Residual Bootstrap Research Articles

Confidence bands for inverse regression models

We construct uniform confidence bands for the regression function in inverse, homoscedastic regression models with convolution-type operators. Here, the convolution is between two non-periodic functions on the whole real line rather than between two periodic functions on a compact interval, since the former situation arguably arises more often in applications. First, following Bickel and Rosenblatt (1973 Ann. Stat. 1 1071–95) we construct asymptotic confidence bands which are based on strong approximations and on a limit theorem for the supremum of a stationary Gaussian process. Further, we propose bootstrap confidence bands based on the residual bootstrap and prove consistency of the bootstrap procedure. A simulation study shows that the bootstrap confidence bands perform reasonably well for moderate sample sizes. Finally, we apply our method to data from a gel electrophoresis experiment with genetically engineered neuronal receptor subunits incubated with rat brain extract.

INFERENCE OF GENETIC NETWORKS USING LPMs: ASSESSMENT OF CONFIDENCE VALUES OF REGULATIONS

When we apply inference methods based on a set of differential equations into actual genetic network inference problems, we often end up with a large number of false-positive regulations. However, as we must check the inferred regulations through biochemical experiments, fewer false-positive regulations are preferable. In order to reduce the number of regulations checked, this study proposes a new method that assigns confidence values to all of the regulations contained in the target network. For this purpose, we combine a residual bootstrap method with the existing method, i.e. the inference method using linear programming machines (LPMs). Through numerical experiments on an artificial genetic network inference problem, we confirmed that most of the regulations with high confidence values are actually present in the target networks. We then used the proposed method to analyze the bacterial SOS DNA repair system, and succeeded in assigning reasonable confidence values to its regulations. Although this study combined the bootstrap method with the inference method using the LPMs, the proposed bootstrap approach could be combined with any method that has an ability to infer a genetic network from time-series of gene expression levels.

Asymptotic properties of the residual bootstrap for Lasso estimators

In this article, we derive the asymptotic distribution of the bootstrapped Lasso estimator of the regression parameter in a multiple linear regression model. It is shown that under some mild regularity conditions on the design vectors and the regularization parameter, the bootstrap approximation converges weakly to a random measure. The convergence result rigorously establishes a previously known heuristic formula for the limit distribution of the bootstrapped Lasso estimator. It is also shown that when one or more components of the regression parameter vector are zero, the bootstrap may fail to be consistent.

Angular diameter estimation of interferometric calibrators

<i>Context. <i/>Accurate long-baseline interferometric measurements require careful calibration with reference stars. Small calibrators with high angular diameter accuracy ensure the true visibility uncertainty to be dominated by the measurement errors.<i>Aims. <i/>We review some indirect methods for estimating angular diameter, using various types of input data. Each diameter estimate, obtained for the test-case calibrator star <i>λ<i/> Gru, is compared with the value 2.71 mas found in the Bordé calibrator catalogue published in 2002.<i>Methods. <i/>Angular size estimations from spectral type, spectral index, in-band magnitude, broadband photometry, and spectrophotometry give close estimates of the angular diameter, with slightly variable uncertainties. Fits on photometry and spectrophotometry need physical atmosphere models with “plausible” stellar parameters. Angular diameter uncertainties were estimated by means of residual bootstrapping confidence intervals. All numerical results and graphical outputs presented in this paper were obtained using the routines developed under PV-WAVE®, which compose the modular software suite SPIDAST, created to calibrate and interprete spectroscopic and interferometric measurements, particularly those obtained with VLTI-AMBER.<i>Results. <i/>The final angular diameter estimate 2.70 mas of <i>λ<i/> Gru, with 68% confidence interval 2.65–2.81 mas, is obtained by fit of the MARCS model on the ISO-SWS 2.38–27.5 <i>μ<i/>m spectrum, with the stellar parameters <i>T<i/><sub>e<sub/> = 4250 K, log <i>g<i/> = 2.0, <i>z<i/> = 0.0 dex, = 1.0 , and = 2.0 km s<sup>-1<sup/>.

