Generating Synthetic Missing Data: A Review by Missing Mechanism <s> VI. CRITICAL ANALYSIS AND DISCUSSION <s> Mutual information is a good indicator of relevance between variables, and have been used as a measure in several feature selection algorithms. However, calculating the mutual information is difficult, and the performance of a feature selection algorithm depends on the accuracy of the mutual information. In this paper, we propose a new method of calculating mutual information between input and class variables based on the Parzen window, and we apply this to a feature selection algorithm for classification problems. <s> BIB001 </s> Generating Synthetic Missing Data: A Review by Missing Mechanism <s> VI. CRITICAL ANALYSIS AND DISCUSSION <s> Increasing the awareness of how incomplete data affects learning and classification accuracy has led to increasing numbers of missing data techniques. This article investigates the robustness and accuracy of seven popular techniques for tolerating incomplete training and test data for different patterns of missing data—different proportions and mechanisms of missing data on resulting tree-based models. The seven missing data techniques were compared by artificially simulating different proportions, patterns, and mechanisms of missing data using 21 complete datasets (i.e., with no missing values) obtained from the University of California, Irvine repository of machine-learning databases (Blake and Merz, 1998). A four-way repeated measures design was employed to analyze the data. The simulation results suggest important differences. All methods have their strengths and weaknesses. However, listwise deletion is substantially inferior to the other six techniques, while multiple imputation, that utilizes the expectation maximization algorithm, represents a superior approach to handling incomplete data. Decision tree single imputation and surrogate variables splitting are more severely impacted by missing values distributed among all attributes compared to when they are only on a single attribute. Otherwise, the imputation—versus model-based imputation procedures gave—reasonably good results although some discrepancies remained. Different techniques for addressing missing values when using decision trees can give substantially diverse results, and must be carefully considered to protect against biases and spurious findings. Multiple imputation should always be used, especially if the data contain many missing values. If few values are missing, any of the missing data techniques might be considered. The choice of technique should be guided by the proportion, pattern, and mechanisms of missing data, especially the latter two. However, the use of older techniques like listwise deletion and mean or mode single imputation is no longer justifiable given the accessibility and ease of use of more advanced techniques, such as multiple imputation and supervised learning imputation. <s> BIB002 </s> Generating Synthetic Missing Data: A Review by Missing Mechanism <s> VI. CRITICAL ANALYSIS AND DISCUSSION <s> Constructing an accurate effort prediction model is a challenge in software engineering. The development and validation of models that are used for prediction tasks require good quality data. Unfortunately, software engineering datasets tend to suffer from the incompleteness which could result to inaccurate decision making and project management and implementation. Recently, the use of machine learning algorithms has proven to be of great practical value in solving a variety of software engineering problems including software prediction, including the use of ensemble (combining) classifiers. Research indicates that ensemble individual classifiers lead to a significant improvement in classification performance by having them vote for the most popular class. This paper proposes a method for improving software effort prediction accuracy produced by a decision tree learning algorithm and by generating the ensemble using two imputation methods as elements. Benchmarking results on ten industrial datasets show that the proposed ensemble strategy has the potential to improve prediction accuracy compared to an individual imputation method, especially if multiple imputation is a component of the ensemble. <s> BIB003 </s> Generating Synthetic Missing Data: A Review by Missing Mechanism <s> VI. CRITICAL ANALYSIS AND DISCUSSION <s> Treatment of missing data has become increasingly significant in scientific research and engineering applications. The classic imputation strategy based on the K nearest neighbours (KNN) has been widely used to solve the plague problem. However, former studies do not give much attention to feature relevance, which has a significant impact on the selection of nearest neighbours. As a result, biased results may appear in similarity measurements. In this paper, we propose a novel method to impute missing data, named feature weighted grey KNN (FWGKNN) imputation algorithm. This approach employs mutual information (MI) to measure feature relevance. We present an experimental evaluation for five UCI datasets in three missingness mechanisms with various missing rates. Experimental results show that feature relevance has a non-ignorable influence on missing data estimation based on grey theory, and our method is considered superior to the other four estimation strategies. Moreover, the classification bias can be significantly reduced by using our approach in classification tasks. <s> BIB004
In this section, we provide a thorough analysis of some details that were left undiscussed in the original papers previously discussed, also referring to non-obvious issues that may arise in each implementation. A. MCAR UNIVA IMPLEMENTATIONS Table 3 refers to some issues/restrictions in MCAR univa implementations. In what concerns MCAR1 univa , three main issues need to be considered: • Definition of MR: By defining the MR over the entire dataset, the possible highest MR that is possible to simulate is dependent on the number of features comprised in the dataset. As an example, if dataset X has 2 features, the highest MR possible is limited to 50%, and ideally, should be lower, since, for 50%, x miss would be completely missing, given the p× MR adjustment. • Usage of Bernoulli trials: To generate the missing values, n Bernoulli trials are performed, each with probability of success p = MR. According to the Law of Large Numbers (LLN), as the number of Bernoulli trials increases (as n, the number of patterns in X increases), the empirical probability of success (the real MR generated) will converge to the theoretical probability of success (the specified MR). As the name implies, the LLN applies when a large number of experiments is performed (large n). Therefore, for small datasets, there is no guarantee that the generated MR will coincide with the desired MR (it will be approximate, though not precise). As an example, for a desired MR of 30%, a certain run of MCAR generation could provide a real MR of 28% while another could return a real MR of 34%. Naturally, there is frequently a small bias in the generated missing percentages in several approaches, due to the rounding performed for the calculation of the number of missing positions to generate. However, this bias seems to be more significant when considering the usage of Bernoulli trials (for small datasets). • Correlation between features: In all of the implementations by Twala BIB002 and Twala and Cartwright BIB003 , VOLUME 7, 2019 x miss is the feature most correlated with the class labels. Furthermore, in some approaches, there is also the need to define pairs of correlated features. In the original papers, Twala et al. consider datasets composed by both quantitative and qualitative features, yet the computation of the correlation between different types of features is not specified. Possible solutions to measure the correlation between different feature types are the computation of mutual information between features or the calculation of different coefficients according to each feature type (e.g. Pearson coefficient for two continuous features, phi coefficient for two binary features, pointbiserial for a continuous and a binary feature, and so forth). The latter solution, however, would have to be looked at as an approximation, since there is no proper way to compare different coefficients. MCAR2 univa implementation allows the definition of MR for the entire dataset or for a single feature and depending on that choice, there are different restrictions to the allowed missing rates (Table 3) . Regarding x miss , it can be randomly chosen or defined by the researcher. To provide a consistent experimental setup, one could choose the same feature x miss to be missing at several MRs (e.g. 5, 10, 20%) and study the effects that higher MRs have in classification performance. Choosing x miss according to the highest mutual information (MI) with the class labels t (MCAR3 univa ) might be problematic for quantitative/continuous features. The MI for two qualitative/categorical features is straightforward since the probability densities can be estimated using a histogram BIB004 . However, for quantitative/continuous features, the estimation of probability densities is more complicated. Frequent solutions include the discretization of continuous features BIB004 or applying Parzen-windows estimation BIB001 , which is the method chosen for MCAR3 univa . The computation of Parzen windows can, however, be computationally expensive. Among all approaches, MCAR2 univa is an efficient method, straightforward to understand and implement, and thus we recommend it for standard MCAR univa experiments.