naturally tractable using only characteristic functions, such as the multivariable linear system with additive Cauchy measurement and process noise. It is shown that even the derivation of the Kalman filter algorithm has advantages when formulated using the characteristic function approach. Finally, in some instances the estimation problem can only be formulated in terms of characteristic functions. This is illustrated by a one-update scalar example for symmetric-α-stable distributions.

Next-to-leading-order corrections to capacity for a nondispersive nonlinear optical fiber channel in the intermediate power region.

We consider the optical fiber channel modeled by the nonlinear Schrödinger equation with zero dispersion and additive Gaussian noise. Using the Feynman path-integral approach for the model, we find corrections to conditional probability density function, output signal distribution, conditional and output signal entropies, and the channel capacity at large signal-to-noise ratio. We demonstrate that the correction to the channel capacity is positive for large signal power. Therefore, this correction increases the earlier calculated capacity for a nondispersive nonlinear optical fiber channel in the intermediate power region.

Cooperative target localization using multiple UAVs with out-of-sequence measurements

PurposeThe time delay would occurs when the measurements of multiple unmanned aerial vehicles (UAVs) are transmitted to the date processing center during cooperative target localization. This problem is often named as the out-of-sequence measurement (OOSM) problem. This paper aims to present a nonlinear filtering based on solving the Fokker–Planck equation to address the issue of OOSM.Design/methodology/approachAccording to the arrival time of measurement, the proposed nonlinear filtering can be divided into two parts. The non-delay measurement would be fused in the first part, in which the Fokker–Planck equation is utilized to propagate the conditional probability density function in the forward form. The time delay measurement is fused in the second part, in which the Fokker–Planck is used in the backward form approximately. The Bayes formula is applied in both parts during the measurement update.FindingsUnder the Bayesian filtering framework, this nonlinear filtering is not only suitable for the Gaussian noise assumption but also for the non-Gaussian noise assumption. The nonlinear filtering is applied to the cooperative target localization problem. Simulation results show that the proposed filtering algorithm is superior to the previous Y algorithm.Practical implicationsIn this paper, the research shows that a better performance can be obtained by fusing multiple UAV measurements and treating time delay in measurement with the proposed algorithm.Originality/valueIn this paper, the OOSM problem is settled based on solving the Fokker–Planck equation. Generally, the Fokker–Planck equation can be used to predict the probability density forward in time. However, to associate the current state with the state related to OOSM, it would be used to propagate the probability density backward either.

Effects of Compound K-Distributed Sea Clutter on Angle Measurement of Wideband Monopulse Radar

The effects of compound K-distributed sea clutter on angle measurement of wideband monopulse radar are investigated in this paper. We apply the conditional probability density function (pdf) of monopulse ratio (MR) error to analyze these effects. Based on the angle measurement procedure of the wideband monopulse radar, this conditional pdf is first deduced in detail for the case of compound K-distributed sea clutter plus noise. Herein, the spatial correlation of the texture components for each channel clutter and the correlation of the texture components between the sum and difference channel clutters are considered, and two extreme situations for each of them are tackled. Referring to the measured sea clutter data, angle measurement performances in various K-distributed sea clutter plus noise circumstances are simulated, and the effects of compound K-distributed sea clutter on angle measurement are discussed.

Probabilistic mapping of flood-induced backscatter changes in SAR time series

The information content of flood extent maps can be increased considerably by including information on the uncertainty of the flood area delineation. This additional information can be of benefit in flood forecasting and monitoring. Furthermore, flood probability maps can be converted to binary maps showing flooded and non-flooded areas by applying a threshold probability value pF=0.5. In this study, a probabilistic change detection approach for flood mapping based on synthetic aperture radar (SAR) time series is proposed. For this purpose, conditional probability density functions (PDFs) for land and open water surfaces were estimated from ENVISAT ASAR Wide Swath (WS) time series containing >600 images using a reference mask of permanent water bodies. A pixel-wise harmonic model was used to account for seasonality in backscatter from land areas caused by soil moisture and vegetation dynamics. The approach was evaluated for a large-scale flood event along the River Severn, United Kingdom. The retrieved flood probability maps were compared to a reference flood mask derived from high-resolution aerial imagery by means of reliability diagrams. The obtained performance measures indicate both high reliability and confidence although there was a slight under-estimation of the flood extent, which may in part be attributed to topographically induced radar shadows along the edges of the floodplain. Furthermore, the results highlight the importance of local incidence angle for the separability between flooded and non-flooded areas as specular reflection properties of open water surfaces increase with a more oblique viewing geometry.

