SMOTE for highdimensional classimbalanced data
 Rok Blagus^{1} and
 Lara Lusa^{1}Email author
DOI: 10.1186/1471210514106
© Blagus and Lusa; licensee BioMed Central Ltd. 2013
Received: 24 July 2012
Accepted: 22 February 2013
Published: 22 March 2013
Abstract
Background
Classification using classimbalanced data is biased in favor of the majority class. The bias is even larger for highdimensional data, where the number of variables greatly exceeds the number of samples. The problem can be attenuated by undersampling or oversampling, which produce classbalanced data. Generally undersampling is helpful, while random oversampling is not. Synthetic Minority Oversampling TEchnique (SMOTE) is a very popular oversampling method that was proposed to improve random oversampling but its behavior on highdimensional data has not been thoroughly investigated. In this paper we investigate the properties of SMOTE from a theoretical and empirical point of view, using simulated and real highdimensional data.
Results
While in most cases SMOTE seems beneficial with lowdimensional data, it does not attenuate the bias towards the classification in the majority class for most classifiers when data are highdimensional, and it is less effective than random undersampling. SMOTE is beneficial for kNN classifiers for highdimensional data if the number of variables is reduced performing some type of variable selection; we explain why, otherwise, the kNN classification is biased towards the minority class. Furthermore, we show that on highdimensional data SMOTE does not change the classspecific mean values while it decreases the data variability and it introduces correlation between samples. We explain how our findings impact the classprediction for highdimensional data.
Conclusions
In practice, in the highdimensional setting only kNN classifiers based on the Euclidean distance seem to benefit substantially from the use of SMOTE, provided that variable selection is performed before using SMOTE; the benefit is larger if more neighbors are used. SMOTE for kNN without variable selection should not be used, because it strongly biases the classification towards the minority class.
Background
The objective of class prediction (classification) is to develop a rule based on a group of samples with known class membership (training set), which can be used to assign the class membership to new samples. Many different classification algorithms (classifiers) exist, and they are based on the values of the variables (features) measured for each sample [1].
Very often the training and/or test data are classimbalanced: the number of observations belonging to each class is not the same. The problem of learning from classimbalanced data has been receiving a growing attention in many different fields [2]. The presence of classimbalance has important consequences on the learning process, usually producing classifiers that have poor predictive accuracy for the minority class and that tend to classify most new samples in the majority class; in this setting the assessment of the performance of the classifiers is also critical [3].
Data are nowadays increasingly often highdimensional: the number of variables is very large and greatly exceeds the number of samples. For example, highthroughput technologies are popular in the biomedical field, where it is possible to measure simultaneously the expression of all the known genes (>20,000) but the number of subjects included in the study is rarely larger than few hundreds. Many papers attempted to develop classification rules using highdimensional gene expression data that were classimbalanced (see for example [46]).
Despite the growing number of applications using highdimensional classimbalanced data, this problem has been seldom addressed from the methodological point of view [2]. It was previously shown for many classifiers that the classimbalance problem is exacerbated when data are highdimensional [7]: the highdimensionality further increases the bias towards the classification into the majority class, even when there is no real difference between the classes. The highdimensionality affects each type of classifier in a different way. A general remark is that large discrepancies between training data and true population values are more likely to occur in the minority class, which has a larger sampling variability: therefore, the classifiers are often trained on data that do not represent well the minority class. The highdimensionality contributes to this problem as extreme values are not exceptional when thousands of variables are considered.
Some of the solutions proposed in the literature to attenuate the classimbalance problem are effective with highdimensional data, while others are not. Generally undersampling techniques, aimed at producing a classbalanced training set of smaller size, are helpful, while simple oversampling is not [7]. The reason is that in most cases simple oversampling does not change the classification rule. Similar results were obtained also for lowdimensional data [8].
The Synthetic Minority Oversampling TEchnique (SMOTE [9]) is an oversampling approach that creates synthetic minority class samples. It potentially performs better than simple oversampling and it is widely used. For example, SMOTE was used for detecting network intrusions [10] or sentence boundary in speech [11], for predicting the distribution of species [12] or for detecting breast cancer [13]. SMOTE is used also in bioinformatics for miRNA gene prediction [14, 15], for the identification of the binding specificity of the regulatory proteins [16] and of photoreceptorenriched genes based on expression data [17], and for histopathology annotation [18].
However, it was recently experimentally observed using lowdimensional data that simple undersampling tends to outperform SMOTE in most situations [8]. This result was further confirmed using SMOTE with SVM as a base classifier [19], extending the observation also to highdimensional data: SMOTE with SVM seems beneficial but less effective than simple undersampling for lowdimensional data, while it performs very similarly to uncorrected SVM and generally much worse than undersampling for highdimensional data. To our knowledge this was the first attempt to investigate explicitly the effect of the highdimensionality on SMOTE, while the performance of SMOTE on highdimensional data was not thoroughly investigated for classifiers other than SVM. Others evaluated the performance of SMOTE on large data sets, focusing on problems where the number of samples, rather than the number of variables was very large [20, 21]. A number of works focused on improving the original SMOTE algorithm [17, 2224] but these modifications were mainly not considered in the highdimensional context.
In this article we investigate the theoretical properties of SMOTE and its performance on highdimensional data. For the sake of simplicity we consider only twoclass classification problems, and limit our attention to Classification and Regression Trees (CART [25]), kNN [26] with k = 1, 3 and 5, linear discriminant analysis methods (diagonal  DLDA, and quadratic  DQDA) [27, 28], random forests (RF [29]), support vector machine (SVM [30]), prediction analysis for microarrays (PAM [31] also known as nearest shrunken centroids classification) and penalized logistic regression (PLR [32]) with the linear (PLRL1) and quadratic penalty (PLRL2). We supplement the theoretical results with empirical results, based on simulation studies and analysis of gene expression microarray data sets.
The rest of the article is organized as follows. In the Results Section we present some theoretical results, a selected series of simulation results and the experimental results. In the Discussion Section we summarize and discuss the most important results of our study. In the Methods Section we briefly describe SMOTE and simple undersampling, the classification algorithms, the variable selection method and the performance measures that we used; we also describe the procedure of data simulation, the breast cancer gene expression data sets and the classification problems addressed.
Results
In this section we present some theoretical properties of SMOTE [9], the simulation results and the experimental data results.
with 0 ≤ u ≤ 1; x^{ R } is randomly chosen among the 5 minority class nearest neighbors of x. We refer the reader to the Methods section for a more detailed description of the method and of the notation used in the paper.
Theoretical properties of SMOTE for highdimensional data
Summary of the theoretical properties of SMOTE for highdimensional data
Property  Consequence of using SMOTE on highdimensional data 