Density Forecasting for Long-Term Peak Electricity Demand

Long-term electricity demand forecasting plays an important role in planning for future generation facilities and transmission augmentation. In a long-term context, planners must adopt a probabilistic view of potential peak demand levels. Therefore density forecasts (providing estimates of the full probability distributions of the possible future values of the demand) are more helpful than point forecasts, and are necessary for utilities to evaluate and hedge the financial risk accrued by demand variability and forecasting uncertainty. This paper proposes a new methodology to forecast the density of long-term peak electricity demand. Peak electricity demand in a given season is subject to a range of uncertainties, including underlying population growth, changing technology, economic conditions, prevailing weather conditions (and the timing of those conditions), as well as the general randomness inherent in individual usage. It is also subject to some known calendar effects due to the time of day, day of week, time of year, and public holidays. A comprehensive forecasting solution is described in this paper. First, semi-parametric additive models are used to estimate the relationships between demand and the driver variables, including temperatures, calendar effects and some demographic and economic variables. Then the demand distributions are forecasted by using a mixture of temperature simulation, assumed future economic scenarios, and residual bootstrapping. The temperature simulation is implemented through a new seasonal bootstrapping method with variable blocks. The proposed methodology has been used to forecast the probability distribution of annual and weekly peak electricity demand for South Australia since 2007. The performance of the methodology is evaluated by comparing the forecast results with the actual demand of the summer 2007-2008.

Identifying patients with diabetic nephropathy based on serum creatinine under zero truncated model

Diabetes mellitus is a clinical syndrome characterized by hyperglycemia due to absolute or relative insufficiency of insulin in the body. Diabetic nephropathy is a generic term referring to deleterious effect on renal structure and/or function caused by diabetes mellitus. In this paper, we estimate the probability of occurrence of diabetic nephropathy, taking serum creatinine as a marker for renal function/dysfunction. We adopted a Zero truncated binomial distribution (ZTBD) with parameters (n i , p), where p, the probability that serum creatinine ≥ cutoff value, was unknown. Maximum likelihood and residual bootstrapping methods were used to estimate p. Retrospective data was collected from 132 patients diagnosed as diabetic as per ADA standards. Out of the available data of 132 patients, 72 patients had no diabetic renal complications with serum creatinine mean ± s.d as 0.9774± 0.12508 and 60 patients had certain diabetic complications with serum creatinine mean ± s.d. as 1.6462 ±0.28827.The mean ± s.d. for duration of disease for 60 patients came out to be 15.46667± 5.54 (yrs) and median as 17.1 yrs. The two groups of patients were found to be significantly different with p<.001. The value of p came out to be 0.4555397 and 0.445545 using maximum likelihood and residual bootstrapping methods respectively. It was found that 60 patients had non -zero probability of renal disease under ZTBD. From this study, it was concluded that the duration of diabetes along with elevated levels of serum creatinine defines a high risk group for the diabetic nephropathy.

Bootstrap quantification of cardiac pulsation artifact in DTI

While several studies have shown the benefit of cardiac gating in diffusion MRI with single-shot EPI acquisition, cardiac gating is still not commonly used. This is probably because it requires additional time and many investigators may not be convinced that cardiac gating is worth the extra effort. Here, we tested a clinically feasible protocol with a minimal increase in scan time, and quantified the effect of cardiac gating under partial or full Fourier acquisition. Eight volunteers were scanned on a 3 T scanner with a SENSE 8-channel head coil. Diffusion-weighted, single-shot spin-echo EPI images were acquired along 32 gradient directions, with or without cardiac gating and with partial or full Fourier acquisition. Vectorcardiography (VCG) was used to trigger acquisition at a minimum delay (30 ms). The uncertainties of DTI derived parameters were estimated using residual bootstrap. With partial Fourier, cardiac gating reduced the uncertainties, and better efficiency in reducing DTI parameter variability was also achieved even allowing for the increase in total scan time. For full Fourier acquisition, minimum time gating slightly decreased the uncertainties but the efficiency was worse. A minimum trigger delay might not be the optimal scheme to avoid the majority of systole but it allows clinically acceptable scan times. We have demonstrated that cardiac gating, especially of partial Fourier acquisitions, can reduce the uncertainties of DTI derived parameters in a time-efficient manner.

Probabilistic Tracking can improve the delineation of cortico-spinal tract for neurosurgical planning

Problem: White matter fiber tracking based on diffusion MRI is the only technique able to localize the white matter pathways in the brain in vivo. There has been tremendous recent interest in utilization of diffusion MRI fiber tracking for mapping eloquent regions during brain surgery. While some manufacturers have begun to address this demand, the algorithms used tend to be the most basic determinsitic tracking. Furthermore, there is very little data supporting the accuracy of any fiber tracking method in this context where the pathways could be damaged or displaced. Intraoperative electrical stimulation (IES) provides a clinical gold standard for the existence of a functional motor pathway that can be used to determine the accuracy of fiber tracking algorithms. Probabilistic fiber tracking algorithms explore other “high probability” pathways. Residual bootstrap is a non-parametric statistical technique based on data resampling that has been showed to be useful in probabilistic tractography and in assessing DTI methods. In this work we used IES data to evaluate the importance of probabilistic versus deterministic tractography, seeding density, and choice of seed regions.

