├── .gitattributes
├── .gitignore
├── Day1
    ├── HMM_Tutorial_Day1.pptx
    ├── Tutorial_HMM_2022_Day1.Rmd
    ├── Tutorial_HMM_2022_Day1.html
    └── data
    │   ├── bathy.grd
    │   ├── bathy.gri
    │   └── seals.csv
├── Day2
    ├── HMM_Tutorial_Day2.pptx
    ├── Tutorial_HMM_2022_Day2.Rmd
    ├── Tutorial_HMM_2022_Day2.html
    └── data
    │   ├── Eclipse_Sound_Map.jpg
    │   ├── dives.csv
    │   ├── land.dbf
    │   ├── land.prj
    │   ├── land.shp
    │   ├── land.shx
    │   └── tracks.csv
├── Day3
    ├── HMMs_DFO_TutorialDay3.pdf
    ├── Tutorial_HMM_2022_Day3.Rmd
    ├── Tutorial_HMM_2022_Day3.html
    └── data
    │   ├── BlacktipB.txt
    │   └── tigershark_depthchange10min.csv
├── LectureSlides
    ├── HMMs_DFO_Day1 - Copy.pptx
    ├── HMMs_DFO_Day1.pptx
    ├── HMMs_DFO_Day2.pdf
    └── HMMs_DFO_Day3.pdf
├── README.md
└── preWorkshopInfo
    ├── 0_Installing_packages.Rmd
    ├── 0_Installing_packages.html
    └── email.docx


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   1 | ---
   2 | title: 'Hidden Markov Models: Basics for animal movement data' 
   3 | author: "Marie Auger-Méthé & Ron Togunov"
   4 | output: 
   5 |   bookdown::html_document2:
   6 |     number_sections: true
   7 |     highlight: tango
   8 | editor_options:
   9 |   chunk_output_type: console
  10 | ---
  11 | 
  12 | <!-- To be able to have continuous line numbers -->
  13 | ```{=html}
  14 | <style>
  15 | body
  16 |   { counter-reset: source-line 0; }
  17 | pre.numberSource code
  18 |   { counter-reset: none; }
  19 | </style>
  20 | ```
  21 | 
  22 | 
  23 | ```{r setup, include=FALSE}
  24 | knitr::opts_chunk$set(echo = TRUE)
  25 | ```
  26 | # Tutorial goals and set up
  27 | 
  28 | The goal of this tutorial is to explore how to fit hidden Markov models
  29 | (HMMs) to movement data. To do so, we will investigate a new R package,
  30 | `momentuHMM`. This package builds on a slightly older package,
  31 | `moveHMM`, that was developed by Théo Michelot , Roland Langrock, and
  32 | Toby Patterson, see associated paper:
  33 | <https://besjournals.onlinelibrary.wiley.com/doi/full/10.1111/2041-210X.12578>.
  34 | `momentuHMM`, was developed by Brett McClintock and Théo Michelot.
  35 | momentuHMM has new features such as allowing for more data streams,
  36 | inclusion of covariates as raster layers, and much more, see associated
  37 | paper:
  38 | <https://besjournals.onlinelibrary.wiley.com/doi/abs/10.1111/2041-210X.12995>.
  39 | 
  40 | The primary learning objectives in this first tutorial are to:
  41 | 
  42 | 1. Fit simple HMMs using `momentuHMM`
  43 | 2. Checking model fit
  44 | 3. Determining number of states to use
  45 | 4. Incorporating and interpreting covariates on behaviour transition probabilities
  46 | 5. Exploring effect of using different temporal resolutions
  47 | 
  48 | ## Setup and data preparation
  49 | 
  50 | First, let's load the packages that we will need to complete the
  51 | analyses. Off course you need to have them installed first.
  52 | 
  53 | ```{r Load packages, echo=TRUE, message=FALSE, attr.source = ".numberLines"}
  54 | library(momentuHMM) # Package for fitting HMMs, builds on moveHMM
  55 | library(raster) # For importing and extracting raster spatial covariates
  56 | ```
  57 | 
  58 | While the `raster` package is being phased out in favour of the new `terra` package, `momentuHMM` still relies on the `raster` package to extract covariates. Therefore, for this tutorial we will still use `raster`, but do start working with `terra` in your work where possible for future compatability. 
  59 | 
  60 | Make sure to set working directory to "Day1" of the HMM workshop folder:
  61 | 
  62 | ```{r, eval=FALSE, message=FALSE, attr.source = ".numberLines"}
  63 | setwd("Day1")
  64 | ```
  65 | 
  66 | One of the main features of the data used with HMMs is that locations
  67 | are taken at regular time steps and that there are negligible position
  68 | error. For example, HMMs are adequate to use on GPS tags that take
  69 | locations at a set temporal frequency (e.g., every 2 hrs). Without
  70 | reprocessing, HMMs are not particularly good for irregular times series
  71 | of locations or locations with large measurement error (e.g., you would
  72 | need to preprocess Argos data before applying HMMs).
  73 | 
  74 | Unfortunately, movement of aquatic
  75 | species is rarely taken at regular time intervals and without
  76 | measurement error. For example, the data set we will work on, is the
  77 | movement track of a grey seal tagged with a Fastlock GPS tag. This is a
  78 | subset of the data set used in the paper from Whoriskey et al. 2017
  79 | (<https://onlinelibrary.wiley.com/doi/abs/10.1002/ece3.2795>), which
  80 | applies a set of different HMMs to the movement data of various aquatic
  81 | species. The original dataset is available in the online supplementary
  82 | information.
  83 | 
  84 | Let's read the data file and peak at it.
  85 | 
  86 | ```{r Load data, warning=FALSE, attr.source = ".numberLines"}
  87 | # Make sure you are in the good directory!
  88 | seal <- read.csv("data/seals.csv", stringsAsFactors = FALSE)
  89 | # Look at the data
  90 | head(seal)
  91 | ```
  92 | 
  93 | The files has three columns: date with the date and time, lon with the
  94 | longitude, and lat with latitude. It's clear from the date column that
  95 | the locations are taken at irregular time intervals. So just like in
  96 | Whoriskey et al. 2017, we will regularize it. Note, that `momentuHMM`
  97 | has functions to preprocess irregular movement time series and/or time
  98 | series with large measurement errors (e.g., `crawlWrap`). 
  99 | However, these
 100 | functions are much more complex. 
 101 | We will cover some of these functions in tomorrow's tutorial.
 102 | For now, as in
 103 | Whoriskey et al. 2017, we will assume that the animal moves in a
 104 | straight line and at constant speed between two locations and
 105 | interpolate the location at regular time intervals based this
 106 | assumption. Note that this kind of interpolation may not be adequate in
 107 | many circumstances and see activities to investigate the effects of the
 108 | choice of the time interval on the analysis.
 109 | 
 110 | ```{r, attr.source = ".numberLines"}
 111 | # First, let's transform the date/time info into a
 112 | # proper time format
 113 | seal$date <- as.POSIXct(seal$date, format = "%Y-%m-%d %H:%M:%S",
 114 | tz = "GMT")
 115 | # Let's find the most appropriate time interval
 116 | # Calculate the time intervals
 117 | tint <- as.numeric(diff(seal$date), units = "hours")
 118 | # Let's look at the time intervals
 119 | summary(tint)
 120 | # Min. 1st Qu. Median Mean 3rd Qu. Max.
 121 | plot(tint, ylab = "Time interval (hr)", pch = 19, cex = 0.5,
 122 | las = 1)
 123 | # Let's go for 2 hrs, look at the activities to see
 124 | # the effect of this choice
 125 | # Regularising Create a 2hr sequence (2*60*60) from
 126 | # the first time to the last time
 127 | ti <- seq(seal$date[1], seal$date[length(seal$date)],
 128 | by = 2 * 60 * 60)
 129 | head(ti)
 130 | # Interpolate the location at the times from the sequence
 131 | iLoc <- as.data.frame(cbind(lon = approx(seal$date, seal$lon,
 132 | xout = ti)$y, lat = approx(seal$date, seal$lat, xout = ti)$y))
 133 | # Create a new object that has the regular time and
 134 | # interpolated locations
 135 | sealreg <- cbind(date = ti, iLoc)
 136 | head(sealreg)
 137 | # Quickly plot the regularized locations
 138 | plot(sealreg$lon, sealreg$lat, pch = 19, cex = 0.5, xlab = "Lon",
 139 | ylab = "Lat", las = 1)
 140 | # Add the original data
 141 | points(seal$lon, seal$lat, pch = 19, cex = 0.5, col = rgb(1,
 142 | 0, 0, 0.5))
 143 | # add legend
 144 | legend(x = -62.4, y = 42.1, pch = c(19, 19),
 145 |        legend = c("Original data", "Regularised locations"),
 146 |        col = c("red", "black"))
 147 | ```
 148 | 
 149 | Here, we interpolated the location data to 2 hours. The resolution we 
 150 | choose can have a significant effect on the results, so the this 
 151 | decision must be made wisely. At the end of this tutorial, in section
 152 | [Effects of interpolation]{#effect_of_int}. In tomorrow's tutorial, we 
 153 | will review a more comprehensive strategy to select an appropriate 
 154 | temporal resolution when we have irregular data. 
 155 | 
 156 | Ok, we have now a regular time series of locations. For the most basic
 157 | movement HMMs, you need to transform the time series of locations into
 158 | two time-series: one which represents the step lengths (distance between
 159 | two locations) and one that represents the turning angle (angle between
 160 | two steps). The package `momentuHMM` has a function that does that:
 161 | `prepData`. The data is in latitude and longitude and I'm assuming using
 162 | WGS84. You can use `prepData` on latitude and longitude data or on
 163 | projected data (e.g., UTM). `prepData` can do much more, and we will see
 164 | some other features later, but for now the arguments that we need to
 165 | think about are:
 166 | 
 167 | -   type: whether it's easting/northing coordinate system (`type="UTM"`)
 168 |     or whether it's longitude/latitude (`type = "LL"`). We are using
 169 |     `"LL"`.
 170 | 
 171 | -   `"coordNames"`: the names of the columns with the coordinates. So in
 172 |     our case `lon` and `lat`.
 173 | 
 174 | ```{r, attr.source = ".numberLines"}
 175 | # Preparing the data: here mainly calculating step
 176 | # lengths and turning angles
 177 | sealPrep <- prepData(sealreg, type = "LL", coordNames = c("lon",
 178 | "lat"))
 179 | # Let's peak at the data
 180 | head(sealPrep)
 181 | # What kind of object is this?
 182 | class(sealPrep)
 183 | ```
 184 | 
 185 | If we take a quick look at the data, we can see that the step lengths
 186 | (column step) and angles (column angle) have been calculated and that
 187 | the names of the columns with the coordinates are now x and y, rather
 188 | than `lon` and `lat`. We also have a new column: `ID`. In cases where
 189 | you have multiple individuals, you would want your original data to have
 190 | an `ID` column before being imputed in the `prepData` function. 
 191 | Note that the object it returns is from a specific S3 class defined in 
 192 | the `momentuHMM` package. This means that we can apply generic functions
 193 | like `plot` to it and it will return specified outputs.
 194 | 
 195 | ```{r, attr.source = ".numberLines"}
 196 | plot(sealPrep)
 197 | ```
 198 | 
 199 | We can see the track of the animal and the times series of the step
 200 | lengths and turning angles, as well as the histograms of the step
 201 | lengths and turning angles. Can you already identify patterns? 
 202 | 
 203 | ## Fitting the HMM
 204 | 
 205 | Now that our data is ready to use, we can fit the HMM. For this we use
 206 | the function `fitHMM`. This is where we need to make many of our
 207 | modelling decisions and most of these decisions will be associated with
 208 | one of the arguments of the function. The minimum arguments `fitHMM`
 209 | requires are: `fitHMM(data, nbState, dist, Par0)`. 
 210 | 
 211 | First, when we fit a HMM, we need to decide the number of behavioural 
 212 | states we are going to model. To start simple, we will only use two 
 213 | behavioural states. These could be, for example, representing one 
 214 | behaviour with fast and directed movement (e.g., travelling) and one 
 215 | behaviour with a more slow and tortuous movement (e.g., foraging). This 
 216 | mean that the argument `nbState` will be set to 2. Second, we need to 
 217 | decide whether we make the transition probabilities between the 
 218 | behavioural states dependent on covariates. Here again, we will start 
 219 | simple and we will not use covariates. That means that our formula 
 220 | argument will be `~1`. Third, we need to select the distribution we will
 221 | use to model the step lengths and the one we will use to model the 
 222 | turning angles. For now, we will use the gamma distribution for the step
 223 | length and the von Mises for the turning angles. This means that the 
 224 | argument dist will be set to: `list(step="gamma", angle="vm")`. Note 
 225 | that `dist` should be a list with an item for each data stream columns 
 226 | in the data that we are modelling (so here the column `step` and 
 227 | `angle`). The gamma distribution is strictly positive (i.e., it does not
 228 | allow for 0s). If you have step lengths that are exactly zero in your 
 229 | data set, you need to use zero-inflated distributions. But in this case,
 230 | we have no zeros. Let's check
 231 | 
 232 | ```{r, attr.source = ".numberLines"}
 233 | sum(sealPrep$step == 0, na.rm = TRUE)
 234 | ```
 235 | 
 236 | Since the sum is 0, we don't have any values of zero and don't need to 
 237 | worry about modelling zero-inflation.
 238 | 
 239 | By default, `fitHMM` will set the argument `estAngleMean` to `NULL`,
 240 | which means that we assume that the mean angle is 0 for both behaviours
 241 | (i.e., the animal has a tendency to continue in the same direction) and
 242 | that only the angle concentration parameters differ. The concentration
 243 | parameters control the extent to which the animal continues forward
 244 | versus turn. Doing so reduces the number of parameters to estimate.
 245 | These are all very important decisions that you must make when you
 246 | construct your model. In addition, we need to specify initial values for
 247 | the parameters to estimate. The HMMs are fit using maximum likelihood
 248 | estimate (MLE) with a numerical optimizer. An unfortunate aspect of
 249 | fitting models using numerical optimizers, is that, to be able to
 250 | explore the parameter space and find the best parameter values for the
 251 | model, the optimizer needs to start somewhere. You need to decide where
 252 | it starts. Unfortunately, choosing bad starting values can result in
 253 | parameter estimates that are not the MLE, but just a local maximum. To
 254 | choose your initial parameter you can take a quick peak at the data
 255 | (e.g., using the plots below) and use some general biological
 256 | information. For example, it's common for animals to have one behaviour
 257 | with long step lengths and small turning angles and one behaviour with
 258 | short step lengths and larger turning angles. From the plots above, it
 259 | looks like the animal has step lengths that are close to 0, 3, 7. As I
 260 | see it, there are probably three behaviours with different mean step
 261 | lengths around these values.
 262 | 
 263 | ```{r, attr.source = ".numberLines"}
 264 | plot(sealPrep$step ~
 265 |        sealPrep$date, ty = "l", ylab = "Step length",
 266 |      xlab = "Date", las = 1)
 267 | abline(h = 0, col = rgb(1, 0, 0, 0.5))
 268 | abline(h = 3, col = rgb(1, 0, 0, 0.5))
 269 | abline(h = 7, col = rgb(1, 0, 0, 0.5))
 270 | ```
 271 | 
 272 | So let's choose 3 and 7 for the means step lengths and use the same
 273 | values for the standard deviations. The turning angles are either very
 274 | close to 0 or pretty spread from $-\pi$ to $\pi$. High concentration
 275 | parameter values ($\kappa$, said kappa) mean that the animal has a
 276 | tendency to move in the same direction. Values close to 0 mean that the
 277 | turning angle distribution is almost uniform (the animal turns in all
 278 | directions). Note that $\kappa$ cannot be exactly 0, so let's choose 0.1
 279 | and 1.
 280 | 
 281 | ```{r, attr.source = ".numberLines"}
 282 | # Setting up the starting values
 283 | mu0 <- c(3, 7) # Mean step length
 284 | sigma0 <- c(3, 7) # Sd of the step length
 285 | kappa0 <- c(0.1, 1) # Turning angle concentration parameter (kappa > 0)
 286 | ```
 287 | 
 288 | Ok, were are ready. Let's fit the HMM
 289 | 
 290 | ```{r message=FALSE, cache=TRUE, attr.source = ".numberLines"}
 291 | # Fit a 2 state HMM
 292 | sealHMM2s <- fitHMM(sealPrep, nbState = 2, dist = list(step = "gamma",
 293 | angle = "vm"), Par0 = list(step = c(mu0, sigma0), angle = kappa0), formula = ~ 1)
 294 | ```
 295 | 
 296 | Let's explore the results.
