Modified Bootstrap Method Research Articles

Smoothed gradient least squares estimator for linear threshold models

In the presence of fixed threshold effects, the least squares (LS) estimator of the threshold parameter poses challenges for statistical inference due to its nonstandard limiting distribution, which also presents challenges for bootstrap methods. To address this issue, we propose a novel estimator: a two-step smoothed gradient least squares (SGLS) estimator. Our proposed method achieves a normal limiting distribution for the threshold parameter with minimal efficiency loss compared to the LS estimator. Furthermore, our modified bootstrap method significantly enhances computational efficiency, leading to improved bootstrap confidence intervals (CIs) for the threshold parameter compared to asymptotic CIs. Our method is validated through a small Monte Carlo study and demonstrated with an empirical application.

Uncertainty quantification of granular computing-neural network model for prediction of pollutant longitudinal dispersion coefficient in aquatic streams

Discharge of pollution loads into natural water systems remains a global challenge that threatens water and food supply, as well as endangering ecosystem services. Natural rehabilitation of contaminated streams is mainly influenced by the longitudinal dispersion coefficient, or the rate of longitudinal dispersion (Dx), a key parameter with large spatiotemporal fluctuations that characterizes pollution transport. The large uncertainty in estimation of Dx in streams limits the water quality assessment in natural streams and design of water quality enhancement strategies. This study develops an artificial intelligence-based predictive model, coupling granular computing and neural network models (GrC-ANN) to provide robust estimation of Dx and its uncertainty for a range of flow-geometric conditions with high spatiotemporal variability. Uncertainty analysis of Dx estimated from the proposed GrC-ANN model was performed by alteration of the training data used to tune the model. Modified bootstrap method was employed to generate different training patterns through resampling from a global database of tracer experiments in streams with 503 datapoints. Comparison between the Dx values estimated by GrC-ANN to those determined from tracer measurements shows the appropriateness and robustness of the proposed method in determining the rate of longitudinal dispersion. The GrC-ANN model with the narrowest bandwidth of estimated uncertainty (bandwidth-factor = 0.56) that brackets the highest percentage of true Dx data (i.e., 100%) is the best model to compute Dx in streams. Considering the significant inherent uncertainty reported in the previous Dx models, the GrC-ANN model developed in this study is shown to have a robust performance for evaluating pollutant mixing (Dx) in turbulent environmental flow systems.

Phylogenetic Analysis of HIV-1 Genomes Based on the Position-Weighted K-mers Method.

HIV-1 viruses, which are predominant in the family of HIV viruses, have strong pathogenicity and infectivity. They can evolve into many different variants in a very short time. In this study, we propose a new and effective alignment-free method for the phylogenetic analysis of HIV-1 viruses using complete genome sequences. Our method combines the position distribution information and the counts of the k-mers together. We also propose a metric to determine the optimal k value. We name our method the Position-Weighted k-mers (PWkmer) method. Validation and comparison with the Robinson–Foulds distance method and the modified bootstrap method on a benchmark dataset show that our method is reliable for the phylogenetic analysis of HIV-1 viruses. PWkmer can resolve within-group variations for different known subtypes of Group M of HIV-1 viruses. This method is simple and computationally fast for whole genome phylogenetic analysis.

A Stochastic View on the Effect of Random Rail Irregularities on Railway Bridge Vibrations

This contribution studies dynamic amplification factors that consider the impact of random rail irregularities on the deflection and acceleration response of simple railway bridges. Response statistics for a set of plane fully coupled bridge–vehicle interaction models with random irregularity profiles are derived by Monte Carlo simulations. The underlying samples of random irregularity profiles are generated from a power spectral density function, assuming that the uneven rail surface is governed by a Gaussian stochastic process in space. A modified bootstrap method is used to identify the required sample size of profiles for sufficiently accurate description of the random response. On the basis of this pilot study, a parametric response examination of short span railway bridges with variable length, fundamental frequency, structural mass and damping, and track quality in a range commonly observed in real structures is conducted. A statistical representation of the uncertain bridge response is employed for analytical modelling of the response amplification due to random rail irregularities considering limit state-based exceedance probabilities. The results are compared to outcomes of existing simplified approaches in design codes used to take into account the effect of rail irregularities on the dynamic bridge response, showing the improvement of the developed framework.

