`ConformalPrediction.jl`

is a package for Predictive Uncertainty Quantification (UQ) through Conformal Prediction (CP) in Julia. It is designed to work with supervised models trained in MLJ (Blaom et al. 2020). Conformal Prediction is easy-to-understand, easy-to-use and model-agnostic and it works under minimal distributional assumptions.

First time here? Take a quick interactive tour to see what this package can do right on JuliaHub (To run the notebook, hit login and then edit).

This `Pluto.jl`

π notebook won the 2nd Price in the JuliaCon 2023 Notebook Competition.

To run the tour locally, just clone this repo and start `Pluto.jl`

as follows:

```
] add Pluto
using Pluto
Pluto.run()
```

All notebooks are contained in `docs/pluto`

.

Donβt worry, weβre not about to deep-dive into methodology. But just to give you a high-level description of Conformal Prediction (CP) upfront:

Conformal prediction (a.k.a. conformal inference) is a user-friendly paradigm for creating statistically rigorous uncertainty sets/intervals for the predictions of such models. Critically, the sets are valid in a distribution-free sense: they possess explicit, non-asymptotic guarantees even without distributional assumptions or model assumptions.

β Angelopoulos and Bates (2022)

Intuitively, CP works under the premise of turning heuristic notions of uncertainty into rigorous uncertainty estimates through repeated sampling or the use of dedicated calibration data.

Conformal Prediction in action: prediction intervals at varying coverage rates. As coverage grows, so does the width of the prediction interval.

The animation above is lifted from a small blog post that introduces Conformal Prediction and this package in the context of regression. It shows how the prediction interval and the test points that it covers varies in size as the user-specified coverage rate changes.

You can install the latest stable release from the general registry:

```
using Pkg
Pkg.add("ConformalPrediction")
```

The development version can be installed as follows:

```
using Pkg
Pkg.add(url="https://github.com/juliatrustworthyai/ConformalPrediction.jl")
```

To illustrate the intended use of the package, letβs have a quick look at a simple regression problem. We first generate some synthetic data and then determine indices for our training and test data using MLJ:

```
using MLJ
# Inputs:
N = 600
xmax = 3.0
using Distributions
d = Uniform(-xmax, xmax)
X = rand(d, N)
X = reshape(X, :, 1)
# Outputs:
noise = 0.5
fun(X) = sin(X)
Ξ΅ = randn(N) .* noise
y = @.(fun(X)) + Ξ΅
y = vec(y)
# Partition:
train, test = partition(eachindex(y), 0.4, 0.4, shuffle=true)
```

We then import Symbolic Regressor (`SymbolicRegression.jl`

) following the standard MLJ procedure.

```
regressor = @load SRRegressor pkg=SymbolicRegression
model = regressor(
niterations=50,
binary_operators=[+, -, *],
unary_operators=[sin],
)
```

To turn our conventional model into a conformal model, we just need to declare it as such by using `conformal_model`

wrapper function. The generated conformal model instance can wrapped in data to create a *machine*. Finally, we proceed by fitting the machine on training data using the generic `fit!`

method:

```
using ConformalPrediction
conf_model = conformal_model(model)
mach = machine(conf_model, X, y)
fit!(mach, rows=train)
```

Predictions can then be computed using the generic `predict`

method. The code below produces predictions for the first `n`

samples. Each tuple contains the lower and upper bound for the prediction interval.

```
show_first = 5
Xtest = selectrows(X, test)
ytest = y[test]
yΜ = predict(mach, Xtest)
yΜ[1:show_first]
```

```
5-element Vector{Tuple{Float64, Float64}}:
(-0.04087262272113379, 1.8635644669554758)
(0.04647464096907805, 1.9509117306456876)
(-0.24248802236397216, 1.6619490673126376)
(-0.07841928163933476, 1.8260178080372749)
(-0.02268628324126465, 1.881750806435345)
```

For simple models like this one, we can call a custom `Plots`

recipe on our instance, fit result and data to generate the chart below:

```
using Plots
zoom = 0
plt = plot(mach.model, mach.fitresult, Xtest, ytest, lw=5, zoom=zoom, observed_lab="Test points")
xrange = range(-xmax+zoom,xmax-zoom,length=N)
plot!(plt, xrange, @.(fun(xrange)), lw=2, ls=:dash, colour=:darkorange, label="Ground truth")
```

