LoRAS: An oversampling approach for imbalanced datasets
Abstract
The Synthetic Minority Oversampling TEchnique (SMOTE) is widelyused for the analysis of imbalanced datasets. It is known that SMOTE frequently overgeneralizes the minority class, leading to misclassifications for the majority class, and effecting the overall balance of the model. In this article, we present an approach that overcomes this limitation of SMOTE, employing Localized Random Affine Shadowsampling (LoRAS) to oversample from an approximated data manifold of the minority class. We benchmarked our algorithm with 12 publicly available imbalaned datasets using three different Machine Learning (ML) algorithms and comparing the performance of LoRAS, SMOTE and several SMOTE extensions, observed that LoRAS, on average generates better ML models in terms of F1Score and Balanced accuracy. Another key observation is that while most of the extensions of SMOTE we have tested, improve the F1Score with respect to SMOTE on an average, they compromise on the Balanced accuracy of a classification model. LoRAS on the contrary, improves both F1 Score and the Balanced accuracy thus produces better classification models. Moreover, to explain the success of the algorithm, we have constructed a mathematical framework to prove that LoRAS oversampling technique provides a better estimate for the mean of the underlying local data distribution of the minority class data space.
font=Small
Index terms— Imbalanced datasets, Oversampling, Synthetic sample generation, Data augmentation, Manifold learning
1 Introduction
Imbalanced datasets are frequent occurrences in a large spectrum of fields, where Machine Learning (ML) has found its applications, including business, finance and banking as well as boimedical science. Oversampling approaches are a popular choice to deal with imbalanced datasets (SMOTE, Han2, He, Bunkhumpornpat2009, Barua2014). We here present Localized Randomized Affine Shadowsampling (LoRAS), which produces better ML models for imbalanced datasets, compared to stateofthe art oversampling techniques such as SMOTE and several of its extensions. We use computational analyses and a mathematical proof to demonstrate that drawing samples from a locally approximated data manifold of the minority class can produce balanced classification ML models. We validated the approach with 12 publicly available imbalanced datasets, comparing the performances of several stateoftheart convexcombination based oversampling techniques with LoRAS. The average performance of LoRAS on all these datasets is better than other oversampling techniques that we investigated. In addition, we have constructed a mathematical framework to prove that LoRAS is a more effective oversampling technique since it provides a better estimate for local mean of the underlying data distribution, in some neighbourhood of the minority class data space.
For imbalanced datasets, the number of instances in one (or more) class(es) is very high (or very low) compared to the other class(es). A class having a large number of instances is called a majority class and one having far fewer instances is called a minority class. This makes it difficult to learn from such datasets using standard ML approaches. Oversampling approaches are often used to counter this problem by generating synthetic samples for the minority class to balance the number of data points for each class. SMOTE is a widely used oversampling technique, which has received various extensions since it was published by SMOTE. The key idea behind SMOTE is to randomly sample artificial minority class data points along line segments joining the minority class data points among of the minority class nearest neighbors of some arbitrary minority class data point. In other words, SMOTE produces oversamples by generationg random convex combinations of two close enough data points.
The SMOTE algorithm, however has several limitations for example: it does not consider the distribution of minority classes and latent noise in a data set (Hu2009). It is known that SMOTE frequently overgeneralizes the minority class, leading to misclassifications for the majority class, and effecting the overall balance of the model (punt). Several other limitations of SMOTE are mentioned in Blagus2013. To overcome such limitations, several algorithms have been proposed as extensions of SMOTE. Some are focusing on improving the generation of synthetic data by combining SMOTE with other oversampling techniques, including the combination of SMOTE with Tomeklinks (ElhassanT2016), particle swarm optimization (Gao, Wang), rough set theory (Ram), kernel based approaches (Mathew), Boosting (Chawla2), and Bagging (Hanifah). Other approaches choose subsets of the minority class data to generate SMOTE samples or cleverly limit the number of synthetic data generated (Narayan). Some examples are Borderline1/2 SMOTE (Han2), ADAptive SYNthetic (ADASYN) (He), Safe Level SMOTE (Bunkhumpornpat2009), Majority Weighted Minority Oversampling TEchnique (MWMOTE) (Barua2014), Modified SMOTE (MSMOTE), and Support Vector MachineSMOTE (SVMSMOTE) (Suh) (see Table 1) (Hu2009). Another recent method, GSMOTE, generates synthetic samples in a geometric region of the input space, around each selected minority instance (GSMOTE). Voronoi diagrams have also been used in recent research for improving classification tasks for imbalanced datasets. Because of properties inherent to Voronoi diagrams, a newly proposed algorithm Vsynth identifies exclusive regions of feature space where it is ideal to create synthetic minority samples (Vor, Vor2).
Related research and novelty: A more recent trend in the research on imbalanced datasets is to generate synthetic samples, aiming to approximate the latent data manifold of the minority class data space. In Belli, a general framework for manifoldbased oversampling, especially for high dimensional datasets, is proposed for synthetic oversampling. The method has been successfully applied in Belli2 to deal with gammaray spectra classification. It produces a synthetic set of instances in the manifoldspace by randomly sampling instances from the PCAtransformed reduced data space. In order to produce unique samples on the manifold, they apply i.i.d. additive Gaussian noise to each sampled instance prior to adding it to the synthetic set , controlling the distribution of the noise through the Gaussian distribution parameters. The synthetic Gaussian instances are then mapped back to the feature space to produce the final synthetic samples (Belli). Another scheme, using autoencoders to oversample from an approximated manifold, has also been discussed in Belli. This approach selects random minority class samples by adding Gaussian noise to them, and using the autoencoder framework first maps them nonorthogonally off the manifold and then maps them back orthogonally on the manifold Belli. It remains unclear from this research how the approach would perform in terms of improving F1Scores of imbalanced classification models as it focuses on relative improvement in the Area Under the (ROC) Curve (AUC) as a performance measure. According to Saito, AUC of the Receiver Operating Characteristic Curve (ROC) curve might not be informative enough for imbalanced datasets. This issue has also been addressed in Davis. Unlike the work of Belli LoRAS relies on locally approximating the manifold by generating random convex combination of noisy minority class data points. Our oversampling strategy LoRAS, rather aims at improving the precisionrecall balance (F1Score) and class wise average accuracy (Balanced accuracy) of the ML models used. The F1Score can measure how well the classification model handled the minority class classification, whereas Balanced accuracy provides us with a measure of how both majority and minority classes were handled by the classification model. Thus, these two measures together can gives us a holistic understanding of a classifier performance on a dataset.
Notably, in the preSMOTE era of research, related to oversampling there has been works aiming to enrich minority classes of imbalanced datasets by adding Gaussian noise noisy and using the noisy data itself, as oversampled data. The strategy of generating oversamples with convex combinations of minority class samples is also well known, SMOTE itself being an example of such a strategy. Our oversampling strategy LoRAS leverages from a combination of these two strategies. Unlike noisy, we generate Gaussian noise in small neighbourhoods around the minority class samples and create our final synthetic data with convex combinations of multiple noisy data points (shadowsamples) as opposed to SMOTE based strategies, that consider combination of only two minority class data points. Adding the shadowsamples allows LoRAS to produce a better estimate for local mean of the latent minority class data distribution.
We also provide a mathematical framework to show that convex combinations of multiple shadowsamples can provide a proper estimate for the local mean of a neighbourhood in the minority class data space. To be specific, an LoRAS oversample is an unbiased estimator of the mean of the underlying local probability distribution, followed by a minority class sample (assuming that it is some random variable) such that the variance of this estimator is significantly less than that of a SMOTE generated oversample, which is also an unbiased estimator of the mean of the underlying local probability distribution, followed by a minority class sample. In addition to this, LoRAS provides an option of choosing the neighbourhood of a minority class data point by performing prior manifold learning over the minority class using tStochastic Neighbourhood Embedding (tSNE) (tsne). tSNE is a stateof the art algorithm used for dimension reduction maintaining the underlying manifold structure in a sense that, in a lower dimension tSNE can cluster points, that are close enough in the latent high dimensional manifold. It uses a symmetric version of the cost function used for it’s predecessor technique Stochastic Neighbourhood Embedding (SNE) and uses a Studentt distribution rather than a Gaussian to compute the similarity between two points in the lowdimensional space. tSNE employs a heavytailed distribution in the lowdimensional space to alleviate both the crowding problem and the optimization problems of SNE (tsne, sne).
Till date there are at least eighty five extension models built on SMOTE (SMOTEVAR). Considering a large number of benchmark datasets explored in our study, it was necessary to shortlist certain oversampling algorithms for a comparative study. We found quite a few studies that have applied or explored SMOTE and extension of SMOTE such as Borderline1/2 SMOTE models, ADASYN, and SVMSMOTE (Suh, AhPine2016, Adisania, Chiama, Wang, Le). Moreover all these oversampling strategies are focused on oversampling from the convex hull of small neighbourhoods in the minority class data space, a similarity that they share with our proposed approach. Considering these factors, we choose to focus on these five oversampling strategies for a comparative study with our oversampling technique LoRAS.
Extension  Description 
Borderline1/2 SMOTE (Han2)  Identifies borderline samples and applies SMOTE on them 
ADASYN (He)  Adaptively changes the weights of different minority samples 
SVMSMOTE (Suh)  Generates new minority samples near borderlines with SVM 
SafeLevelSMOTE (Bunkhumpornpat2009)  Generates data in areas that are completely safe 
MWMOTE (Barua2014)  Identifies and weighs ambiguous minority class samples 
2 LoRAS: Localized Randomized Affine Shadowsampling
In this section we discuss our strategy to approximate the data manifold, given a dataset. A typical dataset for a supervised ML problem consists of a set of features , that are used to characterize patterns in the data and a set of labels or ground truth. Ideally, the number of instances or samples should be significantly greater than the number of features. In order to maintain the mathematical rigor of our strategy we propose the following definition for a small dataset.
Definition 1.
Consider a class or the whole dataset with samples and features. If , then we call the dataset, a small dataset.
The LoRAS algorithm is designed to learn from a dataset by approximating the underlying data manifold. Assuming that is the best possible set of features to represent the data and all features are equally important, we can think of a data oversampling model to be a function , that is, uses parent data points (each with features) to produce an oversampled data point in .
Definition 2.
We define a random affine combination of some arbitrary vectors as the affine linear combination of those vectors, such that the coefficients of the linear combination are chosen randomly. Formally, a vector , , is a random affine combination of vectors , () if , and are the coefficients of the affine combination chosen randomly from a Dirichlet distribution.
The simplest way of augmenting a data point would be to take the average (or random affine combination with positive coefficients as defined in Definition 2) of two data points as an augmented data point. But, when we have features, we can assume that the hypothetical manifold on which our data lies is dimensional. An dimensional manifold can be locally approximated by a collection of dimensional planes.
Given sample points we could exactly derive the equation of an unique dimensional plane containing these sample points. Note that, a small neighbourhood of a dataset can itself be considered as a small dataset. A small neighbourhood of points arount a data point in a dataset, given sufficiently small , satisfies Definition 1, that is and satisfies, . Thus, considering to be sufficiently small we can assume that this small neighbourhood is a small dataset. To enrich this small dataset, we create shadow data points or shadowsamples from our parent data points in the minority class data point neighbourhood. Each shadow data point is generated by adding noise from a normal distribution, for all features , where is some function of the sample variance for the feature . For each of the data points we can generate shadow data points such that, . Now it is possible for us to choose shadow data points from the shadow data points even if . We choose shadow data points as follows: we first choose a random parent data point and then restrict the domain of choice to the shadowsamples generated by the parent data points in .
For high dimensional datasets, choosing knearest neighbours of data point using simple Euclidean, Manhattan or general Minkowski distance measures can be misleading in terms of approximating the latent data manifold. To avoid this, we propose to adopt a manifold learning based strategy. Before choosing the knearest neighbours of a data point, we perform a dimension reduction on the data points of the minority class using the wellknown dimension reduction and manifold learning technique tSNE (tsne). Once we have a two dimensional tembedding of the minority class data, we choose the knearest neighbours of a particular data point consistent to its knearest neighbours (measured as per usual distance metrics) in the 2dimensional tSNE embedding of the minority class.
Once we choose our neighbourhood and generate the shadowsamples, we take a random affine combination with positive coefficients (Convex combination) of the chosen shadowsamples to create one augmented Localized Random Affine Shadowsample or a LoRAS sample as defined in Definition 2. Considering the arbitrary low variance that we can choose for the Normal distribution from which we draw our shadowsamples, we assume that our shadowsamples lie in the latent data manifold itself. It is a practical assumption, considering the stochastic factors leading to small measurement errors. Now, there exists an unique dimensional plane, that contains the shadowsamples, which we assume to be an approximation of the latent data manifold in that small neighbourhood. Thus, a LoRAS sample is an artificially generated sample drawn from an dimensional plane, which locally approximates the underlying hypothetical dimensional data manifold. It is worth mentioning here, that the effective number of features in a dataset is often less than . In high dimensional data there are often correlated features or features with low variance. Thus, for practical use of LoRAS one might consider generating convex combinations of effective number of features which might be less than .
C_maj:  Majority class parent data points 
C_min:  Minority class parent data points 
k:  Number of nearest neighbors to be considered per parent data point (default value : if , otherwise) 
S\textsubscriptp:  Number of generated shadowsamples per parent data point (default value : ) 
L\textsubscript\textsigma:  List of standard deviations for normal distributions for adding noise to each feature (default value : ) 
N\textsubscriptaff:  Number of shadow points to be chosen for a random affine combination (default value : ) 
N\textsubscriptgen:  Number of generated LoRAS points for each nearest neighbors group (default value : ) 
embedding:  Type of Embedding used to choose minority class neighbourhood (regular or tembedding) (default value : ‘regular’ ) 
perplexity:  Perplexity of tembedding (applicable only if embedding=‘tembedding’) (default value : 30) 
In this article, all imbalanced classification problems that we deal with are binary classification problems. For such a problem, there is a minority class containing a relatively less number of samples compared to a majority class . We can thus consider the minority class as a small dataset and use the LoRAS algorithm to oversample. For every data point we can denote a set of shadowsamples generated from as . In practice, one can also choose shadowsamples for an affine combination and choose a desired number of oversampled points to be generated using the algorithm. We can look at LoRAS as an oversampling algorithm as described in Algorithm 1.
The LoRAS algorithm thus described, can be used for oversampling of minority classes in case of highly imbalanced datasets. Note that the input variables for our algorithm are: number of nearest neighbors per sample k, number of generated shadow points per parent data point S\textsubscriptp, list of standard deviations for normal distributions for adding noise to every feature and thus generating the shadowsamples L\textsubscript\textsigma, number of shadowsamples to be chosen for affine combinations N\textsubscriptaff, number of generated points for each nearest neighbors group N\textsubscriptgen and embedding stategy embedding. There is a conditional input variable perplexity which takes a positive numerical value if one chooses a tembedding. The perplexity parameter of the tSNE algorithm is quite crucial. The perplexity parameter can influence the tEmbedding calculated by the tSNE algorithm. There have been several studies that address the issue on finding a right perplexity parameter for a given problem (perp). That is why, we recommend careful choice of this parameter in order to leverage more from our algorithm. Another important parameter of our algorithm is the N\textsubscriptaff. For this parameters an ideal choice would be the number of effective features in a dataset since this number would be a reasonable approximation to the dimension of the underlying data manifold. One could employ a feature selection technique to find out a good estimate for this. A simple random grid search is also very helpful to get reasonably good estimates of these parameters. We have mentioned all the default values of the LoRAS parameters in Algorithm 1, showing the pseudocode for the LoRAS algorithm. As an output, our algorithm generates a LoRAS dataset for the oversampled minority class, which can be subsequently used to train a ML model.
3 Case studies
For testing the potential of LoRAS as an oversampling approach, we designed benchmarking experiments with a total of 12 datasets which are either highly imbalanced or high dimensional. With this number of diverse case studies we should have a comprehensive idea of the advantages of LoRAS over the other oversampling algorithms of our interest.
3.1 Datasets used for validation
Here we provide a brief description of the datasets and the sources that we have used for our studies.
Scikitlearn imbalanced benchmark datasets: The imblearn.datasets package is complementing the sklearn.datasets package. It provides 27 preprocessed datasets, which are imbalanced. The datasets span a large range of realworld problems from several fields such as business, computer science, biology, medicine, and technology. This collection of datasets was proposed in the imblearn.datasets python library by Lema and benchmarked by Ding. Many of these datasets have been used in various research articles on oversampling approaches (Ding, saez). A statistically reliable benchmarking analysis of all 27 datasets in a stratified cross validation framework involves a lot of computational effort. We thus choose 11 datasets out of these two depending on two criteria:

Highly imbalanced: We choose datasets with imbalance ratio more than 25:1. This catrgory includes abalone_19, letter_image, mammography, ozone_level, webpage, wine_quality, yeast_me2 datasets.

High dimensional: We choose the datasets with more than 100 features. This category includes arrhythmia, isolet, scene, webpage and yeast_ml8.
Note that the webpage dataset is common in both the criteria, giving us a total of 11 datasets. We choose these two categories because they are of special interest in research related to imbalanced datasets and have received extensive attention in this research area (Anand, Hooda, Jing, Blagus2013).
Credit card fraud detection dataset: We obtain the description of this dataset from the website. https://www.kaggle.com/mlgulb/creditcardfraud. \sayThe dataset contains transactions made by credit cards in September 2013 by European cardholders. This dataset presents transactions that occurred in two days, where there are 492 frauds out of 284,807 transactions. The dataset is highly unbalanced, the positive class (frauds) account for 0.00172 percent of all transactions. The dataset contains only numerical input variables, which are the result of a PCA transformation. Feature variables are the principal components obtained with PCA, the only features that have not been transformed with PCA are the ‘Time’ and ‘Amount’. The feature ‘Time’ contains the seconds elapsed between each transaction and the first transaction in the dataset. The feature ‘Amount’ consists of the transaction amount. The labels are encoded in the ‘Class’ variable, which is the response variable and takes value 1 in case of fraud and 0 otherwise (cfraud).
Thus, in total we benchmark our oversampling algorithms against the existing algorithms on a total of 12 datasets. We provide relevant statistics on these datasets in Table 2
Dataset  Imbalance ratio  Number of samples  Number of features 

abalone_19  130:1  4177  10 
arrythmia  17:1  452  278 
isolet  12:1  7797  617 
letterimg  26:1  20000  16 
mammography  42:1  11183  6 
scene  13:1  2407  294 
ozone_level  34:1  2536  72 
webpage  33:1  34780  300 
winequality  26:1  4898  11 
yeastme2  28:1  1484  8 
yeastml8  13:1  2417  103 
credit fraud  577:1  284807  28 
3.2 Methodology
For every dataset we have analyzed, we used a consistent workflow. Given a dataset, for every machine learning model, we judge the model performances based on a 510fold stratified cross validation framework. First we randomly schuffle the dataset. For a given dataset, we first split the dataset into 10 folds, each one distinct from the other maintaining the imbalance ratio for every fold. We then train the machine learning models on the dataset without any oversampling with 10fold cross validation. This means that we train and test the model 10 times, each time considering a fold as a test fold and rest 9 folds as training folds. However, while training the ML models with oversampled data, we oversample only on the training folds and leave the test fold as they are for each training session. For each dataset we repeat the whole process five times to avoid the stochastic effects as much as possible.
For the oversampling algorithms, for a given dataset, we chose the same neighbourhood size for every oversampling model. If there were less than 100 data points in the minortiy class the neighbourhood size was chosen to be 5. Otherwise we chose a neighbourhood size of 30. Given a large number of datasets we are analyzing, we did not customize this for every dataset and rather chose to stick to the above mentioned general rule. For LoRAS oversampling however, we performed a preliminary study to find out customized parameter values for every dataset, since the LoRAS algorithm is highly parametrized in nature. We tried several combinations of parameters N\textsubscriptaff, embedding and perplexity employing random grid search. For LoRAS oversampling every dataset we use an unique value for N\textsubscriptaff as presented in Table 3. For indivial ML models we use different settings for the LoRAS parameters embedding and perplexity which we mention explicitly in our supplementary materials while presenting the results for each ML model for each dataset. To ensure fairness of comparison, we oversampled such that the total number of augmented samples generated from the minority class was as close as possible to the number of samples in the majority class as allowed by each oversampling algorithm. Speaking of other parameters of the LoRAS algorithm, for L\textsubscript\textsigma, we chose a list consisting of a constant value of for each dataset and for the parameter N\textsubscriptgen we chose the value as: . We provide a detailed list of parameter settings used by us for the oversampling algorithms in Table 3
Dataset  Minority samples  Oversampling nbd  LoRAS N\textsubscriptaff 
abalone19  32  5  10 
arrythmia  25  5  100 
isolet  600  30  179 
letterimg  734  30  16 
mammography  260  30  6 
scene  177  30  2 
ozone_level  73  5  10 
webpage  981  30  94 
winequality  183  30  2 
yeastme2  51  5  2 
yeastml8  178  30  3 
credit fraud  492  30  30 
To choose ML models for our study we first did a pilot study with ML classifiers such as knearest neighbors (knn), Support Vector Machine (svm) (linear kernel), Logistic regression (lr), Random forest (rf), and Adaboost (ab). As inferred in (Blagus2013) we found that knn was quite effective for the datasets we used. We also noticed that lr and svm performed better compared to rf and ab in most cases. We thus chose knn, svm and lr for our final studies. We used lbfgs solver for our logistic regression model and a linear kernel for our svm models. For our knn models, we choose 10 nearest neighbours for our prediction if there are less than 100 samples in the minority class and 30 nearest neighbours otherwise. For ‘arrhythmia’ and ‘abalone19’, however we use only 5 nearest neighbours for the knn model since it has only 25 and 32 minority class samples respectively. We choose this parameter to be consistent to the neighbourhood size of the oversampling models, since the neighbourhood size directly influences the distribution of the training data and hence the model performance.
In our analysis we take special notice of the credit card fraud detection dataset. This dataset is not included in the imblearn.datasets Python library. However, the main reason why we want to pay a special attention to this dataset is that, it is by far the most imbalanced publicly available dataset that we have come across. The extreme imbalance ratio of 577:1 is uncomparable to any of the datasets in imblearn.datasets. Also, this dataset has received special attention of researchers attempting to use ML in Credit fraud detection (credit). In this article we see that lr and rf have good prediction accuracies on the dataset. Thus we chose these two ML models for the credit fraud dataset. credit has also not provided cross validated analysis of their models, while our models have been trained and tested with the usual 10fold cross validation framework as discussed before.
For computational coding, we used the scikitlearn (V 0.21.2), numpy (V 1.16.4), pandas (V 0.24.2), and matplotlib (V 3.1.0) libraries in Python (V 3.7.4).
4 Results
For imbalanced datasets there are more meaningful performance measures than Accuracy, including Sensitivity or Recall, Precision, and F1Score (FMeasure), and Balanced accuracy that can all be derived from the Confusion Matrix, generated while testing the model. For a given class, the different combinations of recall and precision have the following meanings :