Reliability estimation for drive axle of wheel loader under extreme small sample

It is difficult to have enough samples to implement the full-scale life test on the loader drive axle due to high cost. But the extreme small sample size can hardly meet the statistical requirements of the traditional reliability analysis methods. In this work, the method of combining virtual sample expanding with Bootstrap is proposed to evaluate the fatigue reliability of the loader drive axle with extreme small sample. First, the sample size is expanded by virtual augmentation method to meet the requirement of Bootstrap method. Then, a modified Bootstrap method is used to evaluate the fatigue reliability of the expanded sample. Finally, the feasibility and reliability of the method are verified by comparing the results with the semi-empirical estimation method. Moreover, from the practical perspective, the promising result from this study indicates that the proposed method is more efficient than the semi-empirical method. The proposed method provides a new way for the reliability evaluation of costly and complex structures.

An uncertainty representation based sampling method for metamodeling in auto-motive design applications

Meta-model is frequently employed as the surrogate of high-fidelity finite element models in computer aided engineering. It helps to achieve the trade-off between computational efficiency and predictive accuracy. To improve the predictive capability of meta-models, we developed an uncertainty representation based sampling method to schedule Design of experiment (DOE) for meta-modeling. Several datasets were first generated through a modified Bootstrap method for datasets acquisition, then the influence of the input uncertainty on the output was quantified as weighting factors. The weighting factors were used to integrate the represented distributions into a single one for further sampling. Finally, the sampling results then served as the elements of the DOE matrix to construct meta-models. The proposed method was demonstrated through an analytical case and a real-world vehicle crashworthiness design problem.

The Enhanced Classification for the Stock Index Prediction

It is one of the hardest challenges to predict the movement of the stock price. We propose the modified bootstrap method in random forests to predict the direction of movement of the stock index price. The training set generated by the modified bootstrapping considers the impact of response variable simultaneously and is applied in random forests. The real KOSPI data are used for the experiments and the result shows that the proposed method performs better than the original method in various situations.

간헐적 수요예측을 위한 부트스트랩 시뮬레이션 방법론 개발

Demand forecasting is the basis of management activities including marketing strategy. Especially, the demand of a part is remarkably important in supply chain management (SCM). In the fields of various industries, the part demand usually has the intermittent characteristic. The intermittent characteristic implies a phenomenon that there frequently occurs zero demands. In the intermittent demands, non-zero demands have large variance and their appearances also have stochastic nature. Accordingly, in the intermittent demand forecasting, it is inappropriate to apply the traditional time series models and/or cause-effect methods such as linear regression; they cannot describe the behaviors of intermittent demand. Markov bootstrap method was developed to forecast the intermittent demand. It assumes that first-order autocorrelation and independence of lead time demands. To release the assumption of independent lead time demands, this paper proposes a modified bootstrap method. The method produces the pseudo data having the characteristics of historical data approximately. A numerical example for real data will be provided as a case study.

