Author: Jan-Ole Fischer

# 2 Automatic differentiation via RTMB

#### Jan-Ole Fischer

> Before diving into this vignette, we recommend reading the vignette Introduction to LaMa.

Automatic differentiation (AD) is available in `R` through the `RTMB` package. Using `RTMB`, you can define the likelihood for any statistical model as plain `R` code and have access to fast and exact derivatives of any order (without doing any calculations by hand!). This enables the fitting of very complicated models, potentially with complex random effect structures.

`LaMa` is fully compatible with AD provided by `RTMB`. Estimation of latent Markov models with AD is much faster and more convenient, while model specification is very smooth and less prone to errors. Let’s find out how this works! We begin 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 behavioural 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

The workflow with `RTMB` is always very similar. We need to

1. define the negative log-likelihood function,
2. turn it into an AD objective function, and
3. fit the model by numerical optimisation of the latter.

`RTMB` provides many functions that make the process of specifying a likelihood function and calculating standard deviations of estimates really simple.

We start by fitting a simple stationary HMM with state-dependent gamma distributions for the step lengths. As a first step, we define the initial parameter list `par` containing unconstrained (real-valued) parameters and a `dat` list that contains the data and potential hyperparameters – here \(N\), the number of hidden states. The latter is not strictly necessary but provides a neat way of wrapping everything needed for the model fit.

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

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

Because `par` is a named list, accessing the parameters inside the likelihood is very convenient – annoying and error-prone indexing of parameter vectors is gone! We 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
      # Parameter transformations for strictly positive parameters
      mu = exp(log_mu)
      sigma = exp(log_sigma)
      # Reporting statements for later use
      REPORT(mu); ADREPORT(mu)
      REPORT(sigma); ADREPORT(sigma)
      # Calculating all state-dependent densities
      allprobs = matrix(1, nrow = length(step), ncol = N)
      ind = which(!is.na(step)) # only for non-NA obs.
      for(j in 1:N){
        allprobs[ind,j] = dgamma2(step[ind], mu[j], sigma[j])
      }
      -forward(delta, Gamma, allprobs) # simple forward algorithm
    }

A few points should be made here:

- The 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. It unpacks both the `par` and the `dat` list, making all elements available without the `$` operator.

> Important: `nll` reads `dat` directly from the global environment at the time `MakeADFun()` is called, which bakes a snapshot of `dat` into the compiled objective function. Any changes to `dat` after that point will have no effect on the optimisation. Always finalise `dat` before calling `MakeADFun()`.

- Parameter transformations are still necessary: all parameters in `par` should be unconstrained and transformed to their natural scale inside the likelihood.

- The `REPORT()` function offered by `RTMB` is really convenient. 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.

- For simple parameter transformations, `ADREPORT()` is also great. 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 automatically differentiable objective function. At this point, `RTMB` takes `nll` and generates its own (very fast) version of it, including a gradient. We set `silent = TRUE` to suppress printing of the optimisation process.

    obj = MakeADFun(nll, par, silent = TRUE) # creating the AD 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 transformed 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 call these functions without any argument, we get the corresponding values at the initial parameter vector.

    obj$par
    #>     log_mu     log_mu  log_sigma  log_sigma        eta        eta
    #> -1.2039728  0.0000000 -1.6094379 -0.3566749 -2.0000000 -2.0000000
    obj$fn()
    #> [1] 16378.66
    obj$gr()
    #>          [,1]     [,2]    [,3]      [,4]     [,5]      [,6]
    #> [1,] 609.4426 -2272.55 94.1283 -12144.32 151.3179 -181.5449

We are now ready to fit the model by optimising the objective function. The routine `nlminb()` is very robust and allows us to provide a gradient function. While we also have access to the Hessian function, we do not supply it to `nlminb()` because optimisation is 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) # minimisation

We can check out the estimated parameter and function value by

    opt$par
    #>     log_mu     log_mu  log_sigma  log_sigma        eta        eta
    #> -1.1923951  0.9185717 -1.6018327  0.3993241 -1.6519372 -1.5641927
    opt$objective
    #> [1] 10847.12

Note that the names `par` and `objective` are determined by `nlminb()`. If you use a different optimiser, these may be called differently.

