Negatively correlated spatial data

[Jocelyn Stalker]

Hi Chris,

I have a VHF telemetry reptile tracking dataset with at least one individual whose spatial use is actually negatively correlated (N>n) at multiple scales (annual, cumulative). I suspect this may be due to this animal having a relatively small home range, but making frequent movements that are large relative to its HR. Although this seems suspicious, I've checked the data and can find no issues, and the occurrence at multiple scales implies that this animal's movement may truly be negatively spatially correlated. Is akde() robust to this type of data? ctmm.select fits her with an IID anisotropic model, thus reducing akde to a typical 95% KDE analysis.

Thank you,

Jocelyn

[Christen Fleming]

Hi Jocelyn,

1. If you really mean spatial autocorrelation and not temporal correlation, then that doesn't really matter. Tracking data are location(time), which are time-series data and not so much spatial-field data—location is the dependent variable and time is the independent variable. Spatial-field data are stuff(location)—stuff (e.g., NDVI) is the dependent variable and location is the independent variable. There are contexts where you can think about tracking data as spatial-field data (RSFs), but that doesn't really have anything do to with KDE/AKDE.
   
2. If you mean temporal autocorrelation, then I would look at the variograms before drawing any conclusions. Is the negative autocorrelation significant? Is it persistent? A movement process can only have limited amounts of negative temporal autocorrelation at larger time lags and still maintain continuity (which is physically necessary). The only models in ctmm that have some negative autocorrelation are the oscillatory models, which sometimes show up in central place foragers. Feel free to post examples.
   
3. If these are triangulated VHF data, then might need to consider the error ellipses. Ignoring those can bias the fitted/selected models to appear more IID than they should be.
4. The IID KDE analysis in ctmm is not exactly typical. Its REML-KDEc, which has some bias corrections not present in other KDE methods.
   

Best,

Chris

[Jocelyn Stalker]

Chris,

My apologies for the delayed response. Thank you for your explanation of the independent and dependent variables of tracking data- I found that very helpful. I believe I was mistaken and meant to say that the data are temporally negatively autocorrelated.

The variograms look pretty normal to me, but I do not know what problems to look for.

I made a github page to demonstrate what I'm seeing. I fitted 4 models- one on the entirety of the dataset, and 3 on annual subsets. Fits 1 and 2 are IID with N>n, and fit 4 is IID with n>N. I thought that in IID data N=n. I had to run the code several times to knit, and one of the times fit4 was actually OUf, which then returned to IID upon the following knit. I hadn't changed the code or data at all, so this has confused me even further.

https://jocelynstalker.github.io/Gus_help/index.html

Many thanks,

Jocelyn

[Christen Fleming]

Hi Jocelyn,

That all looks really good to me for home-range estimation.

1. Assuming the data are irregularly sampled in time, fast=FALSE in variogram() is much slower to run, but will depict less false autocorrelation bias in the beginning of the variogram (which happens because of time interpolation). You don't have much data, so that shouldn't take very long to run and it might reveal that your data look even more IID.
2. By N>n, you are talking about DOF[area]>n? I think it should be okay if DOF[area] is a little different in the IID anisotropic case. The area variable here is sqrt(det(COV[location])), so while IID COV[location] would have an effective sample size of exactly n-1, after you take the determinant and square root, it can be a little different. In the isotropic case, sqrt(det(COV[location]))=VAR[location], so the relationship should be direct there. But admittedly, I haven't really though much about the anisotropic IID DOFs, and that does seem weird that it's bigger than n... I just double checked my code and it seems correct.
   
3. There might be a set of OUf parameters that are slightly better than IID but hard to get to because the likelihood function is multi-modal. I can take a look at it if you can send me a copy of that subset. (Also interested if the better variograms help here.)
   

Best,

Chris