# Periodic movement models

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

#### 2025-07-25

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)

    ## DOP values missing. Assuming DOP=1.

### 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 1 0               0.000000   56.39112  182.8373
    ## OUF anisotropic harmonic 2 0             -20.627854   56.80684  178.2975
    ## OUF anisotropic harmonic 1 1               6.968505  100.02511  171.6185
    ## OUF anisotropic harmonic 0 1             107.016321  154.17783  214.5140
    ## OUF anisotropic harmonic 0 0              98.889310  157.45951  205.5584
    ## OUf anisotropic harmonic 1 0              19.201758    0.00000  243.5731
    ## OUf anisotropic harmonic 0 0              99.570859  131.39303  233.3833
    ## OU anisotropic harmonic 1 0               31.032816   59.70672  156.3821
    ## OU anisotropic harmonic 0 0              168.736407  177.67176  163.4152
    ## OUF harmonic 1 0                          92.348707  229.44429  170.3281
    ## OUF harmonic 0 0                         205.376910  323.82900  196.4267
    ## OUF anisotropic circulation harmonic 0 0  99.628111  157.34052  205.5747
    ## OUF circulation harmonic 0 0             206.342545  323.66590  196.4744

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 1 0   0.000000   0.0000000  182.8373
    ## OUF anisotropic harmonic 2 0 -20.627854   0.4157239  178.2975
    ## OUF anisotropic harmonic 1 1   6.968505  43.6339950  171.6185
    ## OUF anisotropic harmonic 0 1 107.016321  97.7867131  214.5140
    ## OUF anisotropic harmonic 0 0  98.889310 101.0683886  205.5584

    # 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   26.06648  205.5584
    ## OUf anisotropic harmonic 0 0               0.6815490    0.00000  233.3833
    ## OUF anisotropic circulation harmonic 0 0   0.7388014   25.94749  205.5747
    ## OU anisotropic harmonic 0 0               69.8470969   46.27874  163.4152
    ## OUF harmonic 0 0                         106.4875996  192.43598  196.4267
    ## OUF circulation harmonic 0 0             107.4532352  192.27287  196.4744

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 2 0               0.00000 106.12269
    ## OUF anisotropic harmonic 1 0              20.62785 108.22504
    ## OUF anisotropic harmonic 1 1              27.59636 105.53444
    ## OUf anisotropic harmonic 1 0              39.82961 152.03911
    ## OU anisotropic harmonic 1 0               51.66067  82.78313
    ## OUF harmonic 1 0                         112.97656  95.20197
    ## OUF anisotropic harmonic 0 0             119.51716 131.12270
    ## OUf anisotropic harmonic 0 0             120.19871 149.16251
    ## OUF anisotropic circulation harmonic 0 0 120.25597 131.61505
    ## OUF anisotropic harmonic 0 1             127.64418 133.20300
    ## OU anisotropic harmonic 0 0              189.36426  88.86910
    ## OUF harmonic 0 0                         226.00476 119.70643
    ## OUF circulation harmonic 0 0             226.97040 120.85055

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 2 0"
    ##
    ## $DOF
    ##      mean      area diffusion     speed
    ##  106.1227  178.2975  307.6169  138.8300
    ##
    ## $CI
    ##                                          low       est      high
    ## period.1 (days)                    0.9991546  1.000011  1.000869
    ## period.2 (days)                    0.0000000 18.778706       Inf
    ## rotation/deviation %              23.9687327 28.903276 33.754619
    ## rotation/speed %                  24.6949227 29.751659 34.718623
    ## area (square kilometers)          41.4466473 48.272397 55.610805
    ## τ[position] (hours)                7.4216923  9.566799 12.331909
    ## τ[velocity] (hours)                1.0061426  1.424067  2.015586
    ## speed (kilometers/day)            14.7941007 16.136116 17.476670
    ## diffusion (square kilometers/day)  8.9797476 10.074195 11.230665

    summary(FITS[[1]])   # harmonic 2 0 # selected by IC/MSPE

    ## $name
    ## [1] "OUF anisotropic harmonic 1 0"
    ##
    ## $DOF
    ##      mean      area diffusion     speed
    ##  108.2250  182.8373  302.8163  123.7805
    ##
    ## $CI
    ##                                          low        est      high
    ## period.1 (days)                    0.9988660  0.9999289  1.000993
    ## period.2 (days)                    0.0000000 17.2778708       Inf
    ## rotation/deviation %              23.1952585 28.2895594 33.297290
    ## rotation/speed %                  21.7242005 26.8305241 31.854304
    ## area (square kilometers)          41.4125296 48.1373907 55.360788
    ## τ[position] (hours)                7.3011104  9.3878346 12.070964
    ## τ[velocity] (hours)                0.9691849  1.3856754  1.981145
    ## speed (kilometers/day)            14.9283532 16.3701651 17.810324
    ## diffusion (square kilometers/day)  9.1621891 10.2884845 11.479125

- `rotation/deviation %` corresponds to \(\eta_P \times 100\%\) from the Péron et al (2017). 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 \(\eta_V \times 100\%\) from the Péron et al (2017). 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[[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!