Inside or Outside: Quantifying Extrapolation Across River Networks

Abstract

Regression techniques are often used to predict responses across landscapes or under scenarios describing changes in climate, management, or land cover. The ability of random forests (RFs) and multivariate adaptive regression splines (MARS) to predict flow variability, low flow, Escherichia coli, and a macroinvertebrate community index was compared. Cross validation was applied to test predictive performance across an induced spectrum of interpolation to extrapolation by splitting each data set into two geographical, environmental, and random groups. RF and MARS both represent nonlinear and interacting patterns but showed contrasting ability to interpolate and extrapolate. RF always performed better than MARS when interpolating within environmental space or extrapolating in geographical space. RF models for all four responses were transferable in geographic space but not to environmental conditions outside the training data. Neither technique was successful when extrapolating across environmental gradients, although RF out‐performed MARS, despite RF predictions being constrained by the training data. New methods to quantify interpolation versus extrapolation for predictions are demonstrated. Degree of extrapolation is calculated by transforming both the training data and new predictors in response turnover space. A decline in cross‐validation performance was related to an increase in degree of extrapolation regardless of whether extrapolating in geographical or environmental space. Degree of extrapolation is valuable. It identifies those predictions that are more reliable because they represent interpolation versus those that are more uncertain that represent extrapolation. For example, high degree of extrapolation under climate or land cover change indicates increased risk of producing misleading predictions from both RF and MARS.

River water temperatures are higher during lower flows after accounting for meteorological variability

“…One benefit of applying a systematic approach was that variance explained and the coefficients relating water temperature to river flow were easily interpretable because their meaning was consistent across sites and meteorological variables. This would not be the case if model reduction or more flexible machine learning approaches such as ANN (e.g., DeWeber & Wagner, 2014) or random forests (e.g., Booker & Whitehead, 2018) were applied. Nevertheless, all sites had a RMSE for mean daily water temperature of less than 1 C (site-mean RMSE across sites was 0.71 C) when seasonal patterns and correlations with earth temperature and runoff were applied.…”

Freeway Short-Term Travel Speed Prediction Based on Data Collection Time-Horizons: A Fast Forest Quantile Regression Approach

“…. , training data samples, with weights w given in Equation (1), [77]. In particular, we consider the conditional distribution function (in Equation (2)) of response variable Y conditioned on the specific values x of the predictor variable X m .…”

Development of a Joint Probabilistic Rainfall‐Runoff Model for High‐to‐Extreme Flow Projections Under Changing Climatic Conditions

“…In fact, this issue not only exists for all types of neural networks (e.g., feedforward, convolutional, and recursive neural networks) and deep learning algorithms but also exists for other regression‐based ML models such as support vector machine (SVM) and linear regression (Kourgialas et al., 2015; Wagena et al., 2020; M. C. Wu et al., 2014). Tree‐structured models, as one important branch in ML, can effectively handle varying skewness of time series (Booker & Whitehead, 2018; Jeong et al., 2016; Taillardat et al., 2017). However, since such models are built based on historical observations, their prediction is thus bounded by the highest and lowest values of the training data range, hindering the prediction of extreme values that lie outside the range of training data.…”