 297 | 
 298 | ```{r, attr.source = ".numberLines"}
 299 | # Let's look at parameter estimates
 300 | sealHMM2s
 301 | # Let's plot it
 302 | plot(sealHMM2s)
 303 | ```
 304 | 
 305 | Based on the mean step parameters, it looks like the first behavioural 
 306 | state (state 1) has really small step lengths compare to state 2. This 
 307 | is particularly easy to see in the step histogram (first figure), where 
 308 | the estimated distribution for each state is overlaid on top of the 
 309 | observed step length frequencies. Surprisingly, the turning angle 
 310 | distributions of the two states both indicate directed movement (second 
 311 | figure). In fact, the concentration parameter of state 1 is bigger than 
 312 | that of state 2. In the last figure with the track, it looks like if we 
 313 | only have state 2. This is strange, since the transition probabilities 
 314 | estimated indicate that the animal remain in each behaviour for multiple
 315 | steps (diagonal values of the transition probability matrix are \> 0.5).
 316 | 
 317 | Part of what's happening may be that the locations of state 1 are hidden
 318 | under state 2. We can see if this is the case by plotting state 2 with a
 319 | higher transparency.
 320 | 
 321 | ```{r, attr.source = ".numberLines"}
 322 | # plot it with transparency
 323 | plot(sealHMM2s, col = c(rgb(230, 160, 20, max = 255), 
 324 |                         rgb(80, 180, 230, alpha = 80, max = 255)))
 325 | ```
 326 | 
 327 | Indeed, we are getting two states, but there is likely more going on. 
 328 | 
 329 | # Identifying behavioural states
 330 | 
 331 | Maybe looking at the state sequence in more detail will help us
 332 | understand. Actually, we are interested in identifying when an animal is
 333 | in each of the behavioural states (here when the animal is in state 1 vs
 334 | state 2), something we call state decoding. To do so, we can use the
 335 | function `viterbi`, which uses the Viterbi algorithm to produce the most
 336 | likely sequence of states according to your fit model and data.
 337 | 
 338 | ```{r, cache=TRUE, attr.source = ".numberLines"}
 339 | # Apply the Viterbi algorithm using your fited model object
 340 | sealStates <- viterbi(sealHMM2s)
 341 | # Let's look at predicted states of the first 20 time steps
 342 | head(sealStates, 20)
 343 | # How many locations in each state do we have?
 344 | table(sealStates)
 345 | ```
 346 | 
 347 | So they are some state 1, we are just not seeing them!
 348 | 
 349 | In many cases, it is more interesting to get the probability of being in
 350 | each state rather than the most likely state. To do so, you can use the
 351 | function `stateProbs`, which returns a matrix of the probability of
 352 | being in each state for each time step.
 353 | 
 354 | ```{r, attr.source = ".numberLines"}
 355 | # Calculate the probability of being in each state
 356 | sealProbs <- stateProbs(sealHMM2s)
 357 | # Let's look at the state probability matrix
 358 | head(sealProbs)
 359 | ```
 360 | 
 361 | We can see here the probability of being in both states for the first 6
 362 | time steps. Here, the probability of being in state 1 is really high for
 363 | these steps, but that may not always be the case. Sometimes you might
 364 | have values closer to 0.5, which would indicate that for that time step,
 365 | it's hard to identify which state you are in (i.e., which step length
 366 | and turning angle distributions fit best).
 367 | 
 368 | You can plot the results from both of these functions using the function
 369 | `plotState`
 370 | 
 371 | ```{r, attr.source = ".numberLines"}
 372 | plotStates(sealHMM2s)
 373 | ```
 374 | 
 375 | We clearly have two behavioural states. The steps from `state` 1 are
 376 | just too small for us to see them in the track.
 377 | 
 378 | # But do we have a good model?
 379 | 
 380 | This is all great, but is this a good model for our data?
 381 | 
 382 | ## Starting values: should we care?
 383 | 
 384 | The first thing we need to look into is whether the parameter estimates
 385 | are reliable. As I mentioned above, the initial parameter values can
 386 | affect the estimating procedures. So it's always good to check if you
 387 | get similar results with different starting values. Here, we will make
 388 | the starting values of the two behavioural states closer to one another.
 389 | 
 390 | ```{r message=FALSE, cache=TRUE, attr.source = ".numberLines"}
 391 | # Setting up the starting values (more similar values)
 392 | mu02 <- c(5, 7) # Mean step length
 393 | sigma02 <- c(5, 7) # Sd of the step length
 394 | kappa02 <- c(1, 1) # Turning angle concentration parameter (kappa > 0)
 395 | 
 396 | # Fit the same 2 state HMM
 397 | sealHMM2s2 <- fitHMM(sealPrep,
 398 |                     nbState = 2,
 399 |                     dist=list(step="gamma", angle="vm"),
 400 |                     Par0 = list(step=c(mu02, sigma02), angle=kappa02))
 401 | ```
 402 | 
 403 | Let's compare the two results. First let's look at which of the two has
 404 | the lowest negative log likelihood (equivalent of highest log
 405 | likelihood, so closer to the real MLE). Let's look also at the parameter
 406 | estimates they each returned.
 407 | 
 408 | ```{r, attr.source = ".numberLines"}
 409 | # Negative log likelihood
 410 | c(original = sealHMM2s$mod$minimum, new = sealHMM2s2$mod$minimum)
 411 | 
 412 | # Parameter estimates
 413 | cbind(sealHMM2s$mle$step, sealHMM2s2$mle$step)
 414 | cbind(sealHMM2s$mle$angle, sealHMM2s2$mle$angle)
 415 | cbind(sealHMM2s$mle$gamma, sealHMM2s2$mle$gamma)
 416 | ```
 417 | 
 418 | Looks like they both returned very close values for everything. So
 419 | that's good! The function `fitHMM` also has the argument `retryFits`
 420 | which perturbs the parameter estimates and retry fitting the model. The
 421 | argument is used to indicate the number of times you want perturb the
 422 | parameters and retry fitting the model (you can choose the size of the
 423 | perturbation by setting the argument `retrySD`).
 424 | 
 425 | Let's try (this will take a few minutes).
 426 | 
 427 | ```{r message=FALSE, cache=TRUE, attr.source = ".numberLines"}
 428 | # Fit the same 2-state HMM with retryFits
 429 | # This is a random pertubation, so setting the seed to get the same results
 430 | set.seed(1234)
 431 | sealHMM2sRF <- fitHMM(sealPrep,
 432 |                     nbState = 2,
 433 |                     dist=list(step="gamma", angle="vm"),
 434 |                     Par0 = list(step=c(mu02, sigma02), angle=kappa02),
 435 |                     retryFits=10)
 436 | ```
 437 | 
 438 | Let's compare the results again.
 439 | 
 440 | ```{r, attr.source = ".numberLines"}
 441 | # Negative log likelihood
 442 | c(original = sealHMM2s$mod$minimum, new = sealHMM2s2$mod$minimum,
 443 |   retryFits = sealHMM2sRF$mod$minimum)
 444 | 
 445 | # Parameter estimates
 446 | cbind(sealHMM2s$mle$step, sealHMM2s2$mle$step, sealHMM2sRF$mle$step)
 447 | cbind(sealHMM2s$mle$angle, sealHMM2s2$mle$angle, sealHMM2sRF$mle$angle)
 448 | cbind(sealHMM2s$mle$gamma, sealHMM2s2$mle$gamma, sealHMM2sRF$mle$gamma)
 449 | ```
 450 | 
 451 | Still looking very similar!
 452 | 
 453 | But let's choose some wild starting values to see the type of problems
 454 | we might run into.
 455 | 
 456 | ```{r message=FALSE, cache=TRUE, attr.source = ".numberLines"}
 457 | # Setting wild starting values
 458 | mu0W <- c(5, 100) # Mean step length
 459 | sigma0W <- c(5, 0.1) # Sd of the step length
 460 | kappa0W <- c(10, 0.01) # Turning angle concentration parameter (kappa > 0)
 461 | sealHMM2sW <- fitHMM(sealPrep, nbState = 2,
 462 |                      dist=list(step="gamma", angle="vm"),
 463 |                      Par0 = list(step=c(mu0W,sigma0W), angle=kappa0W))
 464 | 
 465 | ```
 466 | 
 467 | Let's compare the negative log likelihood (note that we want the minimum
 468 | value here to get the MLE) and parameter estimates.
 469 | 
 470 | ```{r, attr.source = ".numberLines"}
 471 | # Negative log likelihood
 472 | c(original = sealHMM2s$mod$minimum, wild=sealHMM2sW$mod$minimum)
 473 | 
 474 | # Parameter estimates
 475 | cbind(sealHMM2s$mle$step, sealHMM2sW$mle$step)
 476 | cbind(sealHMM2s$mle$angle, sealHMM2sW$mle$angle)
 477 | cbind(sealHMM2s$mle$gamma, sealHMM2sW$mle$gamma)
 478 | ```
 479 | 
 480 | The negative log likelihood here is much larger than the negative log
 481 | likelihood of the model fit with the sensible starting values. Note
 482 | also that some of the parameter estimates are exactly the same values as
 483 | the initial values, which is always a sign of optimization problems.
 484 | These are all signs that the model fit with this new set of starting
 485 | values is not good. Sometimes, you will get error messages saying that
 486 | `nlm` or `optim`, the functions that minimize the negative log
 487 | likelihood, did not converged or that non-finite values were supplied to
 488 | them. Sometimes, you might not be able to plot the data or get the
 489 | states. These are also signs that the starting values are poor or that
 490 | the model is not good for your data.
 491 | 
 492 | We will see later, that the package `momentuHMM` has functions to help
 493 | selecting initial parameters for complex models. However, these
 494 | functions rely on the user choosing initial parameter values for a
 495 | simple model like the one we just explored.
 496 | 
 497 | ## Pseudo-residuals
 498 | 
 499 | Ok, we fit a model to data, looks like the parameter estimates are
 500 | reliable, and we can get state probabilities, but is this a good model
 501 | for our data? Can we use the model results? The best way to investigate
 502 | model fit is through pseudo-residuals. Pseudo-residuals are a type of
 503 | model residuals that account for the interdependence of observations.
 504 | They are calculated for each of the time series (e.g., you will have
 505 | pseudo-residuals for the step length time series and for the turning
 506 | angle time series). If the fit model is appropriate, the
 507 | pseudo-residuals produced by the functions `pseudoRes` should follow a
 508 | standard normal distribution. You can look at pseudo-residuals directly
 509 | via the function `plotPR`, which plots the pseudo-residual times-series,
 510 | the qq-plots, and the autocorrelation functions (ACF).
 511 | 
 512 | ```{r, attr.source = ".numberLines"}
 513 | plotPR(sealHMM2s)
 514 | ```
 515 | 
 516 | Both the qq-plot and the ACF plot indicate that the model does not fit
 517 | the step length time series particularly well. The ACF indicates that
 518 | there is severe autocorrelation remaining in the time series, even after
 519 | accounting for the persistence in the underlying behavioural states.
 520 | 
 521 | This could indicate that there are more hidden behavioural states. Let's
 522 | try a 3-state HMMs.
 523 | 
 524 | ```{r message=FALSE, cache=TRUE, attr.source = ".numberLines"}
 525 | # Setting up the starting values (we need 3 now, because 3 states)
 526 | mu03s <- c(0.1, 3, 7) # Mean step length
 527 | sigma03s <- c(0.1, 3, 7) # Sd of the step length
 528 | kappa03s <- c(0.1, 1, 1) # Turning angle concentration parameter
 529 | 
 530 | # Fit a 3 state HMM
 531 | sealHMM3s <- fitHMM(sealPrep,
 532 |                     nbState = 3,
 533 |                     dist=list(step="gamma", angle="vm"),
 534 |                     Par0 = list(step=c(mu03s, sigma03s), angle=kappa03s))
 535 | ```
 536 | 
 537 | Let's look at it and at the pseudo-residuals.
 538 | 
 539 | ```{r, attr.source = ".numberLines"}
 540 | plot(sealHMM3s)
 541 | plotStates(sealHMM3s)
 542 | plotPR(sealHMM3s)
 543 | ```
 544 | 
 545 | Ah, that's better. Not perfect, but less unexplained autocorrelation,
 546 | especially in the step lengths. Some of the unexplained autocorrelation
 547 | in the turning angles is related to the method we used to get location
 548 | estimates at regular time intervals, see activities to explore this a
 549 | bit more in depth. If we look at the step length distribution, we can
 550 | see that we have still our state with really small steps (maybe
 551 | resting?), a state with steps of medium size (maybe foraging?), and a
 552 | state with large steps (maybe travelling?). Looking at the track, it
 553 | does look like the movement in state 2 is more tortuous and in focal
 554 | areas, while the movement in state 3 is very directed and span larger
 555 | areas.
 556 | 
 557 | ## Simulating data
 558 | 
 559 | We can also use the function `simData` to look at whether the movement
 560 | produced by the models is similar to the observed movement. Let's
 561 | simulate the 3-state HMMs, and then fit it to be able to identify the
 562 | behavioural states. Note that we could just plot directly the simulated
 563 | data and colour code the state, rather than estimating the fitting the model and predicting the
 564 | states and plotting the predicted states. 
 565 | However, plotting the simulated data might require a bit of fiddling (A good thing to know is
 566 | that the colour palette used by the packages is:
 567 | `c("#E69F00", "#56B4E9", "#009E73", "#F0E442", "#0072B2", "#D55E00", "#CC79A7")`).
 568 | One way fitting the model to the simulated data may be useful is in
 569 | identifying whether you have enough data to estimate your parameters
 570 | accurately.
 571 | 
 572 | ```{r message=FALSE, cache=TRUE, attr.source = ".numberLines"}
 573 | # Simulate the model
 574 | sealSim <- simData(model=sealHMM3s, nbAnimals =1, states = TRUE)
 575 | # Fit the model to the simulated data
 576 | sealSimFit <- fitHMM(sealSim,
 577 |                     nbState = 3,
 578 |                     dist=list(step="gamma", angle="vm"),
 579 |                     Par0 = list(step=c(mu03s, sigma03s), angle=kappa03s))
 580 | 
 581 | ```
 582 | 
 583 | ```{r, attr.source = ".numberLines"}
 584 | # Plot the results
 585 | plot(sealSimFit)
 586 | # Compared the parameter estimates
 587 | sealSimFit
 588 | sealHMM3s
 589 | ```
 590 | 
 591 | Looks similar, except maybe that the real seal goes back and forth to
 592 | some of the same locations, something not captured in the simulation.
 593 | The parameter estimates are close to those used to simulate the data,
 594 | which is good.
 595 | 
 596 | ## Confidence intervals
 597 | 
 598 | We can look at the confidence intervals for the parameter estimates.
 599 | This does not necessarily give us an idea of whether the model is good
 600 | or not, but it helps us understand whether the behavioural states have
 601 | different movement characteristics. To compute the confidence intervals,
 602 | we can use the function `CIreal`.
 603 | 
 604 | For example, let's look the step length mean parameters.
 605 | 
 606 | ```{r, attr.source = ".numberLines"}
 607 | # Caculate the CIs
 608 | sealCI <- CIreal(sealHMM3s)
 609 | # Let's look at the step lenth CIs
 610 | rbind(lower=sealCI$step$lower["mean",],
 611 |       upper = sealCI$step$upper["mean",])
 612 | ```
 613 | 
 614 | We can see that the 95% confidence intervals of mean step length does
 615 | not overlap for the three states, indicating that, in term of step
 616 | lengths, they vary quite a bit.
 617 | 
 618 | # Understanding the factors that affect movement: including covariates
 619 | 
 620 | `momentuHMM` allows you to incorporate various covariates in the model
 621 | in many different ways, which allows you to understand how covariates
 622 | affect the behaviour and movement of the animal. There are so many
 623 | different options (including activity centers, spatial covariates, etc),
 624 | that it's impossible to look at them all here. Here are a few examples.
 625 | 
 626 | ## Spatial covariates
 627 | 
 628 | One of the most exciting aspect of `momentuHMM` is that you can
 629 | incorporate spatial covariates that are kept in a raster file. Here we
 630 | will use a raster that contains the bathymetry in the area of the seal.
 631 | I created this file using the package `marmap`. To read the file you
 632 | need the function `raster` from the package with the same name.
 633 | 
 634 | Let's read the file and plot it.
 635 | 
 636 | ```{r, attr.source = ".numberLines"}
 637 | # Read the raster
 638 | bathy <- raster("data/bathy.grd")
 639 | # Let's plot the bathymethy raster
 640 | plot(bathy)
 641 | ```
 642 | 
 643 | This represents the bathymetry in meters. Any value above 0 is on land.
 644 | 
 645 | Here we will use the function `prepData` again, but this time including
 646 | the raster as a spatial covariate. The function will extract the raster
 647 | values at the seal locations. Off course for this to work the spatial
 648 | covariate raster needs to be in the same projection as the movement
 649 | data. It's the case here.
 650 | 
 651 | ```{r, attr.source = ".numberLines"}
 652 | sealPrep <- prepData(sealreg, type="LL", coordNames = c("lon", "lat"),
 653 |                      spatialCovs = list(bathy=bathy))
 654 | head(sealPrep)
 655 | ```
 656 | 
 657 | Now our `sealPrep` object has a new column with the bathymetry value at
 658 | that location.
