The application of deep neural networks in geospatial data has become a trending research problem in the present day. A significant amount of statistical research has already been introduced, such as generalized least square optimization by incorporating spatial variance-covariance matrix, considering basis functions in the input nodes of the neural networks, and so on. However, for lattice data, there is no available literature about the utilization of asymptotic analysis of neural networks in regression for spatial data. This article proposes a consistent localized two-layer deep neural network-based regression for spatial data. We have proved the consistency of this deep neural network for bounded and unbounded spatial domains under a fixed sampling design of mixed-increasing spatial regions. We have proved that its asymptotic convergence rate is faster than that of cite{zhan2024neural}'s neural network and an improved generalization of cite{shen2023asymptotic}'s neural network structure. We empirically observe the rate of convergence of discrepancy measures between the empirical probability distribution of observed and predicted data, which will become faster for a less smooth spatial surface. We have applied our asymptotic analysis of deep neural networks to the estimation of the monthly average temperature of major cities in the USA from its satellite image. This application is an effective showcase of non-linear spatial regression. We demonstrate our methodology with simulated lattice data in various scenarios.

This book is a collection of notes and solved problems about probability theory. The book also contains proposed exercises attached to the solved problems as well as computer codes (in C++ language) added to some of these problems for the purpose of calculation, test and simulation. Illustrations (such as figures and tables) are added when necessary or appropriate to enhance clarity and improve understanding. In most cases intuitive arguments and methods are used to make the notes and solutions natural and instinctive. Like my previous books, maximum clarity was one of the main objectives and criteria in determining the style of writing, presenting and structuring the book as well as selecting its contents.

The frequency response function (FRF) is an established way to describe the outcome of experiments in posture control literature. The FRF is an empirical transfer function between an input stimulus and the induced body segment sway profile, represented as a vector of complex values associated with a vector of frequencies. Having obtained an FRF from a trial with a subject, it can be useful to quantify the likelihood it belongs to a certain population, e.g., to diagnose a condition or to evaluate the human likeliness of a humanoid robot or a wearable device. In this work, a recently proposed method for FRF statistics based on confidence bands computed with Bootstrap will be summarized, and, on its basis, possible ways to quantify the likelihood of FRFs belonging to a given set will be proposed

Improper priors are not allowed for the computation of the Bayesian evidence Z = p(y) (a.k.a., marginal likelihood), since in this case Z is not completely specified due to an arbitrary constant involved in the computation. However, in this work, we remark that they can be employed in a specific type of model selection problem: when we have several (possibly infinite) models belonging to the same parametric family (i.e., for tuning parameters of a parametric model). However, the quantities involved in this type of selection cannot be considered as Bayesian evidences: we suggest to use the name "fake evidences" (or "areas under the likelihood" in the case of uniform improper priors). We also show that, in this model selection scenario, using a use prior and increasing its scale parameter asymptotically to infinity, we cannot recover the value of the area under the likelihood, obtained with a uniform improper prior. We first discuss it from a general point of view. Then we provide, as an applicative example, all the details for Bayesian regression models with nonlinear bases, considering two cases: the use of a uniform improper prior and the use of a Gaussian prior, respectively. A numerical experiment is also provided confirming and checking all the previous statements.

An index of effective number of variables (ENV) is introduced for model selection in nested models. This is the case, for instance, when we have to decide the order of a polynomial function or the number of bases in a nonlinear regression, or choose the number of clusters in a clustering problem, or the number of feature in a variable selection application (to name few examples). It is inspired by the concept of maximum area under the curve (AUC) idea and the Gini index. The interpretation of the ENV index is identical to the effective sample size (ESS) indices with respect to a set of samples. The ENV index improves some drawback the elbow detectors described in the literature, and introduces different measures of uncertainty and reliability of the proposed solution. These novel reliability measures can be employed also jointly with the use different information criteria such as the well-known AIC and BIC. Comparisons with classical and recent schemes are provided in different experiments involving real datasets. Related Matlab code is given.

In this note, we establish a uniform lower bound (w.r.t. the number of players) for the probability of k players tied for first place in the geometric case. To derive this bound, we introduce the concept of supertelescoping series as a generalization of telescoping series. We also provide an insight on the relationship between supertelescopic series and supermartingales.

Volatility estimation and quantile regression are relevant active research areas in statistics, machine learning and econometrics. In this work, we propose two procedures to estimate local variances in generic regression problems by using of kernel smoothers. The proposed schemes can be applied in multidimesional scenarios (not just for time series analysis) and easily in a multi-output framework, as well. Moreover, they allow the possibility of providing uncertainty estimation using a generic kernel smoother technique. Several numerical experiments show the benefits of the proposed methods, even comparing with benchmark techniques. One of these experiment involves a real dataset analysis.

This article analyses the conjecture that excess mortality is underestimated with the pandemic.I use the numbers from the CBS (Dutch Central Bureau for Statistics) as an example. As a baseline we take the expected mortality for 2021 and 2022 from 2019. I correct this expected mortality with the estimated number of people who died in earlier years than expected because of the pandemic. For 2021 this correction is 8K. The CBS expects the mortality to be almost equal to the estimate from 2019. Then the excess mortality increases from 16K (CBS) to 24K.I present the following idea to explain the difference. At the beginning of very year the numbers of people in year groups are usually adjusted by applying a historical determined percentage to the population at January first. Covid hits the weakest the hardest. This changes the distribution of the expected remaining life years in the year group. And thus the average expected remaining life years. Hence the percentage has to be adjusted. Then the expected mortality decreases and the excess mortality increases.The excess mortality within a year are people who for example died in April from covid but who would have died in October without the pandemic. With this number total excess mortality rises with 6K to 30K.Excess mortality is divided in covid and non-covid. De large increase in non-covid deaths is striking.The analysis supports the conjecture that excess mortality is underestimated.Note: The numbers in this article are for the Netherlands. For you own country use the appropriate numbers.

This article discusses the influence of a disturbance like covid on the calculation of life expectancy in year groups etcetera. Life expectancies in year-groups are usually adjusted in the beginning of the year based on the population in the beginning of the year. This is done with a percentage based on previous years. This percentage is a reflection of volume. With the pandemic the weak were hit heavily by covid. A consequence is that the distribution of life expectancy changes in the year groups. This increases the life expectancy and decreases the expected mortality in the year group. Then the calculation for the year groups has to be adjusted accordingly. In this article I give an example of such adjustment. One can accordingly adjust likewise statistics.

The challenges posed by epidemics and pandemics are immense, especially if the causes are novel. This article introduces a versatile open-source simulation framework designed to model intricate dynamics of infectious diseases across diverse population centres. Taking inspiration from historical precedents such as the Spanish flu and COVID-19, and geographical economic theories such as Central place theory, the simulation integrates agent-based modelling to depict the movement and interactions of individuals within different settlement hierarchies. Additionally, the framework provides a tool for decision-makers to assess and strategize optimal distribution plans for limited resources like vaccines or cures as well as to impose mobility restrictions.

We describe models of proportions depending on some independent quantitative variables. An explicit formula for inverse matrices facilitatesinterpolation as a way to calculate the starting values for iterations in nonlinear regression with logistic functions or ratios of exponential functions.

In this paper we address the problem of performing Bayesian inference for the parameters of a nonlinear multi-output model and the covariance matrix of the different output signals. We proposean adaptive importance sampling (AIS) scheme for multivariate Bayesian inversion problems, which is based in two main ideas: the variables of interest are split in two blocks and the inferencetakes advantage of known analytical optimization formulas. We estimate both the unknown parameters of the multivariate non-linear model and the covariance matrix of the noise. In the firstpart of the proposed inference scheme, a novel AIS technique called adaptive target AIS (ATAIS) is designed, which alternates iteratively between an IS technique over the parameters of the nonlinearmodel and a frequentist approach for the covariance matrix of the noise. In the second part of the proposed inference scheme, a prior density over the covariance matrix is considered and the cloud of samples obtained by ATAIS are recycled and re-weighted for obtaining a complete Bayesian study over the model parameters and covariance matrix. ATAIS is the main contribution of the work. Additionally, the inverted layered importance sampling (ILIS) is presented as a possible compelling algorithm (but based on a conceptually simpler idea). Different numerical examples show the benefits of the proposed approaches.

