Symmetric Interval Research Articles

강우빈도해석에서 Bayesian 기법을 이용한 Gumbel 확률분포 매개변수의 불확실성 평가

Paper number: 15-070Received: 7 September 2015; Revised: 20 February 2016/16 Marh 2016; Accepted: 16 Marh 2016AbstractRainfall-runoff modeling in conjunction with rainfall frequency analysis has been widely used for estimating design floods in South Korea. However, uncertainties associated with underlying distribution and sampling error have not been properly addressed. This study applied a Bayesian method to quantify the uncertainties in the rainfall frequency analysis along with Gumbel distribution. For a purpose of comparison, a probability weighted moment (PWM) was employed to estimate confidence interval. The uncertainties associated with design rainfalls were quantitatively assessed using both Bayesian and PWM methods. The results showed that the uncertainty ranges with PWM are larger than those with Bayesian approach. In addition, the Bayesian approach was able to effectively represent asymmetric feature of underlying distribution; whereas the PWM resulted in symmetric confidence interval due to the normal approximation. The use of long period data provided better results leading to the reduction of uncertainty in both methods, and the Bayesian approach showed better performance in terms of the reduction of the uncertainty.Keywords: Uncertainty, Bayesian Analysis, Parameter Uncertainty, Frequency Analysis, Gumbel Distribution

Robust Minimum Variance Lower Bound Estimation by Uncertainty Modeling Using Interval Type‐2 Fuzzy set

AbstractThe Minimum Variance Lower Bound (MVLB) represents the best achievable controller capability in a variance sense. Estimation of the MVLB for nonlinear systems confronts some difficulties. If one simply ignores these nonlinearities, there is the danger of over‐estimating the performance of the control loop in rejecting uncertainties. Assuming that almost all models have uncertainties, in this paper, the MVLB has been estimated considering three types of uncertainties: structural, parametric, and algorithmic. To achieve accurate estimation of the MVLB an interval type‐2 fuzzy set has been utilized. This paper utilizes a strategy for modeling of symmetric interval type‐2 fuzzy sets using their uncertainty degrees. Then, based on this uncertainty measure, one method to construct interval type‐2 fuzzy set models using the uncertain interval data is introduced. Finally, simulation studies demonstrate the effectiveness of the proposed control scheme.

Bit Error Rate Comparison Statistics and Hypothesis Tests for Inverse Sampling (Negative Binomial) Experiments

Inverse, or negative binomial, sampling is often used when the observation of interest occurs extremely infrequently. As this is the case in bit error rate (BER) simulations, especially in high signal-to-noise ratio cases, negative binomial sampling can be advantageously employed in a computationally economic fashion to compare bit error rates between different systems. When the results of two negative binomial sampling tests are compared, point estimates and interval estimates quantify the performance relationship between the results of the tests. This paper derives a new, optimal, logarithmically symmetric confidence interval estimator for the ratio of BER estimates derived from two negative binomial tests. In addition, a three-sided hypothesis test with a single significance level is derived to quantify the confidence of the relationship between the two systems. Low-BER approximations for the confidence interval and decision thresholds are derived based on the F-distribution. The approximation is shown to work with BERs as high as $10^{-2}$ . An example inspired by bit interleaved coded modulation shows how the technique can be used to reduce simulation time by an order of magnitude and facilitate straightforward interpretation and comparison between different systems. Negative binomial sampling is recommended for comparison experiments where BER is the key metric.

Final Distribution of a Diffusion Process: Semi-Markov Approach

A one-dimensional diffusion process is considered. This process is proposed to have the homogeneous Markov property with respect to the first exit time from any open interval (semi-Markov property). A diffusion property of the process is defined as asymptotical equiprobable exit through each of two edges of any symmetric neighborhood interval of an initial point of a sample process trajectory while length of the neighborhood tends to zero. Such a process is proved to have a limit as $t\to\infty$ if probability of the process not leaving this neighborhood is decreased as square of its length. In particular this condition is satisfied for a diffusion Markov process with a break for which the nonexit condition is replaced by the break condition. A semi-Markov method is applied to a derivation of the formula of a conditional final distribution of the diffusion process with a limit at infinity.

