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This paper outlines Binomial and Poisson distributions which are both used to measure the occurrence of a number of random events within a certain period. [ABSTRACT FROM AUTHOR]

The nonlinear chaotic differential/algebraic equation (DAE) has been established to simulate the nonuniform oscillations of the motion of a falling sphere in the non-Newtonian fluid. The DAE is obtained only by learning the experimental data with sparse optimization method. However, the deterministic solution will become increasingly inaccurate for long time approximation of the continuous system. In this paper, we introduce two probabilistic solutions to compute the totally DAE, the Random branch selection iteration (RBSI) and Random switching iteration (RSI). The samples are also taken as the reference trajectory to learn random parameter. The proposed probabilistic solutions can be regarded as the discrete analogues of differential inclusion and switching DAEs, respectively. They have been also compared with the deterministic method, i.e. backward differentiation formula (BDF). The deterministic methods only give limited candidates of all the probability solutions, while the RSI can include all the possible trajectories. The numerical results and statistical information criterion show that RSI can successfully reveal the sustaining instabilities of the motion itself and long time chaotic behavior. [ABSTRACT FROM AUTHOR]

The master equation is used extensively to model chemical reaction systems with stochastic dynamics. However, and despite its phenomenological simplicity, it is not in general possible to compute the solution of this equation. Drawing exact samples from the master equation is possible, but can be computationally demanding, especially when estimating high-order statistical summaries or joint probability distributions. As a consequence, one often relies on analytical approximations to the solution of the master equation or on computational techniques that draw approximative samples from this equation. Unfortunately, it is not in general possible to check whether a particular approximation scheme is valid. The main objective of this paper is to develop an effective methodology to address this problem based on statistical hypothesis testing. By drawing a moderate number of samples from the master equation, the proposed techniques use the well-known Kolmogorov-Smirnov statistic to reject the validity of a given approximation method or accept it with a certain level of confidence. Our approach is general enough to deal with any master equation and can be used to test the validity of any analytical approximation method or any approximative sampling technique of interest. A number of examples, based on the Schlögl model of chemistry and the SIR model of epidemiology, clearly illustrate the effectiveness and potential of the proposed statistical framework. [ABSTRACT FROM AUTHOR]

Abstract Dynamic traffic assignment models rely on a network performance module known as dynamic network loading (DNL), which expresses flow propagation, flow conservation, and travel delay at a network level. The DNL defines the so-called network delay operator , which maps a set of path departure rates to a set of path travel times (or costs). It is widely known that the delay operator is not available in closed form, and has undesirable properties that severely complicate DTA analysis and computation, such as discontinuity, non-differentiability, non-monotonicity, and computational inefficiency. This paper proposes a fresh take on this important and difficult issue, by providing a class of surrogate DNL models based on a statistical learning method known as Kriging. We present a metamodeling framework that systematically approximates DNL models and is flexible in the sense of allowing the modeler to make trade-offs among model granularity, complexity, and accuracy. It is shown that such surrogate DNL models yield highly accurate approximations (with errors below 8%) and superior computational efficiency (9 to 455 times faster than conventional DNL procedures such as those based on the link transmission model). Moreover, these approximate DNL models admit closed-form and analytical delay operators, which are Lipschitz continuous and infinitely differentiable, with closed-form Jacobians. We provide in-depth discussions on the implications of these properties to DTA research and model applications. [ABSTRACT FROM AUTHOR]

Influence maximization has attracted a lot of attention due to its numerous applications, including diffusion of social movements, the spread of news, viral marketing and outbreak of diseases. The objective is to discover a group of users that are able to maximize the spread of influence across a network. The greedy algorithm gives a solution to the Influence Maximization problem while having a good approximation ratio. Nevertheless it does not scale well for large scale datasets. In this paper, we propose Matrix Influence, MATI, an efficient algorithm that can be used under both the Linear Threshold and Independent Cascade diffusion models. MATI is based on the precalculation of the influence by taking advantage of the simple paths in the node’s neighborhood. An extensive empirical analysis has been performed on multiple real-world datasets showing that MATI has competitive performance when compared to other well-known algorithms with regards to running time and expected influence spread. [ABSTRACT FROM AUTHOR]

