# Analysing data with dependence

Now that we are familiar with hierarchical models in the regression setup with independent data, we will now extend these ideas to dependent data situations. We will consider the linear and non-linear time series models used in modeling population dynamics as our examples. The principles, of course, apply to more general situations.

## Discrete time population dynamics models: General setup

We know that next yearâ€™s population size depends on the current population size, recruitment rate and survival rate. In the case of meta-population models, we may also need to take into account immigration and emigration in the models. Almost all of these rates are stochastic, changing from one year to the next depending on the environment. We may also have to consider demographic stochasticity when the populations are closer to extinction. A general form for deterministic models is:

There are a number of assumptions that we make to make statistical analysis feasible. For example, we assume the environmental noise is additive. We can also consider additive noise on the log-scale because we are interested in modeling growth $N_{t+1} / N_{t}$. This is because we tend to use differential equation models of the type: $\frac{1}{N} \frac{dN}{dt} = f(N)$. These models are models of growth.

Let $X_t = log(N_t)$. Then another way to write the population growth models in discrete time is:

Choice of different functional forms for $f(N_t, \theta)$ leads to different growth models. Each one has different dynamical properties. They may also lead to chaotic (or, bifurcation etc.) for different sets of parameters. Please make yourself familiar with these before using the models for estimation and forecasting. Our goal here is to show you why and how hierarchical models are used in this context.

This lecture is more bare-bones than the previous ones. Please feel free to contact the organizers if you have questions.

## Gompertz growth model without observation error

Data generation

set.seed(1234)
a <- 1.5
b <- -0.35
sigma_sq <- 0.01
T <- 30
x <- numeric(T)
x[1] <- log(1)
for (t in 2:T)
x[t] <- x[t - 1] + a + b * x[t - 1] + rnorm(1, 0, sqrt(sigma_sq))
N <- exp(x)
plot(as.ts(N))


Bayesiam analysis

library(dclone)
model <- custommodel("model {
for (k in 1:K) {
x[1,k] ~ dnorm(a / (1 - b), (1 - b^2) / sigma_sq)
for (t in 2:T) {
x[t,k] ~ dnorm(mu[t,k], 1 / sigma_sq)
mu[t,k] <- a + (1 + b) * x[t - 1,k]
}
}
a ~ dnorm(0, 0.01)
b ~ dunif(-0.999, -0.001)
z <- 0.5 * log((1 + b) / (1 - b))
sigma_sq <- exp(log_sigma)^2
log_sigma ~ dnorm(0, 0.01)
}")
dat <- list(x = data.matrix(x), K = 1, T = T)
fit <- jags.fit(dat, c("a", "b", "sigma_sq"), model)
summary(fit)


Data cloning

K <- c(1, 2, 4, 8)
dat <- list(x = dcdim(data.matrix(x)), K = 1, T = T)
dcfit <- dc.fit(dat, c("a", "b", "sigma_sq"), model,
n.clones = K,
unchanged = "T", multiply = "K")
summary(fit)
plot(dcdiag(dcfit))
plot(dctable(dcfit))


Missing data

dat\$x[5,1] <- NA


Bayesian approach to predict missing observations

fit <- jags.fit(dat, c("a", "b", "sigma_sq", "x[5,1]"), model)
summary(fit)
densplot(fit[,"x[5,1]"])
abline(v = x[5], col = 2)
quantile(fit[,"x[5,1]"], probs = c(0.025, 0.5, 0.975))
x[5]