Assessing accuracy of k ‐step‐ahead prediction of non‐linear dynamics with uncertainties

Prediction of k-step ahead requires an accurate model for a non-linear system. If the non-linearities are simply ignored, there will be the danger of overestimating the output value due to uncertainties. Considering that almost all models have uncertainties, in this study, the accuracy of k-step-ahead prediction is assessed considering the structural, parametric, and algorithmic uncertainties. Then, in order to remove uncertainties from data and achieve an accurate k-step-ahead prediction, interval type-2 fuzzy set is utilised. This study proposes a strategy for modelling symmetric interval type-2 fuzzy sets using their uncertainty degrees and centre of gravities. On the basis of these uncertainty measures, a method is introduced for constructing interval type-2 fuzzy set models using the uncertain interval data. The aim is to provide tools for evaluating a model or a set of models on the basis of predictive accuracy or efficiency for non-linear dynamic applications with uncertainties. Simulation studies demonstrate the effectiveness of the proposed control scheme.

The symmetrical interval intuitionistic uncertain linguistic operators and their application to decision making

Interval intuitionistic uncertain linguistic sets are an important generalization of fuzzy sets, which well cope with the experts’ qualitative preferences as well as reflect the interval membership and non-membership degrees of the uncertain linguistic term. This paper first points out the issues of the operational laws on interval intuitionistic uncertain linguistic numbers in the literature, and then defines some alternative ones. To consider the relationship between interval intuitionistic uncertain linguistic sets, the expectation and accuracy functions are defined. To study the application of interval intuitionistic uncertain linguistic sets, two symmetrical interval intuitionistic uncertain linguistic hybrid aggregation operators are defined. Meanwhile, models for the optimal weight vectors are established, by which the optimal weighting vector can be obtained. As a series of development, an approach to multi-attribute decision making under interval intuitionistic uncertain linguistic environment is developed, and the associated example is provided to demonstrate the effectivity and practicality of the procedure.

Asymmetric cycle time bounding in semiconductor manufacturing: an efficient and effective back-propagation-network-based method

Estimating the cycle time of every job in a semiconductor manufacturing factory is essential. However, because cycle times are uncertain, determining the cycle time range is also crucial. Manufactures use the lower bound of a cycle time to promise an attractive due date to customers and the upper bound to indicate the latest possible time that fabrication will be completed. Thus, an interval estimate of cycle time is preferable to a point estimate. In several previous studies, a symmetric interval estimate has been derived from a probabilistic perspective. However, the managerial implications of the upper and lower bounds differ, and an asymmetric interval estimate is more useful. Therefore, this paper proposes a back-propagation-network-based approach to estimating the job cycle time and determining the cycle time range. A real case from a semiconductor manufacturing factory was used to illustrate the proposed method. According to the results, the estimates of the job cycle time obtained using the proposed method were precise and accurate. The established upper and lower bounds (especially the lower bound) were much tighter than those used in existing methods.

ПРАВИЛО СКЛЕИВАНИЯ ДЛЯ ПОЛИНОМОВ БЕРНШТЕЙНА НА СИММЕТРИЧНОМ ОТРЕЗКЕ

ПРАВИЛО СКЛЕИВАНИЯ ДЛЯ ПОЛИНОМОВ БЕРНШТЕЙНА НА СИММЕТРИЧНОМ ОТРЕЗКЕ

Transform Group of Monotonic Functions with the Same Monotonicity on [ $$-$$ - 1, 1] and Operations of Fuzzy Numbers

Operations of fuzzy numbers are the main content of the fuzzy mathematical analysis. This paper defines the transformation of monotonic bounded functions with same monotonicity on the symmetric interval [\(-\)1, 1], and the four fundamental operations of fuzzy numbers based on the fuzzy structured element. It not only make operations of fuzzy numbers easier, but also start a new way for studying on the theory and application of fuzzy analysis.

An interval algorithm of maintenance windows under uncertain parameters

For improving the method of finding maintenance windows under uncertain parameters, an algorithm of maintenance window under uncertainties is presented using interval analysis and sensitivity analysis. Age replacement model is selected to demonstrate how to use this new algorithm. Considered the uncertainties, the optimal maintenance interval of preventive maintenance is not only a single value, but a possible range. The requirement from maintenance engineers is also considered, the maintenance window is made as a symmetrical interval format, like (100 ± 10) day. Comparing with the methods using in the literatures, the new algorithm is without requirement of distribution assumption of uncertain parameter values and computer simulation.