Abstract: The primary focus of this paper is to develop models to estimate link-level crash frequency using land use data extracted and integrated through the use of a distance gradient method. The on-network characteristics were added to the integrated land use characteristics database and were also used in the development and validation of link-level crash frequency estimation models. Both statistical and back- propagation neural network (BPNN)-based approaches were tested and evaluated for modeling. Mean absolute deviation (MAD), median error, 85th percentile error, and root-mean squared error (RMSE) were computed to validate the developed link-level crash frequency es- timation models and compare the two approaches. The results obtained from validation of the link-level crash frequency estimation models indicate that the computed errors are low for models based on both statistical and neural network approaches. Both the approaches have reasonably good predictive capability and can be used to estimate crash frequency. The role of predictor (includes integrated land use) variables on crash frequency along links can be easily understood using outputs from the statistical modeling approach. Also, findings indicate that models based on integrated land use and on-network characteristics (excluding traffic volume) have good predictive capability and can be used as surrogate data to estimate crash frequency if traffic volume data are not available. [ABSTRACT FROM AUTHOR]

In practice, it is often not possible to find an appropriate family of distributions which can be used for fitting the sample distribution with high precision. In these cases, it seems to be opportune to search for the best approximation by a family of distributions instead of an exact fit. In this paper, we consider the Anderson–Darling statistic with plugged-in minimum distance estimator for the parameter vector. We prove asymptotic normality of the Anderson–Darling statistic which is used for a test of goodness of approximation. Moreover, we introduce a measure of discrepancy between the sample distribution and the model class. [ABSTRACT FROM AUTHOR]

Framework of this paper is statistical data editing, specifically how to edit or impute missing or contradictory data and how to merge two independent data sets presenting some lack of information. Assuming a missing at random mechanism, this paper provides an accurate tree-based methodology for both missing data imputation and data fusion that is justified within the Statistical Learning Theory of Vapnik. It considers both an incremental variable imputation method to improve computational efficiency as well as boosted trees to gain in prediction accuracy with respect to other methods. As a result, the best approximation of the structural risk (also known as irreducible error) is reached, thus reducing at minimum the generalization (or prediction) error of imputation. Moreover, it is distribution free, it holds independently of the underlying probability law generating missing data values. Performance analysis is discussed considering simulation case studies and real world applications. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies large and moderate deviation properties of a realized volatility statistic of high frequency financial data. We establish a large deviation principle for the realized volatility when the number of high frequency observations in a fixed time interval increases to infinity. Our large deviation result can be used to evaluate tail probabilities of the realized volatility. We also derive a moderate deviation rate function for a standardized realized volatility statistic. The moderate deviation result is useful for assessing the validity of normal approximations based on the central limit theorem. In particular, it clarifies that there exists a trade-off between the accuracy of the normal approximations and the path regularity of an underlying volatility process. Our large and moderate deviation results complement the existing asymptotic theory on high frequency data. In addition, the paper contributes to the literature of large deviation theory in that the theory is extended to a high frequency data environment. [Copyright &y& Elsevier]

Abstract: The paper reviews and highlights distinctions between function-approximation (FA) and VC theory and methodology, mainly within the setting of regression problems and a squared-error loss function, and illustrates empirically the differences between the two when data is sparse and/or input distribution is non-uniform. In FA theory, the goal is to estimate an unknown true dependency (or ‘target’ function) in regression problems, or posterior probability in classification problems. In VC theory, the goal is to ‘imitate’ unknown target function, in the sense of minimization of prediction risk or good ‘generalization’. That is, the result of VC learning depends on (unknown) input distribution, while that of FA does not. This distinction is important because regularization theory originally introduced under clearly stated FA setting [Tikhonov, N. (1963). On solving ill-posed problem and method of regularization. Doklady Akademii Nauk USSR, 153, 501–504; Tikhonov, N., & V. Y. Arsenin (1977). Solution of ill-posed problems. Washington, DC: W. H. Winston], has been later used under risk-minimization or VC setting. More recently, several authors [Evgeniou, T., Pontil, M., & Poggio, T. (2000). Regularization networks and support vector machines. Advances in Computational Mathematics, 13, 1–50; Hastie, T., Tibshirani, R., & Friedman, J. (2001). The elements of statistical learning: Data mining, inference and prediction. Springer; Poggio, T. and Smale, S., (2003). The mathematics of learning: Dealing with data. Notices of the AMS, 50 (5), 537–544] applied constructive methodology based on regularization framework to learning dependencies from data (under VC-theoretical setting). However, such regularization-based learning is usually presented as a purely constructive methodology (with no clearly stated problem setting). This paper compares FA/regularization and VC/risk minimization methodologies in terms of underlying theoretical assumptions. The control of model complexity, using regularization and using the concept of margin in SVMs, is contrasted in the FA and VC formulations. [Copyright &y& Elsevier]

