Functional Data Analysis for Biomechanics

Abstract

The application of nonlinear tools and advanced statistical methods is becoming more prevalent in biomechanical analyses. In a traditional biomechanics laboratory with motion analysis equipment, large amounts of kinematic data can be collected relatively easily. However, a significant gap exists between all the data that are collected and the data that are actually analyzed. Because movements occur over a period of time, whether seconds or minutes, each movement is represented by a continuous series of kinematic data (e.g., 60 or 120 observations per second). Using standard analytic methods, the continuous data associated with each movement are often reduced to a single discrete number. This reduction to a single summary value, such as a peak flexion, extension, or range of motion, excludes potentially valuable information. Reducing a curve representing hip motion during gait to a single range of motion value, for example, precludes the analysis of the entire movement pattern or the timing of the movement. A handful of investigations have recognized this limitation and have begun using functions to maintain the shape and timing of the movement in the analysis. The primary purpose of this chapter is to introduce an emerging collection of statistical methods called Functional Data Analysis (FDA). FDA is distinct from traditional analytic methods because how data changes continuously over time can be assessed. Therefore, information in continuous signals can be retained, such as changes in joint angles or in landmark positions during a movement task. FDA can be used for both exploratory and hypothesis driven analyses with traditional multivariate statistical methods that have been modified for functional predictor and response variables. Although representing motion data as a set of functions is not new to biomechanics analyses (Chester & Wrigley, 2008; Deluzio & Astephen, 2007; Landry et al., 2007; Lee et al., 2009; Sadeghi et al., 2002; 2000), statistical methods developed specifically for analyzing these functions have not been available. More recently, FDAmethods have been usedwithin biomechanics to studymastication (Crane et al., 2010), back pain (Page et al., 2006), as well as age, gender, and speed effects on walking (Roislien et al., 2009). Given the interest in and need for treating motion data as functions, it is important that methods for analyzing a set a functions using emerging statistical methods are brought to the attention of those in the biomechanics community. Although several excellent references exist for Functional Data Analysis (Ramsay, 2000; Ramsay et al., 2009; Ramsay & Silverman, 2002; 2005) there are important issues for biomechanists to be aware of when implementing this set of statistical tools. Therefore, the aims of this chapter are to provide an overview of the steps associated with FDA, to focus on 4