The Effect of Sampling Rate on Observed Statistics in a Correlated Random Walk

Heberto Mayorquin, Presentation for the Journal Club.

24 / September / 2015

Motivation

Mine

• How the sampling rate affects the statistics of the signals (micro vs macro!)

Tracking in Biology

• Homing beahviour of Rockfish.
• Oscilllations in dive depths of sharks.
• Movements in foraging areas of elks.
• Motion of bacteria.

The Model

• Study the effect of the samplig interval $\tau$.
• Individual moves with constant speed $c_{const}$ in a 2D Correlated Random Walk.
• Reorientations occur with a poisson process with parameter $\tau_R$.
• Important parameter: $\frac{\tau}{\tau_R}$.

Two Models

1. Run only.

   • Reorientations happen instantly.

2. Run and Stop.

   • Reorientations take a finite amount of time (Another $\tau_S$)

Between reorientation events an individual an individual covers a random straight line that id distributed exponentially with parameter $c_{const} \tau_R$

Orientation Change

• The orientation changes are drawn from a Von Misses Distribution.
• Close to normal distribution in a circle but does not require inifite sum.
• Makes the trajectory a correlated random walk.
• $\kappa$ can be expressed in terms of the angular deviation $\sigma_{\delta}$

$$ f_\Phi(\phi, \kappa) = \frac{e^{\kappa\cos(\phi)}}{2 \pi I_0(\kappa)} $$

Where $ I_0 $ denotes the modified Bessel function.

The Parameter $\kappa$ controls the peakedness of it.

Details of Von Misses Distribution

Quantities of Interest

• RAS Relative Apparent Speed
• AAC Apparent Angle Change
• Mean of the RAS: $\overline{c}$
• Standar deviation of RAS: $\sigma_c$
• Angular deviation of AAC $\sigma_{\theta}$

Table of values

Diffusive Limit

• The underlying angular deviation $\sigma_{\delta}$ increases from right to left.
• The bigger the angular deviation the greater the reduction in means speed.
• Two regimes.

• The apparent angular deviation $\sigma_{\theta}$ has an asymptotic limit of $\sqrt{2}$
• The underlying angular deviation $\sigma_{\delta}$ increases from bottom to top.

Table of values

Sampling Distributions (Run Only Model)

• $ \frac{\tau}{\tau_{R}}$ increases from bottom to top (1, 2, 5, 10).
• The speeds are push down with respect to their truth value.
• the y axis is not the same!
• b underlying angular deviation is lower than c.

<img src="figures/figure4.png" width=1000 height=10 00>

Table of values

Sampling Distributions (Run and Stop Model)

• b Takes less reorienting than a.
• Black bars correspond to a higher angular deviation than white.
• $ \frac{\tau}{\tau_{R}}$ increases from bottom to top (1, 2, 5, 10).
• The speeds are push down with respect to their truth value.
• the y axis is not the same!
• When you sample fast enough you see the stop / stationary events and you see lines running in perfect line.

• Diffusive Regime kicks in faster.
• Stationary times $\tau_{s} $ increase from top to bottom.

• $ \frac{\tau}{\tau_{R}}$ increases from bottom to top (1, 2, 5, 10).
• Stationary times $\tau_{s} $ increase from top to bottom.

Analytic Study for High Frequency Sampling

Assumptions

• Black bars far better sampling.
• Zero stop events are removed.

Line of Argumentation

Line of Argumentation

The Jacobian transformation from one set of coordinates to the other

KL Divergence

• Jagged line: AAC.
• Solid Line.

Author's Conclusions

• Low frequency diffusive limits.
• Intermediate levels studied by sampling.
• High frequency has a closed model approximation.

Possible Shortcomings

• Two vs three dimensions.
• Noise?
• Constant Speed?