Author: Jan-Ole Fischer

# 4 Longitudinal data

#### Jan-Ole Fischer

This vignette has not yet been updated to work with RTMB

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

In real-data applications, one will often be faced by a data set consisting of several measurement tracks, that can reasonably be assumed to be mutually independent. Examples for such a longitudinal structure include GPS tracks of several individuals (or several tracks (e.g. days) of one individual), or when analysing sports data, one will often be faced by time series for separate games. In such settings, the researcher of course has to decide whether to pool parameters across tracks or not. Here, we will provide brief examples for complete and partial pooling.

In the situations above, the likelihood function will look slightly different. In case of \(K\) independent tracks, we have \[ L(\theta) = \prod_{k=1}^K L_k(\theta), \] where \(L_k(\theta)\) is the usual HMM likelihood for the \(k\)-th track. Thus the log-likelihood becomes a sum over \(K\) tracks, which we can calculate in a loop. When \(K\) is even moderately large, performing this loop in `R` already leads to severe slowdowns in likelihood evaluation times. Thus, the forward algorithms in `LaMa` allow for the likelihood formulation above, when the indices at which separate tracks begin are specified. Here, we shortly demonstrate how to use this option.

## Complete pooling

### Generating data

We generate \(K\) separate tracks, all from the exact same model:

    # loading the package
    library(LaMa)

    # parameters are shared across individuals
    mu = c(15, 60) # state-dependent means
    sigma = c(10, 40) # state-dependent standard deviations
    Gamma = matrix(c(0.95, 0.05, 0.15, 0.85), nrow = 2, byrow = TRUE) # t.p.m.
    delta = stationary(Gamma) # stationary distribution

    # simulation of all tracks
    set.seed(123)
    K = 200 # number of individuals, for example different animals
    n = 50 # observations per animal only (but many animals)

    s = x = rep(NA, n*K)
    for(k in 1:K){
      sk = xk = rep(NA, n)
      sk[1] = sample(1:2, 1, prob = delta)
      xk[1] = rnorm(1, mu[sk[1]], sigma[sk[1]])
      for(t in 2:n){
        sk[t] = sample(1:2, 1, prob = Gamma[sk[t-1],])
        xk[t] = rnorm(1, mu[sk[t]], sigma[sk[t]])
      }
      s[(k-1)*n + 1:n] = sk
      x[(k-1)*n + 1:n] = xk
    }

    trackID = rep(1:K, each = n)

### Writing the negative log-likelihood function

To calculate the joint log-likelihood of the independent tracks, we slightly modify the standard negative log-likelihood function by adding the additional argument `trackID`. `forward()` now calculates the sum of indivual likelihood contributions, each starting in the respective initial distribution (which we pool here).

    # fast version using trackInd in forward()
    nll_pool = function(par, x, trackID){
      Gamma = tpm(par[1:2])
      delta = stationary(Gamma)
      mu = par[3:4]
      sigma = exp(par[5:6])
      allprobs = matrix(1, length(x), 2)
      for(j in 1:2) allprobs[,j] = dnorm(x, mu[j], sigma[j])

      # here we add trackInd as an argument to forward()
      -forward(delta, Gamma, allprobs, trackID)
    }

    # slow alternative looping over individuals in R
    nll_pool_slow = function(par, x, K){
      n = length(x) / K
      Gamma = tpm(par[1:2])
      delta = stationary(Gamma)
      mu = par[3:4]
      sigma = exp(par[5:6])
      allprobs = matrix(1, length(x), 2)
      for(j in 1:2) allprobs[,j] = dnorm(x, mu[j], sigma[j])

      # here we just loop over individuals in R
      l = 0
      for(k in 1:K){
        l = l + forward(delta, Gamma, allprobs[(k-1)*n + 1:n,])
      }
      -l
    }

### Estimating the model

Now we estimate the model with complete pooling. We compare the fast version using `forward()` with `trackID` with the slow version also using `forward()` but looping over individuals in `R`.

    # initial parameter vector
    par = c(logitgamma = c(-1,-1), # off-diagonals of Gamma (on logit scale)
            mu = c(15, 60), # state-dependent means
            logsigma = c(log(10),log(40))) # state-dependent sds

    # fast version:
    system.time(
      mod <- nlm(nll_pool, par, x = x, trackID = trackID)
    )
    #>    user  system elapsed
    #>   0.288   0.025   0.313

    # slow version
    system.time(
      mod <- nlm(nll_pool_slow, par, x = x, K = K)
    )
    #>    user  system elapsed
    #>   1.504   0.017   1.520

In this example, looping over individuals in `R` already leads to five times longer the estimation time, but this can be much more severe for more complicated models.

## Partial pooling

If some parameters of our model are individual-specific, while the rest is shared, we speak of partial pooling. We demonstrate this here for 5 individuals with their own transition probability matrices. We could estimate a separate transition probability matrix for each individual, but here we opt for a more parsimonious approach, where the transition probabilities depend on an external, individual-specific covariate. We will estimate the effect of this covariate on the transition probabilities.

### Generating data

    K = 5 # number of individuals, for example different animals

    # state-dependent parameters are shared across individuals
    mu = c(15, 60)
    sigma = c(10, 40)

    # but we define a tpm for each individual depending on covariates
    set.seed(123)
    z = rnorm(K) # covariate (e.g. age)
    beta = matrix(c(-2,-2, 1, -1), nrow = 2)
    # we calculate 5 tpms depending on individual-specific covariates:
    Gamma = tpm_g(z, beta)
    # each individual starts in its stationary distribution:
    Delta = matrix(NA, K, 2)
    for(k in 1:K){ Delta[k,] = stationary(Gamma[,,k]) }

    # simulation of all tracks
    set.seed(123)
    n = 200 # observations per animal only (but many animals)
    s = x = rep(NA, n*K)
    for(k in 1:K){
      sk = xk = rep(NA, n)
      sk[1] = sample(1:2, 1, prob = Delta[k, ])
      xk[1] = rnorm(1, mu[sk[1]], sigma[sk[1]])
      for(t in 2:n){
        sk[t] = sample(1:2, 1, prob = Gamma[sk[t-1],,k])
        xk[t] = rnorm(1, mu[sk[t]], sigma[sk[t]])
      }
      s[(k-1)*n + 1:n] = sk
      x[(k-1)*n + 1:n] = xk
    }

### Writing the negative log-likelihood function

Now we write the corresponding negative log-likehood function that incorporates the above structure. As each track has a fixed t.p.m., we can assume stationarity and compute the stationary initial distribution for each track respectively.

    # fast version using trackInd in forward()
    nll_partial = function(par, x, z, trackID){
      # individual-specific tpms
      beta = matrix(par[1:4], nrow = 2)
      Gamma = tpm_g(z, beta)
      Delta = t(sapply(1:k, function(k) stationary(Gamma[,,k])))
      mu = par[5:6]
      sigma = exp(par[7:8])
      allprobs = matrix(1, length(x), 2)
      for(j in 1:2) allprobs[,j] = dnorm(x, mu[j], sigma[j])
      # just handing a Delta matrix and Gamma array for all individuals to forward()
      -forward(Delta, Gamma, allprobs, trackID)
    }

### Estimating the model

    # again defining all the indices where a new track begins
    trackID = rep(1:K, each = n)

    # initial parameter vector
    par = c(beta = c(-2, -2, 0, 0), # beta
            mu = c(15, 60), # state-dependent means
            log(10), log(40)) # state-dependent sds

    system.time(
      mod_partial <- nlm(nll_partial, par, x = x, z = z, trackID = trackID)
    )
    #>    user  system elapsed
    #>   0.256   0.003   0.259