Animal-Human Discrimination Track-level Discrimination

Analysis of Movement Data Generally intended to delineate paths through terrain. Generally descriptive – not usually predictive or intended for classification. Examples Random walks Correlated random walks Problem for present work – positional data not precise enough.

Analysis of Movement Data Goal of Phase 1 – develop algorithms that would be sensitive to qualitative differences in movement patterns of non-human animals, indigenous peoples, and dismounts. Key feature of animals’ use of their habitat – the home range.

Home Range Concept Territory = area that animal actively defends – by marking or actual physical combat Home range – area that an animal samples on a more-or-less regular basis. Home range may overlap with territories/home ranges of conspecifics. Habitat ‘quality’ is key determinant of home range size, location and utilization patterns

Home Range Concept – Key Features Animal movements restricted by (often cryptic) boundaries that define the home range Home range size (area) is a nonlinear function of animal size. Animals on home range tend to return to specific points on a more-or-less regular basis.

Home Range Size in Animals

Home Range Concept – Key Features Movement patterns within home range often show characteristics of Lévy flights. – Somewhat controversial – Postdicted by optimal foraging theory – Uniform or Brownian motion-based search strategies result in oversampling of an area  increased cost:benefit ratio

Lévy flights vs. Brownian Motion Brownian MotionLévy Flight

Red-knobbed Hornbill

Giant argus

Animal-Human Discrimination Analysis of Movement Data We’re going to argue for a combined approach based on a number of algorithms An eclectic mix of algorithms will better meet your programmatic needs that a single approach based solely on FFT.

Animal-Human Discrimination Analysis of Movement Data Wavelet Analysis (Discrete) Differential Geometry Nonlinear algorithms

Animal-Human Discrimination Analysis of Movement Data Wavelet Analysis (Discrete) Differential Geometry Nonlinear algorithms

Wavelets Conceptually similar to windowed FFT, but the basis set is different. Daubechies 4 wavelet: Both amplitude and the width of the basis set (i.e., the wavelet) can vary.

Wavelets To a large extent, wavelet analysis gets around the problem of signal non-stationarity – FFT doesn’t handle non-stationary signals well Additional benefit of wavelets – If necessary, we can develop new wavelets that are more specific and appropriate for our purposes

Wavelet Analysis of Animal Movement Data Data are in form of (x, y) = f(t) position coordinates. Convert to single time series by computing Euclidean distances of each (x, y) pair from arbitrary starting point. Compute wavelet transform of time series

Wavelet Analysis of Dismount Movement Simulated a path with varying tortuousity Dismount moved at constant velocity along path Wavelet analysis as for animal movement data

Animal-Human Discrimination Analysis of Movement Data Wavelet Analysis Differential Geometry Nonlinear algorithms

Differential Geometry of Plane Curves Theorems from differential geometry address the question of how normal vectors to an animal’s path should ‘behave’. Specifically, the behavior of normal vectors differs for open paths (the most likely path taken by dismounts passing through an area) and closed paths, whether simple or complex Nascent subdiscipline of Discrete Differential Geometry demonstrates that concepts apply to discrete systems. » Intermittent targeting of sectors by FOPEN GMTI radar system would provide discrete data.

Differential Geometry of Plane Curves Nascent subdiscipline of Discrete Differential Geometry demonstrates that concepts apply to discrete systems. » Intermittent targeting of sectors by FOPEN GMTI radar system would provide discrete data. Useful for producing null models against which data can be compared. Osculating circles and curvature are one approach. » Other features, such as turning angle and winding number might be useful.

Gauss-Bonnet Theorem The integral of the signed curvature around a simple closed smooth curve on a flat, planar surface is equal to 2  :  C  G (s)ds = 2n  where n is the crossing number. Leads to prediction that most normals to a simple closed smooth curve point towards the center. – Same prediction for a complex curve that crosses itself Simulate the movement pattern of an animal on its home range with a complex closed curve:

Gauss-Bonnet Theorem

Animal-Human Discrimination Conclusions Behavioral ecology of animals and indigenous humans leads to home ranges w/ dimensional scale of < 10 to 30 kilometers Movements within a home range result in: » Periodic or quasiperiodic movement patterns that are revealed by wavelet analysis. » Closed curves (tracks) for which large proportion of normal vectors point towards points that lie close to or within home range boundaries. Dismounts should exhibit qualitatively different movement patterns from those of indigenous humans and non-human animals

Animal-Human Discrimination Analysis of Movement Data Wavelet Analysis Differential Geometry Nonlinear algorithms

Animal Movement is Fractal Limb movement scales in fractal way in humans and other mammals Don’t know about limb behavior in birds. Movement speed scales differently for small and large mammals = multifractal? Need to account for these two facts when modeling micro-Doppler signals.