Abstract: In this paper three near-exact distributions are developed for the sphericity test statistic. The exact probability density function of this statistic is usually represented through the use of the Meijer G function, which renders the computation of quantiles impossible even for a moderately large number of variables. The main purpose of this paper is to obtain near-exact distributions that lie closer to the exact distribution than the asymptotic distributions while, at the same time, correspond to density and cumulative distribution functions practical to use, allowing for an easy determination of quantiles. In addition to this, two asymptotic distributions that lie closer to the exact distribution than the existing ones were developed. Two measures are considered to evaluate the proximity between the exact and the asymptotic and near-exact distributions developed. As a reference we use the saddlepoint approximations developed by Butler et al. [1993. Saddlepoint approximations for tests of block independence, sphericity and equal variances and covariances. J. Roy. Statist. Soc., Ser. B 55, 171–183] as well as the asymptotic distribution proposed by Box. [Copyright &y& Elsevier]

In dealing with large data sets, the reduced support vector machine (RSVM) was proposed for the practical objective to overcome some computational difficulties as well as to reduce the model complexity. In this paper, we study the RSVM from the viewpoint of sampling design, its robustness, and the spectral analysis of the reduced kernel. We consider the nonlinear separating surface as a mixture of kernels. Instead of a full model, the RSVM uses a reduced mixture with kernels sampled from certain candidate set. Our main results center on two major themes. One is the robustness of the random subset mixture model. The other is the spectral analysis of the reduced kernel. The robustness is judged by a few criteria as follows: 1) model variation measure; 2) model bias (deviation) between the reduced model and the full model; and 3) test power in distinguishing the reduced model from the full one. For the spectral analysis, we compare the eigenstructures of the full kernel matrix and the approximation kernel matrix. The approximation kernels are generated by uniform random subsets. The small discrepancies between them indicate that the approximation kernels can retain most of the relevant information for learning tasks in the full kernel. We focus on some statistical theory of the reduced set method mainly in the context of the RSVM. The use of a uniform random subset is not limited to the RSVM. This approach can act as a supplemental algorithm on top of a basic optimization algorithm, wherein the actual optimization takes place on the subset-approximated data. The statistical properties discussed in this paper are still valid. [ABSTRACT FROM AUTHOR]

In many cyclical processes it is of interest to have a system for on-line detection of turning points. This could be detection of the turns in leading economic indicators in order to predict the next turn of the business cycle. In natural family planning we want to detect the peak in the human menstrual cycle in order to predict the most fertile phase. Another example is detection of an influenza outbreak, by monitoring the weekly reports of the number of patients showing influenza-like symptoms. The methodology of statistical surveillance is used here to construct alarm systems, consisting of an alarm statistic and an alarm limit. At each time point a new observation becomes available and the alarm system is used to make a decision as to whether the turning point has occurred or not. The same methodology is used in control charts. Optimal alarm systems are based on the likelihood ratio, the ratio between the likelihood functions for the in-control process and the out-of-control process. The optimal likelihood ratio method for surveillance is based on the assumption that the parametric model for the cyclical process is known. This rather strong assumption can be avoided by using an approximation, the maximum likelihood ratio method, where a non-parametric estimation procedure is used. The estimation is made using only monotonicity restrictions, no parametric restrictions are placed on the cycles. Hereby, mis-specification of the parameters is avoided and thus we are sure that the false alarms are really controlled at the nominal level. The aim of this paper is to evaluate how two likelihood ratio based methods for on-line detection(the likelihood ratio method and the maximum likelihood ratio method) react to different assumption about the intensity, i.e., different assumption regarding how often we can expect a turn. Information about the previous intensity of the change point process is included in the methods by the empirical density of the change point times. This paper compares the performances of the likelihood ratio method and the maximum likelihood ratio method, when an empirical intensity is used(based on a bell shaped density) and when a constant intensity is used. In addition, these methods are compared and evaluated for the situation when no information about the time of the turn is used. The results of the study in this paper show that the likelihood ratio method with the empirical intensity works well except for early turns, when the time until detection is very long. For early turns the likelihood ratio method with a constant intensity without any information about the intensity, gives quicker detection. The maximum likelihood ratio method with the empirical intensity only works well for those turning point times that are the most likely according to the empirical density. Early turns take a long time to detect and late alarms have low predictive value. For the maximum likelihood ratio approach, the constant intensity gives the shortest expected delay and high predictive values. [ABSTRACT FROM AUTHOR]

Many idealized problems in signal processing, machine learning, and statistics can be reduced to the problem of finding the symmetric canonical decomposition of an underlying symmetric and orthogonally decomposable (SOD) tensor. Drawing inspiration from the matrix case, the successive rank-one approximation (SROA) scheme has been proposed and shown to yield this tensor decomposition exactly, and a plethora of numerical methods have thus been developed for the tensor rank-one approximation problem. In practice, however, the inevitable errors (say) from estimation, computation, and modeling necessitate that the input tensor can only be assumed to be a nearly SOD tensor--i.e., a symmetric tensor slightly perturbed from the underlying SOD tensor. This paper shows that even in the presence of perturbation, SROA can still robustly recover the symmetric canonical decomposition of the underlying tensor. It is shown that when the perturbation error is small enough, the approximation errors do not accumulate with the iteration number. Numerical results are presented to support the theoretical findings. [ABSTRACT FROM AUTHOR]