E(SMOTE) = E(X)  Little impact on classifiers that depend on mean values (DLDA); 
$\mathit{\text{var}}\left(\text{SMOTE}\right)=\frac{2}{3}\mathit{\text{var}}\left(X\right)$  Minority class variability is underestimated; negative impact on classifiers that use classspecific variances (DQDA); inflated statistical significance of statistical tests for comparing classes (ttest); 
d(SMOTE, TEST) < d(X, TEST)d: Euclidean distance  Test samples are classified mostly in the minority class for classifiers based on Euclidean distance (kNN); variable selection is helpful in reducing this problem; 
cor(SMOTE, X) ≥ 0; cor(SMOTE^{ s }, SMOTE^{ t }) ≥ 0  Training set samples are no longer independent; independence of samples is assumed by most classifiers (DLDA, PLR,...) and variable selection methods (ttest, MannWhitney,...) 
Most of the proofs require the assumptions that x^{ R } and x are independent and have the same expected value (E(·)) and variance (var(·)). We conducted a limited set of simulations in which we showed that in practice these assumptions are valid for highdimensional data, while they do not hold for lowdimensional data (Additional file 1), where the samples are positively correlated. Similar results were described by others [33, 34].
The proofs and details of the results presented in this section are given in Additional file 1, where most of the results are derived also without assuming the independence and equal distribution of the original and nearest neighbor samples.
SMOTE does not change the expected value of the (SMOTEaugmented) minority class and it decreases its variability
SMOTE samples have the same expected value as the original minority class samples ($E\left({X}_{j}^{\mathit{\text{SMOTE}}}\right)=E\left({X}_{j}\right)$), but smaller variance ($\mathit{\text{var}}\left({X}_{j}^{\mathit{\text{SMOTE}}}\right)=\frac{2}{3}\mathit{\text{var}}\left({X}_{j}\right)$).
Practical consequences
The overall expected value of the SMOTEaugmented minority class is equal to the expected value of the original minority class, while its variance is smaller. Therefore, SMOTE has little impact on the classifiers that base their classification rules on classspecific mean values and overall variances (as DLDA), while it has some (harmful) impact on the classifiers that use classspecific variances (as DQDA), because they use biased estimates.
SMOTE impacts also variable selection. For example, the pvalues obtained comparing two classes with a ttest after SMOTEaugmenting the data are smaller than those obtained using the original data (SMOTE reduces the standard error increasing the sample size and decreasing the variance, while the difference between the sample means does not change much). This can misleadingly indicate that many variables are differentially expressed between the classes. SMOTE does not substantially alter the ranking of the variables by their t statistics: the overlap between the variables selected using original or SMOTEaugmented data is substantial when the number of selected variables is kept fixed.
SMOTE introduces correlation between some samples, but not between variables
SMOTE does not introduce correlation between different variables. The SMOTE samples are strongly positively correlated with the samples from the minority class used to generate them (x and x^{ R } from Eq. 1) and with the SMOTE samples obtained using the same original samples.
Practical consequences
SMOTE can be problematic for the classifiers that assume independence among samples, as for example penalized logistic regression or discriminant analysis methods. Also, performing variable selection after using SMOTE should be done with some care because most variable selection methods assume that the samples are independent.
SMOTE modifies the Euclidean distance between test samples and the (SMOTEaugmented) minority class
When data are highdimensional and the similarity between samples is measured using the Euclidean distance, the test samples are on average more similar to SMOTE samples than to the original samples from the minority class.
Practical consequences
