Periodic movement models

Guillaume Péron, Christen H. Fleming, and Justin M. Calabrese


In this vignette, we walk through periodogram for movement data and periodic movement model fitting and selection. It is assumed that you are already familiar with data preparation with ctmm, as well as the maximum likelihood procedure described in the variogram vignette. Our example maned wolf data is already prepared into a telemetry object.

Gamba <- wolf$Gamba

Non-parametric approach: Periodograms

Before anything else, we want to plot the data in a way that makes periodic patterns apparent. This is the periodogram.

LSP <- periodogram(Gamba,fast=2,res.time=2)

The fast=2 option requests the use of the (much) faster FFT-based algorithm and furthermore samples a highly composite number of times. Set the argument to FALSE to revert to Scargle’s original algorithm, which involves fewer numerical approximations. The res.time argument increases the resolution of the temporal grid when fast>0. The algorithm defaults to adequate resolution for regularly scheduled data (permitting gaps), while variable sampling rates require res.time>1 to resolve the fine scale spectrum correctly.


The max=TRUE option keeps only local maxima and often yields periodograms that are easier to interpret, especially when the resolution of the periodogram has been increased from the default.

The diagnostic=TRUE option draws the periodogram of the sampling schedule with red symbols. If the periodogram of the sampling schedule exhibits peaks, this indicates that the corresponding peaks in the movement data could be simply caused by irregularities in the sampling schedule and not by periodicity in the movement. Here the periodogram exhibits a clear peak for the period of one day, but this is also present in the (red) diagnostic.

Periodograms are a great data exploration tool, but they do not detect everything. In particular circulation processes, those that induce circulatory patterns via a stochastic rotational effect, are often not visible in periodograms. Periodograms are also difficult to compare from one individual to the next, if the aperiodic stochastic movements of these individuals are not the same.

Parametric approach: Fitting movement models with circulatory patterns

Circulatory patterns can be incorporated into stochastic continuous time movement models in two ways: via the mean of the process, modelling that the animal reverts to a point that moves periodically through time, or via the stochastic component, by including a rotation effect. The first is called a periodic-mean process, and the second is called a circulation process.

For periodic-mean processes, you need to specify one or more period values that you want ctmm to consider.

In our example, we are going to specify a period of a day, which is what the periodogram was saying. Other individuals in this population were also exhibiting a weekly periodicity, and in similar species, lunar cycles are well-known to affect ranging behaviors.

# circle=TRUE takes longer to fit and gives optim more problems
# PROTO <- ctmm(mean="periodic",period=c(1 %#% "day",1 %#% "month"),circle=TRUE)
PROTO <- ctmm(mean="periodic",period=c(1 %#% "day",1 %#% "month"))

The mean="periodic" option tells ctmm that you want a periodic-mean process. The default is mean="stationary". The period values are all specified in seconds (SI unit) by the %#% function (see help("%#%") for more information).

The circle=TRUE option tells ctmm that you want to try and fit a model in which there is stochastic circulation on top of the periodicity in the process mean. For circulation processes, you do not need to specify candidate period values. The period of the circulation is estimated from the data.

As with other uses of ctmm, the next step is to generate an initial guess of the parameter values to feed to the likelihood optimization routine. You can do this using your intuition, by visually examining the variogram, or by letting ctmm do it for you.

SVF <- variogram(Gamba,res=3)
GUESS <- ctmm.guess(Gamba,PROTO,variogram=SVF,interactive=FALSE)

ctmm.guess is a generalization of that can estimate other quantites from the data that are not apparent in the variogram, such as the circulation period and location correlations. A variogram argument is not necessary, but here we used the res option to increase the FFT variogram’s temporal resolution and counteract sampling variability. The interactive argument works just as with

Potentially, the most complex model based on our prototype could have both circulation and multiple harmonics of daily periodicity. In addition, we also do not know whether the velocities are autocorrelated in time (OUF model) and whether the animal moves amounts in different directions (anisotropy). All of this makes for a large number of effects. Forcing all these effects upon data that do not support them would inflate the risk of overfitting or convergence issues. We thereby conduct a model selection.

# ctmm beta optimizer is more reliable here
# control <- list(method="pNewton",cores=-1) # use all but 1 core
control <- list(method="pNewton",cores=2) # CRAN policy limits to 2 processes
FITS <-,GUESS,verbose=TRUE,control=control)
## Nyquist frequency estimated at harmonic 3 88.5917638888889 of the period.

The non-default control list here specifies our prototype optimization code that has been tested to be faster and more accurate on this example. With verbose=TRUE, we obtain a list fitted ctmm objects, one for each relevant combination of effects, in order of increasing AICc. The first element of that list is the preferred model.

