Introduction

Illustrations of the (perceived) problems and offered remedies appear in numerous parts of the ML literature, ranging from Web tutorials to the research literature. Major packages, such as caret, parsnip and mlr3, also offer remedies.

All of these sources recommend that you artificially equalize the class counts in your data, via various resampling methods. Say for instance we are in the two-class case, and have class sizes of 20000 and 80000 for class 1 and class 0, respectively. Here are some ways to rebalance.

• Randomly discard 60000 data points from class 0.

• Resample 80000 data points from the 20000, with replacement.

• Using distribution approximation methods such as SMOTE, resample 80000 from the 20000.

Arguably, though, this is generally inadvisable, indeed harmful, for several reasons:

• Undersampling is clearly problematic: Why throw away data? Discarding data weakens our ability to predict new cases.

• The data may be unbalanced for a reason. Thus the imbalance itself is useful information, again resulting in reduced predictive power if it is ignored.

• There are principled alternatives to resampling, including an adjustment formula to be presented here.

In other words:

Resampling methods are both harmful and unnecessay.

Motivating examples

Terminology

We refer to the class probabilities for given feature values as conditional class probabilities, because they are probabilities subject to conditions. If say we wish to classify a patient as to whether she has a certain disease or not, and we know her blood test value on some measure is 2.39, the latter is the condition. Among all patients with that blood test value, what proportion of them have the disease?

The overall class probabilities are unconditional class probabilities.

The conditional and unconditional class probabilities are often referred to as the posterior and prior probabilities. This sounds Bayesian, and Bayes’ Rule, but there is no subjectivity involved; these are real probabilities. E.g. in a disease classification application, there is a certain proportion of people in the population who have the disease. This is a prior probability, which we will estimate from our training data.

Notation

• c = number of classes

• Y = class label, 0,1,…,c-1

• X = vector of features

• p_i = P(Y = i) (prior probs.)

• q_i(t) = P(Y = i | X = t) (posterior probs.)

• Y_pred = the value we predict for Y

• (X_i,Y_i), i=1,2,…,n: training data

• f_i: (conditional) density of X within class i

• l_ij: loss incurred by guessing class i when the true class is j

2-class case (Y = 0,1):

• p: P(Y = 1) (probability of class 1)

• r_i: estimated q₁(X_i)

Key issue: How were the data generated?

The examples above illustrate two important cases:

• The fraud data is imbalanced, but naturally so. Assuming the two-day data collection period was typical, the population class probability for fraud will be about what we see in the data, 0.172%.

• The LetterRecognition data is balanced, but only artificially so. The curator of the dataset wanted the data to have about the same number of instances of each letter. But in general English usage, letters occur with quite different frequencies:

E 12.02%
T 9.10%
A 8.12%
O 7.68%
I 7.31%
N 6.95%
…
…
Q 0.11
J 0.10
Z 0.07

One can obtain these numbers in the regtools package, included by ‘qeML’. :

> data(ltrfreqs)
> lf <- ltrfreqs
> lf <- ltrfreqs[,2] / 100
> names(lf) <- ltrfreqs[,1]
# example
> lf['A']
     A 
0.0812 

What your ML algorithm is thinking

ML algorithms take your data literally. Say you have a two-class ⊕The term population is sometimes only conceptual, but in all cases we believe our data are generated randomly in some manner, and the term refers to the associated distribution. setting, for Classes 0 and 1. If about 1/2 your data is Class 1, then the algorithm, whether directly or indirectly, operates under the assumption that the true population class probabilities are each about 0.5.

In the LetterRecognition data, since the sampling was actually designed to have about the same number of instances for each letter, the algorithm you use will then assume the true probabilities of the letters are about 1/26 each. We know that is false, as the table shown earlier illustrates.

So, if your sampling scheme artificially creates balanced data, as in the LetterRecognition data, or if you do resampling to make your data balanced, as is commonly recommended, you are fooling your ML algorithm.

In the following, let’s focus on the 2-class case for simplicity.

Artificial balance will not achieve our goals

In fooling your algorithm, it will generate the wrong conditional class-1 probabilities r_i in our notation above. And whether we wish to minimize the overall probability of misclassification, or expected loss, or any other criterion, the algorithm will (again, directly or indirectly) rely on the values of r_i.

Consider the fraud example, in which the data are highly imbalanced but in which wrongly guessing the large class carries heavy penalties for us. Recall our notation l₀₁ from above. As noted, this may be difficult to quantify, but for the moment, let’s suppose we can do so. What are the implications in terms of artificially balancing the data?

Actually, if we are going to be doing any adjustment of class sizes in our data, we should make the fraud class larger than the non-fraud class, not of equal size. How much larger will depend on the value of l₀₁, but in any case, balancing the data will be wrong.

