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Over-sampling for imbalanced regression using Chebyshev's inequality.

Chebyshev's inequality can be used to define the probability of target observations being frequent values (w.r.t. the distribution mean).

Let \(Y\) be a random variable with finite expected value \(\overline{y}\) and non-zero variance \(\sigma^2\). For any real number \(t > 0\), the Chebyshev's inequality states that, for a wide class of unimodal probability distributions: \(Pr(|y-\overline{y}| \ge t\sigma) \le \dfrac{1}{t^2}\).

Taking \(t=\dfrac{|y-\overline{y}|}{\sigma}\), and assuming \(t > 1\), the Chebyshev’s inequality for an observation \(y\) becomes: \(P(|y - \overline{y}|=t) = \dfrac{\sigma^2}{|y-\overline{y}|}\).

Alternatively, one can use \(t\) directly to estimate a frequency weight \(\kappa = \lceil t\rceil\) and define an over-sampling strategy for extreme and rare target values1. Each incoming instance is used \(\kappa\) times to update the underlying regressor. Frequent target values contribute only once to the underlying regressor, whereas rares cases are used multiple times for training.


  • regressor (base.Regressor)

    The regression model that will receive the biased sample.


>>> from river import datasets
>>> from river import evaluate
>>> from river import imblearn
>>> from river import metrics
>>> from river import preprocessing
>>> from river import rules

>>> model = (
...     preprocessing.StandardScaler() |
...     imblearn.ChebyshevOverSampler(
...         regressor=rules.AMRules(
...             n_min=50, delta=0.01
...         )
...     )
... )

>>> evaluate.progressive_val_score(
...     datasets.TrumpApproval(),
...     model,
...     metrics.MAE(),
...     print_every=500
... )
[500] MAE: 1.152726
[1,000] MAE: 0.954873
MAE: 0.954049



Return a fresh estimator with the same parameters.

The clone has the same parameters but has not been updated with any data. This works by looking at the parameters from the class signature. Each parameter is either - recursively cloned if it's a River classes. - deep-copied via copy.deepcopy if not. If the calling object is stochastic (i.e. it accepts a seed parameter) and has not been seeded, then the clone will not be idempotent. Indeed, this method's purpose if simply to return a new instance with the same input parameters.


Fits to a set of features x and a real-valued target y.


  • x
  • y
  • kwargs




Predicts the target value of a set of features x.


  • x


The prediction.


  1. Aminian, Ehsan, Rita P. Ribeiro, and JoΓ£o Gama. "Chebyshev approaches for imbalanced data streams regression models." Data Mining and Knowledge Discovery 35.6 (2021): 2389-2466.