Mean Integrated Squared Error Research Articles

Fault Diagnosis of Complex Chemical Processes Based on Enhanced Naive Bayesian Method

Fault diagnosis is crucial for the stable and reliable operation of chemical processes. However, faced with the complexity of chemical processes, conventional diagnosis methods suffer from the expertise of feature extraction and classifier design. They also lack extracting effective features from raw data whose attributes are correlated and coupled. This article proposes an enhanced naive Bayesian classifier (ENBC) for fault diagnosis of the complex chemical processes. Assuming that all attributes are related, ENBC can utilize a joint probability density function (pdf) based on multivariate Gaussian kernel function and adaptively extract the related information from the raw data. ENBC can estimate the class-conditional pdf based on the optimal smoothing parameter, which is the crucial part of the joint pdf estimation. ENBC seeks an optimal smoothing parameter by minimizing the mean integrated squared error (MISE) between the true pdf and the estimated pdf. The benchmark Tennessee Eastman (TE) process is used to illustrate the performance of ENBC. The experimental results demonstrate that ENBC is more effective than other methods.

Density and Risk Function of the Circular Kernel Study

This article is based on the works of [1], [2] and [3] on the estimation of the survival function and the function of risk in independent cases and identically distributed with and without censorship from which we established the bias and variance of the density of the circular kernel. In addition, we determined the optimal window b ∗ n of this estimator after having first established the mean square error (MSE) and mean integrated squared error (MISE) which are necessary conditions for obtaining the optimal window. Finally, we have established the asymptotic expression of the bias of the risk function of the circular kernel estimator

Doubly robust kernel density estimation when group membership is missing at random

When there are subjects with subpopulation memberships missing, the kernel density estimates of the subpopulations based on the subjects with verified memberships may not be valid unless the missingness of the memberships satisfies the stringent missing completely at random assumption. Under more plausible missing at random (MAR) assumptions, the inverse probability weighting (IPW) kernel smoothing method can be employed if the missing mechanism is well understood. On the other hand, if the memberships can be well predicted by a prediction model, one may replace the memberships (either all or only the missing ones), with the probabilities for the memberships to deal with the missing values. By integrating the information of the missing mechanism and the prediction model, we develop a doubly robust kernel density estimate which is valid as long as either of the models is correctly specified. Asymptotic properties of the new estimates are derived and the asymptotic orders of the mean integrated square errors (MISE) are compared with existing methods. We prove that the doubly robust estimate can be more efficient than the IPW estimate when the model for the missing mechanism is correctly specified. We also prove that the prediction-model based approaches may have smaller MISEs than the complete data when the prediction model is correctly specified, suggesting that those methods can be applied even when there are no missing values in the memberships. Simulation studies are carried out to assess the performances of methods and a real study example is used for illustrative purposes.

Discrete multivariate associated kernel estimators using two multiplicative bias correction methods

Two multiplicative bias correction (MBC) approaches for nonparametric multivariate associated kernel estimators for joint probability mass functions in the context of discrete supported data are proposed. Both techniques reach an optimal rate of convergence of the mean integrated squared error. We show some properties of the MBC multivariate associated kernel estimators like bias, variance and mean integrated square error. A simulation study and an application on a real count data set illustrate the performance of the MBC estimators based on the Dirac Discrete Uniform associated kernel in terms of the integrated squared error and integrated squared bias.

Data Dependent Asymmetric Kernels for Estimating the Density Function

In the nonparametric setup, smooth density estimators are obtained by using a kernel function and it is used to account for contribution of each observation to the estimator. Generally, these kernel functions are symmetric about zero. We propose an estimator by using data dependent asymmetric kernels being generated by an unimodal symmetric kernel. To account for the contribution of the ith order statistic to the estimator, a suitable asymmetric kernel is used. Asymptotic Bias, Mean Square Error (MSE) and Mean Integrated Square Error (MISE) of the estimator are obtained with higher accuracies by using some properties of the densities of order statistics. The convergence rates depend on both n, the sample size and hn, the band width. Expressions for the optimum band width minimizing the asymptotic MISE and the rate of convergence of the optimum MISE are obtained. Simulation study and data analysis indicate that the proposed estimator not only reduces bias outside the support significantly but also it is closer to the true density function as compared to that of the usual estimator based on a common symmetric kernel. Scope for the multivariate generalization of the method is given. Some methods of generating multivariate skew symmetric kernels are described. As an illustration one of them has been used in the simulation study and estimation of density in bivariate case.

Wavelet estimation of copula function based on censored data

In this paper, we consider wavelet analysis to obtain an estimator of a copula function based on censored data. We show that optimal convergence rates for the mean integrated squared error (MISE) of linear wavelet-based function estimators are exact under right censoring model. Moreover, we derive asymptotic formulae for MISE. Finally, the simulation results and the analysis of real data validate the proposed procedure.

