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Approximations of the critical region of the Friedman statistic
Communications in Statistics - Theory and Methods
Abstract
The Friedman (1937) test for the randomized complete block design is used to test the hypothesis of no treatment effect among k treatments with b blocks. Difficulty in determination of the size of the critical region for this hypothesis is com¬pounded by the facts that (1) the most recent extension of exact tables for the distribution of the test statistic by Odeh (1977) go up only to the case with k6 and b6, and (2) the usual chi-square approximation is grossly inaccurate for most commonly used combinations of (k,b). The purpose of this paper 2 is to compare two new approximations with the usual x and F large sample approximations. This work represents an extension to the two-way layout of work done earlier by the authors for the one-way Kruskal-Wallis test statistic.
... But it is not specified which FS models are different from the other FS models. So, to identify which of the FS models are different from the other models, post-hoc analysis is After the acceptance of the H 1 and the rejection of the H 0 , the post-hoc analysis is conducted by using the Holm Test (or Holm procedure) [73,74]. For the comparison of the performance of one model with another model with the help of Holm procedure, the p value and z value are considered. ...
Software fault prediction (SFP) is a cost-effective method that helps to detect software faults at an early stage and improves the quality of software. Different machine learning models were used for the prediction of software defects. These algorithms use previous software metrics and fault data as independent features to detect the defects in the software. Early detection of software faults saves the costs required to rectify those vulnerabilities. But as the amount of data (independent features) is very large, feature selection (FS) has become a crucial process to get the most useful information from the data. In this work, a novel and economical approach for SFP has been proposed namely, FS using Coati Optimization Algorithm (FSCOA). It selects the relevant and most useful subset of features from the data. This FS algorithm was embedded with four popular classifiers (KNN, QDA, DT, and NB) and the results were compared with four well-known FS algorithms such as, FS using PSO, FS using ACO, FS using GA, and FS using DE. Twelve open-source software defect datasets were used for conducting the experiments. The experimental results indicated that the FSCOA outperformed the compared algorithms in more than 90% of the test cases. FSCOA gave higher accuracies of classification and the number of optimal features selected were less than 50% of the original features. Moreover, the superiority of the proposed FSCOA in comparison to the other FS models has been statistically validated with the help of Friedman Test and Holm Test where, FSCOA held a Friedman rank 1 in all the test cases. The results from these statistical tests supports the experimental results showing the enhanced performance of FSCOA compared to the other FS models taken into consideration.
... The rank of the j-th algorithm on the i-th dataset is denoted as r j i , and the average rank of the algorithm is R 2 j = 1 N N i=1 r j i . However, Iman and Davenport's research indicated that the Friedman statistic is overly conservative and proposed the F statistic (Iman and Davenport 1980). ...
The complementarity of ensemble learning primarily stems from the diversity among its classifier members, which is typically assessed by calculating the similarity or distance between classifiers. However, traditional methods for similarity or distance calculation are limited by the representation of classifiers, which may not accurately reflect the true relationships between them, leading to inferior ensemble pruning performance. To address this issue, this study proposes a new classifier representation paradigm and ensemble pruning algorithm(Vec2Q-CEEP). To begin with, we transform the representation of classifiers through spatial mapping, overcoming the limitations of traditional similarity calculation methods, enriching the ways of calculating classifier diversity, and enhancing the distinguishability between classifiers. Furthermore, we introduce a novel similarity measurement method that integrates classifier information from both the original and mapped spaces, enabling a more comprehensive evaluation of the relationships between classifiers. After conducting extensive experiments on 12 UCI datasets, the experimental results demonstrate that this algorithm outperforms several state-of-the-art ensemble pruning algorithms, thus validating its effectiveness and feasibility.
... The performance of three final predictors is remarkable on PCA. Moreover, considering the HOG features, the combination of HOG with NB outperformed the combination with KNN and SVM in five types pose-aware stacked ensemble model [28][29][30][31][32][33][34][35]. ...
In this paper, we developed a pose-aware facial expression recognition technique. The proposed technique employed K nearest neighbor for pose detection and a neural network-based extended stacking ensemble model for pose-aware facial expression recognition. For pose-aware facial expression classification, we have extended the stacking ensemble technique from a two-level ensemble model to three-level ensemble model: base-level, meta-level and predictor. The base-level classifier is the binary neural network. The meta-level classifier is a pool of binary neural networks. The outputs of binary neural networks are combined using probability distribution to build the neural network ensemble. A pool of neural network ensembles is trained to learn the similarity between multi-pose facial expressions, where each neural network ensemble represents the presence or absence of a facial expression. The predictor is the Naive Bayes classifier, it takes the binary output of stacked neural network ensembles and classifies the unknown facial image as one of the facial expressions. The facial concentration region was detected using the Voila-Jones face detector. The Radboud faces database was used for stacked ensembles’ training and testing purpose. The experimental results demonstrate that the proposed technique achieved 90% accuracy using Eigen features with 160 stacked neural network ensembles and Naive Bayes classifier. It demonstrates that the proposed techniques performed significantly as compare to state of the art pose-ware facial expression recognition techniques.
... Furthermore, the Friedman test and Wilcoxon's rank-sum test were employed to evaluate the overall performance of the algorithms and the significance of the results. The Friedman test is a non-parametric statistical approach that assesses the significance of differences in average algorithm rankings [80]. The algorithm that ranks first has the best overall performance. ...