We study the properties of regression coefficients when the sum of the dependent variables is one,ie, the dependent variables are compositional.We show that the sum of intercepts is equal toone and the sum of other corresponding regressioncoefficients is zero. We do it for simple linearregressions and also for a more general case usingmatrix notation. The last part treats the casewhen the dependent variables do not sum up to one. We simplify the well known formula derived by theuse of Lagrange multipliers.

Counting immunopositive cells on biological tissues generally requires either manual annotation or (when available) automatic rough systems, for scanning signal surface and intensity in whole slide imaging. In this work, we tackle the problem of counting microglial cells in biomedical images that represent lumbar spinal cord cross-sections of rats. Note that counting microglial cells is typically a time-consuming task, and additionally entail extensive personnel training. We skip the task of detecting the cells and we focus only on the counting problem. Firstly, a linear predictor is designed based on the information provided by filtered images, obtained applying color threshold values to the labelled images in thedataset. Non-linear extensions and other improvements are presented. The choice of the threshold values is also discussed. Different numerical experiments show the capability of the proposed algorithms. Furthermore, the proposed schemes could be applied to different counting problems of small objects in other types of images (from satellites, telescopes, and/or drones, to name a few).

The L1 distance between two points in a square lattice is the sum of horizontal and vertical absolute differences of the Cartesian coordinates and - as in graph theory - also the minimumnumber of edges to walk to reach one point from the other. The manuscript contains a Java program that computes in a finite square grid of fixed shapethe number of point pairs as a function of that distance.

Substances representing tier (Iron, Bronze, Silver, Gold, Platinum, Diamond) and its typical price (USD/gram) in several games using a tier system have a positive correlation [1, 2, 5].

We design a Universal Automatic Elbow Detector (UAED) for deciding the effective number of components in model selection problems. The relationship with the information criteria widely employed in the literature is also discussed. The proposed UAED does not require the knowledge of a likelihood function and can be easily applied in diverse applications, such as regression and classification, feature and/or order selection, clustering, and dimension reduction. Several experiments involving synthetic and real data show the advantages of the proposed scheme with benchmark techniques in the literature.

We introduce a generalized information criterion that contains other well-known information criteria, such as Bayesian information Criterion (BIC) and Akaike information criterion (AIC), as special cases. Furthermore, the proposed spectral information criterion(SIC) is also more general than the other information criteria, e.g., since the knowledge of a likelihood function is not strictly required. SIC extracts geometric features of the error curve and, as a consequence, it can be considered an automatic elbow detector. SIC provides a subset of all possible models, with a cardinality that often is much smaller than the total number of possible models. The elements of this subset are elbows" of the error curve. A practical rule for selecting a unique model within the sets of elbows is suggested as well. Theoretical invariance properties of SIC are analyzed. Moreover, we test SIC in ideal scenarios where provides always the optimal expected results. We also test SIC inseveral numerical experiments: some involving synthetic data, and two experiments involving real datasets. They are all real-world applications such as clustering, variable selection, or polynomial order selection, to name a few. The results show the benefits of the proposed scheme. Matlab code related to the experiments is also provided. Possible future research lines are finally discussed.

This article try to unified the four basic forces by Maxwell equations, the only experimental theory. Self-consistent Maxwell equations with the e-current coming from matter current is proposed, and is solved to electrons and the structures of particles and atomic nucleus. The static properties and decay are reasoned, all meet experimental data. The equation of general relativity sheerly with electromagnetic field is discussed as the base of this theory. In the end the conformation elementarily between this theory and QED and weak theory is discussed.

We present a method of minimizing an objective function subject to an inequality constraint. It enables us to minimize the sum of squares of deviations in linear regression under inequality restrictions. We demonstrate how to calculate the coefficients of cubic function under the restriction that it is increasing, we also mention how to fit a convex quartic polynomial. We use such results for interpolation as a method for calculation of starting values for iterative methods of fitting some specific functions, such as four-parameter logistic, positive bi-exponential, or Gomperz functions. Curvature-driven interpolation enables such calculations for otherwise solutions to interpolation equations may not exist or may not be unique. We also present examples to illustrate how it works and compare our approach with that of Zhang (2020).

The idea of using a path of tempered posterior distributions has been widely applied in the literature for the computation of marginal likelihoods (a.k.a., Bayesian evidence). Thermodynamic integration, path sampling and annealing importance sampling are well-known examples of algorithms belonging to this family of methods. In this work, we introduce a generalized thermodynamic integration (GTI) scheme which is able to perform a complete Bayesian inference, i.e., GTI can approximate generic posterior exceptions (not only the marginal likelihood). Several scenarios of application of GTI are discussed and different numerical simulations are provided.

In this work, we analyze alternative e ective sample size (ESS) measures for importance sampling algorithms. More specifically, we study a family of ESS approximations introduced in [11]. We show that all the ESS functions included in this family (called Huggins-Roy's family) satisfy all the required theoretical conditions introduced in [17]. We also highlight the relationship of this family with the Renyi entropy. By numerical simulations, we study the performance of different ESS approximations introducing also an optimal linear combination of the most promising ESS indices introduced in literature. Moreover, we obtain the best ESS approximation within the Huggins-Roy's family, that provides almost a perfect match with the theoretical ESS values.

Many applications in signal processing and machine learning require the study of probability density functions (pdfs) that can only be accessed through noisy evaluations. In this work, we analyze the noisy importance sampling (IS), i.e., IS working with noisy evaluations of the target density. We present the general framework and derive optimal proposal densities for noisy IS estimators. The optimal proposals incorporate the information of the variance of the noisy realizations, proposing points in regions where the noise power is higher. We also compare the use of the optimal proposals with previous optimality approaches considered in a noisy IS framework.

An initial value boundary problem for the linear Schr ˙odinger equation with nonlinear functional boundary conditions is considered. It is shown that attractor of problem contains periodic piecewise constant functions on the complex plane with finite points of discontinuities on a period. The method of reduction of the problem to a system of integro-difference equations has been applied. Applications to optical resonators with feedback has been considered. The elements of the attractor can be interpreted as white and black solitons in nonlinear optics.

Data processing and data cleaning are essential steps before applying statistical or machine learning procedures. R provides a flexible way for data processing using vectors. R packages also provide other ways for manipulating data such as using SQL and using chained functions. I present yet another way to process data in a row by row manner using data manipulation oriented script, DataSailr script. This article introduces datasailr package, and shows potential benefits of using domain specific language for data processing.

In this communication a short and straightforward algorithm, written in Octave (version 6.1.0 (2020-11-26))/Matlab (version '9.9.0.1538559 (R2020b) Update 3'), is proposed for brute-force computation of the principal constants $d_{2}$ and $d_{3}$ used to calculate control limits for various types of variables control charts encountered in statistical process control (SPC).

In many inference problems, the evaluation of complex and costly models is often required. In this context, Bayesian methods have become very popular in several fields over the last years, in order to obtain parameter inversion, model selection or uncertainty quantification. Bayesian inference requires the approximation of complicated integrals involving (often costly) posterior distributions. Generally, this approximation is obtained by means of Monte Carlo (MC) methods. In order to reduce the computational cost of the corresponding technique, surrogate models (also called emulators) are often employed. Another alternative approach is the so-called Approximate Bayesian Computation (ABC) scheme. ABC does not require the evaluation of the costly model but the ability to simulate artificial data according to that model. Moreover, in ABC, the choice of a suitable distance between real and artificial data is also required. In this work, we introduce a novel approach where the expensive model is evaluated only in some well-chosen samples. The selection of these nodes is based on the so-called compressed Monte Carlo (CMC) scheme. We provide theoretical results supporting the novel algorithms and give empirical evidence of the performance of the proposed method in several numerical experiments. Two of them are real-world applications in astronomy and satellite remote sensing.

