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TL;DR

See title.


Motivation

I am hoping for a canonical answer along the lines of "(1) No, (2) Not applicable, because (1)", which we can use to close many wrong questions about unbalanced datasets and oversampling. I would be quite as happy to be proven wrong in my preconceptions. Fabulous Bounties await the intrepid answerer.


My argument

I am baffled by the many questions we get in the tag. Unbalanced classes seem to be self-evidently bad. And the minority class(es) is quite as self-evidently seen as helping to address the self-evident problems. Many questions that carry both tags proceed to ask how to perform oversampling in some specific situation.

I understand neither what problem unbalanced classes pose, nor how oversampling is supposed to address these problems.

In my opinion, unbalanced data do not pose a problem at all. One should model class membership probabilities, and these may be small. As long as they are correct, there is no problem. One should, of course, not use accuracy as a KPI to be maximized in a classification problem. Or calculate classification thresholds. Instead, one should assess the quality of the entire predictive distribution using proper . Tetlock's Superforecasting serves as a wonderful and very readable introduction to predicting unbalanced classes, even if this is nowhere explicitly mentioned in the book.


Related

The discussion in the comments has brought up a number of related threads.

IcannotFixThis' answer, seems to presume (1) that the KPI we attempt to maximize is accuracy, and (2) that accuracy is an appropriate KPI for classification model evaluation. It isn't. This may be one key to the entire discussion.

AdamO's answer focuses on the low precision of estimates from unbalanced factors. This is of course a valid concern and probably the answer to my titular question. But oversampling does not help here, any more than we can get more precise estimates in any run-of-the-mill regression by simply duplicating each observation ten times.


Summary

The threads above can apparently be summarized as follows.

  • Rare classes (both in the outcome and in predictors) are a problem, because parameter estimates and predictions have high variance/low precision. This cannot be addressed through oversampling. (In the sense that it is always better to get more data that is representative of the population, and selective sampling will induce bias per my and others' simulations.)
  • Rare classes are a "problem" if we assess our model by accuracy. But accuracy is not a good measure for assessing classification models. (I did think about including accuracy in my simulations, but then I would have needed to set a classification threshold, which is a closely related wrong question, and the question is long enough as it is.)

An example

Let's simulate for an illustration. Specifically, we will simulate ten predictors, only a single one of which actually has an impact on a rare outcome. We will look at two algorithms that can be used for probabilistic classification: and .

In each case, we will apply the model either to the full dataset, or to an oversampled balanced one, which contains all the instances of the rare class and the same number of samples from the majority class (so the oversampled dataset is smaller than the full dataset).

For the logistic regression, we will assess whether each model actually recovers the original coefficients used to generate the data. In addition, for both methods, we will calculate probabilistic class membership predictions and assess these on holdout data generated using the same data generating process as the original training data. Whether the predictions actually match the outcomes will be assessed using the Brier score, one of the most common proper scoring rules.

We will run 100 simulations. (Cranking this up only makes the beanplots more cramped and makes the simulation run longer than one cup of coffee.) Each simulation contains $n=10,000$ samples. The predictors form a $10,000\times 10$ matrix with entries uniformly distributed in $[0,1]$. Only the first predictor actually has an impact; the true DGP is

$$ \text{logit}(p_i) = -7+5x_{i1}. $$

This makes for simulated incidences for the minority TRUE class between 2 and 3%:

training_incidence

Let's run the simulations. Feeding the full dataset into a logistic regression, we (unsurprisingly) get unbiased parameter estimates (the true parameter values are indicated by the red diamonds):

logistic_coefficients

However, if we feed the oversampled dataset to the logistic regression, the intercept parameter is heavily biased:

logistic_coefficients_oversampled

Let's compare the Brier scores between models fitted to the "raw" and the oversampled datasets, for both the logistic regression and the Random Forest. Remember that smaller is better:

logistic_regression_Brier_scores

Random_Forest_Brier_score

In each case, the predictive distributions derived from the full dataset are much better than those derived from an oversampled one.

I conclude that unbalanced classes are not a problem, and that oversampling does not alleviate this non-problem, but gratuitously introduces bias and worse predictions.

Where is my error?


A caveat

I'll happily concede that oversampling has one application: if

  1. we are dealing with a rare outcome, and
  2. assessing the outcome is easy or cheap, but
  3. assessing the predictors is hard or expensive

A prime example would be genome-wide association studies (GWAS) of rare diseases. Testing whether one suffers from a particular disease can be far easier than genotyping their blood. (I have been involved with a few GWAS of PTSD.) If budgets are limited, it may make sense to screen based on the outcome and ensure that there are "enough" of the rarer cases in the sample.

However, then one needs to balance the monetary savings against the losses illustrated above - and my point is that the questions on unbalanced datasets at CV do not mention such a tradeoff, but treat unbalanced classes as a self-evident evil, completely apart from any costs of sample collection.