PARTIALLY LINEAR MODEL SELECTION BY THE BOOTSTRAP

SummaryWe propose a new approach to the selection of partially linear models based on the conditional expected prediction square loss function, which is estimated using the bootstrap. Because of the different speeds of convergence of the linear and the nonlinear parts, a key idea is to select each part separately. In the first step, we select the nonlinear components using an ‘m‐out‐of‐n’ residual bootstrap that ensures good properties for the nonparametric bootstrap estimator. The second step selects the linear components from the remaining explanatory variables, and the non‐zero parameters are selected based on a two‐level residual bootstrap. We show that the model selection procedure is consistent under some conditions, and our simulations suggest that it selects the true model most often than the other selection procedures considered.

Smooth Residual Bootstrap for Empirical Processes of Non‐parametric Regression Residuals

Abstract. The aim of this paper is to prove the validity of smooth residual bootstrap versions of procedures that are based on the empirical process of residuals estimated from a non‐parametric regression model. From this result, consistency of various model tests in non‐parametric regression is deduced, such as goodness‐of‐fit tests for the regression and variance function, tests for equality of regression functions and tests concerning the error distribution.

Asymptotic normality and confidence intervals for inverse regression models with convolution-type operators

We consider inverse regression models with convolution-type operators which mediate convolution on R d ( d ≥ 1 ) and prove a pointwise central limit theorem for spectral regularisation estimators which can be applied to construct pointwise confidence regions. Here, we cope with the unknown bias of such estimators by undersmoothing. Moreover, we prove consistency of the residual bootstrap in this setting and demonstrate the feasibility of the bootstrap confidence bands at moderate sample sizes in a simulation study.

Robustness of residual-based bootstrap to the composition of serially correlated errors

In a simple autoregressive model with serially correlated errors, we evaluate size distortions resulting from the residual bootstrap when the Wold innovation is serially dependent and hence is expected to contaminate the inference. The small distortions caused by the presence of strong conditional heteroskedasticity or other nonlinearities can be partly removed further by using the wild bootstrap.

Challenges in developing confidence intervals on modal parameters estimated for large civil infrastructure with stochastic subspace identification

This paper examines current methods for estimating uncertainty and confidence intervals on modal parameters estimated using stochastic subspace identification. It is found that most perturbation methods have been limited to estimating variance and not higher moments of the identified modal parameters. It is shown that the modal parameters may exhibit non-normal distributions and in such cases the variance is insufficient in estimating confidence intervals. A current perturbation method is extended to estimate higher moments but it is shown that the inaccuracy in estimating the covariance of the covariance function limits the accuracy of this approach. A residual bootstrapping approach is then investigated; however, it is found with both numerical and measured data that the residuals are not all reduced to white series. It is also found that the excitation of the suspension bridge investigated is non-Gaussian distributed. These are serious difficulties in applying a residual bootstrapping procedure reliably. Based on these investigations, some current challenges in obtaining accurate confidence intervals on estimated modal parameters for large civil infrastructure are summarized. Copyright © 2009 John Wiley & Sons, Ltd.

Yule–Walker type estimators in periodic bilinear models: strong consistency and asymptotic normality

This paper studies the covariance structure and the asymptotic properties of Yule–Walker (YW) type estimators for a bilinear time series model with periodically time-varying coefficients. We give necessary and sufficient conditions ensuring the existence of moments up to eighth order. Expressions of second and third order joint moments, as well as the limiting covariance matrix of the sample moments are given. Strong consistency and asymptotic normality of the YW estimator as well as hypotheses testing via Wald’s procedure are derived. We use a residual bootstrap version to construct bootstrap estimators of the YW estimates. Some simulation results will demonstrate the large sample behavior of the bootstrap procedure.

Using the Model-Based Residual Bootstrap to Quantify Uncertainty in Fiber Orientations From $Q$-Ball Analysis

Bootstrapping of repeated diffusion-weighted image datasets enables nonparametric quantification of the uncertainty in the inferred fiber orientation. The wild bootstrap and the residual bootstrap are model-based residual resampling methods which use a single dataset. Previously, the wild bootstrap method has been presented as an alternative to conventional bootstrapping for diffusion tensor imaging. Here we present a study of an implementation of model-based residual bootstrapping using q -ball analysis and compare the outputs with conventional bootstrapping. We show that model-based residual bootstrap q-ball generates results that closely match the output of the conventional bootstrap. Both the residual and conventional bootstrap of multifiber methods can be used to estimate the probability of different numbers of fiber populations existing in different brain tissues. Also, we have shown that these methods can be used to provide input for probabilistic tractography, avoiding existing limitations associated with data calibration and model selection.

