Principal Component Analysis

Principal component analysis (PCA) is a multi-variate statistical analysis that reduces the high-dimensional matrix of correlated, time-varying signals into a low-dimensional and statistically uncorrelated set of principal components (PCs). These PCs explain the variance found in the original signals and represent the most important features of the data, e.g., the overall magnitude or the shape of the time series at a particular point in the stride cycle. The value of each particular subject’s score for the individual PCs represents how strongly that feature was present in the data. More detail on the mathematics behind PCA can be found on our page: The Math of Principal Component Analysis (PCA).

The Utility of PCA

When we analyse biomechanical signals, we could identify many isolated quantities (e.g. maximum and minimum values) and compare the signals based on these metrics. Given that there is a common underlying shape to all of these signals, however, it can be more informative to use a multivariate statistical technique that can capture this basic shape and compare the shape of each signal to this underlying shape.

Visualizing PCA Results

Inspect3D provides a number of ways to visualize and interact with the results of PCA. An overview of all PCA visualizes can be found in Inspect3D PCA Graphs.

This page includes:

• Visualizing the variance explained by each PC individually;
• Visualizing the variance explained by each PC at each point in the signal's cycle;
• Scatter-plotting workspace scores in PC-space;
• Showing the distribution of scores by group for each PC;
• Visualizing the mean and extreme values that result from reconstructing the underlying data with each PC; and
• Visualizing how the signals can be reconstructed from the computed PCs.

Tutorials

For a step-by-step example of how to use Inspect3D to perform PCA on your data, see the PCA Tutorial.

For a step-by-step example of how to use Inspect3D to perform further statistical testing on PCA results, see the Tutorial: Run K-Means.

For a step-by-step example of processing and analyzing large data sets in Inspect3D and using PCA to distinguish between groups, see the Tutorial: Treadmill Walking In Healthy Individuals and the Tutorial: Analysis of Baseball Hitters.

Reference

Our implementation of Principal Component Analysis is based on the article:

Deluzio KJ and Astephen JL (2007) Biomechanical features of gait waveform data associated with knee osteoarthritis. An application of principal component anslysis. Gait & Posture 23. 86-93 (pdf)

Abstract

This study compared the gait of 50 patients with end-stage knee osteoarthritis to a group of 63 age-matched asymptomatic control subjects. The analysis focused on three gait waveform measures that were selected based on previous literature, demonstrating their relevance to knee osteoarthritis (OA): the knee flexion angle, flexion moment, and adduction moment. The objective was to determine the biomechanical features of these gait measures, related to knee osteoarthritis. Principal component analysis was used as a data reduction tool, as well as a preliminary step for further analyses to determine gait pattern differences between the OA and the control groups. These further analyses included statistical hypothesis testing to detect group differences, and discriminant analysis to quantify overall group separation and to establish a hierarchy of discriminatory ability among the gait waveform features at the knee. The two groups were separated with a misclassification rate (estimated by cross-validation) of 8%. The discriminatory features of the gait waveforms were, in order of their discriminatory ability: the amplitude of the flexion moment, the range of motion of the flexion angle, the magnitude of the flexion moment during early stance, and the magnitude of the adduction moment during stance.

The fundamentals of Principal Component Analysis for our implementation and the presented article are derived from the Research Methods in Biomechanics textbook.