High Precision & High Recall: The model handled the classification task properly

High Precision & Low Recall: The model cannot classify the data points of the particular class properly, but is highly reliable when it does so

Low Precision & High Recall: The model classifies the data points of the particular class well, but misclassifies high number of data points from other classes as the class in consideration

Low Precision & Low Recall: The model handled the classification task poorly
F1Score, calculated as the harmonic mean of precision and recall and, therefore, balances a model in terms of precision and recall. These measures have been defined and discussed thoroughly by AbdElrahman2013. Balanced accuracy is the mean of the individual class accuracies and in this context, it is more informative than the usual accuracy score. High Balanced accuracy ensures that the ML algorithm learns adequately for each individual class.
In our experiments we have noticed an interesting behaviour of oversampling models in terms of their average F1Score and Balanced accuracy. Once we present our experiment results, we will discuss why considering F1Score and Balanced accuracy can give us a clearer idea about model performances. We will use the above mentioned performance measures wherever applicable in this article.
Selected model performances for all datasets: We provide the detailed results of our experiments for all machine learning models as supplementary material. To be precise, for every combination of datasets, ML models and oversampling strategies we provide the mean and variance of the 10fold cross validation process over 5 repetitions. For judging the performance of the oversampling models we follow the following scheme:

First, for a given dataset, we choose the ML model trained on that dataset that provides the highest average F1Score over all the oversampling models and training without oversampling. The F1Score reflects the balance between precision and recall and considered as a reliable metric for imbalanced classification task.

We then consider the Balanced accuracy and F1 score of the chosen model as an evaluation of how well the oversampling model performs on the considered dataset. Following this evaluation scheme we present our results in Table 4.
Dataset  ML  Baseline  SMOTE  Bl1  Bl2  SVM  ADASYN  LoRAS 
abalone19  knn  .534/.000  .644/.054  .552/.044  .552/.044  .556/.045  .571/.055  .675/.059 
arrythmia  lr  .679/.37  .666/.345  .672/.352  .709/.307  .679/.350  .667/.362  .694/.380 
isolet  lr  .900/.826  .898/.806  .899/.802  .906/.693  .911/.799  .898/.806  .904/.809 
letterimg  knn  .927/.915  .988/.781  .984/.768  .977/.687  .986/.724  .985/.732  .989/.833 
mammography  knn  .703/.549  .911/.413  .909/.414  .899/.326  .909/.467  .905/.353  .896/.511 
scene  lr  .551/.168  .616/.222  .619/.230  .620/.223  .616/.235  .620/.224  .616/.226 
ozone_level  lr  .517/.062  .800/.190  .777/.212  .781/.183  .738/.215  .803/.192  .809/.207 
webpage  knn  .805/.711  .906/.267  .901/.274  .903/.287  .904/.267  .903/.264  .923/.613 
winequality  lr  .517/.067  .718/.179  .715/.182  .711/.171  .712/.216  .721/.180  .734/.197 
yeastml8  knn  .500/.000  .558/.152  .561/.153  .563/.153  .572/.158  .558/.151  .559/.152 
yeastme2  knn  .523/.074  .834/.331  .797/.373  .79/.304  .785/.388  .825/.315  .842/.354 
credit fraud  rf  .669/.775  .922/.359  .919/.645  .919/.556  .913/.741  .923/.350  .904/.820 
Average    .672/.401  .801/.367  .795/.400  .798/.353  .793/.414  .800/.357  .806/.463 
Calculating average performances over all datasets, LoRAS has the best Balanced accuracy and F1Score. As expected, SMOTE improved Balanced accuracy compared to model training without any oversampling. Surprisingly, it lags behind in F1Score, for quite a few datasets with high baseline F1Score such as letter_image, isolet, mammography, webpage and credit fraud. Interestingly, the oversampling approaches SVMSMOTE and Borderline1 SMOTE also improved the average F1Score compared to SMOTE, but compromised for a lower Balanced accuracy. On the other hand, applying ADASYN increased the Balanced accuracy compared to SMOTE, but again compromises on the F1Score. In contrast, our LoRAS approach produces the best Balanced accuracy on average by maintaining the highest average F1Score among all oversampling techniques. We want to emphasize that, even considering stochastic factors, LoRAS can improve both the Balanced accuracy and F1Score of ML models significantly compared to SMOTE, which makes it unique.
Datasets with high imbalance ratio: To verify the performance of LoRAS on highly imbalanced datasets we present the selected model performances for the datasets with highest imbalance ratios (among the ones we have tested) in Table 5
Dataset  Imbalance Ratio  Baseline  SMOTE  Bl1  Bl2  SVM  ADASYN  LoRAS 
abalone19  130:1  .534/.000  .644/.054  .552/.044  .552/.044  .556/.045  .571/.055  .675/.059 
letterimg  26:1  .927/.915  .988/.781  .984/.768  .977/.687  .986/.724  .985/.732  .989/.833 
mammography  42:1  .703/.549  .911/.413  .909/.414  .899/.326  .909/.467  .905/.353  .896/.511 
ozone_level  34:1  .517/.062  .800/.190  .777/.212  .781/.183  .738/.215  .803/.192  .809/.207 
webpage  33:1  .805/.711  .906/.267  .901/.274  .903/.287  .904/.267  .903/.264  .923/.613 
winequality  26:1  .517/.067  .718/.179  .715/.182  .711/.171  .712/.216  .721/.180  .734/.197 
yeastme2  28:1  .523/.074  .834/.331  .797/.373  .79/.304  .785/.388  .825/.315  .842/.354 
credit fraud  577:1  .669/.775  .922/.359  .919/.645  .919/.556  .913/.741  .923/.350  .904/.820 
Average    .662/.381  .840/.321  .819/.364  .817/.319  .814/.382  .841/.305  .846/.449 
From our results we observe that LoRAS oversampling can significantly improve model performances for highly imbalanced datasets. LoRAS provides the highest F1Score and Balanced accuracy among all the oversampling models. The results here show similar properties for SMOTE, Borderline1 SMOTE, SVM SMOTE, ADASYN and LoRAS as discussed before. Note that, for the credit fraud dataset, which is the most imbalanced among all, LoRAS has significant success over the other oversampling models in terms of Balanced accuracy. For the webpage dataset as well it improves the Balanced accuracy significantly, compromising minimally on the baseline F1Score. The same trend follows for the letter_image dataset. Notably, these three datasets have the highest number of overall samples as well, implying that with more data LoRAS can significantly outperform compared convex combination based oversampling models.
High dimensional datasets: It is also of interest to us to check how LoRAS performs on high dimensional datasets. We therefore select five datasets with highest number of features among our tested datasets and present the performances of the selected ML methods in Table 6
Dataset  Features num.  Baseline  SMOTE  Bl1  Bl2  SVM  ADASYN  LoRAS 
arrythmia  278  .679/.37  .666/.345  .672/.352  .709/.307  .679/.350  .667/.362  .694/.380 
isolet  617  .900/.826  .898/.806  .899/.802  .906/.693  .911/.799  .898/.806  .904/.809 
scene  294  .551/.168  .616/.222  .619/.230  .620/.223  .616/.235  .620/.224  .616/.226 
webpage  300  .805/.711  .906/.267  .901/.274  .903/.287  .904/.267  .903/.264  .923/.613 
yeastml8  103  .500/.000  .558/.152  .561/.153  .563/.153  .572/.158  .558/.151  .559/.152 
Average    .687/.415  .728/.358  .730/.362  .740/.332  .736/.361  .729/.361  .739/.436 
From our results for high dimensional datasets, we observe that LoRAS produces the best F1Score and second best Balanced accuracy on average among all oversampling models as Borderline2 SMOTE beats LoRAS marginally. SMOTE improves both F1Score and Balanced accuracy with respect to the baseline score here. Borderline1 SMOTE and SVM SMOTE further increases SMOTE’s performance both in terms of F1Score and Balanced accuracy. Borderline2 SMOTE, although improves the Balanced accuracy of SMOTE compromises on the F1Score. Note that, even excluding the webpage dataset, where LoRAS has an overwhelming success, LoRAS still has the best average F1Score and third highest Balanced accuracy marginally behind SVMSMOTE and Borederline2 SMOTE. We thus conclude, that for high dimensional datasets LoRAS can outperform the compared oversampling models in terms of F1Score, while compromising marginally for Balanced accuracy.
5 Discussion
We have constructed a mathematical framework to prove that LoRAS is a more effective oversampling technique since it provides a better estimate for the mean of the underlying local data distribution of the minority class data space. Let be an arbitrary minority class sample. Let be the set of the knearest neighbors of , which will consider the neighborhood of . Both SMOTE and LoRAS focus on generating augmented samples within the neighborhood at a time. We assume that a random variable follows a shifted tdistribution with degrees of freedom, location parameter , and scaling parameter . Note that here is not referring to the standard deviation but sets the overall scaling of the distribution (Jackman), which we choose to be the sample variance in the neighborhood of . A shifted tdistribution is used to estimate population parameters, if there are less number of samples (usually, 30) and/or the population variance is unknown. Since in SMOTE or LoRAS we generate samples from a small neighborhood, we can argue in favour of our assumption that locally, a minority class sample as a random variable, follows a tdistribution. Following Blagus2013, we assume that if then and are independent. For , we also assume:
(1) 
where, and denote the expectation and variance of the random variable respectively. However, the mean has to be estimated by an estimator statistic (i.e. a function of the samples). Both SMOTE and LoRAS can be considered as an estimator statistic for the mean of the tdistribution that follows locally.
Theorem 1.
Both SMOTE and LoRAS are unbiased estimators of the mean of the tdistribution that follows locally. However, the variance of the LoRAS estimator is less than the variance of SMOTE given that .
Proof.
A shadowsample is a random variable where , the neighborhood of some arbitrary and follows .
(2) 
assuming and are independent. Now, a LoRAS sample , where are shadowsamples generated from the elements of the neighborhood of , , such that . The affine combination coefficients follow a Dirichlet distribution with all concentration parameters assuming equal values of 1 (assuming all features to be equally important). For arbitrary ,
where denotes the covariance of two random variables and . Assuming and to be independent,
(3) 
Thus is an unbiased estimator of . For ,
(4) 
since is independent of . For an arbitrary , th component of a LoRAS sample
(5) 
For LoRAS, we take an affine combination of shadowsamples and SMOTE considers an affine combination of two minority class samples. Note, that since a SMOTE generated oversample can be interpreted as a random affine combination of two minority class samples, we can consider, for SMOTE, independent of the number of features. Also, from Equation 3, this implies that SMOTE is an unbiased estimator of the mean of the local data distribution. Thus, the variance of a SMOTE generated sample as an estimator of would be (since for SMOTE). But for LoRAS as an estimator of , when , the variance would be less than that of SMOTE. ∎
This implies that, locally, LoRAS can estimate the mean of the underlying tdistribution better than SMOTE. To visualize the key aspects of LoRAS oversampling, we provide the PCA plots for oversampled data from the ozone_level dataset several oversampling methods we have studied in Figure 2.
From Figure 2 we can observe that SMOTE and ADASYN oversamples highly on the neighbourhood of the outliers, depicted by a blue box in each subplot. While this is somewhat controlled in Borderline1SMOTE and SVM SMOTE, they still generate some synthetic samples in this neighbourhood. LoRAS on the other hand refrains, leveraging on its strategy to produce a better estimate for local mean of the underlying local data distribution. This enables LoRAS to ignore the outliers and to oversample more uniformly resulting in a better approximation of the data manifold. Note that, the average F1Scores of the oversampling models as presented in Table 4 has a direct correlation to how the oversampling strategy oversamples in this neighbourhood. SMOTE and ADASYN generates the lowest F1Scores and show a tendency of oversampling excessively from this neighbourhood. BorderlineSMOTE and SVM improves the F1Score compared to SMOTE and ADASYN, again, consistent to their behaviour of oversampling lesser in this neighbourhood. LoRAS, has the highest average F1Score and oversampling very sparsely from this neighbourhood.
6 Conclusions
Oversampling with LoRAS produces comparatively balanced ML model performances on average, in terms of Balanced Accuracy and F1Score among the compared convexcombination strategy based oversampling techniques. This is due to the fact that, in most cases LoRAS produces lesser misclassifications on the majority class with a reasonably small compromise for misclassifications on the minority class. From our study we infer that for tabular high dimensional and highly imbalanced datasets our LoRAS oversampling approach can better estimate the mean of the underlying local distribution for a minority class sample (considering it a random variable) and can improve Balanced accuracy and F1Score of ML classification models. However, the scope of such convex combination based strategies including LoRAS, might be limited for heterogeneous image based imbalanced datasets.
The distribution of both the minority and majority class data points is considered in the oversampling techniques such as Borderline1 SMOTE, Borderline2 SMOTE, SVMSMOTE, and ADASYN (Gosain2017). SMOTE and LoRAS are the only two techniques, among the oversampling techniques we explored, that deal with the problem of imbalance just by generating new data points, independent of the distribution of the majority class data points. Thus, comparing LoRAS and SMOTE gives a fair impression about the performance of our novel LoRAS algorithm as an oversampling technique, without considering any aspect of the distributions of the minority and majority class data points and relying just on resampling. Other extensions of SMOTE such as Borderline1 SMOTE, Borderline2 SMOTE, SVMSMOTE, and ADASYN can also be built on the principle of LoRAS oversampling strategy. According to our analyses LoRAS already reveals great potential on a broad variety of applications and evolves as a true alternative to SMOTE, while processing highly unbalanced datasets.
Availability of code: A preliminary implementation of the algorithm in Python (V 3.7.4) and an example Jupyter Notebook for the credit card fraud detection dataset is provided on the GitHub repository https://github.com/sbirostock/LoRAS. This version does not yet include the tembedding parameter. In our computational code, S\textsubscriptp corresponds to num_shadow_points, L\textsubscript\textsigma corresponds to list_sigma_f, N\textsubscriptaff corresponds to num_aff_comb, N\textsubscriptgen corresponds to num_generated_points.
Acknowledgements: We thank Prof. Ria Baumgrass from Deutsches RheumaForschungszentrum (DRFZ), Berlin for enlightening discussions on small datasets occuring in her research related to cancer therapy, that led us to the current work. We thank the German Network for Bioinformatics Infrastructure (de.NBI) and Establishment of Systems Medicine Consortium in Germany e:Med for their support, as well as the German Federal Ministry for Education and Research (BMBF) programs (FKZ 01ZX1709C) for funding us.
References
Supplementary data
We provide the detailed individual results for each dataset and each ML model for our analysis as supplementary data. Here, we use the acronyms bl1, bl2, SVM and ADA for the oversampling models Borderline1 SMOTE, Borderline2 SMOTE, SVMSMOTE and ADASYN respectively. We mark in bold for each dataset, the ML model with the highest average F1Score, a criteria by which we select the model results to include in our further analysis.
Dataset: abalone_19
Oversampling models  lr1  lr2  lr3  lr4  lr5  mean  sd 