Adaptive kernels in approximate filtering of state‐space models

SummaryStandard Bayesian algorithms used for online filtering of states of hidden Markov models from noisy measurements assume an accurate knowledge of the measurement model in the form of a conditional probability density function. However, this knowledge is often unreachable in practice, and the used models are more or less misspecified, or it is too complex, making the resulting models intractable. This paper focuses on these issues from the particle filtering perspective. It adopts the principles of the approximate Bayesian filtering, where the particle weights are based on the (dis)similarity of the true measurements and the pseudo‐measurements obtained by plugging the state particles directly into the measurement equation. Specifically, a new robust method for online tuning of the weighting kernel is proposed. Copyright © 2016 John Wiley & Sons, Ltd.

Inference of reaction rate parameters based on summary statistics from experiments

We present the results of an application of Bayesian inference and maximum entropy methods for the estimation of the joint probability density for the Arrhenius rate parameters of the rate coefficient of the H2/O2-mechanism chain branching reaction H+O2→OH+O. Available published data is summary statistics in terms of nominal values and error bars of the rate coefficient of this reaction at a number of temperature values obtained from shock-tube experiments. Our approach relies on generating data, in this case OH concentration profiles, consistent with the given summary statistics, using Approximate Bayesian Computation methods and a Markov chain Monte Carlo procedure. The approach permits the forward propagation of parametric uncertainty through the computational model in a manner that is consistent with the published statistics. A consensus joint posterior on the parameters is obtained by pooling the posterior parameter densities given each consistent data set. To expedite this process, we construct efficient surrogates for the OH concentration using a combination of Padé and polynomial approximants. These surrogate models adequately represent forward model observables and their dependence on input parameters and are computationally efficient to allow their use in the Bayesian inference procedure. We also utilize Gauss–Hermite quadrature with Gaussian proposal probability density functions for moment computation resulting in orders of magnitude speedup in data likelihood evaluation. Despite the strong non-linearity in the model, the consistent data sets all result in nearly Gaussian conditional parameter probability density functions. The technique also accounts for nuisance parameters in the form of Arrhenius parameters of other rate coefficients with prescribed uncertainty. The resulting pooled parameter probability density function is propagated through stoichiometric hydrogen–air auto-ignition computations to illustrate the need to account for correlation among the Arrhenius rate parameters of one reaction and across rate parameters of different reactions.

Estimation-Guided Guidance and Its Implementation via Sequential Monte Carlo Computation

Existing missile guidance strategies are traditionally based on the separation theorem, which has never been proven valid in realistic guidance scenarios. In such cases, only the general separation theorem may be applied, implying a separately designed estimator and a guidance law accounting for the conditional probability density function. A new general-separation-theorem-compliant geometry-based approach is proposed to fusion of estimation and guidance. The conventional notion of reachability sets is extended, facilitating the introduction of miss-sets. A stochastic guidance strategy is proposed that aims at maximizing the pursuer’s single-shot kill probability by driving its own miss-set to optimally cover the evader’s miss-set (in a probabilistic sense). Information-based trajectory shaping is employed, when applicable, to enhance the scenario’s observability, thereby reducing the evader’s miss-set uncertainty. Computationally efficient sequential Monte Carlo methods are employed to estimate the evader’s miss-set and implement the guidance scheme. A numerical simulation study is used to demonstrate the proposed methodology’s robustness and accuracy in a realistic stochastic engagement. The performance of the proposed strategy is compared with that of an advanced perfect-information guidance law, addressing (in particular) real-time implementation issues and constraints. Proof-of-concept simulations demonstrate that the method is real-time amenable using present-day technology.

Non-equilibrium by molecular dynamics: a dynamical approach

Some time ago, Ciccotti and Jacucci [Phys. Rev. Lett. 1975;35:789-792] – on the basis of Onsager regression hypothesis as explained by Kubo – suggested and implemented an original approach to study both stationary and time-dependent situations in non-equilibrium systems. The key idea of their approach was to tackle separately the dynamical evolution from the problem of sampling the initial condition. In this review, we discuss the evolution of this approach dubbed D-NEMD, the dynamical approach to non-equilibrium molecular dynamics, to differentiate it from the stationary NEMD methods. We will go through the D-NEMD theoretical framework and illustrate how it allows to compute time-dependent macroscopic dynamical behaviours by averaging on a large sample of non-equilibrium trajectories starting from an ensemble of initial conditions associated to a suitable distribution (either equilibrium or non-equilibrium) at time zero. We discuss a few case studies where the D-NEMD method is illustrated: first for ‘historical’ applications, like the study of transport properties in the linear and non-linear regimes, including a divertissement on the calculation of time correlation functions in the Gran Canonical ensemble. We finally briefly illustrate D-NEMD applications to the study of problems arising in key hydrodynamic processes such as the formation of convective cells and the relaxation of a non-uniform density profile in a fluid, giving attention to the problem of sampling the conditional probability density ensemble associated with the initial states.

