Conditional Probability Density Research Articles

Predicting the probability distribution of bus travel time to measure the reliability of public transport services

An important aspect of the quality of a public transport service is its reliability, which is defined as the invariability of the service attributes. In order to measure reliability during the service planning phase, a key piece of information is the long-term prediction of the density of the travel time, which conveys the uncertainty of travel times. This work empirically compares probabilistic models for the prediction of the conditional probability density function (PDF) of the travel time and proposes a simulation framework taking as input the latter distributions to approximate the expected secondary delays, a measure of the reliability of public transport services. Two types of probabilistic models, namely similarity-based density estimation models and a smoothed logistic regression for probabilistic classification model, are compared on a dataset of more than 41,000 trips and 50 bus routes of the city of Montréal. A similarity-based density estimation model using a k-nearest neighbors method and a log-logistic distribution predicted the best estimate of the true conditional PDF of the travel time and generated the most accurate approximations of the expected secondary delays on this dataset. This model reduced the mean squared error of the expected secondary delay by approximately 9% compared to the benchmark model, namely a random forest. This result highlights the added value of modeling the conditional PDF of the travel time with probabilistic models.

Bayesian‐MLE signal detection for multi‐antenna ambient backscatter communication

Multi-antenna signal detection is one of the most critical and challenging issues for ambient backscatter communication (AmBC) systems. This paper proposes an efficient multi-antenna AmBC signal detection method, called Bayesian-MLE (maximum likelihood estimation). It shows good performance on high transmission rate, detection accuracy and low energy consumption. Particularly, a practical multi-antenna AmBC system model is developed to offer transmit-receive diversity, and then an efficient multi-antenna AmBC signal detection method is presented using Bayesian optimization and MLE theory. Furthermore, to maximize the detection performance, an optimal detection threshold selection scheme is developed. Particularly, a non-central chi-square distribution conditional probability density function (PDF) is considered instead of the conventional Gaussian PDF. Extensive qualitative and quantitative experiments are performed, showing that the proposed Bayesian-MLE detector achieves the state-of-the-art signal detection performance.

Optimal input signal distribution for a nonlinear optical fiber channel with small Kerr nonlinearity

We consider the information channel described by the Schrödinger equation with additive Gaussian noise. We introduce the model of the input signal and the model of the output signal receiver. For this channel, using perturbation theory for the small nonlinearity parameter, we calculate the first three terms of the expansion of the conditional probability density function in the nonlinearity parameter. At a large signal-to-noise power ratio, we calculate the conditional entropy, output signal entropy, and mutual information in the leading and next-to-leading order in the nonlinearity parameter and in the leading order in the parameter 1/SNR. Using the mutual information, we find the optimal input signal distribution and channel capacity in the leading and next-to-leading order in the nonlinearity parameter. Finally, we present the method of the construction of the input signal with the optimal statistics for the given shape of the signal.

Radar Cross Section Based Statistical Recognition of UAVs at Microwave Frequencies

This article presents a radar cross-section (RCS)-based statistical recognition system for identifying/classifying unmanned aerial vehicles (UAVs) at microwave frequencies. First, the article presents the results of the vertical (VV) and horizontal (HH) polarization RCS measurement of six commercial UAVs at 15 and 25 GHz in a compact range anechoic chamber. The measurement results show that the average RCS of the UAVs depends on shape, size, material composition of the target UAV as well as the azimuth angle, frequency, and polarization of the illuminating radar. Afterward, radar characterization of the target UAVs is achieved by fitting the RCS measurement data to 11 different statistical models. From the model selection analysis, we observe that the lognormal, generalized extreme value, and gamma distributions are most suitable for modeling the RCS of the commercial UAVs while the Gaussian distribution performed relatively poorly. The best UAV radar statistics forms the class conditional probability densities for the proposed UAV statistical recognition system. The performance of the UAV statistical recognition system is evaluated at different signal noise ratio (SNR) with the aid of Monte Carlo analysis. At an SNR of 10 dB, the average classification accuracy of 97.60% or better is achievable.

Kernel learning backward SDE filter for data assimilation

In this paper, we develop a kernel learning backward SDE filter method to estimate the state of a stochastic dynamical system based on its partial noisy observations. A system of forward backward stochastic differential equations is used to propagate the state of the target dynamical model, and Bayesian inference is applied to incorporate the observational information. To characterize the dynamical model in the entire state space, we introduce a kernel learning method to learn a continuous global approximation for the conditional probability density function of the target state by using discrete approximated density values as training data. Numerical experiments demonstrate that the kernel learning backward SDE is highly effective.