Multi-UAV path planning presents a critical challenge in Unmanned Aerial Vehicle (UAV) applications, particularly in environments with various obstacles and restrictions. These conditions transform multi-UAV path planning into a complex optimization problem with multiple constraints, significantly reducing the number of feasible solutions and complicating the generation of optimal flight trajectories. Although the slime mold algorithm (SMA) has proven effective in optimization missions, it still suffers from limitations such as inadequate exploration capacity, premature convergence, and a propensity to become stuck in local optima. These drawbacks degrade its performance in intricate multi-UAV scenarios. This study proposes a self-adaptive improved slime mold algorithm called AI-SMA to address these issues. Firstly, AI-SMA incorporates a novel search mechanism to balance exploration and exploitation by integrating ranking-based differential evolution (rank-DE). Then, a self-adaptive switch operator is introduced to increase population diversity in later iterations and avoid premature convergence. Finally, a self-adaptive perturbation strategy is implemented to provide an effective escape mechanism, facilitating faster convergence. Extensive experiments were conducted on the CEC 2017 benchmark test suite and multi-UAV path-planning scenarios. The results show that AI-SMA improves the quality of optimal fitness by approximately 7.83% over the original SMA while demonstrating superior robustness and effectiveness in generating collision-free trajectories.
... The calculation of the statistical measure F ID , as proposed by Iman and Davenport [75], is computed as: ...
Segmentation of dermoscopic images is an important phase of computer-aided diagnosis and classification of skin cancers. The varied shapes, sizes, and colors of the lesion, differences in the intensity of pixels within a lesion, presence of hair, and fuzziness at the lesion boundary make precise segmentation more difficult. To overcome all these problems, we propose a novel weighted fuzzy-based clustering method named WFC_AS_KL , which involves three terms in its objective function. The first term is a weighted combination of Fuzzy C-Mean and Fuzzy k-Plane Clustering, which is suitable to handle asymmetrical data distribution and changes in the color of dermoscopic images. To mitigate the effect of noise, weighted region-level and local-level spatial information is used as the second term. The third term is Kullback–Leibler information, which reduces the fuzziness of the fuzzy partition matrix. It also handles the cluster-center overlapping problem and unbalanced cluster size problem. An adaptive constraint is also used to control the original and spatial information of the image. We demonstrate the superior performance of the proposed WFC_AS_KL method on two publicly available medical image datasets against 13 fuzzy-based clustering methods in terms of sensitivity, specificity, precision, false positive rate, accuracy, dice similarity coefficient, and jaccard index. We also compared the performance of the proposed method against twelve traditional skin lesion segmentation methods and twelve existing deep learning methods. We also performed a non-parametric statistical test to demonstrate the statistically superior performance of the proposed WFC_AS_KL method in comparison to 13 fuzzy-based segmentation methods.
Several approximations to the exact distribution of the Kruskal-Wallis test' statistic presently exist. There approximations can roughly be grouped into two classes: (i) computationally difficult with good accuracy, and (ii) easy to compute but not as accurate as the first class. The purpose of this paper is to introduce two nev approximations (one in the latter class and one which is computationally more involved)y and to compare these with other popular approximations. These comparisons use exact probabilities where available and Monte Carlo simulation otherwise.
Error bounds depending explicitly on parameters of the problem are given for large-sample approximations to the central and noncentral distributions ofFriedman's [1937]x
r
2 statistic,Steel's [1959] statistics for comparingr treatments with a control, andNemenyi's [1963] statistics for pair-wise comparisons among all treatments. These bounds are of ordero(N
−1/2); an improved bound of ordero (N
−r/(r+1)) is given forFriedman's statistic in the central case. Applications yield bounds on the actual type 1 and type 2 error probabilities in terms of their normal-theory approximations.
Exact tables for Spearman’s rho are available only for n ≤ 16 and no ties in the data. Some accurate methods of approximating the distribution with no ties present have been used to obtain approximate tables for larger values of n. Often ties are present in the data so these tables are no longer exact. Also sometimes the tables are not conveniently available to the user. In such situations an approximation that is both simple and accurate would be useful. Such an approximation is presented in this paper. Comparisons are made with other standard approximations for all cases where exact tables (no ties) are available, and for one case where exact tables were generated for a situation with ties. The results show the approximation presented here to be the most accurate of the approximations examined. Also it is simple enough to be readily understood by the average user of Spearman’s rho.
Friedman’s (1937, 1940) S-statistic is designed to test the hypothesis that there is no treatment effect in a randomized-block design with k treatments and n blocks. In this paper we give tables of the null distribution of S for k = 5, n = 6(1)8, and for k = 6, n = 2(1)6. Computational details are discussed.
An approximation to the exact distribution of the Wilcoxon signed ranks test statistic based on the one-sample t-test applied to ranks is compared with the usual normal approximation. A second approximation based on a linear combination of the normal statistic and the t-statistic is introduced. The normal approximation tends to result in a conservative test in the tails, while the Student's t approximation tends to be liberal. The average of the two statistics provides a test that usually has an alpha level as close as possible to the α-levels .05, .025, .01 and .005, for values of n < 50 (only cases examined).
An approximation to the exact distribution of the Wilcoxon rank sum test (Mann-Whitney U-test) and the Siegel-Tukey test based on a linear combination of the two-sample t-test applied to ranks and the normal approximation is compared with the usual normal approximation. The normal approximation results in a conservative test in the tails while the linear combination of the test statistics provides a test that has a very high percentage of agreement with tables of the exact distribution. Sample sizes 3≤m, n≤50 were considered.
The distribution of a linear combination of two statistics distributed as is Chi-square is studied. The degree of approximation involved in assuming a Chi-square distribution is illustrated for several representative cases. It is concluded that the approximation is sufficiently accurate to use in many practical applications. Illustrations are given of its use in extending the Chi-square, the Student t and the Fisher z tests to a wider range of problems.