Interpolation for Trajectories

1. Interpolation for Trajectories Marc van Kreveld (UU) reporting on master thesis research ofBart Liefers (UU) also in collaboration with Emiel van Loon (UvA)

2. Sampling and trajectories • Usually we get movement data from a set of measured locations at known times (x3,y3), t3 (x3,y3), t3 (x1,y1), t1 (x2,y2), t2

3. Sampling and trajectories • Usually we get movement data from a set of measured locations at known times (x3,y3), t3 at time (t3+t2)/2 (x3,y3), t3 (x1,y1), t1 (x2,y2), t2 (piecewise) linear interpolation

4. Sampling and trajectories • Sometimes linear interpolation is not called for

5. Sampling and trajectories • Sometimes linear interpolation is not called for

6. Sampling and trajectories • Interpolated location implies velocity (speed and heading) (x3,y3), t3 at time (t3+t2)/2 (x3,y3), t3 (x1,y1), t1 (x2,y2), t2 linear interpolation  constant speed

7. The case of gulls • Lesser black-backed gull • Five gulls, colony on Texel • Sampling intervals irregular;3 sec – 30 min • Also velocity measurementsData from the UvA, computational geo-ecology,with Emiel van Loon and Judy Shamoun-Baranes

8. 30 measurements 5:22 hours

9. Interpolation affects basic properties • Location at any time • Speed at any time • Trajectory length • Average speed piecewise linear interpolation spline interpolation

10. Interpolation affects basic properties • Location at any time • Speed at any time • Trajectory length • Average speed • Availability of velocityallows new interpolations piecewise linear interpolation spline interpolation

11. Interpolation affects basic properties • Location at any time • Speed at any time • Trajectory length • Average speed • Availability of velocityallows new interpolations piecewise linear interpolation spline interpolation consistent spline interpolation

12. Interpolation issues • Consistency location and velocity: • Position correct at the measured locations • Velocity correct at the measured locations • Integral of velocity = path between any two measured locations • Scale-invariance for trajectory length: • fewer sampled locations should not result in a smaller trajectory length

13. Interpolation issues: gull specific 11:10, velocity 0 m/s 11:00, velocity 12 m/s What velocity at 11:05?

14. What probably happened between 11:00 and 11:10 …

15. Interpolation issues: gull specific • Speed constancy between measurements t0 t1 1 m/s 12 m/s speed speed profile t0 t1 time

16. Interpolation issues: gull specific • Speed constancy between measurements t0 t1 1 m/s 12 m/s speed speed profile t0 t1 time

17. Interpolation models • Linear model (basic, ignores velocity) • Cubic Bezier models • Use measured velocity • Infer velocity from adjacent samples • Speed constancy models • Linear interpolation for path (ignores heading) • Piecewise linear interpolation for path • Path from interpolation of heading

18. Interpolation models • Extrapolation model (use velocity of nearest sample, location is not continuous) • Brownian bridges model half-time

20. Properties of the models consistency scale invariant - - - - yes - yes - continuity C0 C1 C1 C0 C0 C1 C-1 C0 speed - yes - yes yes yes yes - heading - yes - - yes yes yes - linear cubic Bezier measured cubic Bezier inferred speed constancy, linear speed constancy, PL speed constancy, heading extrapolation Brownian bridges

21. Analysis using densely sampled trajectories • Triples with 3-second intervals (exclude stationary birds) • Predict location & speed at middle sample from the outer samples • Analyze coarser and coarser sampled trajectories

22. Analysis using densely sampled trajectories • Triples with 3-second intervals (exclude stationary birds) • Predict location & speed at middle sample from the outer samples • Analyze coarser and coarser sampled trajectories Linear model

23. Analysis using densely sampled trajectories • Triples with 3-second intervals (exclude stationary birds) • Predict location & speed at middle sample from the outer samples • Analyze coarser and coarser sampled trajectories Speed constancy model, linear

24. Analysis using densely sampled trajectories • Triples with 3-second intervals (exclude stationary birds) • Predict location & speed at middle sample from the outer samples • Analyze coarser and coarser sampled trajectories Cubic Bezier model using velocity

25. Analysis of location • At high resolution several models are best, about 20% better than linear interpolation • At lower resolution the speed constancy model, linear, is best, about 30% better than linear interpolation at lower resolutions, speed helps but heading doesn’t

26. Analysis of speed and heading • At high resolutions, several models are best for speed, including linear interpolation • At lower resolutions the speed constancy model, linear, is best for speed • For heading, linear interpolation is best, especially for lower resolutions

27. Analysis of trajectory length • The extrapolation model and piecewise linear speed constancy model are not biased by sampling rate, unlike all other models • Simple integration of speed works best

28. Conclusions • For interpolation, scale matters • The particular application matters • At lower resolutions, speed helps but heading doesn’t • Speed consistency models appear to work well for location and velocity

29. The end