Components of measurement uncertainty from a measurement model with two stages involving two output quantities

The GUM uncertainty framework and the Monte Carlo Method can be extended to include multistage measurement models and measurement models having more than one output quantity. A typical multistage measurement model, that also includes a measurement stage with more than one output quantity is the calculation of the coefficients of a calibration curve, constructed by means of least squares regression and the subsequent use this calibration curve for the estimation of a quantity value. In the present work the Monte Carlo Method was used to evaluate the component of measurement uncertainty from a calibration curve used for the determination of the total nitrogen found in a gasoline sample. The slope and the intercept of the calibration curve were treated as a vector output quantity characterized by a joint probability distribution (bivariate Gaussian) and two types of coverage regions were estimated (rectangular and ellipsoid). The Monte Carlo Method algorithm employing fixed number of trials (106) gave a predicted value of 2.09mgL−1, a standard measurement uncertainty of 0.04mgL−1 and a 95% symmetrical coverage interval [2.01–2.17] mgL−1. The standard measurement uncertainty obtained by MCM agrees well with the outcome of the equation giving the standard error of the estimate.

Confidence Interval for <inline-formula><tex-math notation="LaTeX">${\bm {F_1}}$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="wang-ieq1-2359667.gif"/></alternatives></inline-formula> Measure of Algorithm Performance Based on Blocked 3<inline-formula><tex-math notation="LaTeX">$\bm {\times}$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="wang-ieq2-2359667.gif"/></alternatives></inline-formula>2 Cross-Validation

In studies on the application of machine learning such as Information Retrieval (IR), the focus is typically on the estimation of the F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> measure of algorithm performance. Approximate symmetrical confidence intervals constructed by the F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> value based on cross-validated L distribution are commonly used in the literature. However, theoretical analysis on the distribution of F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> values shows that such distribution is actually non-symmetrical. Thus, simply using symmetrical distribution to approximate non-symmetrical distribution may be inappropriate and may result in a low degree of confidence and long interval length for the confidence interval. In the present study, a non-symmetrical confidence interval of the F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> measure based on Beta prime distribution is constructed by using the F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> value computed based on the average confusion matrix of a blocked 3 x 2 cross-validation. Experimental results show that in most cases, our method has high degrees of confidence. With an acceptable degree of confidence, our method has a shorter interval length than the approximate symmetrical confidence intervals based on the blocked 3 x 2 and 5 x 2 cross-validated L distributions. The approximate symmetrical confidence interval based on the 10-fold cross-validated L distribution has the shortest interval length of the four confidence intervals but with low degrees of confidence in all cases. Taking these two factors into consideration, our method is recommended.

Eigenvalues bounds for symmetric interval matrices

The aim of this paper is to approximate the eigenvalues bounds of real symmetric interval matrices as sharp as possible. Using the concepts of interval analysis, this is done by solving the standard interval eigenvalue problem by applying the interval extension of the single step eigen perturbation method. The deviation amplitude of the interval matrix is considered as a perturbation around the nominal value of the interval matrix. The mathematical formulation of the method is described. Two numerical examples are worked out and the results obtained are compared with the results of existing methods. It is observed that our method is reliable, efficient and gives better results for all examples considered.

Complexity issues for the symmetric interval eigenvalue problem

Abstract We study the problem of computing the maximal and minimal possible eigenvalues of a symmetric matrix when the matrix entries vary within compact intervals. In particular, we focus on computational complexity of determining these extremal eigenvalues with some approximation error. Besides the classical absolute and relative approximation errors, which turn out not to be suitable for this problem, we adapt a less known one related to the relative error, and also propose a novel approximation error. We show in which error factors the problem is polynomially solvable and in which factors it becomes NP-hard.

Bipolar semicopulas

The concept of semicopula plays a fundamental role in the aggregation theory on interval [0,1]. Semicopulas are applied, for example, in the definition of universal integrals. We present an extension of the notion of semicopula to the case of symmetric bipolar interval [−1,1]. We call this extension bipolar semicopula. The last definition can be used to obtain a simplified definition of the bipolar universal integral. Moreover, bipolar semicopulas allow for an extension of theory of quasi-copulas to the interval [−1,1].