 659 | 
 660 | There are two ways in which the bathymetry can influence the movement of
 661 | the animal. It can influence the behavioural state the animal is in or
 662 | it can influence the movement characteristics in some of the state (e.g.
 663 | change the mean step length value in one of the behavioural states).
 664 | 
 665 | Let's explore the first option. So here we are assuming that the
 666 | probability to switch between behavioural states is affected by the
 667 | bathymetry. Because of the way the HMMs are constructed, we have to
 668 | estimate one more parameter per switching probability ($\gamma_{ij}$
 669 | said gamma i j), excluding the probability of remaining in the same
 670 | state, which can be deducted based on the switching probabilities.
 671 | Please see the mathy section below to understand the details. So if $N$
 672 | represents the number of states, it means you need to estimate $N(N-1)$
 673 | new parameters. For an HMM with 3 states, we would estimate 6 new
 674 | parameters.
 675 | 
 676 | The main argument of the `fitHMM` function that we need to change are
 677 | the `formula` and the starting values (because we have 6 new parameters
 678 | to estimate). The default value for the `formula` is $\sim 1$, which
 679 | means that the transition probability are not affected by covariates.
 680 | Here, we want to indicate that the transition probabilities are affected
 681 | by the bathymetry. We can use the function `getPar0` to help us find
 682 | starting values based on our simpler 3-state HMMs. This might be a bit
 683 | of an overkill for this model, as `getPar0` will set the starting values
 684 | for the covariate effects on the transition probabilities to 0. But it
 685 | can make a small difference to use good starting values for the other
 686 | parameters (might be worth looking if it makes indeed a difference).
 687 | Note that to be able to use `getPar0` in this case, we have to refit the
 688 | simpler model with the new data that has the bathymetry data.
 689 | 
 690 | ```{r message=FALSE, attr.source = ".numberLines"}
 691 | # New formula
 692 | bathyForm <- ~bathy
 693 | 
 694 | # Refit the simple 3-states HMM - so it's associated with the data with bathymetry
 695 | sealHMM3s <- fitHMM(sealPrep,
 696 |                     nbState = 3,
 697 |                     dist=list(step="gamma", angle="vm"),
 698 |                     Par0 = list(step=c(mu03s, sigma03s), angle=kappa03s))
 699 | 
 700 | # Get new starting values based on simpler model, here the 3-states HMM
 701 | par0b <- getPar0(model=sealHMM3s, formula=bathyForm)
 702 | ```
 703 | 
 704 | ```{r, attr.source = ".numberLines"}
 705 | # Look at the starting values
 706 | par0b
 707 | ```
 708 | 
 709 | ```{r message=FALSE, attr.source = ".numberLines"}
 710 | # Now we fit the new model with bathymetry
 711 | sealHMMb <- fitHMM(sealPrep, nbState = 3,
 712 |                    dist=list(step="gamma", angle="vm"),
 713 |                    Par0 = par0b$Par, beta0=par0b$beta,
 714 |                    formula=bathyForm)
 715 | ```
 716 | 
 717 | ```{r, attr.source = ".numberLines"}
 718 | sealHMMb
 719 | ```
 720 | 
 721 | The first thing to note is that the model now has a new row of
 722 | parameters in the section
 723 | `Regression coeffs for the transition probabilities`. This row
 724 | represents the effects of bathymetry on the transition probabilities. A
 725 | positive value means that increasing values of the covariate increase
 726 | the switching probability, while a negative value means the reverse
 727 | relationship. So for example, as bathymetry increases, the probability
 728 | of switching from state 1 to state 2 only slightly increases, but the
 729 | probability of switching from state 3 to state 1 increases much more.
 730 | Note that the relationship between these coefficients and the final
 731 | transition probabilities are dependent on each other and on the
 732 | intercept, see math section for details. Keep in mind here that
 733 | bathymetry is represented in negative values, so as bathymetry value
 734 | increases the seal is in shallower water.
 735 | 
 736 | The row with the intercept represents the base probabilities before we
 737 | incorporate the bathymetry effects. In model without covariates, you
 738 | would only get the intercept values and would use them alone to
 739 | calculate the transition probabilities, see the math section below for
 740 | more details.
 741 | 
 742 | Let's look at it visually. First by using the generic `plot` function
 743 | and, then, by using the function `plotStationary`, which plots the
 744 | stationary state probabilities.
 745 | 
 746 | ```{r, attr.source = ".numberLines"}
 747 | plot(sealHMMb)
 748 | ```
 749 | 
 750 | <!-- # plotStationary(sealHMMb) -->
 751 | 
 752 | We can see that the bathymetry mainly affects the transition probability
 753 | in high bathymetry values (so, close/on land). One of the most obvious
 754 | patterns is that it increases the probability of switching to state 1
 755 | from both states 2 and 3 and decreases the probability of staying in
 756 | state 2 and staying in state 3. Remember, state 1 is associated with the
 757 | very small step lengths. This is likely seal hauling out on land or
 758 | resting in shallow waters! In fact, one of the places where we have a
 759 | lot of state 1 is on small islands, where I'm pretty sure they are known
 760 | to haul out.
 761 | 
 762 | Here a quick plot to convince you.
 763 | 
 764 | ```{r, attr.source = ".numberLines"}
 765 | # Create a new file, so we can see land better
 766 | water <- bathy
 767 | water[water >= 0] <- NA
 768 | # Create color ramp to show bathymetry in blue gradients 
 769 | bluePal <- colorRampPalette(c(rgb(0.1,0.1,1), rgb(0.9,0.9,1)))
 770 | 
 771 | # plot depth with land (NA) as grey
 772 | plot(water, col = bluePal(50), colNA = "grey30")
 773 | # Let's plot only the state 1
 774 | points(sealPrep[viterbi(sealHMMb) == 1, c("x","y")],
 775 |        col = rgb(230, 160, 20, 12, max = 255),
 776 |        pch = 19)
 777 | # add legend 
 778 | legend(x = -60.8, y = 40.3, pch=c(19, NA),
 779 |        legend = c("State 1", "Land"), col = c("#E69F00", NA), 
 780 |        fill = c(NA, "grey20"), border = NA)
 781 | ```
 782 | 
 783 | Ok, but it is a better model?
 784 | 
 785 | ```{r, attr.source = ".numberLines"}
 786 | AIC(sealHMM3s, sealHMMb)
 787 | ```
 788 | 
 789 | Yes.
 790 | 
 791 | What about the pseudo-residuals?
 792 | 
 793 | ```{r, attr.source = ".numberLines"}
 794 | plotPR(sealHMM3s)
 795 | ```
 796 | 
 797 | Still some issues, but not too too bad.
 798 | 
 799 | ## Non spatial covariates
 800 | 
 801 | Could there be diurnal patterns explaining the autocorrelation in
 802 | turning angles? We are not limited to incorporate spatial covariates. We
 803 | can incorporate any covariates as long as we can relate each of the
 804 | locations to a covariate value (that's in essence what the `prepData`
 805 | function does for the spatial covariates).
 806 | 
 807 | So here we are going to create a new column that keeps track of the hour
 808 | of the day for each observation. We are going to set it in the time zone
 809 | where the seals are.
 810 | 
 811 | ```{r, attr.source = ".numberLines"}
 812 | sealPrep$hour <- as.integer(strftime(sealPrep$date, format = "%H", tz="Etc/GMT+4"))
 813 | ```
 814 | 
 815 | Here again we can make the covariate affect the transition
 816 | probabilities. The difference is that we have to use a more complex
 817 | function that represents the cyclical nature of time of day. This is the
 818 | `cosinor` function. For more information on the `cosinor` function, see
 819 | mathy section below.
 820 | 
 821 | ```{r message = FALSE, attr.source = ".numberLines", cache=TRUE}
 822 | # New formula that says that the transition probabilties are affected by the time of day
 823 | diurnForm <- ~ cosinor(hour, period = 24)
 824 | 
 825 | # Refit model with new data
 826 | sealHMM3s <- fitHMM(sealPrep,
 827 |                     nbState = 3,
 828 |                     dist = list(step = "gamma", angle = "vm"),
 829 |                     Par0 = list(step = c(mu03s, sigma03s), angle = kappa03s))
 830 | 
 831 | # Get initial parameters
 832 | par0diurn <- getPar0(model = sealHMM3s, formula = diurnForm)
 833 | 
 834 | # Now we fit the diurnal model with the starting values returned by getPar0
 835 | sealHMMdiurn <- fitHMM(sealPrep,nbState = 3,
 836 |                    dist = list(step = "gamma", angle = "vm"),
 837 |                    Par0 = par0diurn$Par, beta0 = par0diurn$beta,
 838 |                    formula = diurnForm)
 839 | ```
 840 | 
 841 | ```{r, attr.source = ".numberLines"}
 842 | # Let's look at the results
 843 | sealHMMdiurn
 844 | plot(sealHMMdiurn)
 845 | AIC(sealHMMb, sealHMMdiurn)
 846 | ```
 847 | 
 848 | I'm not a 100% sure how to interpret the results biologically. But again
 849 | it appears to be affecting state 1, which is likely resting. According
 850 | to AIC it's less good than the bathymetry model.
 851 | 
 852 | To help understand the relationship further we can also use the function
 853 | `plotStationary`, which display the relationship between the covariate
 854 | and the stationary probability of being in each state rather than the
 855 | relationship between the covariates and the switching probability.
 856 | 
 857 | ```{r, attr.source = ".numberLines"}
 858 | plotStationary(sealHMMdiurn)
 859 | ```
 860 | 
 861 | So it looks like there is a higher probability of being in state 2 (the
 862 | foraging type of behaviour) at night and a peak in each of the other
 863 | behaviours during different period of the day. Some datasets are very 
 864 | high resolution, and it's important to note that daily effects can only 
 865 | be used when the data spans at least several days.
 866 | 
 867 | Covariates can affect the state-dependent movement, by which I mean the
 868 | characteristics of the movement in a given state. So for example, the
 869 | animal might make all the behaviours at night and during the day, but at
 870 | night it might do them with smaller steps. Here, we are going to look at
 871 | whether the seal is more or less directed at night. To do this, we are
 872 | going to change the argument `DM`. For this kind of model, it's
 873 | particular useful to use `getPar0`, because it helps set starting values
 874 | for the parameters associated with the effects of covariates on the
 875 | movement parameters.
 876 | 
 877 | ```{r, attr.source = ".numberLines"}
 878 | # Use the formula to describe the changes in the design matrix
 879 | diurnDM <- list(angle = list(concentration = ~ cosinor(hour, period = 24)))
 880 | 
 881 | # Get parameter starting values
 882 | par0diurnDM <- getPar0(model=sealHMM3s, DM=diurnDM)
 883 | # Look at starting values
 884 | par0diurnDM
 885 | ```
 886 | 
 887 | ```{r message=FALSE, attr.source = ".numberLines", cache=TRUE}
 888 | # Now we fit the new diurnal model with the starting values returned by getPar0
 889 | sealHMMdiurnDM <- fitHMM(sealPrep, nbState = 3,
 890 |                         dist=list(step="gamma", angle="vm"),
 891 |                         Par0 = par0diurnDM$Par, beta0=par0diurnDM$beta,
 892 |                         DM=diurnDM)
 893 | ```
 894 | 
 895 | ```{r, attr.source = ".numberLines"}
 896 | # Let's look at the results
 897 | sealHMMdiurnDM
 898 | plot(sealHMMdiurnDM, plotCI=TRUE)
 899 | AIC(sealHMMb, sealHMMdiurn, sealHMMdiurnDM)
 900 | ```
 901 | 
 902 | So here it looks like the seal makes more directed movement in state 1
 903 | at night, makes more directed movement in state 2 early in the morning,
 904 | and makes more directed movement in state 3 early in the morning.
 905 | But the CIs are wide and according to AIC, the bathymetry model outperform this model. 
 906 | 
 907 | Note that we can combine various covariates. For example, here we are
 908 | going to combine bathymetry and time of day.
 909 | 
 910 | ```{r message=FALSE, attr.source = ".numberLines", cache=TRUE}
 911 | # Get parameter starting values
 912 | par0bd <- getPar0(model=sealHMM3s, formula=bathyForm, DM=diurnDM)
 913 | 
 914 | # Now we fit the new model with the starting values returned by getPar0
 915 | sealHMMbd <- fitHMM(sealPrep, nbState = 3,
 916 |                         dist=list(step="gamma", angle="vm"),
 917 |                         Par0 = par0bd$Par, beta0=par0bd$beta,
 918 |                         formula=bathyForm,
 919 |                         DM=diurnDM)
 920 | ```
 921 | 
 922 | ```{r, attr.source = ".numberLines"}
 923 | # Let's look at the results
 924 | sealHMMbd
 925 | plot(sealHMMbd)
 926 | AIC(sealHMMb, sealHMMdiurn, sealHMMdiurnDM, sealHMMbd)
 927 | plotPR(sealHMMbd)
 928 | ```
 929 | 
 930 | While this combined model is better, given the complexity, I think I
 931 | would stick to the bathymetry model for the moment and look into better
 932 | ways to try to explain the data, especially since incorporating the time
 933 | of day doesn't fix the angle pseudo-residual problem.
 934 | 
 935 | This is where we stop the main tutorial. There are two other sections
 936 | below. One that describes some of the mathematical details and one with
 937 | activities that you can do on your own to further explore HMMs.
 938 | 
 939 | # For the statistically/mathematically inclined
 940 | 
 941 | One of the most exciting aspect of the `momentuHMM` package is that you
 942 | can incorporate the potential effects of environmental covariates on the
 943 | probability of switching behavioural states. To understand how the
 944 | covariates can affect the behaviour, it's useful to spend time
 945 | understanding the transition probabilities. This section is a bit more
 946 | technical, but I think is needed to fully understand what the estimated
 947 | covariate relationships mean.
 948 | 
 949 | If you have a model that has two behavioural states (state 1 and 2) and
 950 | no covariates, you would have a transition probability matrix like this:
 951 | 
 952 | \begin{equation}
 953 | \mathbf{\Gamma} =
 954 | \begin{pmatrix}
 955 | \gamma_{11} & \gamma_{12}\\
 956 | \gamma_{21} & \gamma_{22} (\#eq:gamma)
 957 | \end{pmatrix},
 958 | \end{equation}
 959 | 
 960 | where $\gamma_{ij}$ represent the probability of switching from i to j.
 961 | For example, $\gamma_{11}$ represents the probability of staying in
 962 | state 1 if you are already in state 1. Similarly, $\gamma_{12}$
 963 | represents the probability of switching to state 2 if you are in state
 964 | 1. By definition the row entries need to sum to 1. Which simply means
 965 | that if you are in state 1 at time $t-1$ you will end up in a state at
 966 | time $t$ (i.e., you don't go in behavioural limbo).
 967 | 
 968 | Let's look again at the transition probability matrix that was estimated
 969 | in our seal example.
 970 | 
 971 | ```{r, attr.source = ".numberLines"}
 972 | sealHMM2s
 973 | ```
 974 | 
 975 | We can see that the probability of switching in state 2 if you are in
 976 | state 1 is $\approx `r round(sealHMM2s$mle$gamma[1,2],2)`$ and so the probability
 977 | of staying in state 1 is
 978 | $\approx `r round(sealHMM2s$mle$gamma[1,1],2)`$.
 979 | The same logic applies to the probability of staying or switching from
 980 | state 2. This means that for a model with $N$ behavioural states, we
 981 | only need to estimate $N (N-1)$ transition probabilities (so 2 for the
 982 | example here), because for each state at time $t-1$ we will be able to
 983 | calculate one of the transition probabilities based on the difference
 984 | with the sum of the transition probabilities estimated (so here we can
 985 | deduce $\gamma_{11}$ based on $\gamma_{12}$).
 986 | 
 987 | To incorporate the covariates we are saying that the transition
 988 | probabilities are changing through time as a function of the covariate
 989 | values at that time step. Specifically,
 990 | 
 991 | 
 992 | \begin{equation}
 993 | \gamma^{(t)}_{ij} = \frac{\exp(\eta_{ijt})}{\sum^N_{k=1}\exp(\eta_{ikt})}, (\#eq:linkFx)
 994 | \end{equation}
 995 | 
 996 | where
 997 | 
 998 | 
 999 | \begin{equation}
1000 | \eta_{ijt} =\begin{cases}\beta_{0ij} + \sum^p_{l=1} \beta_{lij} w_{lt} & \text{if} \ i \neq j,\\0, & \text{otherwise},\end{cases}(\#eq:covRel)
1001 | \end{equation}
1002 | 
1003 | 
1004 | where $w_{lt}$ is the $l$-th covariate at time $t$ and $p$ is the number
1005 | of covariates considered. Equation \@ref(eq:linkFx) is only a link
1006 | function to ensure that the transitions probabilities are between 0 and
1007 | 1 and that the row sum to 1. The important relationship between the
1008 | covariates and the transition probabilities are in equation
1009 | \@ref(eq:covRel), where the $\beta_{0ij}$ is the base transition probability
1010 | for the switch from state $i$ to state $j$ that is not affected by the
1011 | covariates. The $\beta_{lij}$ are the coefficient relating the
1012 | transition probability to the $l$-th covariate. Note that $\eta_{ii}$
1013 | (the probability of staying in state $i$, the diagonal values in the
1014 | transition probability matrix) are fixed to 0 because we only want to
1015 | $N (N-1)$ transition probabilities to be estimated and the row sums to
1016 | one.