Frank Harrell says it well:

For this reason the odd practice of subsampling the controls is used in an attempt to balance the frequencies and get some variation that will lead to sensible looking classifiers (users of regression models would never exclude good data to get an answer). Then they have to, in some ill-defined way, construct the classifier to make up for biasing the sample. It is simply the case that a classifier trained to a 1⁄2 [q = 1/2] prevalence situation will not be applicable to a population with a 1⁄1000 [p = 1/1000] prevalence.

So, what SHOULD be done?

We will focus here on the 2-class class. Let r_new denote the estimated conditional class probability q_new = P(Y_new = 1 | X_new) for a new case to be classified.

Note that (a) as always, the r values are only sample estimates, subject to sampling variability, and (b) not all qe-series functions are capable of calculating the r values.

Clearly, one’s course of action should center around the value of ⊕Both approaches will be informal. The reader is urged to avoid “automatic” solutions, as proper action is highly dependent on the specific application. r_new. Specifically, how should we use it? We will discuss two approaches:

1. Use of the ROC curve, which is derived from the r_i values.

2. Direct consideration of the r_new value.

Approach 1: use the ROC curve

Here one considers various threshhold values h, where we guess class 1 if r_new > h, 0 otherwise. The value one chooses for h will determine the True Positive Rate and False Positive Rate:

TPR(h) = P(Y_pred = 1 | Y = 1) = ∫_{t > h} f₁(t) dt

FPR(h) = P(Y_pred = 1 | Y = 0) = ∫_{t > h} f₀(t) dt

The ROC curve is then a graph of TPR vs. FPR. As we vary h, it traces out the ROC curve. Some R packages, such as ROCR, colorize the curve, with the color showing the value of h. The qeML function qeROC (which wraps the ROCR package) does this.

For instance, let’s look at the Missed Appointments Dataset. Here are the code and result:

> data(MissedAppts) 
> z <- qeXGBoost(MissedAppts,'No.show')
> qeROC(MissedAppts,z,'Yes')

Missed Appts ROC

Often a 45-degree line is superimposed on the ROC graph for comparison. To see why, say we are classifying patients as having a certain disease or not, say on the basis of a blood test. For a value of h associated with a point on the line, diseased and disease-free patients would have the same probability of testing positive, indicating that the test has no discriminatory value. Thus, the further above the 45-line our ROC curve is, the better our predictive power.

This can be used as an aid for the analyst to try to find a “happy medium” point for deciding whether to take action on a suspicion that the client will not show up.

For instance, if we were to set h = 0.24, then FPR and TPR would be about 0.2 and 0.4, respectively. Is this an acceptable tradeoff? The answer depends on the severity of trouble that a missed appointment brings the business and many other factors. Taking all aspects of the situation into account, the analyst can choose a value of h to use in all future such decisions.

A note on AUC: AUC is the total area under the ROC curve. As such, it is a measure of the general predictive ability of your algorithm on this data. This may seem attractive at first, but it is probably irrelevant in most applications, as it places equal weight on all possible TPR/FPR scenarios; usually we are far more interested in some settings than others. Note too that AUC values for original data vs. the artificially balanced data are not comparable.

Adjusting for incorrect/changed p_i

As noted in the LetterRecognition data example, in some cases the data are artificially balanced to begin with, due to the sampling design. Or we may have a situation like that of the Mt. Sinai Hospital radiology data, in which the class probabilities may change from one site to another, or may change over time as in the cell phone fraud example. Thus, our estimated values of the p_i will be wrong. Can we fix that?

classadjust <- function (econdprobs, wrongprob1, trueprob1) 
{
    wrongratio <- (1 - wrongprob1)/wrongprob1
    fratios <- (1/econdprobs - 1) * (1/wrongratio)
    trueratios <- (1 - trueprob1)/trueprob1
    1/(1 + trueratios * fratios)
}

}

Summary

• There’s really no need to artificially balance your data, and doing so could be harmful.

• Please don’t throw away data.

• Conversely, creating extra data by up-sampling distorts the analysis and may undermine our predictive power.

• If your data is “naturally” unbalanced, meaning it reflects population structure as in the credit card fraud example, don’t artificially force balance. Instead, choose a threshhold for conditional probabilities, and flag new cases that exceed it.

• If your data is unrealistically balanced, as in the LettersRecognition example, and the true unconditional class probabilities are known, use the adjustment formula to convert the reported unconditional probabilities to realistic ones, and classify using them.

• If your data is unrealistically balanced but the true unconditional class probabilities are unknown, recognize that your ML analysis may have only a very restricted interpretation and value.

