A Turing Test of Bootstrap Scenarios

Scenario generation can be an important part of the training of human decision makers. A good scenario generation method should be able to generate large numbers of realistic scenarios. When the scenarios take the form of univariate stationary time series, the moving blocks bootstrap has the potential to be a good automatic scenario generator. However, one must determine the proper bootstrap block length. We have developed a method of setting the block length based on the distribution of a statistic computed from zero crossing counts. To test whether this way of setting the block length results in realistic scenarios, we performed two Turing tests. These visualization experiments confirmed that, when a bootstrap is optimally tuned, it is difficult for sophisticated subjects to identify a bootstrap sample plotted among several real samples. The main visual defects in moving blocks bootstrap samples are sudden jumps at block boundaries and repeating patterns.     

Estimating critical values for testing the i.i.d. in standardized residuals from GARCH models in finite samples

Taking into account that the BDS test—which is used as a misspecification test applied to standardized residuals from the GARCH(1,1) model—is characterized by size distortion and departure from normality in finite samples, this paper obtains the critical values for the finite sample distribution of the BDS test. We focus on bootstrap simulation to avoid the sampling uncertainty of parameter estimation and make use of estimated response surface regressions (RSR) derived from the experimental results. We consider an extensive grid of models to obtain critical values with the results of the bootstrap experiments. The RSR used to estimate them is an artificial neural network (ANN) model, instead of the traditional linear regression models. Specifically, we estimate critical values by using a bootstrap aggregated neural network (BANN) and by employing functions of the sample size and parameters used in the experiment as the embedding dimension and proximity parameters in the BDS statistic, GARCH parameters and even the q-quantiles of the BDS distributions. The main results confirm that the sample size and BDS parameters play a role in size distortion. Finally, an empirical application to three price indexes is performed, to highlight the differences between decisions made using the asymptotic or our predicted critical values for the BDS test in finite samples.     

On the recoverable robust traveling salesman problem

We consider an uncertain traveling salesman problem, where distances between nodes are not known exactly, but may stem from an uncertainty set of possible scenarios. This uncertainty set is given as intervals with an additional bound on the number of distances that may deviate from their expected, nominal values. A recoverable robust model is proposed, that allows a tour to change a bounded number of edges once a scenario becomes known. As the model contains an exponential number of constraints and variables, an iterative algorithm is proposed, in which tours and scenarios are computed alternately. While this approach is able to find a provably optimal solution to the robust model, it also needs to solve increasingly complex subproblems. Therefore, we also consider heuristic solution procedures based on local search moves using a heuristic estimate of the actual objective function. In computational experiments, these approaches are compared.     

Divisors,measures and critical functions

In [4] we have introduced a new distance between Galois orbits over ℚ. Using generalized divisors, we have extended the notion of trace of an algebraic number to other transcendental quantities. In this article we continue the work started in [4]. We define the critical function for a class of transcendental numbers, that generalizes the notion of minimal polynomial of an algebraic number. Our results extend the results obtained by Popescu et al [5].     

An Estimate of the Odds Ratio That Always Exists

This article proposes an estimate of the odds ratio in a (2 × 2) table obtained from studies in which the row totals are fixed by design, such as a phase II clinical trial. Our estimate, based on the median unbiased estimate of the probabilities of success in the (2× 2) table, will always be in the interval (0, ∞). Another estimate of the odds ratio which has such properties is obtained when adding .5 to each cell of the table. Using simulations, we compared our proposed estimate to that obtained by adding .5 to every cell, and found that our estimate had smaller finite sample bias, and larger mean square error. We also propose the use of the bootstrap to form a confidence interval for the odds ratio based on our proposed estimate. Instead of a Monte Carlo bootstrap, one can easily calculate the “exact” bootstrap distribution of our estimate of the odds ratio, and use this distribution to calculate confidence intervals.     

A branch and bound algorithm for locating input and output points of departments on the block layout

We consider the problem of locating input and output (I/O) points of each department for a given layout. The objective of the problem is to minimise the total distance of material flows between the I/O points. Here, distances between the I/O points are computed as the lengths of the shortest path (not the rectilinear distances) between the I/O points. We developed a procedure to eliminate dominated candidate positions of I/O points that do not need to be considered. With this procedure, a large number of dominated candidate positions can be eliminated. A linear programming (LP) model for minimising the total rectilinear distance of flows is used to obtain a lower bound. Using the elimination procedure and the LP model, a branch and bound algorithm is developed to find an optimal location of the I/O points. Results from computational experiments show that the suggested algorithm finds optimal solutions in a very short time even for large-sized problems.     