A Large-Sample Confidence Interval for the Inverse Prediction of Quantile Differences in Logistic Regression for Two Independent Tests

ABSTRACTLogistic regression is a nonlinear method used for modeling a dichotomous (i.e., binary) response variable as a function of covariates. Such models have wide applicability and have proved especially useful in the health sciences where the question of effective dose (ED) or lethal dose (LD) is a central issue. For example, in toxicology studies, LD50 traditionally denotes the dose at which 50% of treated individuals perish. More generally, is the dose level corresponding to an average given percentage, 100p (i.e., the 100th quantile) of individuals responding to a treatment. In our field of flight test, logistic regression has a similar wide applicability. It is common to record a response as {hit, miss} or {success, failure} and to count the number of response successes at each level of input; the response is thus a quantal variable. In the example we develop here, the interest is in an application of blip-scan radar where the response is Y = {detect, no detect} and the covariate is range from target. In particular, we are interested in obtaining a confidence interval for the difference in range between two same-percentile values, one from each of two independent flights. The difference may be due to a different radar equipment configuration on each of the two flights and engineers are interested in quantifying the size of this difference in the detection performance. We approach the problem analytically and derive a symmetric confidence interval approximation for the average difference that is straightforward to compute and does not require simulation. Our results are based on the large-sample properties of maximum likelihood estimates and extend a result in nonlinear modeling given by Schwenke and Milliken (1991). The confidence interval so constructed is shown to give good probability coverage. Monte Carlo simulation is used to evaluate the procedure.

Percentile and Percentile-t Bootstrap Confidence Intervals: A Practical Comparison

Abstract This paper employs a Monte Carlo study to compare the performance of equal-tailed bootstrap percentile-t, symmetric bootstrap percentile-t, bootstrap percentile, and standard asymptotic confidence intervals in two distinct heteroscedastic regression models. Bootstrap confidence intervals are constructed with both the XY and wild bootstrap algorithm. Theory implies that the percentile-t methods will outperform the other methods, where performance is based on the convergence rate of empirical coverage to the nominal level. Results are consistent across models, in that in the case of the XY bootstrap algorithm the symmetric percentile-t method outperforms the other methods, but in the case of the wild bootstrap algorithm the two percentile-t methods perform similarly and outperform the other methods. The implication is that practitioners that employ the XY algorithm should utilize the symmetric percentile-t interval, while those who opt for the wild algorithm should use either of the percentile-t methods.

Investigation of the spectrum of the Sturm–Liouville problem with a nonlocal integral condition

This paper presents some new results on the spectrum for the second order dif-ferential problem with one integral type nonlocal boundary condition (NBC). We investigate how the spectrum of this problem depends on the integral nonlocal boundary condition pa-rameters γ, ξ and the symmetric interval in the integral. Some new results are given on the complex spectra of this problem. Many results are presented as graphs of real and complex characteristic functions.

Confidence Level Based on Ridge Estimator in Process Measurement and Its Application

Ordinary least squares (OLS) algorithm is widely applied in process measurement, because the sensor model used to estimate unknown parameters can be approximated through multivariate linear model. However, with few or noisy data or multi-collinearity, unbiased OLS leads to large variance. Biased estimators, especially ridge estimator, have been introduced to improve OLS by trading bias for variance. Ridge estimator is feasible as an estimator with smaller variance. At the same confidence level, with additive noise as the normal random variable, the less variance one estimator has, the shorter the two-sided symmetric confidence interval is. However, this finding is limited to the unbiased estimator and few studies analyze and compare the confidence levels between ridge estimator and OLS. This paper derives the matrix of ridge parameters under necessary and sufficient conditions based on which ridge estimator is superior to OLS in terms of mean squares error matrix, rather than mean squares error. Then the confidence levels between ridge estimator and OLS are compared under the condition of OLS fixed symmetric confidence interval, rather than the criteria for evaluating the validity of different unbiased estimators. We conclude that the confidence level of ridge estimator can not be directly compared with that of OLS based on the criteria available for unbiased estimators, which is verified by a simulation and a laboratory scale experiment on a single parameter measurement.

Bounds on eigenvalues of real and complex interval matrices

We present a cheap and tight formula for bounding real and imaginary parts of eigenvalues of real or complex interval matrices. It outperforms the classical formulae not only for the complex case but also for the real case. In particular, it generalizes and improves the results by Rohn (1998) [5] and Hertz (2009) [19]. The main idea behind is to reduce the problem to enclosing eigenvalues of symmetric interval matrices, for which diverse methods can be utilized.The result helps in analysing stability of uncertain dynamical systems since the formula gives sufficient conditions for testing Schur and Hurwitz stability of interval matrices. It may also serve as a starting point for some iteration methods.