However, `RTMB` provides a nicer way to get the estimates. `obj` is automatically updated after the optimisation. Note that calling `obj$gr()` after optimisation now gives the gradient at the optimum, while `obj$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 = report(obj) # runs the reporting from the negative log-likelihood once
    (delta = mod$delta) # stationary distribution
    #>        S1        S2
    #> 0.4817329 0.5182671
    (Gamma = mod$Gamma) # estimated tpm
    #>           S1        S2
    #> S1 0.8269542 0.1730458
    #> S2 0.1608473 0.8391527
    (mu = mod$mu) # estimated state-dependent means
    #> [1] 0.3034935 2.5057089
    (sigma = mod$sigma) # estimated state-dependent sds
    #> [1] 0.2015268 1.4908167

This 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 downstream tasks like state decoding or residuals calculation. The `viterbi` and `stateprobs` functions either take these as arguments or, more conveniently, can also take the whole reported `mod` object as an input. Because the state-dependent parameters depend on the specific model formulation, these need to be reported manually by the user specifying the likelihood. Having all the parameters, we can plot the decoded time series

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

    # or much 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.

    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")

Note that because we have used `report()`, `mod` now has class “LaMaModel”, which permits the use of `AIC()` and `BIC()`:

    AIC(mod)
    #> [1] 21706.25
    BIC(mod)
    #> [1] 21749.51

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) # see ?summary.sdreport for more info
    #>             Estimate  Std. Error
    #> log_mu    -1.1923951 0.011461828
    #> log_mu     0.9185717 0.009076411
    #> log_sigma -1.6018327 0.016858309
    #> log_sigma  0.3993241 0.013479572
    #> eta       -1.6519372 0.042684391
    #> eta       -1.5641927 0.041157505
    #> mu         0.3034935 0.003478590
    #> mu         2.5057089 0.022742845
    #> sigma      0.2015268 0.003397402
    #> sigma      1.4908167 0.020095571

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

    # estimated parameter in list format
    as.list(sdr, "Estimate")
    # 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(obs = trex$step, # observation sequence
                           dist = "gamma2", # which parametric CDF to use
                           par = list(mean = mu, sd = sigma), # estimated pars for that CDF
                           mod = mod) # model object
    plot(pres_step, hist = TRUE)

## Tips and common issues

There are some commonly occuring issues with `RTMB` to keep in mind. I list the main ones I have encountered here – please let me know if you encounter more so they can be added.

#### Non-standard distributions

If you need a non-standard probability distribution, existing implementations will likely fail due to AD incompatibility. Check whether the distribution is available in `RTMBdist`, which provides a library of AD-compatible distributions for use inside likelihood functions.

#### Operator overloading

A typical issue is that some operators may need to be overloaded to allow for automatic differentiation. In typical model setups `LaMa` functions handle this automatically, but if you take a more individualistic route and encounter 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.

#### Initialising with `NA`

A common problem occurs when initialising objects with `NA` values and then filling them with `numeric` values. `NA` is logical, which causes type mismatches that break automatic differentiation. Always initialise with `numeric` or `NaN` values instead. For example, avoid

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

and use

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

#### Wrapping arrays in `AD()`

If you create an array and fill it with something parameter-dependent, wrap it in `AD()`:

    X <- AD(array(...))

This ensures `X` is an AD object from the start and that classes are compatible. Wrapping in `AD()` is generally a good idea to resolve the above error, as it introduces no meaningful overhead when not needed.

#### No `if` statements on parameters

You cannot use `if` statements on parameters directly, as these are not differentiable. `RTMB` builds the tape (computational graph) at the initial parameter value, so `if` branches depending on a parameter will produce gradients that are wrong elsewhere. `RTMB` will fail, likely without a helpful error message. `if` statements that do not involve parameters are typically fine, since the branch is fixed throughout optimisation.

#### Byte compiler side effects

Some unfortunate side effects of `R`’s byte compiler (enabled by default) can interfere with `RTMB`. If you encounter an error not covered above, try disabling it with

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

#### Further reading

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

> Continue reading with Penalised splines.