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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: 1.787162
[1,000] MAE: 1.515711
MAE: 1.515236



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


  • x
  • y
  • kwargs




Predict the output 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.