Conformal Prediction for Regression

In this chapter, we will cover conformal prediction for regression problems.

Regression is a cornerstone of machine learning, enabling us to predict continuous outcomes from given data. However, as with many predictive tasks, the predictions are never free from uncertainty. Traditional regression techniques give us a point estimate but fail to measure the uncertainty. This is where the power of conformal prediction comes into play, extending our regression models to produce well-calibrated prediction intervals.

This chapter delves deep into conformal prediction tailored specifically for regression problems. By understanding and appreciating the importance of quantifying uncertainty, we will explore how conformal prediction augments regression to provide not just a point prediction but an entire interval or even a distribution where the actual outcome will likely fall with pre-specified confidence. This is invaluable in many real-world scenarios...

Uncertainty quantification for regression problems

After completing this chapter, whenever you predict any continuous variable, you’ll be equipped to add a layer of robustness and reliability to your predictions. Understanding and quantifying this uncertainty is crucial for several reasons:

• Model interpretability and trust: Uncertainty quantification helps us understand the reliability of our model predictions. By providing a range of possible outcomes, we can build trust in our model’s predictions and interpret them more effectively.
• Decision-making: In many practical applications of regression analysis, decision-makers must rely on something other than point estimates. They often need to know the range within which the actual value will likely fall with a certain probability. This range, or prediction interval, provides crucial information about the uncertainty of the prediction and aids in risk management.
• Model improvement: Uncertainty can highlight...

Conformal prediction for regression problems

In the preceding chapters, we investigated the numerous advantages that conformal prediction provides. These include the following:

• Validity and calibration: Conformal prediction maintains its validity and calibration, irrespective of the dataset’s size. This makes it a robust method for prediction across different dataset sizes.
• Distribution-free nature: One of the significant benefits of conformal prediction is its distribution-free nature. It makes no specific assumptions about the underlying data distribution, making it a flexible and versatile tool for many prediction problems.
• Compatibility with various predictors: Conformal prediction can seamlessly integrate with any point predictor, irrespective of its nature. This property enhances its adaptability and widens its scope of application in diverse domains.
• Non-intrusiveness: The conformal prediction framework is non-intrusive, implying that it does not interfere...

Building prediction intervals and predictive distributions using conformal prediction

ICP is a computationally efficient variant of the original transductive conformal prediction framework. Like all other models from the conformal prediction family, ICP is model-agnostic in terms of the underlying point prediction model and data distribution and comes with automatic validity guarantees for final samples of any size.

The key advantage of ICP compared to the original variant of conformal prediction (transductive conformal prediction) is that ICP requires training the underlying regression model only once, leading to efficient computations during the calibration and prediction phases. ICP is highly computationally efficient as the conformal layer requires very little additional computation overhead compared to training the underlying model.

The ICP process involves splitting the dataset into a proper training set and a calibration set. The training set is used to create the initial...

Mechanics of CQR

In the previous section, we observed that ICP generates prediction intervals of uniform width. Consequently, it doesn’t adjust adaptively to heteroscedastic data, where the variability of the response variable isn’t constant across different regions of the data.

In many cases, not only it is crucial to ensure valid coverage in final samples but it is also beneficial to generate the most concise prediction intervals for each point within the input space. This helps maintain the informativeness of these intervals. When dealing with heteroscedastic data, the model should be capable of adjusting the length of prediction intervals to match the local variability associated with each point in the feature space.

CQR (developed by Yaniv Romano, Evan Patterson, and Emmanuel Candes and published in the paper Conformalized Quantile Regression (https://arxiv.org/abs/1905.03222)) is one of the most popular and widely adopted conformal prediction models. It was...

Summary

This chapter explored uncertainty quantification for regression problems, a critical aspect of data science and machine learning. It highlighted the importance of uncertainty and the methods to handle it effectively to make more reliable predictions and decisions.

One of the significant sections of this chapter was dedicated to various approaches that can be used to produce prediction intervals. It systematically broke down and explained diverse methods, elucidating how each works and their advantages and disadvantages. This detailed analysis aids in understanding the mechanisms behind these approaches and their practical application in real-world regression problems.

Furthermore, this chapter discussed building prediction intervals and predictive distributions using conformal prediction. We provided a step-by-step guide to constructing these intervals and distributions. This chapter also offered practical insights and tips for effectively utilizing conformal prediction...