ondil: Online Distributional Learning

Introduction

This package provides an online estimation of distributional regression and linear regression models in Python. We provide:

• Online linear regression models including regularization (Lasso, Ridge, Elastic Net).
• An online implementation of the generalized additive models for location, shape and scale (GAMLSS, see Rigby & Stasinopoulos, 2005) developed in Hirsch, Berrisch & Ziel, 2024.
• The multivariate extension for online distributional regression models developed in Hirsch, 2025.

All models are implemented in a way that they are fully compatible with scikit-learn estimators and transformers. The main advantage of the online approach is that the model can be updated incrementally using model.update(X, y) without the need to refit the whole model from scratch. This is especially useful for large datasets or streaming data.

Please have a look at the documentation or the example files. We're actively working on the package and welcome contributions from the community. Have a look at the Release Notes and the Issue Tracker.

Distributional Regression

The main idea of distributional regression (or regression beyond the mean, multiparameter regression) is that the response variable $Y$ is distributed according to a specified distribution $\mathcal{F}(\theta)$, where $\theta$ is the parameter vector for the distribution. In the Gaussian case, we have $\theta = (\theta_1, \theta_2) = (\mu, \sigma)$. We then specify an individual regression model for all parameters of the distribution of the form

$$g_k(\theta_k) = \eta_k = X_k\beta_k$$

where $g_k(\cdot)$ is a link function, which ensures that the predicted distribution parameters are in a sensible range (we don't want, e.g. negative standard deviations), and $\eta_k$ is the predictor. For the Gaussian case, this would imply that we have two regression equations, one for the mean (location) and one for the standard deviation (scale) parameters. Distributions other than the normal distribution are possible, and we have already implemented them, e.g., Student's $t$-distribution and Johnson's $S_U$ distribution. If you are interested in another distribution, please open an Issue.

This allows us to specify very flexible models that consider the conditional behaviour of the variable's volatility, skewness and tail behaviour. A simple example for electricity markets is wind forecasts, which are skewed depending on the production level - intuitively, there is a higher risk of having lower production if the production level is already high since it cannot go much higher than "full load" and if, the turbines might cut-off. Modelling these conditional probabilistic behaviours is the key strength of distributional regression models.

Features

• 🚀 First native Python implementation of generalized additive models for location, shape and scale (GAMLSS).
• 🚀 Online-first approach, which allows for incremental updates of the model using model.update(X, y).
• 🚀 Support for various distributions, including Gaussian, Student's $t$, Johnson's $S_U$, Gamma, Log-normal, Exponential, Beta, Gumbel, Inverse Gaussian and more. Implementing new distributions is straight-forward.
• 🚀 Flexible link functions for each distribution, allowing for custom transformations of the parameters.
• 🚀 Support for regularization methods like Lasso, Ridge and Elastic Net.
• 🚀 Fast and efficient implementation using numba for just-in-time compilation.
• 🚀 Full compatibility with scikit-learn estimators and transformers.
• 🚀 Directly score distributional forecasts using the CRPS from scoringrules (optional dependency) - either via fine-grid quantile approximation or closed-form solutions.
• 🚀 Online pre-processing using incremental scaling and pipeline support.

Example

Basic estimation and updating procedure:

import numpy as np
from sklearn.datasets import load_diabetes
from ondil.estimators import OnlineDistributionalRegression
from ondil.distributions import StudentT

X, y = load_diabetes(return_X_y=True)

# Model coefficients
equation = {
    0: "all",  # Can also use "intercept" or np.ndarray of integers / booleans
    1: "all",
    2: "all",
}

# Create the estimator
online_gamlss_lasso = OnlineDistributionalRegression(
    distribution=StudentT(),
    method="lasso",
    equation=equation,
    fit_intercept=True,
    ic="bic",
)

# Initial Fit
online_gamlss_lasso.fit(
    X=X[:-11, :],
    y=y[:-11],
)
print("Coefficients for the first N-11 observations \n")
print(online_gamlss_lasso.beta)

# Update call
online_gamlss_lasso.update(X=X[[-11], :], y=y[[-11]])
print("\nCoefficients after update call \n")
print(online_gamlss_lasso.beta)

# Prediction for the last 10 observations
prediction = online_gamlss_lasso.predict_distribution_parameters(X=X[-10:, :])

print("\n Predictions for the last 10 observations")
# Location, scale and shape (degrees of freedom)
print(prediction)

Installation & Dependencies

The package is available from pypi - do pip install ondil and enjoy.

ondil is designed to have minimal dependencies. We rely on python>=3.10, numpy, numba, scipy and scikit-learn in a reasonably up-to-date versions.

I was looking for rolch but I found ondil?

rolch (Regularized Online Learning for Conditional Heteroskedasticity) was the original name of this package, but we decided to rename it to ondil (Online Distributional Learning) to better reflect its purpose and functionality, since conditional heteroskedasticity (=non constant variance) is just one of the many applications for distributional regression models that can be estimated with this package.

Contributing

We welcome every contribution from the community. Feel free to open an issue if you find bugs or want to propose changes. We're still in an early phase and welcome feedback, especially on the usability and "look and feel" of the package. Secondly, we're working to port distributions from the R-GAMLSS package and welcome according PRs.

Contributors

ondil was developed by Simon Hirsch, Jonathan Berrisch and Florian Ziel. We're grateful for contributions below (sorted alphabetically by GitHub username).

Acknowledgements & Disclosure

Simon Hirsch is employed at Statkraft and gratefully acknowledges support received from Statkraft for his Ph.D. studies. This work contains the author's opinion and not necessarily reflects Statkraft's position.

Citation

If you use ondil in your research, please cite the following paper(s), depending on what you've used. Thank you!

@article{hirsch2024online,
  title={Online distributional regression},
  author={Hirsch, Simon and Berrisch, Jonathan and Ziel, Florian},
  journal={arXiv preprint arXiv:2407.08750},
  year={2024}
}

@article{hirsch2025online,
  title={Online Multivariate Regularized Distributional Regression for High-dimensional Probabilistic Electricity Price Forecasting},
  author={Hirsch, Simon},
  journal={arXiv preprint arXiv:2504.02518},
  year={2025}
}

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