Monte Carlo Integration Method Research Articles

Investigation of the functional relationship between antecedent rainfall and the probability of debris flow occurrence in Jiangjia Gully, China

Abstract. A larger antecedent effective precipitation (AEP) indicates a higher probability of a debris flow (Pdf) being triggered by subsequent rainfall. Scientific topics surrounding this qualitative conclusion that can be raised include what kinds of variation rules they follow and whether there is a boundary limit. To answer these questions, Jiangjia Gully in Dongchuan, Yunnan Province, China, is chosen as the study area, and numerical calculation, a rainfall scenario simulation, and the Monte Carlo integration method have been used to calculate the occurrence probability of debris flow under different AEP conditions and derive the functional relationship between Pdf and AEP. The relationship between Pdf and AEP can be quantified by a piecewise function. Pdf is equal to 15.88 %, even when AEP reaches 85 mm, indicating that debris flow by nature has an extremely small probability compared to the rainfall frequency. Data from 1094 rainfall events and 37 historical debris flow events are collected to verify the reasonability of the functional relationship. The results indicate that the piecewise functions are highly correlated with the observation results. Our study confirms the correctness of the qualitative description of the relationship between AEP and Pdf, clarifies that debris flow is a small-probability event compared to rainfall frequency, and quantitatively reveals the evolution law of debris flow occurrence probability with AEP. All the above discoveries can provide a clear reference for the early warning of debris flows.

ALGORITHM FOR RECOGNITION TECHNOLOGY OF ACOUSTIC SIGNALS

An algorithm for processing the reflected acoustic signal for the technology of object recognition against the background of noise interference is presented. The considered mathematical model takes into account that the input data (acoustic signals) have a previously unknown probability distribution. To build an acoustic signal processing technology the Markov chain based probabilistic models, the Monte Carlo integration method and the Bootstrap filter method built on it were used. Based on them, an algorithm for processing an acoustic signal reflected from obstacles was created. The signal processing technology implements the transition from the input data to the representation of the input data in the form of a grid, where a characteristic-intensity normally distributed according to the central limit theorem is assigned to each cell. A likelihood function is used to detect noise at a random point. The algorithm is ready for practical use, which is confirmed by examples. The result of the algorithm application for the recognition of two moving objects, which have a reflected acoustic signal close to the noise, is graphically presented. The dependence of the processing speed on the amount of data is noted. Several constructive approaches to solving this problem are given. It is noted that by using the methods of storing sparse matrices the estimate of the memory used is reduced from quadratic to linear with deterioration of the speed estimate by a logarithm.

A numerical optimization approach for pricing components in customer defined bundles in a B2B market

This research examines the problem of pricing products in multi-product, multi-quantity bundles in a business-to-business market setting in which customers’ valuation of products diminishes with the size of the bundle. The framework developed in this paper consists of three steps: (i) estimation of diminished valuation of each product based on their stand-alone valuations and bundle composition, (ii) maximization of expected profit of the firm to obtain the optimal bundle price, and (iii) maximization of customer satisfaction to obtain the optimal split of the bundle price into component prices. A Nelder–Mead optimization based method is proposed as a solution approach, which leverages an importance sampling based Monte Carlo integration method to approximate the quantities of interests in the model. The framework is also used to analyze the behavior of the model for quantity increase. Results indicate that the price per unit of each product decreases when the quantity of any of the products is increased. However, the rate of decrement is the least for the product whose quantity is incremented. Additional results show that fixing prices (and not re-optimizing), when the bundle composition changes, can significantly reduce the firm’s profit.

Ronkin function and nD-BIBO stability test: a Monte-Carlo algorithm

We relate the BIBO stability of a rational nD-system with irreducible transfer function G(z1, ..., zn)/F(z1,..., zn), with the geometry of the amœba of F. The amœba of a polynomial is the image of its zero set under the logarithmic modulus map. A BIBO stability criterion is derived from the Ronkin function of F, which characterizes the complement of the amœba. The criterion is then numerically implemented using a Monte-Carlo integration method. The resulting BIBO stability test does not stuck from the curse of dimensionality, it can be applied for large n. The test is illustrated through an example with n = 10.