1017 | 
1018 | So in a model with no covariates, only the $\beta_{0ij}$ for $i \neq j$
1019 | (switching probabilities from state $i$ to another state $j$) are
1020 | estimated. If you look in the code box above you'll see that we only
1021 | have \emph{regression coeffs for the transition probabilities} for 1
1022 | $->$ 2 and 2 $->$ 1, and that the transition probabilities for
1023 | corresponding states are
1024 | $\gamma_{ij} = \frac{\exp(\eta_{ij})}{\sum^N_{k=1}\exp(\eta_{ik})}$.
1025 | 
1026 | For example, for the transition probability from state 1 to state 2.
1027 | 
1028 | ```{r, attr.source = ".numberLines"}
1029 | # gamma_{12}
1030 | sealHMM2s$mle$gamma[1,2]
1031 | # gamma_{12} calculated based on the equations above
1032 | exp(sealHMM2s$mle$beta[1])/(exp(0)+exp(sealHMM2s$mle$beta[1]))
1033 | ```
1034 | 
1035 | So the way we interpret the $\beta$ values, is that if they are
1036 | positive, it means that they increase the transition probability. Since
1037 | the total value of $\eta$ is what affects the transition probability,
1038 | the magnitude and sign of the intercept ($\beta_0$) will affect when the
1039 | transition probability goes from above or below 0.5.
1040 | 
1041 | This is a great way to incorporate covariates, but it means that it add
1042 | $N(N-1)$ parameters to your model, where $N$ is the number of states. So
1043 | if you have many behavioural states, it will results in a very complex
1044 | model.
1045 | 
1046 | ## cosinor
1047 | 
1048 | The function `cosinor` internally creates the covariates
1049 | cos$(2 \pi \; \text{value}/\text{period})$ and
1050 | sin$(2 \pi \; \text{value}/\text{period})$. So for example
1051 | cos$(2 \pi \; \text{hour}/24)$.
1052 | 
1053 | ```{r, attr.source = ".numberLines"}
1054 | plot(cos(2*pi*(1:24)/24), pch=19, ylab="cosinorCos", xlab="Time of day")
1055 | ```
1056 | 
1057 | ```{r, attr.source = ".numberLines"}
1058 | plot(sin(2*pi*(1:24)/24), pch=19, ylab="cosinorSin", xlab="Time of day")
1059 | ```
1060 | 
1061 | <!-- WOULD NOT PUT THAT MUCH FAITH IN INITIAL DIST ESTIMATE (ASSUME BY -->
1062 | 
1063 | <!-- DEFAULT THAT NON-STATIONARY) % % TO DO: standardize? % % TO DO: model -->
1064 | 
1065 | <!-- selection, AIC, think Ben Bolker, model complexity. % % TO DO: use known -->
1066 | 
1067 | <!-- states -->
1068 | 
1069 | # Activities
1070 | 
1071 | ## Try it on your own data
1072 | 
1073 | Give it a try on your own data! If you don't have data try the other
1074 | activities.
1075 | 
1076 | ## Activity centers
1077 | 
1078 | This is fun but could be hard. Skip to the next activity if you find the
1079 | material challenging.
1080 | 
1081 | Let's look whether adding bathymetry improved the simulations. Here,
1082 | it's easy to have some problem, where the animal moves outside of the
1083 | raster. So to help, we are setting the initial position as the initial
1084 | position observed. We are also limiting the simulation to a time series
1085 | of 200 steps. We are also setting the seed, because the animal will
1086 | often wonder off map by chance.
1087 | 
1088 | ```{r message=FALSE, attr.source = ".numberLines", cache=TRUE}
1089 | set.seed(1)
1090 | sealSimB <- simData(model=sealHMMb, nbAnimals =1, spatialCovs = list(bathy=bathy),
1091 |                     initialPosition = as.matrix(sealreg[1,c("lon","lat")]),
1092 |                     obsPerAnimal = 100, retrySims = 10)
1093 | sealSimFitB <- fitHMM(sealSimB,
1094 |                     nbState = 3,
1095 |                    dist=list(step="gamma", angle="vm"),
1096 |                    Par0 = par0b$Par, beta0=par0b$beta,
1097 |                    formula=bathyForm)
1098 | ```
1099 | 
1100 | ```{r, attr.source = ".numberLines"}
1101 | # Plot simulation
1102 | plot(water, col=bluePal(50))
1103 | points(sealSimB[,c("x","y")],
1104 |        col=c("#E69F00", "#56B4E9", "#009E73")[viterbi(sealHMMb)],
1105 |        pch=19, cex=0.5)
1106 | ```
1107 | 
1108 | I'm not sure. The animal definitely wonders off in regions we wouldn't
1109 | expect it to go. But the state 1 is only in shallow waters.
1110 | 
1111 | Look at the `momentuHMM` vignette and see whether you can include
1112 | activity centers.
1113 | 
1114 | ## Modelling step length and turning angles
1115 | 
1116 | You can model the observations with a set of different distributions. To
1117 | change the distribution simply requires changing the argument `dist` in
1118 | the function `fitHMM`. In the example above, we used the gamma
1119 | distribution (using the name `"gamma"`) for the step length. However,
1120 | you can use a variety of alternative distribution, including: Weibull
1121 | (`"weibull"`) and exponential (`"exp"`). Similarly, we used the von
1122 | Mises (named `"vm"`) for the turning angle distribution, but one could
1123 | use the wrapped Cauchy (`"wrpcauchy"`).
1124 | 
1125 | Explore whether changing the distribution affects the fit of the model.
1126 | Use AIC to compare the different models. Use the `plot` and `viterbi`
1127 | functions to see whether it has a meaningful biological effect.
1128 | 
1129 | For a description of the parameters needed for each of the
1130 | distributions, please see `momentuHMM` vignette.
1131 | 
1132 | ```{r message=FALSE, attr.source = ".numberLines"}
1133 | 
1134 | #### Wrapped Cauchy for turning angle
1135 | kappa0wp <- c(0.9, 0.9) # Turning angle concentration parameter, kappa close
1136 | sealHMMwp <- fitHMM(sealPrep, nbState = 2,
1137 |                     dist=list(step="gamma", angle="wrpcauchy"),
1138 |                     Par0 = list(step=c(mu0,sigma0), angle=kappa0wp))
1139 | ```
1140 | 
1141 | ```{r, attr.source = ".numberLines"}
1142 | AIC(sealHMM2s, sealHMMwp)# Wrapped Cauchy better
1143 | 
1144 | #### Weibull for step length
1145 | mu0 <- c(0.6, 1) # Shape
1146 | sigma0 <- c(1, 5) # Rate
1147 | kappa0 <- c(0.9, 0.9) # Turning angle concentration parameter, kappa close
1148 | sealHMMwb <- fitHMM(sealPrep, nbState = 2, dist=list(step="weibull", angle="vm"),
1149 |        Par0 = list(step=c(mu0,sigma0), angle=kappa0))
1150 | 
1151 | AIC(sealHMM2s,sealHMMwb) # Weibull is better
1152 | plot(sealHMMwb) # Doesn't have a big effect
1153 | sum(abs(viterbi(sealHMM2s) - viterbi(sealHMMwb)))/nrow(sealPrep) # Only 2 different states (< 1% difference)
1154 | 
1155 | #### Exponential for step length
1156 | mu0 <- c(2, 1) # Rate - has one less parameters
1157 | kappa0 <- c(0.9, 0.9) # Turning angle concentration parameter, kappa close
1158 | ```
1159 | 
1160 | ```{r message=FALSE, attr.source = ".numberLines", cache=TRUE}
1161 | sealHMMexp <- fitHMM(sealPrep, nbState = 2,
1162 |                      dist=list(step="exp", angle="vm"),
1163 |                      Par0 = list(step=c(mu0), angle=kappa0))
1164 | ```
1165 | 
1166 | ```{r, attr.source = ".numberLines"}
1167 | AIC(sealHMM2s, sealHMMwb,sealHMMexp) # Exponential is worst
1168 | plot(sealHMMexp) # It has a bit more of an effect
1169 | sum(abs(viterbi(sealHMM2s) - viterbi(sealHMMexp)))/nrow(sealPrep) # 4 different states (< 1%)
1170 | ```
1171 | 
1172 | Many of the distributions, including the gamma distribution, are
1173 | strictly positive. This means for example that you cannot have a step
1174 | length of exactly 0 if you simply use the gamma distribution. You can
1175 | however, use a zero-inflated distribution. Such a distribution assumes
1176 | that there is a probability $z$ to observe a 0 and a probability $1-z$
1177 | to observe a positive value distributed according the the strictly
1178 | positive distribution of your choice (e.g., the gamma). This can be
1179 | achieved by adding a zero-mass starting value for each state in the step
1180 | parameters (see the help page of `fitHMM` under the argument `Par0`).
1181 | 
1182 | ## Effects of interpolation {#effect_of_int}
1183 | 
1184 | The seal data we are using is not regular, by which we mean that the
1185 | locations are not taken at regular time intervals. As a quick trick we
1186 | regularised the data to 2 hours intervals using linear interpolation,
1187 | but this is likely to affect the analysis. Here, explore the effects of
1188 | using different time intervals. For example, look at the results when
1189 | using 1, 4, 8 hours intervals. Look at both the fit of the model and the
1190 | pseudo residuals. We commented out the plots for space, but do run them 
1191 | to visualise the effect of resolution on the data and results.
1192 | 
1193 | ```{r, attr.source = ".numberLines", cache=TRUE}
1194 | # Let's try different time intervals
1195 | # 1 hr
1196 | ti1 <- seq(seal$date[1], seal$date[length(seal$date)], by=60*60)
1197 | # 4 hr
1198 | ti4 <- seq(seal$date[1], seal$date[length(seal$date)], by=4*60*60)
1199 | ti8 <- seq(seal$date[1], seal$date[length(seal$date)], by=8*60*60)
1200 | 
1201 | # Interpolate the location at the times from the sequence
1202 | iLoc1 <- as.data.frame(cbind(lon=approx(seal$date, seal$lon,
1203 |                                         xout = ti1)$y,
1204 |                              lat=approx(seal$date, seal$lat,
1205 |                                         xout = ti1)$y))
1206 | iLoc4 <- as.data.frame(cbind(lon=approx(seal$date, seal$lon,
1207 |                                         xout = ti4)$y,
1208 |                              lat=approx(seal$date, seal$lat,
1209 |                                         xout = ti4)$y))
1210 | iLoc8 <- as.data.frame(cbind(lon=approx(seal$date, seal$lon,
1211 |                                         xout = ti8)$y,
1212 |                              lat=approx(seal$date, seal$lat,
1213 |                                         xout = ti8)$y))
1214 | 
1215 | # Create a new object that has the regular time and interpolated locations
1216 | sealreg1 <- cbind(date=ti1, iLoc1)
1217 | sealreg4 <- cbind(date=ti4, iLoc4)
1218 | sealreg8 <- cbind(date=ti8, iLoc8)
1219 | 
1220 | # # Quickly plot the regularized locations
1221 | # plot(seal$lon, seal$lat,
1222 | #      pch=19, cex=0.5, xlab="Lon", ylab="Lat", las=1)
1223 | # points(sealreg1$lon, sealreg1$lat,  pch=19, cex=0.5, col=rgb(1,0,0,0.5))
1224 | # points(sealreg$lon, sealreg$lat,  pch=19, cex=0.5,
1225 | #        col=rgb(0.8,0.8,0.8,0.5))
1226 | # points(sealreg4$lon, sealreg4$lat, pch=19, cex=0.5, col=rgb(0,0,1,0.5))
1227 | # points(sealreg8$lon, sealreg8$lat, pch=19, cex=0.5, col=rgb(0,1,1,0.5))
1228 | 
1229 | # Create the prep data
1230 | sealPrep1 <- prepData(sealreg1, type="LL", coordNames = c("lon", "lat"))
1231 | sealPrep4 <- prepData(sealreg4, type="LL", coordNames = c("lon", "lat"))
1232 | sealPrep8 <- prepData(sealreg8, type="LL", coordNames = c("lon", "lat"))
1233 | 
1234 | hist(sealPrep1$angle, breaks=40,
1235 |      col=rgb(1,0,0,1), freq=FALSE,
1236 |      ylim=c(0,2), las=1, border = NA)
1237 | hist(sealPrep$angle, breaks=40, add=TRUE,
1238 |      col=rgb(0.8,0.8,0.8,0.8), freq=FALSE, border = NA)
1239 | hist(sealPrep4$angle, breaks=40, add=TRUE,
1240 |      col=rgb(0,0,1,0.3), freq=FALSE, border = NA)
1241 | hist(sealPrep8$angle, breaks=40, add=TRUE,
1242 |      col=rgb(0,1,1,0.3), freq=FALSE, border = NA)
1243 | 
1244 | # None is exactly 0 == surprising...
1245 | sum(sealPrep1$angle == 0, na.rm = TRUE)
1246 | sum(sealPrep$angle == 0, na.rm = TRUE)
1247 | sum(sealPrep4$angle == 0, na.rm = TRUE)
1248 | sum(sealPrep8$angle == 0, na.rm = TRUE)
1249 | 
1250 | # Ok let's fit the 3 state HMM again
1251 | # Fit a 3 state HMM
1252 | sealHMM3s1 <- fitHMM(sealPrep1,
1253 |                     nbState = 3,
1254 |                     dist=list(step="gamma", angle="vm"),
1255 |                     Par0 = list(step=c(mu03s, sigma03s), angle=kappa03s))
1256 | sealHMM3s4 <- fitHMM(sealPrep4,
1257 |                     nbState = 3,
1258 |                     dist=list(step="gamma", angle="vm"),
1259 |                     Par0 = list(step=c(mu03s, sigma03s), angle=kappa03s))
1260 | sealHMM3s8 <- fitHMM(sealPrep8,
1261 |                     nbState = 3,
1262 |                     dist=list(step="gamma", angle="vm"),
1263 |                     Par0 = list(step=c(mu03s, sigma03s), angle=kappa03s))
1264 | 
1265 | # plot(sealHMM3s1)
1266 | # plot(sealHMM3s)
1267 | # plot(sealHMM3s4)
1268 | # plot(sealHMM3s8)
1269 | #
1270 | # plotPR(sealHMM3s1)
1271 | # plotPR(sealHMM3s)
1272 | # plotPR(sealHMM3s4)
1273 | # plotPR(sealHMM3s8)
1274 | ```
1275 | 
1276 | # Thanks
1277 | 
1278 | Thanks to Théo Michelot for input and help!
