Skip to content


Under-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: \(P(|y - \overline{y}|=t) = \dfrac{\sigma^2}{|y-\overline{y}|}\). The reciprocal of this probability is used for under-sampling1 the most frequent cases. Extreme valued or rare cases have higher probabilities of selection, whereas the most frequent cases are likely to be discarded. Still, frequent cases have a small chance of being selected (controlled via the sp parameter) in case few rare instances were observed.


  • regressor (base.Regressor)

    The regression model that will receive the biased sample.

  • sp (float) – defaults to 0.15

    Second chance probability. Even if an example is not initially selected for training, it still has a small chance of being selected in case the number of rare case observed so far is small.

  • seed (int) – defaults to None

    Random seed to support reproducibility.


>>> 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.ChebyshevUnderSampler(
...         regressor=rules.AMRules(
...             n_min=50, delta=0.01
...         ),
...         seed=42
...     )
... )

>>> evaluate.progressive_val_score(
...     datasets.TrumpApproval(),
...     model,
...     metrics.MAE(),
...     print_every=500
... )
[500] MAE: 1.633571
[1,000] MAE: 1.460907
MAE: 1.4604



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.