DC approach to predict missing observations

pmodel <- custommodel("model {
for (k in 1:K) {
x[1,k] ~ dnorm(a / (1 - b), (1 - b^2) / sigma_sq)
for (t in 2:T) {
x[t,k] ~ dnorm(mu[t,k], 1 / sigma_sq)
mu[t,k] <- a + (1 + b) * x[t - 1,k]
}
}
param[1:3] ~ dmnorm(cf[], Vinv[,])
a <- param[1]
z <- param[2]
b <- (exp(2 * z) - 1) / (exp(2 * z) + 1)
log_sigma <- param[3]
sigma_sq <- exp(log_sigma)^2
}")
dcfit2 <- dcfit <- dc.fit(dat, c("a", "z", "log_sigma"), model,
n.clones = 8,
unchanged = "T", multiply = "K")
(V <- vcov(dcfit2)[c("a", "z", "log_sigma"), c("a", "z", "log_sigma")])
(cf <- coef(dcfit2)[c("a", "z", "log_sigma")])
prdat <- c(dat, list(cf = cf, Vinv = solve(V)))
pred <- jags.fit(prdat, "x[5,1]", pmodel)

densplot(pred)
abline(v = x[5], col = 2)
quantile(pred, probs = c(0.025, 0.5, 0.975))
x[5]


Predicting future states, Bayesian

T2 <- 10
dat2 <- list(x = data.matrix(c(x, rep(NA, T2))),
K = 1, T = T + T2)
fit <- jags.fit(dat2, c("x[31:40,1]"), model)
summary(fit)
pi1 <- quantile(fit, probs = c(0.025, 0.5, 0.975))


Predicting future states, DC

prdat2 <- c(dat2, list(cf = cf, Vinv = solve(V)))
pred2 <- jags.fit(prdat2, "x[31:40,1]", pmodel)
summary(pred2)
pi2 <- quantile(pred2, probs = c(0.025, 0.5, 0.975))

plot(1:T2 - 0.2, pi1[2,], type = "p",
col = 2, pch = 19, cex = 0.5,
ylim = range(pi1, pi2))
segments(x0 = 1:T2 - 0.2, x1 = 1:T2 - 0.2,
y0 = pi1[1,], y1 = pi1[3,], col = 2)
points(1:T2 + 0.2, pi2[2,], type = "p",
col = 4, pch = 19, cex = 0.5)
segments(x0 = 1:T2 + 0.2, x1 = 1:T2 + 0.2,
y0 = pi2[1,], y1 = pi2[3,], col = 4)


## Gompertz model with observation error

Poisson error

set.seed(1234)
a <- 1.5
b <- -0.35
sigma_sq <- 0.01
T <- 30
x <- numeric(T)
x[1] <- log(1)
for (t in 2:T)
x[t] <- x[t - 1] + a + b * x[t - 1] + rnorm(1, 0, sqrt(sigma_sq))
N <- exp(x)
Y <- rpois(T, lambda = N)
plot(as.ts(Y))
lines(as.ts(N), col=2)

model <- custommodel("model {
for (k in 1:K) {
Y[1,k] ~ dpois(exp(x[1,k]))
x[1,k] ~ dnorm(a / (1 - b), (1 - b^2) / sigma_sq)
for (t in 2:T) {
Y[t,k] ~ dpois(exp(x[t,k]))
x[t,k] ~ dnorm(mu[t,k], 1 / sigma_sq)
mu[t,k] <- a + (1 + b) * x[t - 1,k]
}
}
a ~ dnorm(0, 0.01)
b ~ dunif(-0.999, -0.001)
z <- 0.5 * log((1 + b) / (1 - b))
sigma_sq <- exp(log_sigma)^2
log_sigma ~ dnorm(0, 0.01)
}")
dat <- list(Y = data.matrix(Y), K = 1, T = T)
fit <- jags.fit(dat, c("a", "b", "sigma_sq"), model)
summary(fit)


Normal error

set.seed(1234)
a <- 1.5
b <- -0.35
sigma_sq <- 0.01
tau_sq <- 0.1
T <- 30
x <- numeric(T)
x[1] <- log(1)
for (t in 2:T)
x[t] <- x[t - 1] + a + b * x[t - 1] + rnorm(1, 0, sqrt(sigma_sq))
Y <- rnorm(T, mean = x, sd = sqrt(tau_sq))
N <- exp(Y)
plot(as.ts(N))
lines(as.ts(exp(x)), col=2)