Higher‐Order Accurate Spectral Density Estimation of Functional Time Series

Under the frequency domain framework for weakly dependent functional time series, a key element is the spectral density kernel which encapsulates the second‐order dynamics of the process. We propose a class of spectral density kernel estimators based on the notion of a flat‐top kernel. The new class of estimators employs the inverse Fourier transform of a flat‐top function as the weight function employed to smooth the periodogram. It is shown that using a flat‐top kernel yields a bias reduction and results in a higher‐order accuracy in terms of optimizing the integrated mean square error (IMSE). Notably, the higher‐order accuracy of flat‐top estimation comes at the sacrifice of the positive semi‐definite property. Nevertheless, we show how a flat‐top estimator can be modified to become positive semi‐definite (even strictly positive definite) in finite samples while retaining its favorable asymptotic properties. In addition, we introduce a data‐driven bandwidth selection procedure realized by an automatic inspection of the estimated correlation structure. Our asymptotic results are complemented by a finite‐sample simulation where the higher‐order accuracy of flat‐top estimators is manifested in practice.

Exact mean integrated squared error and bandwidth selection for kernel distribution function estimators

An exact, closed form, and easy to compute expression for the mean integrated squared error (MISE) of a kernel estimator of a normal mixture cumulative distribution function is derived for the class of arbitrary order Gaussian-based kernels. Comparisons are made with MISE of the empirical distribution function, the infeasible minimum MISE, and the uniform kernel. A simple plug-in method of simultaneously selecting the optimal bandwidth and kernel order is proposed based on a non asymptotic approximation of the unknown distribution by a normal mixture. A simulation study shows that the method provides a viable alternative to existing bandwidth selection procedures.

Investigating effects of bandwidth selection in local polynomial regression model with applications

Cross-validation (CV) and direct plug-in (DPI) are two commonly used bandwidth selection methods in nonparametric estimation. In this paper, we compare the performance of CV and DPI methods in local polynomial kernel regression through simulation study. We consider continuous response and binary re sponse cases, with local constant, local linear and local quadratic kernel estimators, respectively. Furthermore, we investigate the first derivatives of local quadratic kernel estimators. Our results show that CV and DPI methods excel in different cases, in terms of minimizing the Mean Integrated Squared Errors (MISE); the results are also verified in empirical studies.

The Comparison Between the MLE and Standard Bayes Estimators of the Reliability Function of Exponential Distribution

In this paper, a Monte Carlo Simulation technique is used to compare the performance of MLE and the standard Bayes estimators of the reliability function of the one parameter exponential distribution.Two types of loss functions are adopted, namely, squared error loss function (SELF) and modified square error loss function (MSELF) with informative and non- informative prior. The criterion integrated mean square error (IMSE) is employed to assess the performance of such estimators .

Penalized log-density estimation using Legendre polynomials

In this article, we present a penalized log-density estimation method using Legendre polynomials with penalty to adjust estimate’s smoothness. Re-expressing the logarithm of the density estimator via a linear combination of Legendre polynomials, we can estimate parameters by maximizing the penalized log-likelihood function. Besides, we proposed an implementation strategy that builds on the coordinate decent algorithm, together with the Bayesian information criterion (BIC). In particular, we derive a numerical solution to the maximum tuning parameter which leads to all zero coefficients and practically facilitates searching the optimal tuning parameter. Extensive simulation studies clearly show that our proposed estimator is computationally competitive with other existing nonparametric density estimators (e.g., kernel, kernel smooth and logspline estimators) benchmarked by the mean integrated squared errors (MISE) and the mean integrated absolute error (MIAE) under the experiment scenario of separated bimodal models in regard to the true density function. With an application to Old Faithful geyser data, our proposed method is found to effectively perform density estimation.

Nonparametric density estimation based on beta prime kernel

In this work, we propose beta prime kernel estimator for estimation of a probability density functions defined with nonnegative support. For the proposed estimator, beta prime probability density function used as a kernel. It is free of boundary bias and nonnegative with a natural varying shape. We obtained the optimal rate of convergence for the mean squared error (MSE) and the mean integrated squared error (MISE). Also, we use adaptive Bayesian bandwidth selection method with Lindley approximation for heavy tailed distributions and compare its performance with the global least squares cross-validation bandwidth selection method. Simulation studies are performed to evaluate the average integrated squared error (ISE) of the proposed kernel estimator against some asymmetric competitors using Monte Carlo simulations. Moreover, real data sets are presented to illustrate the findings.

The Comparison Between Standard Bayes Estimators of the Reliability Function of Exponential Distribution

In this paper, a Monte Carlo Simulation technique is used to compare the performance of the standard Bayes estimators of the reliability function of the one parameter exponential distribution .Three types of loss functions are adopted, namely, squared error loss function (SELF) ,Precautionary error loss function (PELF) andlinear exponential error loss function(LINEX) with informative and non- informative prior .The criterion integrated mean square error (IMSE) is employed to assess the performance of such estimators

A study on methods for estimating the PDF and the CDF in the exponentiated gamma distribution

The exponentiated gamma distribution is a two parameters lifetime distribution with monotone and non-monotone failure rates. In this article, some estimation methods of the probability density function and the cumulative distribution function of the exponentiated gamma distribution such as uniformly minimum variance unbiased (UMVU), maximum likelihood (ML), least squares, weighted least squares and Minimum distance estimators are studied and their performances through numerical simulations are compared. By the mean integrated squared error (MISE), the UMVU and ML are approximately equivalent and more efficient than the others based on simulation studies when the sample size is more than 40. A real data set is analyzed for illustrative purposes.

