TrackMeasures: Track Measures

In MotilityLab: Quantitative Analysis of Motion

Description

Statistics that can be used to quantify tracks. All of these functions take a single track as input and give a single number as output.

Usage

trackLength(x)

duration(x)

speed(x)

displacement(x, from = 1, to = nrow(x))

squareDisplacement(x, from = 1, to = nrow(x))

displacementVector(x)

maxDisplacement(x)

displacementRatio(x)

outreachRatio(x)

straightness(x)

overallAngle(x, from = 1, to = nrow(x), xdiff = diff(x))

meanTurningAngle(x)

overallDot(x, from = 1, to = nrow(x), xdiff = diff(x))

asphericity(x)

hurstExponent(x)

fractalDimension(x)

Arguments

x	a single input track; a matrix whose first column is time and whose remaining columns are a spatial coordinate.
from	index, or vector of indices, of the first row of the track.
to	index, or vector of indices, of last row of the track.
xdiff	row differences of x.

Details

Some track measures consider only the first and last position (or steps) of a track, and are most useful in conjunction with aggregate.tracks; for instance, squareDisplacement combined with aggregate.tracks gives a mean square displacement plot, and overallAngle combined with aggregate.tracks gives a turning angle plot (see the examples for aggregate.tracks). To speed up computation of these measures on subtracks of the same track, the arguments from, to and possibly xdiff are exploited by aggregate.tracks.

trackLength sums up the distances between subsequent positsion; in other words, it estimates the length of the underlying track by linear interpolation (usually an underestimation). The estimation could be improved in some circumstances by using interpolateTrack.

duration returns the time elapsed between x's first and last positions.

speed simply divides trackLength by duration.

displacement returns the Euclidean distance between the track endpoints and squareDisplacement returns the squared Euclidean distance.

displacementVector returns the vector between the track endpoints.

maxDisplacement computes the maximal Euclidean distance of any position on the track from the first position.

displacementRatio divides the displacement by the maxDisplacement; outreachRatio divides the maxDisplacement by the trackLength (Mokhtari et al, 2013). Both measures return values between 0 and 1, where 1 means a perfectly straight track. If the track has trackLength 0, then NaN is returned.

straightness divides the displacement by the trackLength. This gives a number between 0 and 1, with 1 meaning a perfectly straight track. If the track has trackLength 0, then NaN is returned.

asphericity is a different appraoch to measure straightness (Mokhtari et al, 2013): it computes the asphericity of the set of positions on the track _via_ the length of its principal components. Again this gives a number between 0 and 1, with higher values indicating straighter tracks. Unlike straightness, however, asphericity ignores back-and-forth motion of the object, so something that bounces between two positions will have low straightness but high asphericity. We define the asphericity of every track with two or fewer positions to be 1. For one-dimensional tracks with one or more positions, NA is returned.

overallAngle Computes the angle (in radians) between the first and the last segment of the given track. Angles are measured symmetrically, thus the return values range from 0 to pi; for instance, both a 90 degrees left and right turns yield the value pi/2. This function is useful to generate autocorrelation plots (together with aggregate.tracks).

meanTurningAngle averages the overallAngle over all adjacent segments of a given track; a low meanTurningAngle indicates high persistence of orientation, whereas for an uncorrelated random walk we expect 90 degrees. Note that angle measurements will yield NA values for tracks in which two subsequent positions are identical.

overallDot computes the dot product between the first and the last segment of the given track. This function is useful to generate autocovariance plots (together with aggregate.tracks).

hurstExponent computes the corrected empirical Hurst exponent of the track. This uses the function hurstexp from the 'pracma' package. If the track has less than two positions, NA is returned. fractalDimension estimates the fractal dimension of a track using the function fd.estim.boxcount from the 'fractaldim' package. For self-affine processes in n dimensions, fractal dimension and Hurst exponent are related by the formula H=n+1-D. For non-Brownian motion, however, this relationship need not hold. Intuitively, while the Hurst exponent takes a global approach to the track's properties, fractal dimension is a local approach to the track's properties (Gneiting and Schlather, 2004).

References

Zeinab Mokhtari, Franziska Mech, Carolin Zitzmann, Mike Hasenberg, Matthias Gunzer and Marc Thilo Figge (2013), Automated Characterization and Parameter–Free Classification of Cell Tracks Based on Local Migration Behavior. PLoS ONE 8(12), e80808. doi:10.1371/journal.pone.0080808

Tillmann Gneiting and Martin Schlather (2004), Stochastic Models That Separate Fractal Dimension and the Hurst Effect. SIAM Review 46(2), 269–282. doi:10.1137/S0036144501394387

Examples

## show a turning angle plot with error bars for the T cell data.
with( (aggregate(BCells,overallDot,FUN="mean.se",na.rm=TRUE)),{
  plot( mean ~ i, xlab="time step", 
  	ylab="turning angle (rad)", type="l" )
  segments( i, lower, y1=upper )
} )

MotilityLab documentation built on May 2, 2019, 8:31 a.m.