Functional Data Analysis in Ecosystem Research: The Decline of Oweekeno Lake Sockeye Salmon and Wannock River Flow

Abstract

Functional regression is a natural tool for exploring the potential impact of the physical environment (continuously monitored) on biological processes (often only assessed annually). This paper explores the potential use of functional regression analysis and the closely related functional principal component analysis for studying the relationship between river flow (continuously monitored) and salmon abundance (measured annually). The specific example involves a depressed sockeye salmon population in Rivers Inlet, BC. Particular attention is given to (i) the role of subject matter expertise and cross-validation techniques in guiding decisions on basis functions and smoothing parameters, and (ii) the importance of restricting the time domain for the continuously monitored variable to a scientifically meaningful period of time. In addition, we derive a joint confidence region for the functional regression coefficient function and discuss its use relative to the more commonly used pointwise confidence intervals. The analysis points to a substantial negative correlation between early spring river flow and marine survival of the sockeye salmon that subsequently migrate down the inlet.

Quantifying flow–ecology relationships with functional linear models

“…In such cases a B-spline curve can be used to define the hydrograph as a functional predictor variable (e.g. Ainsworth et al, 2011). B-spline curves, which are a series of joined polynomial functions, offer great flexibility in quantifying the functional form of time series data (Ramsay and Silverman, 2006).…”

“…A functional linear model provides an efficient alternative to summarise the time series into a smaller number of basis functions, however, the potential for overfitting still exists with functional linear models. Therefore, the challenge is to use a functional regression coefficient that is sufficiently complex to capture relationships between the response variable and the time series of the predictor, yet is simple enough to interpret and not result in an overfit model (Ramsay et al, 2010;Ainsworth et al, 2011). As is common in most statistical models, this amounts to finding the trade-off between detecting signal from noise.…”

“…We tested 29 different values for the smoothing parameter, λ (10 1 , 10 1.5 , 10 2 , 10 2.5 … 10 14.5 , 10 15 ), and identified the best fitting, smoothed regression coefficient via generalised cross-validation (Ramsay et al, 2010). Ainsworth et al, (2011) provide a useful discussion on selection of smoothing parameters with respect to generalised cross-validation.…”

“…Building on these analytical approaches A c c e p t e d M a n u s c r i p t is the field of functional data analysis, which has provided a framework to incorporate such spectral decompositions of functional data into statistical models (Ramsay and Silverman, 2006). Functional data can have many forms but typically consist of continuous, or effectively continuous observations of a process over space or time that are derived from mathematical functions (Ramsay and Silverman, 2002), such as temperature or river discharge (e.g., Ainsworth et al, 2011). Quantifying and understanding the functional data series may itself be the goal of the analysis or they may be used as predictor variables in a functional linear model (Ramsay and Silverman, 2002;Müller and Stadtmüller, 2005).…”

Modelling Animal Activity as Curves: An Approach Using Wavelet-Based Functional Data Analysis

“…In this note, we use wavelets to model temporal activity data as curves in the context of functional data analysis (FDA [32] [33] [34] [35]). One may argue the case for treating these temporal activity data as vector-valued data.…”