# LaMa and RTMB

> Before diving into this vignette, we recommend reading the vignettes Introduction to LaMa, Inhomogeneous HMMs and Periodic HMMs.

The recently introduced `R` package `RTMB` conveniently allows for automatic differentiation for non-standard statistical models written in plain `R` code. This enables the estimation of very complicated models, potentially with complex random effect structures. The process feels like magic because you have access to analytic gradients – drastically increasing accuracy and speed – without doing any calculations!

`LaMa` is now also fully compatible with AD provided by `RTMB`. Hence, estimation of latent Markov models is now faster and more convenient, while model specification is very smooth and less prone to errors – which at the current state tend to happen when one is not experienced with `RTMB`.

Here we demonstrate how to use `LaMa` and `RTMB` to fit hidden Markov models and their extensions. We start by loading the `LaMa` package, which automatically loads `RTMB` as well.

    library(LaMa)

For the purpose of this vignette, we will analyse the `trex` data set contained in the package. It contains hourly step lengths of a Tyrannosaurus rex, living 66 million years ago, and we aim to understand its behavoural process using HMMs.

    head(trex, 5)
    #>   tod      step     angle state
    #> 1   9 0.3252437        NA     1
    #> 2  10 0.2458265  2.234562     1
    #> 3  11 0.2173252 -2.262418     1
    #> 4  12 0.5114665 -2.958732     1
    #> 5  13 0.3828494  1.811840     1

### Basic workflow

The workflow with `RTMB` is basically always the same. We need to

- define the negative log-likelihood function,
- create an automatically differentiable objective function from it and
- fit the model by numerical minimisation of the latter.

`RTMB` also provides many functions that make this process very convenient.

### Simple HMM

We start by fitting a super simple stationary HMM with state-dependent gamma distributions for the step lengths and von Mises distributions for the turning angles. As a first step, we define the initial parameter list `par` and a `dat` list that contains the data and potential hyperparameters – here \(N\), the number of hidden states. The names `par` and `dat` are of course arbitrary.

    par = list(
      logmu = log(c(0.3, 1)),      # initial means for step length (log-transformed)
      logsigma = log(c(0.2, 0.7)), # initial sds for step length (log-transformed)
      logkappa = log(c(0.2, 0.7)), # initial concentration for turning angle (log-transformed)
      eta = rep(-2, 2)             # initial t.p.m. parameters (on logit scale)
      )

    dat = list(
      step = trex$step,   # hourly step lengths
      angle = trex$angle, # hourly turning angles
      N = 2
      )

As `par` is a named list of initial parameter values, accessing the parameters later on is much more convenient than indexing. You can also use a parameter vector with `RTMB`, but using a named list makes our life so much easier.

We can now define the negative log-likelihood function in a similar fashion to basic numerical ML

    nll = function(par) {
      getAll(par, dat) # makes everything contained available without $
      Gamma = tpm(eta) # computes transition probability matrix from unconstrained eta
      delta = stationary(Gamma) # computes stationary distribution
      # exponentiating because all parameters strictly positive
      mu = exp(logmu)
      sigma = exp(logsigma)
      kappa = exp(logkappa)
      # reporting statements for later use
      REPORT(mu); ADREPORT(mu)
      REPORT(sigma); ADREPORT(sigma)
      REPORT(kappa); ADREPORT(kappa)
      # calculating all state-dependent densities
      allprobs = matrix(1, nrow = length(step), ncol = N)
      ind = which(!is.na(step) & !is.na(angle)) # only for non-NA obs.
      for(j in 1:N){
        allprobs[ind,j] = dgamma2(step[ind],mu[j],sigma[j])*dvm(angle[ind],0,kappa[j])
      }
      -forward(delta, Gamma, allprobs) # simple forward algorithm
    }

but a few points should be made here:

- Most prominently, the negative log-likelihood is a function of the parameters to be estimated only while data and other parameters are not passed as an argument at this stage. This is something to get used to (I know), but just the way `RTMB` works.
- The `getAll()` function is very useful and you should use it in the first line to unpack both the `par` and the `dat` list, making all elements available without the `$` operator. At this stage, `nll` just takes the dat object from the global environment.
- Parameter transformations are of course still necessary, i.e. all parameters in `par` should be unconstrained.
- You might wonder how on earth `RTMB` can calculate the gradient of parameters in distributions like the gamma or von Mises distribution. The answer is: It can’t but provides its own version of all standard distributions like `dnorm()`, `dbinom()`, etc. In this case both `dgamma2()` and `dvm()` come from `LaMa` as these are non-standard, but under the hood build on `RTMB` functions (`dgamma2()` is actually just a convenience function that reparametrises the gamma distribution in terms of mean and standard deviation).
- Actually, most of the standard functions (e.g. `sum()`), operators (e.g. `%*%`) and methods (e.g. `matrix`) are “overwritten” when called inside `MakeADFun()` but you typically don’t notice that and should not care.
- the `REPORT()` function offered by `RTMB` is really convenient as any quantities calculated in the likelihood function (for which you have written the code anyway), if reported, will be available after optimisation, while the report statements are ignored during optimisation. So no annoying backtransformations anymore, wohoo!
- for simple parameter transformations, `ADREPORT()` is also great, because it calculates standard deviations for `ADREPORT()`ed quantities, based on the delta method. Just note that the delta method is not advisable for complex non-linear and multivariate transformations.

Having defined the negative log-likelihood, we can now create the autmatically differentiable objective function and fit the model. This needs a little explanation: At this point, `RTMB` takes the negative log-likelihood function and generates its own (very fast) version of it, including a gradient. `MakeADFun()` now also grabs whatever is saved as `dat` in the global environment and bakes it into the objective function. Therefore, changes to `dat` after this point will have no effect on the optimisation result. We set `silent = TRUE` to suppress printing of the optimisation process.

    obj = MakeADFun(nll, par, silent = TRUE) # creating the objective function

Let’s check out `obj`:

    names(obj)
    #>  [1] "par"          "fn"           "gr"           "he"           "hessian"
    #>  [6] "method"       "retape"       "env"          "report"       "simulate"
    #> [11] "force.update"

It contains the initial parameter `par` (now tranformed to a named vector), the objective function `fn` (which in this case just evaluates `nll` but faster), its gradient `gr` and Hessian `he`.

If we now call these functions without any argument, we get the corresponding values at the initial parameter vector.

    obj$par
    #>      logmu      logmu   logsigma   logsigma   logkappa   logkappa        eta
    #> -1.2039728  0.0000000 -1.6094379 -0.3566749 -1.6094379 -0.3566749 -2.0000000
    #>        eta
    #> -2.0000000
    obj$fn()
    #> [1] 33293.84
    obj$gr()
    #>          [,1]      [,2]     [,3]      [,4]     [,5]      [,6]     [,7]
    #> [1,] 573.7198 -2467.274 95.35893 -12045.97 55.92507 -807.9504 134.0732
    #>           [,8]
    #> [1,] -181.2148

We are now ready to optimise the objective function. The optimisation routine `nlminb()` is very robust and conveniently allows us to provide a gradient function. Alternatively, you can also use `optim()` or any other optimiser you like that allows you to pass a gradient function.

Indeed, we do not provide the Hessian to `nlminb()` because, while evaluating the Hessian is very fast with `RTMB`, optimisation is still much faster if we use a quasi-Newton algorithm that approximates the current Hessian based on previous gradient evaluations, compared to using full Newton-Raphson.

    opt = nlminb(obj$par, obj$fn, obj$gr) # optimization

We can check out the estimated parameter and function value by

    opt$par
    #>      logmu      logmu   logsigma   logsigma   logkappa   logkappa        eta
    #> -1.1916144  0.9182131 -1.5995349  0.3999258 -2.2872716  0.4019563 -1.6621910
    #>        eta
    #> -1.5735921
    opt$objective
    #> [1] 27248.59

Note that the naming here is determined by `nlminb()`. If you use a different optimiser, these may be called differently.

Much nicer however, is that `obj` (yes `obj` not `opt`) is automatically updated after the optimisation. Note that calling `obj$gr()` after optimisation now gives the gradient at the optimum, while `obj$fn()` still gives the objective at the starting value and `obj$par` is not updated but still the initial parameter vector (kind of confusing).