1279 | 


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108 | 2014-02-26 01:24:48,-66.0691833496094,42.2390594482422
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110 | 2014-02-26 06:26:51,-66.0214233398438,42.2238883972168
111 | 2014-02-26 08:27:14,-66.0623397827148,42.2094993591309
112 | 2014-02-26 09:28:15,-66.0995788574219,42.199821472168
113 | 2014-02-26 10:28:33,-66.1304016113281,42.1834983825684
114 | 2014-02-26 11:29:00,-66.1504364013672,42.1638717651367
115 | 2014-02-26 12:33:46,-66.1571807861328,42.1404685974121
116 | 2014-02-26 13:39:05,-66.1564331054688,42.11376953125
117 | 2014-02-26 14:39:28,-66.1530990600586,42.0839385986328
118 | 2014-02-26 15:44:54,-66.1493377685547,42.0518112182617
119 | 2014-02-26 16:44:57,-66.1593399047852,42.0392303466797
120 | 2014-02-26 17:45:00,-66.1732635498047,42.0174789428711
121 | 2014-02-26 18:48:48,-66.2029266357422,41.9938812255859
122 | 2014-02-26 19:54:29,-66.2205581665039,41.9822807312012
123 | 2014-02-26 22:02:04,-66.3026504516602,41.9777488708496
124 | 2014-02-26 23:02:04,-66.305419921875,41.9834785461426
125 | 2014-02-27 00:02:59,-66.32470703125,41.9753303527832
126 | 2014-02-27 01:03:31,-66.3317337036133,41.972469329834
127 | 2014-02-27 03:05:47,-66.3102188110352,41.9481391906738
128 | 2014-02-27 04:06:18,-66.3082122802734,41.9112205505371
129 | 2014-02-27 05:06:40,-66.316650390625,41.8748588562012
130 | 2014-02-27 07:14:44,-66.3306274414063,41.8185691833496
131 | 2014-02-27 09:23:51,-66.3972015380859,41.8138885498047
132 | 2014-02-27 10:31:04,-66.4090576171875,41.8277397155762
133 | 2014-02-27 11:32:25,-66.4393997192383,41.8371315002441
134 | 2014-02-27 12:38:13,-66.4575805664063,41.8502006530762
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143 | 2014-02-28 06:00:20,-66.6767807006836,41.7723388671875
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154 | 2014-02-28 21:35:36,-66.5141525268555,41.8184814453125
155 | 2014-02-28 23:38:13,-66.5329818725586,41.8546485900879
156 | 2014-03-01 00:38:46,-66.5355911254883,41.8852882385254
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189 | 2014-03-02 19:30:42,-66.4157028198242,41.859130859375
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199 | 2014-03-03 11:53:16,-66.2940521240234,41.8311195373535
200 | 2014-03-03 13:55:13,-66.3498229980469,41.8574485778809
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202 | 2014-03-03 15:58:32,-66.4177780151367,41.9118194580078
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223 | 2014-03-04 23:53:23,-66.2253036499023,41.8344306945801
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225 | 2014-03-05 05:01:26,-66.3557510375977,41.8551902770996
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228 | 2014-03-05 11:12:42,-66.203498840332,41.7617301940918
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239 | 2014-03-05 22:34:39,-66.3691711425781,41.741870880127
240 | 2014-03-05 23:37:49,-66.3809509277344,41.7058982849121
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246 | 2014-03-06 10:00:22,-66.3308563232422,41.7361793518066
247 | 2014-03-06 11:03:29,-66.3452529907227,41.7313690185547
248 | 2014-03-06 15:06:37,-66.4409637451172,41.7308883666992
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251 | 2014-03-06 18:14:36,-66.4386901855469,41.7923316955566
252 | 2014-03-06 21:16:11,-66.4762191772461,41.8047103881836
253 | 2014-03-06 23:18:53,-66.4688415527344,41.8095588684082
254 | 2014-03-07 01:22:42,-66.4500885009766,41.8075485229492
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256 | 2014-03-07 03:27:36,-66.4726028442383,41.826099395752
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259 | 2014-03-07 06:30:59,-66.4253692626953,41.8892784118652
260 | 2014-03-07 08:32:52,-66.4065399169922,41.9289817810059
261 | 2014-03-07 11:36:15,-66.3897094726563,41.9789009094238
262 | 2014-03-07 12:40:14,-66.3886566162109,41.9798583984375
263 | 2014-03-07 13:40:20,-66.3914413452148,41.9822387695313
264 | 2014-03-07 14:43:04,-66.3992462158203,41.9920387268066
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267 | 2014-03-07 18:48:32,-66.4045867919922,42.1007614135742
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270 | 2014-03-07 22:54:57,-66.2777786254883,42.246410369873
271 | 2014-03-07 23:55:25,-66.2669982910156,42.2690391540527
272 | 2014-03-08 00:58:11,-66.265251159668,42.2932090759277
273 | 2014-03-08 02:02:11,-66.2688217163086,42.3193702697754
274 | 2014-03-08 03:08:56,-66.2751998901367,42.3540687561035
275 | 2014-03-08 04:13:09,-66.2833404541016,42.3811988830566
276 | 2014-03-08 05:16:29,-66.2900924682617,42.411018371582
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279 | 2014-03-08 10:27:29,-66.2874221801758,42.5416793823242
280 | 2014-03-08 12:28:52,-66.2559204101563,42.5474395751953
281 | 2014-03-08 13:32:35,-66.2616806030273,42.5566291809082
282 | 2014-03-08 14:32:49,-66.2801208496094,42.5773696899414
283 | 2014-03-08 15:32:52,-66.2878875732422,42.6055297851563
284 | 2014-03-08 16:34:06,-66.2861785888672,42.6358795166016
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286 | 2014-03-08 18:39:00,-66.2616424560547,42.7099494934082
287 | 2014-03-08 21:46:58,-66.2158889770508,42.8342514038086
288 | 2014-03-09 00:50:51,-66.1405334472656,42.910888671875
289 | 2014-03-09 01:55:07,-66.1178970336914,42.9314498901367
290 | 2014-03-09 02:55:50,-66.1024169921875,42.9502105712891
291 | 2014-03-09 03:57:23,-66.0984725952148,42.9736404418945
292 | 2014-03-09 05:03:00,-66.0969467163086,42.9938888549805
293 | 2014-03-09 09:12:06,-66.1147384643555,43.1446113586426
294 | 2014-03-09 10:12:12,-66.0935592651367,43.1786308288574
295 | 2014-03-09 11:13:04,-66.0550918579102,43.2109718322754
296 | 2014-03-09 12:13:13,-66.0091934204102,43.2401885986328
297 | 2014-03-09 13:13:21,-65.9849319458008,43.2747611999512
298 | 2014-03-09 14:15:27,-65.976188659668,43.3183212280273
299 | 2014-03-09 15:15:42,-65.9868927001953,43.3578987121582
300 | 2014-03-09 17:17:05,-66.0408935546875,43.4135818481445
301 | 2014-03-09 18:17:06,-66.0435104370117,43.4582786560059
302 | 2014-03-09 19:17:42,-66.0268478393555,43.5063896179199
303 | 2014-03-09 20:17:48,-66.0077590942383,43.5050392150879
304 | 2014-03-10 08:17:48,-66.007926940918,43.5048217773438
305 | 2014-03-10 20:17:48,-66.0077514648438,43.5048904418945
306 | 2014-03-11 08:03:04,-66.0113677978516,43.504768371582
307 | 2014-03-11 09:06:42,-66.0707321166992,43.5102310180664
308 | 2014-03-11 10:10:09,-66.1294326782227,43.5191802978516
309 | 2014-03-11 11:11:48,-66.1911087036133,43.5127410888672
310 | 2014-03-11 12:12:38,-66.2472381591797,43.499568939209
311 | 2014-03-11 13:15:35,-66.3031921386719,43.4843902587891
312 | 2014-03-11 14:15:36,-66.3430480957031,43.4470901489258
313 | 2014-03-11 17:22:18,-66.4512023925781,43.3196105957031
314 | 2014-03-11 18:27:15,-66.4982528686523,43.2820091247559
315 | 2014-03-11 19:29:12,-66.543701171875,43.2527885437012
316 | 2014-03-11 21:34:42,-66.629280090332,43.214111328125
317 | 2014-03-11 22:40:13,-66.6536483764648,43.1937217712402
318 | 2014-03-11 23:46:30,-66.6819534301758,43.1777305603027
319 | 2014-03-12 01:56:27,-66.7158889770508,43.1239891052246
320 | 2014-03-12 03:59:47,-66.7416076660156,43.0896186828613
321 | 2014-03-12 05:05:32,-66.7640533447266,43.068359375
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323 | 2014-03-12 07:12:04,-66.7528762817383,43.0177192687988
324 | 2014-03-12 08:19:00,-66.7592697143555,42.9945106506348
325 | 2014-03-12 09:19:46,-66.7751007080078,42.9773292541504
326 | 2014-03-12 10:20:30,-66.7827606201172,42.9606094360352
327 | 2014-03-12 11:23:31,-66.7931594848633,42.9391403198242
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329 | 2014-03-12 14:34:41,-66.7983093261719,42.8636894226074
330 | 2014-03-12 15:34:44,-66.7945175170898,42.8352088928223
331 | 2014-03-12 16:39:37,-66.7822723388672,42.8070983886719
332 | 2014-03-12 17:47:21,-66.7671890258789,42.7778816223145
333 | 2014-03-12 18:50:44,-66.7615814208984,42.7642593383789
334 | 2014-03-12 19:52:50,-66.7453765869141,42.7675285339355
335 | 2014-03-12 20:53:59,-66.7329483032227,42.759220123291
336 | 2014-03-12 21:55:29,-66.7200469970703,42.7634201049805
337 | 2014-03-12 22:58:38,-66.7193984985352,42.7581596374512
338 | 2014-03-13 01:03:05,-66.737419128418,42.753101348877
339 | 2014-03-13 02:04:00,-66.717041015625,42.7337188720703
340 | 2014-03-13 03:05:51,-66.6930236816406,42.716609954834
341 | 2014-03-13 07:17:49,-66.6443710327148,42.6006889343262
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344 | 2014-03-13 12:25:04,-66.6765594482422,42.5537796020508
345 | 2014-03-13 13:25:04,-66.6747970581055,42.5386390686035
346 | 2014-03-13 14:27:04,-66.6876373291016,42.5324592590332
347 | 2014-03-13 17:28:39,-66.6751327514648,42.500171661377
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350 | 2014-03-13 22:36:59,-66.6369934082031,42.5733604431152
351 | 2014-03-14 00:43:11,-66.6603393554688,42.5884399414063
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353 | 2014-03-14 05:53:04,-66.6635513305664,42.5202789306641
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363 | 2014-03-14 22:30:29,-66.779411315918,42.5843200683594
364 | 2014-03-14 23:30:30,-66.7766571044922,42.6048583984375
365 | 2014-03-15 00:32:04,-66.7750396728516,42.619270324707
366 | 2014-03-15 01:33:04,-66.7701034545898,42.6281204223633
367 | 2014-03-15 02:38:26,-66.7453994750977,42.6220207214355
368 | 2014-03-15 03:39:43,-66.7279281616211,42.6049194335938
369 | 2014-03-15 06:46:15,-66.6325607299805,42.5211181640625
370 | 2014-03-15 07:48:38,-66.6045532226563,42.5112991333008
371 | 2014-03-15 08:51:41,-66.6008605957031,42.5105590820313
372 | 2014-03-15 09:54:30,-66.5931396484375,42.5223808288574
373 | 2014-03-15 11:55:54,-66.6462707519531,42.5457801818848
374 | 2014-03-15 14:59:17,-66.6907501220703,42.5522804260254
375 | 2014-03-15 18:04:45,-66.6393280029297,42.5403518676758
376 | 2014-03-15 19:04:46,-66.6079483032227,42.5339202880859
377 | 2014-03-15 20:05:46,-66.5715408325195,42.5329284667969
378 | 2014-03-15 21:08:00,-66.5424423217773,42.5366897583008
379 | 2014-03-16 00:12:38,-66.5511474609375,42.5824089050293
380 | 2014-03-16 01:16:10,-66.5737991333008,42.5995597839355
381 | 2014-03-16 03:21:54,-66.5942535400391,42.6138305664063
382 | 2014-03-16 04:22:08,-66.5849990844727,42.5986785888672
383 | 2014-03-16 05:22:09,-66.5817337036133,42.5864601135254
384 | 2014-03-16 08:25:31,-66.5327224731445,42.5267181396484
385 | 2014-03-16 09:27:10,-66.5232620239258,42.5161590576172
386 | 2014-03-16 10:29:31,-66.5164337158203,42.5150108337402
387 | 2014-03-16 13:29:46,-66.5446395874023,42.5196189880371
388 | 2014-03-16 14:30:22,-66.5523223876953,42.5138893127441
389 | 2014-03-16 16:30:23,-66.5350723266602,42.5029106140137
390 | 2014-03-16 17:30:28,-66.5175628662109,42.493091583252
391 | 2014-03-16 18:32:47,-66.4850463867188,42.5033798217773
392 | 2014-03-16 19:32:52,-66.4506301879883,42.5064697265625
393 | 2014-03-16 21:33:09,-66.4038467407227,42.5088691711426
394 | 2014-03-16 23:36:49,-66.4191665649414,42.5464401245117
395 | 2014-03-17 01:36:50,-66.4727096557617,42.5875587463379
396 | 2014-03-17 03:40:50,-66.5107421875,42.6231307983398
397 | 2014-03-17 04:40:52,-66.5237426757813,42.6198310852051
398 | 2014-03-17 05:41:30,-66.5369186401367,42.600959777832
399 | 2014-03-17 06:43:51,-66.5420227050781,42.5710105895996
400 | 2014-03-17 08:52:42,-66.5433502197266,42.520320892334
401 | 2014-03-17 10:54:16,-66.5733337402344,42.513729095459
402 | 2014-03-17 12:00:39,-66.6072311401367,42.5289192199707
403 | 2014-03-17 13:05:45,-66.642822265625,42.527889251709
404 | 2014-03-17 14:07:40,-66.6515808105469,42.5108299255371
405 | 2014-03-17 15:16:12,-66.6509628295898,42.5089302062988
406 | 2014-03-17 18:17:16,-66.5912628173828,42.5042610168457
407 | 2014-03-17 20:19:12,-66.5239562988281,42.4908790588379
408 | 2014-03-17 23:19:22,-66.4628601074219,42.4978790283203
409 | 2014-03-18 00:19:26,-66.4787216186523,42.5079917907715
410 | 2014-03-18 03:20:07,-66.5380783081055,42.5511512756348
411 | 2014-03-18 05:24:53,-66.5422668457031,42.5354499816895
412 | 2014-03-18 06:32:49,-66.5327301025391,42.512378692627
413 | 2014-03-18 08:40:27,-66.4854431152344,42.4927597045898
414 | 2014-03-18 11:46:17,-66.4532699584961,42.4815483093262
415 | 2014-03-18 14:54:07,-66.5419235229492,42.4866790771484
416 | 2014-03-18 16:59:07,-66.5598983764648,42.4892196655273
417 | 2014-03-18 18:03:18,-66.5649490356445,42.4699287414551
418 | 2014-03-18 19:13:20,-66.5472564697266,42.4581108093262
419 | 2014-03-18 20:21:20,-66.523567199707,42.439640045166
420 | 2014-03-18 21:21:24,-66.4935913085938,42.4346618652344
421 | 2014-03-18 22:21:46,-66.4660873413086,42.4362602233887
422 | 2014-03-18 23:25:25,-66.4600830078125,42.4383087158203
423 | 2014-03-19 00:26:36,-66.4628982543945,42.4420013427734
424 | 2014-03-19 01:29:04,-66.4740524291992,42.4381484985352
425 | 2014-03-19 04:40:59,-66.5576095581055,42.4057388305664
426 | 2014-03-19 06:42:30,-66.5512619018555,42.3696594238281
427 | 2014-03-19 09:51:20,-66.4588165283203,42.3096199035645
428 | 2014-03-19 13:13:15,-66.4714431762695,42.306339263916
429 | 2014-03-19 16:24:37,-66.5824432373047,42.339168548584
430 | 2014-03-19 18:26:04,-66.6110763549805,42.3819999694824
431 | 2014-03-19 20:44:40,-66.563591003418,42.4159889221191
432 | 2014-03-19 22:44:44,-66.5090484619141,42.4460983276367
433 | 2014-03-19 23:44:50,-66.4902496337891,42.460090637207
434 | 2014-03-20 00:53:26,-66.4797821044922,42.4773902893066
435 | 2014-03-20 01:54:38,-66.4766387939453,42.4914207458496
436 | 2014-03-20 02:56:12,-66.4833374023438,42.5061416625977
437 | 2014-03-20 03:56:16,-66.4890365600586,42.5251007080078
438 | 2014-03-20 06:04:44,-66.5259780883789,42.5495300292969
439 | 2014-03-20 07:04:46,-66.5055694580078,42.5410194396973
440 | 2014-03-20 08:07:59,-66.4812622070313,42.5414810180664
441 | 2014-03-20 10:22:05,-66.4055404663086,42.5199584960938
442 | 2014-03-20 11:23:43,-66.3849868774414,42.513729095459
443 | 2014-03-20 12:23:46,-66.3756103515625,42.5237503051758
444 | 2014-03-20 15:26:36,-66.449592590332,42.5710792541504
445 | 2014-03-20 16:30:04,-66.4717178344727,42.584831237793
446 | 2014-03-20 20:45:56,-66.4187622070313,42.5330390930176
447 | 2014-03-20 23:53:21,-66.3545913696289,42.500659942627
448 | 2014-03-21 01:55:48,-66.3874969482422,42.4983711242676
449 | 2014-03-21 04:57:27,-66.4865417480469,42.5498695373535
450 | 2014-03-21 05:57:34,-66.5059509277344,42.5634918212891
451 | 2014-03-21 07:01:37,-66.5050201416016,42.5554504394531
452 | 2014-03-21 08:10:26,-66.4961471557617,42.5377616882324
453 | 2014-03-21 09:17:37,-66.471076965332,42.5128517150879
454 | 2014-03-21 10:22:21,-66.4316329956055,42.4929695129395
455 | 2014-03-21 11:22:24,-66.4011917114258,42.4911613464355