This phenomenon is present also when there are some differences between classes but few variables truly discriminate the classes. This is often the case for highdimensional data and it has important practical implications. For example, when the number of variables is large, SMOTE is likely to bias the classification towards the minority class for kNN classifiers that measure the similarity between samples using the Euclidean distance. Conversely, SMOTE does not bias the classification towards the minority class if the number of variables is small, as the Euclidean distance of new samples from both classes is similar for the null variables (Figure 1). For these reasons SMOTE seems useful in reducing the classimbalance problem for kNN when the number of variables is small or if the number of variables is reduced using variable selection methods (see simulation results and the analyses of empirical data for further insights).
Results on simulated data
Simulations were used to systematically explore the behavior of SMOTE with highdimensional data and to show empirically the consequences of the theoretical results. Under the null case the class membership was randomly assigned, while in the alternative case the classmembership depended on some of the variables. If not stated otherwise, the results refer to simulations where the variables were correlated (ρ = 0.8), the samples (but not the variables) were normalized and SMOTE was used before variable selection. In the alternative case we present the results where the difference between classes was moderate (μ^{(2)} = 1).
Classification of lowdimensional data (p = G = 5, n_{ train } = 40, 80, 200, k_{1}= 0.10)
Classification of highdimensional data (p = 1, 000, G = 1, 000 or 40, n_{ train }= 80)
Adjusting the classification threshold substantially decreased the classimbalance bias of 5NN, RF and SVM (more effectively when variable selection was not performed), and was helpful to some extent also for PAM, provided that variable selection was performed. A slight improvement was observed also for PLRL1 (more obvious when variable selection was not performed) and PLRL2, while this strategy was not effective for the other classifiers. The peculiar behavior of 5NN with classification threshold is expected, as under the null hypothesis the class specific probabilities are piecewise monotone functions of classimbalance with breakpoints at k_{1} = 1 / 5, 2 / 5, 3 / 5, 4 / 5.
SMOTE had only a small impact on the classspecific PA of all the classifiers other than kNN and PAM: SMOTE either further increased the probability of classification in the majority class (DQDA and SVM, and almost imperceptibly for DLDA) or slightly decreased it (RF, PLRL1, PLRL2 and CART). However, the overall effect of SMOTE was almost negligible.
SMOTE had the most dramatic effect on kNN classifiers but the effectiveness of its use depended on the variable selection strategy. SMOTE classified most of the new samples in the minority class for any level of classimbalance when all the variables were used, while it reduced the bias observed in the uncorrected analyses when used with variable selection: the classspecific PA of the two classes were approximately equal for a wide range of classimbalance levels, especially for 3NN and 5NN, both in the null and in the alternative case.
To a lesser extent, SMOTE with variable selection was beneficial also in reducing the classimbalance problem of PAM, decreasing the number of samples classified in the majority class, both in the null and in the alternative case; this was not the case when PAM was used without prior variable selection. A possible explanation of this behavior is given in the Additional file 2.
Similar conclusions would be obtained using AUC and Gmean to interpret the results (Additional file 3). SMOTE without variable selection reduced the Gmean for kNN, DQDA and SVM, it increased it for RF, PLRL1, PLRL2 and PAM (when the classimbalance was large) and did not change it for DLDA and CART. The AUC values were very similar using SMOTE or uncorrected analysis, but SMOTE with variable selection increased AUC and Gmean values for kNN and PAM.
Performing variable selection before or after SMOTE did not significantly impact the performance of the classification methods (data not shown). In general, the results observed in the alternative case were similar to those observed in the null case, suggesting that our theoretical findings are relevant also in the situations where the classmembership depends on some of the variables. When the differences between the classes were larger, the classimbalance problem was less severe, therefore using SMOTE was less helpful (data not shown).
Results from the experiments on gene expression data sets
Experimental data sets
Data set  Prediction task  Number of features  Number of samples  n _{ min }  k _{ min }  Minority class 

