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.

library(ctmm)
data(wolf)
Gamba <- wolf$Gamba
plot(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.

plot(LSP,max=TRUE,diagnostic=TRUE,cex=0.5)

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.

Periodicities in the data cause peaks at their respective periods on the horizontal axis. To visually assess significance, these peaks should be compared to the normal variation around the smooth trend of the periodogram. Important natural periods like the stellar day, synodic month and tropical year are labeled on the horizontal axes and given dark vertical lines in the plot. Harmonics of these natural periods have unlabeled tick marks on the horizontal axis and are are given lighter vertical lines in the plot. E.g., the two tick marks under a day represent the second and third harmonic of the day, or (24/2) 12 hours and (24/3) 8 hours, respectively Peaks at the harmonics of a period indicate that fine details of the periodicity are resolved by the data.

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 periodic and 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.

PROTO <- ctmm(mean="periodic",period=c(24 %#% "hours",1 %#% "month"),circle=TRUE)

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 variogram.fit 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 variogram.fit.

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 <- ctmm.select(Gamba,GUESS,verbose=TRUE,control=control)
## Nyquist frequency estimated at harmonic 3 88.5917639444444 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, where the first element of that list is the preferred model.

summary(FITS)
##                                               ΔAICc  ΔRMSPE (m) DOF[mean]
## OUF anisotropic harmonic 2 0               0.000000    0.000000  106.1204
## OUF anisotropic harmonic 1 0              20.554301    3.829218  108.2142
## OUF anisotropic harmonic 2 1               5.562484    6.341045  107.9071
## OUF anisotropic harmonic 1 1              25.975607    9.790403  110.0647
## OUF anisotropic harmonic 0 0             123.457393  101.908218  132.4812
## OUF anisotropic harmonic 0 1             128.194996  104.472521  134.7361
## OUF anisotropic harmonic 3 0             -17.064596 2416.027179  107.1803
## OUF anisotropic circulation harmonic 0 0 124.199987  101.825991  132.9631
## OUF isotropic harmonic 0 0               229.937343  266.697224  120.9668
## OUF isotropic circulation harmonic 0 0   230.906720  266.639995  121.3771

From the first model, We see that the velocity autocorrelation (OUF), anisotropy, and 2 harmonics of daily periodicity were all selected. Given how we specified the prototype, harmonic 2 0 means that this preferred model has no lunar periodicity, but has two 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.

The sorting of our candidate models is more complex than in previous stationary examples. For a given autocovariance model, the different non-stationary models are sorted by mean square predictive error (MSPE) and not the information criteria. As we will demonstrate, likelihood-based model selection can badly overfit with these types of models. For sorting between the autocovariance models (each with best non-stationary model), the information criteria is used. MSPE is not valid for general purpose selection.

# these are sorted by MSPE
summary(FITS[1:7])
##                                   ΔAICc  ΔRMSPE (m) DOF[mean]
## OUF anisotropic harmonic 2 0   0.000000    0.000000  106.1204
## OUF anisotropic harmonic 1 0  20.554301    3.829218  108.2142
## OUF anisotropic harmonic 2 1   5.562484    6.341045  107.9071
## OUF anisotropic harmonic 1 1  25.975607    9.790403  110.0647
## OUF anisotropic harmonic 0 0 123.457393  101.908218  132.4812
## OUF anisotropic harmonic 0 1 128.194996  104.472521  134.7361
## OUF anisotropic harmonic 3 0 -17.064596 2416.027179  107.1803
# these are sorted by IC
summary(FITS[c(5,8:10)])
##                                                ΔAICc  ΔRMSPE (cm)
## OUF anisotropic harmonic 0 0               0.0000000     0.000000
## OUF anisotropic circulation harmonic 0 0   0.7425933    -8.222723
## OUF isotropic harmonic 0 0               106.4799500 16478.900545
## OUF isotropic circulation harmonic 0 0   107.4493271 16473.177657
##                                          DOF[mean]
## OUF anisotropic harmonic 0 0              132.4812
## OUF anisotropic circulation harmonic 0 0  132.9631
## OUF isotropic harmonic 0 0                120.9668
## OUF isotropic circulation harmonic 0 0    121.3771

If only the information criteria was used, we would have selected harmonic 3 0 over harmonic 2 0.

# sorting by IC only
summary(FITS,MSPE=NA)
##                                              ΔAICc DOF[mean]
## OUF anisotropic harmonic 3 0               0.00000  107.1803
## OUF anisotropic harmonic 2 0              17.06460  106.1204
## OUF anisotropic harmonic 2 1              22.62708  107.9071
## OUF anisotropic harmonic 1 0              37.61890  108.2142
## OUF anisotropic harmonic 1 1              43.04020  110.0647
## OUF anisotropic harmonic 0 0             140.52199  132.4812
## OUF anisotropic circulation harmonic 0 0 141.26458  132.9631
## OUF anisotropic harmonic 0 1             145.25959  134.7361
## OUF isotropic harmonic 0 0               247.00194  120.9668
## OUF isotropic circulation harmonic 0 0   247.97132  121.3771

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[[7]]) # harmonic 3 0 # selected by IC
## $name
## [1] "OUF anisotropic harmonic 3 0"
## 
## $DOF
##        mean        area       speed 
## 107.1803473 178.3530926   0.8606009 
## 
## $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  55.605890
## τ[position] (hours)       6.984418   9.267908  12.297962
## τ[velocity] (hours)       1.126099   1.617410   2.323076
## speed (kilometers/day)   12.674828 103.001923 204.382857
summary(FITS[[1]]) # harmonic 2 0 # selected by MSPE
## $name
## [1] "OUF anisotropic harmonic 2 0"
## 
## $DOF
##     mean     area    speed 
## 106.1204 179.5625 139.5356 
## 
## $CI
##                                 low        ML      high
## rotation/deviation %     23.6228690 29.019474 35.648925
## rotation/speed %         24.3431625 29.868660 36.648355
## area (square kilometers) 41.1021783 47.844462 55.091282
## τ[position] (hours)       7.2925137  9.568355 12.554439
## τ[velocity] (hours)       0.9699886  1.423753  2.089789
## speed (kilometers/day)   14.7247951 16.056839 17.387435
  • rotation/deviation % corresponds to \(100 \eta_P\) from the paper. It is interpreted as the proportion of the variance in the animal`s location that is caused by the periodicity in the mean.
  • rotation/speed % corresponds to \(100 \eta_V\) from the paper. It is interpreted as the proportion of the variance in the animal`s velocity that is caused by the periodicity in the mean.
  • circulation period is the period of the stochastic circulations. On average, the animal re-pass through the same neighborhoods every estimated number of months (or days, or hours, depending on the automated unit specification).

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"
plot(SVF,CTMM=FITS[[7]],xlim=xlim)
title("3 Harmonics")
plot(SVF,CTMM=FITS[[1]],xlim=xlim)
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.

Home-range estimation

This is not coded yet!