Sampling for Imbalanced Data in Regression

Question

There have been good questions on handling imbalanced data in the classification context, but I am wondering what people do to sample for regression.

Say the problem domain is very sensitive to the sign but only somewhat sensitive to the magnitude of the target. However the magnitude is important enough that the model should be regression (continuous target) not classification (positive vs. negative classes). And say in this problem domain that any set of training data will have 10x more negative than positive targets.

In this scenario, I might oversample the positive-target examples to match the count of negative-target examples, and then train a model to differentiate the two cases. Obviously the training approach does badly on imbalanced data, so I need to do sampling of some sort. What would be a decent way to "undo" this oversampling when making predictions? Perhaps translating by the (negative) mean or median of the target of the natural training data?

Answer 1

Imbalance is not necessarily a problem, but how you get there can be. It is unsound to base your sampling strategy on the target variable. Because this variable incorporates the randomness in your regression model, if you sample based on this you will have big problems doing any kind of inference. I doubt it is possible to "undo" those problems.

You can legitimately over- or under-sample based on the predictor variables. In this case, provided you carefully check that the model assumptions seem valid (eg homoscedasticity one that springs to mind as important in this situation, if you have an "ordinary" regression with the usuals assumptions), I don't think you need to undo the oversampling when predicting. Your case would now be similar to an analyst who has designed an experiment explicitly to have a balanced range of the predictor variables.

Edit - addition - expansion on why it is bad to sample based on Y

In fitting the standard regression model $y=Xb+e$ the $e$ is expected to be normally distributed, have a mean of zero, and be independent and identically distributed. If you choose your sample based on the value of the y (which includes a contribution of $e$ as well as of $Xb$) the e will no longer have a mean of zero or be identically distributed. For example, low values of y which might include very low values of e might be less likely to be selected. This ruins any inference based on the usual means of fitting such models. Corrections can be made similar to those made in econometrics for fitting truncated models, but they are a pain and require additional assumptions, and should only be employed whenm there is no alternative.

Consider the extreme illustration below. If you truncate your data at an arbitrary value for the response variable, you introduce very significant biases. If you truncate it for an explanatory variable, there is not necessarily a problem. You see that the green line, based on a subset chosen because of their predictor values, is very close to the true fitted line; this cannot be said of the blue line, based only on the blue points.

This extends to the less severe case of under or oversampling (because truncation can be seen as undersampling taken to its logical extreme).

# generate data
x <- rnorm(100)
y <- 3 + 2*x + rnorm(100)

# demonstrate
plot(x,y, bty="l")
abline(v=0, col="grey70")
abline(h=4, col="grey70")
abline(3,2, col=1)
abline(lm(y~x), col=2)
abline(lm(y[x>0] ~ x[x>0]), col=3)
abline(lm(y[y>4] ~ x[y>4]), col=4)
points(x[y>4], y[y>4], pch=19, col=4)
points(x[x>0], y[x>0], pch=1, cex=1.5, col=3)
legend(-2.5,8, legend=c("True line", "Fitted - all data", "Fitted - subset based on x",
    "Fitted - subset based on y"), lty=1, col=1:4, bty="n")