数学代写|理论计算机代写theoretical computer science代考|Our Proposed Algorithm

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|理论计算机代写theoretical computer science代考|Our Proposed Algorithm

The proposed framework and algorithm is described in this section. The classifier works better with more labeled data, unlabeled data is much more than labeled data in semi supervised algorithms. If unlabeled data is labeled correctly and added to the labeled set, performance of the classifier improves.

Our proposed algorithm is a semi-supervised self-training algorithm. Figure 4 shows the steps of the proposed framework. In the first step, we find $\delta$ and $\rho$ for all training data set (labeled and unlabeled) and then the high density peaks are found in the labeled data set. The number of peaks is the number of classes. Then, for each peak we find corresponding far point. Step 2 consists of two parts. One section is about selecting a set of unlabeled data which is the candidate for labeling. The distance of all unlabeled data are calculated from decision boundary. Next unlabeled data are selected for labeling that are closer to the decision boundary. Threshold (distance of decision boundary) is considered the mean distance of all unlabeled data from decision boundary.

Another part of this step is how to label unlabeled data. Our proposed method predicts the label of unlabeled data by high density peak and Apollonius circle concepts. The label of peak $k_{i}$ is assigned to the unlabeled data that are inside the Apollonius circle corresponding $p e a k_{i}$ and in parallel the same unlabeled subset is given to SVM to label them. The agreement based on the classifier predictions and Apollonius circle is used to select a newly-labeled set The labeled set is updated and is used to retrain the classifier. The process is repeated until it reaches the stopping condition. Figure 4 represents the overview of proposed algorithm.

数学代写|理论计算机代写theoretical computer science代考|Experimental Setup

In this section, we perform several experiments to compare the classification performance of our proposed algorithm to the state-of-the-art semi-supervised methods using several different datasets. We also setup several experiments to show, the impact of selecting data close the decision boundary for improving the classification performance.

In the experiment, some UCI datasets are used. Table 1 summarizes the specification of 8 benchmark datasets from the UCI data repository which are used in our experiments.For each dataset, $30 \%$ of data are kept as test set randomly and the rest are used for training set. The training set is divided into two sets of labeled data and unlabeled data. Classes are selected at a proportions equal to the original dataset for all sets. Initially, we have assigned $5 \%$ to the labeled data and $95 \%$ to the unlabeled data. We have repeated each experiment ten times with different subsets of training and testing data. We report the mean accuracy rate (MAR) of 10 times repeating the experiments.

Table 2 shows the comparison of our algorithm with some other algorithms when labeled data is $10 \%$. The second and third columns in this table give respectively the performance of the supervised SVM and self-training SVM. The fourth col$u m n$ is the performance of state-of-the-art algorithm that is called STC-DPC algorithm [7]. The last column is the performance of our algorithm. Base learner for all of algorithms is SVM. Cut of distance parameter (dc) for our algorithm and STC-DPC is $0.05$. From Table 2 , we observe that Our algorithm works well for datasets that have a separate data density such as Iris, Seeds, Wine. Our proposed algorithm doesn’t work very well if dataset is very mixed, such as banknote Fig. 5 and Fig. 6. We also investigate the behavior of the algorithms based on increasing ratio of labeled data. Fig. 7 is a comparison of the three algorithms with the increase ratio of label data from $5 \%$ to $50 \%$.

数学代写|理论计算机代写theoretical computer science代考|Impact of Selecting Data Close the Decision Boundary

In most datasets, labeling all unlabeled data can not improve the performance but also reduces the accuracy. In addition to decreasing accuracy, runtime is also increased. Although the unlabeled data that are far from the decision boundary are more reliable, they are not informative. They play little role in improving decision boundary. That’s why we haven’t added all the unlabeled data to the

training set, rather, we add those that are closer to the decision boundary than a certain value.

We show the results on a number of datasets in Table 3 . The second column is the accuracy of our algorithm when we have added all the unlabeled data and the third column is the accuracy of our algorithm when we add only the data point closes to the decision boundary. As can be seen from Table 3 , the accuracy of the algorithm increases when we only add unlabeled data closer to the decision boundary instead of all the points.

In this paper, we proposed a semi-supervised self-training method based on Apollonius, named SSApolo. First candidate data are selected from among the unlabeled data to be labeled in the self training process, then, using the density peak clustering, the peak points are found and by making an Apollonius circle of each peak points, their neighbors are found and labeled. Support vector machine is used for classification. A series of experiments was performed on some datasets and the performance of the proposed algorithm was evaluated. According to the experimental results, we conclude that our algorithm performs better than STCDPC algorithm and supervised SVM and self-training SVM, especially when classes of dataset are not very mixed. In addition, the impact of selecting data are close to decision boundary was investigated. We find that selecting data are close to decision boundary can improves the performance.

有限元方法代写

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MATLAB代写

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