R code

    library(randomForest)
    library(beanplot)
    
    nn_train <- nn_test <- 1e4
    n_sims <- 1e2
    
    true_coefficients <- c(-7, 5, rep(0, 9))
    
    incidence_train <- rep(NA, n_sims)
    model_logistic_coefficients <- 
         model_logistic_oversampled_coefficients <- 
         matrix(NA, nrow=n_sims, ncol=length(true_coefficients))
    
    brier_score_logistic <- brier_score_logistic_oversampled <- 
      brier_score_randomForest <- 
    brier_score_randomForest_oversampled <- 
      rep(NA, n_sims)
    
    pb <- winProgressBar(max=n_sims)
    for ( ii in 1:n_sims ) {
        setWinProgressBar(pb,ii,paste(ii,"of",n_sims))
        set.seed(ii)
        while ( TRUE ) {    # make sure we even have the minority 
                            # class
            predictors_train <- matrix(
              runif(nn_train*(length(true_coefficients) - 1)), 
                  nrow=nn_train)
            logit_train <- 
             cbind(1, predictors_train)%*%true_coefficients
            probability_train <- 1/(1+exp(-logit_train))
            outcome_train <- factor(runif(nn_train) <= 
                     probability_train)
            if ( sum(incidence_train[ii] <- 
               sum(outcome_train==TRUE))>0 ) break
        }
        dataset_train <- data.frame(outcome=outcome_train, 
                          predictors_train)
        
        index <- c(which(outcome_train==TRUE),  
          sample(which(outcome_train==FALSE),   
                sum(outcome_train==TRUE)))
        
        model_logistic <- glm(outcome~., dataset_train, 
                    family="binomial")
        model_logistic_oversampled <- glm(outcome~., 
              dataset_train[index, ], family="binomial")
        
        model_logistic_coefficients[ii, ] <- 
               coefficients(model_logistic)
        model_logistic_oversampled_coefficients[ii, ] <- 
          coefficients(model_logistic_oversampled)
        
        model_randomForest <- randomForest(outcome~., dataset_train)
        model_randomForest_oversampled <- 
          randomForest(outcome~., dataset_train, subset=index)
        
        predictors_test <- matrix(runif(nn_test * 
            (length(true_coefficients) - 1)), nrow=nn_test)
        logit_test <- cbind(1, predictors_test)%*%true_coefficients
        probability_test <- 1/(1+exp(-logit_test))
        outcome_test <- factor(runif(nn_test)<=probability_test)
        dataset_test <- data.frame(outcome=outcome_test, 
                         predictors_test)
    
        prediction_logistic <- predict(model_logistic, dataset_test, 
                                        type="response")
        brier_score_logistic[ii] <- mean((prediction_logistic - 
               (outcome_test==TRUE))^2)
    
        prediction_logistic_oversampled <-      
               predict(model_logistic_oversampled, dataset_test, 
                        type="response")
        brier_score_logistic_oversampled[ii] <- 
          mean((prediction_logistic_oversampled - 
                (outcome_test==TRUE))^2)
        
        prediction_randomForest <- predict(model_randomForest, 
            dataset_test, type="prob")
        brier_score_randomForest[ii] <-
          mean((prediction_randomForest[,2]-(outcome_test==TRUE))^2)
    
        prediction_randomForest_oversampled <-   
                         predict(model_randomForest_oversampled, 
                                  dataset_test, type="prob")
        brier_score_randomForest_oversampled[ii] <- 
          mean((prediction_randomForest_oversampled[, 2] - 
                (outcome_test==TRUE))^2)
    }
    close(pb)
    
    hist(incidence_train, breaks=seq(min(incidence_train)-.5, 
            max(incidence_train) + .5),
      col="lightgray",
      main=paste("Minority class incidence out of", 
                    nn_train,"training samples"), xlab="")
    
    ylim <- range(c(model_logistic_coefficients, 
                   model_logistic_oversampled_coefficients))
    beanplot(data.frame(model_logistic_coefficients),
      what=c(0,1,0,0), col="lightgray", xaxt="n", ylim=ylim,
      main="Logistic regression: estimated coefficients")
    axis(1, at=seq_along(true_coefficients),
      c("Intercept", paste("Predictor", 1:(length(true_coefficients) 
             - 1))), las=3)
    points(true_coefficients, pch=23, bg="red")
    
    beanplot(data.frame(model_logistic_oversampled_coefficients),
      what=c(0, 1, 0, 0), col="lightgray", xaxt="n", ylim=ylim,
      main="Logistic regression (oversampled): estimated 
              coefficients")
    axis(1, at=seq_along(true_coefficients),
      c("Intercept", paste("Predictor", 1:(length(true_coefficients) 
             - 1))), las=3)
    points(true_coefficients, pch=23, bg="red")
    