PARS - Path recycling and sorting for efficient cloud tomography

Inverse rendering estimates scene characteristics from image data. We derive an efficient framework for inverse rendering and specifically computed tomography (CT) of volumetric scattering objects. We focus on clouds, which have a key role in the climate system and require efficient analysis at a huge scale. Data for such reconstruction are multiview images of each cloud taken simultaneously. This acquisition mode is expected by upcoming future spaceborne imagers, such as CloudCT. Prior art shows that scattering CT can rely on Monte–Carlo (MC) light transport. This approach usually iterates differentiable radiative transfer, requiring many sampled paths per iteration. We present an acceleration approach: path recycling and sorting (PARS). It efficiently uses paths from previous iterations for estimating a loss gradient at the current iteration. This reduces the iteration run time. PARS enables further efficient realizations. Specifically, sorting paths according to their size accelerates implementations on a graphical processing unit (GPU). PARS, however, requires a correction operation for unbiased gradient estimation. This can be achieved by utilizing a well-established concept from MC integration methods, as we show in this paper. We derive the theory of PARS and demonstrate its efficiency on cloud tomography of both synthetic and real-world scenes. Moreover, we demonstrate PARS on simple reflectometry examples.

A Novel Ice Cloud Retrieval Algorithm for Submillimeter Wave Radiometers: Simulations and Application to an Airborne Experiment

A retrieval methodology based on the Bayesian neural network (BNN) is presented that inverts the ice water path (IWP), mean mass-weighted diameter (D <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">me</sub> ), and cloud height of ice clouds from sub-millimeter radiometer observations. The training dataset was created using collecting cloud profiles from the DARDAR (raDAR/liDAR) database and running simulations by the Atmospheric Radiative Transfer Simulator (ARTS) model. Since the effective radius (r <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> ) is the size descriptor of ice particles in the DARDAR database, a look-up table of ice water content (IWC), D <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">me</sub> , and r <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> was constructed to convert r <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> profiles into D <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">me</sub> profiles. In addition, random noises corresponding to the measurement uncertainties of the Compact Scanning Submillimeter-wave Imaging Radiometer (CoSSIR) during the TC4 experiment were added to the simulated brightness temperatures before training the BNN. The proposed retrieval method was first applied to the simulated testing database, and then to the observations of CoSSIR. Moreover, the retrieved IWP and D <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">me</sub> were compared to the retrievals of the Bayesian Monte Carlo Integration (BMCI) method. The retrieved cloud height was assessed by cloud height extracted from the reflectivity data of the Cloud Radar System (CPS) flow on the same aircraft with CoSSIR. The comparison showed that the correlation coefficients of the retrieved IWP and D <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">me</sub> from the two methods are above 0.8, and the retrieved cloud height also showed good agreement with that extracted from the CPS.

A physics-informed learning approach to Bernoulli-type free boundary problems

• A PINN methodology for a Bernoulli free boundary variational problem is proposed. • Quasi Monte-Carlo integration methods are used in the learning process of the PINN. • A redistribution strategy of domain points based on solution's Laplacian is developed. • Analysis on convex and non-convex domains assess the PINN reliability. Bernoulli free boundary problems (BFBP) govern many real applications, from chemistry to fluid dynamics. BFBP are overdetermined differential models in which the boundary of the solution's domain appears as an unknown. This work provides a variational formulation for BFBP, and a computational method focused on the novel methodology of Physics Informed Neural Networks (PINNs) is proposed. It consists in training a neural network to approximate the solution of the differential problem, minimizing a suitable cost function built, taking into account the physics constraint given by the model. In particular, since physics laws are injected through a loss function containing integral terms in our case, an approach based on quasi Monte-Carlo integration methods has been implemented. Moreover, an adaptive strategy to increase the model accuracy, exploiting topological information related to the shape of the solution, has been developed. Finally, when available, comparisons with the analytical solutions and studies in the case of non-convex domains assess the reliability of the PINN approach in the examined context.

Configurational entropy of colloidal particles in a confined space.

