Zeni, Gianluca, Fontana, Matteo, Vantini, Simone

In this work we provide a review of basic ideas and novel developments about Conformal Prediction -- an innovative distribution-free, non-parametric forecasting method, based on minimal assumptions -- that is able to yield in a very straightforward way predictions sets that are valid in a statistical sense also in in the finite sample case. The in-depth discussion provided in the paper covers the theoretical underpinnings of Conformal Prediction, and then proceeds to list the more advanced developments and adaptations of the original idea.

Gauraha, Niharika, Spjuth, Ola

The conformalClassification package implements Transductive Conformal Prediction (TCP) and Inductive Conformal Prediction (ICP) for classification problems. Conformal Prediction (CP) is a framework that complements the predictions of machine learning algorithms with reliable measures of confidence. TCP gives results with higher validity than ICP, however ICP is computationally faster than TCP. The package conformalClassification is built upon the random forest method, where votes of the random forest for each class are considered as the conformity scores for each data point. Although the main aim of the conformalClassification package is to generate CP errors (p-values) for classification problems, the package also implements various diagnostic measures such as deviation from validity, error rate, efficiency, observed fuzziness and calibration plots. In future releases, we plan to extend the package to use other machine learning algorithms, (e.g. support vector machines) for model fitting.

Papadopoulos, H., Vovk, V., Gammerman, A.

In this paper we apply Conformal Prediction (CP) to the k-Nearest Neighbours Regression (k-NNR) algorithm and propose ways of extending the typical nonconformity measure used for regression so far. Unlike traditional regression methods which produce point predictions, Conformal Predictors output predictive regions that satisfy a given confidence level. The regions produced by any Conformal Predictor are automatically valid, however their tightness and therefore usefulness depends on the nonconformity measure used by each CP. In effect a nonconformity measure evaluates how strange a given example is compared to a set of other examples based on some traditional machine learning algorithm. We define six novel nonconformity measures based on the k-Nearest Neighbours Regression algorithm and develop the corresponding CPs following both the original (transductive) and the inductive CP approaches. A comparison of the predictive regions produced by our measures with those of the typical regression measure suggests that a major improvement in terms of predictive region tightness is achieved by the new measures.

Spjuth, Ola, Brännström, Robin Carrión, Carlsson, Lars, Gauraha, Niharika

Conformal Prediction is a framework that produces prediction intervals based on the output from a machine learning algorithm. In this paper we explore the case when training data is made up of multiple parts available in different sources that cannot be pooled. We here consider the regression case and propose a method where a conformal predictor is trained on each data source independently, and where the prediction intervals are then combined into a single interval. We call the approach Non-Disclosed Conformal Prediction (NDCP), and we evaluate it on a regression dataset from the UCI machine learning repository using support vector regression as the underlying machine learning algorithm, with varying number of data sources and sizes. The results show that the proposed method produces conservatively valid prediction intervals, and while we cannot retain the same efficiency as when all data is used, efficiency is improved through the proposed approach as compared to predicting using a single arbitrarily chosen source.

Messoudi, Soundouss, Destercke, Sébastien, Rousseau, Sylvain

The most common supervised task in machine learning is to learn a single-task, single-output prediction model. However, such a setting can be ill-adapted to some problems and applications. On the one hand, producing a single output can be undesirable when data is scarce and when producing reliable, possibly set-valued predictions is important (for instance in the medical domain where examples are very hard to collect for specific targets, and where predictions are used for critical decisions). Such an issue can be solved by using conformal prediction approaches [1]. It was initially proposed as a transductive online learning approach to provide set predictions (in the classification case) or interval predictions (in the case of regression) with a statistical guarantee depending on the probability of error tolerated by the user, but was then extended to handle inductive processes [2]. On the other hand, there are many situations where there are multiple, possibly correlated output variables to predict at once, and it is then natural to try to leverage such correlations to improve predictions. Such learning tasks are commonly called Multi-task in the literature [3]. Most research work on conformal prediction for multi-task learning focuses on the problem of multi-label prediction [4, 5], where each task is a binary classification one. Conformal prediction for multi-target regression has been less explored, with only a few studies dealing with it: Kuleshov et al. [6] provide a theoretical framework to use conformal predictors within manifold (e.g., to provide a mono-dimensional embedding of the multi-variate output), while Neeven and Smirnov [7] use a straightforward multi-target extension of a conformal single-output k-nearest neighbor regressor [8] to provide weather forecasts.