To get our estimated parameters on their natural scale, we don’t have to do the backtransformation manually. We can just run the reporting:

    mod = obj$report() # runs the reporting from the negative log-likelihood once
    (delta = mod$delta)
    #>       S1       S2
    #> 0.481525 0.518475
    (Gamma = mod$Gamma)
    #>           S1        S2
    #> S1 0.8282951 0.1717049
    #> S2 0.1594681 0.8405319
    (mu = mod$mu)
    #> [1] 0.3037305 2.5048106
    (sigma = mod$sigma)
    #> [1] 0.2019904 1.4917139
    (kappa = mod$kappa)
    #> [1] 0.1015431 1.4947460

which works because of the `REPORT()` statements in the likelihood function. Note that `delta`, `Gamma` and `allprobs` are always reported by default when using `forward()` which is very useful for e.g. state decoding with `viterbi()`, because many downstream `LaMa` functions take these arguments as inputs. Functions of the `viterbi` and `stateprobs` family can also take the reported list object as an input. As the state-dependent parameters depend on the specific model formulation, these need to be reported manually by the user specifying the negative log-likelihood. Having all the parameters, we can plot the decoded time series

    # manually
    mod$states = viterbi(mod$delta, mod$Gamma, mod$allprobs)

    # or simpler
    mod$states = viterbi(mod = mod)

    # defining color vector
    color = c("orange", "deepskyblue")

    plot(trex$step[1:200], type = "h", xlab = "time", ylab = "step length",
         col = color[mod$states[1:200]], bty = "n")
    legend("topright", col = color, lwd = 1, legend = c("state 1", "state 2"), bty = "n")

or the estimated state-dependent distributions.

    oldpar = par(mfrow = c(1,2))
    hist(trex$step, prob = TRUE, breaks = 40,
         bor = "white", main = "", xlab = "step length")
    for(j in 1:2) curve(delta[j] * dgamma2(x, mu[j], sigma[j]),
                        lwd = 2, add = T, col = color[j])
    curve(delta[1]*dgamma2(x, mu[1], sigma[1]) + delta[2]*dgamma2(x, mu[2], sigma[2]),
          lwd = 2, lty = 2, add = T)
    legend("top", lwd = 2, col = color, legend = c("state 1", "state 2"), bty = "n")

    hist(trex$angle, prob = TRUE, breaks = 40,
         bor = "white", main = "", xlab = "turning angle")
    for(j in 1:2) curve(delta[j] * dvm(x, 0, kappa[j]),
                        lwd = 2, add = T, col = color[j])
    curve(delta[1]*dvm(x, 0, kappa[1]) + delta[2]*dvm(x, 0, kappa[2]),
          lwd = 2, lty = 2, add = T)

    par(oldpar) # resetting to default

Moreover, we can also use the `sdreport()` function to directly give us standard errors for our unconstrained parameters and everything we `ADREPORT()`ed.

    sdr = sdreport(obj)

We can then get an overview of the estimated parameters and `ADREPORT()`ed quantities as well as their standard errors by

    summary(sdr)
    #>            Estimate  Std. Error
    #> logmu    -1.1916144 0.011067932
    #> logmu     0.9182131 0.008875692
    #> logsigma -1.5995349 0.016232361
    #> logsigma  0.3999258 0.013272894
    #> logkappa -2.2872716 0.207126331
    #> logkappa  0.4019563 0.019299344
    #> eta      -1.6621910 0.041754277
    #> eta      -1.5735921 0.040795512
    #> mu        0.3037305 0.003361669
    #> mu        2.5048106 0.022231928
    #> sigma     0.2019904 0.003278782
    #> sigma     1.4917139 0.019799361
    #> kappa     0.1015431 0.021032256
    #> kappa     1.4947460 0.028847617

To get the estimated parameters or their standard errors in list format, type

    # estimated parameter in list format
    as.list(sdr, "Estimate")
    # parameter standard errors in list format
    as.list(sdr, "Std")

and to get the estimates and standard errors for `ADREPORT()`ed quantities in list format, type

    # adreported parameters as list
    as.list(sdr, "Estimate", report = TRUE)
    # their standard errors
    as.list(sdr, "Std", report = TRUE)

Lastly, the automatic reporting with `LaMa` and `RTMB` together makes calculating pseudo-residuals really convenient:

    pres_step = pseudo_res(trex$step, "gamma2", list(mean = mu, sd = sigma), mod = mod)
    plot(pres_step, hist = TRUE)

    pres_angle = pseudo_res(trex$angle, "vm", list(mu = 0, kappa = kappa), mod = mod)
    plot(pres_angle, hist = TRUE)

### Covariate effects

We can now generalise the previous model to include covariate effects. In our example, we might be interested how the T-rex’s behaviour varies with the time of day. Hence, we add diel variation to the state process. For example, we can model the transition probabilities as a function of the time of day using a trigonometric basis expansion to ensure diurnal continuity. The transition probabilities are given by \[ \text{logit}(\gamma_{ij}^{(t)}) = \beta_0^{(ij)} + \beta_1^{(ij)} \sin \bigl(\frac{2 \pi t}{24}\bigr) + \beta_2^{(ij)} \cos \bigl(\frac{2 \pi t}{24}\bigr) + \beta_3^{(ij)} \sin \bigl(\frac{2 \pi t}{12}\bigr) + \beta_4^{(ij)} \cos \bigl(\frac{2 \pi t}{12}\bigr), \] where \(t\) is the time of day.