Analyzing short time series data from periodically fluctuating rodent populations by threshold models: A nearest block bootstrap approach

The study of the rodent fluctuations of the North was initiated in its modern form with Elton’s pioneering work. Many scientific studies have been designed to collect yearly rodent abundance data, but the resulting time series are generally subject to at least two “problems”: being short and non-linear. We explore the use of the continuous threshold autoregressive (TAR) models for analyzing such data. In the simplest case, the continuous TAR models are additive autoregressive models, being piecewise linear in one lag, and linear in all other lags. The location of the slope change is called the threshold parameter. The continuous TAR models for rodent abundance data can be derived from a general prey-predator model under some simplifying assumptions. The lag in which the threshold is located sheds important insights on the structure of the prey-predator system. We propose to assess the uncertainty on the location of the threshold via a new bootstrap called the nearest block bootstrap (NBB) which combines the methods of moving block bootstrap and the nearest neighbor bootstrap. The NBB assumes an underlying finite-order time-homogeneous Markov process. Essentially, the NBB bootstraps blocks of random block sizes, with each block being drawn from a non-parametric estimate of the future distribution given the realized past bootstrap series. We illustrate the methods by simulations and on a particular rodent abundance time series from Kilpisjärvi, Northern Finland.     

A posteriori error estimate for computing tr(f(A)) by using the Lanczos method

An outstanding problem when computing a function of a matrix, f(A), by using a Krylov method is to accurately estimate errors when convergence is slow. Apart from the case of the exponential function that has been extensively studied in the past, there are no well‐established solutions to the problem. Often, the quantity of interest in applications is not the matrix f(A) itself but rather the matrix–vector products or bilinear forms. When the computation related to f(A) is a building block of a larger problem (e.g., approximately computing its trace), a consequence of the lack of reliable error estimates is that the accuracy of the computed result is unknown. In this paper, we consider the problem of computing tr(f(A)) for a symmetric positive‐definite matrix A by using the Lanczos method and make two contributions: (a) an error estimate for the bilinear form associated with f(A) and (b) an error estimate for the trace of f(A). We demonstrate the practical usefulness of these estimates for large matrices and, in particular, show that the trace error estimate is indicative of the number of accurate digits. As an application, we compute the log determinant of a covariance matrix in Gaussian process analysis and underline the importance of error tolerance as a stopping criterion as a means of bounding the number of Lanczos steps to achieve a desired accuracy.     

On the asymptotic accuracy of the bootstrap under arbitrary resampling size

We study the order of convergence of the Kolmogorov-Smirnov distance for the bootstrap of the mean and the bootstrap of quantiles when an arbitrary bootstrap sample size is used. We see that for the bootstrap of the mean, the best order of the bootstrap sample is of the order ofn, wheren is the sample size. In the case of non-lattice distributions and the bootstrap of the sample mean; the bootstrap removes the effect of the skewness of the distribution only when the bootstrap sample equals the sample size. However, for the bootstrap of quantiles, the preferred order of the bootstrap sample isn ^2/3. For the bootstrap of quantiles, if the bootstrap sample is of ordern ² or bigger, the bootstrap is not consistent.     

线性回归模型Bootstrap LM-Error检验的水平扭曲

基于OLS估计残差,本文将Bootstrap方法用于空间误差相关性LM-Error检验,综合考虑Bootstrap模拟抽样次数、空间衔接结构以及样本量,研究并比较空间误差相关Bootstrap LM-Error检验与渐近检验的水平扭曲。大量Monte Carlo实验结果显示,当模型误差不满足独立正态分布的假设条件时,空间误差相关LM-Error渐近检验的水平扭曲较大,采用Bootstrap方法可以较好地降低该水平扭曲;不管模型误差是否满足独立正态分布的假设条件,Bootstrap方法均能够有效地降低LMError渐近检验的水平扭曲。     

Cross-validated bagged learning

Many applications aim to learn a high dimensional parameter of a data generating distribution based on a sample of independent and identically distributed observations. For example, the goal might be to estimate the conditional mean of an outcome given a list of input variables. In this prediction context, bootstrap aggregating (bagging) has been introduced as a method to reduce the variance of a given estimator at little cost to bias. Bagging involves applying an estimator to multiple bootstrap samples and averaging the result across bootstrap samples. In order to address the curse of dimensionality, a common practice has been to apply bagging to estimators which themselves use cross-validation, thereby using cross-validation within a bootstrap sample to select fine-tuning parameters trading off bias and variance of the bootstrap sample-specific candidate estimators. In this article we point out that in order to achieve the correct bias variance trade-off for the parameter of interest, one should apply the cross-validation selector externally to candidate bagged estimators indexed by these fine-tuning parameters. We use three simulations to compare the new cross-validated bagging method with bagging of cross-validated estimators and bagging of non-cross-validated estimators.     