Appendix A: derivation of the unequal-loss rule

Say the random variable W takes on the values 0,1, with P(W = 1) = r. (In the above, W plays the role of Y, conditioned on X.) Let’s compute the expected loss under two strategies:

• We guess W = 0.

Then our loss is 0 P(W = 0) + l₀₁ P(W = 1) = l₀₁ r.

• We guess W = 1.

Then our loss is 1 P(W = 0) + 0 P(W = 1) = 1-r.

So our optimal stragegy is to guess W = 1 if and only if 1-r < l₀₁ r. In other words,

Guess W = 1 if and only if r > 1/(1+l₀₁).

Appendix B: derivation of the adjustment formula

(For ease of notation etc., no distinction will be made here between sample and population quantities.)

Say there are two classes, labeled 1 and 0. Let Y denote the label and X denote the features, and say we have a new case with X = t. Then the true equation is

P(Y = 1 | X = t) = p f₁(t) / [p f₁t) + (1-p) f₀(t)]

(Eqn. 1)

where p is P(Y = 1), the true unconditional probability of Class 1, and f_i(t) is the conditional density of X within Class i. (If X is a discete random variable, substitute a probability for f.)

Rewrite the above as

P(Y = 1 | X = t) = 1 / [1 + {(1-p)/p} f₀(t) / f₁(t)]
(Eqn. 2)

Now suppose the analyst artificially changes the class counts in the data (or, as in the LetterRecognition example, the data is artificially sampled by design), with proportions q and 1-q for the two classes. In the case of artificially equalizing the class proportions, we have q = 0.5. Then the above becomes, in the eyes of your ML algorithm,

P(Y = 1 | X = t) = 1 / [1 + {(1-q)/q} f₀(t) / f₁(t)]
(Eqn. 3)

As noted earlier, what the ML algorithm is computing, directly or indirectly, is P(Y = 1 | X = t). Moreover, as also noted earlier, even caret and mlr3 do make these quantities available for the various t, so we can solve for f₀(t) / f₁(t):

f₀(t) / f₁(t) = [g(t)-1] q/(1-q)
(Eqn. 4)

where

g(t) = 1 / P(Y = 1 | X = t)

Keep in mind that the g values are the ones output by our ML algorithm. Setting t to the matrix of our feature values, we can plug t into Eqn. 4 to get the vector of f_0/f1 values, and plug the result into Eqn. 2 to obtain the vector of correct conditional probabilities.

We can now substitute in (Eqn. 2) from (Eqn. 4) to get the proper conditional probability.

The general m-class case. classes 0,1,…,m-1, actually reduces to the 2-class case, because

P(Y = i | X = t)

can be viewed as the conditional probability of class i vs. all other classes.

For each j = 1,…,c, denote by q_j and p_j the probabilities assumed by our ML algorithm, and the actual probabilities, respecitvely. For example, if the data are artificially balanced, either by resampling or by the original sampling design, we have q_j = 1/c for all j.

To guide our intuition, note first that Eqn. 2 is now

P(Y = j | X = X_i) = 1 / [1 + {(1-p_j)/p_j} [Σ_{m ≠ j} f_m(X_i)] / f_j(X_i)
Eqn. 5

Now to handle the multiclass case, for each j, j = 1,…,c and each i = 1,…,n, do:

1. Set g_ij to the reciprocal of the outputted estimated value of P(Y = j | X = X_i),

g_ij = 1 / P(Y = j | X = X_i),

2. Set

[Σ_{m ≠ j} f_m(X_i)] / f_j(X_i) = [g_ij - 1] q_j / (1 - q_j)
Eqn. 6

3. Plug the result of Eqn. 6, along with the true p_j ,into Eqn. 5 to obtain the true estimated conditional class probabilities.

Appendix C: What is really happening if you use equal class probabilities?

Say we have balanced data, so the q_i = 1/m for each i. Then in predicting the class of a new case having X = t, Equation 1 becomes

P(Y = i | X = t) = f_i(t) / [ Σ_j f_jt) ]

for i = 0,1,…,m-1, since all the 1/m factors cancel.

This shows that our guessed class is

j = arg max_i f_i(t)

In other words, you are in effect asking, “Within which class j would our data X = t be most likely?” That is, we are maximizing (say in the discrete case)

P(X = t | Y = j)

over j. That’s completely different from the question we really are interested in, “Which class j is most likely, given X = t?”, i.e. maximizing

P(Y = j | X = t )

But it does show that if we artificially equalize the class sizes, we are finding the Maximum Likelihood Estimate of j, if the p_i are unknown.

If we really don’t know the true class probabilities p_i, and artificially equalize the class sizes, we are at least getting a kind of MLE. However, what then is the practical meaning? Unless you are a subjective Bayesian (I am not), setting a flat prior, there is not much here.