Bayesian Probabilistic Decline-Curve Analysis Reliably Quantifies Uncertainty in Shale-Well-Production Forecasts

Summary Several analytical decline-curve models have been developed recently for shale-gas wells (Ilk et al. 2008; Anderson et al. 2010; Valko and Lee 2010). However, these authors did not quantify the uncertainty in production forecasts and reserves estimates. This is important because most shale plays are in the early stages of production and virtually any method will have large uncertainty when there are limited production data available. Jochen and Spivey (1996) and Cheng et al. (2010) developed bootstrap methods that can generate probabilistic decline forecasts and quantify reserves uncertainty. Hindcasts with the modified bootstrap method (MBM) (Cheng et al. 2010) provide good coverage of the true cumulative production. However, the authors did not show they can quantify reserves uncertainty with limited production data in unconventional plays. In this paper, we introduce a Bayesian probabilistic methodology using Markov-chain Monte Carlo (MCMC) combined with Arps' decline-curve analysis. We tested this model on two data sets: Barnett shale horizontal-well gas production with more than 7 years of history and Eagle Ford shale horizontal-well oil production with more than 1 year of history. In both cases, P50 hindcasts were very close to true cumulative production and P90 and P10 hindcasts quantified the cumulative production uncertainty reliably with as little as 6 months of production available for matching. In this Bayesian methodology, the decline-curve parameters qi, Di, and b are assumed to be random variables instead of parameters to be modified to obtain a best fit. A Markov chain of the decline-curve parameters is constructed by use of MCMC with the Metropolis algorithm (random walk). We developed the model by performing hindcasts with the Barnett case study consisting of 197 horizontal gas wells with more than 7 years of production. The prior distribution, proposal distribution, and likelihood function were calibrated so the probabilistic decline curves quantified the cumulative-production uncertainty reliably with as little as 6 months of data. The same model was then tested with analysis of Eagle Ford shale oil production from 536 wells; the probabilistic decline curves quantified the cumulative-production uncertainty reasonably well by changing only the prior distribution. The proposed Bayesian methodology provides a means and a workflow to generate probabilistic decline-curve forecasts and quantify reserves uncertainty in shale plays quickly and reliably. This Bayesian methodology can also be applied with other analytical decline-curve models if desired.

Short-Term Load Forecasting Based on a Semi-Parametric Additive Model

Short-term load forecasting is an essential instrument in power system planning, operation, and control. Many operating decisions are based on load forecasts, such as dispatch scheduling of generating capacity, reliability analysis, and maintenance planning for the generators. Overestimation of electricity demand will cause a conservative operation, which leads to the start-up of too many units or excessive energy purchase, thereby supplying an unnecessary level of reserve. On the other hand, underestimation may result in a risky operation, with insufficient preparation of spinning reserve, causing the system to operate in a vulnerable region to the disturbance. In this paper, semi-parametric additive models are proposed to estimate the relationships between demand and the driver variables. Specifically, the inputs for these models are calendar variables, lagged actual demand observations, and historical and forecast temperature traces for one or more sites in the target power system. In addition to point forecasts, prediction intervals are also estimated using a modified bootstrap method suitable for the complex seasonality seen in electricity demand data. The proposed methodology has been used to forecast the half-hourly electricity demand for up to seven days ahead for power systems in the Australian National Electricity Market. The performance of the methodology is validated via out-of-sample experiments with real data from the power system, as well as through on-site implementation by the system operator.

Bootstrapping Lasso Estimators

In this article, we consider bootstrapping the Lasso estimator of the regression parameter in a multiple linear regression model. It is known that the standard bootstrap method fails to be consistent. Here, we propose a modified bootstrap method, and show that it provides valid approximation to the distribution of the Lasso estimator, for all possible values of the unknown regression parameter vector, including the case where some of the components are zero. Further, we establish consistency of the modified bootstrap method for estimating the asymptotic bias and variance of the Lasso estimator. We also show that the residual bootstrap can be used to consistently estimate the distribution and variance of the adaptive Lasso estimator. Using the former result, we formulate a novel data-based method for choosing the optimal penalizing parameter for the Lasso using the modified bootstrap. A numerical study is performed to investigate the finite sample performance of the modified bootstrap. The methodology proposed in the article is illustrated with a real data example.

