Periodic movement models

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

2020-12-08

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.

# CRAN policy limits to 2 processes (cores)
FITS <- ctmm.select(Gamba,GUESS,verbose=TRUE,cores=2)
## Nyquist frequency estimated at harmonic 3 88.5917638888889 of the period.
# if you get errors on your platform, then try cores=1

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[area]
## OUF anisotropic harmonic 2 0               0.000000   60.815820  175.8871
## OUF anisotropic harmonic 1 0              20.689425   63.785895  179.2759
## OUF anisotropic harmonic 2 1               5.493885   89.639598  169.4066
## OUF anisotropic harmonic 1 1              26.103584   91.796921  172.8667
## OUF anisotropic harmonic 0 0             123.673098  157.298557  205.5584
## OUF anisotropic harmonic 0 1             128.448440  177.809096  200.7891
## OUF anisotropic harmonic 3 0             -17.231893 2496.228990  174.7865
## OUf anisotropic harmonic 2 0              19.477244    0.000000  237.2510
## OUf anisotropic harmonic 1 0              39.906174    5.147749  240.7653
## OUf anisotropic harmonic 0 0             124.354647  131.232070  233.3833
## OU anisotropic harmonic 2 0               34.440063   67.757512  146.6522
## OU anisotropic harmonic 1 0               51.739611   69.857170  150.1455
## OU anisotropic harmonic 0 0              193.520195  177.510802  163.4152
## OUF isotropic harmonic 2 0                88.869590  236.982415  163.4619
## OUF isotropic harmonic 1 0               113.063969  238.575574  166.0153
## OUF isotropic harmonic 0 0               230.160697  323.668045  196.4267
## OUF anisotropic circulation harmonic 0 0 124.426467  160.554277  204.4379
## OUF isotropic circulation harmonic 0 0   231.141563  327.779305  195.1978

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 criterion. 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 criterion is used. MSPE is not valid for general purpose selection.

# these are sorted by MSPE
SUB <- grepl("OUF anisotropic harmonic",names(FITS))
summary(FITS[SUB])
##                                   ΔAICc  ΔRMSPE (m) DOF[area]
## OUF anisotropic harmonic 2 0   0.000000    0.000000  175.8871
## OUF anisotropic harmonic 1 0  20.689425    2.970074  179.2759
## OUF anisotropic harmonic 2 1   5.493885   28.823778  169.4066
## OUF anisotropic harmonic 1 1  26.103584   30.981100  172.8667
## OUF anisotropic harmonic 0 0 123.673098   96.482737  205.5584
## OUF anisotropic harmonic 0 1 128.448440  116.993276  200.7891
## OUF anisotropic harmonic 3 0 -17.231893 2435.413170  174.7865
# these are sorted by IC
SUB <- grepl('0 0',names(FITS))
summary(FITS[SUB])
##                                                ΔAICc ΔRMSPE (m) DOF[area]
## OUF anisotropic harmonic 0 0               0.0000000   26.06649  205.5584
## OUf anisotropic harmonic 0 0               0.6815490    0.00000  233.3833
## OUF anisotropic circulation harmonic 0 0   0.7533689   29.32221  204.4379
## OU anisotropic harmonic 0 0               69.8470969   46.27873  163.4152
## OUF isotropic harmonic 0 0               106.4875996  192.43598  196.4267
## OUF isotropic circulation harmonic 0 0   107.4684652  196.54723  195.1978

If only the information criterion 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 106.67983
## OUF anisotropic harmonic 2 0              17.23189 105.76775
## OUF anisotropic harmonic 2 1              22.72578 103.79740
## OUf anisotropic harmonic 2 0              36.70914 149.25890
## OUF anisotropic harmonic 1 0              37.92132 107.66572
## OUF anisotropic harmonic 1 1              43.33548 105.71558
## OU anisotropic harmonic 2 0               51.67196  79.90435
## OUf anisotropic harmonic 1 0              57.13807 151.46287
## OU anisotropic harmonic 1 0               68.97150  81.79362
## OUF isotropic harmonic 2 0               106.10148  93.38189
## OUF isotropic harmonic 1 0               130.29586  94.69244
## OUF anisotropic harmonic 0 0             140.90499 131.12270
## OUf anisotropic harmonic 0 0             141.58654 149.16251
## OUF anisotropic circulation harmonic 0 0 141.65836 131.20178
## OUF anisotropic harmonic 0 1             145.68033 129.27965
## OU anisotropic harmonic 0 0              210.75209  88.86910
## OUF isotropic harmonic 0 0               247.39259 119.70643
## OUF isotropic circulation harmonic 0 0   248.37346 119.73740

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.

SUB <- rownames(summary(FITS,MSPE=NA))[1]
summary(FITS[[SUB]]) # harmonic 3 0 # selected by IC
## $name
## [1] "OUF anisotropic harmonic 3 0"
## 
## $DOF
##        mean        area       speed 
## 106.6798345 174.7864587   0.8479798 
## 
## $CI
##                                low        est       high
## rotation/deviation %     74.360524  91.679710 100.000000
## rotation/speed %         96.735689  98.970180 100.000000
## area (square kilometers) 41.910582  48.890071  56.399203
## τ[position] (hours)       7.019750   9.335749  12.415855
## τ[velocity] (hours)       1.108651   1.599096   2.306504
## speed (kilometers/day)   12.330893 103.010397 205.063003
summary(FITS[[1]])   # harmonic 2 0 # selected by IC/MSPE
## $name
## [1] "OUF anisotropic harmonic 2 0"
## 
## $DOF
##     mean     area    speed 
## 105.7677 175.8871 139.8130 
## 
## $CI
##                                 low       est      high
## rotation/deviation %     23.4517770 28.855685 35.504796
## rotation/speed %         24.2563566 29.787735 36.580479
## area (square kilometers) 41.5574968 48.453880 55.871898
## τ[position] (hours)       7.2915577  9.600382 12.640280
## τ[velocity] (hours)       0.9719415  1.428103  2.098356
## speed (kilometers/day)   14.7661450 16.100485 17.433375

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[[SUB]],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!