Sotiriou  ER  7,650  99  34  0.34  ER 
Grade  7,650  99  45  0.45  Grade 3  
Pittman  ER  12,625  158  48  0.30  ER 
Grade  12,625  158  63  0.40  Grade 1 or 2  
Miller  ER  22,283  247  34  0.14  ER 
Grade  22,283  249  54  0.22  Grade 3 
The experimental results were very consistent with the simulation results. Most uncorrected classifiers seemed to be sensitive to classimbalance, even when the classimbalance was moderate. With few exceptions, the majority class had a better classspecific PA (most notably for kNN, RF, PLRL1, PLRL2 and CART); the larger differences were seen when the classimbalance was large (Miller’s and Pittman’s data) and for harder classification tasks (grade). The classspecific PA of DLDA and DQDA were about the same for all the classification tasks; these classifiers, together with PAM, had the largest AUC and Gmean values and seemed the least sensitive to classimbalance. SMOTE, cutoff adjustment and undersampling had little or no effect on their classification results.
Changing the cutoff point decreased the classimbalance bias of RF, SVM, PAM, PLRL1 and PLRL2 and 5NN (with the exception of the results obtained on the Sotiriou’s data) and outperformed undersampling, while it was inefficient with the other classifiers.
Performance of the classifiers on the Miller data set without feature selection
ER  Grade  

