Robustness of classification rules that incorporate additional information

The discrimination problem for two normal populations with the same covariance matrix when additional information on the population is available is considered. A study of the robustness properties against training sample contamination of classification rules that incorporate this additional information is performed. These rules have received recently attention where their total misclassification probability (TMP) is proved to be lower than Fisher's linear discriminant rule. The results of a simulation study on the TMP which compares the behaviour of the new rules against Fisher's rule and some of its robustified versions under different types of contamination are presented. These results show that the rules that incorporate the additional information not only have lower TMP, but they also prevent against some types of contamination. In order to achieve prevention from all types of contamination a robustifed version of these rules is recommended.     

基于人工蜂群与模糊C均值的自适应小波变换的噪声图像分割

针对传统模糊C均值（FCM）聚类算法在处理噪声图像时易受到噪声影响的问题,提出了基于FCM的小波域特征增强的噪声图像分割方法。首先,将噪声图像进行二维小波分解;其次,对近似系数进行边缘增强,同时利用人工蜂群（ABC）优化算法对细节系数进行阈值处理,并将处理后的系数进行小波重构;最后,对重构后的图片使用FCM算法来进行图像分割。选取5幅典型的灰度图像,分别添加高斯噪声和椒盐噪声,使用多种方法进行分割,以分割后图像的峰值信噪比（PSNR）和误分率（ME）作为性能指标,实验结果表明,所提方法分割后的图片相较于传统FCM聚类算法分割方法和粒子群优化（PSO）分割方法分割后的图片在PSNR上最多分别有281%和54%的提升,在ME上最多分别有55%和41%的降低。可见所提出的分割方法较好地保留了图像边缘纹理信息,其抗噪性能与分割性能得到了提升。     

Estimating the Expected and the Optimal Probabilities of Misclassification

The problem is to estimate the expected and the optimal probabilities of misclassification in the context of the two group p-dimensional normal classification problem with means and common covariance matrix unknown and a rule based on the linear discriminant function. Performance of several estimators is compared by means of a computer sampling study. For larger p (p = 20) certain estimators are definitely superior for each of the probabilities, while for small p there is less differentiation in performance.     

Note on a Method for Calculating Corrected Sums of Squares and Products

Two models of non-random initial misclassifications are studied. In these models, observations which are closer to the mean of the “wrong” population have a greater chance of being misclassified than others. Sampling studies show that (a) the actual error rates of the rules from samples with initial misclassification are only slightly affected; (b) the apparent error rates, obtained by resubstituting the observations into the calculated discriminant function, are drastically affected, and cannot be used; and (c) the Mahalanobis D ² is greatly inflated.     

Further results on the effect of intraclass correlation among training samples in discriminant analysis

The effect of intraclass correlation among training samples on the misclasification probabilities of Bayes' procedure has been recently studied by Basu and Odell.⁽¹³⁾ This paper investigates the effect further by expressing the misclassification probabilities in the form of asymptotic expansions. It is subsequently shown that contrary to previous conclusions the misclassification probabilities do change in the presence of simple equicorrelation among the training samples.     

Effect of intraclass correlation among training samples on the misclassification probabilities of bayes procedure

The sample mean and sample covariance matrix are unbiased and consistent estimates of population mean and covariance matrix only if the samples are independent. In practical applications of Bayes' procedure these estimates are used in place of population means and covariance matrices on the assumption of independence among the training samples. This practice has often given, especially in remote sensing data analysis, misclassification probabilities much higher than that can be accounted for theoretically. The reason may be that the assumption of independence may not be valid. In reality, the samples are rarely independent, they are rather dependent, at best equicorrelated. This paper investigates how such intraclass correlation among the training samples affects the misclassification probabilities of the Bayes' procedure.     

Multiple costs based decision making with back-propagation neural networks

The current research investigates a single cost for cost-sensitive neural networks (CNN) for decision making. This may not be feasible for real cost-sensitive decisions which involve multiple costs. We propose to modify the existing model, the traditional back-propagation neural networks (TNN), by extending the back-propagation error equation for multiple cost decisions. In this multiple-cost extension, all costs are normalized to be in the same interval (i.e. between 0 and 1) as the error estimation generated in the TNN. A comparative analysis of accuracy dependent on three outcomes for constant costs was performed: (1) TNN and CNN with one constant cost (CNN-1C), (2) TNN and CNN with two constant costs (CNN-2C), and (3) CNN-1C and CNN-2C. A similar analysis for accuracy was also made for non-constant costs; (1) TNN and CNN with one non-constant cost (CNN-1NC), (2) TNN and CNN with two non-constant costs (CNN-2NC), and (3) CNN-1NC and CNN-2NC. Furthermore, we compared the misclassification cost for CNNs for both constant and non-constant costs (CNN-1C vs. CNN-2C and CNN-1NC vs. CNN-2NC). Our findings demonstrate that there is a competitive behavior between the accuracy and misclassification cost in the proposed CNN model. To obtain a higher accuracy and lower misclassification cost, our results suggest merging all constant cost matrices into one constant cost matrix for decision making. For multiple non-constant cost matrices, our results suggest maintaining separate matrices to enhance the accuracy and reduce the misclassification cost.     

Likelihood-based confidence intervals for the risk ratio using double sampling with over-reported binary data

In this article we derive likelihood-based confidence intervals for the risk ratio using over-reported two-sample binary data obtained using a double-sampling scheme. The risk ratio is defined as the ratio of two proportion parameters. By maximizing the full likelihood function, we obtain closed-form maximum likelihood estimators for all model parameters. In addition, we derive four confidence intervals: a naive Wald interval, a modified Wald interval, a Fieller-type interval, and an Agresti-Coull interval. All four confidence intervals are illustrated using cervical cancer data. Finally, we conduct simulation studies to assess and compare the coverage probabilities and average lengths of the four interval estimators. We conclude that the modified Wald interval, unlike the other three intervals, produces close-to-nominal confidence intervals under various simulation scenarios examined here and, therefore, is preferred in practice.     

Bayesian multiscale smoothing in supervised and semi-supervised kernel discriminant analysis

In kernel discriminant analysis, it is common practice to select the smoothing parameter (bandwidth) based on the training data and use it for classifying all unlabeled observations. But this method of selecting a single scale of smoothing ignores the major issue of model uncertainty. Moreover, in addition to depending on the training sample, a good choice of bandwidth may also depend on the observation to be classified, and a fixed level of smoothing may not work well in all parts of the measurement space. So, instead of using a single smoothing parameter, it may be more useful in practice to study classification results for multiple scales of smoothing and judiciously aggregate them to arrive at the final decision. This paper adopts a Bayesian approach to carry out one such multiscale analysis using a probabilistic framework. This framework also helps us to extend our multiscale method for semi-supervised classification, where, in addition to the training sample, one uses unlabeled test set observations to form the decision rule. Some well-known benchmark data sets are analyzed to show the utility of these proposed methods.     

Estimating the Expected Probability of Misclassification for a Rule Based on the Linear Discriminant Function: Univariate Normal Case

The problem is to estimate the average probability of misclassifying an observation from a given population in the context of the two group classification problem when populations are univariate normal with unknown means and common known variance, and the rule is based on the linear discriminant function. Several estimators are compared with respect to asymptotic MSE and with respect to the distribution of the absolute error between estimator and parameter, and conclusions drawn about the best estimators.