This paper develops the theory of the kth power expectile estimation and considers its relevant hypothesis tests for coefficients of linear regression models. We prove that the asymptotic covariance matrix of kth power expectile regression converges to that of quantile regression as k converges to one, and hence provide a moment estimator of asymptotic matrix of quantile regression. The kth power expectile regression is then utilized to test for homoskedasticity and conditional symmetry of the data. Detailed comparisons of the local power among the kth power expectile regression tests, the quantile regression test, and the expectile regression test have been provided. When the underlying distribution is not standard normal, results show that the optimal k are often larger than 1 and smaller than 2, which suggests the general kth power expectile regression is necessary.

In this note, we give a proof of a theorem of Linnik concerning the theory of errors, stated in his book "Least squares method and the mathematical bases of the statistical theory of the treatment of observations", without proof.

This paper introduces an automatic methodology to construct emulators for costly radiative transfer models (RTMs). The proposed method is sequential and adaptive, and it is based on the notion of the acquisition function by which instead of optimizing the unknown RTM underlying function we propose to achieve accurate approximations. The Automatic Gaussian Process Emulator (AGAPE) methodology combines the interpolation capabilities of Gaussian processes (GPs) with the accurate design of an acquisition function that favors sampling in low density regions and flatness of the interpolation function. We illustrate the good capabilities of the method in toy examples and for the construction of an optimal look-up-table for atmospheric correction based on MODTRAN5.

Monte Carlo (MC) methods are widely used for Bayesian inference and optimization in statistics, signal processing and machine learning. Two well-known class of MC methods are the Importance Sampling (IS) techniques and the Markov Chain Monte Carlo (MCMC) algorithms. In this work, we introduce the Group Importance Sampling (GIS) framework where different sets of weighted samples are properly summarized with one summary particle and one summary weight. GIS facilitates the design of novel efficient MC techniques. For instance, we present the Group Metropolis Sampling (GMS) algorithm which produces a Markov chain of sets of weighted samples. GMS in general outperforms other multiple try schemes as shown by means of numerical simulations.

Solving inverse problems is central in geosciences and remote sensing. Very often a mechanistic physical model of the system exists that solves the forward problem. Inverting the implied radiative transfer model (RTM) equations numerically implies, however, challenging and computationally demanding problems. Statistical models tackle the inverse problem and predict the biophysical parameter of interest from radiance data, exploiting either in situ data or simulated data from an RTM. We introduce a novel nonlinear and nonparametric statistical inversion model which incorporates both real observations and RTM-simulated data. The proposed Joint Gaussian Process (JGP) provides a solid framework for exploiting the regularities between the two types of data, in order to perform inverse modeling. Advantages of the JGP method over competing strategies are shown on both a simple toy example and in leaf area index (LAI) retrieval from Landsat data combined with simulated data generated by the PROSAIL model.

We introduce a Particle Metropolis-Hastings algorithm driven by several parallel particle filters. The communication with the central node requires the transmission of only a set of weighted samples, one per filter. Furthermore, the marginal version of the previous scheme, called Distributed Particle Marginal Metropolis-Hastings (DPMMH) method, is also presented. DPMMH can be used for making inference on both a dynamical and static variable of interest. The ergodicity is guaranteed, and numerical simulations show the advantages of the novel schemes.

Professor Rolin Zhang kindly invited in The 6th Int'l Conference on Probability and Stochastic Analysis (ICPSA 2021), January 5-7, 2021 in Sanya, China as a Keynote speaker and so, we will state the basic interrelations with reproducing kernels and division by zero from the viewpoint of the conference topics. The connection with reproducing kernels and Probability and Stochastic Analysis are already fundamental and well-known, and so, we will mainly refer to the basic relations with our new division by zero $1/0=0/0=z/0=\tan(\pi/2) =\log 0 =0, [(z^n)/n]_{n=0} = \log z$, $[e^{(1/z)}]_{z=0} = 1$.　

The expressive power of Bayesian kernel-based methods has led them to become an important tool across many different facets of artificial intelligence, and useful to a plethora of modern application domains, providing both power and interpretability via uncertainty analysis. This article introduces and discusses two methods which straddle the areas of probabilistic Bayesian schemes and kernel methods for regression: Gaussian Processes and Relevance Vector Machines. Our focus is on developing a common framework with which to view these methods, via intermediate methods a probabilistic version of the well-known kernel ridge regression, and drawing connections among them, via dual formulations, and discussion of their application in the context of major tasks: regression, smoothing, interpolation, and filtering. Overall, we provide understanding of the mathematical concepts behind these models, and we summarize and discuss in depth different interpretations and highlight the relationship to other methods, such as linear kernel smoothers, Kalman filtering and Fourier approximations. Throughout, we provide numerous figures to promote understanding, and we make numerous recommendations to practitioners. Benefits and drawbacks of the different techniques are highlighted. To our knowledge, this is the most in-depth study of its kind to date focused on these two methods, and will be relevant to theoretical understanding and practitioners throughout the domains of data-science, signal processing, machine learning, and artificial intelligence in general.

It is a digital version of a manuscript of a course about the theory of errors given by the Engineer-in-Chief Philippe Hottier at the '80s, at the French National School of Geographic Sciences. The course gives the foundation of the method of the least squares for the case of linear models.

Aims: Different processes or events which are objectively given and real are equally one of the foundations of human life (necessary conditions) too. However, a generally accepted, logically consistent (bio)-mathematical description of these natural processes is still not in sight. Methods: Discrete random variables are analysed. Results: The mathematical formula of the necessary condition is developed. The impact of study design on the results of a study is considered. Conclusion: Study data can be analysed for necessary conditions.

The life expectancy of the currently living German population is calculated per age and as weighted average. The same calculation is repeated after considering everyone infected with and potentially killed by SARS-CoV-2 within one year, given the current age-dependent lethality estimates from a study at London Imperial College [1]. For an average life expectancy of 83.0 years in the current population, the reduction due to SARS-CoV-2 infection amounts to 2.0 (1.1-3.9) months. The individual values show a maximum of 7.7 (4.4-15.2) months for a 70-year-old. People below age 50 loose less than 1 month in average.

Aim: The relationship between Epstein-Barr virus and multiple sclerosis is assessed once again in order to gain a better understanding of this disease. Methods: A systematic review and meta-analysis is provided aimed to answer among other the following question. Is there a cause effect relationship between Epstein-Barr virus and multiple sclerosis? The conditio sine qua non relationship proofed the hypothesis without an Epstein-Barr virus infection no multiple sclerosis. The mathematical formula of the causal relationship k proofed the hypothesis of a cause effect relationship between Epstein-Barr virus infection and multiple sclerosis. Significance was indicated by a p-value of less than 0.05. Results: The data of the studies analysed provide evidence that an Epstein-Barr virus infection is a necessary condition (a conditio sine qua non) of multiple sclerosis. In particular and more than that. The data of the studies analysed provided impressive evidence of a cause-effect relationship between Epstein-Barr virus infection and multiple sclerosis. Conclusion: Multiple sclerosis is caused by an Epstein-Barr virus infection.

We propose a novel adaptive importance sampling scheme for Bayesian inversion problems where the inference of the variables of interest and the power of the data noise is split. More specifically, we consider a Bayesian analysis for the variables of interest (i.e., the parameters of the model to invert), whereas we employ a maximum likelihood approach for the estimation of the noise power. The whole technique is implemented by means of an iterative procedure, alternating sampling and optimization steps. Moreover, the noise power is also used as a tempered parameter for the posterior distribution of the the variables of interest. Therefore, a sequence of tempered posterior densities is generated, where the tempered parameter is automatically selected according to the actual estimation of the noise power. A complete Bayesian study over the model parameters and the scale parameter can be also performed. Numerical experiments show the benefits of the proposed approach.