456 | 2014-03-21 12:22:24,-66.3796691894531,42.4873008728027
457 | 2014-03-21 14:26:05,-66.4200134277344,42.48583984375
458 | 2014-03-21 15:27:51,-66.4417724609375,42.5061416625977
459 | 2014-03-21 16:30:24,-66.4667892456055,42.5253295898438
460 | 2014-03-21 17:35:09,-66.4944686889648,42.5426597595215
461 | 2014-03-21 19:41:56,-66.5073013305664,42.5297508239746
462 | 2014-03-21 20:41:56,-66.4891204833984,42.5206489562988
463 | 2014-03-21 23:50:34,-66.4159927368164,42.5258102416992
464 | 2014-03-22 01:50:41,-66.4164123535156,42.5178985595703
465 | 2014-03-22 02:51:48,-66.4341278076172,42.5208511352539
466 | 2014-03-22 07:00:07,-66.537483215332,42.557201385498
467 | 2014-03-22 08:04:56,-66.5489883422852,42.5474586486816
468 | 2014-03-22 09:05:00,-66.5456008911133,42.5288887023926
469 | 2014-03-22 10:14:16,-66.5221786499023,42.513069152832
470 | 2014-03-22 11:14:21,-66.4931106567383,42.5122299194336
471 | 2014-03-22 13:15:35,-66.4590377807617,42.5244102478027
472 | 2014-03-22 14:19:04,-66.4826126098633,42.5336494445801
473 | 2014-03-22 16:19:36,-66.5151977539063,42.563159942627
474 | 2014-03-22 19:23:22,-66.5438613891602,42.6050910949707
475 | 2014-03-22 20:24:36,-66.5428466796875,42.6228904724121
476 | 2014-03-22 23:28:33,-66.4767303466797,42.7019805908203
477 | 2014-03-23 00:28:57,-66.4418334960938,42.7164306640625
478 | 2014-03-23 01:29:53,-66.4032363891602,42.7388000488281
479 | 2014-03-23 02:30:54,-66.3818893432617,42.7511291503906
480 | 2014-03-23 03:31:06,-66.3774719238281,42.7628288269043
481 | 2014-03-23 05:36:47,-66.3644180297852,42.8162384033203
482 | 2014-03-23 08:41:25,-66.3245086669922,42.8274688720703
483 | 2014-03-23 09:44:57,-66.3008804321289,42.8545417785645
484 | 2014-03-23 10:47:05,-66.2809295654297,42.8759117126465
485 | 2014-03-23 11:47:09,-66.2565612792969,42.8874893188477
486 | 2014-03-23 12:47:33,-66.2292175292969,42.9006881713867
487 | 2014-03-23 13:47:35,-66.2004928588867,42.9129104614258
488 | 2014-03-23 14:48:57,-66.1824798583984,42.9334716796875
489 | 2014-03-23 15:49:00,-66.1861572265625,42.9694290161133
490 | 2014-03-23 16:51:27,-66.1911926269531,43.0107002258301
491 | 2014-03-23 18:56:05,-66.2030715942383,43.0953903198242
492 | 2014-03-23 19:58:27,-66.2048110961914,43.1451683044434
493 | 2014-03-23 21:01:04,-66.1954193115234,43.1907615661621
494 | 2014-03-23 22:03:32,-66.1588897705078,43.225269317627
495 | 2014-03-23 23:03:33,-66.1100311279297,43.2487297058105
496 | 2014-03-24 00:06:57,-66.0567779541016,43.2591285705566
497 | 2014-03-24 01:10:29,-66.0204391479492,43.2763404846191
498 | 2014-03-24 02:10:32,-66.0033493041992,43.3075714111328
499 | 2014-03-24 04:14:50,-66.0322570800781,43.3981399536133
500 | 2014-03-24 05:14:50,-66.0266723632813,43.4243392944336
501 | 2014-03-24 06:17:45,-66.0271377563477,43.464729309082
502 | 2014-03-24 07:20:26,-66.0045928955078,43.4933090209961
503 | 2014-03-25 08:20:32,-66.0083465576172,43.5047302246094
504 | 2014-03-25 13:18:08,-66.0086822509766,43.5033912658691
505 | 2014-03-25 14:18:52,-66.0335235595703,43.4842681884766
506 | 2014-03-25 15:23:45,-66.0698776245117,43.4524116516113
507 | 2014-03-25 16:28:24,-66.1076431274414,43.4163818359375
508 | 2014-03-25 17:31:41,-66.1584320068359,43.3917198181152
509 | 2014-03-25 19:43:22,-66.270751953125,43.3606109619141
510 | 2014-03-25 21:47:32,-66.3533020019531,43.3315582275391
511 | 2014-03-25 22:51:04,-66.3767318725586,43.3035583496094
512 | 2014-03-25 23:54:18,-66.3894805908203,43.266960144043
513 | 2014-03-26 00:54:47,-66.4004516601563,43.2280693054199
514 | 2014-03-26 01:54:52,-66.4024505615234,43.1844215393066
515 | 2014-03-26 02:56:23,-66.4096603393555,43.1446800231934
516 | 2014-03-26 04:00:19,-66.4109802246094,43.1117210388184
517 | 2014-03-26 05:06:52,-66.4167709350586,43.0841102600098
518 | 2014-03-26 07:19:46,-66.4372482299805,43.0390586853027
519 | 2014-03-26 08:24:39,-66.4592514038086,43.0263595581055
520 | 2014-03-26 10:36:04,-66.472541809082,42.9726104736328
521 | 2014-03-26 11:39:05,-66.4802474975586,42.938060760498
522 | 2014-03-26 12:39:49,-66.4846496582031,42.9032402038574
523 | 2014-03-26 13:39:49,-66.4835433959961,42.869571685791
524 | 2014-03-26 14:39:49,-66.4750213623047,42.856258392334
525 | 2014-03-26 15:41:54,-66.4749526977539,42.8450698852539
526 | 2014-03-26 17:48:21,-66.4689025878906,42.8081512451172
527 | 2014-03-26 20:55:09,-66.4930572509766,42.7645683288574
528 | 2014-03-26 21:55:12,-66.4862213134766,42.7667617797852
529 | 2014-03-26 22:57:10,-66.4856414794922,42.7572402954102
530 | 2014-03-27 01:02:28,-66.4674911499023,42.7387084960938
531 | 2014-03-27 03:04:56,-66.4294967651367,42.6913681030273
532 | 2014-03-27 06:11:18,-66.3688507080078,42.6209182739258
533 | 2014-03-27 07:17:13,-66.3665237426758,42.6034698486328
534 | 2014-03-27 08:17:58,-66.3800506591797,42.5882110595703
535 | 2014-03-27 09:19:26,-66.3977203369141,42.5706481933594
536 | 2014-03-27 10:19:32,-66.4270629882813,42.5553283691406
537 | 2014-03-27 11:26:35,-66.4399185180664,42.5337600708008
538 | 2014-03-27 12:26:40,-66.4347381591797,42.5128898620605
539 | 2014-03-27 13:34:08,-66.4266662597656,42.4877815246582
540 | 2014-03-27 14:39:22,-66.3937225341797,42.4633483886719
541 | 2014-03-27 15:43:14,-66.383186340332,42.4336814880371
542 | 2014-03-27 21:01:35,-66.3262481689453,42.4419212341309
543 | 2014-03-27 22:03:58,-66.373420715332,42.467700958252
544 | 2014-03-27 23:05:18,-66.4225082397461,42.4953918457031
545 | 2014-03-28 00:06:59,-66.4534912109375,42.5174407958984
546 | 2014-03-28 01:14:43,-66.4667663574219,42.5329895019531
547 | 2014-03-28 02:17:14,-66.4646987915039,42.5406188964844
548 | 2014-03-28 03:23:28,-66.4449462890625,42.5486717224121
549 | 2014-03-28 04:26:46,-66.4398193359375,42.5449104309082
550 | 2014-03-28 05:27:19,-66.4343719482422,42.5149116516113
551 | 2014-03-28 06:33:10,-66.4406280517578,42.4891700744629
552 | 2014-03-28 07:39:56,-66.4673004150391,42.468391418457
553 | 2014-03-28 09:52:15,-66.5511093139648,42.4580307006836
554 | 2014-03-28 10:56:27,-66.5905914306641,42.4574089050293
555 | 2014-03-28 12:00:14,-66.6362609863281,42.4643096923828
556 | 2014-03-28 13:03:24,-66.6938095092773,42.4736595153809
557 | 2014-03-28 17:06:06,-66.7465972900391,42.5107917785645
558 | 2014-03-28 21:17:22,-66.7301635742188,42.5743103027344
559 | 2014-03-29 00:25:24,-66.7617492675781,42.6344299316406
560 | 2014-03-29 01:26:15,-66.7729797363281,42.6401786804199
561 | 2014-03-29 02:32:24,-66.7770233154297,42.6293487548828
562 | 2014-03-29 03:33:33,-66.7842483520508,42.6046104431152
563 | 2014-03-29 04:39:29,-66.7734985351563,42.5706214904785
564 | 2014-03-29 05:47:27,-66.7555770874023,42.5390586853027
565 | 2014-03-29 07:49:21,-66.7402191162109,42.5195388793945
566 | 2014-03-29 09:53:48,-66.7604904174805,42.5415191650391
567 | 2014-03-29 10:54:43,-66.7739410400391,42.5612983703613
568 | 2014-03-29 11:55:58,-66.7956314086914,42.5621109008789
569 | 2014-03-29 13:58:44,-66.8058929443359,42.5289916992188
570 | 2014-03-29 15:02:56,-66.7978210449219,42.5151405334473
571 | 2014-03-29 16:03:00,-66.7751007080078,42.4987182617188
572 | 2014-03-29 17:04:10,-66.7352066040039,42.4984512329102
573 | 2014-03-29 20:13:59,-66.6720886230469,42.4865989685059
574 | 2014-03-29 23:28:27,-66.7120208740234,42.5177383422852
575 | 2014-03-30 00:34:58,-66.7082595825195,42.5299987792969
576 | 2014-03-30 01:35:30,-66.7069473266602,42.5255889892578
577 | 2014-03-30 02:35:33,-66.7056884765625,42.5114707946777
578 | 2014-03-30 03:35:36,-66.685546875,42.5054817199707
579 | 2014-03-30 04:35:36,-66.6574935913086,42.4984397888184
580 | 2014-03-30 05:40:39,-66.6261215209961,42.4838905334473
581 | 2014-03-30 09:51:13,-66.5473785400391,42.4657402038574
582 | 2014-03-30 10:51:16,-66.550407409668,42.4766807556152
583 | 2014-03-30 11:55:48,-66.595817565918,42.4770698547363
584 | 2014-03-30 12:57:49,-66.6331481933594,42.4890213012695
585 | 2014-03-30 13:59:45,-66.6443023681641,42.5136795043945
586 | 2014-03-30 15:07:19,-66.6425476074219,42.5265083312988
587 | 2014-03-30 16:08:16,-66.6245269775391,42.5337181091309
588 | 2014-03-30 18:14:24,-66.5577011108398,42.5299110412598
589 | 2014-03-30 19:14:28,-66.513786315918,42.5353088378906
590 | 2014-03-30 22:17:04,-66.4886703491211,42.5477981567383
591 | 2014-03-30 23:17:04,-66.5135803222656,42.5681686401367
592 | 2014-03-31 00:20:56,-66.5387725830078,42.5905914306641
593 | 2014-03-31 01:21:00,-66.5693283081055,42.6079216003418
594 | 2014-03-31 03:23:48,-66.6140899658203,42.6452293395996
595 | 2014-03-31 05:31:39,-66.5952682495117,42.6184692382813
596 | 2014-03-31 06:32:08,-66.5691680908203,42.5928382873535
597 | 2014-03-31 07:34:46,-66.5526885986328,42.5598411560059
598 | 2014-03-31 10:47:26,-66.5596694946289,42.5205116271973
599 | 2014-03-31 11:50:51,-66.5790481567383,42.5421714782715
600 | 2014-03-31 12:50:56,-66.6156997680664,42.5555000305176
601 | 2014-03-31 15:00:17,-66.6616821289063,42.5822486877441
602 | 2014-03-31 17:00:21,-66.657356262207,42.5726890563965
603 | 2014-03-31 19:00:28,-66.6555633544922,42.5627708435059
604 | 2014-03-31 21:04:32,-66.6360321044922,42.5653495788574
605 | 2014-03-31 22:06:36,-66.6515808105469,42.564811706543
606 | 2014-04-01 00:06:48,-66.6975173950195,42.5909996032715
607 | 2014-04-01 01:06:48,-66.718620300293,42.6029281616211
608 | 2014-04-01 02:06:52,-66.7417984008789,42.6130714416504
609 | 2014-04-01 04:06:54,-66.7799911499023,42.6118202209473
610 | 2014-04-01 05:11:04,-66.7929306030273,42.6011505126953
611 | 2014-04-01 06:15:14,-66.8158569335938,42.592098236084
612 | 2014-04-01 11:30:49,-66.7959976196289,42.5603790283203
613 | 2014-04-01 16:39:46,-66.8376922607422,42.5803413391113
614 | 2014-04-01 19:45:04,-66.7559661865234,42.5513687133789
615 | 2014-04-01 20:52:31,-66.7179107666016,42.5291595458984
616 | 2014-04-01 21:52:37,-66.6853332519531,42.518310546875
617 | 2014-04-01 22:55:13,-66.6652374267578,42.5142211914063
618 | 2014-04-01 23:56:04,-66.6466522216797,42.5228500366211
619 | 2014-04-02 00:59:04,-66.6595993041992,42.5250701904297
620 | 2014-04-02 04:06:04,-66.6750793457031,42.5106391906738
621 | 2014-04-02 06:11:08,-66.6526336669922,42.4819793701172
622 | 2014-04-02 07:17:51,-66.6201019287109,42.465461730957
623 | 2014-04-02 08:25:46,-66.5784378051758,42.4577102661133
624 | 2014-04-02 09:29:31,-66.5353164672852,42.450870513916
625 | 2014-04-02 10:35:19,-66.4901580810547,42.4537200927734
626 | 2014-04-02 11:38:11,-66.4648208618164,42.4646987915039
627 | 2014-04-02 14:49:18,-66.4859313964844,42.5199012756348
628 | 2014-04-02 15:53:29,-66.5163116455078,42.5471305847168
629 | 2014-04-02 17:53:44,-66.5363464355469,42.5861396789551
630 | 2014-04-02 18:54:12,-66.533073425293,42.5793304443359
631 | 2014-04-02 19:54:56,-66.4957275390625,42.5838088989258
632 | 2014-04-02 20:55:23,-66.4550170898438,42.6033782958984
633 | 2014-04-02 23:00:12,-66.3878479003906,42.6469802856445
634 | 2014-04-03 01:00:31,-66.3949813842773,42.6890602111816
635 | 2014-04-03 02:02:29,-66.4081878662109,42.721851348877
636 | 2014-04-03 03:04:12,-66.4152603149414,42.7651786804199
637 | 2014-04-03 05:04:16,-66.3905029296875,42.823429107666
638 | 2014-04-03 06:04:16,-66.3857116699219,42.8364486694336
639 | 2014-04-03 08:58:25,-66.3114700317383,42.8520584106445
640 | 2014-04-03 09:58:28,-66.2808532714844,42.860279083252
641 | 2014-04-03 11:01:06,-66.2563705444336,42.8677215576172
642 | 2014-04-03 12:06:52,-66.2493515014648,42.878101348877
643 | 2014-04-03 13:06:53,-66.2497634887695,42.8956298828125
644 | 2014-04-03 14:08:23,-66.2773666381836,42.9193382263184
645 | 2014-04-03 15:08:53,-66.2955780029297,42.9488792419434
646 | 2014-04-03 16:13:20,-66.313850402832,42.9863815307617
647 | 2014-04-03 17:13:35,-66.3260192871094,43.0230712890625
648 | 2014-04-03 18:15:31,-66.3258590698242,43.055850982666
649 | 2014-04-03 19:19:35,-66.3119812011719,43.0810585021973
650 | 2014-04-03 20:22:32,-66.2968063354492,43.1058006286621
651 | 2014-04-03 21:22:55,-66.2774276733398,43.1279716491699
652 | 2014-04-03 23:23:47,-66.2645111083984,43.158390045166
653 | 2014-04-04 03:31:55,-66.2836303710938,43.3379516601563
654 | 2014-04-04 05:37:26,-66.2104187011719,43.4353485107422
655 | 2014-04-04 06:42:04,-66.1509780883789,43.4554786682129
656 | 2014-04-04 07:43:32,-66.0870590209961,43.4603118896484
657 | 2014-04-04 08:46:04,-66.0100173950195,43.4541702270508
658 | 2014-04-04 09:46:29,-65.9934005737305,43.4765586853027
659 | 2014-04-04 10:46:32,-66.0084228515625,43.5041084289551
660 | 2014-04-04 11:46:32,-66.0085296630859,43.5043716430664
661 | 2014-04-04 15:38:56,-66.0115814208984,43.5057182312012
662 | 2014-04-04 16:39:00,-66.0084991455078,43.5046691894531
663 | 2014-04-05 04:32:54,-66.0094833374023,43.5044403076172
664 | 2014-04-05 05:36:15,-66.0149078369141,43.5065383911133
665 | 2014-04-05 06:36:20,-66.0125503540039,43.5042381286621
666 | 2014-04-05 07:43:00,-66.0042419433594,43.5026206970215
667 | 2014-04-05 08:43:15,-65.985710144043,43.5110893249512
668 | 2014-04-05 09:45:10,-65.9736785888672,43.4858207702637
669 | 2014-04-05 10:45:16,-65.9865570068359,43.4637908935547
670 | 2014-04-05 12:45:48,-65.9874267578125,43.4641990661621
671 | 2014-04-05 14:56:58,-65.9879302978516,43.4623489379883
672 | 2014-04-05 16:00:06,-66.0271301269531,43.4346389770508


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  1 | ---
  2 | title: 'HMM in Marine Sciences: model selection, covariates, hierarchical structures' 
  3 | author: "Marco Gallegos Herrada and Vianey Leos Barajas"
  4 | output: 
  5 |   bookdown::html_document2:
  6 |     number_sections: true
  7 |     highlight: tango
  8 | editor_options:
  9 |   chunk_output_type: console
 10 | ---
 11 | 
 12 | <!-- To be able to have continuous line numbers -->
 13 | ```{=html}
 14 | <style>
 15 | body
 16 |   { counter-reset: source-line 0; }
 17 | pre.numberSource code
 18 |   { counter-reset: none; }
 19 | </style>
 20 | ```
 21 | 
 22 | ```{r setup, include=FALSE}
 23 | knitr::opts_chunk$set(echo = TRUE)
 24 | knitr::opts_chunk$set(warning = FALSE)
 25 | knitr::opts_chunk$set(cache = TRUE)
 26 | knitr::opts_chunk$set(message = FALSE)
 27 | ```
 28 | 
 29 | 
 30 | 
 31 | ```{r}
 32 | library(readr)
 33 | library(momentuHMM)
 34 | library(ggplot2)
 35 | library(dplyr)
 36 | library(lubridate)
 37 | library(data.tree)
 38 | library(DiagrammeR)
 39 | ```
 40 | 
 41 | # Tutorial objectives
 42 | 
 43 | The goal of this tutorial is to explore how to fit hidden Markov models to accelerometer data, how to incorporate covariates into the transition probabilities, and the implementation of hierarchical Markov models to animal movement data. For the first two objectives, we will use 4 days of acceleration data obtained from a free-ranging blacktip reef shark at Palmyra Atoll in the central Pacific Ocean (data taken from Leos-Barajas et al. 2017). 