Baseline  0  0  0  0  0  0  0 
SMOTE  0.045  0.046  0.048  0.049  0.049  0.0474  0.00181659 
bl1  0.048  0.05  0.04  0.051  0.043  0.0464  0.00472229 
bl2  0.044  0.047  0.04  0.056  0.051  0.0476  0.0061887 
SVM  0.057  0.062  0.049  0.057  0.051  0.0552  0.00521536 
ADA  0.045  0.046  0.049  0.05  0.048  0.0476  0.00207364 
LoRAS (Em=t,p=10)  0.055  0.055  0.056  0.057  0.057  0.056  0.001 
F1Score average  0.04288571 
Oversampling model  lr1  lr2  lr3  lr4  lr5  mean  sd 

Baseline  0.5  0.5  0.499  0.499  0.499  0.4994  0.00054772 
SMOTE  0.709  0.73  0.725  0.734  0.729  0.7254  0.00971082 
bl1  0.677  0.686  0.623  0.696  0.659  0.6682  0.02870017 
bl2  0.657  0.687  0.64  0.733  0.701  0.6836  0.03661694 
SVM  0.662  0.689  0.631  0.667  0.659  0.6616  0.02075572 
ADA  0.71  0.729  0.727  0.735  0.726  0.7254  0.00928978 
LoRAS (Em=t,p=10)  0.728  0.747  0.733  0.735  0.743  0.7372  0.00769415 
Oversampling models  svm1  svm2  svm3  svm4  svm5  mean  sd 

Baseline  0  0  0  0  0  0  0 
SMOTE  0.032  0.03  0.028  0.028  0.031  0.0298  0.00178885 
bl1  0.023  0.025  0.024  0.027  0.027  0.0252  0.00178885 
bl2  0.023  0.025  0.024  0.027  0.027  0.0252  0.00178885 
SVM  0.035  0.039  0.035  0.039  0.025  0.0346  0.00572713 
ADA  0.031  0.03  0.028  0.028  0.031  0.0296  0.00151658 
LoRAS (Em=t,p=10))  0.032  0.033  0.029  0.033  0.032  0.0318  0.00164317 
F1Score average  0.02517143 
Oversampling models  svm1  svm2  svm3  svm4  svm5  mean  sd 

Baseline  0.5  0.5  0.5  0.5  0.5  0.5  0 
SMOTE  0.766  0.733  0.701  0.727  0.74  0.7334  0.02343715 
bl1  0.615  0.662  0.635  0.703  0.684  0.6598  0.03563285 
bl2  0.615  0.662  0.635  0.703  0.684  0.6598  0.03563285 
SVM  0.686  0.728  0.684  0.737  0.702  0.7074  0.02416195 
ADA  0.757  0.733  0.699  0.726  0.753  0.7336  0.02334095 
LoRAS (Em=t,p=10)  0.745  0.758  0.703  0.768  0.752  0.7452  0.02505394 
Oversampling models  knn1  knn2  knn3  knn4  knn5  mean  sd 

Baseline  0  0  0  0  0  0  0 
SMOTE  0.056  0.05  0.053  0.047  0.066  0.0544  0.00730068 
bl1  0.044  0.059  0.035  0.052  0.034  0.0448  0.01080278 
bl2  0.044  0.062  0.042  0.039  0.031  0.0436  0.0114149 
SVM  0.046  0.062  0.037  0.045  0.035  0.045  0.01065364 
ADA  0.057  0.049  0.053  0.054  0.063  0.0552  0.00521536 
LoRAS (Em=r,p=NA)  0.058  0.062  0.055  0.061  0.063  0.0598  0.00327109 
F1Score average  0.04325714 
Oversampling models  knn1  knn2  knn3  knn4  knn5  mean  sd 

Baseline  0.677  0.5  0.5  0.5  0.5  0.5354  0.07915681 
SMOTE  0.555  0.652  0.661  0.635  0.72  0.6446  0.05943316 
bl1  0.572  0.569  0.522  0.564  0.537  0.5528  0.02210656 
bl2  0.555  0.594  0.541  0.543  0.545  0.5556  0.02213143 
SVM  0.678  0.57  0.523  0.548  0.538  0.5714  0.06199032 
ADA  0.656  0.651  0.668  0.669  0.71  0.6708  0.02323144 
LoRAS (Em=r,p=NA)  0.666  0.683  0.664  0.667  0.698  0.6756  0.01463899 
Dataset: arrhythmia
Oversampling models  lr1  lr2  lr3  lr4  lr5  mean  sd 

Baseline  0.36  0.39  0.354  0.35  0.396  0.37  0.02140093 
SMOTE  0.346  0.34  0.249  0.35  0.44  0.345  0.06762396 
bl1  0.356  0.347  0.38  0.309  0.372  0.3528  0.0277074 
bl2  0.334  0.362  0.317  0.27  0.252  0.307  0.04540925 
SVM  0.403  0.426  0.244  0.326  0.351  0.35  0.07141078 
ADA  0.398  0.306  0.336  0.336  0.435  0.3622  0.05270863 
LoRAS (Em=t,p=1)  0.429  0.43  0.288  0.353  0.403  0.3806  0.06045908 
F1Score average  0.35251429 
Oversampling models  lr1  lr2  lr3  lr4  lr5  mean  sd 

Baseline  0.688  0.701  0.646  0.673  0.687  0.679  0.02094039 
SMOTE  0.674  0.673  0.624  0.673  0.686  0.666  0.02411431 
bl1  0.686  0.676  0.676  0.654  0.672  0.6728  0.01171324 
bl2  0.729  0.766  0.697  0.68  0.673  0.709  0.03850325 
SVM  0.713  0.723  0.621  0.671  0.668  0.6792  0.04074555 
ADA  0.696  0.642  0.649  0.656  0.695  0.6676  0.02594802 
LoRAS (Em=t,p=1)  0.718  0.725  0.639  0.695  0.694  0.6942  0.03377425 
Oversampling models  svm1  svm2  svm3  svm4  svm5  mean  sd 

Baseline  0.259  0.381  0.387  0.376  0.321  0.3448  0.05475582 
SMOTE  0.259  0.381  0.387  0.376  0.321  0.3448  0.05475582 
bl1  0.259  0.381  0.387  0.376  0.321  0.3448  0.05475582 
bl2  0.259  0.381  0.387  0.376  0.321  0.3448  0.05475582 
SVM  0.259  0.381  0.387  0.376  0.321  0.3448  0.05475582 
ADA  0.259  0.381  0.387  0.376  0.321  0.3448  0.05475582 
LoRAS (Em=t,p=1)  0.259  0.381  0.387  0.376  0.321  0.3448  0.05475582 
F1Score average  0.3448 
Oversampling models  svm1  svm2  svm3  svm4  svm5  mean  sd 