A novel method for estimating utility harmonic impedance based probabilistic evaluation

Here, a novel method for estimating utility harmonic impedance (UHI) based on probabilistic evaluation is proposed. By analysing harmonic data, it is determined that UHI tends to obey a particular distribution, that varies with the amplitude of the background harmonics. Therefore, calculating the probability distribution of the UHI is considered to estimate its value. The UHI can be estimated by calculating the conditional probability density function of the background harmonics and prior distribution of UHI. The estimation results are not affected by the correlation of harmonic impedances between the two sides of the point of common coupling. The validity and accuracy of the proposed method were further verified through simulations and the analysis of field testing data.

A new gamma degradation process with random effect and state-dependent measurement error

In this paper, a new gamma-based degradation process with random effect is proposed that allows to account for the presence of measurement error that depends in stochastic sense on the measured degradation level. This new model extends a perturbed gamma model recently suggested in the literature, by allowing for the presence of a unit to unit variability. As the original one, the extended model is not mathematically tractable. The main features of the proposed model are illustrated. Maximum likelihood estimation of its parameters from perturbed degradation measurements is addressed. The likelihood function is formulated. Hence, a new maximization procedure that combines a particle filter and an expectation-maximization algorithm is suggested that allows to overcome the numerical issues posed by its direct maximization. Moreover, a simple algorithm based on the same particle filter method is also described that allows to compute the cumulative distribution function of the remaining useful life and the conditional probability density function of the hidden degradation level, given the past noisy measurements. Finally, two numerical applications are developed where the model parameters are estimated from two sets of perturbed degradation measurements of carbon-film resistors and fuel cell membranes. In the first example the presence of random effect is statistically significant while in the second example it is not significant. In the applications, the presence of random effect is checked via appropriate statistical procedures. In both the examples, the influence of accounting for the presence of random effect on the estimates of the cumulative distribution function of the remaining useful life of the considered units is also discussed. Obtained results demonstrate the affordability of the proposed approach and the usefulness of the proposed model.

Semi and Nonparametric Conditional Probability Density, a Case Study of Pedestrian Crashes

Background: Kernel-based methods have gained popularity as employed model residual’s distribution might not be defined by any classical parametric distribution. Kernel-based method has been extended to estimate conditional densities instead of conditional distributions when data incorporate both discrete and continuous attributes. The method often has been based on smoothing parameters to use optimal values for various attributes. Thus, in case of an explanatory variable being independent of the dependent variable, that attribute would be dropped in the nonparametric method by assigning a large smoothing parameter, giving them uniform distributions so their variances to the model’s variance would be minimal. Objectives: The objective of this study was to identify factors to the severity of pedestrian crashes based on an unbiased method. Especially, this study was conducted to evaluate the applicability of kernel-based techniques of semi- and nonparametric methods on the crash dataset by means of confusion techniques. Methods: In this study, two non- and semi-parametric kernel-based methods were implemented to model the severity of pedestrian crashes. The estimation of the semi-parametric densities is based on the adoptive local smoothing and maximization of the quasi-likelihood function, which is similar somehow to the likelihood of the binary logit model. On the other hand, the nonparametric method is based on the selection of optimal smoothing parameters in estimation of the conditional probability density function to minimize mean integrated squared error (MISE). The performances of those models are evaluated by their prediction power. To have a benchmark for comparison, the standard logistic regression was also employed. Although those methods have been employed in other fields, this is one of the earliest studies that employed those techniques in the context of traffic safety. Results: The results highlighted that the nonparametric kernel-based method outperforms the semi-parametric (single-index model) and the standard logit model based on the confusion matrices. To have a vision about the bandwidth selection method for removal of the irrelevant attributes in nonparametric approach, we added some noisy predictors to the models and a comparison was made. Extensive discussion has been made in the content of this study regarding the methodological approach of the models. Conclusion: To summarize, alcohol and drug involvement, driving on non-level grade, and bad lighting conditions are some of the factors that increase the likelihood of pedestrian crash severity. This is one of the earliest studies that implemented the methods in the context of transportation problems. The nonparametric method is especially recommended to be used in the field of traffic safety when there are uncertainties regarding the importance of predictors as the technique would automatically drop unimportant predictors.