The article discusses moments of the distribution of sample size, N in a sequential probability ratio test (SPRT). The present paper provides variance, the third and the fourth moments of N. The details are worked out in five common applications of the SPRIT. The relation of the variance of N to the truncation of a SPRT is discussed is also discussed in the paper. Scholar A. Wald indicated in passing how one can obtain the moments of N, but the only published literature where the author encountered a general expression for the variance of N. However, their expression is incorrect. Using scholar J. Wolfowitz's results, which they do, or differentiating Wald's, fundamental identity twice one gets provided. In many practical applications of the SPRT, μ and moments in an equation derived are differentiable functions of a real-valued parameter. The limiting expressions for the moments can then be determined by standard methods of mathematical analysis. However, for the third and fourth moments the actual technique may involve an excessive amount of arithmetic.

The emerging collective motions of swarms of interacting agents are a subject of great interest in application areas ranging from biology to physics and robotics. In this paper, we conduct a careful analysis of the collective dynamics of a swarm of self-propelled heterogeneous, delay-coupled agents. We show the emergence of collective motion patterns and segregation of populations of agents with different dynamic properties; both of these behaviors (pattern formation and segregation) emerge naturally in our model, which is based on self-propulsion and attractive pairwise interactions between agents. We derive the bifurcation structure for emergence of different swarming behaviors in the mean field as a function of physical parameters and verify these results through simulation. [ABSTRACT FROM PUBLISHER]

We consider φ-divergence statistics Cϕ for the test of one factor independence from the other two in an r × s × t contingency table. Statistics Cϕ include the statistics Ra based on the power divergence as a special case. Statistic R0 is the log likelihood ratio statistic and R1 is Pearson's X2 statistic. Statistic R2/3 corresponds to the statistic for the goodness-of-fit test recommended by Cressie and Read (1984). Statistics Cϕ have the same chi-square limiting distribution under the hypothesis that one factor and the other two are independent. In this paper, when we assume that the distribution of Cϕ is continuous, we show the derivation of an expression of approximation based on a multivariate Edgeworth expansion for the distribution of Cϕ under the hypothesis that one factor and the other two are independent. Using the expression, we propose a new approximation of the distribution of Cϕ. In addition, on the basis of the approximation, we obtain transformed statistics that improve the speed of convergence to a chi-square limiting distribution of Cϕ. By numerical comparison in the case of Ra, we show that the transformed statistics perform well for a small sample. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper, we introduce generalized Baskakov-Szász type operators and study some approximation properties of these operators e.g., rate of convergence in ordinary and simultaneous approximation, statistical convergence and the estimate of the rate of convergence for absolutely continuous functions having a derivative coinciding a.e. with a function of bounded variation. [Copyright &y& Elsevier]

Abstract: The concept of statistical convergence was introduced by H. Fast, and studied by various authors. Recently, by using the idea of statistical convergence, M. Balcerzak, K. Dems and A. Komisarski introduced a new type of convergence for sequences of functions called equistatistical convergence. In the present paper we introduce the concepts of -statistical convergence and -statistical convergence of order . We show that -statistical convergence is a non-trivial extension of ordinary and statistical convergences. Moreover we show that -statistical convergence includes statistical convergence, -statistical convergence, and lacunary statistical convergence. We also introduce the concept of -equistatistical convergence which is a non-trivial extension of equistatistical convergence. Moreover, we prove that -equistatistical convergence lies between -statistical pointwise convergence and -statistical uniform convergence. Finally we prove Korovkin type approximation theorems via -statistical uniform convergence of order and -equistatistical convergence of order . [Copyright &y& Elsevier]

Let F be a set system over an underlying finite set X, and let μ be a nonnegative measure over X; i.e., for every S ⊆ X, μ(S) = Σx∈S μ(x). A measure μ* on X is called a multiplicative λ-approximation of μ on (F,X) if for every S ∈ F it holds that aμ(S) ≤ μ*(S) ≤ bμ(S), and b/a = λ ≥ 1. The central question raised and partially answered in the present paper is about the existence of meaningful structural properties of F implying that for any μ on X there exists an 1+∈/1-∈-approximation μ* supported on a small subset of X. It turns out that the parameter that governs the support size of a multiplicative approximation is the triangular rank of F, trk(F). It is defined as the maximal length of a sequence of sets {Si}i=1t in F such that for all 1 < i ≤ t, Si … ∪j