    beanplot(data.frame(Raw=brier_score_logistic, 
            Oversampled=brier_score_logistic_oversampled),
      what=c(0,1,0,0), col="lightgray", main="Logistic regression: 
             Brier scores")
    beanplot(data.frame(Raw=brier_score_randomForest, 
      Oversampled=brier_score_randomForest_oversampled),
      what=c(0,1,0,0), col="lightgray", 
              main="Random Forest: Brier scores")
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    $\begingroup$ I've also ran the same simulation, with an even wider selection of models, and a wider range of prior class probabilities, and observed the same results. Additionally, if you measure the AUC of your models, you'll notice that they are all the same, regardless of the class balance of your training data. I wonder about the source of this wide conception on the evils of class balance, where did it come from, how did we get to this point? $\endgroup$ Jul 16, 2018 at 21:44
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    $\begingroup$ Honestly, knowing there is someone else out there that is mystified by the endless class balancing questions is comforting. $\endgroup$ Jul 16, 2018 at 21:52
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    $\begingroup$ "How did we get here?" is a great question. I don't know the definitive answer. But my hunch is that this all started when the machine learning community was only concerned with accuracy. Eventually someone pointed out that stupidly high accuracy can be achieved if (1) your classes are severely imbalanced and (2) you predict the majority class. Instead of measuring model quality with a metric other than accuracy, oversampling/SMOTE/etc were all invented to "solve" this problem. This isn't a history, just a story I made up based on my impressions and observable evidence. $\endgroup$
    – Sycorax
    Jul 16, 2018 at 21:55
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    $\begingroup$ I think a large part of this comes from "big data." For rare events, you need a lot of data, and perhaps before (say, 20 years ago), we saw less class imbalance because you'd have laughably few positive examples in your dataset, hence wouldn't even try using it. Nowedays you might easily have a dataset with millions of rows and say, a few hundred positive examples. $\endgroup$
    – Alex R.
    Jul 16, 2018 at 22:04
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    $\begingroup$ @Sycorax Don't get me started on the sklearn's developers decision to map the predict method on models to the hard decision rule thresholding the probabilities at 0.5. $\endgroup$ Jul 16, 2018 at 22:11

2 Answers 2

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I'd like to start by seconding a statement in the question:

... my point is that the questions on unbalanced datasets at CV do not mention such a tradeoff, but treat unbalanced classes as a self-evident evil, completely apart from any costs of sample collection.

I also have the same concern, my questions here and here are intended to invite counter-evidence that it is a "self-evident evil" the lack of answers (even with a bounty) suggests it isn't. A lot of blog posts and academic papers don't make this clear either. Classifiers can have a problem with imbalanced datasets, but only where the dataset is very small, so my answer is concerned with exceptional cases, and does not justify resampling the dataset in general.

There is a class imbalance problem, but it is not caused by the imbalance per se, but because there are too few examples of the minority class to adequately describe it's statistical distribution. As mentioned in the question, this means that the parameter estimates can have high variance, which is true, but that can give rise to a bias in favour of the majority class (rather than affecting both classes equally). In the case of logistic regression, this is discussed by King and Zeng,

3 Gary King and Langche Zeng. 2001. “Logistic Regression in Rare Events Data.” Political Analysis, 9, Pp. 137–163. https://j.mp/2oSEnmf

[In my experiments I have found that sometimes there can be a bias in favour of the minority class, but that is caused by wild over-fitting where the class-overlap dissapears due to random sampling, so that doesn't really count and (Bayesian) regularisation ought to fix that]

The good thing is that MLE is asymptotically unbiased, so we can expect this bias against the minority class to go away as the overall size of the dataset increases, regardless of the imbalance.

As this is an estimation problem, anything that makes estimation more difficult (e.g. high dimensionality) seems likely to make the class imbalance problem worse.

Note that probabilistic classifiers (such as logistic regression) and proper scoring rules will not solve this problem as "popular statistical procedures, such as logistic regression, can sharply underestimate the probability of rare events" 3. This means that your probability estimates will not be well calibrated, so you will have to do things like adjust the threshold (which is equivalent to re-sampling or re-weighting the data).

So if we look at a logistic regression model with 10,000 samples, we should not expect to see an imbalance problem as adding more data tends to fix most estimation problems.

So an imbalance might be problematic, if you have an extreme imbalance and the dataset is small (and/or high dimensional etc.), but in that case it may be difficult to do much about it (as you don't have enough data to estimate how big a correction to the sampling is needed to correct the bias). If you have lots of data, the only reason to resample is because operational class frequencies are different to those in the training set or different misclassification costs etc. (if either are unknown or variable, your really ought to use a probabilistic classifier).

This is mostly a stub, I hope to be able to add more to it later.