Effect of heat source on statistics and scaling in compressible homogeneous shear turbulence

The effects of heat sources on the velocity and pressure spectra, Mach number scaling of one-point statistics, and small-scale structures of compressible homogeneous shear turbulence are numerically studied. The dilatational components of flow fields are significantly enhanced by a strong heat source at low turbulent Mach numbers Mt and are dominated by an acoustic mode, leading to a strong acoustic equilibrium between the dilatational velocity and pressure. As the magnitude of the heat source increases, the scaling behaviors of the dilatational components of kinetic energy and dissipation rate change from Mt4 and approach a state that is nearly independent of the turbulent Mach number. Furthermore, a strong heat source has a significant effect on small-scale structures at low turbulent Mach numbers. The conditional probability density functions of the normalized eigenvalues of a strain rate tensor become more dependent on the dilatation owing to the effect of the heat source. For low turbulent Mach numbers with strong heat sources, the ratio of the normalized eigenvalues of the strain rate tensor tends to −1:0:0 and −0.2:0.25:1 in the strong compression and strong expansion regions, respectively, and the dilatational vortex stretching term can significantly enhance the enstrophy production.

Efficient Moment-Independent Sensitivity Analysis of Uncertainties in Seismic Demand of Bridges Based on a Novel Four-Point-Estimate Method

Moment-independent importance (MII) analysis is known as a global sensitivity measurement in qualifying the influence of uncertainties, which is taken as a crucial step towards seismic performance analysis. Most MII analysis is based on Monte Carlo simulation, which leads to a high computational cost since a large number of nonlinear time history analyses are required to obtain the probability density function. To address this limitation, this study presents a computational efficient MII analysis to investigate the uncertain parameters in the seismic demands of bridges. A modified four-point-estimate method is derived from Rosenblueth’s two-point-estimate method. Thus, the statistical moments of a bridge’s seismic demands can be obtained by several sampling points and their weights. Then, the shifted generalized lognormal distribution method is adopted to estimate the unconditional and conditional probability density functions of seismic demands, which are used for the MII analysis. The analysis of seismic demands based on piers and bearings in a finite element model of a continuous girder bridge is taken as a validation example. The MII measures of the uncertain parameters are estimated by just several nonlinear time history analyses at the point-estimate sampling points, and the results by the proposed method are compared with those found by Monte Carlo simulation.

Bayesian Approach for Multigamma Radionuclide Quantification Applied on Weakly Attenuating Nuclear Waste Drums

Gamma spectrometry is a passive nondestructive assay method used to quantify radionuclides present in nuclear objects. Basic methods using empirical calibration with a standard to quantify the activity of nuclear materials by determining the calibration coefficient are ineffective on nonreproducible nuclear objects such as waste packages. Package specifications such as composition or geometry change from one package to another and exhibit large variability of objects. The current standard quantification process uses numerical modeling of the measured scene with few available data such as geometry or composition, in particular density, material, screen, geometric shape, matrix composition, matrix, and source distribution. Some of them are strongly dependent on package data knowledge and operator backgrounds. The French Atomic Energy Commission (CEA) is developing a methodology to quantify nuclear materials in waste packages and waste drums without operator adjustment and internal package configuration knowledge. This method suggests combining a stochastic approach which uses, among others, surrogate models available to simulate the gamma attenuation behavior, a Bayesian approach considering conditional probability densities and prior information of problem inputs, and Markov Chain Monte Carlo (MCMC) algorithms which solve inverse problems, with gamma ray emission radionuclide spectra, and the outside dimensions of the objects of interest. The methodology has been tested to quantify actinide activity with a low bulk density matrix, weakly attenuating compositions, without information on the distribution of the source in terms of actinide masses and materials composing the drums. Activity uncertainties are taken into account.

Approximations of conditional probability density functions in Lebesgue spaces via mixture of experts models

Mixture of experts (MoE) models are widely applied for conditional probability density estimation problems. We demonstrate the richness of the class of MoE models by proving denseness results in Lebesgue spaces, when inputs and outputs variables are both compactly supported. We further prove an almost uniform convergence result when the input is univariate. Auxiliary lemmas are proved regarding the richness of the soft-max gating function class, and their relationships to the class of Gaussian gating functions.

