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.
We generate \(K\) separate tracks, all from the exact same model:
# 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)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
}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
#> 0.234 0.019 0.253
# slow version
system.time(
mod <- nlm(nll_pool_slow, par, x = x, K = K)
)
#> user system elapsed
#> 1.117 0.019 1.137In 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.
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.
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
}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)
}# 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
#> 0.138 0.003 0.140