The Math of Principal Component Analysis (PCA)

New technology allows researchers to generate large quantities of data, whether it be time-series data, or any other waveform data that we commonly see in biomechanics research. And what is common amongst all these forms of data is that they are highly dimensional. With new technologies like [Markerless], we are collecting so much more data that trying to make meaningful comparisons is difficult to understand and to do computationally.

In the research methods in biomechanics text, it is emphasized that the basic point of looking at methods of processing waveform data is that no matter the type of research questions that a biomechanist is trying to answer, transforming our data into smaller forms makes it more manageable and useful for analysis. Prior to being able to make meaningful comparisons, we want to reduce the amount of data and not lose any important discriminatory information that may be crucial to our comparisons.

Traditional approaches of extracting features from waveform data (max, min, range, etc.), mentioned in the text, result in a lose of so much temporal information in the waveform, leading to such a large reduction and discarded info. Additionally, varying definitions of these discrete parameters across the field lead to inconsistent conclusions, and these measures may not always be definitive in certain populations. One such example in the text found that knee adduction angles in patients with OA don’t have definitive peaks, and thus finding max and min angles is meaningless.

PCA

PCA is an orthogonal decomposition technique that computes and extracts a unique set of basis functions from the waveforms based on the variation that is present in the waveform data.

Waveform Data

Waveform data can be represented in matrix form , where the rows are the time series waveforms for each subject, and the columns are particular points within the waveform (n = subjects, p = points).

Covariance Matrix

To define variation in the waveform matrix X based on how the data changes over time, and between subjects (n) at each point in time (p), we look at the covariance matrix S.