A framework for fatigue life prediction of materials under the multi-level cyclic loading

Fatigue life prediction of materials under the cyclic loading is still a particularly challenging task remains to be resolved. This study aims to develop a generic framework for fatigue life prediction under the multi-level cyclic (MLC) loading with consideration of the loading sequence effects. Firstly, a conditional probability density function (PDF) model is presented to quantify the fatigue life distributions of materials under any constant amplitude cyclic (CAC) loading. Subsequently, a fatigue damage accumulation model is proposed from the perspective of cumulative failure probability, which is capable of accounting for the nonlinear characteristics of damage accumulation and the loading sequence effects. Finally, a generic framework for fatigue life prediction of materials under the MLC loading is developed based on the proposed models. The fatigue life data of fiber reinforced composites under the two-level and three-level cyclic loading are utilized to verify the proposed framework. The results show that the predicted fatigue lives agree well with the fatigue test data of fiber reinforced composites. Moreover, the proposed framework can be easily extended to calculate fatigue life of materials under any MLC loading.

Interval estimation of conditional probabilities in Bayesian Belief Network

An interval estimation of conditional probabilities in Bayesian belief network is considered, taking into account the uncertainty of the initial data. To account for errors in the values of functions and arguments, it is proposed to use confluent analysis methods. It is assumed that the conditional probability densities correspond to the normal distribution law. Formulas for interval estimation of conditional probability densities are given, taking into account the errors of their parameters. The results of mathematical modeling of obtaining point and interval estimates of conditional and a posteriori probabilities of the Bayesian belief network are shown.

Short and Medium Range Irrigation Scheduling Using Stochastic Simulation‐Optimization Framework With Farm‐Scale Ecohydrological Model and Weather Forecasts

AbstractDespite the remarkable improvements in skillful weather forecasts, their uses in irrigation water management at a farm scale are still limited. This is attributable to the scale mismatch between weather and hydrologic models as well complexities in farmscale ecohydrological processes. Here, we have developed a simulation‐optimization algorithm for minimizing irrigation water application using short to medium range forecasts by determining the conditional probability density functions of the rainfall and subsequently the soil moisture for the days in forecast range. With the forecasted soil moisture information, the model ensures that the probability of crops undergoing water stress is less than a prescribed threshold. The optimization model includes a farmscale ecohydrological model, which computes daily evapotranspiration, runoff, and leakage based on a probabilistic description of rainfall through Monte‐Carlo simulations. After calibrating and validating the ecohydrological model with past data obtained from the farmers, this proposed optimization framework was employed to test the outcome for two pilot sites in Nashik, Maharashtra, India. We found that using the proposed framework, irrigation water use can be reduced by 10%–30% as compared to that resulting from the conventional strategies used by the farmers, without significant loss in crop yields, by almost always maintaining the soil moisture at or above the prescribed threshold. Considering that irrigation accounts for over 80% of the total water use worldwide, the value of such an approach as a decision‐support tool for irrigation optimization is self‐evident. We also posit that the co‐production of this tool with the farmers increases its usability and credibility.

Computational aspects of the kNN local linear smoothing for some conditional models in high dimensional statistics

We consider the problem of nonparametric estimation of certain conditional models when the explanatory variable is functional. The models studied are the Conditional Expectation (CE), the Conditional Distribution Function (CDF) and the Conditional Probability Density (CPD). The estimators are constructed by combining the local linear method and the k-Nearest Neighbors (kNN) smoothing approach. For each of the three models, we define an optimal estimator associated with the best number of neighbors chosen using two bandwidth selection procedures which are the cross-validation criterion and the bootstrap smoothing rule. The asymptotic properties of these optimal kNN Local Linear Estimators ( NN- LLE) are established under some standard conditions. The performances of the finite samples of these estimators are then examined and compared through a Monte-carlo study. In addition, an application on the riboflavin content in the yogurt using the Near-infrared curves is carried out to demonstrate the usefulness of the models proposed in the point-wise prediction as well as for the predictive interval.