A Random Effect Block Bootstrap for Clustered Data

Random effects models for hierarchically dependent data, for example, clustered data, are widely used. A popular bootstrap method for such data is the parametric bootstrap based on the same random effects model as that used in inference. However, it is hard to justify this type of bootstrap when this model is known to be an approximation. In this article, we describe a random effect block bootstrap approach for clustered data that is simple to implement, free of both the distribution and the dependence assumptions of the parametric bootstrap, and is consistent when the mixed model assumptions are valid. Results based on Monte Carlo simulation show that the proposed method seems robust to failure of the dependence assumptions of the assumed mixed model. An application to a realistic environmental dataset indicates that the method produces sensible results. Supplementary materials for the article, including the data used for the application, are available online.     

Spatial distributions of islands in fractal surfaces and natural surfaces

The size and spatial distributions of islands in random fractal surfaces and natural islands are investigated. Multi-scale island distributions from a fractal surface are analyzed to determine the size distribution of islands, such as the number of islands greater than a particular size, and this was found to exhibit a Korcak-type empirical relation. In the same multi-scale analysis, the spatial distribution of islands, which is the number of islands less than or equal to a particular distance between a pair of islands, is also investigated to determine the effects of differences in scale. The spatial distribution of distance between uniformly distributed islands is analytically calculated and compared with the island distributions of the fractal surface. We seek to determine the degree of resemblance between simulated spatial distributions and the natural island distribution of French Polynesia. We also estimate the size and spatial distributions of islands according to different sea levels.     

Asymptotic properties of sample quantiles from a finite population

In this paper we consider the problem of estimating quantiles of a finite population of size N on the basis of a finite sample of size n selected without replacement. We prove the asymptotic normality of the sample quantile and show that the scaled variance of the sample quantile converges to the asymptotic variance under a slight moment condition. We also consider the performance of the bootstrap in this case, and find that the usual (Efron’s) bootstrap method fails to be consistent, but a suitably modified version of the bootstrapped quantile converges to the same asymptotic distribution as the sample quantile. Consistency of the modified bootstrap variance estimate is also proved under the same moment conditions.     

Profit,Directional Distance Functions,and Nerlovian Efficiency

The directional technology distance function is introduced, given an interpretation as a min-max, and compared with other functional representations of the technology including the Shephard input and output distance functions and the McFadden gauge function. A dual correspondence is developed between the directional technology distance function and the profit function, and it is shown that all previous dual correspondences are special cases of this correspondence. We then show how Nerlovian (profit-based) efficiency measures can be computed using the directional technology distance function.     

L_1 regression estimate and its bootstrap

We study the asymptotic distribution of the L1 regression estimator under general condi-tions with matrix norming and possibly non i.i.d.errors.We then introduce an appropriate bootstrap procedure to estimate the distribution of this estimator and study its asymptotic properties.It is shown that this bootstrap is consistent under suitable conditions and in other situations the bootstrap limit is a random distribution.     

Music and timbre segmentation by recursive constrained K-means clustering

Clustering of features generated of musical sound recordings proved to be beneficial for further classification tasks such as instrument recognition (Ligges and Krey in Comput Stat 26(2):279–291, 2011). We propose to use order constrained solutions in K-means clustering to stabilize the results and improve the interpretability of the clustering. With this method a further improvement of the misclassification error in the aforementioned instrument recognition task is possible. Using order constrained K-means the musical structure of a whole piece of popular music can be extracted automatically. Visualizing the distances of the feature vectors through a self distance matrix allows for an easy visual verification of the result. For the estimation of the right number of clusters, we propose to calculate the adjusted Rand indices of bootstrap samples of the data and base the decision on the minimum of a robust version of the coefficient of variation. In addition to the average stability (measured through the adjusted Rand index) this approach takes the variation between the different bootstrap samples into account.     

Estimate the shortest paths on fractal m-gons

The self-similar sets seem to be a class of fractals which is most suitable for mathematical treatment. The study of their structural properties is important. In this paper, we estimate the formula for the mean geodesic distance of self-similar set (denote fractal m-gons). The quantity is computed precisely through the recurrence relations derived from the self-similar structure of the fractal considered. Out of result, obtained exact solution exhibits that the mean geodesic distance approximately increases as a exponential function of the number of nodes (small copies with the same size) with exponent equal to the reciprocal of the fractal dimension.     

Trace class multipliers and spectral variation of normal matrices

In this article we show how to estimate the trace multiplier norm of a rank 2 matrix. As an application, an alternative proof of a theorem of Holbrook et al. (Maximal spectral distance, Linear Algebra Appl., 249 (1996) 197–205) on the maximal spectral distance between two normal matrices with prescribed eigenvalues is given.     

L1 regression estimate and its bootstrap

We study the asymptotic distribution of the L ₁ regression estimator under general conditions with matrix norming and possibly non i.i.d. errors. We then introduce an appropriate bootstrap procedure to estimate the distribution of this estimator and study its asymptotic properties. It is shown that this bootstrap is consistent under suitable conditions and in other situations the bootstrap limit is a random distribution. This work was supported by J.C. Bose National Fellowship, Government of India