Baseline  0.659  0.676  0.686  0.697  0.694  0.6824  0.01540454 
SMOTE  0.659  0.676  0.686  0.697  0.694  0.6824  0.01540454 
bl1  0.659  0.676  0.686  0.697  0.694  0.6824  0.01540454 
bl2  0.659  0.676  0.686  0.697  0.694  0.6824  0.01540454 
SVM  0.659  0.676  0.686  0.697  0.694  0.6824  0.01540454 
ADA  0.659  0.676  0.686  0.697  0.694  0.6824  0.01540454 
LoRAS (Em=t,p=1)  0.659  0.676  0.686  0.697  0.694  0.6824  0.01540454 
Oversampling models  knn1  knn2  knn3  knn4  knn5  mean  sd 

Baseline  0  0  0  0  0.5  0.1  0.2236068 
SMOTE  0.185  0.191  0.202  0.192  0.22  0.198  0.01372953 
bl1  0.187  0.195  0.182  0.194  0.158  0.1832  0.01505656 
bl2  0.176  0.194  0.163  0.169  0.154  0.1712  0.01508973 
SVM  0.213  0.224  0.178  0.206  0.191  0.2024  0.01814663 
ADA  0.196  0.196  0.202  0.169  0.206  0.1938  0.01449828 
LoRAS (Em=t,p=1)  0.179  0.179  0.168  0.192  0.207  0.185  0.01494992 
F1Score average  0.17622857 
Oversampling models  knn1  knn2  knn3  knn4  knn5  mean  sd 

Baseline  0.5  0.5  0.5  0.5  0.5  0.5  0 
SMOTE  0.674  0.688  0.72  0.728  0.755  0.713  0.03234192 
bl1  0.665  0.677  0.66  0.691  0.658  0.6702  0.01377316 
bl2  0.666  0.689  0.642  0.666  0.658  0.6642  0.01697645 
SVM  0.684  0.681  0.609  0.69  0.689  0.6706  0.03463091 
ADA  0.712  0.693  0.719  0.727  0.73  0.7162  0.01475466 
LoRAS (Em=t,p=1)  0.664  0.673  0.67  0.728  0.715  0.69  0.02930017 
Dataset: isolet
Oversampling models  lr1  lr2  lr3  lr4  lr5  mean  sd 

Base  0.901  0.792  0.819  0.811  0.807  0.826  0.0430581 
SMOTE  0.799  0.798  0.808  0.817  0.809  0.8062  0.00785493 
bl1  0.792  0.797  0.805  0.812  0.804  0.802  0.00771362 
bl2  0.695  0.698  0.694  0.7  0.681  0.6936  0.0074364 
SVM  0.796  0.795  0.813  0.795  0.8  0.7998  0.00766159 
ADA  0.806  0.793  0.815  0.813  0.803  0.806  0.00877496 
LoRAS (Em=t,p=30)  0.809  0.793  0.821  0.816  0.81  0.8098  0.01056882 
F1Score average  0.79191429 
Ovesampling models  lr1  lr2  lr3  lr4  lr5  mean  sd 

baseline  0.901  0.89  0.906  0.903  0.903  0.9006  0.0061887 
SMOTE  0.894  0.892  0.901  0.904  0.902  0.8986  0.00527257 
bl1  0.892  0.894  0.903  0.906  0.9  0.899  0.00591608 
bl2  0.906  0.907  0.905  0.912  0.902  0.9064  0.00364692 
SVM  0.907  0.908  0.92  0.909  0.911  0.911  0.00524404 
ADA  0.899  0.891  0.901  0.904  0.897  0.8984  0.00487852 
LoRAS (Em=t,p=30)  0.905  0.894  0.909  0.908  0.906  0.9044  0.00602495 
Oversampling models  svm1  svm2  svm3  svm4  svm5  mean  variance 

Base  0.791  0.788  0.797  0.798  0.793  0.7934  0.00415933 
SMOTE  0.457  0.459  0.458  0.463  0.46  0.4594  0.00230217 
bl1  0.49  0.489  0.486  0.494  0.498  0.4914  0.00466905 
bl2  0.49  0.489  0.486  0.494  0.498  0.4914  0.00466905 
SVM  0.476  0.479  0.473  0.474  0.475  0.4754  0.00230217 
ADA  0.483  0.477  0.48  0.483  0.486  0.4818  0.00342053 
LoRAS (Em=t,p=30)  0.505  0.501  0.501  0.486  0.507  0.5  0.00824621 
F1Score average  0.52754286 
Oversampling models  svm1  svm2  svm3  svm4  svm5  mean  sd 

Baseline  0.893  0.893  0.906  0.9  0.904  0.8992  0.00605805 
SMOTE  0.894  0.893  0.895  0.897  0.895  0.8948  0.00148324 
bl1  0.905  0.902  0.904  0.906  0.908  0.905  0.00223607 
bl2  0.905  0.902  0.904  0.906  0.908  0.905  0.00223607 
SVM  0.899  0.901  0.899  0.899  0.897  0.899  0.00141421 
ADA  0.903  0.898  0.902  0.903  0.904  0.902  0.00234521 
LoRAS (Em=t,p=30)  0.91  0.906  0.909  0.908  0.911  0.9088  0.00192354 
Oversampling models  knn1  knn2  knn3  knn4  knn5  mean  sd 

Base  0.874  0.881  0.87  0.873  0.871  0.8738  0.00432435 
SMOTE  0.496  0.496  0.498  0.492  0.495  0.4954  0.00219089 
bl1  0.506  0.507  0.508  0.508  0.508  0.5074  0.00089443 
bl2  0.46  0.464  0.463  0.462  0.468  0.4634  0.00296648 
SVM  0.525  0.526  0.527  0.528  0.53  0.5272  0.00192354 
ADA  0.486  0.487  0.488  0.485  0.487  0.4866  0.00114018 
LoRAS (Em=t,p=30)  0.421  0.424  0.42  0.422  0.422  0.4218  0.00148324 
F1Score average  0.53937143 
Oversampling models  knn1  knn2  knn3  knn4  knn5  mean  sd 

Baseline  0.914  0.92  0.91  0.915  0.913  0.9144  0.00364692 
SMOTE  0.915  0.915  0.915  0.913  0.913  0.9142  0.00109545 
bl1  0.917  0.918  0.918  0.918  0.918  0.9178  0.00044721 
bl2  0.905  0.903  0.902  0.902  0.904  0.9032  0.00130384 
SVM  0.923  0.923  0.924  0.924  0.925  0.9238  0.00083666 
ADA  0.911  0.912  0.912  0.911  0.911  0.9114  0.00054772 
LoRAS (Em=t,p=30)  0.883  0.884  0.882  0.884  0.883  0.8832  0.00083666 
Dataset: letter_image
Oversampling models  lr1  lr2  lr3  lr4  lr5  mean  sd 

Baseline  0.743  0.745  0.745  0.744  0.742  0.7438  0.00130384 
SMOTE  0.581  0.587  0.582  0.585  0.583  0.5836  0.00240832 
bl1  0.477  0.499  0.497  0.488  0.487  0.4896  0.00882043 
bl2  0.331  0.337  0.337  0.339  0.339  0.3366  0.00328634 
SVM  0.463  0.468  0.47  0.467  0.468  0.4672  0.00258844 
ADA  0.52  0.52  0.521  0.522  0.518  0.5202  0.00148324 
LoRAS(Em=r,p=NA)  0.666  0.643  0.636  0.638  0.644  0.6454  0.01199166 
F1Score average  0.54091429 
Oversampling models  lr1  lr2  lr3  lr4  lr5  mean  sd 

Baseline  0.831  0.833  0.833  0.831  0.83  0.8316  0.00134164 
SMOTE  0.953  0.956  0.954  0.955  0.953  0.9542  0.00130384 
bl1  0.928  0.921  0.926  0.926  0.92  0.9242  0.00349285 
bl2  0.912  0.907  0.905  0.91  0.906  0.908  0.00291548 
SVM  0.945  0.947  0.945  0.944  0.944  0.945  0.00122474 
ADA  0.948  0.949  0.948  0.949  0.948  0.9484  0.00054772 
LoRAS (Em=r,p=NA)  0.915  0.942  0.942  0.944  0.946  0.9378  0.01285302 
Oversampling models  svm1  svm2  svm3  svm4  svm5  mean  sd 

Baseline  0.753  0.757  0.757  0.762  0.761  0.758  0.00360555 
SMOTE  0.166  0.166  0.165  0.165  0.165  0.1654  0.00054772 
bl1  0.174  0.173  0.174  0.174  0.174  0.1738  0.00044721 
bl2  0.174  0.169  0.174  0.174  0.174  0.173  0.00223607 
SVM  0.168  0.177  0.169  0.169  0.17  0.1706  0.00364692 
ADA  0.177  0.186  0.177  0.177  0.177  0.1788  0.00402492 
LoRAS (Em=t,p=1)  0.204  0.206  0.204  0.204  0.203  0.2042  0.00109545 
F1Score average  0.26054286 
Oversampling models  svm1  svm2  svm3  svm4  svm5  mean  sd 
Base  0.829  0.831  0.831  0.833  0.832  0.8312  0.00148324 
SMOTE  0.809  0.809  0.808  0.807  0.807  0.808  0.001 
bl1  0.819  0.819  0.819  0.819  0.819  0.819  0 
bl2  0.819  0.819  0.819  0.819  0.819  0.819  0 
SVM  0.812  0.813  0.813  0.812  0.814  0.8128  0.00083666 
ADA  0.823  0.823  0.823  0.823  0.823  0.823  1.2413E16 
LoRAS (Em=t,p=1)  0.851  0.852  0.851  0.85  0.849  0.8506  0.00114018 
Oversampling models  knn1  knn2  knn3  knn4  knn5  mean  sd 

Base  0.913  0.914  0.914  0.916  0.919  0.9152  0.00238747 
SMOTE  0.78  0.782  0.782  0.782  0.782  0.7816  0.00089443 
bl1  0.757  0.769  0.773  0.769  0.775  0.7686  0.0069857 
bl2  0.757  0.675  0.659  0.674  0.672  0.6874  0.03943729 
SVM  0.662  0.741  0.738  0.741  0.738  0.724  0.0346915 
ADA  0.732  0.729  0.732  0.735  0.734  0.7324  0.00230217 
LoRAS (Em=r,p=NA)  0.839  0.831  0.832  0.83  0.833  0.833  0.00353553 
F1Score average  0.77745714 
Oversampling models  knn1  knn2  knn3  knn4  knn5  mean  sd 

Base  0.926  0.927  0.926  0.928  0.93  0.9274  0.00167332 
SMOTE  0.989  0.989  0.988  0.989  0.989  0.9888  0.00044721 
bl1  0.984  0.983  0.984  0.986  0.985  0.9844  0.00114018 
bl2  0.978  0.977  0.973  0.98  0.977  0.977  0.00254951 
SVM  0.986  0.986  0.986  0.986  0.986  0.986  0 
ADA  0.985  0.985  0.986  0.986  0.986  0.9856  0.00054772 
LoRAS (Em=r,p=NA)  0.99  0.989  0.989  0.99  0.989  0.9894  0.00054772 
Dataset: mammography
Oversampling models  lr1  lr2  lr3  lr4  lr5  mean  sd 