##                                   dAICc DOF[mean] method
## OUF anisotropic harmonic 3 0   0.000000  107.1803     ML
## OUF anisotropic harmonic 4 0   4.818792  107.1268     ML
## OUF anisotropic harmonic 3 1   5.599727  108.9836     ML
## OUF anisotropic harmonic 2 0  17.064596  106.1204     ML
## OUF anisotropic harmonic 2 1  22.627079  107.9071     ML
## OUF anisotropic harmonic 1 0  37.618897  108.2142     ML
## OUF anisotropic harmonic 1 1  43.040203  110.0647     ML
## OUF anisotropic harmonic 0 0 140.521989  132.4811     ML
## OUF anisotropic harmonic 0 1 145.259592  134.7361     ML

We see that the velocity autocorrelation (OUF), anisotropy, and circulation were unambiguously selected using the level option and were present in all the candidate models. In this maned wolf, there is rather strong support (Delta AIC > 4) for the preferred model (the first of the list). Given how we specified the prototype, harmonic 3 0 means that this preferred model has no lunar periodicity, but has three harmonics of the one-day periodicity. This fittingly corresponds to what the periodogram was saying. If there was some moon-related pattern of space use and given how we specified our prototype, we would have had a non-zero value as the second harmonic value. Circulation would also be selected if included.

Next we do some sanity checking on our results. The sampling interval for Gamba is fairly steady at

"hour" %#% stats::median(diff(Gamba$t))
## [1] 4

4 hours, which (ideally) corresponds to a Nyquist period of 8 (2 \(\times\) 4) hours, or 3 (24/8) times per day, or 3 harmonics of the day. The Nyquist period/frequency is an information limit on discretely sampled data. We expect to be able to extract a maximum of 3 harmonics from uniformly sampled data. Therefore, we should limit our consideration to harmonics of the day \(\leq\) 3, while for lower quality data, we might have to limit our consideration even further.

Consistent with these considerations, let us look at harmonics 3 and 2 of the day.

summary(FITS[[1]]) # harmonic 3 0
## $DOF
##     mean     area 
## 107.1803 178.3531 
## $CI
##                                low         ML       high
## rotation/deviation %     74.795669  91.774630 100.000000
## rotation/speed %         96.807431  98.987421 100.000000
## area (square kilometers) 41.444887  48.269156  55.605889
## tau position (hours)      6.984418   9.267907  12.297961
## tau velocity (hours)      1.126099   1.617410   2.323076
## speed (kilometers/day)   12.674828 103.001923 204.382857
summary(FITS[[4]]) # harmonic 2 0
## $DOF
##     mean     area 
## 106.1204 179.5625 
## $CI
##                                 low        ML      high
## rotation/deviation %     23.6228690 29.019474 35.648925
## rotation/speed %         24.3431626 29.868660 36.648355
## area (square kilometers) 41.1021785 47.844463 55.091282
## tau position (hours)      7.2925137  9.568355 12.554439
## tau velocity (hours)      0.9699886  1.423753  2.089789
## speed (kilometers/day)   14.7247950 16.056839 17.387435

Note that, aside from the rotational indices, these two models are largely consistent, and the 3 harmonic model has an extrordinarily uncertain speed estimate. We can get a better idea of what is happening by comparing the variograms.

xlim <- c(0,1/2) %#% "month"
title("3 Harmonics")
title("2 Harmonics")

While the confidence bands encompass the empirical variogram, the 3 harmonic model has clearly overfit in attempting to match the Nyquist period (3/day) with less than ideal data. A more stringent infromation criteria would not necessarily select the 3 harmonic model.

##                                      dBIC DOF[mean] method
## OUF anisotropic harmonic 3 0   0.00000000  107.1803     ML
## OUF anisotropic harmonic 2 0   0.02420079  106.1204     ML
## OUF anisotropic harmonic 1 0   3.47828338  108.2142     ML
## OUF anisotropic harmonic 4 0  21.79870746  107.1268     ML
## OUF anisotropic harmonic 3 1  22.57964253  108.9836     ML
## OUF anisotropic harmonic 2 1  22.62707927  107.9071     ML
## OUF anisotropic harmonic 1 1  25.99980779  110.0647     ML
## OUF anisotropic harmonic 0 0  89.22197845  132.4811     ML
## OUF anisotropic harmonic 0 1 111.11897842  134.7361     ML

But still, it is troubling that a bad model has the lowest BIC. Last, we turn to the mean square predictive error (MSPE), which explicitly penalizes highly-variable trend predictions like the ones we see here.

##                              dRMSPE (m) DOF[mean] method
## OUF anisotropic harmonic 2 0   0.000000  106.1204     ML
## OUF anisotropic harmonic 3 0   2.508577  107.1803     ML
## OUF anisotropic harmonic 4 0   2.531323  107.1268     ML
## OUF anisotropic harmonic 1 0   3.832049  108.2142     ML
## OUF anisotropic harmonic 2 1   6.085733  107.9071     ML
## OUF anisotropic harmonic 3 1   8.285109  108.9836     ML
## OUF anisotropic harmonic 1 1   9.542700  110.0647     ML
## OUF anisotropic harmonic 0 0 101.913159  132.4811     ML
## OUF anisotropic harmonic 0 1 104.265404  134.7361     ML

Finally, a good model wins out, with the 2 harmonic model beating the 3 harmonic model by 2.5 meters of RMS predictive error (RMSPE). Unlike likelihood based ICs (AIC & BIC), MSPE has units and an absolute scale, though in summary we only show differences in the RMSPE for simplicity. Note that MSPE should only be used to select among models with the same autocorrelation structure.

Home-range estimation

This is not coded yet!