1NN  3NN  5NN  1NN  3NN  5NN  
NC (CUTOFF)  PA  0.838  0.862  0.874 (0.777)  0.779  0.839  0.835 (0.835) 
PA_{1}  0.925  0.953  0.972 (0.789)  0.897  0.954  0.949 (0.897)  
PA_{2}  0.294  0.294  0.265 (0.706)  0.352  0.426  0.426 (0.611)  
AUC  0.610  0.692  0.772 (0.772)  0.625  0.769  0.816 (0.816)  
Gmean  0.522  0.529  0.507 (0.746)  0.562  0.637  0.636 (0.741)  
SMOTE  PA  0.271  0.249  0.249  0.364  0.373  0.384 
(0.012)  (0.013)  (0.012)  (0.014)  (0.015)  (0.016)  
PA_{1}  0.156  0.130  0.132  0.194  0.209  0.223  
(0.014)  (0.014)  (0.013)  (0.018)  (0.020)  (0.020)  
PA_{2}  0.996  0.992  0.984  0.979  0.966  0.966  
(0.010)  (0.015)  (0.017)  (0.012)  (0.013)  (0.011)  
AUC  0.576  0.632  0.671  0.586  0.680  0.736  
(0.009)  (0.014)  (0.013)  (0.011)  (0.013)  (0.010)  
Gmean  0.393  0.359  0.360  0.435  0.449  0.464  
(0.018)  (0.020)  (0.019)  (0.020)  (0.021)  (0.021)  
UNDER  PA  0.625  0.685  0.691  0.766  0.836  0.840 
(0.065)  (0.056)  (0.049)  (0.017)  (0.012)  (0.012)  
PA_{1}  0.742  0.841  0.863  0.798  0.871  0.878  
(0.017)  (0.013)  (0.012)  (0.016)  (0.011)  (0.012)  
PA_{2}  0.761  0.866  0.890  0.649  0.709  0.700  
(0.017)  (0.013)  (0.010)  (0.051)  (0.039)  (0.028)  
AUC  0.693  0.822  0.861  0.723  0.833  0.850  
(0.033)  (0.021)  (0.021)  (0.027)  (0.015)  (0.008)  
Gmean  0.689  0.770  0.784  0.719  0.786  0.784  
(0.036)  (0.031)  (0.029)  (0.029)  (0.022)  (0.017) 
SMOTE reduced the discrepancy in classspecific PA for the other classifiers (RF, SVM, PAM, PLRL1, PLRL2 and CART), but simple undersampling performed very similarly (PAM) or better (RF, SVM, PLRL1, PLRL2 and CART).
Results obtained modifying the classimbalance of Sotiriou’s data
For the uncorrected classifiers the PA of the minority class markedly decreased as the classimbalance increased, despite of the fact that the sample size of the training set was larger. This effect was more pronounced when the differences between classes were smaller (grade classification) or for smaller sample sizes (n_{1} = 5).
For most classifiers SMOTE improved the PA of the minority class, compared to the uncorrected analyses. The classifiers that benefited the most from the use of SMOTE were the kNN classifiers, especially 5NN (note that variable selection was performed); SMOTE was somehow beneficial also for PAM, PLRL1 and PLRL2, while the minority class PA improved only moderately for DLDA, RF, SVM and CART, and decreased for DQDA. However, SMOTE did not remove the classimbalance problem and, even if it was beneficial compared to the uncorrected analysis, it generally performed worse than undersampling. The exceptions were PAM and 5NN for ER classification (but not for grade), where the drop in the PA of the minority class was very moderate. Overall, the classification results were in line with the simulation results and confirmed our theoretical findings.
Discussion
The classifiers that we considered in this study were previously shown to be sensitive to classimbalance: the predictive accuracy of the minority class tends to be poor and they tend to classify most test samples in the majority class, even when there are no differences between the classes. The highdimensionality further increases the bias towards the classification in the majority class; undersampling techniques seem to be helpful in reducing the classimbalance problem for highdimensional data, while simple oversampling [2] is not [7].
In this article we focused on highdimensional data and investigated the performance of SMOTE, an oversampling approach that creates synthetic samples. We explored the properties of SMOTE on highdimensional data from a theoretical and empirical point of view, using simulation studies and breast cancer gene expression microarray data. The performance of the classifiers was evaluated with overall and class specific predictive accuracies, area under the ROC curve (AUC) and Gmean.
Most of the classifiers that we considered benefit from SMOTE if data are lowdimensional: SMOTE reduces the bias towards the classification in the majority class for kNN, SVM, PAM, PLRL1, PLRL2, CART and, to some extent, for RF, while it hardly affects the discriminant analysis classifiers (DLDA and DQDA). On the other hand, for highdimensional data SMOTE is not beneficial in most circumstances: it performs similarly to uncorrected classimbalanced classification and worse than cutoff adjustment or simple undersampling.
In practice, only kNN classifiers seem to benefit substantially from the use of SMOTE in the highdimensional setting, provided that variable selection is performed before using SMOTE; the benefit is larger if more neighbors are used. SMOTE for kNN without variable selection should not be used, because it surprisingly biases the classification towards the minority class: we showed that the reason lies in the way SMOTE modifies the Euclidean distance between the new samples and the minority class. Our theoretical proofs made many assumptions; however, analyzing the simulated and real data, where the assumptions were violated, we observed that our results were valid in practice.
We showed that for highdimensional data SMOTE does not change the mean value of the SMOTEaugmented minority class, while it reduces its variance; the practical consequence of these results is that SMOTE hardly affects the classifiers that base their classification rules on class specific means and overall variances; such classifiers include the widely used DLDA. Additionally, SMOTE harms the classifiers that use classspecific variances (as DQDA), as it produces biased estimates: our experimental data confirmed these finding, showing that SMOTE further increased the bias towards the majority class. SMOTE should therefore not be used with these types of classifiers.
For the other classifiers it is more difficult to isolate the reasons why SMOTE might or might not work on highdimensional data. SMOTE has a very limited impact on SVM and CART. PLRL1, PLRL2 and RF seem to benefit from SMOTE in some circumstances, however the improvements in the predictive accuracy of the minority class seem moderate when compared to the results obtained using the original data and can be probably attributed to the balancing of the training set. The apparent benefit of SMOTE for PAM is limited to situations where variable selection is performed before using PAM, which is not a normally used procedure, and can be explained as the effect of removing the PAMembedded classimbalance correction, which increases the probability of classifying a sample in the majority class.