Using the bootstrap for finite sample confidence intervals of the log periodogram regression

Log periodogram regression is widely applied in empirical applications to estimate the memory parameter, d , of long memory time series. This estimator is consistent for d < 1 and pivotal asymptotically normal for d < 3 / 4 . However, the asymptotic distribution is a poor approximation of the (unknown) finite sample distribution if the sample size is small. Finite sample improvements in the construction of confidence intervals can be achieved by different nonparametric bootstrap procedures based on the residuals of log periodogram regression. In addition to the basic residual bootstrap, the local and block bootstraps seem adequate for replicating the structure that may arise in the errors of the regression when the series shows weak dependence in addition to long memory. The performances of different bias correcting bootstrap techniques and a bias reduced log periodogram regression are also analyzed with a view to adjusting the bias caused by that structure. Finally, an application to the Nelson and Plosser US macroeconomic data is included.

A bootstrap version of the residual-based smooth empirical distribution function

In this paper, we consider estimating the error distribution in a non-parametric regression model by a smooth version of the empirical distribution function of residuals. We show that a classical residual bootstrap version of the resulting residual-based empirical process joins the same limiting distribution. From this result, consistency of various goodness-of-fit tests in non-parametric regression models is obtained.

Probabilistic streamline q-ball tractography using the residual bootstrap

Q-ball imaging has the ability to discriminate multiple intravoxel fiber populations within regions of complex white matter architecture. This information can be used for fiber tracking; however, diffusion MR is susceptible to noise and multiple other sources of uncertainty affecting the measured orientation of fiber bundles. The proposed residual bootstrap method utilizes a spherical harmonic representation for high angular resolution diffusion imaging (HARDI) data in order to estimate the uncertainty in multimodal q-ball reconstructions. The accuracy of the q-ball residual bootstrap technique was examined through simulation. The residual bootstrap method was then used in combination with q-ball imaging to construct a probabilistic streamline fiber tracking algorithm. The residual bootstrap q-ball fiber tracking algorithm is capable of following the corticospinal tract and corpus callosum through regions of crossing white matter tracts in the centrum semiovale. This fiber tracking algorithm is an improvement upon prior diffusion tensor methods and the q-ball data can be acquired in a clinically feasible time frame.

Bootstrapping Clustered Data

SummaryVarious bootstraps have been proposed for bootstrapping clustered data from one-way arrays. The simulation results in the literature suggest that some of these methods work quite well in practice; the theoretical results are limited and more mixed in their conclusions. For example, McCullagh reached negative conclusions about the use of non-parametric bootstraps for one-way arrays. The purpose of this paper is to extend our understanding of the issues by discussing the effect of different ways of modelling clustered data, the criteria for successful bootstraps used in the literature and extending the theory from functions of the sample mean to include functions of the between and within sums of squares and non-parametric bootstraps to include model-based bootstraps. We determine that the consistency of variance estimates for a bootstrap method depends on the choice of model with the residual bootstrap giving consistency under the transformation model whereas the cluster bootstrap gives consistent estimates under both the transformation and the random-effect model. In addition we note that the criteria based on the distribution of the bootstrap observations are not really useful in assessing consistency.

Current Control by ECCD for W7-X

The magnetic configuration of the Wendelstein 7-X (W7-X) stellarator is optimized following a set of criteria including a rotational transform profile with low shear and minimized bootstrap current that must be controlled for proper functioning of the island divertor. This paper studies the compensation of residual bootstrap current by using electron cyclotron current drive (ECCD). The modeling shows that the loop voltage induced by ECCD leads to a redistribution of the current density with a diffusion time of ~2 s. The relaxation time of the total current is much longer, however - for W7-X plasma parameters the total toroidal current reaches steady state after several L/R times requiring hundreds of seconds. In order to keep the toroidal current and its profile in the acceptable range, a feed-forward or predictive control method using ECCD as actuator is proposed. The main steps are as follows: (a) calculate the bootstrap current distribution using plasma parameters measured in the online transport analysis and (b) determine and apply ECCD as needed. For the current control to work properly and to avoid long relaxation times, the reaction time of the control loop must be less than the current skin time.