Many different measures of association are used by medical literature, the relative risk is one of these measures. However, to judge whether results of studies are reliable, it is essential to use among other measures of association which are logically consistent. In this paper, we will present how to deal with one of the most commonly used measures of association, the relative risk. The conclusion is inescapable that the relative risk is logically inconsistent and should not be used any longer.

This is an up-to-date introduction to, and overview of, marginal likelihood computation for model selection and hypothesis testing. Computing normalizing constants of probability models (or ratio of constants) is a fundamental issue in many applications in statistics, applied mathematics, signal processing and machine learning. This article provides a comprehensive study of the state-of-the-art of the topic. We highlight limitations, benefits, connections and differences among the different techniques. Problems and possible solutions with the use of improper priors are also described. Some of the most relevant methodologies are compared through theoretical comparisons and numerical experiments.

Generalized Maxwell distribution is an extension of the classic Maxwell distribution. In this paper, we concentrate on the joint distributional asymptotics of normalized maxima and minima. Under optimal normalizing constants, asymptotic expansions of joint distribution and density for normalized partial maxima and minima are established. These expansions are used to educe speeds of convergence of joint distribution and density of normalized maxima and minima tending to its corresponding ultimate limits. Numerical analysis are provided to support our results.

This paper presents an approach to modelling passive forever churn (i.e., the probability that a user never returns to a game that does not require them to cancel it). The approach is based on parametric mixture models (Weibull, Gamma, and Log-normal) for return times. The model and data are inverted using Bayesian methods (MCMC and DIC) to get parameter estimates, uncertainties, as well as determine the return time distribution for retained users. The inversion scheme is tested on three groups of simulated data sets and one observed data set. The simulated data are generated with each of the parametric models. Each data set is censored to six time horizons, creating 18 data sets. All data sets are inverted with all three parametric models and the DIC is used to select the return time distribution. For all data sets the true return time distribution (i.e., the one that is used to simulate the data) has the best DIC value; for 16 inversions the true return time distribution is found to be significantly better than the other options. For the observed data set inversion, the scheme is able to accurately estimate the \% of users that did return (before the game transitioned into open beta) to given 14 days of observations.

It has been stated in the literature that for finding uniformly minimum-variance unbiased estimator through the theorems of Rao-Blackwell and Lehmann-Scheffe, the sufficient statistic should be complete; otherwise the discussion and the way of finding uniformly minimum-variance unbiased estimator should be changed, since the sufficiency assumption in the Rao-Blackwell and Lehmann-Scheffe theorems limits its applicability. So, it seems that the sufficiency assumptions should be expressed in a way that the uniformly minimum-variance unbiased estimator be derivable via the Rao-Blackwell and Lehmann-Scheffe theorems.

Bayesian models have become very popular over the last years in several fields such as signal processing, statistics and machine learning. Bayesian inference needs the approximation of complicated integrals involving the posterior distribution. For this purpose, Monte Carlo (MC) methods, such as Markov Chain Monte Carlo (MCMC) and Importance Sampling (IS) algorithms, are often employed. In this work, we introduce theory and practice of a Compressed MC (C-MC) scheme, in order to compress the information contained in a could of samples. CMC is particularly useful in a distributed Bayesian inference framework, when cheap and fast communications with a central processor are required. In its basic version, C-MC is strictly related to the stratification technique, a well-known method used for variance reduction purposes. Deterministic C-MC schemes are also presented, which provide very good performance. The compression problem is strictly related to moment matching approach applied in different filtering methods, often known as Gaussian quadrature rules or sigma-point methods. The connections to herding algorithms and quasi-Monte Carlo perspective are also discussed. Numerical results confirm the benefit of the introduced schemes, outperforming the corresponding benchmark methods.

In this paper, asymptotic expansions of the distributions and densities of powered extremes for Maxwell samples are considered. The results show that the convergence speeds of normalized partial maxima relies on the powered index. Additionally, compared with previous result, the convergence rate of the distribution of powered extreme from Maxwell samples is faster than that of its extreme. Finally, numerical analysis is conducted to illustrate our findings.

It is a short lectures of Geostatistics giving some elements of this field for third-year students of the Geomatics license of the Faculty of Sciences of Tunis.

In this paper, we propose a novel nonconvex penalty function for compressed sensing using integral convolution approximation. It is well known that an unconstrained optimization criterion based on $\ell_1$-norm easily underestimates the large component in signal recovery. Moreover, most methods either perform well only under the measurement matrix satisfied restricted isometry property (RIP) or the highly coherent measurement matrix, which both can not be established at the same time. We introduce a new solver to address both of these concerns by adopting a frame of the difference between two convex functions with integral convolution approximation. What's more, to better boost the recovery performance, a weighted version of it is also provided. Experimental results suggest the effectiveness and robustness of our methods through several signal reconstruction examples in term of success rate and signal-to-noise ratio (SNR).

Latent Dirichlet Allocation (LDA) is a generative model describing the observed data as being composed of a mixture of underlying unobserved topics, as introduced by Blei et al. (2003). A key hyperparameter of LDA is the number of underlying topics <i>k</i>, which must be estimated empirically in practice. Selecting the appropriate value of <i>k</i> is essentially selecting the correct model to represent the data; an important issue concerning the goodness of fit. We examine in the current work a series of metrics from literature on a quantitative basis by performing benchmarks against a generated dataset with a known value of <i>k</i> and evaluate the ability of each metric to recover the true value, varying over multiple levels of topic resolution in the Dirichlet prior distributions. Finally, we introduce a new metric and heuristic for estimating kand demonstrate improved performance over existing metrics from the literature on several benchmarks.

Viewing the random motions of objects, an observer might think it is 50-50 chances that an object would move toward or away. It might be intuitive, however, it is far from the truth. This study derives the probability functions of Doppler blueshift and redshift effect of signal detection. The fact is, Doppler redshift detection is highly dominating in space, surface, and linear observation. Under the conditions of no quality loss of radiation over distance, and the observer has perfect vision; It is more than 92% probability of detecting redshift, in three-dimensional observation, 87% surface, and 75\% linear. In cosmic observation, only 7.81% of the observers in the universe will detect blueshift of radiations from any object, on average. The remaining 92.19% of the observers in the universe will detect redshift. It it universal for all observers, aliens or Earthlings at all locations of the universe.

Many applications in signal processing require the estimation of some parameters of interest given a set of observed data. More specifically, Bayesian inference needs the computation of a-posteriori estimators which are often expressed as complicated multi-dimensional integrals. Unfortunately, analytical expressions for these estimators cannot be found in most real-world applications, and Monte Carlo methods are the only feasible approach. A very powerful class of Monte Carlo techniques is formed by the Markov Chain Monte Carlo (MCMC) algorithms. They generate a Markov chain such that its stationary distribution coincides with the target posterior density. In this work, we perform a thorough review of MCMC methods using multiple candidates in order to select the next state of the chain, at each iteration. With respect to the classical Metropolis-Hastings method, the use of multiple try techniques foster the exploration of the sample space. We present different Multiple Try Metropolis schemes, Ensemble MCMC methods, Particle Metropolis-Hastings algorithms and the Delayed Rejection Metropolis technique. We highlight limitations, benefits, connections and dierences among the different methods, and compare them by numerical simulations.

We present a review of data types and statistical methods often encountered in astronomy. The aim is to provide an introduction to statistical applications in astronomy for statisticians and computer scientists. We highlight the complex, often hierarchical, nature of many astronomy inference problems and advocate for cross-disciplinary collaborations to address these challenges.