We calculate the configurational entropy of colloidal particles in a confined geometry interacting as hard disks using Monte Carlo integration. In particular, we consider systems with three kinds of boundary conditions: hard, periodic, and spherical. For small to moderate packing fraction ϕ values, we find the entropies per particle for systems with the periodic and the spherical boundary conditions tend to reach the same value with the increase of the particle number N, while that for the system with the hard boundary conditions still has obvious differences compared to them within the studied N range. Surprisingly, despite the small system sizes, the estimated entropies per particle at infinite system size from extrapolations in the periodic and spherical systems are in reasonable agreement with that calculated using thermodynamic integration. Besides, as N increases we find the pair correlation function begins to exhibit similar features as that of a thermally equilibrated hard-disk fluid at the same packing fraction. Our findings may contribute to a better understanding of how the configurational entropy changes with the system size and the influence of boundary conditions, and provide insights relevant to engineering particles in confined spaces.

Type-II progressive censoring with GLM-based random removal mechanism dependent on the experimental conditions

This article presents a novel stochastic removal mechanism under Type-II progressive random censoring in which removal probabilities are allowed to be dependent on the lifetime conditions through Generalized Linear Models (GLM). These conditions potentially include failure distances (the time required to observe the next failure) or other covariate information available in the experiment. The proposed GLM-based random removal mechanism includes a set of tuning parameters that are determined by the researcher according to the possible failure distance category. These parameters allow flexible determination of the removal probabilities leading to necessary experimental cost and time reductions. To establish the proposed mechanism, the Proportional Hazard Rate (PHR) family of distributions is considered. Also, the maximum likelihood estimators of parameters and their asymptotic variances are derived for the Weibull distributed lifetime data. A simple simulation algorithm for generating Type-II progressive censoring samples with GLM-based dependent removal probabilities is also presented. The expected experiment time required to complete the life test under this censoring scheme is also investigated using the Monte Carlo integration method. Several simulation studies are conducted to evaluate and compare the performance of the proposed mechanism. A sensitivity analysis is also considered to study the effect of misspecification of removal mechanism coefficients. Finally, two real data sets are analyzed for illustrative purposes.

Sparse Bayesian deep learning for dynamic system identification

This paper proposes a sparse Bayesian treatment of deep neural networks (DNNs) for system identification. Although DNNs show impressive approximation ability in various fields, several challenges still exist for system identification problems. First, DNNs are known to be too complex that they can easily overfit the training data. Second, the selection of the input regressors for system identification is nontrivial. Third, uncertainty quantification of the model parameters and predictions are necessary. The proposed Bayesian approach offers a principled way to alleviate the above challenges by marginal likelihood/model evidence approximation and structured group sparsity-inducing priors construction. The identification algorithm is derived as an iterative regularised optimisation procedure that can be solved as efficiently as training typical DNNs. Remarkably, an efficient and recursive Hessian calculation method for each layer of DNNs is developed, turning the intractable training/optimisation process into a tractable one. Furthermore, a practical calculation approach based on the Monte-Carlo integration method is derived to quantify the uncertainty of the parameters and predictions. The effectiveness of the proposed Bayesian approach is demonstrated on several linear and nonlinear system identification benchmarks by achieving good and competitive simulation accuracy. The code to reproduce the experimental results is open-sourced and available online.

Single‐pass stratified importance resampling

AbstractResampling is the process of selecting from a set of candidate samples to achieve a distribution (approximately) proportional to a desired target. Recent work has revisited its application to Monte Carlo integration, yielding powerful and practical importance sampling methods. One drawback of existing resampling methods is that they cannot generate stratified samples. We propose two complementary techniques to achieve efficient stratified resampling. We first introduce bidirectional CDF sampling which yields the same result as conventional inverse CDF sampling but in a single pass over the candidates, without needing to store them, similarly to reservoir sampling. We then order the candidates along a space‐filling curve to ensure that stratified CDF sampling of candidate indices yields stratified samples in the integration domain. We showcase our method on various resampling‐based rendering problems.

Analysis of UV Multiple Scattering Based on Improved MCI-PIS Model

Wireless ultraviolet communication uses the atmospheric scattering effect for signal transmission, and this paper improves the partially important sampling model (MCI-PIS) in the Monte Carlo integration method, and the improved model is used to simulate the influence of multiple system parameters such as communication distance, transceiver tilt angle, transceiver divergence angle on the path loss and impulse response of ultraviolet light communication.The simulation results show that the path loss of the improved model for multiple scattering of ultraviolet light is increased by 0.2-2 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dB</i> , and the impulse response is almost the same as that of MCI-PIS, but the calculation effect is increased by 20% compared with the MCI-PIS model.