 44 | 
 45 | 
 46 | # Accelerometer data
 47 | 
 48 | Accelerometer devices measure up to three axes, which can be described relative to the body of the animal: longitudinal (surge), lateral (sway) and dorsoventral (heave). These devices are becoming more prevalent in the fields of animal biologging data as they provide a means of measuring activity in a meaningful and quantitative way. From tri-axial acceleration data, we can also derive several measures that summarize effort or exertion and relate acceleration  to activity levels such as overall dynamic body acceleration (ODBA) and vectorial dynamic body acceleration (VeDBA). These metrics can be used to reduce the dimensionality of three-dimension acceleration data while retaining important information. Further, because acceleration data is often at high temporal resolutions over time, it also naturally exhibits a large degree of autocorrelation, making it impossible to assume independence between sequential observations. As we have learned, HMMs can account for the autocorrelation present in the data while assuming that the data were generated according to a finite set of (unobserved) behaviors making them a good candidate model for this type of data structure. Today, we will fit an HMM to the ODBA of a blacktip shark, calculated every second.
 49 | 
 50 | For the blacktip shark, we have time of day, water temperature, depth and ODBA. Since one of our goals is to use the time of observations (second of the day) as a covariate, we also need to extract this information from the time variable.
 51 | 
 52 | 
 53 | ```{r}
 54 | # Reading the data
 55 | BlacktipB <- read_delim("data/BlacktipB.txt", 
 56 |                         delim = "\t", escape_double = FALSE, 
 57 |                         trim_ws = TRUE)
 58 | # Let's transform the date/time info into a proper time format
 59 | # In this case we need to extract the second of the day correspondent to each observation 
 60 | BlacktipB = BlacktipB %>% 
 61 |   mutate(Time = as.POSIXct(Time,format = "%m/%d/%Y %H:%M")) %>% 
 62 |   mutate(hour_to_sec =  as.integer(seconds(hm(format(Time, format = "%H:%M"))))) %>% 
 63 |   group_by(Time) %>% mutate(sec = row_number()) %>% ungroup() %>% 
 64 |   mutate(sec = case_when(hour_to_sec == 53160 ~ as.integer(sec + 55),
 65 |                          TRUE ~ sec),
 66 |          hour_to_sec = hour_to_sec + sec)
 67 | 
 68 | head(BlacktipB)
 69 | 
 70 | ```
 71 | 
 72 | 
 73 | # Fitting our model
 74 | 
 75 | Looking at the ODBA values through the observed period, we find ODBA is unusually high at some times -- for this shark we assumed that values between 0 and 2 were consistent with what we expected. Because accommodating extreme values can pose a problem for identification of an adequate state-dependent distribution in our HMM, we removed them from the data set. However, note that in general, deciding whether to remove extreme values or not will more likely depend on whether we find appropriate distributional forms that can accommodate them. Generally, we need to make sure that extreme values are in fact some artefact of the data collection process, not representative of a behavior of interest, or inconsistent with what we are trying to capture as well. Removing data is not good general practice but instead we can assess on a case-by-case basis. 
 76 | 
 77 | We can see the original time series here across the four days: 
 78 | 
 79 | ```{r}
 80 | BlacktipB %>% ggplot(aes(Time,ODBA)) + geom_line()
 81 | 
 82 | ```
 83 | 
 84 | 
 85 | ```{r}
 86 | 
 87 | BlacktipB = BlacktipB %>% filter(ODBA <= 2.0)
 88 | 
 89 | ```
 90 | 
 91 | And now, the modified time series with values above 2.0 removed. Note that here we ignore the fact that we do not have data for these time points.
 92 | 
 93 | Now, we are ready to start to look for models for this data!
 94 | 
 95 | ```{r, echo=F}
 96 | BlacktipB %>% ggplot(aes(Time,ODBA)) + geom_line()
 97 | 
 98 | ```
 99 | 
100 | 
101 | Given our data, we are interested in finding possible behaviours through the observation process (ODBA). Let's take a quick look at the histogram of the observations.
102 | 
103 | ```{r, echo=F}
104 | 
105 | hist(BlacktipB$ODBA[BlacktipB$ODBA < .4],breaks=80,main="Histogram of ODBA", 
106 |      xlab = "ODBA")
107 | 
108 | ```
109 | 
110 | 
111 | As we have indicated before, the best way to start when fitting a hidden Markov model is to keep things simple. In this case, we will be considering 2 behavioral states, with gamma state-dependent distributions, and no covariates for the transition probability matrix. As mentioned in previous tutorials, now is time to implement the decisions that we have made so far. For the choice of initial parameter values we can take a quick peak at the data (e.g., using the plots above). From the plots above, we specify the means of our state-dependent distributions as 0.1 and 0.3. 
112 | 
113 | 
114 | Now that the data is ready for modeling, we choose to fit a 2-state hidden Markov model. For this purpose, we first need to assign the class `momentuHMMData` to the data in order to be presentable for the functions related to `momentuHMM`.
115 | 
116 | ```{r}
117 | BlacktipBData = prepData(BlacktipB,coordNames = NULL,
118 |                    covNames = "hour_to_sec")
119 | 
120 | ```
121 | 
122 | Let's fit our model and take a look at the output of the fitted model. 
123 | 
124 | ```{r, cache=TRUE}
125 | fit1 = fitHMM(BlacktipBData,nbStates=2,dist=list(ODBA="gamma"),Par0 = list(ODBA=c(.1,.3,1,1)))
126 | 
127 | fit1
128 | ```
129 | 
130 | We can also plot the results to obtain a visual representation of the fitted model.
131 | 
132 | ```{r}
133 | plot(fit1,breaks = 80)
134 | ```
135 | 
136 | Let's look at the pseudo-residuals.
137 | 
138 | ```{r}
139 | plotPR(fit1)
140 | ```
141 | 
142 | We can also compute the most likely sequence of states.
143 | 
144 | ```{r, cache=TRUE}
145 | # identify most likely state using the Viterbi algorithm
146 | BlacktipB$state <- viterbi(fit1)
147 | 
148 | # proportion of the behaviour states during the observed period
149 | table(BlacktipB$state)/length(BlacktipB$state)
150 | 
151 | BlacktipB %>% mutate(day = day(Time)) %>% ggplot(aes(Time,state)) + facet_wrap(~day,scales = "free_x") + geom_point()
152 | ```
153 | 
154 | Let's include retryFits in the fitHMM function. This can take time to run.
155 | 
156 | ```{r, cache=TRUE}
157 | set.seed(147)
158 | fit1_s2 <- fitHMM(BlacktipBData,
159 |                   nbState = 2,
160 |                   dist=list(ODBA="gamma"),
161 |                   Par0 = list(ODBA=c(.1,.3,1,1)),
162 |                   retryFits=10)
163 | fit1_s2
164 | ```
165 | 
166 | Seems nothing changed at all!
167 | 
168 | Now let's go further and include a high perturbation in one of the initial values (instead of .1 and .3, let's do .1 and 2). Do we still have similar estimated coefficients and log likelihood? (no matter the initial values, the coefficients should be similar)
169 | 
170 | ```{r, cache=TRUE}
171 | fit1_s2_long <- fitHMM(BlacktipBData,
172 |                   nbState = 2,
173 |                   dist=list(ODBA="gamma"),
174 |                   Par0 = list(ODBA=c(.1,2,1,1)))
175 | 
176 | fit1_s2_long
177 | ```
178 | 
179 | Let's look at the pseudo-residuals.
180 | 
181 | ```{r}
182 | plotPR(fit1_s2_long)
183 | ```
184 | 
185 | You may get warnings.
186 | 
187 | We can see that there is high autocorrelation and some deviation from normality.
188 | 
189 | We can also compute the most likely sequence of states. What can we infer from this? Is there something else we can say from this? According to fitted model, can we see if there is any interesting pattern?
190 | 
191 | ```{r, cache=TRUE}
192 | # identify most likely state using the Viterbi algorithm
193 | BlacktipB$state_wildPar0 <- viterbi(fit1_s2_long)
194 | 
195 | # proportion of the behaviour states during the observed period
196 | table(BlacktipB$state_wildPar0)/length(BlacktipB$state_wildPar0)
197 | 
198 | BlacktipB %>% mutate(day = day(Time)) %>% ggplot(aes(Time,state_wildPar0)) + facet_wrap(~day,scales = "free_x") + geom_point()
199 | 
200 | ```
201 | 
202 | Here we can see that there is only one state when we use these new starting values, this is an indication that there may be problems.
203 | 
204 | # Incorporating covariates
205 | 
206 | As in Leos-Barajas et al. 2017, we can incorporate other information that may help explain the values of ODBA. In this case, we consider the second of the day of every observation. Time of day is represented by two trigonometric functions with period 24 h, $cos(2\pi t/86,400)$ and $sin(2\pi t/86,400)$ (86 400 is the number of seconds in a day). Using the function cosinor, we can convert our data stream to something that is useful for us. As well, we need to provide the formula corresponding to the regression that will be stored in the transition probability values.
207 | 
208 | ```{r, cache=TRUE}
209 | # formula corresponding to the regression coefficients for the transition probabilities
210 | formula = ~ cosinor(hour_to_sec, period = 86400)
211 | Par0_fit2 <- getPar0(model=fit1, formula=formula)
212 | 
213 | fit2 = fitHMM(BlacktipBData,nbStates=2,dist=list(ODBA="gamma"),Par0 = Par0_fit2$Par,formula=formula)
214 | 
215 | fit2
216 | ```
217 | 
218 | 
219 | ```{r, cache=TRUE}
220 | plot(fit2,breaks=80)
221 | ```
222 | 
223 | Let's explore the results. Do the coefficients vary much? What about the ACF? Did the autocorrelation decrease with this innovation?
224 | 
225 | ```{r}
226 | plotPR(fit2)
227 | ```
228 | 
229 | We can also take a quick look at the Akaike information criteria (AIC) for the two models to do a comparison. 
230 | 
231 | ```{r}
232 | 
233 | AIC(fit1)
234 | AIC(fit2)
235 | 
236 | ```
237 | 
238 | # Depth Data
239 | 
240 | Tiger shark data were collected in Oahu, Hawai'i. The shark's depth was recorded in 0.5 m intervals every 2 s over a 23 day period, from March 9 to March 31, 2009. On some days, the tiger shark inhabited varied ranges of depth levels and performed many dives whereas other days it remained relatively constant in the water column and dove less. For sharks, movement in the water column is not easily segmented into types of dives. However, dives, or simply ascending or descending, can be an important part of a shark's behavior. We extracted a depth position every ten minutes from the available data record and sequentially computed the absolute change in depth position, $y^*_t = |d_t - d_{t-1}|$, for the tiger shark in order to understand how the depths inhabited by the shark differ across days. On some days the tiger shark tends to remain in the same part of the water column for long periods of time while on others, it tends to move up and down more frequently throughout the day. We processed the data to produce $M=144$ observations per day, across $K=21$ days, excluding the first and last days for which only partial data records were available. 
241 | 
242 | ```{r}
243 | 
244 | # Read the data
245 | tigerShark <- read_csv("data/tigershark_depthchange10min.csv")
246 | tigerShark = tigerShark %>% mutate(abs_change_depth = abs(Depth - lag(Depth,default = 0)))
247 | head(tigerShark)
248 | 
249 | ```
250 | 
251 | ```{r, cache=TRUE}
252 | tigerShark %>%
253 |   filter(days != 9) %>%
254 |   ggplot(aes(x=HST,y=Depth)) + facet_wrap(~days,scales = "free_x") + geom_line() +
255 |   scale_x_datetime(breaks= "8 hour", date_labels = "%H:%M") + theme_minimal()
256 | 
257 | 
258 | tigerShark %>% filter(days != 9) %>% 
259 |   ggplot(aes(x=HST,y=2*abs_change_depth)) + facet_wrap(~days,scales = "free_x") + geom_line() +
260 |   scale_x_datetime(breaks= "8 hour", date_labels = "%H:%M") + theme_minimal()
261 | 
262 | ```
263 | 
264 | # Hierarchical Hidden Markov models
265 | 
266 | The general idea of a hiearchical HMM is that there are (at least) two behavioral processes of interest but they manifest at different temporal scales. For instance, here we will look at both fine-scale vertical movement behavior and across day movement behavior as well. These are referred to as fine state-level (10 min) and coarse state-level behaviors (day). 
267 | 
268 | To fit these models in momentuHMM, we will introduce a new column called "level".
269 | 
270 | ```{r}
271 | 
272 | days_range = unique(tigerShark$days)
273 | tigerSharkData <- NULL
274 | 
275 | for(i in days_range){
276 |   coarseInd <- data.frame(tigerShark %>% filter(days == i) %>% 
277 |                                    filter(row_number() == 1) %>% select(HST),
278 |                                  days = i,
279 |                                  level=c("1","2i"),
280 |                                  abs_change_depth=NA)
281 |   tmp <- rbind(coarseInd,tigerShark %>% filter(days == i) %>% mutate(level = "2") %>% 
282 |                  select(HST,days,level,abs_change_depth))
283 |   tigerSharkData <- rbind(tigerSharkData,tmp)
284 | }
285 | 
286 | head(tigerSharkData)
287 | 
288 | tigerSharkData = prepData(tigerSharkData,
289 |                           coordNames = NULL, hierLevels = c("1","2i","2"))
290 | 
291 | 
292 | # summarize prepared data
293 | summary(tigerSharkData, dataNames = names(tigerSharkData)[-1])
294 | 
295 | ```
296 | 
297 | Note that 2i indicates the initial distribution in the fine state level for every time in the coarse state level, and 2 indicates the observations for the fine state level.
298 | 
299 | However, one of the difficulties with hierarchical HMMs is the construction of the number of states, just like in regular HMMs! Here we choose three fine-scale states as that has been tested for this data to produce the best compromise between model fit and model complexity, and two coarse-scale states. 
300 | 
301 | ```{r}
302 | 
303 | ### define hierarchical HMM
304 | ### states 1-3 = coarse state 1 (nontravelling)
305 | ### states 4-6 = coarse state 2 (travelling)
306 | hierStates <- data.tree::Node$new("tiger shark HHMM states")
307 | hierStates$AddChild(name="nontravelling")
308 | hierStates$nontravelling$AddChild(name="nt1", state=1)
309 | hierStates$nontravelling$AddChild(name="nt2", state=2)
310 | hierStates$nontravelling$AddChild(name="nt3", state=3)
311 | hierStates$AddChild(name="travelling")
312 | hierStates$travelling$AddChild(name="t1", state=4)
313 | hierStates$travelling$AddChild(name="t2", state=5)
314 | hierStates$travelling$AddChild(name="t3", state=6)
315 | 
316 | plot(hierStates)
317 | 
318 | ```
319 | 
320 | ```{r,eval=F}
321 | 
322 | #Alternative way for specifying
323 | hierStates <- data.tree::as.Node(list(name="tiger shark HHMM states",
324 |                                       nontravelling=list(nt1=list(state=1),
325 |                                                 nt2=list(state=2),
326 |                                                 nt3=list(state=3)),
327 |                                       travelling=list(t1=list(state=4),
328 |                                                     t2=list(state=5),
329 |                                                     t3=list(state=6))))
330 | 
331 | ```
332 | 
333 | 
334 | The name for any of the “children” added to a node are user-specified and are akin
335 | to the stateNames argument in fitHMM for a standard HMM. While these names are
336 | arbitrary, the name and state attributes must be unique.