Monte Carlo (MC) methods have become very popular in signal processing during the past decades. The adaptive rejection sampling (ARS) algorithms are well-known MC technique which draw efficiently independent samples from univariate target densities. The ARS schemes yield a sequence of proposal functions that converge toward the target, so that the probability of accepting a sample approaches one. However, sampling from the proposal pdf becomes more computationally demanding each time it is updated. We propose the Parsimonious Adaptive Rejection Sampling (PARS) method, where an efficient trade-off between acceptance rate and proposal complexity is obtained. Thus, the resulting algorithm is faster than the standard ARS approach.

Bayesian methods and their implementations by means of sophisticated Monte Carlo techniques have become very popular in signal processing over the last years. Importance Sampling (IS) is a well-known Monte Carlo technique that approximates integrals involving a posterior distribution by means of weighted samples. In this work, we study the assignation of a single weighted sample which compresses the information contained in a population of weighted samples. Part of the theory that we present as Group Importance Sampling (GIS) has been employed implicitly in dierent works in the literature. The provided analysis yields several theoretical and practical consequences. For instance, we discuss theapplication of GIS into the Sequential Importance Resampling framework and show that Independent Multiple Try Metropolis schemes can be interpreted as a standard Metropolis-Hastings algorithm, following the GIS approach. We also introduce two novel Markov Chain Monte Carlo (MCMC) techniques based on GIS. The rst one, named Group Metropolis Sampling method, produces a Markov chain of sets of weighted samples. All these sets are then employed for obtaining a unique global estimator. The second one is the Distributed Particle Metropolis-Hastings technique, where dierent parallel particle lters are jointly used to drive an MCMC algorithm. Dierent resampled trajectories are compared and then tested with a proper acceptance probability. The novel schemes are tested in dierent numerical experiments such as learning the hyperparameters of Gaussian Processes, two localization problems in a wireless sensor network (with synthetic and real data) and the tracking of vegetation parameters given satellite observations, where they are compared with several benchmark Monte Carlo techniques. Three illustrative Matlab demos are also provided.

During the last decades, Power-law distributions played significant roles in analyzing the topology of scale-free (SF) networks. However, in the observation of degree distributions of practical networks and other unequal distributions such as wealth distribution, we uncover that, instead of monotonic decreasing, there exists a peak at the beginning of most real distributions, which cannot be accurately described by a Power-law. In this paper, in order to break the limitation of the Power-law distribution, we provide detailed derivations of a novel distribution called Subnormal distribution from evolving networks with variable elements and its concrete statistical properties. Additionally, imulations of fitting the subnormal distribution to the degree distribution of evolving networks, real social network, and personal wealth distribution are displayed to show the fitness of proposed distribution.

Monte Carlo methods are essential tools for Bayesian inference. Gibbs sampling is a well-known Markov chain Monte Carlo (MCMC) algorithm, extensively used in signal processing, machine learning, and statistics, employed to draw samples from complicated high-dimensional posterior distributions. The key point for the successful application of the Gibbs sampler is the ability to draw efficiently samples from the full-conditional probability density functions. Since in the general case this is not possible, in order to speed up the convergence of the chain, it is required to generate auxiliary samples whose information is eventually disregarded. In this work, we show that these auxiliary samples can be recycled within the Gibbs estimators, improving their efficiency with no extra cost. This novel scheme arises naturally after pointing out the relationship between the standard Gibbs sampler and the chain rule used for sampling purposes. Numerical simulations involving simple and real inference problems confirm the excellent performance of the proposed scheme in terms of accuracy and computational efficiency. In particular we give empirical evidence of performance in a toy example, inference of Gaussian processes hyperparameters, and learning dependence graphs through regression.

In this article, the high-order asymptotic expansions of cumulative distribution function and probability density function of extremes for generalized Maxwell distribution are established under nonlinear normalization. As corollaries, the convergence rates of the distribution and density of maximum are obtained under nonlinear normalization.

Mobile crowdsensing can facilitate environmental surveys by leveraging sensor-equipped mobile devices that carry out measurements covering a wide area in a short time, without bearing the costs of traditional field work. In this paper, we examine statistical methods to perform an accurate estimate of the mean value of an environmental parameter in a region, based on such measurements. The main focus is on estimates produced by considering the mobile device readings at a random instant in time. We compare stratified sampling with different stratification weights to sampling without stratification, as well as an appropriately modified version of systematic sampling. Our main result is that stratification with weights proportional to stratum areas can produce significantly smaller bias, and gets arbitrarily close to the true area average as the number of mobiles increases, for a moderate number of strata. The performance of the methods is evaluated for an application scenario where we estimate the mean area temperature in a linear region that exhibits the so-called <i>Urban Heat Island</i> effect, with mobile users moving in the region according to the Random Waypoint Model.

In this work, we design an efficient Monte Carlo scheme for a node-specific inference problem where a vector of global parameters and multiple vectors of local parameters are involved. This scenario often appears in inference problems over heterogeneous wireless sensor networks where each node performs observations dependent on a vector of global parameters as well as a vector of local parameters. The proposed scheme uses parallel local MCMC chains and then an importance sampling (IS) fusion step that leverages all the observations of all the nodes when estimating the global parameters. The resulting algorithm is simple and flexible. It can be easily applied iteratively, or extended in a sequential framework.

The Sequential Importance Resampling (SIR) method is the core of the Sequential Monte Carlo (SMC) algorithms (a.k.a., particle filters). In this work, we point out a suitable choice for weighting properly a resampled particle. This observation entails several theoretical and practical consequences, allowing also the design of novel sampling schemes. Specifically, we describe one theoretical result about the sequential estimation of the marginal likelihood. Moreover, we suggest a novel resampling procedure for SMC algorithms called partial resampling, involving only a subset of the current cloud of particles. Clearly, this scheme attenuates the additional variance in the Monte Carlo estimators generated by the use of the resampling.

The Effective Sample Size (ESS) is an important measure of efficiency of Monte Carlo methods such as Markov Chain Monte Carlo (MCMC) and Importance Sampling (IS) techniques. In the IS context, an approximation $\widehat{ESS}$ of the theoretical ESS definition is widely applied, involving the inverse of the sum of the squares of the normalized importance weights. This formula, $\widehat{ESS}$, has become an essential piece within Sequential Monte Carlo (SMC) methods, to assess the convenience of a resampling step. From another perspective, the expression $\widehat{ESS}$ is related to the Euclidean distance between the probability mass described by the normalized weights and the discrete uniform probability mass function (pmf). In this work, we derive other possible ESS functions based on different discrepancy measures between these two pmfs. Several examples are provided involving, for instance, the geometric mean of the weights, the discrete entropy (including the {\it perplexity} measure, already proposed in literature) and the Gini coefficient among others. We list five theoretical requirements which a generic ESS function should satisfy, allowing us to classify different ESS measures. We also compare the most promising ones by means of numerical simulations.

Many signal processing applications require performing statistical inference on large datasets, where computational and/or memory restrictions become an issue. In this big data setting, computing an exact global centralized estimator is often unfeasible. Furthermore, even when approximate numerical solutions (e.g., based on Monte Carlo methods) working directly on the whole dataset can be computed, they may not provide a satisfactory performance either. Hence, several authors have recently started considering distributed inference approaches, where the data is divided among multiple workers (cores, machines or a combination of both). The computations are then performed in parallel and the resulting distributed or partial estimators are finally combined to approximate the intractable global estimator. In this paper, we focus on the scenario where no communication exists among the workers, deriving efficient linear fusion rules for the combination of the distributed estimators. Both a Bayesian perspective (based on the Bernstein-von Mises theorem and the asymptotic normality of the estimators) and a constrained optimization view are provided for the derivation of the linear fusion rules proposed. We concentrate on minimum mean squared error (MMSE) partial estimators, but the approach is more general and can be used to combine any kind of distributed estimators as long as they are unbiased. Numerical results show the good performance of the algorithms developed, both in simple problems where analytical expressions can be obtained for the distributed MMSE estimators, and in a wireless sensor network localization problem where Monte Carlo methods are used to approximate the partial estimators.