New Monte Carlo Integration Models for Underwater Wireless Optical Communication

Underwater wireless optical communications are knowing a high interest in recent years, with many military and civil applications. The design of efficient systems relies on powerful simulation tools. The most used simulation algorithms are based on a paper from 1989, written by Prahl and designed for light propagation through human tissue. This method relies on Monte Carlo Simulation, following the path of a photon in the participating media. As such, it is difficult to propose evolution or optimization. In addition, a large amount of photons are needed to obtain good results with low error, <i>e.g</i>. for turbid water. With Monte Carlo Integration method, there exist a lot of optimization techniques to reduce the variance. This paper proposes a mathematical formalization of the propagation of light in water or any other participating medium as an integral problem. Therefore, it opens the way for a large number of future optimizations. A very straightforward and simple variance reduction technique is proposed as an example. Our simulation results show that this new technique has a lower sample variance as expected, and thus better convergence rates.

Artificial stochastic neural network on the base of double quantum wells

In this paper, we consider a model of an artificial neural network based on quantum-mechanical particles in [Formula: see text] potential. These particles play the role of neurons in our model. To simulate such a quantum-mechanical system, the Monte Carlo integration method is used. A form of the self-potential of a particle as well as two interaction potentials (exciting and inhibiting) are proposed. Examples of simplest logical elements (such as AND, OR and NOT) are shown. Further, we show an implementation of the simplest convolutional network in framework of our model.

Anomalous diffusion of cosmic rays: A geometric approach

In this work, we analyze the process leading to the occurrence of anomalous transport phenomena associated with galactic cosmic rays propagating through the interstellar space. The origin of non-conventional transport is found in the dynamics of cosmic rays dominated by long uninterrupted flights interspersed with interactions with magnetized scattering clouds. The process is analyzed via a geometric Monte Carlo model that is able to mimic the intrinsic non-local character of the investigated transport phenomena. Numerical results show the existence of ranges of density and re-emission strengths of the interstellar magnetic clouds leading to heavy-tailed (Lévy α-stable) distributions of the propagating cosmic rays denoting a marked superdiffusive character. The correspondence between the geometric Monte Carlo integration method proposed in this work and the fractional Green’s functions suggests a novel approach to efficiently performing integration in higher dimensional spaces.

Numerical calculation of 5-loop QED contributions to the electron anomalous magnetic moment

Preliminary numerical results of calculating the 5-loop QED contributions to the electron (and muon) anomalous magnetic moment are presented. The results include the total contribution of the 5-loop Feynman diagrams without lepton loops and the contributions of nine gauge-invariant classes that form that set. A discrepancy with known results is revealed. The contributions of the gauge-invariant classes are presented for the first time. The method of the computation is briefly described (with the corresponding references). The calculation is based on the following elements:(i)a subtraction procedure for removing infrared (IR) and ultraviolet (UV) divergences point by point in Feynman parametric representation before integration;(ii)a nonadaptive Monte Carlo integration method that is founded on probability density functions (PDF) that are constructed for each Feynman diagram individually using its combinatorial structure;(iii)a GPU-based numerical integration with the help of the supercomputer “Govorun” (JINR, Dubna, Russia).

Maximum Likelihood and Bayesian Estimation of Repeatability Index: Application of Estimating Ratio of Variance Components

An index of repeatability is constructed to evaluate the relative magnitude of measurement error. This index is constructed as a ratio of two variance components. Estimation of the index is derived under the one-way random effects model. We compare the well-known maximum likelihood estimator to the Bayesian estimation procedure using non-informative prior. Large sample variance of the of the maximum likelihood estimator are obtained using the inverse of Fisher’s information matrix and the delta method. Inference procedure using the. We also construct a test statistic on the equality of two repeatability indices using the Monte Carlo integration and sampling Importance re-sampling method. We illustrate the methodologies on the estimation of the index of repeatability of Gamma-glutamyl-transferase, an enzyme found in many organs all over the human body, with the highest concentrations found in the liver. This enzyme’s level is raised in the blood in most diseases that cause damage to the liver or bile ducts and is considered an essential serum marker for alcohol-related liver disease.

