Animal-Human Discrimination Track-level Discrimination

Organization of Presentation Patterns of movement among non-human animals Home range concept

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. – 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 – animals on home range tend to return to specific points on a more-or-less regular basis.

Home Range Size in Mammals

Lévy flights vs. Brownian Motion Brownian MotionLévy Flight

Red-knobbed Hornbill

Giant argus

Analysis of Movement Data We’re going to argue for a combined approach based on a number of algorithms My work so far – and for foreseeable future – is aimed at producing an eclectic mix of algorithms that do the best job of meeting your needs.

Analysis of Movement Data Wavelet Analysis (Discrete) Differential Geometry Nonlinear algorithms

Analysis of Movement Data Wavelet Analysis (Discrete) Differential Geometry

Wavelets Signal analysis types refer to this as space-time analysis, everybody else calls it frequency-time analysis. – Actually, it can be either, or both. Conceptually similar to fft, but the basis set is different. Rather than the ***sin-cos*** basis set of fft, where amplitude- determining coefficients do the fitting, both amplitude (as with fft) and the frequency width of the basis set (i.e., the wavelet) can vary. To a large extent, wavelet analysis gets around the problem of signal non- stationarity (i.e., when frequency of the signal can vary with time and/or space. – FFT not appropriate in this case Additional benefit – can develop new wavelets that are more specific/appropriate for our purposes

Example Wavelet – the Daubechies 4 Wavelet Scaling Function

Analysis of Movement Data Wavelet Analysis (Discrete) Differential Geometry

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 and. Useful for producing null models against which data can be compared. Osculating circles and curvature are one approach.

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 = 2  Leads to prediction that most normals to a simple closed smooth curve point towards the center.

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

***The following slides would be optional***

Animal-Human Discrimination Alternative approaches Nonlinear Time-series Analysis Sensitive to signal complexity Stochastic Resonance Can dramatically enhance SNR

Nonlinear Algorithms Entropy measures – old & new » Shannon – » Kolmogorov – » Spectral Entropy – basically a Shannon entropy of the power spectrum » Approximate Entropy (AppEn) – suffers from significant statistical bias, supplanted by Sample Entropy » Sample Entropy (SampEn) – used in various applications » Permutation Entropy (PermEn) –

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)

Chaotic Dynamics

Chaotic Dynamics – 2D

Chaotic Dynamics – 3D

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

Algorithms Based on Chaos Theory Significant controversy about the applicability of chaos theory-based algorithms to analysis of most time series data. Algorithms for computing each work well…but only if applied to mathematical models. Results are in doubt 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, 1 and D2 appear to Pragmatic approach

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. Recommend a pragmatic approach – – Apply many algorithms to our data – Assess which work best in a cost-based classification scheme.

Multiscale measures Exciting relatively new conceptual approach Lyapunov Exponents ( 1 ) – so-called Scale-dependent Lyapunov Exponents (SDLE) may be highly sensitive to weak signals embedded in a return pulse corrupted by clutter Entropy measures » Multiscale Sample Entropy (MSE) – used in various applications, including analysis of human gait » Multiscale Permutation Entropy – not yet applied to human locomation Detrended Fluctuation Analysis ***

Stochastic Resonance 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 applied to the micro-Doppler problem as of yet.

Have to decide whether to use the following slides and, if so, where to put?

Animal-Human Discrimination Limb movement scales in fractal way in 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.