model <- custommodel("model {
for (k in 1:K) {
Y[1,k] ~ dnorm(x[1,k], 1 / tau_sq)
x[1,k] ~ dnorm(a / (1 - b), (1 - b^2) / sigma_sq)
for (t in 2:T) {
Y[t,k] ~ dnorm(x[t,k], 1 / tau_sq)
x[t,k] ~ dnorm(mu[t,k], 1 / sigma_sq)
mu[t,k] <- a + (1 + b) * x[t - 1,k]
}
}
a ~ dnorm(0, 0.01)
b ~ dunif(-0.999, -0.001)
z <- 0.5 * log((1 + b) / (1 - b))
sigma_sq <- exp(log_sigma)^2
log_sigma ~ dnorm(0, 0.01)
tau_sq <- exp(log_tau)^2
log_tau ~ dnorm(0, 0.01)
}")
dat <- list(Y = data.matrix(Y), K = 1, T = T)
fit <- jags.fit(dat, c("a", "b", "sigma_sq", "tau_sq"), model)
summary(fit)


## Ricker without observation error

a-b parametrization

set.seed(1234)
a <- 0.5
b <- -0.1
sigma_sq <- 0.05
T <- 30
x <- numeric(T)
x[1] <- log(1)
for (t in 2:T)
x[t] <- x[t - 1] + a + b * exp(x[t - 1]) +
rnorm(1, 0, sqrt(sigma_sq))
N <- exp(x)
plot(as.ts(N))

library(dclone)
model_ab <- custommodel("model {
for (k in 1:kk) {
N[1,k] <- exp(x[1,k])
for (t in 2:T) {
x[t,k] ~ dnorm(mu[t,k], 1 / sigma_sq)
mu[t,k] <- a + b * N[t - 1,k] + x[t - 1,k]
N[t,k] <- min(exp(x[t,k]), 10000)
}
}
a ~ dnorm(1, 0.01)
b ~ dnorm(0, 0.1)
sigma_sq <- exp(log_sigma)^2
log_sigma ~ dnorm(0, 1)
}")
dat <- list(x = data.matrix(x), kk = 1, T = T)
fit <- jags.fit(dat, c("a", "b", "log_sigma"), model_ab)
summary(fit)


r-K (logistic) parametrization (r=a, K=-a/b)

model_rK <- custommodel("model {
for (k in 1:kk) {
N[1,k] <- exp(x[1,k])
for (t in 2:T) {
x[t,k] ~ dnorm(mu[t,k], 1 / sigma_sq)
mu[t,k] <- r * (1 - N[t - 1,k] / K) + x[t - 1,k]
N[t,k] <- min(exp(x[t,k]), 10000)
}
}
r ~ dnorm(0, 0.1)
K <- exp(log_K)
log_K ~ dnorm(0, 0.1)
sigma_sq <- exp(log_sigma)^2
log_sigma ~ dnorm(0, 1)
}")
dat <- list(x = data.matrix(x), kk = 1, T = T)
fit <- jags.fit(dat, c("r", "K", "sigma_sq"), model_rK)
summary(fit)


## Ricker with observation error

a-b parametrization

set.seed(2345)
a <- 0.5
b <- -0.1
sigma_sq <- 0.05
T <- 30
x <- numeric(T)
x[1] <- log(1)
for (t in 2:T)
x[t] <- x[t - 1] + a + b * exp(x[t - 1]) +
rnorm(1, 0, sqrt(sigma_sq))
O <- rpois(T, lambda = exp(x))
plot(as.ts(O))

library(dclone)
model_ab <- custommodel("model {
for (k in 1:kk) {
x[1,k] <- log(O[1,k])
N[1,k] <- O[1,k]
for (t in 2:T) {
x[t,k] ~ dnorm(mu[t,k], 1 / sigma_sq)
mu[t,k] <- a + b * N[t - 1,k] + x[t - 1,k]
N[t,k] <- min(exp(x[t,k]), 10000)
O[t,k] ~ dpois(N[t,k])
}
}
a ~ dnorm(1, 0.01)
b ~ dnorm(0, 1)
sigma_sq <- exp(log_sigma)^2
log_sigma ~ dnorm(0, 1)
}")
dat <- list(O = data.matrix(O), kk = 1, T = T)
fit <- jags.fit(dat, c("a", "b", "sigma_sq"), model_ab)
summary(fit)