337 | 
338 | ```{r}
339 | 
340 | # data stream distributions
341 | # level 1 = coarse scale (no data streams)
342 | # level 2 = fine scale (dive_duration, maximum_depth, dive_wiggliness)
343 | hierDist <- data.tree::Node$new("tiger shark HHMM dist")
344 | hierDist$AddChild(name="level1")
345 | hierDist$AddChild(name="level2")
346 | hierDist$level2$AddChild(name="abs_change_depth", dist="gamma")
347 | plot(hierDist)
348 | 
349 | ```
350 | 
351 | The Node attribute dist is required in hierDist and specifies the probability distribution for each data stream at each level of the hierarchy (Figure 14. In this case,
352 | level1 (corresponding to coarse-scale observations with level=1) has no data streams,
353 | and each of the data streams for level2 (corresponding to fine-scale observations with
354 | level=2) is assigned a gamma distribution with an additional point-mass on zero to account for zero changes in depth. 
355 | 
356 | We did not include any covariates on the t.p.m. or initial distribution for either level of the hierarchy, but, for demonstration purposes, here is
357 | how we would use the hierFormula and hierFormulaDelta arguments to specify the
358 | t.p.m. and initial distribution formula for each level of the hierarchy in fitHMM:
359 | 
360 | ```{r}
361 | 
362 | # define hierarchical t.p.m. formula(s)
363 | hierFormula <- data.tree::Node$new("harbor porpoise HHMM formula")
364 | hierFormula$AddChild(name="level1", formula=~1)
365 | hierFormula$AddChild(name="level2", formula=~1)
366 | 
367 | # define hierarchical initial distribution formula(s)
368 | hierFormulaDelta <- data.tree::Node$new("harbor porpoise HHMM formulaDelta")
369 | hierFormulaDelta$AddChild(name="level1", formulaDelta=~1)
370 | hierFormulaDelta$AddChild(name="level2", formulaDelta=~1)
371 | 
372 | ```
373 | 
374 | We can assume the data stream probability distributions do not depend on the coarse-scale state, so we can constrain the state-dependent parameters for states 1 (“nt1”) and 4 (“t1”), states 2 (“nt2”) and 5 (“t2”), and states 3 (“nt3”) and 6 (“t3”) to be equal using the DM argument:
375 | 
376 | ```{r}
377 | 
378 | # defining starting values
379 | cd.mu0 = rep(c(5,50,100),hierStates$count)
380 | cd.sigma0 = rep(c(5,15,40),hierStates$count)
381 | cd.pi0 = rep(c(0.2,0.01,0.01),hierStates$count)
382 | 
383 | Par0 = list(abs_change_depth = c(cd.mu0,cd.sigma0,cd.pi0))
384 | 
385 | nbStates <- length(hierStates$Get("state",filterFun=data.tree::isLeaf))
386 | 
387 | 
388 | ```
389 | 
390 | ```{r}
391 | 
392 | # constrain fine-scale data stream distributions to be same
393 | cd_DM <- matrix(cbind(kronecker(c(1,1,0,0,0,0),diag(3)),
394 |                       kronecker(c(0,0,1,1,0,0),diag(3)),
395 |                       kronecker(c(0,0,0,0,1,1),diag(3))),
396 |                 nrow=nbStates*3,
397 |                 ncol=9,
398 |                 dimnames=list(c(paste0("mean_",1:nbStates),
399 |                                 paste0("sd_",1:nbStates),
400 |                                 paste0("zeromass_",1:nbStates)),
401 |                               paste0(rep(c("mean","sd","zeromass"),each=3),
402 |                                      c("_14:(Intercept)",
403 |                                        "_25:(Intercept)",
404 |                                        "_36:(Intercept)"))))
405 | DM = list(abs_change_depth = cd_DM)
406 | 
407 | ```
408 | 
409 | ```{r}
410 | 
411 | # get initial parameter values for data stream probability distributions
412 | Par <- getParDM(tigerSharkData,hierStates=hierStates,hierDist=hierDist,
413 |                 Par=Par0,DM=DM)
414 | 
415 | # check hierarchical model specification and parameters
416 | checkPar0(tigerSharkData,hierStates=hierStates,hierDist=hierDist,Par0=Par,
417 |           hierFormula=hierFormula,hierFormulaDelta=hierFormulaDelta,
418 |           DM=DM)
419 | 
420 | ```
421 | 
422 | When interpreting the output, we are looking at two parts: 
423 | 
424 | - the estimated state-dependent distribution parameters
425 | - the different transition probability matrices that define the differences at the coarse-scale
426 | 
427 | ```{r, cache=TRUE}
428 | 
429 | # fit hierarchical HMM
430 | hhmm <- fitHMM(data=tigerSharkData,hierStates=hierStates,hierDist=hierDist,
431 |                #hierFormula=hierFormula,#hierFormulaDelta=hierFormulaDelta,
432 |                Par0=Par,#hierBeta=hierBeta,hierDelta=hierDelta,
433 |                DM=DM,nlmPar=list(hessian=FALSE))
434 | hhmm
435 | 
436 | ```
437 | 
438 | We can use the same functions as before to visualize our model results.
439 | 
440 | ```{r}
441 | plot(hhmm)
442 | plotPR(hhmm)
443 | ```
444 | 
445 | # Exercises
446 | 
447 | Generate new data streams from original datasets (overall mean in different time periods). Does the ACF changes between these data sets? How are they compared to the original data set? What can you say about this?
448 | 
449 | - Instead of considering observations every second, create a new data set for 10 sec, another for 30 s and 1 min. Code below will generate such data sets. Fit a basic HMM for the each one of these new datasets. Did the ACF changed? does they seems to not break the conditional independence? (i.e., the ACF values are lower?). 
450 |   
451 | ```{r, eval=F}
452 | 
453 | BlacktipB = BlacktipB %>% 
454 |   mutate(group_10s = case_when(sec <= 10 ~ 1,
455 |                                10 < sec & sec <= 20 ~ 2,
456 |                                20 < sec & sec <= 30 ~ 3,
457 |                                30 < sec & sec <= 40~ 4,
458 |                                40 < sec & sec <= 50 ~ 5,
459 |                                50 < sec & sec <= 60 ~ 6,
460 |                                TRUE ~ -1),
461 |          group_30s = case_when(sec <= 30 ~ 1,
462 |                                30 < sec & sec <= 60 ~ 2,
463 |                                TRUE ~ -1)) %>% 
464 |   group_by(Time,group_10s) %>% mutate(ODBA_mean_10s = mean(ODBA)) %>% ungroup() %>% 
465 |   group_by(Time,group_30s) %>% mutate(ODBA_mean_30s = mean(ODBA)) %>% ungroup() %>% 
466 |   group_by(Time) %>% mutate(ODBA_mean_60s = mean(ODBA)) %>%
467 |   mutate(hour_to_sec_10s = floor((hour_to_sec-1)/10+1),
468 |          hour_to_sec_30s = floor((hour_to_sec-1)/30+1),
469 |          hour_to_sec_60s = floor((hour_to_sec-1)/60+1)) 
470 | 
471 | BlacktipB_10s = BlacktipB %>% group_by(Time,group_10s) %>% filter(row_number() == 1) %>%
472 |   ungroup() %>% select(Time,Depth,ODBA_mean_10s,hour_to_sec_10s)
473 | BlacktipB_30s = BlacktipB %>% group_by(Time,group_30s) %>% filter(row_number() == 1) %>% 
474 |   ungroup() %>% select(Time,Depth,ODBA_mean_30s,hour_to_sec_30s)
475 | BlacktipB_60s = BlacktipB %>% group_by(Time) %>% filter(row_number() == 1) %>% 
476 |   ungroup() %>% select(Time,Depth,ODBA_mean_60s,hour_to_sec_60s)
477 | 
478 | BlacktipB = BlacktipB %>% select(Time,Depth,ODBA,hour_to_sec)
479 | 
480 | 
481 | ```
482 | 
483 | Fit a 3-state HMM with no covariates and fit another one using time as a covariate. What happents to the ACF? According to loglikelihood and AIC, which model is better?
484 | 
485 | - Reproduce the analysis from the blacktip shark data, but now include three states. What about the ACF? Did something change compared to the 2-state HMM? Now include hour_to_sec as a covariate and fit a new model. What happen with the AIC values and the loglikelihood? According to these, what seems to be a better model?


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/README.md:
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 1 | # DFO 2022 Hidden Markov Model Workshop
 2 | 
 3 | ## Description and format
 4 | 
 5 |   This workshop was developed by Marie Auger-Méthé, Vianey Leos Barajas, Ron Togunov, and Marco Antonio Gallegos Herrada for the Department of Fisheries and Oceans, Canada in October 2022. The goal of the workshop is to illustrate the application of hidden Markov models (HMMs) to movement data from marine animals to classify behaviours and identify behaviour-specific habitat associations. 
 6 | 
 7 | The workshop will be composed of three one-hour lectures and three three-hour tutorials in R. These lecture can be found on our youtube channel: https://www.youtube.com/playlist?list=PLYkxNfKA95Pzcp5xfJdQKl6xLQoXrP39x  
 8 | Each tutorial will be completed together over zoom and will begin with a 10-minute introduction over Zoom that will provide an overview of the tutorial objectives. 
 9 | 
10 | ## Workshop learning objectives
11 | 
12 | - Statistical framework for HMMs and their application to animal movement data
13 | - Selecting appropriate temporal resolution for HMM analysis
14 | - Interpolating missing locations (linear, crw, path segmentation, and multiple imputation)
15 | - Fit HMMs to animal movement data to identify behaviours
16 | - Integrating diving data-streams to identify more complex behaviours
17 | - Incorporate covariates on state transition probability to identify conditions that promote different behaviours
18 | - Integrating covariates on emission probabilities to model biased random walks 
19 | - Modelling behaviour using accelerometer data
20 | - Fitting hierarchical HMMs
21 | 
22 | ## Prerequisite experience
23 | 
24 | - Intermediate R coding
25 | - Familiarity with animal movement/telemetry data
26 | 
27 | ## Preparation/instructions
28 | 
29 | - Download and unzip workshop zip from Github
30 | - Install all the required packages as described in 0_Installing_packages.Rmd
31 | - Make sure all packages are up-to-date as older versions may not work
32 | - View each lecture in advance of their respective workshop date
33 | - Create new R script in the DFO_HMM_2022 directory
34 | - Follow along with each days tutorial html files (e.g., "Day1/HMM_Tutorial_Day1.html")
35 | 


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/preWorkshopInfo/0_Installing_packages.Rmd:
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  1 | ---
  2 | title: "Installing packages"
  3 | subtitle: "Tips to install some of the packages"
  4 | author: "Joe Watson, Ron Togunov, & Marie Auger-Méthé"
  5 | date: "26/07/2022"
  6 | output: 
  7 |     html_document:
  8 |     number_sections: true
  9 | ---
 10 | 
 11 | 
 12 | ```{r setup, include=FALSE}
 13 | knitr::opts_chunk$set(echo = TRUE)
 14 | ```
 15 | 
 16 | ## Installing packages: tips to install some of the packages
 17 | 
 18 | Some of the packages we will use in the workshop are not available on CRAN and some require the installation of additional software and compilers.
 19 | 
 20 | ### Compilers
 21 | 
 22 | C++ compilers are needed. 
 23 | 
 24 | For Mac, installing Xcode is an easy way to install compilers. 
 25 | You can install Xcode directly from the App store (it's free).
 26 | 
 27 | You can also install it via terminal, but because it is a very large program it's worth checking if you already have it installed. If not, open terminal and run:
 28 | 
 29 | ```{r, eval=FALSE}
 30 | xcode-select --install
 31 | ```
 32 | 
 33 | For Windows, installing Rtools is an easy way to install compilers. Rtools is found [cran.r-project.org/bin/windows/Rtools](https://cran.r-project.org/bin/windows/Rtools/index.html)
 34 | 
 35 | For Linux, the necessary compilers are installed by default with R. To double check they are installed run:
 36 | 
 37 | ```{r, eval=FALSE}
 38 | gcc --version
 39 | ```
 40 | 
 41 | If nothing is returned, then they can be installed manually as follows:
 42 | 
 43 | ```{r, eval=FALSE}
 44 | sudo apt update
 45 | sudo apt install build-essential
 46 | ```
 47 | 
 48 | 
 49 | ### Required GIS software
 50 | 
 51 | **The workshop is developed with latest version of gdal and geos. The workshop material will not work without them or if you are using outdated versions of these.** 
 52 | 
 53 | If you do not have these software already installed, please install them (see below). **If you already have `rgdal` and gdal installed, verify that the version that's already installed is adequate. Load the `rgdal` package and use the function rgdal function `GDALis3ormore()` in R. It should return TRUE. We want gdal to use the appropriate PROJ transformation software ([proj.org](https://proj.org/)), so in addition the rgdal function `PROJis6ormore()` should return TRUE. If either return FALSE, update your gdal.**
 54 | *Note new version of some of these GIS software and packages (e.g., raster) have made significant changes and might affect other code.*
 55 | 
 56 | 
 57 | On Mac, installing these is easiest done using homebrew ([brew.sh](https://brew.sh/)). With homebrew installed, open terminal and run:
 58 | 
 59 | ```{r, eval=FALSE}
 60 | brew tap osgeo/osgeo4mac && brew tap --repair
 61 | brew install pkg-config
 62 | brew install proj
 63 | brew install geos
 64 | brew install gdal
 65 | ```
 66 | 
 67 | On Windows, express install osgeo4w by following the instruction found at  [trac.osgeo.org/osgeo4w](https://trac.osgeo.org/osgeo4w/). This will install the PROJ, GDAL, and GEOS libraries.
 68 | 
 69 | On Linux, run the following in the terminal:
 70 | ```{r, eval=FALSE}
 71 | sudo add-apt-repository ppa:ubuntugis/ubuntugis-unstable
 72 | sudo apt-get update
 73 | sudo apt-get install libudunits2-dev libgdal-dev libgeos-dev libproj-dev 
 74 | ```
 75 | 
 76 | 
 77 | ### Installing the latest version of the spatial packages
 78 | 
 79 | The latest version of sf and terra are **required**. The workshop material **will not work without these**. These can be installed **after** the latest GIS software has been installed.
 80 | 
 81 | ```{r, eval=FALSE}
 82 | install.packages(c('sf', 'terra', 'tmap'), dep=TRUE)
 83 | ```
 84 | 
 85 | We are trying to remove any usage of the package raster (this package is now replaced by terra), but since momentuHMM still uses raster, it is sometimes easier to use functions from the raster package.
 86 | Thus, please install raster.
 87 | 
 88 | ```{r, eval=FALSE}
 89 | install.packages('raster', dep=TRUE)
 90 | ```
 91 | 
 92 | 
 93 | ### Packages available on CRAN without special dependencies
 94 | 
 95 | The remaining packages do not have special dependencies, and should be easily installed or updated via CRAN (using install.packages()):
 96 | 
 97 | + momentuHMM
 98 | + dplyr
 99 | + tidyr
100 | + stringr
101 | + data.table
102 | + lubridate
103 | + units
104 | + diveMove
105 | + earthtide
106 | + corrplot
107 | + conicfit
108 | + car
109 | + mitools
110 | + doFuture
111 | + kableExtra
112 | + data.tree
113 | 
114 | ### Check that all packages are installed
115 | 
116 | Once all the packages are installed, you can check that they are properly installed by loading them.
117 | 
118 | ```{r, warning=FALSE, message=FALSE}
119 | library(sf)        # spatial data processing
120 | library(terra)     # raster data processing
121 | library(tmap)      # mapping spatial data
122 | library(sp)        # old spatial package
123 | library(raster)    # old raster package
124 | library(momentuHMM)# fitting HMMs
125 | library(dplyr)     # data management
126 | library(tidyr)     # data management
127 | library(stringr)   # character string management
128 | library(data.table)# data management
129 | library(lubridate) # date management
130 | library(units)     # physical measurement units management
131 | library(diveMove)  # calculating dive metrics 
132 | library(earthtide) # estimating tidal data 
133 | library(corrplot)  # calculating Pearson's correlation matrices
134 | library(conicfit)  # fitting error ellipses (required by momentuHMM)
135 | library(car)       # additional operations to Applied Regression
136 | library(mitools)   # parallel processing
137 | library(doFuture)  # parallel processing
138 | library(kableExtra)# produce visually appealing tables
139 | library(data.tree)
140 | library(DiagrammeR)
141 | ```
142 | 


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