Base  0.532  0.535  0.524  0.533  0.533  0.5314  0.00427785 
SMOTE  0.283  0.286  0.282  0.282  0.288  0.2842  0.00268328 
bl1  0.244  0.245  0.243  0.243  0.244  0.2438  0.00083666 
bl2  0.218  0.217  0.216  0.216  0.217  0.2168  0.00083666 
SVM  0.32  0.315  0.314  0.312  0.314  0.315  0.003 
ADA  0.207  0.21  0.21  0.209  0.209  0.209  0.00122474 
LoRAS (Em=t,p=.01)  0.366  0.362  0.355  0.363  0.366  0.3624  0.00450555 
F1Score average  0.30894286 
Oversampling models  lr1  lr2  lr3  lr4  lr5  mean  sd 

Base  0.702  0.704  0.7  0.702  0.702  0.702  0.00141421 
SMOTE  0.885  0.884  0.88  0.881  0.886  0.8832  0.00258844 
bl1  0.881  0.881  0.878  0.879  0.88  0.8798  0.00130384 
bl2  0.872  0.87  0.868  0.87  0.872  0.8704  0.00167332 
SVM  0.883  0.88  0.88  0.882  0.88  0.881  0.00141421 
ADA  0.864  0.865  0.867  0.867  0.868  0.8662  0.00164317 
LoRAS (Em=t,p=.01)  0.853  0.853  0.863  0.856  0.859  0.8568  0.00426615 
Oversampling models  svm1  svm2  svm3  svm4  svm5  mean  sd 

Base  0.458  0.435  0.412  0.437  0.418  0.432  0.01806931 
SMOTE  0.097  0.096  0.097  0.0096  0.096  0.07912  0.03886608 
bl1  0.098  0.102  0.103  0.103  0.103  0.1018  0.00216795 
bl2  0.098  0.102  0.103  0.103  0.103  0.1018  0.00216795 
SVM  0.1  0.096  0.096  0.096  0.096  0.0968  0.00178885 
ADA  0.1  0.096  0.095  0.095  0.095  0.0962  0.00216795 
LoRAS (Em=t,p=.01)  0.108  0.106  0.108  0.106  0.106  0.1068  0.00109545 
F1Score average  0.14493143 
Oversampling models  svm1  svm2  svm3  svm4  svm5  mean  sd 

Base  0.66  0.649  0.637  0.647  0.639  0.6464  0.00915423 
SMOTE  0.751  0.754  0.759  0.756  0.756  0.7552  0.00294958 
bl1  0.748  0.768  0.771  0.771  0.771  0.7658  0.01003494 
bl2  0.748  0.768  0.771  0.771  0.771  0.7658  0.01003494 
SVM  0.763  0.754  0.756  0.755  0.753  0.7562  0.00396232 
ADA  0.764  0.755  0.754  0.753  0.753  0.7558  0.00465833 
LoRAS (Em=t,p=.01)  0.78  0.774  0.78  0.773  0.775  0.7764  0.00336155 
Oversampling models  knn1  knn2  knn3  knn4  knn5  mean  sd 

Base  0.557  0.554  0.545  0.544  0.54  0.548  0.00717635 
SMOTE  0.408  0.417  0.411  0.416  0.416  0.4136  0.00391152 
bl1  0.417  0.413  0.416  0.417  0.411  0.4148  0.00268328 
bl2  0.324  0.33  0.331  0.326  0.323  0.3268  0.00356371 
SVM  0.473  0.469  0.463  0.463  0.467  0.467  0.00424264 
ADA  0.356  0.354  0.353  0.352  0.354  0.3538  0.00148324 
LoRAS (Em=r,p=NA)  0.515  0.512  0.505  0.511  0.512  0.511  0.00367423 
F1Score average  0.43357143 
Oversampling models  knn1  knn2  knn3  knn4  knn5  mean  sd 

Base  0.706  0.706  0.701  0.702  0.7  0.703  0.00282843 
SMOTE  0.909  0.914  0.913  0.912  0.91  0.9116  0.00207364 
bl1  0.91  0.908  0.912  0.908  0.908  0.9092  0.00178885 
bl2  0.898  0.9  0.901  0.9  0.899  0.8996  0.00114018 
SVM  0.913  0.91  0.91  0.906  0.908  0.9094  0.00260768 
ADA  0.903  0.91  0.905  0.904  0.906  0.9056  0.00270185 
LoRAS (Em=r,p=NA)  0.9  0.894  0.899  0.896  0.894  0.8966  0.00279285 
Dataset: scene
Oversampling models  lr1  lr2  lr3  lr4  lr5  mean  sd 

Base  0.178  0.181  0.173  0.172  0.138  0.1684  0.01738678 
SMOTE  0.236  0.231  0.222  0.216  0.205  0.222  0.01226784 
bl1  0.246  0.242  0.244  0.224  0.198  0.2308  0.02032732 
bl2  0.227  0.24  0.226  0.221  0.205  0.2238  0.01263725 
SVM  0.234  0.243  0.237  0.24  0.224  0.2356  0.00730068 
ADA  0.231  0.242  0.233  0.216  0.2  0.2244  0.01653179 
LoRAS (Em=t,p=30)  0.239  0.234  0.233  0.22  0.206  0.2264  0.01339029 
F1Score average  0.21877143 
Oversampling models  lr1  lr2  lr3  lr4  lr5  mean  sd 

Base  0.554  0.559  0.553  0.553  0.536  0.551  0.00874643 
SMOTE  0.628  0.629  0.613  0.612  0.6  0.6164  0.01217785 
bl1  0.632  0.632  0.631  0.613  0.589  0.6194  0.01882286 
bl2  0.625  0.638  0.621  0.616  0.603  0.6206  0.01277889 
SVM  0.611  0.622  0.616  0.618  0.607  0.6148  0.00589067 
ADA  0.625  0.64  0.626  0.613  0.596  0.62  0.01647726 
LoRAS (Em=t,p=30)  0.631  0.628  0.625  0.61  0.6  0.6188  0.01325519 
Oversampling models  svm1  svm2  svm3  svm4  svm5  mean  sd 

Base  0.333  0.189  0.149  0.169  0.158  0.1996  0.0760513 
SMOTE  0.226  0.198  0.203  0.188  0.194  0.2018  0.01460137 
bl1  0.222  0.214  0.204  0.199  0.193  0.2064  0.01163185 
bl2  0.222  0.214  0.204  0.199  0.193  0.2064  0.01163185 
SVM  0.269  0.203  0.208  0.189  0.189  0.2116  0.03317831 
ADA  0.241  0.205  0.195  0.191  0.195  0.2054  0.0205621 
LoRAS (Em=t,p=30)  0.239  0.211  0.209  0.191  0.197  0.2094  0.01851486 
F1Score average  0.2058 
Oversampling models  svm1  svm2  svm3  svm4  svm5  mean  sd 

Base  0.507  0.516  0.539  0.552  0.545  0.5318  0.01935717 
SMOTE  0.633  0.63  0.639  0.613  0.624  0.6278  0.00988433 
bl1  0.622  0.657  0.64  0.631  0.619  0.6338  0.01535252 
bl2  0.622  0.657  0.64  0.631  0.619  0.6338  0.01535252 
SVM  0.644  0.64  0.652  0.617  0.615  0.6336  0.01665233 
ADA  0.649  0.643  0.624  0.619  0.624  0.6318  0.01329286 
LoRAS (Em=t,p=30)  0.639  0.651  0.645  0.616  0.627  0.6356  0.01409965 
Oversampling models  knn1  knn2  knn3  knn4  knn5  mean  sd 

Base  0.011  0  0.011  0  0.011  0.0066  0.00602495 
SMOTE  0.217  0.209  0.217  0.215  0.218  0.2152  0.00363318 
bl1  0.232  0.235  0.234  0.238  0.236  0.235  0.00223607 
bl2  0.234  0.235  0.238  0.233  0.232  0.2344  0.00230217 
SVM  0.247  0.26  0.259  0.248  0.263  0.2554  0.00736885 
ADA  0.208  0.211  0.213  0.214  0.21  0.2112  0.00238747 
LoRAS (Em=t,p=30)  0.222  0.222  0.223  0.225  0.223  0.223  0.00122474 
F1Score average  0.19725714 
Oversampling models  knn1  knn2  knn3  knn4  knn5  mean  sd 

Base  0.502  0.5  0.502  0.5  0.502  0.5012  0.00109545 
SMOTE  0.698  0.68  0.701  0.7  0.7  0.6958  0.00889944 
bl1  0.71  0.715  0.714  0.719  0.716  0.7148  0.00327109 
bl2  0.711  0.717  0.722  0.714  0.711  0.715  0.00463681 
SVM  0.704  0.724  0.722  0.702  0.725  0.7154  0.01139298 
ADA  0.684  0.69  0.697  0.695  0.69  0.6912  0.00506952 
LoRAS (Em=t,p=30)  0.7  0.701  0.705  0.71  0.706  0.7044  0.00403733 
Dataset: ozone_level
Oversampling models  lr1  lr2  lr3  lr4  lr5  mean  sd 

Base  0.091  0.063  0.069  0.041  0.05  0.0628  0.01918854 
SMOTE  0.199  0.191  0.19  0.191  0.18  0.1902  0.00676018 
bl1  0.218  0.202  0.219  0.213  0.211  0.2126  0.00680441 
bl2  0.18  0.172  0.194  0.186  0.186  0.1836  0.00817313 
SVM  0.217  0.216  0.211  0.219  0.212  0.215  0.00339116 
ADA  0.2  0.19  0.191  0.197  0.186  0.1928  0.00563028 
LoRAS (Em=t,p=10)  0.216  0.204  0.209  0.205  0.205  0.2078  0.00496991 
F1Score average  0.18068571 
Oversampling models  lr1  lr2  lr3  lr4  lr5  mean  sd 

Base  0.525  0.52  0.52  0.511  0.513  0.5178  0.00571839 
SMOTE  0.812  0.795  0.805  0.804  0.785  0.8002  0.01042593 
bl1  0.785  0.771  0.778  0.775  0.776  0.777  0.00514782 
bl2  0.767  0.768  0.795  0.795  0.781  0.7812  0.013755 
SVM  0.735  0.737  0.733  0.747  0.739  0.7382  0.0054037 
ADA  0.816  0.8  0.8  0.808  0.793  0.8034  0.00882043 
LoRAS (Em=t,p=10)  0.813  0.808  0.807  0.809  0.808  0.809  0.00234521 
Oversampling models  svm1  svm2  svm3  svm4  svm5  mean  sd 

Base  0  0.02  0.022  0.018  0.02  0.016  0.00905539 
SMOTE  0.121  0.122  0.123  0.122  0.121  0.1218  0.00083666 
bl1  0.131  0.135  0.142  0.13  0.14  0.1356  0.00531977 
bl2  0.169  0.135  0.142  0.13  0.14  0.1432  0.01515586 
SVM  0.122  0.169  0.174  0.175  0.177  0.1634  0.02333024 
ADA  0.133  0.121  0.124  0.122  0.122  0.1244  0.0049295 
LoRAS (Em=t,p=30)  0.132  0.136  0.136  0.135  0.136  0.135  0.00173205 
F1Score average  0.11991429 
Oversampling models  svm1  svm2  svm3  svm4  svm5  mean  sd 

Base  0.498  0.505  0.506  0.503  0.504  0.5032  0.00311448 
SMOTE  0.753  0.756  0.758  0.748  0.754  0.7538  0.00376829 
bl1  0.756  0.775  0.785  0.746  0.787  0.7698  0.01810249 
bl2  0.756  0.775  0.785  0.746  0.787  0.7698  0.01810249 
SVM  0.787  0.791  0.79  0.792  0.803  0.7926  0.00610737 
ADA  0.755  0.751  0.76  0.749  0.759  0.7548  0.00481664 
LoRAS (Em=t,p=30)  0.778  0.782  0.782  0.774  0.788  0.7808  0.00521536 
Oversampling models  knn1  knn2  knn3  knn4  knn5  mean  sd 