An extension of Chao’s estimator of population size based on the first three capture frequency counts

A new estimator for estimating the size of an elusive target population is presented using frequency counts from capture–recapture sampling. The proposed estimator is developed by extending the idea of Chao’s estimator using monotonicity of ratios of neighbouring frequency counts under a specific Poisson mixture sampling framework, the Poisson–Gamma mixture or negative binomial. The new estimator is achieved using a simple linear model on the basis of the log-ratio of neighbouring frequency counts as dependent variable which is valid under the Poisson–Gamma mixture. A simulation study is provided to study the performance of the proposed estimator under a variety of heterogeneous Poisson capture probabilities. Confidence interval estimation is done by means of an approximating normal approach and a modified bootstrap method, and was found to perform well. A variety of real data sets were also examined in order to illustrate the use of the proposed method.

Measuring Evolutionary Constraints Through the Dimensionality of the Phenotype: Adjusted Bootstrap Method to Estimate Rank of Phenotypic Covariance Matrices

The potential and direction of phenotypic evolution is constrained by the distribution of genetic variation for the traits as described by the phenotypic (P) and genetic covariance matrices (G). The rank of the covariance matrix reflects the number of independent variational dimensions of the phenotype. Covariance matrices with less than full rank indicate lack of variation in some directions of the phenotype space and thus are an indication of absolute evolutionary constraints. Because selection acts upon phenotypic variation, the rank of P represents the upper limit of the dimensionality in G, relevant for selection response. The limitations of current methods to estimate matrix rank motivated us to analyze and adjust a bootstrap method and evaluate its performance by simulation. The results show that the modified bootstrap method (ABRE) gives reliable and rather conservative rank estimates when the sample size is sufficient for the number of variables studied (the sample size is at least five-fold the number of variables). Applying the method to various datasets suggests high phenotypic dimensionality in all cases. The analysis thus provides no evidence for absolute evolutionary constraints.

Quantile Regression: On Inferences about the Slopes Corresponding to One, Two or Three Quantiles

The problem of testing hypotheses about the slope of a quantile regression line when the sample size is small is considered. A modified bootstrap method is suggested that is found to have certain advantages over the inverse rank method recommended by Koenker (1994). A method is suggested that simultaneously controls the probability of at least one Type I error when performing two or three tests corresponding to two or three specific quantiles. Using data from actual studies, it is illustrated that the new method can yield substantially shorter confidence intervals than the rank inverse method and, even with a large sample size, the choice of method can matter.

A comparison between the design-based and model-based approaches using longitudinal survey data

Survey data analysis using complex sampling designs ought to account for clustering, stratification and unequal probability of selection. Design-based and model-based methods are two commonly used routes taken to account for such survey designs. Several studies of cross-sectional survey designs have shown that these two approaches provide similar results when the model fits the data well. The present paper aims at comparing these two approaches for longitudinal survey design using the National Population Health Survey (NPHS) dataset. A marginal modeling approach proposed by Rao and modified bootstrap method for longitudinal data were used by way of a design-based method. The Generalized Estimating Equation (GEE) method, proposed by Liang and Zeger was used as a typical model-based approach. The parameter estimates obtained using the design-based and model-based methods were similar. However, the standard errors and the 95% confidence interval were different. Rao's method produced the most conservative standard errors. In conclusion, design-based methods should be preferred over model-based methods, as this method provides reliable results.

Development of species sensitivity distributions and estimation of HC5 of organochlorine pesticides with five statistical approaches