Population Monte Carlo (PMC) sampling methods are powerful tools for approximating distributions of static unknowns given a set of observations. These methods are iterative in nature: at each step they generate samples from a proposal distribution and assign them weights according to the importance sampling principle. Critical issues in applying PMC methods are the choice of the generating functions for the samples and the avoidance of the sample degeneracy. In this paper, we propose three new schemes that considerably improve the performance of the original PMC formulation by allowing for better exploration of the space of unknowns and by selecting more adequately the surviving samples. A theoretical analysis is performed, proving the superiority of the novel schemes in terms of variance of the associated estimators and preservation of the sample diversity. Furthermore, we show that they outperform other state of the art algorithms (both in terms of mean square error and robustness w.r.t. initialization) through extensive numerical simulations.

We design a sequential Monte Carlo scheme for the dual purpose of Bayesian inference and model selection. We consider the application context of urban mobility, where several modalities of transport and different measurement devices can be employed. Therefore, we address the joint problem of online tracking and detection of the current modality. For this purpose, we use interacting parallel particle filters, each one addressing a different model. They cooperate for providing a global estimator of the variable of interest and, at the same time, an approximation of the posterior density of each model given the data. The interaction occurs by a parsimonious distribution of the computational effort, with online adaptation for the number of particles of each filter according to the posterior probability of the corresponding model. The resulting scheme is simple and flexible. We have tested the novel technique in different numerical experiments with artificial and real data, which confirm the robustness of the proposed scheme.

Let fXn; n 1g be a strictly stationary sequence of negatively associated random variables, with common continuous and bounded distribution function F. We consider the estimation of the two-dimensional distribution function of (X1;Xk+1) based on kernel type estimators as well as the estimation of the covariance function of the limit empirical process induced by the sequence fXn; n 1g where k 2 IN0. Then, we derive uniform strong convergence rates for the kernel estimator of two-dimensional distribution function of (X1;Xk+1) which were not found already and do not need any conditions on the covari- ance structure of the variables. Furthermore assuming a convenient decrease rate of the covariances Cov(X1;Xn+1); n 1, we prove uniform strong convergence rate for covari- ance function of the limit empirical process based on kernel type estimators. Finally, we use a simulation study to compare the estimators of distribution function of (X1;Xk+1).

Importance Sampling methods are broadly used to approximate posterior distributions or some of their moments. In its standard approach, samples are drawn from a single proposal distribution and weighted properly. However, since the performance depends on the mismatch between the targeted and the proposal distributions, several proposal densities are often employed for the generation of samples. Under this Multiple Importance Sampling (MIS) scenario, many works have addressed the selection or adaptation of the proposal distributions, interpreting the sampling and the weighting steps in different ways. In this paper, we establish a general framework for sampling and weighting procedures when more than one proposal is available. The most relevant MIS schemes in the literature are encompassed within the new framework, and, moreover novel valid schemes appear naturally. All the MIS schemes are compared and ranked in terms of the variance of the associated estimators. Finally, we provide illustrative examples which reveal that, even with a good choice of the proposal densities, a careful interpretation of the sampling and weighting procedures can make a significant difference in the performance of the method.

The adaptive rejection sampling (ARS) algorithm is a universal random generator for drawing samples eciently from a univariate log-concave target probability density function (pdf). ARS generates independent samples from the target via rejection sampling with high acceptance rates. Indeed, ARS yields a sequence of proposal functions that converge toward the target pdf, so that the probability of accepting a sample approaches one. However, sampling from the proposal pdf becomes more computational demanding each time it is updated. In this work, we propose a novel ARS scheme, called Cheap Adaptive Rejection Sampling (CARS), where the computational effort for drawing from the proposal remains constant, decided in advance by the user. For generating a large number of desired samples, CARS is faster than ARS.

The multiple Try Metropolis (MTM) algorithm is an advanced MCMC technique based on drawing and testing several candidates at each iteration of the algorithm. One of them is selected according to certain weights and then it is tested according to a suitable acceptance probability. Clearly, since the computational cost increases as the employed number of tries grows, one expects that the performance of an MTM scheme improves as the number of tries increases, as well. However, there are scenarios where the increase of number of tries does not produce a corresponding enhancement of the performance. In this work, we describe these scenarios and then we introduce possible solutions for solving these issues.

Monte Carlo (MC) methods are widely used for Bayesian inference and optimization in statistics, signal processing and machine learning. A well-known class of MC methods are Markov Chain Monte Carlo (MCMC) algorithms. In order to foster better exploration of the state space, specially in high-dimensional applications, several schemes employing multiple parallel MCMC chains have been recently introduced. In this work, we describe a novel parallel interacting MCMC scheme, called {\it orthogonal MCMC} (O-MCMC), where a set of ``vertical'' parallel MCMC chains share information using some "horizontal" MCMC techniques working on the entire population of current states. More specifically, the vertical chains are led by random-walk proposals, whereas the horizontal MCMC techniques employ independent proposals, thus allowing an efficient combination of global exploration and local approximation. The interaction is contained in these horizontal iterations. Within the analysis of different implementations of O-MCMC, novel schemes in order to reduce the overall computational cost of parallel multiple try Metropolis (MTM) chains are also presented. Furthermore, a modified version of O-MCMC for optimization is provided by considering parallel simulated annealing (SA) algorithms. Numerical results show the advantages of the proposed sampling scheme in terms of efficiency in the estimation, as well as robustness in terms of independence with respect to initial values and the choice of the parameters.

Monte Carlo methods represent the \textit{de facto} standard for approximating complicated integrals involving multidimensional target distributions. In order to generate random realizations from the target distribution, Monte Carlo techniques use simpler proposal probability densities to draw candidate samples. The performance of any such method is strictly related to the specification of the proposal distribution, such that unfortunate choices easily wreak havoc on the resulting estimators. In this work, we introduce a \textit{layered} (i.e., hierarchical) procedure to generate samples employed within a Monte Carlo scheme. This approach ensures that an appropriate equivalent proposal density is always obtained automatically (thus eliminating the risk of a catastrophic performance), although at the expense of a moderate increase in the complexity. Furthermore, we provide a general unified importance sampling (IS) framework, where multiple proposal densities are employed and several IS schemes are introduced by applying the so-called deterministic mixture approach. Finally, given these schemes, we also propose a novel class of adaptive importance samplers using a population of proposals, where the adaptation is driven by independent parallel or interacting Markov Chain Monte Carlo (MCMC) chains. The resulting algorithms efficiently combine the benefits of both IS and MCMC methods.

We introduce logarithmic generalized Maxwell distribution which is an extension of the generalized Maxwell distribution. Some interesting properties of this distribution are studied and the asymptotic distribution of the partial maximum of an independent and identically distributed sequence from the logarithmic generalized Maxwell distribution is gained. The expansion of the limit distribution from the normalized maxima is established under the optimal norming constants, which shows the rate of convergence of the distribution for normalized maximum tending to the extreme limit.

In this paper, with optimal normalized constants, the asymptotic expansions of the distribution and density of the normalized maxima from generalized Maxwell distribution are derived. For the distributional expansion, it shows that the convergence rate of the normalized maxima to the Gumbel extreme value distribution is proportional to $1/\log n.$ For the density expansion, on the one hand, the main result is applied to establish the convergence rate of the density of extreme to its limit. On the other hand, the main result is applied to obtain the asymptotic expansion of the moment of maximum.

In this paper, the higher-order asymptotic expansion of the moment of extreme from generalized Maxwell distribution is gained, by which one establishes the rate of convergence of the moment of the normalized partial maximum to the moment of the associate Gumbel extreme value distribution.

The first prime number with the special property that its addition with its reversal gives as result a prime number too is 299. The prime numbers with this property will be called Luhn prime numbers. In this article we intend to present a performing algorithm for determining the Luhn prime numbers.