Base  0  0  0  0  0  0  0 
SMOTE  0.095  0.098  0.117  0.112  0.129  0.1102  0.01398928 
bl1  0.099  0.081  0.118  0.126  0.123  0.1094  0.01903418 
bl2  0.108  0.089  0.11  0.107  0.105  0.1038  0.00846759 
SVM  0.102  0.09  0.11  0.101  0.127  0.106  0.01372953 
ADA  0.094  0.101  0.113  0.11  0.11  0.1056  0.00789303 
LoRAS (Em=t,p=30)  0.136  0.124  0.113  0.114  0.142  0.1258  0.01296919 
F1Score average  0.0944 
Oversampling models  knn1  knn2  knn3  knn4  knn5  mean  sd 

Base  0.5  0.5  0.5  0.5  0.5  0.5  0 
SMOTE  0.598  0.603  0.64  0.633  0.664  0.6276  0.02733679 
bl1  0.574  0.552  0.602  0.617  0.606  0.5902  0.02659323 
bl2  0.616  0.584  0.621  0.621  0.614  0.6112  0.01551451 
SVM  0.57  0.557  0.578  0.572  0.603  0.576  0.01692631 
ADA  0.595  0.61  0.635  0.627  0.624  0.6182  0.0158019 
LoRAS (Em=t,p=30)  0.67  0.659  0.63  0.641  0.676  0.6552  0.01938298 
Dataset: webpage
Oversampling models  lr1  lr2  lr3  lr4  lr5  mean  sd 

Base  0.747  0.744  0.757  0.751  0.74  0.7478  0.00653452 
SMOTE  0.913  0.093  0.093  0.093  0.092  0.2568  0.36682721 
bl1  0.112  0.11  0.11  0.107  0.11  0.1098  0.00178885 
bl2  0.079  0.081  0.082  0.082  0.079  0.0806  0.00151658 
SVM  0.118  0.116  0.118  0.117  0.117  0.1172  0.00083666 
ADA  0.093  0.096  0.095  0.094  0.093  0.0942  0.00130384 
LoRAS (Em=r,p=NA)  0.098  0.101  0.098  0.102  0.095  0.0988  0.00277489 
F1Score average  0.21502857 
Oversampling models  lr1  lr2  lr3  lr4  lr5  mean  sd 

Base  0.83  0.832  0.836  0.839  0.828  0.833  0.00447214 
SMOTE  0.709  0.714  0.715  0.717  0.713  0.7136  0.00296648 
bl1  0.768  0.763  0.763  0.756  0.762  0.7624  0.00427785 
bl2  0.663  0.672  0.675  0.676  0.66  0.6692  0.00725948 
SVM  0.78  0.776  0.781  0.778  0.778  0.7786  0.00194936 
ADA  0.717  0.724  0.721  0.718  0.716  0.7192  0.00327109 
LoRAS (Em=r,p=NA)  0.729  0.738  0.729  0.738  0.719  0.7306  0.00789303 
Oversampling models  svm1  svm2  svm3  svm4  svm5  mean  sd 

Base  0.729  0.732  0.749  0.747  0.737  0.7388  0.00889944 
SMOTE  0.087  0.088  0.088  0.088  0.089  0.088  0.00070711 
bl1  0.106  0.107  0.106  0.106  0.107  0.1064  0.00054772 
bl2  0.106  0.107  0.106  0.106  0.107  0.1064  0.00054772 
SVM  0.118  0.118  0.119  0.117  0.118  0.118  0.00070711 
ADA  0.091  0.083  0.092  0.091  0.091  0.0896  0.00371484 
LoRAS (Em=r,p=NA)  0.09  0.091  0.093  0.095  0.09  0.0918  0.00216795 
F1Score average  0.19128571 
Oversampling models  svm1  svm2  svm3  svm4  svm5  mean  sd 

Base  0.817  0.814  0.832  0.823  0.825  0.8222  0.00704982 
SMOTE  0.693  0.699  0.697  0.697  0.702  0.6976  0.00328634 
bl1  0.752  0.754  0.752  0.752  0.757  0.7534  0.00219089 
bl2  0.752  0.754  0.752  0.752  0.757  0.7534  0.00219089 
SVM  0.779  0.78  0.782  0.778  0.78  0.7798  0.00148324 
ADA  0.708  0.715  0.714  0.708  0.71  0.711  0.00331662 
LoRAS (Em=r,p=NA)  0.7  0.708  0.712  0.85  0.702  0.7344  0.06479815 
Oversampling models  knn1  knn2  knn3  knn4  knn5  mean  sd 

Base  0.71  0.709  0.715  0.707  0.714  0.711  0.00339116 
SMOTE  0.268  0.27  0.264  0.268  0.269  0.2678  0.00228035 
bl1  0.274  0.275  0.272  0.278  0.274  0.2746  0.00219089 
bl2  0.291  0.287  0.285  0.287  0.287  0.2874  0.00219089 
SVM  0.269  0.268  0.266  0.267  0.268  0.2676  0.00114018 
ADA  0.266  0.267  0.261  0.265  0.265  0.2648  0.00228035 
LoRAS (Em=r,p=NA)  0.62  0.614  0.609  0.61  0.616  0.6138  0.00449444 
F1Score average  0.38385714 
Oversampling models  knn1  knn2  knn3  knn4  knn5  mean  sd 

Base  0.804  0.804  0.808  0.806  0.806  0.8056  0.00167332 
SMOTE  0.906  0.908  0.904  0.907  0.907  0.9064  0.00151658 
bl1  0.9  0.901  0.9  0.903  0.903  0.9014  0.00151658 
bl2  0.905  0.903  0.902  0.901  0.905  0.9032  0.00178885 
SVM  0.904  0.905  0.904  0.905  0.906  0.9048  0.00083666 
ADA  0.903  0.905  0.901  0.904  0.906  0.9038  0.00192354 
LoRAS (Em=r,p=NA)  0.924  0.919  0.921  0.924  0.928  0.9232  0.00342053 
Dataset: wine_quiality
Oversampling models  lr1  lr2  lr3  lr4  lr5  mean  sd 

Base  0.069  0.069  0.064  0.069  0.068  0.0678  0.00216795 
SMOTE  0.181  0.174  0.187  0.182  0.175  0.1798  0.00535724 
bl1  0.183  0.18  0.181  0.188  0.178  0.182  0.00380789 
bl2  0.169  0.17  0.173  0.175  0.168  0.171  0.00291548 
SVM  0.214  0.215  0.207  0.217  0.228  0.2162  0.00759605 
ADA  0.181  0.18  0.181  0.182  0.18  0.1808  0.00083666 
LoRAS (Em=r,p=NA)  0.2  0.198  0.199  0.196  0.194  0.1974  0.00240832 
F1Score average  0.17071429 
Oversampling models  lr1  lr2  lr3  lr4  lr5  mean  sd 

Base  0.518  0.518  0.517  0.518  0.517  0.5176  0.00054772 
SMOTE  0.723  0.709  0.731  0.724  0.707  0.7188  0.01035374 
bl1  0.72  0.712  0.715  0.721  0.709  0.7154  0.00512835 
bl2  0.709  0.708  0.715  0.715  0.709  0.7112  0.00349285 
SVM  0.72  0.711  0.704  0.71  0.718  0.7126  0.00646529 
ADA  0.723  0.722  0.718  0.726  0.718  0.7214  0.00343511 
LoRAS (Em=r,p=NA)  0.739  0.732  0.736  0.733  0.731  0.7342  0.00327109 
Oversampling models  svm1  svm2  svm3  svm4  svm5  mean  sd 

Base  0  0  0  0.009  0  0.0018  0.00402492 
SMOTE  0.123  0.121  0.123  0.123  0.126  0.1232  0.00178885 
bl1  0.125  0.119  0.12  0.122  0.118  0.1208  0.00277489 
bl2  0.125  0.119  0.12  0.122  0.118  0.1208  0.00277489 
SVM  0.125  0.12  0.135  0.139  0.128  0.1294  0.00763544 
ADA  0.125  0.119  0.123  0.123  0.125  0.123  0.00244949 
LoRAS (Em=t,p=30)  0.128  0.126  0.13  0.129  0.126  0.1278  0.00178885 
F1Score average  0.10668571 
Oversampling models  svm1  svm2  svm3  svm4  svm5  mean  sd 

Base  0.5  0.499  0.499  0.502  0.499  0.4998  0.00130384 
SMOTE  0.689  0.68  0.687  0.687  0.696  0.6878  0.00571839 
bl1  0.695  0.675  0.68  0.687  0.668  0.681  0.01046422 
bl2  0.695  0.675  0.68  0.683  0.668  0.6802  0.01003494 
SVM  0.692  0.671  0.687  0.691  0.67  0.6822  0.01084896 
ADA  0.695  0.673  0.688  0.687  0.694  0.6874  0.00879204 
LoRAS (Em=t,p=30)  0.703  0.697  0.706  0.707  0.696  0.7018  0.00506952 
Oversampling models  knn1  knn2  knn3  knn4  knn5  mean  sd 

Base  0  0  0  0  0  0  0 
SMOTE  0.155  0.147  0.158  0.159  0.153  0.1544  0.00477493 
bl1  0.186  0.182  0.172  0.18  0.183  0.1806  0.00527257 
bl2  0.165  0.171  0.167  0.167  0.166  0.1672  0.00228035 
SVM  0.216  0.23  0.217  0.222  0.218  0.2206  0.00572713 
ADA  0.152  0.147  0.158  0.157  0.156  0.154  0.00452769 
LoRAS (Em=t,p=30)  0.156  0.156  0.157  0.163  0.154  0.1572  0.00342053 
F1Score average  .  0.14771429 
Oversampling models  knn1  knn2  knn3  knn4  knn5  mean  sd 

Base  0.5  0.5  0.5  0.5  0.5  0.5  0 
SMOTE  0.694  0.676  0.702  0.708  0.691  0.6942  0.01217374 
bl1  0.704  0.702  0.685  0.7  0.701  0.6984  0.00763544 
bl2  0.698  0.714  0.704  0.71  0.705  0.7062  0.00609918 
SVM  0.696  0.711  0.698  0.7  0.696  0.7002  0.00626099 
ADA  0.693  0.684  0.708  0.709  0.701  0.699  0.01055936 
LoRAS (Em=t,p=30)  0.693  0.693  0.701  0.711  0.693  0.6982  0.00794984 
Dataset: yeast_ml8
Oversampling models  lr1  lr2  lr3  lr4  lr5  mean  sd 

Base  0.023  0.031  0.02  0.032  0.011  0.0234  0.00861974 
SMOTE  0.155  0.154  0.152  0.15  0.153  0.1528  0.00192354 
bl1  0.142  0.147  0.15  0.175  0.155  0.1538  0.01275539 
bl2  0.146  0.147  0.151  0.157  0.159  0.152  0.00583095 
SVM  0.131  0.144  0.139  0.155  0.156  0.145  0.01065364 
ADA  0.14  0.156  0.148  0.154  0.139  0.1474  0.00779744 
LoRAS (Em=r,p=NA)  0.138  0.14  0.158  0.154  0.141  0.1462  0.0091214 
F1Score average  0.13151429 
Oversampling models  lr1  lr2  lr3  lr4  lr5  mean  sd 

