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By
Jagadeeswaran, R.; Hickernell, Fred J.
1 Citations
Automatic cubatures approximate integrals to userspecified error tolerances. For highdimensional problems, it is difficult to adaptively change the sampling pattern, but one can automatically determine the sample size, n, given a reasonable, fixed sampling pattern. We take this approach here using a Bayesian perspective. We postulate that the integrand is an instance of a Gaussian stochastic process parameterized by a constant mean and a covariance kernel defined by a scale parameter times a parameterized function specifying how the integrand values at two different points in the domain are related. These hyperparameters are inferred or integrated out using integrand values via one of three techniques: empirical Bayes, full Bayes, or generalized crossvalidation. The sample size, n, is increased until the halfwidth of the credible interval for the Bayesian posterior mean is no greater than the error tolerance. The process outlined above typically requires a computational cost of
$$O(N_{\text {opt}}n^3)$$
, where
$$N_{\text {opt}}$$
is the number of optimization steps required to identify the hyperparameters. Our innovation is to pair low discrepancy nodes with matching covariance kernels to lower the computational cost to
$$O(N_{\text {opt}} n \log n)$$
. This approach is demonstrated explicitly with rank1 lattice sequences and shiftinvariant kernels. Our algorithm is implemented in the Guaranteed Automatic Integration Library (GAIL).
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By
Takagishi, Mariko ; Yadohisa, Hiroshi
2 Citations
In functional data analysis, it is often of interest to discover a general common pattern, or shape, of the function. When the subjectspecific amplitude and phase variation of data are not of interest, curve registration can be used to separate the variation from the data. Shapeinvariant models (SIM), one of the registration methods, aim to estimate the unknown sharedshape function. However, the use of SIM and of general registration methods assumes that all curves have the sharedshape in common and does not consider the existence of outliers, such as a curve, whose shape is inconsistent with the remainder of the data. Therefore, we propose using the t distribution to robustify SIMs, allowing outliers of amplitude, phase, and other errors. Our SIM can identify and classify the three types of outliers mentioned above. We use simulation and an empirical data set to evaluate the performance of our robust SIM.
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By
Hornung, Roman
The ordinal forest method is a random forest–based prediction method for ordinal response variables. Ordinal forests allow prediction using both lowdimensional and highdimensional covariate data and can additionally be used to rank covariates with respect to their importance for prediction. An extensive comparison study reveals that ordinal forests tend to outperform competitors in terms of prediction performance. Moreover, it is seen that the covariate importance measure currently used by ordinal forest discriminates influential covariates from noise covariates at least similarly well as the measures used by competitors. Several further important properties of the ordinal forest algorithm are studied in additional investigations. The rationale underlying ordinal forests of using optimized score values in place of the class values of the ordinal response variable is in principle applicable to any regression method beyond random forests for continuous outcome that is considered in the ordinal forest method.
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By
Jorgensen, Terrence D.; GarnierVillarreal, Mauricio; Pornprasermanit, Sunthud; Lee, Jaehoon
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We simulated Bayesian CFA models to investigate the power of PPP to detect model misspecification by manipulating sample size, strongly and weakly informative priors for nontarget parameters, degree of misspecification, and whether data were generated and analyzed as normal or ordinal. Rejection rates indicate that PPP lacks power to reject an inappropriate model unless priors are unrealistically restrictive (essentially equivalent to fixing nontarget parameters to zero) and both sample size and misspecification are quite large. We suggest researchers evaluate global fit without priors for nontarget parameters, then search for neglected parameters if PPP indicates poor fit.
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By
Giordano, Francesco; Coretto, Pietro
Signaltonoise ratio (SNR) statistics play a central role in many applications. A common situation where SNR is studied is when a continuous time signal is sampled at a fixed frequency with some noise in the background. While estimation methods exist, little is known about its distribution when the noise is not weakly stationary. In this paper we develop a nonparametric method to estimate the distribution of an SNR statistic when the noise belongs to a fairly general class of stochastic processes that encompasses both short and longrange dependence, as well as nonlinearities. The method is based on a combination of smoothing and subsampling techniques. Computations are only operated at the subsample level, and this allows to manage the typical enormous sample size produced by modern data acquisition technologies. We derive asymptotic guarantees for the proposed method, and we show the finite sample performance based on numerical experiments. Finally, we propose an application to electroencephalography data.
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By
Chernoyarov, O. V.; Dachian, S.; Kutoyants, Yu. A.
1 Citations
This work is devoted to the problem of estimation of the localization of Poisson source. The observations are inhomogeneous Poisson processes registered by more than three detectors on the plane. We study the behavior of the Bayes estimators in the asymptotic of large intensities. It is supposed that the intensity functions of the signals arriving in the detectors have cusptype singularity. We show the consistency, limit distributions, the convergence of moments and asymptotic efficiency of these estimators.
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By
Li, KimHung ; Li, Cheuk Ting
In this paper we prove a recursive identity for the cumulative distribution function of a linear combination of independent exponential random variables. The result is then extended to probability density function, expected value of functions of a linear combination of independent exponential random variables, and other functions. Our goal is on the exact and approximate calculation of the above mentioned functions and expected values. We study this computational problem from different views, namely as a Hermite interpolation problem, and as a matrix function evaluation problem. Examples are presented to illustrate the applicability and performance of the methods.
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By
Lima, Italo R.; Cao, Guanqun; Billor, Nedret
Estimating and constructing a simultaneous confidence band for the mean function in the presence of outliers is an important problem in the framework of functional data analysis. In this paper, we propose a robust estimator and a robust simultaneous confidence band for the mean function of functional data using Mestimation and Bsplines. The robust simultaneous confidence band is also extended to the difference of mean functions of two populations. Further, the asymptotic properties of the Mbased mean function estimator, such as the asymptotic consistency and asymptotic normality, are studied. The performance of the proposed robust methods and their robustness are demonstrated with an extensive simulation study and two real data examples.
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By
Krickeberg, Klaus; Van Trong, Pham; Thi My Hanh, Pham
In most if not all countries of the world Public Health is being built and operated in a piecemeal way. Its various parts are rarely linked to each other by an allembracing concept and often not by their practice either. The reasons are largely historical. A particular Public Health Service was created when there seemed to be a more or less urgent need. Screening in schools for tooth decay, bad eyesight and other ailments is a typical example. This practice started about one and a half centuries ago. Preventive measures such as hygiene, clean drinking water, sewage systems, and recommendations about nutrition and physical activities are documented from ancient Egypt, China, Greece and Rome but not as parts of a general systematic effort. Immunizations followed in the nineteenth Century after some forerunners; see Sect.
5.1
.
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