We can evaluate the conformal model using the standard MLJ workflow with a custom performance measure. You can use either `emp_coverage`

for the overall empirical coverage (correctness) or `ssc`

for the size-stratified coverage rate (adaptiveness).

```
_eval = evaluate!(mach; measure=[emp_coverage, ssc], verbosity=0)
display(_eval)
println("Empirical coverage: $(round(_eval.measurement[1], digits=3))")
println("SSC: $(round(_eval.measurement[2], digits=3))")
```

```
PerformanceEvaluation object with these fields:
model, measure, operation, measurement, per_fold,
per_observation, fitted_params_per_fold,
report_per_fold, train_test_rows, resampling, repeats
Extract:
ββββββββββββββββββββββββββββββββββββββββββββββββ¬ββββββββββββ¬ββββββββββββββ¬ββββββ
β measure β operation β measurement β 1.9 β―
ββββββββββββββββββββββββββββββββββββββββββββββββΌββββββββββββΌββββββββββββββΌββββββ
β ConformalPrediction.emp_coverage β predict β 0.953 β 0.0 β―
β ConformalPrediction.size_stratified_coverage β predict β 0.953 β 0.0 β―
ββββββββββββββββββββββββββββββββββββββββββββββββ΄ββββββββββββ΄ββββββββββββββ΄ββββββ
2 columns omitted
Empirical coverage: 0.953
SSC: 0.953
```

If after reading the usage example above you are just left with more questions about the topic, thatβs normal. Below we have have collected a number of further resources to help you get started with this package and the topic itself:

- Blog post introducing conformal classifiers: [Quarto], [TDS], [Forem].
- Blog post applying CP to a deep learning image classifier: [Quarto], [TDS], [Forem].
- The package docs and in particular the FAQ.

*A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification*by Angelopoulos and Bates (2022) (pdf).*Predictive inference with the jackknife+*by Barber et al. (2021) (pdf)*Awesome Conformal Prediction*repository by Valery Manokhin (repo).- Documentation for the Python package MAPIE.

This package is in its early stages of development and therefore still subject to changes to the core architecture and API.

The following CP approaches have been implemented:

**Regression**:

- Inductive
- Naive Transductive
- Jackknife
- Jackknife+
- Jackknife-minmax
- CV+
- CV-minmax

**Classification**:

- Inductive
- Naive Transductive
- Adaptive Inductive

The package has been tested for the following supervised models offered by MLJ.

**Regression**:

`keys(tested_atomic_models[:regression])`

```
KeySet for a Dict{Symbol, Expr} with 8 entries. Keys:
:ridge
:lasso
:evo_tree
:nearest_neighbor
:decision_tree_regressor
:quantile
:random_forest_regressor
:linear
```

**Classification**:

`keys(tested_atomic_models[:classification])`

```
KeySet for a Dict{Symbol, Expr} with 5 entries. Keys:
:nearest_neighbor
:evo_tree
:random_forest_classifier
:logistic
:decision_tree_classifier
```

To evaluate conformal predictors we are typically interested in correctness and adaptiveness. The former can be evaluated by looking at the empirical coverage rate, while the latter can be assessed through metrics that address the conditional coverage (Angelopoulos and Bates 2022). To this end, the following metrics have been implemented:

`emp_coverage`

(empirical coverage)`ssc`

(size-stratified coverage)

There is also a simple `Plots.jl`

recipe that can be used to inspect the set sizes. In the regression case, the interval width is stratified into discrete bins for this purpose:

`bar(mach.model, mach.fitresult, X)`

Contributions are welcome! A good place to start is the list of outstanding issues. For more details, see also the Contributorβs Guide. Please follow the SciML ColPrac guide.

To build this package I have read and re-read both Angelopoulos and Bates (2022) and Barber et al. (2021). The Awesome Conformal Prediction repository (Manokhin 2022) has also been a fantastic place to get started. Thanks also to @aangelopoulos, @valeman and others for actively contributing to discussions on here. Quite a few people have also recently started using and contributing to the package for which I am very grateful. Finally, many thanks to Anthony Blaom (@ablaom) for many helpful discussions about how to interface this package to `MLJ.jl`

.

Angelopoulos, Anastasios N., and Stephen Bates. 2022. βA Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification.β https://arxiv.org/abs/2107.07511.

Barber, Rina Foygel, Emmanuel J. CandΓ¨s, Aaditya Ramdas, and Ryan J. Tibshirani. 2021. βPredictive Inference with the Jackknife+.β *The Annals of Statistics* 49 (1): 486β507. https://doi.org/10.1214/20-AOS1965.

Blaom, Anthony D., Franz Kiraly, Thibaut Lienart, Yiannis Simillides, Diego Arenas, and Sebastian J. Vollmer. 2020. βMLJ: A Julia Package for Composable Machine Learning.β *Journal of Open Source Software* 5 (55): 2704. https://doi.org/10.21105/joss.02704.

Manokhin, Valery. 2022. βAwesome Conformal Prediction.β https://doi.org/10.5281/zenodo.6467205; Zenodo. https://doi.org/10.5281/zenodo.6467205.