Eighteen organochlorine pesticides (OCPs) were studied to develop species sensitivity distributions (SSDs) and calculate hazardous concentration thresholds for 5% of species (HC5), using both parametric (log-normal and log-logistic) and nonparametric bootstrap methods. In order to avoid picking repetitive values in each resample when performing bootstrap, and to determine the influence of fluctuation of toxicity data of single species on the SSDs and HC5, a modified bootstrap method was introduced, which can generate unrepetitive sampling data other than original elements in datasets. This method can enlarge a dataset without any assumption of a special distribution. Combined with parametric methods, modified bootstrap was also used to develop SSDs and determine HC5. The HC5 estimated by five approaches coincide well with each other with good positive correlation. Even if there is intra-species variation in a certain range of toxicity data; SSDs and HC5 are not very sensitive to the local fluctuation of toxicity of single species. The studied OCPs were classified according to their estimated HC5. A lower HC5 indicates higher ecological toxicity potentials. Endrin, DDTs and Endosulfan are OCPs with very high ecological toxicity potential. alpha-HCH has the lowest ecological toxicity potential in the studied OCPs. For OCPs with high ecological potential, more attention should be paid to their ecological risk.

Quantile Processes in the Presence of Auxiliary Information

We employ the empirical likelihood method to propose a modified quantile process under a nonparametric model in which we have some auxiliary information about the population distribution. Furthermore, we propose a modified bootstrap method for estimating the sampling distribution of the modified quantile process. To explore the asymptotic behavior of the modified quantile process and to justify the bootstrapping of this process, we establish the weak convergence of the modified quantile process to a Gaussian process and the almost-sure weak convergence of the modified bootstrapped quantile process to the same Gaussian process. These results are demonstrated to be applicable, in the presence of auxiliary information, to the construction of asymptotic bootstrap confidence bands for the quantile function. Moreover, we consider estimating the population semi-interquartile range on the basis of the modified quantile process. Results from a simulation study assessing the finite-sample performance of the proposed semi-interquartile range estimator are included.

A modified bootstrap for autoregression without stationarity

In this paper, we consider a bootstrap approximation to the distribution of the least squares estimator \\ ̂ gb of the autoregressive parameter β in a first-order autoregressive process which may or may not have a stationary solution. Our bootstrap procedure is a modification of the standard bootstrap and employs a data based shrinkage towards the critical values β = ±1. The error in estimation of the sampling distribution of \\ ̂ gb by the above procedure converges to zero as the sample size grows, irrespective of the value of the autoregressive parameter. This result is in sharp contrast with the behavior of the standard bootstrap for which a random limiting distribution emerges at the critical values β = ±1, resulting in a failure of the bootstrap procedure. The asymptotic validity of our procedure enables us, inter alia, to construct a confidence interval for β which will have the correct coverage probability (asymptotically) under any real β. The theoretical validity result for the modified bootstrap is supported by appropriate simulations. Finally, we also indicate an extension of the proposed modified bootstrap method to the pth order autoregressive processes.

Bootstrapping general first order autoregression

In this paper we consider general first order autoregression, including the stationary, the explosive and the unstable cases. It is well-known in the literature that the usual bootstrap method for the least squares parameter estimator is asymptotically consistent for the stationary and the explosive cases, but does not work in the unstable case, where the parameter value is equal to +1 and or −1. We propose a modified bootstrap method, which turns out to be asymptotically consistent in all possible situations. Furthermore, we derive tests for stationarity and nonstationarity for first order autoregressions. The bootstrap method is used to obtain critical values. Some simulation results are also enclosed.

Bootstrapping shear-wave splitting errors

Abstract We have developed a bootstrap method to estimate errors associated with inverting SKS waveforms for shear-wave splitting parameters. Although presented for shear-wave splitting inversions, this method is suitable for any waveform inversion procedure. The bootstrap error estimation method consists of multiple inversions of simulated data that imitate the original data with differing noise sequences. The results of the bootstrap inversions are used to directly calculate variances and covariances for all model parameters. We employ a bootstrap error estimation technique to nonlinear inversion for shear-wave splitting parameters. Since seismic data have correlated errors, the bootstrap method was modified for stationary bandlimited time series. This modified bootstrap method was applied to shear-wave splitting measurements from over 60 pairs of horizontal seismograms. The method is stable under a large range of noise conditions. By using this bootstrap method, we can distinguish among data with no apparent splitting, data with splitting, and noisy data.