# Identifying and characterizing extrapolation in multivariate response data

## TL;DR

This paper extends methods for detecting extrapolation from univariate to multivariate ecological data, using predictive variance and classification trees to identify regions where predictions go beyond the data range.

## Contribution

It introduces a novel approach for identifying extrapolation in multivariate response data using predictive variance and classification trees, addressing a gap in ecological modeling.

## Key findings

- Applied the method to lake nutrient and algal biomass data from over 7000 lakes.
- Identified regions of covariate space prone to extrapolation using classification and regression trees.
- Demonstrated the approach's effectiveness in ecological data analysis.

## Abstract

Extrapolation is defined as making predictions beyond the range of the data used to estimate a statistical model. In ecological studies, it is not always obvious when and where extrapolation occurs because of the multivariate nature of the data. Previous work on identifying extrapolation has focused on univariate response data, but these methods are not directly applicable to multivariate response data, which are more and more common in ecological investigations. In this paper, we extend previous work that identified extrapolation by applying the predictive variance from the univariate setting to the multivariate case. We illustrate our approach through an analysis of jointly modeled lake nutrients and indicators of algal biomass and water clarity in over 7000 inland lakes from across the Northeast and Mid-west US. In addition, we illustrate novel exploratory approaches for identifying regions of covariate space where extrapolation is more likely to occur using classification and regression trees.