Generating random variates for stable sub-Gaussian processes with memory

We present a computationally efficient method to generate random variables from a univariate conditional probability density function (PDF) derived from a multivariate α-sub-Gaussian (αSG) distribution. The approach may be used to sequentially generate variates for sliding-window models that constrain immediately adjacent samples to be αSG random vectors. We initially derive and establish various properties of the conditional PDF and show it to be equivalent to a Student's t-distribution in an asymptotic sense. As the αSG PDF does not exist in closed form, we use these insights to develop a method based on the rejection sampling (accept-reject) algorithm that allows generating random variates with computational ease.

The electron localization as the information content of the conditional pair density.

In the present work, the information gained by an electron for "knowing" about the position of another electron with the same spin is calculated using the Kullback-Leibler divergence (DKL) between the same-spin conditional pair probability density and the marginal probability. DKL is proposed as an electron localization measurement, based on the observation that regions of the space with high information gain can be associated with strong correlated localized electrons. Taking into consideration the scaling of DKL with the number of σ-spin electrons of a system (N(σ)), the quantity χ = (N(σ) - 1) DKLfcut is introduced as a general descriptor that allows the quantification of the electron localization in the space. fcut is defined such that it goes smoothly to zero for negligible densities. χ is computed for a selection of atomic and molecular systems in order to test its capability to determine the region in space where electrons are localized. As a general conclusion, χ is able to explain the electron structure of molecules on the basis of chemical grounds with a high degree of success and to produce a clear differentiation of the localization of electrons that can be traced to the fluctuation in the average number of electrons in these regions.

A Multi-view Learning Approach to Foreground Detection for Traffic Surveillance Applications

This paper proposes an effective multi-view learning approach to foreground detection for traffic surveillance applications. This approach involves three main steps. First, a reference background image is generated via temporal median filtering, and multiple heterogeneous features (including brightness variation, chromaticity variation, and texture variation, each of which represents a unique view) are extracted from the video sequence. Then, a multi-view learning strategy is devised to online estimate the conditional probability densities for both the foreground and the background. The probability densities of three features are approximately conditionally independent and are estimated with kernel density estimation. Pixel soft labeling is conducted by using Bayes rule, and the pixelwise foreground posteriors are computed. Finally, a Markov random field is constructed to incorporate the spatiotemporal context into the foreground/background decision model. The belief propagation algorithm is used to label each pixel of the current frame. Experimental results verify that the proposed approach is effective to detect foreground objects from challenging traffic environments and outperforms some state-of-the-art methods.

Prediction of reliability of the corroded pipeline considering the randomness of corrosion damage and its stochastic growth

Abstract The paper deals with exploration of the probabilistic characteristics of the deformed state and stress concentration of a pipeline section with corrosion damage considering its random spatial shape. The investigation was done based on 3D FE models. A special algorithm is developed for the direct solid modeling of the corroded area in the upper surface of the pipe. A series of 3D models of a pipe with a volumetric surface defect, which has a random shape, is created based on the developed algorithm. An analysis of a probable stress concentration values for different statistical parameters of corrosion defect was carried out in the frame work of Monte-Carlo simulations. The statistical analysis of the system response allowed us to obtain the conditional probability density functions of the maximum stress concentration variation. It was done taking into account the level of the maximum corrosion depth. This information together with statistical data on the corrosion rate obtained from the references gave us a possibility to propose a method to estimate a reliability of a corroded pipe in operation.

Technical Note: Statistical dependences between channels in radiochromic film readings. Implications in multichannel dosimetry.

This note studies the statistical relationships between color channels in radiochromic film readings with flatbed scanners. The same relationships are studied for noise. Finally, their implications for multichannel film dosimetry are discussed. Radiochromic films exposed to wedged fields of 6 MV energy were read in a flatbed scanner. The joint histograms of pairs of color channels were used to obtain the joint and conditional probability density functions between channels. Then, the conditional expectations and variances of one channel given another channel were obtained. Noise was extracted from film readings by means of a multiresolution analysis. Two different dose ranges were analyzed, the first one ranging from 112 to 473 cGy and the second one from 52 to 1290 cGy. For the smallest dose range, the conditional expectations of one channel given another channel can be approximated by linear functions, while the conditional variances are fairly constant. The slopes of the linear relationships between channels can be used to simplify the expression that estimates the dose by means of the multichannel method. The slopes of the linear relationships between each channel and the red one can also be interpreted as weights in the final contribution to dose estimation. However, for the largest dose range, the conditional expectations of one channel given another channel are no longer linear functions. Finally, noises in different channels were found to correlate weakly. Signals present in different channels of radiochromic film readings show a strong statistical dependence. By contrast, noise correlates weakly between channels. For the smallest dose range analyzed, the linear behavior between the conditional expectation of one channel given another channel can be used to simplify calculations in multichannel film dosimetry.