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  • $\begingroup$ Thank you, I am looking forward to your expanding this. If I understand you correctly, the class imbalance problem you see is high variance of parameter estimates, right? It seems to me that oversampling etc. would not address this, correct? $\endgroup$ Jan 5 at 14:00
  • $\begingroup$ @StephanKolassa in principle it can, but you need to know the right amount of oversampling to apply (or equivalently reweighting or threshold adjustment), which is going to be difficult if you don't have enough data to estimate the model in the first place. King takes the threshold adjustment approach, as it can be analytically approximated for logistic regression, but it is not clear it has great practical utility. $\endgroup$ Jan 5 at 14:05
  • $\begingroup$ @StephanKolassa I think I didn't give a very direct answer to your question. I think what is happening is that the variance in the parameter estimates causes the undue bias against the minority class because of the structure of the problem. Reducing the variance ought to reduce the bias, but resampling (or better re-weighting the data in the loss) can address the bias directly. However, it will be difficult to estimate how much regularisation or re-sampling/re-weighting is required in practice. $\endgroup$ Jan 8 at 9:26
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    $\begingroup$ I just finished reading the King & Zeng paper, thank you for drawing my attention to it! It was most illuminating, and I have learned something today. (To add to your comments, they add one additional reason in favor of nonrandom sampling: the costs of collecting data.) $\endgroup$ Jan 9 at 13:09
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    $\begingroup$ My intuition is that logistic regression is likely to be more robust to this sort of bias than most, and that some classifiers may be more susceptible to problems in practical applications (but still with small datasets) and that is perhaps why there is some perception that class imbalance is a problem. $\endgroup$ Jan 9 at 13:15
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I fully agree with a marked answer "No problem of imbalance per se" - the problem is Lack of data - when minority class do not form Gaussian distribution & thus minority class just remains to be the outlier in the overall distribution == this is problem for Unsupervised methods.

But if are making Supervised learning, knowing target classes & their characteristics (features) in advance -- imbalanced ds could not be a problem IF deal it correctly. -- at least each Cross-Validated subsample (e.g. in ensemble bagging & boosting) should include representatives of both (source) classes for comprehensive approximation of decision boundary

Main problem of minority class: "the model cannot model the boundary of these low-density regions well during the learning, resulting in ambiguity and poor generalization". Solution was found in (1) semi-supervised learning method, which generates pseudo-labels on unlabeled data and then train together, OR (2) (when extreme imbalance, as medical) self-supervised pre-training with further main training

So, decrease of IMBALANCE Problem can be achieved by means of programming logics or (better) by increasing qty of samples == even without oversampling (artificial injection of fake minority class samples)

P.S. some hints:

first - "Exactly like we should do feature selection inside the cross validation loop, we should also oversample inside the loop." (source)

second "Re-balancing makes sense only in the training set, so as to prevent the classifier from simply and naively classifying all instances as negative for a perceived accuracy of 99%."

third "when comparing two binary classifiers, the AUC is one of the criteria that should not fooled by the imbalancedness of the data."

forth as alternative - Weighted Cost/Loss Function - "Thanks to the Sklearn, there is a built-in parameter called class_weight in most of the ML algorithms which helps you to balance the contribution of each class." - e.g weighted Sigmoid Cross-Entropy loss for binary CLF

fifth see 8th at link - "be creative"

sixth Threshold moving & Searching optimal value from a grid

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    $\begingroup$ I'm afraid I do not understand this answer at all. The unusual punctuation doesn't help, but I suspect I wouldn't follow it even otherwise. Could you edit it to make your point more clearly? $\endgroup$
    – mkt
    Jul 30 at 5:54
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    $\begingroup$ The point about only resampling the training data is a good one. I have often wondered whether papers promoting SMOTE give a false impression of how well it works because of information leaking from test set to training set via the synthetic examples. We want to know how well the model will work in operational conditions, so the test set should be representative of test conditions (where you won't be using SMOTE) $\endgroup$ Jul 30 at 7:05
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    $\begingroup$ I am confused by the comment about the minority class not being Gaussian. $\endgroup$
    – Dave
    Jul 30 at 11:07
  • $\begingroup$ when exploring rare events & having lack of data - of course you cannot see on KDE-charts normal distribution for both classes, only for majority class. $\endgroup$
    – JeeyCi
    Jul 30 at 12:10
  • $\begingroup$ But if you cannot even draw your samples && see the decision boundary as is on figure - there is no sence for further automatization of its probabilistic estimation with mathematical tools (such as Bayesian logics). If there is no statistically meaningfull data for your research - Bayes will not give you trustfull probability of boundary between true & false cases. I prefer to see apriory: If not seen in KDE-charts - there is nothing for binary division mathematically further... (suppose rare medical disease) $\endgroup$
    – JeeyCi
    Jul 30 at 12:12

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