An efficient method by nesting adaptive Kriging into Importance Sampling for failure-probability-based global sensitivity analysis

Failure-probability-based global sensitivity (FP-GS) analysis can measure the effect of the input uncertainty on the failure probability. The state-of-the-art for estimating the FP-GS are less efficient for the rare failure event and the implicit performance function case. Thus, an adaptive Kriging nested Importance Sampling (AK-IS) method is proposed in this work to efficiently estimate the FP-GS. For eliminating the dimensionality dependence in the calculation, an equivalent form of the FP-GS transformed by the Bayes’ formula is employed by the proposed method. Then the AK model is nested into IS for recognizing the failure samples. After all the failure samples are correctly identified from the IS sample pool, the failure samples are transformed into those subjected to the original conditional probability density function (PDF) on the failure domain by the Metropolis–Hastings algorithm, on which the conditional PDF of the input on the failure domain can be estimated for the FP-GS finally. The proposed method highly improves the efficiency of estimating the FP-GS comparing with the state-of-the-art, which is illustrated by the results of several examples in this paper.

Analysis of rain fade characteristics from experimental satellite measurements at Ka/Q bands for a temperate location Vigo, Spain

Rain is one of the most dominant factors for satellite communication systems link performance at frequencies above 10 GHz, increasing rapidly at higher frequencies. The prediction of time dynamics of the rain attenuation is required to improve the efficiency of various fade mitigation techniques (FMTs) to reduce signal fading. Fade duration and fade slope statistics are used to describe the dynamic behavior of attenuation experienced by radio communication links. The present study is based on experimental measurements from a Ka band (KaSAT) and Q band (AlphaSAT) beacon receiver at the frequencies 19. 7 GHz and 39. 4 GHz respectively. In this paper, annual and monthly variation of rain attenuation statistics for a temperate location Vigo, Spain is presented. Monthly variation of rain attenuation statistics is much larger than the yearly variation for the present location. This paper also analyzes the fade duration and fade slope statistics for both the bands. The conditional probability density function (PDF) of fade slope shows general symmetry between positive and negative slope values for both Ka and Q bands. All the measured statistics (1st order and 2nd order) are compared with the existing ITU-R and COST-205 models. Predictions of rain attenuation by both the models are close to the measured data. For fade duration statistics, as attenuation threshold increases, performance of both ITU-R and COST-205 models improves for Ka band. For Q band, COST model performs better compared to ITU-R model. Measured probability density function of fade slope fits with the ITU-R model more closely for Q band compared to Ka band.

Are basic actors brainbound agents? Narrowing down solutions to the problem of probabilistic content for predictive perceivers

Clark (Journal of Consciousness Studies, 25(3–4), 71–87, 2018) worries that predictive processing (PP) accounts of perception introduce a puzzling disconnect between the content of personal-level perceptual states and their underlying subpersonal representations. According to PP, in perception, the brain encodes information about the environment in conditional probability density distributions over causes of sensory input. But it seems perceptual experience only presents us with one way the world is at a time. If perception is at bottom probabilistic, shouldn’t this aspect of subpersonally represented content show up in consciousness? To address this worry, Clark argues that representations underlying personal-level content are constrained by the need to provide a single action-guiding take on the environment. However, this proposal rests a conception of the nature of agency, famously articulated by Davidson (1980a, b), that is inconsistent with a view of the mind as embodied-extended. Since Clark and other enactivist PP theorists present the extended mind as an important consequence of the predictive framework, the proposal is in tension with his complete view. I claim that this inconsistency could be resolved either by retaining the Davidsonian view of action and abandoning the extended-embodied approach, or by adopting a more processual, world-involving account of agency and perceptual experience than Clark currently endorses. To solve the puzzle he raises, Clark must become a radical enactivist or a consistent internalist.

The single-loop Kriging model combined with Bayes’ formula for time-dependent failure probability based global sensitivity

Time-dependent failure probability based global sensitivity (TDFP-GS) analysis quantifies the effect of input uncertainty on the failure probability for the time-dependent structure, which is usually time-consuming to be performed. For alleviating this issue, a method by combining the single-loop Kriging (SILK) model with Bayes’ formula is proposed in this work. After transformed by Bayes’ formula, the estimation of the conditional failure probability in the TDFP-GS expression is converted to that of conditional probability density function (PDF) of the input on the failure domain, which eliminates the dimensionality dependence in the calculation process. The SILK model is established to estimate the failure probability of the structure. When all the failure samples are accurately recognized by the converged SILK model, the conditional PDF can be estimated by the kernel density estimation (KDE), which doesn’t require any performance function evaluation. The TDFP-GS can be estimated by the difference between the original PDF and the conditional PDF of the input on the failure domain. Four examples are presented for validating the effectiveness and the efficiency of the proposed method.