Confidence Corridors Research Articles

Global Inference and Test for Eigensystems of Imaging Data Over Complicated Domains

A nonparametric approach for analyzing eigensystems of image data over a complex domain is novelly developed. The proposed estimators, which are based on bivariate splines, have both oracle efficiency of O p ( n − 1 / 2 ) , meaning they are asymptotically indistinguishable from estimators computed with true trajectories, and computational efficiency with more convenient spectrum decomposition forms. Under mild conditions, we derive consistency and weak convergence of our proposed estimators, which can be used to construct confidence intervals and simultaneous confidence corridors for any individual eigenvalue and eigenfunction, as well as design uniform inference procedures for eigensystems with a diverging number of components. In addition, we extend our method to two-sample test problems. Extensive simulation results provide strong support for the asymptotic theory, and the procedures are applied to two potential seawater temperature data sets to demonstrate its validity.

Functional Data Analysis for Imaging Mean Function Estimation: Computing Times and Parameter Selection

In the field of medical imaging, one of the most extended research setups consists of the comparison between two groups of images, a pathological set against a control set, in order to search for statistically significant differences in brain activity. Functional Data Analysis (FDA), a relatively new field of statistics dealing with data expressed in the form of functions, uses methodologies which can be easily extended to the study of imaging data. Examples of this have been proposed in previous publications where the authors settle the mathematical groundwork and properties of the proposed estimators. The methodology herein tested allows for the estimation of mean functions and simultaneous confidence corridors (SCC), also known as simultaneous confidence bands, for imaging data and for the difference between two groups of images. FDA applied to medical imaging presents at least two advantages compared to previous methodologies: it avoids loss of information in complex data structures and avoids the multiple comparison problem arising from traditional pixel-to-pixel comparisons. Nonetheless, computing times for this technique have only been explored in reduced and simulated setups. In the present article, we apply this procedure to a practical case with data extracted from open neuroimaging databases; then, we measure computing times for the construction of Delaunay triangulations and for the computation of mean function and SCC for one-group and two-group approaches. The results suggest that the previous researcher has been too conservative in parameter selection and that computing times for this methodology are reasonable, confirming that this method should be further studied and applied to the field of medical imaging.

Assessment of the electrical conductivity of thermally activated water jets containing injections of inhibiting salt used to extinguish electrical equipment at gas compressor stations

Introduction. One of the main problems of fire extinguishing at gas compressor stations is fires in confined spaces where electrical installations can be under voltage. To extinguish these fires, the co-authors propose to use jets of thermally activated water containing salts that inhibit burning. It is necessary to assess the electrical conductivity of water jets to prevent the exposure of fire-fighting units to electrical injuries.Materials and methods. In the experimental study, the principal electricity measuring instrument is digital dc megaohmmeter M4122U, which takes readings if operated from a laptop. The flow rate was measured using IT 2518 measurement complex and TDR14–2–3 primary flow converter of the turbine type. Termit T-35 hardness converter performed electromagnetic water treatment.Theoretical fundamentals. Measurements of leakage currents in jets of thermally activated water, fed through fire nozzles mounted onto a multi-purpose fire truck, were taken, and the maximum permissible distance to an electrical installation was determined experimentally and analytically using an experimental stand.Results and discussions. Averaged values of resistance of a jet of thermally activated water fed from a long-range nozzle at control points do not leave the confidence corridor with a probability of 0.95, which means that they do not cause a significant change in dc resistance. Inhibiting salt, injected into the water, reduces the resistance of jets of thermally activated water by no more than 2–3 % compared to under-heated water. The treatment of the inhibiting salt, injected into the under-heated water, using Termit T-35 hardness converter, does not significantly change the resistance of thermally activated water jets.Conclusions. It is theoretically and experimentally proved that the injection of inhibiting salts into the underheated water, as well as the electromagnetic treatment of water using low-power stiffness converters, do not significantly affect the electrical conductivity of jets of thermally activated water if compared to jets that contain no inhibiting admixtures. The extinguishing of surfaces of electrical installations using jets of thermally activated water containing flame inhibiting injections of salts, as well as the development of an inerting concentration in confined spaces of gas compressor stations is electrically safe for any person engaged in fire extinguishing.

Estimation of Uncertainty of a Volumetric Reservoir Model and Optimization of Numerical Inversion in Well Log Data Interpretation

This study considers an approach to joint inversion of logging data in oil and gas wells for finding the volume fractions of reservoir components. The inversion is reduced to a linear least squares problem with additional iterations to account for physical constraints. To analyse the quality of inversion, uncertainty evaluation tools are developed for both the model responses and inversion results – volume fractions of the reservoir components. The tools include algorithms for construction of depth-related “confidence corridors” for individual properties and standard deviation ellipses to analyse cross-correlations. An optimization problem of improving the quality of inversion by tuning specific readings of logging methods for individual reservoir layers (or facies/lithotypes) is also posed and solved by minimizing a mismatch criterion between the actual and simulated log curves. Thus, the developed toolkit provides a means for optimized joint inversion of logging data with visual control of uncertainty and cross-correlation in the inversion results.