Using the gene expression data we compared SMOTE with simple undersampling, the method that obtains a balanced training set by removing some of the samples from the majority class. Our results show that for RF, SVM, PLR, CART and DQDA simple undersampling seems to be more useful than SMOTE in improving the predictive accuracy of the minority class without largely decreasing the predictive accuracy of the majority class. SMOTE and simple undersampling perform similarly for PAM (with variable selection) and DLDA; similar results were obtained by others also for lowdimensional data [8]. Sometimes SMOTE performs better than simple undersampling for kNN (with variable selection). Our results are in agreement with the finding that SMOTE had little or no effect on SVM when data were highdimensional [19].
The results showing that simple undersampling ourperforms SMOTE might seem surprising, as this method uses only a small subset of the data. In practice undersampling is effective in removing the gap between the classspecific predictive accuracies for highdimensional data [7] and it is often used as a reasonable baseline for algorithmic comparison [35]. One of its shortcomings is the large variability of its estimates, which can be reduced by bagging techniques that use multiple undersampled training sets. We previously observed that bagged undersampling techniques outperform simple undersampling for highdimensional data, especially when the classimbalance is extreme [7]. Others showed that bagged undersampling techniques outperformed SMOTE for SVM with highdimensional data [19]. Therefore, we expect that the classification results presented in this paper could be further improved by the use of bagged undersampling methods.
We devoted a lot of attention to studying the performance of SMOTE in the situation where there was no difference between the classes or where most of the variables did not differ between classes. We believe that in this context these situations are extremely relevant. It is well known that most of the problems arising from learning on classimbalanced data arise in the region where the two classspecific densities overlap. When the difference between the classspecific densities is large enough, the classimbalance does not cause biased classification for the classifiers that we considered, even in the highdimensional setting [7]. The other reason is that when a very large number of variables is measured for each subject, in most situations the vast majority of variables do not differentiate the classes and the signaltonoise ratio can be extreme. For example, Sotiriou et al. [36] identified 606 out of the 7,650 measured genes as discriminating ER+ from ER samples in their gene expression study; at the same time ER status was the known clinicopathological breast cancer phenotype for which the largest number of variables was identified (137 out of the 7,650 genes discriminated grade, 11 out of the 7,650 node positivity, 3 out of the 7,650 tumor size and 13 out of the 7,650 menopausal status). Similar results can be found in most gene expression microarray studies, where rarely more than few hundreds of genes differentiate the classes of interest. Furthermore, the results from the simulation studies where all the variables were differentially expressed were consistent with those obtained when only few variables differentiated the classes, indicating that our conclusions are not limited to sparse highdimensional data.
Variable selection is generally advisable for highdimensional data, as it removes some of the noise from the data [37]. SMOTE does not affect the ranking of variables if the variable selection method is based on classspecific means and variances. For example, when variable selection is based on a twosample ttest and a fixed number of variables are selected, as in our simulations, the same results are obtained if variable selection is performed before or after using SMOTE. However, the results obtained by performing variable selection on SMOTEaugmented data must be interpreted with great care. For example, the pvalues of a twosample ttest are underestimated and should not be interpreted other than for ranking purposes: if the number of variables to select depends on a threshold on the pvalues it will appear that many variables are significantly different between the classes. Another reason of concern is that SMOTE introduces some correlation between the samples and most variable selection methods (as well as some classifiers) assume the independence among samples.
Many variants of the original version of SMOTE exist, however in this paper we only considered the original version of SMOTE. The variants of SMOTE are very similar in terms of the expected value and variance of the SMOTE samples, as well as the expected value and variance of the Euclidean distance between new samples and samples from the SMOTEaugmented data set. Under the null hypothesis all the theoretical results presented in this paper would apply also for BorderlineSMOTE [22] and SafeLevelSMOTE [23]. Further research would be needed to assess the performance of these algorithms for highdimensional data when there is some difference between the classes.
We considered only a limited number of simple classification methods, which are known to perform well in the highdimensional setting, where the use of simple classifiers is generally recommended [37]. Our theoretical and empirical results suggest that many different types of classifiers do not benefit from SMOTE if data are highdimensional; the only exception that we identified are the kNN classifiers. It is however possible that also in the highdimensional setting SMOTE might be more beneficial for some classifiers that were not included in our study.
Conclusions
SMOTE is a very popular method for generating synthetic samples that can potentially diminish the classimbalance problem. We applied SMOTE to highdimensional classimbalanced data (both simulated and real) and used also some theoretical results to explain the behavior of SMOTE. The main findings of our analysis are:

in the lowdimensional setting SMOTE is efficient in reducing the classimbalance problem for most classifiers;

SMOTE has hardly any effect on most classifiers trained on highdimensional data;

when data are highdimensional SMOTE is beneficial for kNN classifiers if variable selection is performed before SMOTE;

SMOTE is not beneficial for discriminant analysis classifiers even in the lowdimensional setting;

undersampling or, for some classifiers, cutoff adjustment are preferable to SMOTE for highdimensional classprediction tasks.
Even though SMOTE performs well on lowdimensional data it is not effective in the highdimensional setting for the classifiers considered in this paper, especially in the situations where signaltonoise ratio in the data is small.
Methods
Notation
Let x_{ ij } be the value of jth variable (j = 1, ..., p) for the ith sample (i = 1, ..., n) that belongs to Class c (c = 1 or 2), k_{ c } = n_{ c } / n is the proportion of samples from Class c and n_{ c } is the number of samples in class c. Let the sample size of the minority class be denoted by n_{ min }. Let us say we limit our attention to G ≤ p variables that are the most informative about the class distinction. Capital letters (as X) denote random variables while lowercase letters (as x) denote observations; bold letters (x) indicate set of variables. The Gaussian distribution with mean μ and standard deviation σ is indicated with N(μ, σ) and the uniform distribution defined on [0,1] with U(0, 1).
SMOTE
where u was randomly chosen from U(0, 1). u was the same for all variables, but differed for each SMOTE sample; this choice guarantees that the SMOTE sample lies on the line joining the two original samples used to generate it [2, 9]. By SMOTEaugmenting the minority class we obtained a classbalanced training set, as suggested in [8].
Simple undersampling
Simple undersampling (downsizing) consists of obtaining a classbalanced training set by removing a subset of randomly selected samples from the larger class [2]. The undersampled training set can be considerably smaller than the original training set if the classimbalance is large. Simple undersampling was used only for the analysis of the experimental data sets.
Cutoff adjustment
We attempted to adjust for the classimbalance by changing the classification threshold of the classifiers. For each classifier we estimated the posterior probability of classification in Class 1 for the new samples ($\hat{p}(c=1{\mathbf{x}}^{\ast})$). The classification rule was then defined as: classify at random if p(c = 1x^{∗}) = k_{1}, classify to Class 1 when $\hat{p}(c=1{\mathbf{x}}^{\ast})>{k}_{1}$ and to Class 2 otherwise. (Note that the uncorrected classifiers use the threshold value of 0.5 for any level of class imbalance.)
Data simulation of highdimensional data
We simulated p = 1, 000 variables for each of n = 100 samples. The variables were simulated under a block exchangeable correlation structure, in which the 10 variables within each block had a pairwise correlation of ρ = 0.8, 0.5, 0.2 or 0 (independence case), while the variables from different blocks were independent [38]. The data set was split into a training set (n_{ train } = 80) and a balanced test set (n_{ test } = 20). Different levels of classimbalance were considered for the training sets, varying the proportion of samples from Class 1 from k_{1} = 0.05 to 0.95.
Under the null case the class membership was randomly assigned and all the variables were simulated from N(0, 1). Under the alternative case, the class membership was dependent on the values of p_{ DE } = 20 nonnull variables, generated from N(0, 1) in Class 1 and from N(μ^{(2)}, 1) in Class 2 (μ^{(2)} = 0.5, 0.7, 1, 2); the remaining variables were simulated as in the null case. We considered also a situation where all variables were differentially expressed. In this setting we used μ^{(2)} = 0.2, which assured a similar predictive power as in the situation where we used sparse data and moderate differences between the classes (p_{ DE } = 20 and μ^{(2)} = 1).
We performed also a limited set of simulations where all the variables were simulated from the exponential distribution with rate equal to one. In the alternative case a number randomly generated from U(1, 1.5) was added to the p_{ DE } = 20 nonnull variables in Class 2.
Each simulation was repeated 1,000 times and overall more than 11 million classifiers were trained.
Data simulation of lowdimensional data
We performed also a limited number of simulations where data were lowdimensional. We simulated and used p = G = 5 or 10 variables and varied the size of the training set (n_{ train } = 40, 80 and 200), keeping the level of classimbalance fixed (k_{1} = 0.10). The test sets were balanced (n_{ test } = 40). All the variables were correlated (ρ = 0.8) and simulated as described for the highdimensional data (μ^{(2)} = 1 for the alternative case).
Data normalization, variable selection and derivation of the classifiers