To practically achieve this, we compute the trigonometric basis design matrix `Z` corresponding to above predictor and add the time of day to the `dat` list for indexing inside the likelihood function. The `LaMa` function `cosinor()` does this very conveniently. We can either call it directly to build the design matrix, or use it in a `formula` passed to `make_matrices()`. The latter is preferable when dealing with more complicated models. Note that the first option does not include an intercept column, while the second does. When used with `tpm_g()` this does not matter as it automatically checks if an intercept column is included.

    Z = cosinor(1:24, period = c(24, 12))

    modmat = make_matrices(~ cosinor(tod, period = c(24, 12)),
                           data = data.frame(tod = 1:24))
    Z = modmat$Z

    # only compute the 24 unique values and index later for entire time series
    dat$Z = Z # adding design matrix to dat
    dat$tod = trex$tod # adding time of day to dat for indexing

We also need to change the parameter list `par` to include the regression parameters for the time of day. The regression parameters for the state process will typically have the form of a \(N (N-1) \times p+1\) matrix, where \(N\) is the number of states and \(p\) is the number of regressors – this format is also expected by `tpm_g()` which computes the array of transition matrices based on the design and parameter matrix. Another lovely convenience that `RTMB` allows for is that, in our parameter list, we can have matrices, making reshaping of vectors to matrices inside the likelihood function unnessesary.

    par = list(logmu = log(c(0.3, 1)),
               logsigma = log(c(0.2, 0.7)),
               logkappa = log(c(0.2, 0.7)),
               beta = matrix(c(rep(-2, 2),
                               rep(0, 2*4)), nrow = 2)) # 2 times 4+1 matrix
    # replacing eta with regression parameter matrix, initializing slopes at zero

We can now define a more general likelihood function with the main difference being the use of `tpm_g()` instead of `tpm()` and the inclusion of the time of day in the transition matrix calculation. This leads to us using `stationary_p()` instead of `stationary()` to calculate the initial distribuion and `forward_g()` instead of `forward()` to calculate the log-likelihood.

    nll2 = function(par) {
      getAll(par, dat) # makes everything contained available without $
      Gamma = tpm_g(Z, beta) # covariate-dependent tpms (in this case only 24 unique)
      # tpm_g() automatically checks if intercept column is included
      ADREPORT(Gamma) # adreporting
      Delta = stationary_p(Gamma) # all periodically stationary distributions
      ADREPORT(Delta)
      delta = Delta[tod[1],] # initial periodically stationary distribution
      # exponentiating because all parameters strictly positive
      mu = exp(logmu); REPORT(mu)
      sigma = exp(logsigma); REPORT(sigma)
      kappa = exp(logkappa); REPORT(kappa)
      # calculating all state-dependent densities
      allprobs = matrix(1, nrow = length(step), ncol = N)
      ind = which(!is.na(step) & !is.na(angle)) # only for non-NA obs.
      for(j in 1:N){
        allprobs[ind,j] = dgamma2(step[ind],mu[j],sigma[j])*dvm(angle[ind],0,kappa[j])
      }
      -forward_g(delta, Gamma[,,tod], allprobs) # indexing 24 unique tpms by tod in data
    }

Having done this, the model fit is then essentially the same:

    obj2 = MakeADFun(nll2, par, silent = TRUE) # creating the objective function
    opt2 = nlminb(obj2$par, obj2$fn, obj2$gr) # optimisation

and we can look at the reported results. In this case, for simplicity I get standard errors for `Gamma` with the delta method while, in general, this is not advisable.

    mod2 = obj2$report()

    sdr = sdreport(obj2)
    Gamma = as.list(sdr, "Estimate", report = TRUE)$Gamma
    Gammasd = as.list(sdr, "Std", report = TRUE)$Gamma