Multivariate Spline Estimation and Inference for Image-on-Scalar Regression

Motivated by recent data analyses in biomedical imaging studies, we consider a class of image-on-scalar regression models for imaging responses and scalar predictors. We propose using flexible multivariate splines over triangulations to handle the irregular domain of the objects of interest on the images, as well as other characteristics of images. The proposed estimators of the coefficient functions are proved to be root-n consistent and asymptotically normal under some regularity conditions. We also provide a consistent and computationally efficient estimator of the covariance function. Asymptotic pointwise confidence intervals and data-driven simultaneous confidence corridors for the coefficient functions are constructed. Our method can simultaneously estimate and make inferences on the coefficient functions while incorporating spatial heterogeneity and spatial correlation. A highly efficient and scalable estimation algorithm is developed. Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed method, which is then applied to the spatially normalized positron emission tomography data of the Alzheimer's Disease Neuroimaging Initiative.

Certainty of Linear Interrelation of Technical and Economic Parameters of Power Units of 300 МWt Thermal Power Stations

For 300 MWt gas-and-oil-burning power units which durability exceeds the estimated one, when comparing and ranking, it is necessary to take into account not only efficiency indicators (for example, the specific consumption of conventional fuel), but also indicators of reliability and safety. In practice, this is exactly what is being done. However, it is being done just by intuition. An intuitive approach solves this problem, but not always certainly. It would seem that there is a quite fully developed mathematical apparatus for regression and correlation analysis, a set of algorithms and calculation programs. But there is one specific feature that, if it is not taken into account, further increases the risk of an erroneous decision. The fact is that the average monthly data on the technical and economic indicators of power units do not belong to the samples from the general population that correspond to the normal distribution law, the latter being a necessary condition for using such methods. This is a non-random sample from a finite set of multidimensional data. Naturally, the methods of classification of multidimensional data are not simple; they require the development of special calculation programs that recommend solutions for maintenance and repair, load distribution, etc. The article presents only one, but very important issue of the problem, viz. the assessment of the reliability of the assumption about the linear relationship of technical and economic indicators. Its solution will simultaneously demonstrate difficulties in comparing the efficiency of aging power units. It is noted that the known and practically used method for assessing the reliability of the linear regression equation, based on the construction of a “confidence corridor” or “uncertainty band”, does not allow one to answer the main question, viz. whether the relationship of the considered technical and economic indicators corresponds to a linear one. A new method for evaluating this relationship is proposed, based on constructing a fiducial domain of possible regression line implementations. It is shown that for small values of the number of sample implementations, a significant part of independent samples has a correlation coefficient exceeding 0.9.

Simultaneous confidence corridors for mean functions in functional data analysis of imaging data.

Motivated by recent work involving the analysis of biomedical imaging data, we present a novel procedure for constructing simultaneous confidence corridors for the mean of imaging data. We propose to use flexible bivariate splines over triangulations to handle an irregular domain of the images that is common in brain imaging studies and in other biomedical imaging applications. The proposed spline estimators of the mean functions are shown to be consistent and asymptotically normal under some regularity conditions. We also provide a computationally efficient estimator of the covariance function and derive its uniform consistency. The procedure is also extended to the two-sample case in which we focus on comparing the mean functions from two populations of imaging data. Through Monte Carlo simulation studies, we examine the finite sample performance of the proposed method. Finally, the proposed method is applied to analyze brain positron emission tomography data in two different studies. One data set used in preparation of this article was obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database.

Extreme Conditional Tail Moment Estimation under Serial Dependence

Abstract A wide range of risk measures can be written as functions of conditional tail moments (CTMs) and value-at-risk (VaR), for instance the expected shortfall (ES). In this paper, we derive joint central limit theory for semi-parametric estimates of CTMs, including in particular ES, at arbitrarily small risk levels. We also derive confidence corridors for VaR at different levels far out in the tail, which allows for simultaneous inference. We work under a semi-parametric Pareto-type assumption on the distributional tail of the observations and only require an extremal-near epoch dependence assumption on the serial dependence. In simulations, our semi-parametric ES estimate is often shown to be more accurate in terms of mean absolute deviation than extant non- and semi-parametric estimates. An empirical application to the extreme swings in Volkswagen log-returns during the failed takeover attempt by Porsche illustrates the proposed methods.