Base  0.5  0.5  0.505  0.507  0.501  0.5026  0.00320936 
SMOTE  0.554  0.551  0.549  0.545  0.547  0.5492  0.00349285 
bl1  0.533  0.541  0.541  0.576  0.551  0.5484  0.01669731 
bl2  0.541  0.54  0.543  0.552  0.558  0.5468  0.00785493 
SVM  0.525  0.537  0.531  0.548  0.548  0.5378  0.0102323 
ADA  0.532  0.553  0.541  0.55  0.529  0.541  0.0106066 
LoRAS (Em=r,p=NA)  0.528  0.531  0.553  0.548  0.531  0.5382  0.01143241 
Oversampling models  svm1  svm2  svm3  svm4  svm5  mean  sd 

Base  0  0  0  0  0  0  0 
SMOTE  0.149  0.14  0.15  0.143  0.137  0.1438  0.00563028 
bl1  0.137  0.15  0.157  0.155  0.141  0.148  0.0087178 
bl2  0.137  0.15  0.157  0.155  0.141  0.148  0.0087178 
SVM  0.135  0.154  0.147  0.156  0.156  0.1496  0.00896103 
ADA  0.144  0.143  0.146  0.138  0.147  0.1436  0.00350714 
LoRAS (Em=r,p=NA)  0.141  0.152  0.15  0.151  0.148  0.1484  0.00439318 
F1Score average  0.12591429 
Oversampling models  svm1  svm2  svm3  svm4  svm5  mean  sd 

Base  0.5  0.5  0.5  0.5  0.5  0.5  0 
SMOTE  0.544  0.525  0.549  0.531  0.52  0.5338  0.01235718 
bl1  0.52  0.545  0.561  0.559  0.529  0.5428  0.01808867 
bl2  0.52  0.545  0.561  0.559  0.529  0.5428  0.01808867 
SVM  0.527  0.55  0.539  0.55  0.551  0.5434  0.01040673 
ADA  0.533  0.53  0.539  0.521  0.541  0.5328  0.00794984 
LoRAS (Em=r,p=NA)  0.528  0.549  0.546  0.547  0.543  0.5426  0.00844393 
Oversampling models  knn1  knn2  knn3  knn4  knn5  mean  sd 

Base  0  0  0  0  0  0  0 
SMOTE  0.153  0.153  0.15  0.154  0.15  0.152  0.00187083 
bl1  0.155  0.149  0.153  0.153  0.157  0.1534  0.00296648 
bl2  0.156  0.154  0.15  0.15  0.157  0.1534  0.00328634 
SVM  0.158  0.155  0.162  0.156  0.162  0.1586  0.00328634 
ADA  0.152  0.15  0.153  0.151  0.153  0.1518  0.00130384 
LoRAS (Em=r,p=NA)  0.152  0.151  0.152  0.153  0.154  0.1524  0.00114018 
F1Score average  0.13165714 
Oversampling models  knn1  knn2  knn3  knn4  knn5  mean  sd 

Base  0.5  0.5  0.5  0.5  0.5  0.5  0 
SMOTE  0.56  0.56  0.552  0.564  0.558  0.5588  0.00438178 
bl1  0.566  0.548  0.562  0.558  0.571  0.561  0.0087178 
bl2  0.57  0.562  0.56  0.549  0.574  0.563  0.00969536 
SVM  0.573  0.563  0.581  0.563  0.581  0.5722  0.0090111 
ADA  0.56  0.553  0.563  0.553  0.563  0.5584  0.00507937 
LoRAS (Em=r,p=NA)  0.558  0.555  0.558  0.561  0.565  0.5594  0.00378153 
Dataset: yeast_me2
Oversampling models  lr1  lr2  lr3  lr4  lr5  mean  sd 

Base  0.206  0.171  0.214  0.205  0.254  0.21  0.0296395 
SMOTE  0.261  0.267  0.267  0.255  0.259  0.2618  0.00521536 
bl1  0.324  0.337  0.322  0.328  0.32  0.3262  0.00672309 
bl2  0.276  0.279  0.28  0.284  0.274  0.2786  0.00384708 
SVM  0.366  0.373  0.361  0.358  0.36  0.3636  0.00602495 
ADA  0.25  0.245  0.243  0.241  0.241  0.244  0.00374166 
LoRAS (Em=t,p=100)  0.287  0.285  0.29  0.286  0.286  0.2868  0.00192354 
F1Score average  0.28157143 
Oversampling models  lr1  lr2  lr3  lr4  lr5  mean  sd 

Base  0.572  0.562  0.576  0.567  0.585  0.5724  0.00879204 
SMOTE  0.793  0.803  0.801  0.791  0.799  0.7974  0.00517687 
bl1  0.811  0.823  0.81  0.813  0.819  0.8152  0.0055857 
bl2  0.805  0.807  0.814  0.817  0.82  0.8126  0.00642651 
SVM  0.812  0.814  0.819  0.804  0.811  0.812  0.00543139 
ADA  0.802  0.802  0.8  0.793  0.792  0.7978  0.00491935 
LoRAS (Em=t,p=100)  0.809  0.809  0.808  0.81  0.808  0.8088  0.00083666 
Oversampling models  svm1  svm2  svm3  svm4  svm5  mean  sd 

Base  0  0  0  0  0  0  0 
SMOTE  0.295  0.296  0.285  0.282  0.279  0.2874  0.00770065 
bl1  0.329  0.332  0.32  0.329  0.324  0.3268  0.00476445 
bl2  0.329  0.332  0.32  0.329  0.324  0.3268  0.00476445 
SVM  0.346  0.345  0.362  0.358  0.348  0.3518  0.00769415 
ADA  0.27  0.258  0.268  0.277  0.257  0.266  0.00845577 
LoRAS (Em=r,p=NA)  0.301  0.299  0.284  0.296  0.291  0.2942  0.00683374 
F1Score average  0.26471429 
Oversampling models  svm1  svm2  svm3  svm4  svm5  mean  sd 

Base  0.5  0.5  0.5  0.5  0.5  0.5  0 
SMOTE  0.819  0.821  0.815  0.806  0.907  0.8336  0.04143429 
bl1  0.82  0.822  0.81  0.821  0.819  0.8184  0.00482701 
bl2  0.82  0.822  0.81  0.821  0.819  0.8184  0.00482701 
SVM  0.816  0.809  0.81  0.82  0.816  0.8142  0.00460435 
ADA  0.81  0.8  0.809  0.812  0.799  0.806  0.00604152 
LoRAS (Em=r,p=NA)  0.812  0.837  0.799  0.817  0.811  0.8152  0.01386362 
Oversampling models  knn1  knn2  knn3  knn4  knn5  mean  sd 

Base  0.061  0.066  0.061  0.061  0.123  0.0744  0.02725436 
SMOTE  0.32  0.342  0.319  0.329  0.345  0.331  0.01210372 
bl1  0.379  0.391  0.372  0.379  0.348  0.3738  0.01595932 
bl2  0.296  0.306  0.343  0.295  0.282  0.3044  0.02320129 
SVM  0.393  0.385  0.388  0.398  0.379  0.3886  0.00730068 
ADA  0.299  0.318  0.316  0.318  0.328  0.3158  0.01049762 
LoRAS (Em=r,p=NA)  0.355  0.347  0.355  0.36  0.357  0.3548  0.00481664 
F1Score average  0.30611429 
Oversampling models  knn1  knn2  knn3  knn4  knn5  mean  sd 

Base  0.517  0.524  0.518  0.519  0.537  0.523  0.00827647 
SMOTE  0.809  0.855  0.819  0.839  0.849  0.8342  0.01962651 
bl1  0.797  0.809  0.799  0.791  0.793  0.7978  0.00701427 
bl2  0.781  0.791  0.815  0.783  0.784  0.7908  0.01404279 
SVM  0.782  0.783  0.785  0.786  0.789  0.785  0.00273861 
ADA  0.804  0.841  0.817  0.818  0.845  0.825  0.01739253 
LoRAS (Em=r,p=NA)  0.833  0.84  0.836  0.853  0.852  0.8428  0.00920326 
Dataset: credit fraud
Oversampling models  lr1  lr2  lr3  lr4  lr5  mean  sd 

Base  0.674  0.693  0.682  0.687  0.683  0.6838  0.00499166 
SMOTE  0.113  0.13  0.133  0.143  0.15  0.1338  0.00920145 
bl1  0.229  0.254  0.241  0.219  0.228  0.2342  0.01528616 
bl2  0.173  0.161  0.174  0.19  0.187  0.177  0.0132916 
SVM  0.282  0.305  0.276  0.262  0.273  0.2796  0.01834848 
ADA  0.109  0.132  0.125  0.127  0.123  0.1232  0.00386221 
LoRAS (Em=t,p=30)  0.56  0.544  0.558  0.595  0.539  0.5592  0.02531139 
F1Score average  0.31297143 
Oversampling models  lr1  lr2  lr3  lr4  lr5  mean  sd 

Base  0.83  0.846  0.833  0.84  0.838  0.8374  0.00622896 
SMOTE  0.923  0.93  0.928  0.934  0.934  0.9298  0.00460435 
bl1  0.927  0.93  0.928  0.93  0.928  0.9286  0.00134164 
bl2  0.926  0.932  0.93  0.932  0.931  0.9302  0.00248998 
SVM  0.927  0.924  0.927  0.925  0.924  0.9254  0.00151658 
ADA  0.922  0.932  0.932  0.93  0.927  0.9286  0.004219 
LoRAS (Em=t,p=30)  0.904  0.905  0.904  0.906  0.904  0.9046  0.00089443 
Oversampling models  rf1  rf2  rf3  rf4  rf5  mean  

Base  0.67  0.669  0.664  0.667  0.675  0.669  0.00464579 
SMOTE  0.36  0.366  0.355  0.359  0.357  0.3594  0.00478714 
bl1  0.644  0.639  0.662  0.644  0.64  0.6458  0.01071992 
bl2  0.552  0.545  0.571  0.55  0.562  0.556  0.01174734 
SVM  0.743  0.741  0.745  0.741  0.739  0.7418  0.00251661 
ADA  0.35  0.354  0.348  0.348  0.351  0.3502  0.00287228 
LoRAS (Em=t,p=30)  0.821  0.823  0.82  0.818  0.82  0.8204  0.00206155 
F1Score average  0.5918 
Oversampling models  rf1  rf2  rf3  rf4  rf5  mean  sd 

Base  0.775  0.775  0.772  0.774  0.779  0.775  0.00254951 
SMOTE  0.922  0.923  0.922  0.922  0.925  0.9228  0.00130384 
bl1  0.92  0.92  0.919  0.918  0.918  0.919  0.001 
bl2  0.919  0.919  0.919  0.92  0.92  0.9194  0.00054772 
SVM  0.914  0.914  0.913  0.914  0.911  0.9132  0.00130384 
ADA  0.922  0.925  0.922  0.924  0.925  0.9236  0.00151658 
LoRAS (Em=t,p=30)  0.905  0.906  0.904  0.904  0.904  0.9046  0.00089443 