We evaluated the effect of data normalization, developing classification rules (i) using raw data (x_{ ij }), (ii) normalizing the samples (${x}_{\mathit{\text{ij}}}^{s}={x}_{\mathit{\text{ij}}}\frac{1}{p}\sum _{k=1}^{p}{x}_{\mathit{\text{ik}}}$) and (iii) normalizing the variables (${x}_{\mathit{\text{ij}}}^{v}={x}_{\mathit{\text{ij}}}\frac{1}{n}\sum _{k=1}^{n}{x}_{\mathit{\text{kj}}})$. Normalization was performed separately on the training and test set, before variable selection or augmentation of the training set. Data normalizatoin was not performed when all the variables were differentially expressed.
We used all the variables (p = G) or selected G = 40 variables with the largest absolute tstatistics derived from the two sample ttest with assumed equal variances; variable selection was performed on the training set, either before or after using SMOTE but only after using undersampling (this strategy outperforms variable selection before undersampling [7]).
The classification rules were derived completely on the training set, using seven types of classification methods: kNN with k = 1, 3 or 5, discriminant analysis (DLDA and DQDA), RF, SVM, PAM, penalized logistic regression (PLR) with linear penalty (PLRL1) and quadratic penalty (PLRL2) and CART. For CART we used pruning, the maximum depth of any node of the final tree was set to 5 and the complexity parameter was 0.01. We used the penalized package to fit PLR; the penalization coefficient was optimized based on crossvalidated likelihood. The parameters used for the other classifiers were the same as in [7], where the classifiers are shortly described.
Evaluation of the performance of the classifiers
The classifiers were evaluated on the independent test sets, using five performance measures: (i) overall predictive accuracy (PA, the number of correctly classified samples from the test set divided by the total number of samples in the test set), (ii) predictive accuracy of Class 1 (P A_{1}), (iii) predictive accuracy of Class 2 (P A_{2}), (iv) Area Under the ReceiverCharacteristicOperating Curve (AUC [39]) and (v) Gmean ($\sqrt{P{A}_{1}\xb7P{A}_{2}}$). We used the function sommers2 in the Hmisc package to compute the AUC.
Experimental data sets
We considered three breast cancer gene expression data sets [36, 40, 41] and two classification tasks for each of them: prediction of estrogen receptor status (ER+ or ER) and prediction of grade of tumors (grade 1 and 2 or grade 3). Data were preprocessed as described in the original publications. The number of variables varied from 7,650 to 22,283, the number of samples from 99 to 249, and the proportion of minority class samples from 0.14 to 0.45 (Table 2).
The classifiers were trained with G=40 variables, using SMOTE, simple undersampling, the uncorrected classifiers or adjusted classification threshold. Their performance was assessed with leaveoneout cross validation. To take the sampling variability into account, each classifier was trained using 50 different SMOTEaugmented or undersampled training sets. Overall, 10,878 classifiers were trained, and their performance was assessed training about one million classifiers on crossvalidated training sets.
Additionally, to isolate the effect of classimbalance, we used the Sotiriou data and obtained different levels of classimbalance in the training set by including a randomly chosen subset of the samples in the analyses. The training sets contained a fixed number of samples in the minority class (5 or 10 ER or grade 3 samples), while the number of samples of the majority class varied; the classimbalance of the training sets ranged from k_{1}=0.50 to 0.90 at most, while the test sets were classbalanced. The analysis was replicated 500 times for each level of classimbalance, randomly selecting the samples to include in the training and test set and using SMOTE or no correction; G=40 variables were selected at each iteration. The results were presented as average overall and classspecific PA.
Analysis
Analyses and simulations were carried out using R 2.8.1 [42].
Abbreviations
 SMOTE:

Synthetic minority oversampling technique
 CART:

Classification and regression trees
 PA:

Predictive accuracy
 PA1:

Predictive accuracy for Class 1
 PA2:

Predictive accuracy for Class 2
 kNN:

Nearest neighbor classifier with k neighbors
 DLDA:

Diagonal linear discriminant analysis
 DQDA:

Diagonal quadratic discriminant analysis
 RF:

Random forests
 SVM:

Support vector machines
 PAM:

Prediction analysis of microarrays
 PLR:

Penalized logistic regression
 LOOCV:

Leaveoneout crossvalidation
 ER:

Estrogen receptor
 ER+:

Positive estrogen receptor
 ER:

Negative estrogen receptor.
Declarations
Acknowledgements
The highperformance computation facilities were kindly provided by Bioinformatics and Genomics Unit at Department of Molecular Biotechnology and Heath Sciences, University of Torino, Italy.
Authors’ Affiliations
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