Reducing noise for PIC simulations using kernel density estimation algorithm

Noise is a major concern for Particle-In-Cell (PIC) simulations. We propose a new theoretical and algorithmic framework to evaluate and reduce the noise level for PIC simulations based on the Kernel Density Estimation (KDE) theory, which has been widely adopted in machine learning and big data science. According to this framework, the error on particle density estimation for PIC simulations can be characterized by the Mean Integrated Square Error (MISE), which consists of two parts, systematic error and noise. A careful analysis shows that in the standard PIC methods, noise is the dominant error, and the noise level can be reduced if we select different shape functions that are capable of balancing the systematic error and the noise. To improve, we use the von Mises distribution as the shape function and seek an optimal particle width that minimizes the MISE, represented by a cross-validation function. It is shown that this procedure significantly reduces the noise and the MISE for PIC simulations. A particle-wise width adjustment algorithm and a width update algorithm are also developed to further reduce the MISE. Simulations using the examples of Langmuir wave and Landau damping demonstrate that relative to the standard PIC methods, the KDE algorithm developed in the present study reduces the noise level on density estimation by 98% and gives a much more accurate result on the linear damping rate. To achieve the same accuracy, the KDE algorithm is 40% faster.

Wavelet density estimation for mixing and size-biased data

This paper considers wavelet estimation for a multivariate density function based on mixing and size-biased data. We provide upper bounds for the mean integrated squared error (MISE) of wavelet estimators. It turns out that our results reduce to the corresponding theorem of Shirazi and Doosti (Stat. Methodol. 27:12–19, 2015), when the random sample is independent.

"Optimized binning technique in decision tree model for predicting the Helicoverpa armigera (H�bner) incidence on cotton"

The data mining technique decision tree induction model is a popular method used for prediction and classification problems. The most suitable model in pest forewarning systems is decision tree analysis since pest surveillance data contains biotic, abiotic and environmental variables and IF-THEN rules can be easily framed. The abiotic factors like maximum and minimum temperature, rainfall, relative humidity, etc. are continuous numerical data and are important in climate-change studies. The decision tree model is implemented after pre-processing the data which are suitable for analysis. Data discretization is a pre-processing technique which is used to transform the continuous numerical data into categorical data resulting in interval as nominal values. The most commonly used binning methods are equal-width partitioning and equal-depth partitioning. The total number of bins created for the variable is important because either large number of bins or small number of bins affects the accuracy in results of IF-THEN rules. Hence, optimized binning technique based on Mean Integrated Squared Error (MISE) method is proposed for forming accurate IF-THEN rules in predicting the pest Helicoverpa armigera incidence on cotton crop based on decision tree analysis.

Shape-preserving wavelet-based multivariate density estimation

Wavelet estimators for a probability density f enjoy many good properties, however they are not ‘shape-preserving’ in the sense that the final estimate may not be non-negative or integrate to unity. A solution to negativity issues may be to estimate first the square root of f and then square this estimate up. This paper proposes and investigates such an estimation scheme, generalizing to higher dimensions some previous constructions which are valid only in one dimension. The estimation is mainly based on nearest-neighbor-balls. The theoretical properties of the proposed estimator are obtained, and it is shown to reach the optimal rate of convergence uniformly over large classes of densities under mild conditions. Simulations show that the new estimator performs as well in terms of Mean Integrated Squared Error as the classical wavelet estimator and better than it in terms of Mean Squared Hellinger Distance between the estimator and the truth, while automatically producing estimates which are bona fide densities.

On the Adaptive Nadaraya-Watson Kernel Estimator for the Discontinuity in the Presence of Jump Size

In this paper, we studied an Adaptive Nadaraya Watson kernel estimator to check the bias effect on both side of the discontinuity in the presence of jump size for regression discontinuity model. We have proposed the modified Adaptive Nadaraya Watson kernel estimator and derived its normality and variance. We have also compared with the asymptotic normality of the Mean Integrated Square Error (MISE) of Adaptive Nadaraya Watson kernel estimator and Nadaraya Watson kernel estimator. The results obtained from the simulation study have showed that Adaptive Nadaraya Watson estimator has better performance than the Nadaraya Watson Kernel estimator.

Polygonal smoothing of the empirical distribution function

We present two families of polygonal estimators of the distribution function: the first family is based on the knowledge of the support while the second addresses the case of an unknown support. Polygonal smoothing is a simple and natural method for regularizing the empirical distribution function $$F_n$$ but its properties have not been studied deeply. First, consistency and exponential type inequalities are derived from well-known convergence properties of $$F_n$$ . Then, we study their mean integrated squared error (MISE) and we establish that polygonal estimators may improve the MISE of $$F_n$$ . We conclude by some numerical results to compare these estimators globally, and also together with the integrated kernel distribution estimator.