r-K (logistic) parametrization (r=a, K=-a/b)

model_rK <- custommodel("model {
for (k in 1:kk) {
x[1,k] <- log(O[1,k])
N[1,k] <- O[1,k]
for (t in 2:T) {
x[t,k] ~ dnorm(mu[t,k], 1 / sigma_sq)
mu[t,k] <- r * (1 - N[t - 1,k] / K) + x[t - 1,k]
N[t,k] <- min(exp(x[t,k]), 10000)
O[t,k] ~ dpois(N[t,k])
}
}
r ~ dnorm(0, 0.1)
K <- exp(log_K)
log_K ~ dnorm(0, 0.1)
sigma_sq <- exp(log_sigma)^2
log_sigma ~ dnorm(0, 1)
}")
dat <- list(O = data.matrix(O), kk = 1, T = T)
fit <- jags.fit(dat, c("r", "K", "sigma_sq"), model_rK)
summary(fit)


## Compare predictive distributions

a-b parametrization

model_ab_ppi <- custommodel("model {
x[1] <- log(O[1])
N[1] <- O[1]
for (t in 2:T) {
x[t] ~ dnorm(mu[t], 1 / sigma_sq)
mu[t] <- a + b * N[t - 1] + x[t - 1]
N[t] <- min(exp(x[t]), 10000)
O[t] ~ dpois(N[t])
}
for (i in (T + 1):(T + T2)) {
x[i] ~ dnorm(mu[i], 1 / sigma_sq)
mu[i] <- a + b * min(exp(x[i - 1]), 10000) + x[i - 1]
}
a ~ dnorm(1, 0.01)
b ~ dnorm(0, 1)
sigma_sq <- exp(log_sigma)^2
log_sigma ~ dnorm(0, 1)
}")
T2 <- 10
dat_ppi <- list(O = O, T = T, T2 = T2)
ppi_ab_b <- jags.fit(dat_ppi, "x", model_ab_ppi)

model_rK_ppi <- custommodel("model {
x[1] <- log(O[1])
N[1] <- O[1]
for (t in 2:T) {
x[t] ~ dnorm(mu[t], 1 / sigma_sq)
mu[t] <- r * (1 - N[t - 1] / K) + x[t - 1]
N[t] <- min(exp(x[t]), 10000)
O[t] ~ dpois(N[t])
}
for (i in (T + 1):(T + T2)) {
x[i] ~ dnorm(mu[i], 1 / sigma_sq)
mu[i] <- r * (1 - min(exp(x[i - 1]), 10000) / K) + x[i - 1]
}
r ~ dnorm(0, 0.1)
K <- exp(log_K)
log_K ~ dnorm(0, 0.1)
sigma_sq <- exp(log_sigma)^2
log_sigma ~ dnorm(0, 1)
}")
dat_ppi <- list(O = O, T = T, T2 = T2)
ppi_rK_b <- jags.fit(dat_ppi, "x", model_rK_ppi)

qppi_ab_b <- quantile(ppi_ab_b, probs = c(0.025, 0.5, 0.975))
qppi_rK_b <- quantile(ppi_rK_b, probs = c(0.025, 0.5, 0.975))

plot(1:(T + T2) - 0.2, qppi_ab_b[2,], type = "p",
col = 2, pch = 19, cex = 0.5,
ylim = range(qppi_ab_b, qppi_rK_b))
segments(x0 = 1:(T + T2) - 0.2, x1 = 1:(T + T2) - 0.2,
y0 = qppi_ab_b[1,], y1 = qppi_ab_b[3,], col = 2)
points(1:(T + T2) + 0.2, qppi_rK_b[2,], type = "p",
col = 4, pch = 19, cex = 0.5)
segments(x0 = 1:(T + T2) + 0.2, x1 = 1:(T + T2) + 0.2,
y0 = qppi_rK_b[1,], y1 = qppi_rK_b[3,], col = 4)