Are speed and Tau position independent?

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Vilis Nams

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Jun 17, 2021, 12:20:21 PM6/17/21
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The estimate for speed by ctmm is one of instantaneous speed. Tau position is affected by large-scale speed and size of home range.  If instantaneous speed were to double, but the rest of the movement path structure remain the same, would tau position remain the same, or decrease?
Thanks, Vilis

Christen Fleming

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Jun 17, 2021, 9:47:21 PM6/17/21
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Hi Vilis,

In the OUF model, the mean square speed = VAR[location] / (tau[position]*tau[velocity])

so if you double the average speed, then some other parameters would have to change as well. If you want to fix the static structure of the movement paths, then you would want to shrink tau-position and tau-velocity by the same proportion. This is the same as having a link function on time, which I want to introduce to ctmm soon.

Best,
Chris

Vilis Nams

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Jun 21, 2021, 6:27:25 PM6/21/21
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Chris,

I am comparing individuals with very different home range sizes. I can see 4 ways (not mutually exclusive) that animals can increase home range size: Individuals could (1) spend more time traveling, (2) travel faster (3) travel in a straighter path at a small scale, and (4) travel in a straighter path at a larger scale (of course, I'm assuming that there are only 2 scales of interest).

I can estimate (1) by classifying active vs inactive time periods, I can estimate (2) with speed, and (3) with Tau velocity. But what would be a good estimate for (4)? It would have to be independent of the other 3.

Thanks, Vilis

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Christen Fleming

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Jun 22, 2021, 2:16:23 PM6/22/21
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Hi Vilis,

I'm not exactly sure how you are thinking about 3-4, but in the IOU and OUF models the paths appear straight as you resolve finer than tau[velocity] and they appear fractal as you resolve coarser than tau[velocity]. You could imagine a more complex multi-scale process, though. These are some of the parameterizations that I've thought about, if that helps:

tau[position] = home-range crossing timescale / location autocorrelation timescale / location time-to-independence
tau[velocity] = persistence-of-motion timescale / velocity autocorrelation timescale / velocity time-to-independence
f[velocity] = 1/tau[velocity] = tortuosity (frequency that ranges from 0 to ∞, with 0 being ballistic and ∞ being fractal)
VAR[location] = home-range area or length^2 scale
VAR[velocity] = VAR[location]/(tau[position]*tau[velocity]) = speed^2 scale
VAR[velocity] * tau[velocity]^2 = VAR[location]*tau[velocity]/tau[position]= persistence-of-motion length^2 scale
VAR[location]/tau[position] = diffusion rate

As for active and inactive periods, I would think about that as non-stationary behavior and I wouldn't naively model that in a way that would have any impact on the home-range area. I would either model that as a switching process (Michelot, Parton, Blackwell, etc.) that modify the home-range area or as a link function on time that also wouldn't modify the home-range area. The latter is going in to ctmm soon-ish.

Best,
Chris

Vilis Nams

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Jun 23, 2021, 7:17:32 PM6/23/21
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Chris,

Thanks, these are helpful.

Also, what you said about the the structure of the movement models is interesting. If the OUF is fractal as you resolve coarser than tau[velocity], then what happens at >= tau[position]?

Thanks, Vilis

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Vilis O Nams
Dept of PFES
Faculty of Agriculture
Dalhousie University
vilis.nams @ dal.ca
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Christen Fleming

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Jun 23, 2021, 10:03:19 PM6/23/21
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Hi Vilis,

At coarser than tau[position], the process appears IID, which for movement is like a teleporting process.

Best,
Chris

Vilis Nams

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Jun 25, 2021, 9:18:02 AM6/25/21
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Chris,

What parameter(s) determine the tortuousity of the fractal path between the resolutions of tau[velocity] and tau[position]?

Thanks, Vilis
Hi Vilis,

Vilis Nams

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Jun 25, 2021, 9:28:50 AM6/25/21
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Chris,

tau[velocity] affects  tortuousity, but it is also the timescale for persistence of motion. These two seem to be different things - i.e. could there be a specific tau[velocity], below which the path is straight, but above which there are different values of path toruousity?

Thanks, Vilis

Christen Fleming

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Jun 25, 2021, 2:12:55 PM6/25/21
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Hi Vilis,

If you observe for some period of time <tau[velocity], then the animal will likely be traveling in the same direction and at the same speed for that entire period, but if you wait for a longer period >tau[velocity], then the velocity will likely change. Velocity correlation implies persistence of motion (or the lack motion).

If you sample at an interval >>tau[velocity], then you can't distinguish the process from a fractal process. As you decrease the sampling interval, you keep getting more complexity/detail in the track (like a fractal) as long as you remain above tau[velocity]. But eventually, as you approach and surpass tau[velocity], the finer details of the track will emerge as smooth. So tau[velocity] is the scale at which the process can be distinguished from fractal. The process is only truly fractal if tau[velocity] is 0, but it can't be distinguished from fractal if tau[velocity] is smaller than what you can resolve.

Best,
Chris
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