Bayesian optimal designs for cox regression model with random and nonrandom intercept based on type I censored data

In this article, we aim to develop A and D-optimal designs for censored data via Bayesian strategy. In this regard, based on some different sorts of prior distributions for the unknown parameters, the appropriate designs are calculated. In the sequel, we define the random intercept model for the Cox regression model assuming the random intercept distributed normally. It is observed that the likelihood function of the observations cannot be acquired in a closed form. Thus, we obtain the quasi-information matrix based on the quasi-likelihood function. We then, use this matrix to obtain the A and D-optimal designs. It is seen that the Fisher information matrix depends on the unknown parameters as a result the locally optimal designs will be inefficient when the true guess about the parameters is failed. To overcome this problem, we propose the Bayesian strategy to obtain more robust optimal designs. Some numerical techniques such as Monte Carlo integration and numerical optimization methods are implemented to calculate the designs. Further, the generalized equivalence theorem is adapted to confirm the optimality of the proposed designs. All of the numerical results are done with R software. The results of this article will help to remove the weakness of the locally optimal designs that have been addressed in Schmidt and Schwabe when the parameters are misspecfied.

A computational note on maximum likelihood estimation in random effects panel probit model

Panel data sets have recently been developed in various areas, and many recent studies have analyzed panel, or longitudinal data sets. Often a dichotomous dependent variable occur in survival analysis, biomedical and epidemiological studies that is analyzed by a generalized linear mixed effects model (GLMM). The most common estimation method for the binary panel data may be the maximum likelihood (ML). Many statistical packages provide ML estimates; however, the estimates are computed from numerically approximated likelihood function. For instance, R packages, pglm (Croissant, 2017) approximate the likelihood function by the Gauss–Hermite quadratures, while Rchoice (Sarrias, Journal of Statistical Software, 74, 1–31, 2016) use a Monte Carlo integration method for the approximation. As a result, it can be observed that different packages give different results because of different numerical computation methods. In this note, we discuss the pros and cons of numerical methods compared with the exact computation method.

Modelling Spatio-Temporal Variation in Sparse Rainfall Data Using a Hierarchical Bayesian Regression Model

Rainfall is a critical component of climate governing vegetation growth and production, forage availability and quality for herbivores. However, reliable rainfall measurements are not always available, making it necessary to predict rainfall values for particular locations through time. Predicting rainfall in space and time can be a complex and challenging task, especially where the rain gauge network is sparse and measurements are not recorded consistently for all rain gauges, leading to many missing values. Here, we develop a flexible Bayesian model for predicting rainfall in space and time and apply it to Narok County, situated in southwestern Kenya, using data collected at 23 rain gauges from 1965 to 2015. Narok County encompasses the Maasai Mara ecosystem, the northern-most section of the Mara–Serengeti ecosystem, famous for its diverse and abundant large mammal populations and spectacular migration of enormous herds of wildebeest, zebra and Thomson’s gazelle. The model incorporates geographical and meteorological predictor variables, including elevation, distance to Lake Victoria and minimum temperature. Salient features of our model are the use of non-stationary covariance structures and the facility to handle excess zeros and many missing observations. We assess the efficiency of the model by comparing it empirically with the established Gaussian process, Kriging, simple linear and Bayesian linear models. We use the model to predict total monthly rainfall and its standard error for all 5 $$\times $$ 5 km grid cells in Narok County for each year from 1965 to 2015. Using the Monte Carlo integration method, we estimate seasonal and annual rainfall and their standard errors for 29 sub-regions of Narok for each of the 51 years spanning 1965–2015. The non-stationary model can handle data from a sparse network of observations with many missing values and performs at least as well as or better than four established and widely used models on the Narok rainfall data set. The model produces rainfall predictions consistent with expectation and in good agreement with blended station and satellite rainfall values. The predictions are precise enough for most practical purposes. The model is very general and applicable to other variables besides rainfall. Supplementary materials accompanying this paper appear online