Biphasic movement pattern: High frequencies of slow movements Lower frequencies of higher-speed movements Usual interpretation – animals move rapidly between favorable patches in their home range, slowly within favorable patches Favorable = suitable for energy/nutrient acquisition

The ‘break’ in the pattern is often interpreted as a critical scale parameter that reflects the dimensional scale of the distribution of a critical resource, e.g., food, within the home range.

The frequency distribution is fat-tailed and reflects optimal sampling of habitats. Movement speeds consequently exhibit fractal patterning. In turn, limb movements scale in a fractal way. Possibility of multi-fractal scaling FFT doesn’t deal with fractal patterns because of nonstationarity.

Animal-Human Discrimination Micro-Doppler Scale Analysis of Movement Data Wavelet Analysis (Discrete) Differential Geometry Nonlinear algorithms

Animal-Human Discrimination Micro-Doppler Scale Analysis of Movement Data Alternative approaches Nonlinear Time-series Analysis Sensitive to signal complexity Stochastic Resonance Can dramatically enhance SNR

Nonlinear Algorithms Entropy measures – old & new » Shannon – the standard from Information theory; measures ‘eveness’ in a probability distribution of values in a time series » Spectral Entropy – basically a Shannon entropy of the power spectrum » Approximate Entropy (AppEn) – widely used, but suffers from significant statistical bias, supplanted by Sample Entropy » Sample Entropy (SampEn) – used in various applications » Permutation Entropy (PermEn) – looks at pattern of relative values of sequence of points

Animal-Human Discrimination Algorithms Based on Chaos Theory Largest Lyapunov exponent ( 1 ) – reflects time-dependent evolution of the initial nearest-neighbor distance. Specifically d 12 (t) = d 12 (0)e 1 – 1 > 0 implies chaotic (complex) dynamics. Correlation dimension (D2) – Not commonly employed, but may be useful.

Chaotic Dynamics

Chaotic Dynamics – 2D

Chaotic Dynamics – 3D

Chaotic Dynamics Orbits of Planetary BodiesElectroencephalogram (EEG) From Hinse et al., 2008

Animal-Human Discrimination Algorithms Based on Chaos Theory Significant controversy about the applicability of chaos theory to analysis of time series data. Algorithms for computing each work well…but only if applied to mathematical models. Results are questionable if signal is: » Of limited duration. » Nonstationary – statistical properties change with time. » Corrupted by noise – leads to high estimates of 1 and low estimates of D2, both suggestive of “low-dimensional chaotic dynamics” which may not, if fact, be present. However…

Animal-Human Discrimination Algorithms Based on Chaos Theory 1 and D2 are easily computed from time series data 1 and D2 appear to be sensitive to different features in the data than FFT, wavelets, entropy measures, etc.

Animal-Human Discrimination Multiscale measures Exciting, relatively new conceptual approach Compute measures such as entropy or Lyapunov Exponents at different scales of the data Entropy measures » Multiscale Sample Entropy (MSE) – used in various applications, including analysis of human gait » Multiscale Permutation Entropy – not yet applied to human locomotion

Electroencephalogram Analysis Multiscale measures

Animal-Human Discrimination Various measures of signal complexity are sensitive to different features in the data THEREFORE… We recommend a pragmatic approach – – Apply many algorithms to our data – Assess which work best in a cost-based classification scheme.

Animal-Human Discrimination Detrended Fluctuation Analysis – best way to estimate Hurst parameter (fractal dimensionality) » Animal movements on home range tend to be fractal Potentially really exciting: Scale-dependent Lyapunov Exponents (SDLE) may be highly sensitive to weak signals embedded in a return pulse corrupted by clutter Reason to expect that other multi-scale measures are as well.

Animal-Human Discrimination Miscellaneous Thoughts Hilbert-Huang Empirical Mode Decomposition Stochastic resonance

Stochastic Resonance First reported (1992) in neuromuscular system of crayfish. Conceptually simple Sub-threshold signal may become detectable if augmented with appropriate amount of noise.

Stochastic Resonance Applications in radar detection of large aerial targets Not aware of its application to the micro- Doppler problem.