    Delta = as.list(sdr, "Estimate", report = TRUE)$Delta
    Deltasd = as.list(sdr, "Std", report = TRUE)$Delta

    tod_seq = seq(0, 24, length = 200) # sequence for plotting
    Z_pred = trigBasisExp(tod_seq, degree = 2) # design matrix for prediction

    Gamma_plot = tpm_g(Z_pred, mod2$beta) # interpolating transition probs

    plot(tod_seq, Gamma_plot[1,2,], type = "l", lwd = 2, ylim = c(0,1),
         xlab = "time of day", ylab = "transition probability", bty = "n")
    segments(x0 = 1:24, y0 = Gamma[1,2,]-1.96*Gammasd[1,2,],
             y1 = Gamma[1,2,]+1.96*Gammasd[1,2,])
    segments(x0 = 1:24, y0 = Gamma[2,1,]-1.96*Gammasd[2,1,],
             y1 = Gamma[2,1,]+1.96*Gammasd[2,1,])
    lines(tod_seq, Gamma_plot[2,1,], lwd = 2, lty = 3)
    legend("topleft", lwd = 2, lty = c(1,3), bty = "n",
           legend = c(expression(gamma[12]^(t)), expression(gamma[21]^(t))))

    plot(Delta[,2], type = "b", lwd = 2, xlab = "time of day", ylab = "Pr(active)",
         col = "deepskyblue", bty = "n", xaxt = "n")
    segments(x0 = 1:24, y0 = Delta[,2]-1.96*Deltasd[,2], lwd = 2,
             y1 = Delta[,2]+1.96*Deltasd[,2], col = "deepskyblue")
    axis(1, at = seq(0,24,by=4), labels = seq(0,24,by=4))

### Common issues with `RTMB`

There are some problems with `RTMB` one has to keep in mind. They can be a bit annoying, but in my opinion the benefits of automatic differentiation far outweigh the drawbacks. I list the main ones I have encountered here, but please tell me if you encounter more, such that they can be added.

A typical issue with `RTMB` is that some operators might need to be overloaded to allow for automatic differentiation which cannot be done by default. In typical model setups `LaMa` functions do this themselves, but if you go a very individualistic route and get an error like

    stop("Invalid argument to 'advector' (lost class attribute?)")

you might have to overload the operator yourself. To do this put

    "[<-" <- ADoverload("[<-")

as the first line of your likelihood function. If the error still prevails also add

    "c" <- ADoverload("c")
    "diag<-" <- ADoverload("diag<-")

which should hopefully fix the error.

Another common problem occurs when initiating objects with `NA` values and then trying to fill them with `numeric` values. This is because `NA` is logical which screws up the automatic differentiation due to the mismatching types. To avoid this, always initiate with `numeric` or `NaN` values. For example, don’t do

    X = array(dim = c(1,2,3))
    # which is the same as
    X = array(NA, dim = c(1,2,3))

but rather

    X = array(NaN, dim = c(1,2,3))
    # or
    X = array(0, dim = c(1,2,3))

to avoid the error.

Importantly, you cannot use `if` or `max`/ `min` statements on the parameter itself as these are not differentiable. If you do so, `RTMB` will fail and probably does not produce a helpful error message. The problem here results from `RTMB` building the tape (computational graph) of the function at the initial parameter value. When you have `if` statements, the resulting gradient will be different from the one at a different parameter value. Often, you can remedy this behaviour by exploiting the fact that `abs()` is differentiable (in code). For example, you can create the differentiable `max` alternative:

    max2 = function(x,y){
      (x + y + abs(x - y)) / 2
    }

So you might be able to solve such problems by finding a clever alternative. If the `if` statement does not involve the parameter, it will typically be fine because it does not change during the optimisation.

Furthermore, there are some unfortunate side effects of R’s ‘byte compiler’ (enabled by default in R). So if you encounter an error not matching the previous ones, try disabling the byte compiler with

    compiler::enableJIT(0)
    #> [1] 3

and see if the error is resolved.

Some more minor things:

- if you’re used to `expm::expm()` that won’t work with AD. Use `Matrix::expm()` instead.
- `CircStats::dvm()` also isn’t compatible with AD. Use `LaMa::dvm()` instead.
- most of the standard distributions are available in `RTMB`. If you need a non-standard one, try implementing the density function yourself using plain R code. `RTMB` also provides AD versions of many building-block functions (like the Gamma or Bessel function) which might help with this.

For more information on `RTMB`, check out its documentation or the TMB users Google group.

> Continue reading with Penalised splines.