Statistical inference for generalized additive partially linear models

The Generalized Additive Model (GAM) is a powerful tool and has been well studied. This model class helps to identify additive regression structure. Via available test procedures one may identify the regression structure even sharper if some component functions have parametric form. The Generalized Additive Partially Linear Models (GAPLM) enjoy the simplicity of the GLM and the flexibility of the GAM because they combine both parametric and nonparametric components. We use the hybrid spline-backfitted kernel estimation method, which combines the best features of both spline and kernel methods for making fast, efficient and reliable estimation under alpha-mixing condition. In addition, simultaneous confidence corridors (SCCs) for testing overall trends and empirical likelihood confidence region for parameters are provided under independent condition. The asymptotic properties are obtained and simulation results support the theoretical properties. For the application, we use the GAPLM to improve the accuracy ratio of the default predictions for $19610$ German companies. The quantlets for this paper are available on https://github.com.

A Confidence Corridor for Expectile Functions

Let (X1; Y1), …, (Xn; Yn) be i.i.d. rvs and denote v(x) be the unknown τ - expectile regression curve of Y conditional on X. An expectile-smoother vn(x) is a localized, nonlinear estimator of v(x). The strong uniform consistency rate is established under general conditions. In many applications it is necessary to know the stochastic fluctuation of the process {vn(x) – v(x)}. Using strong approximations of the empirical process and extreme value theory, we consider the asymptotic maximal deviation sup0≤x≤1 |vn(x) – v(x)|. The derived result helps in the construction of a uniform confidence band for the expectile curve v(x). This paper considers fitting a simultaneous confidence corridor (SCC) around the estimated expectile function of the conditional distribution of Y given x based on the observational data generated according to a nonparametric regression model. Moreover, we construct the simultaneous confidence corridors around the expectiles of the residuals from the temperature models to investigate the temperature risk drivers.

A Confidence Corridor for Sparse Longitudinal Data Curves

Longitudinal data analysis is a central piece of statistics. The data are curves and they are observed at random locations. This makes the construction of a simultaneous confidence corridor (SCC) (confidence band) for the mean function a challenging task on both the theoretical and the practical side. Here we propose a method based on local linear smoothing that is implemented in the sparse (i.e., low number of nonzero coefficients) modelling situation. An SCC is constructed based on recent results obtained in applied probability theory. The precision and performance is demonstrated in a spectrum of simulations and applied to growth curve data. Technically speaking, our paper intensively uses recent insights into extreme value theory that are also employed to construct a shoal of confidence intervals (SCI).

Simultaneous Confidence Corridors and Variable Selection for Generalized Additive Models

In spite of the widespread use of generalized additive models (GAMs), there is no well established methodology for simultaneous inference and variable selection for the components of GAM. There is no doubt that both, inference on the marginal component functions and their selection, are essential in this additive statistical models. To this end, we establish simultaneous confidence corridors (SCCs) and a variable selection criteria through the spline-backfitted kernel smoothing techniques. To characterize the global features of each component, SCCs are constructed for testing their shapes. By extending the BIC to additive models with identity/trivial link, an asymptotically consistent BIC approach for variable selection is proposed. Our procedures are examined in simulations for its theoretical accuracy and performance, and used to forecast the default probability of listed Japanese companies.

Confidence Corridors for Multivariate Generalized Quantile Regression

We focus on the construction of confidence corridors for multivariate nonparametric generalized quantile regression functions. This construction is based on asymptotic results for the maximal deviation between a suitable nonparametric estimator and the true function of interest, which follow after a series of approximation steps including a Bahadur representation, a new strong approximation theorem, and exponential tail inequalities for Gaussian random fields. As a byproduct we also obtain multivariate confidence corridors for the regression function in the classical mean regression. To deal with the problem of slowly decreasing error in coverage probability of the asymptotic confidence corridors, which results in meager coverage for small sample sizes, a simple bootstrap procedure is designed based on the leading term of the Bahadur representation. The finite-sample properties of both procedures are investigated by means of a simulation study and it is demonstrated that the bootstrap procedure considerably outperforms the asymptotic bands in terms of coverage accuracy. Finally, the bootstrap confidence corridors are used to study the efficacy of the National Supported Work Demonstration, which is a randomized employment enhancement program launched in the 1970s. This article has supplementary materials online.

On the search for a (n,f) cross-section reference at intermediate energies

The (n,f) cross-sections proposed as references by the IAEA for 235U, 238U and 209Bi are compared with a new analysis that combines the measurements performed at CERN-n_TOF of their cross-section ratios with new calculations done using Monte Carlo codes based on phenomenological models INCL+ +, GEMINI+ +, and ABLA07. The calculations are cross-checked with those for the (p,f) reactions, where experimental values are available. We have evaluated in this way the (n,f) cross sections for 238U, 235U and 209Bi, in the intermediate energy region going from 190 MeV to 2 GeV. Our results definitively discard the JENDL/HE-2007 evaluations above 300 MeV, falling inside the confidence corridor proposed by IAEA but for the points around 300–400 MeV where a discrepancy is to be noticed.

