Principal-component analysis of particle motion

We demonstrate the application of principal-component analysis (PCA) to the analysis of particle motion data in the form of a time series of images. PCA has the ability to resolve and isolate spatiotemporal patterns in the data. Using simulated data, we show that this translates into the ability to separate individual frequency components of the particle motion. We also show that PCA can be used to extract the fluid viscosity from images of particles undergoing Brownian motion. PCA thus provides an efficient alternative to more traditional particle-tracking methods for the analysis of microrheological data.

Figure 1

A schematic illustration of the decomposition of an $m\times T$ data matrix $\mathbf{X}$ into its principal components ${\mathbf{w}}_i$. The vectors ${\mathbf{y}}_i$ are the time courses of the corresponding principal components.

Figure 2

The results of PCA of simulated data for particles undergoing simple harmonic motion. The left-hand panels of (a)–(e) show the first, second, fourth, fifth, and ninth principal components, respectively. The inset on the corresponding right-hand panel shows the time course of the principal component, and the power spectrum is plotted in the main graph of the right-hand panel, with all $y$-axis units arbitrary. (a) The first principal component is the time average of the data. Its time course is roughly constant, and there is no strong peak (other than that at zero frequency) in the power spectrum. The other principal components shown in (b)–(e) selectively pick out oscillations at frequencies 0.5 Hz, 0.167 Hz, 0.2 Hz, and 0.1 Hz respectively. Oscillations at 0.056 Hz were not isolated by any principal component.

Figure 3

(a) ${\delta }^{\left(1\right)}$, the fraction of the total variance in the data accounted for by the first principal component as a function of $\sigma$ for a simulation of 25 particles undergoing Brownian motion. The inset shows the same data with linear axes. In both cases the line is the prediction of Eq. (18). Panel (b) shows the fraction of the total variance in the first $s$ principal components for $\sigma =0.01$ (solid diamonds), 0.5 (open diamonds), 1 (solid squares), 2 (open squares), 4 (solid circles), and 20 (open circles) pixels.

Figure 4

(a) The first two principal components and (b) their time courses, calculated from simulated data on particles undergoing Brownian motion. The insets at the bottom right of the images in (a) are magnified views of the small regions outlined by rectangles.

Figure 5

PCA of simulations in which diffusing particles become progressively more confined over time. (a) The first principal component ${\mathbf{w}}_1$. The inset at the bottom left is a magnified view of the small region outlined by the rectangle. (b) Its time course ${\mathbf{y}}_1$ (solid line) and the rms displacement $\sigma$ of the particles in a single time step over the course of the simulations (dashed line). ${\mathbf{y}}_1$ increases over time as the particles become less mobile, while the image of ${\mathbf{w}}_1$ shows the areas to which the particles become confined.