Derivation of the recurrence relation for orthogonal polynomials and usage. Вывод рекуррентного соотношения ортогональных многочленов из процесса ортогонализации Грама-Шмидта, а также схема применения полученного рекуррентного соотношения

The purpose of this paper is to present a general method to estimate the probability of transitions of a system between phases. The system must be represented in a quantitative model, with vectorial variables depending on time, satisfying general conditions which are usually met. The method can be implemented in Physics, Economics or Finances.

This work is based on the short course “A Metodologia AMMI: Com Aplicacão ao Melhoramento Genético” taught during the 58a RBRAS and 15o SEAGRO held in Campina Grande - PB and aim to introduce the AMMI method for those that have and no have the mathematical training. We do not intend to submit a detailed work, but the intention is to serve as a light for researchers, graduate and postgraduate students. In other words, is a work to stimulate research and the quest for knowledge in an area of statistical methods. For this propose we make a review about the genotype-by-environment interaction, definition of the AMMI models and some selection criteria and biplot graphic. More details about it can be found in the material produced for the short course.

Let $\{X_n,~n\geq1\}$ be independent and identically distributed random variables with each $X_n$ following skew normal distribution. Let $M_n=\max\{X_k,~1\leq k\leq n\}$ denote the partial maximum of $\{X_n,~n\geq1\}$. Liao et al. (2014) considered the convergence rate of the distribution of the maxima for random variables obeying the skew normal distribution under linear normalization. In this paper, we obtain the asymptotic distribution of the maximum under power normalization and normalizing constants as well as the associated pointwise convergence rate under power normalization.

Let $\{X_n,n\geq1\}$ be an independent and identically distributed random sequence with common distribution $F$ obeying the lognormal distribution. In this paper, we obtain the exact uniform convergence rate of the distribution of maxima to its extreme value limit under power normalization.

Motivated by Finner et al. (2008), the asymptotic behavior of the probability density function (pdf) and the cumulative distribution function (cdf) of the generalized exponential and Maxwell distributions are studied. Specially, we consider the asymptotic behavior of the ratio of the pdfs (cdfs) of the generalized exponential and Student's $t$-distributions (likewise for the Maxwell and Student's $t$-distributions) as the degrees of freedom parameter approach infinity in an appropriate way. As by products, Mills' ratios for the generalized exponential and Maxwell distributions are gained. Moreover, we illustrate some examples to indicate the application of our results in extreme value theory.

Markov Chain Monte Carlo (MCMC) algorithms and Sequential Monte Carlo (SMC) methods (a.k.a., particle filters) are well-known Monte Carlo methodologies, widely used in different fields for Bayesian inference and stochastic optimization. The Multiple Try Metropolis (MTM) algorithm is an extension of the standard Metropolis- Hastings (MH) algorithm in which the next state of the chain is chosen among a set of candidates, according to certain weights. The Particle MH (PMH) algorithm is another advanced MCMC technique specifically designed for scenarios where the multidimensional target density can be easily factorized as multiplication of conditional densities. PMH combines jointly SMC and MCMC approaches. Both, MTM and PMH, have been widely studied and applied in literature. PMH variants have been often applied for the joint purpose of tracking dynamic variables and tuning constant parameters in a state space model. Furthermore, PMH can be also considered as an alternative particle smoothing method. In this work, we investigate connections, similarities and differences among MTM schemes and PMH methods. This study allows the design of novel efficient schemes for filtering and smoothing purposes in state space models. More specially, one of them, called Particle Multiple Try Metropolis (P-MTM), obtains very promising results in different numerical simulations.

Multipath fading is one of the most common distortions in wireless communications. The simulation of a fading channel typically requires drawing samples from a Rayleigh, Rice or Nakagami distribution. The Nakagami-m distribution is particularly important due to its good agreement with empirical channel measurements, as well as its ability to generalize the well-known Rayleigh and Rice distributions. In this paper, a simple and extremely efficient rejection sampling (RS) algorithm for generating independent samples from a Nakagami-m distribution is proposed. This RS approach is based on a novel hat function composed of three pieces of well-known densities from which samples can be drawn easily and efficiently. The proposed method is valid for any combination of parameters of the Nakagami distribution, without any restriction in the domain and without requiring any adjustment from the final user. Simulations for several parameter combinations show that the proposed approach attains acceptance rates above 90% in all cases, outperforming all the RS techniques currently available in the literature.

Monte Carlo (MC) methods are well-known computational techniques, widely used in different fields such as signal processing, communications and machine learning. An important class of MC methods is composed of importance sampling (IS) and its adaptive extensions, such as population Monte Carlo (PMC) and adaptive multiple IS (AMIS). In this work, we introduce a novel adaptive and iterated importance sampler using a population of proposal densities. The proposed algorithm, named adaptive population importance sampling (APIS), provides a global estimation of the variables of interest iteratively, making use of all the samples previously generated. APIS combines a sophisticated scheme to build the IS estimators (based on the deterministic mixture approach) with a simple temporal adaptation (based on epochs). In this way, APIS is able to keep all the advantages of both AMIS and PMC, while minimizing their drawbacks. Furthermore, APIS is easily parallelizable. The cloud of proposals is adapted in such a way that local features of the target density can be better taken into account compared to single global adaptation procedures. The result is a fast, simple, robust and high-performance algorithm applicable to a wide range of problems. Numerical results show the advantages of the proposed sampling scheme in four synthetic examples and a localization problem in a wireless sensor network.

Bayesian inference often requires efficient numerical approximation algorithms, such as sequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC) methods. The Gibbs sampler is a well-known MCMC technique, widely applied in many signal processing problems. Drawing samples from univariate full-conditional distributions efficiently is essential for the practical application of the Gibbs sampler. In this work, we present a simple, self-tuned and extremely efficient MCMC algorithm which produces virtually independent samples from these univariate target densities. The proposal density used is self-tuned and tailored to the specific target, but it is not adaptive. Instead, the proposal is adjusted during an initial optimization stage, following a simple and extremely effective procedure. Hence, we have named the newly proposed approach as FUSS (Fast Universal Self-tuned Sampler), as it can be used to sample from any bounded univariate distribution and also from any bounded multi-variate distribution, either directly or by embedding it within a Gibbs sampler. Numerical experiments, on several synthetic data sets (including a challenging parameter estimation problem in a chaotic system) and a high-dimensional financial signal processing problem, show its good performance in terms of speed and estimation accuracy.

This paper aims to offer a testing framework for the structural properties of the Brownian motion of the underlying stochastic process of a time series. In particular, the test can be applied to financial time-series data and discriminate among the lognormal random walk used in the Black-Scholes-Merton model, the Gaussian random walk used in the Ornstein-Uhlenbeck stochastic process, and the square-root random walk used in the Cox, Ingersoll and Ross process. Alpha-level hypothesis testing is provided. This testing framework is helpful for selecting the best stochastic processes for pricing contingent claims and risk management.

Local association analysis, such as local similarity analysis and local shape analysis, of biological time series data helps elucidate the varying dynamics of biological systems. However, their applications to large scale high-throughput data are limited by slow permutation procedures for statistical signicance evaluation. We developed a theoretical approach to approximate the statistical signicance of local similarity and local shape analysis based on the approximate tail distribution of the maximum partial sum of independent identically distributed (i.i.d) and Markovian random variables. Simulations show that the derived formula approximates the tail distribution reasonably well (starting at time points > 10 with no delay and > 20 with delay) and provides p-values comparable to those from permutations. The new approach enables ecient calculation of statistical signicance for pairwise local association analysis, making possible all-to-all association studies otherwise prohibitive. As a demonstration, local association analysis of human microbiome time series shows that core OTUs are highly synergetic and some of the associations are body-site specic across samples. The new approach is implemented in our eLSA package, which now provides pipelines for faster local similarity and shape analysis of time series data. The tool is freely available from eLSA's website: http://meta.usc.edu/softs/lsa.