Statistical inference for generalized additive models: simultaneous confidence corridors and variable selection

In spite of widespread use of generalized additive models (GAMs) to remedy the “curse of dimensionality”, there is no well-grounded methodology developed for simultaneous inference and variable selection for GAM in existing literature. However, both are essential in enhancing the capability of statistical models. To this end, we establish simultaneous confidence corridors (SCCs) and a type of Bayesian information criterion (BIC) through the spline-backfitted kernel smoothing techniques proposed in recent articles. To characterize the global features of each non-parametric components, SCCs are constructed for testing their overall trends and entire shapes. By extending the BIC in additive models with identity/trivial link, an asymptotically consistent BIC approach for variable selection is built up in GAM to improve the parsimony of model without loss of prediction accuracy. Simulations and a real example corroborate the above findings.

A simultaneous confidence corridor for varying coefficient regression with sparse functional data

We consider a varying coefficient regression model for sparse functional data, with time varying response variable depending linearly on some time-independent covariates with coefficients as functions of time-dependent covariates. Based on spline smoothing, we propose data-driven simultaneous confidence corridors for the coefficient functions with asymptotically correct confidence level. Such confidence corridors are useful benchmarks for statistical inference on the global shapes of coefficient functions under any hypotheses. Simulation experiments corroborate with the theoretical results. An example in CD4/HIV study is used to illustrate how inference is made with computable p values on the effects of smoking, pre-infection CD4 cell percentage and age on the CD4 cell percentage of HIV infected patients under treatment.

A Smooth Simultaneous Confidence Corridor for the Mean of Sparse Functional Data

Functional data analysis (FDA) has become an important area of statistics research in the recent decade, yet a smooth simultaneous confidence corridor (SCC) does not exist in the literature for the mean function of sparse functional data. SCC is a powerful tool for making statistical inference on an entire unknown function, nonetheless classic “Hungarian embedding” techniques for establishing asymptotic correctness of SCC completely fail for sparse functional data. We propose a local linear SCC and a shoal of confidence intervals (SCI) for the mean function of sparse functional data, and establish that it is asymptotically equivalent to the SCC of independent regression data, using new results from Gaussian process extreme value theory. The SCC procedure is examined in simulations for its superior theoretical accuracy and performance, and used to analyze growth curve data, confirming findings with quantified high significance levels. Supplementary materials for this article are available online.

Simultaneous Confidence Corridors and Variable Selection for Generalized Additive Models

In spite of the widespread use of generalized additive models (GAMs), there is no well established methodology for simultaneous inference and variable selection for the components of GAM. There is no doubt that both, inference on the marginal component functions and their selection, are essential in this additive statistical models. To this end, we establish simultaneous confidence corridors (SCCs) and a variable selection criteria through the spline-backfitted kernel smoothing techniques. To characterize the global features of each component, SCCs are constructed for testing their shapes. By extending the BIC to additive models with identity/trivial link, an asymptotically consistent BIC approach for variable selection is proposed. Our procedures are examined in simulations for its theoretical accuracy and performance, and used to forecast the default probability of listed Japanese companies.

A Simultaneous Confidence Corridor for Varying Coefficient Regression with Sparse Functional Data

We consider a varying coefficient regression model for sparse functional data, with time varying response variable depending linearly on some time independent covariates with coefficients as functions of time dependent covariates. Based on spline smoothing, we propose data driven simultaneous confidence corridors for the coefficient functions with asymptotically correct confidence level. Such confidence corridors are useful benchmarks for statistical inference on the global shapes of coefficient functions under any hypotheses. Simulation experiments corroborate with the theoretical results. An example in CD4/HIV study is used to illustrate how inference is made with computable p-values on the effects of smoking, preinfection CD4 cell percentage and age on the CD4 cell percentage of HIV infected patients under treatment.

A Confidence Corridor for Expectile Functions

Let (X1; Y1), ..., (Xn; Yn) be i.i.d. rvs and let v(x) be the unknown tau - expectile regression curve of Y conditional on X. An expectile-smoother vn(x) is a localized, nonlinear estimator of v(x). The strong uniform consistency rate is established under general conditions. In many applications it is necessary to know the stochastic fluctuation of the process {vn(x)-v(x)}. Using strong approximations of the empirical process and extreme value theory, we consider the asymptotic maximal deviation sup06x61 jvn(x)o€€€v(x)j. The derived result helps in the construction of a uniform confidence band for the expectile curve v(x). This paper considers fitting a simultaneous confidence corridor (SCC) around the estimated expectile function of the conditional distribution of Y given x based on the observational data generated according to a nonparametric regression model. Moreover, we construct the simultaneous confidence corridors around the expectiles of the residuals from the temperature models to investigate the temperature risk drivers.