rcalibration

Estimation and hypothesis tests of calibration in R using CalibrationErrors.jl and CalibrationTests.jl.

rcalibration is a package for estimating calibration of probabilistic models in R. It is an R interface for CalibrationErrors.jl and CalibrationTests.jl. As such, the package allows the estimation of calibration errors (ECE and SKCE) and statistical testing of the null hypothesis that a model is calibrated.

Installation

You can install rcalibration with devtools:

> library(devtools)
> devtools::install_github("devmotion/rcalibration")

The use of rcalibration requires that its dependency JuliaCall (installed automatically) and itself are configured correctly.

For JuliaCall, you have to install Julia. The configuration process is described in the JuliaCall documentation.

When JuliaCall is configured correctly, you can install the Julia packages required by rcalibration:

> library(rcalibration)
> rcalibration::install()

Crash on MacOS with Julia 1.6

Due to a problem in Julia 1.6, JuliaCall and rcalibration crash on MacOS with this Julia version. Please use Julia 1.5 on MacOS until this issue is fixed.

Custom Julia environment

With the default settings, JuliaCall and rcalibration install all Julia dependencies in the default environment. In particular, if you use Julia for other projects as well, a separate project environment can simplify package management and ensure that the state of the Julia dependencies is reproducible. In JuliaCall and rcalibration, a custom project environment is used if you set the environment variable JULIA_PROJECT:

export JULIA_PROJECT="path/to/the/environment/"

Usage

Import and setup calibration analysis tools from CalibrationErrors.jl and CalibrationTests.jl with

> ca <- rcalibration::load()

You can then do the same as would be done in Julia, except you have to add ca$ in front for functionality from the Julia packages. Most of the commands will work without any modification. Thus the documentation of the Julia packages is the main in-depth documentation for this package.

Callable objects

R does not support the callable object syntax that is a common idiom in Julia. JuliaCall supports the syntax f$.(x) in R for the function call f(x) with callable object f in Julia.

Calibration errors

Let us estimate the squared kernel calibration error (SKCE) with the tensor product kernel

$$k((p, y), (p̃, ỹ)) = exp(-|p - p̃|) δ(y - ỹ)$$

from a set of predictions and corresponding observed outcomes.

> skce <- ca$SKCE(ca$tensor(ca$ExponentialKernel(), ca$WhiteKernel()))

Other estimators of the SKCE and estimators of other calibration errors such as the expected calibration error (ECE) are available as well. The Julia package KernelFunctions.jl supports a variety of kernels, all compositions and transformations of kernels available there can be used.

Probabilities

Predictions can be provided as probabilities. In this case, the predictions correspond to Bernoulli distributions with these parameters and the targets are boolean values.

> set.seed(1234)
> predictions <- runif(100)
> outcomes <- sample(c(TRUE, FALSE), 100, replace=TRUE)
> skce$.(predictions, outcomes)
[1] 0.01518318

Probability vectors

Predictions can be provided as probability vectors (i.e., vectors in the probability simplex) as well. In this case, the predictions correspond to categorical distributions with these class probabilities and the targets are integers in {1,...,n}. The probability vectors can be given as a matrix. However, it is required to specify if the probability vectors correspond to rows or columns of the matrix by wrapping them in ca.RowVecs and ca.ColVecs, respectively. These wrappers are defined in KernelFunctions.jl.

> library(extraDistr)
> set.seed(1234)
> predictions <- rdirichlet(100, c(3, 2, 5))
> outcomes <- sample(1:3, 100, replace=TRUE)
> skce$.(ca$RowVecs(predictions), outcomes)
[1] 0.02585344

Probability distributions

Predictions can also be provided as probability distributions defined in the Julia package Distributions.jl. Currently, analytical formulas for the estimators of the SKCE and unnormalized calibration mean embedding (UCME) are implemented for uni- and multivariate normal distributions ca$Normal and ca$MvNormal with squared exponential kernels on the target space and Laplace distributions ca$Laplace with exponential kernels on the target space.

In this example we use the tensor product kernel

$$k((p, y), (p̃, ỹ)) = exp(-W₂(p, p̃)) exp(-(y - ỹ)²/2),$$

where W₂(p, p̃) is the 2-Wasserstein distance of the two normal distributions p and p̃. It is given by

$$W₂(p, p̃) = √((μ - μ̃)² + (σ - σ̃)²),$$

where p = N(μ, σ) and p̃ = N(μ̃, σ̃).

> set.seed(1234)
> predictions <- replicate(100, ca$Normal(rnorm(1), runif(1)))
> outcomes <- rnorm(100)
> skce <- ca$SKCE(ca$tensor(ca$ExponentialKernel(metric=ca$Wasserstein()), ca$SqExponentialKernel()))
> skce$.(predictions, outcomes)
[1] 0.02301165

Calibration tests

rcalibration provides different calibration tests that estimate the p-value of the null hypothesis that a model is calibrated, based on a set of predictions and outcomes:

• ca$ConsistencyTest estimates the p-value with consistency resampling for a given calibration error estimator
• ca$DistributionFreeSKCETest computes distribution-free (and therefore usually quite weak) upper bounds of the p-value for different estimators of the SKCE
• ca$AsymptoticBlockSKCETest estimates the p-value based on the asymptotic distribution of the unbiased block estimator of the SKCE
• ca$AsymptoticSKCETest estimates the p-value based on the asymptotic distribution of the unbiased estimator of the SKCE
• ca$AsymptoticCMETest estimates the p-value based on the asymptotic distribution of the UCME

> library(extraDistr)
> set.seed(1234)
> predictions <- rdirichlet(100, c(3, 2, 5))
> outcomes <- sample(1:3, 100, replace=TRUE)
> test <- ca$AsymptoticSKCETest(kernel, ca$RowVecs(predictions), outcomes)
> print(test)
Julia Object of type AsymptoticSKCETest{KernelTensorProduct{Tuple{ExponentialKernel{TotalVariation}, WhiteKernel}}, Float64, Float64, Matrix{Float64}}.
Asymptotic SKCE test
--------------------
Population details:
    parameter of interest:   SKCE
    value under h_0:         0.0
    point estimate:          0.0259434

Test summary:
    outcome with 95% confidence: reject h_0
    one-sided p-value:           0.0100

Details:
    test statistic: -0.007291403994633658
> ca$pvalue(test)
[1] 0.004

Citing

If you use rcalibration as part of your research, teaching, or other activities, please consider citing the following publications:

Widmann, D., Lindsten, F., & Zachariah, D. (2019). Calibration tests in multi-class classification: A unifying framework. In Advances in Neural Information Processing Systems 32 (NeurIPS 2019) (pp. 12257–12267).

Widmann, D., Lindsten, F., & Zachariah, D. (2021). Calibration tests beyond classification. International Conference on Learning Representations (ICLR 2021).

Acknowledgements

This work was financially supported by the Swedish Research Council via the projects Learning of Large-Scale Probabilistic Dynamical Models (contract number: 2016-04278), Counterfactual Prediction Methods for Heterogeneous Populations (contract number: 2018-05040), and Handling Uncertainty in Machine Learning Systems (contract number: 2020-04122), by the Swedish Foundation for Strategic Research via the project Probabilistic Modeling and Inference for Machine Learning (contract number: ICA16-0015), by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation, and by ELLIIT.