Recent developments in experimental molecular techniques, such as microarray, next generation sequencing technologies, have led molecular biology into a high-throughput era with emergent omics research areas, including metagenomics and transcriptomics. Massive-size omics datasets generated and being generated from the experimental laboratories put new challenges to computational biologists to develop fast and accurate quantitative analysis tools. We have developed two statistical and algorithmic methods, GRAMMy and eLSA, for metagenomics and microbial community time series analysis. GRAMMy provides a unied probabilistic framework for shotgun metagenomics, in which maximum likelihood method is employed to accurately compute Genome Relative Abundance of microbial communities using the Mixture Model theory (GRAMMy). We extended the Local Similarity Analysis technique (eLSA) to time series data with replicates, capturing statistically signicant local and potentially time-delayed associations. Both methods are validated through simulation studies and their capability to reveal new biology is also demonstrated through applications to real datasets. We implemented GRAMMy and eLSA as C++ extensions to Python, with both superior computational eciency and easy-to-integrate programming interfaces. GRAMMy and eLSA methods will be increasingly useful tools as new omics researches accelerating their pace.http://meta.usc.edu/softs/lsa.

For free electromagnetic field, there are two kinds of the wave equation, one is Maxwell wave equation, another is generalized wave equation. In the paper, according to the matrix transformation the author transform the general quadratic form into diagonal matrix. Then this can obtain both forms of wave equation. One is the Maxwell wave equation, another is the second form of the wave equation. In the half latter of the paper the author establish other two vibrator differential equations.

Many decision making problems that arise in Finance, Economics, Inventory etc. can be formulated as Markov Decision Problems (MDPs) and solved using Dynamic Programming techniques. Further, to mitigate the statistical errors in estimating the underlying transition matrix or to exercise optimal control under adverserial setup led to the study of robust formulations of the same problems in Ghaoui and Nilim~\cite{ghaoui} and Iyengar~\cite{garud}. In this work, we study the computational methodologies to develop and validate feasible control policies for the Robust Dynamic Programming Problem. In terms of developing control policies, the current work can be seen as generalizing the existing literature on Approximate Dynamic Programming (ADP) to its robust counterpart. The work also generalizes the Information Relaxation and Dual approach of Brown, Smith and Sun~\cite{bss} to robust multi period problems. While discussing this framework we approach it both from a discrete control perspective and also as a set of conditional continous measures as in Ghaoui and Nilim~\cite{ghaoui} and Iyengar~\cite{garud}. We show numerical experiments on applications like ... In a nutshell, we expand the gamut of problems that the dual approach can handle in terms of developing tight bounds on the value function.

In this current work, we generalize the recent Pathwise Optimization approach of Desai et al.~\cite{desai2010pathwise} to Multiple stopping problems. The approach also minimizes the dual bound as in Desai et al.~\cite{desai2010pathwise} to find the best approximation architecture for the Multiple stopping problem. Though, we establish the convexity of the dual operator, in this setting as well, we cannot directly take advantage of this property because of the computational issues that arise due to the combinatorial nature of the problem. Hence, we deviate from the pure martingale dual approach to \emph{marginal} dual approach of Meinshausen and Hambly~\cite{meinshausenhambly2004} and solve each such optimal stopping problem in the framework of Desai et al.~\cite{desai2010pathwise}. Though, this Pathwise Optimization approach as generalized to the Multiple stopping problem is computationally intensive, we highlight that it can produce superior dual and primal bounds in certain settings.

The Hawkes process having a kernel in the form of a linear combination of exponential functions ν(t)=sum_(j=1)^Pα_j*e^(-β_j*t) has a nice recursive structure that lends itself to tractable likelihood expressions. When P=1 the kernel is ν(t)=α e^(-β t) and the inverse of the compensator can be expressed in closed-form as a linear combination of exponential functions and the LambertW function having arguments which can be expressed as recursive functions of the jump times.

A common problem in multi-environment trials arises when some genotype-by-environment combinations are missing. The aim of this paper is to propose a new deterministic imputation algorithm using a modification of the Gabriel cross-validation method. The method involves the singular value decomposition (SVD) of a matrix and was tested using three alternative component choices of the SVD in simulations based on two complete sets of real data, with values deleted randomly at different rates. The quality of the imputations was evaluated using the correlations and the mean square deviations between these estimates and the true observed values. The proposed methodology does not make any distributional or structural assumptions and does not have any restrictions regarding the pattern or mechanism of the missing data.

In this paper, we use statistical data mining techniques to analyze a multivariate data set of career batting performances in Major League Baseball. Principal components analysis (PCA) is used to transform the high-dimensional data to its lower-dimensional principal components, which retain a high percentage of the sample variation, hence reducing the dimensionality of the data. From PCA, we determine a few important key factors of classical and sabermetric batting statistics, and the most important of these is a new measure, which we call Offensive Player Grade (OPG), that efficiently summarizes a player’s offensive performance on a numerical scale. The determination of these lower-dimensional principal components allows for accessible visualization of the data, and for segmentation of players into groups using clustering, which is done here using the K-means clustering algorithm. We provide illuminating visual displays from our statistical data mining procedures, and we also furnish a player listing of the top 100 OPG scores which should be of interest to those that follow baseball.

In this paper, we propose a new test of uniformity on the circle based on the Gini mean difference of the sample arc-lengths. These sample arc-lengths, which are the gaps between successive observations on the circumference of the circle, are analogous to sample spacings on the real line. The Gini mean difference, which compares these arc-lengths between themselves, is analogous to Rao's spacings statistic, which has been used to test the uniformity of circular data. We obtain both the exact and asymptotic distributions of the Gini mean difference arc-lengths test, under the null hypothesis of circular uniformity. We also provide a table of upper percentile values of the exact distribution for small to moderate sample sizes. Illustrative examples in circular data analysis are also given. It is shown that a generalized Gini mean difference test has better asymptotic efficiency than the corresponding generalized Rao's test in the sense of Pitman asymptotic relative efficiency.

In this paper, we investigate the asymptotic theory for U-statistics based on sample spacings, i.e. the gaps between successive observations. The usual asymptotic theory for U-statistics does not apply here because spacings are dependent variables. However, under the null hypothesis, the uniform spacings can be expressed as conditionally independent Exponential random variables. We exploit this idea to derive the relevant asymptotic theory both under the null hypothesis and under a sequence of close alternatives. The generalized Gini mean difference of the sample spacings is a prime example of a U-statistic of this type. We show that such a Gini spacings test is analogous to Rao's spacings test. We find the asymptotically locally most powerful test in this class, and it has the same efficacy as the Greenwood statistic.

In this paper, we will investigate the problem of obtaining confidence intervals for a baseball team's Pythagorean expectation, i.e. their expected winning percentage and expected games won. We study this problem from two different perspectives. First, in the framework of regression models, we obtain confidence intervals for prediction, i.e. more formally, prediction intervals for a new observation, on the basis of historical binomial data for Major League Baseball teams from the 1901 through 2009 seasons, and apply this to the 2009 MLB regular season. We also obtain a Scheffé-type simultaneous prediction band and use it to tabulate predicted winning percentages and their prediction intervals, corresponding to a range of values for log(RS=RA). Second, parametric bootstrap simulation is introduced as a data-driven, computer-intensive approach to numerically computing confidence intervals for a team's expected winning percentage. Under the assumption that runs scored per game and runs allowed per game are random variables following independent Weibull distributions, we numerically calculate confidence intervals for the Pythagorean expectation via parametric bootstrap simulation on the basis of each team's runs scored per game and runs allowed per game from the 2009 MLB regular season. The interval estimates, from either framework, allow us to infer with better certainty as to which teams are performing above or below expectations. It is seen that the bootstrap confidence intervals appear to be better at detecting which teams are performing above or below expectations than the prediction intervals obtained in the regression framework.