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Expand Up @@ -27,35 +27,6 @@ pip install git+https://github.com/dsgibbons/shap.git
## How to use
We refer to the [example section](https://github.com/StatMixedML/XGBoostLSS/tree/master/examples) for example notebooks.

## Available Distributions
XGBoostLSS currently supports the following distributions.

| Distribution | Usage |Type | Support | Number of Parameters |
| :----------------------------------------------------------------------------------------------------------------------------------: |:------------------------: |:-------------------------------------: | :-----------------------------: | :-----------------------------: |
| [Beta](https://pytorch.org/docs/stable/distributions.html#beta) | `Beta()` | Continuous <br /> (Univariate) | $y \in (0, 1)$ | 2 |
| [Cauchy](https://pytorch.org/docs/stable/distributions.html#cauchy) | `Cauchy()` | Continuous <br /> (Univariate) | $y \in (-\infty,\infty)$ | 2 |
| [Dirichlet](https://pytorch.org/docs/stable/distributions.html#dirichlet) | `Dirichlet(D)` | Continuous <br /> (Multivariate) | $y_{D} \in (0, 1)$ | D |
| [Expectile](https://epub.ub.uni-muenchen.de/31542/1/1471082x14561155.pdf) | `Expectile()` | Continuous <br /> (Univariate) | $y \in (-\infty,\infty)$ | Number of expectiles |
| [Gamma](https://pytorch.org/docs/stable/distributions.html#gamma) | `Gamma()` | Continuous <br /> (Univariate) | $y \in (0, \infty)$ | 2 |
| [Gaussian](https://pytorch.org/docs/stable/distributions.html#normal) | `Gaussian()` | Continuous <br /> (Univariate) | $y \in (-\infty,\infty)$ | 2 |
| [Gumbel](https://pytorch.org/docs/stable/distributions.html#gumbel) | `Gumbel()` | Continuous <br /> (Univariate) | $y \in (-\infty,\infty)$ | 2 |
| [Laplace](https://pytorch.org/docs/stable/distributions.html#laplace) | `Laplace()` | Continuous <br /> (Univariate) | $y \in (-\infty,\infty)$ | 2 |
| [LogNormal](https://pytorch.org/docs/stable/distributions.html#lognormal) | `LogNormal()` | Continuous <br /> (Univariate) | $y \in (0,\infty)$ | 2 |
| [Multivariate Normal (Cholesky)](https://pytorch.org/docs/stable/distributions.html#multivariatenormal) | `MVN(D)` | Continuous <br /> (Multivariate) | $y_{D} \in (-\infty,\infty)$ | D(D + 3)/2 |
| [Multivariate Normal (Low-Rank)](https://pytorch.org/docs/stable/distributions.html#lowrankmultivariatenormal) | `MVN_LoRa(D, rank)` | Continuous <br /> (Multivariate) | $y_{D} \in (-\infty,\infty)$ | D(2+rank) |
| [Multivariate Student-T](https://docs.pyro.ai/en/stable/distributions.html#multivariatestudentt) | `MVT(D)` | Continuous <br /> (Multivariate) | $y_{D} \in (-\infty,\infty)$ | 1 + D(D + 3)/2 |
| [Negative Binomial](https://pytorch.org/docs/stable/distributions.html#negativebinomial) | `NegativeBinomial()` | Discrete Count <br /> (Univariate) | $y \in (0, 1, 2, 3, \ldots)$ | 2 |
| [Poisson](https://pytorch.org/docs/stable/distributions.html#poisson) | `Poisson()` | Discrete Count <br /> (Univariate) | $y \in (0, 1, 2, 3, \ldots)$ | 1 |
| [Spline Flow](https://docs.pyro.ai/en/stable/distributions.html#pyro.distributions.transforms.Spline) | `SplineFlow()` | Continuous \& Discrete Count <br /> (Univariate) | $y \in (-\infty,\infty)$ <br /> <br /> $y \in [0, \infty)$ <br /> <br /> $y \in [0, 1]$ <br /> <br /> $y \in (0, 1, 2, 3, \ldots)$ | 2xcount_bins + (count_bins-1) (order=quadratic) <br /> <br /> 3xcount_bins + (count_bins-1) (order=linear) |
| [Student-T](https://pytorch.org/docs/stable/distributions.html#studentt) | `StudentT()` | Continuous <br /> (Univariate) | $y \in (-\infty,\infty)$ | 3 |
| [Weibull](https://pytorch.org/docs/stable/distributions.html#weibull) | `Weibull()` | Continuous <br /> (Univariate) | $y \in [0, \infty)$ | 2 |
| [Zero-Adjusted Beta](https://github.com/pyro-ppl/pyro/blob/dev/pyro/distributions/zero_inflated.py) | `ZABeta()` | Discrete-Continuous <br /> (Univariate) | $y \in [0, 1)$ | 3 |
| [Zero-Adjusted Gamma](https://github.com/pyro-ppl/pyro/blob/dev/pyro/distributions/zero_inflated.py) | `ZAGamma()` | Discrete-Continuous <br /> (Univariate) | $y \in [0, \infty)$ | 3 |
| [Zero-Adjusted LogNormal](https://github.com/pyro-ppl/pyro/blob/dev/pyro/distributions/zero_inflated.py) | `ZALN()` | Discrete-Continuous <br /> (Univariate) | $y \in [0, \infty)$ | 3 |
| [Zero-Inflated Negative Binomial](https://github.com/pyro-ppl/pyro/blob/dev/pyro/distributions/zero_inflated.py#L150) | `ZINB()` | Discrete-Count <br /> (Univariate) | $y \in [0, 1, 2, 3, \ldots)$ | 3 |
| [Zero-Inflated Poisson](https://github.com/pyro-ppl/pyro/blob/dev/pyro/distributions/zero_inflated.py#L121) | `ZIPoisson()` | Discrete-Count <br /> (Univariate) | $y \in [0, 1, 2, 3, \ldots)$ | 2 |


## Some Notes
### Stabilization
Since XGBoostLSS updates the parameter estimates by optimizing Gradients and Hessians, it is important that these are comparable in magnitude for all distributional parameters. Due to variability regarding the ranges, the estimation of Gradients and Hessians might become unstable so that XGBoostLSS might not converge or might converge very slowly. To mitigate these effects, we have implemented a stabilization of Gradients and Hessians.
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