allen.Rmd

# Comparison of Nonparametric Bayesian Gaussian Process estimates to standard the Parametric Bayesian approach

```{r runsetup}
setwd("~/Documents/code/nonparametric-bayes/inst/examples/BUGS/")
```



Plotting and knitr options, (can generally be ignored)

```{r plotting-options}
opts_chunk$set(tidy=FALSE, warning=FALSE, message=FALSE, cache=FALSE, comment=NA,
               fig.width=6, fig.height=4)

library(knitcitations)

library(ggplot2) # plotting 
opts_knit$set(upload.fun = socialR::flickr.url)
theme_set(theme_bw(base_size=10))
theme_update(panel.background = element_rect(fill = "transparent", colour = NA),
             plot.background = element_rect(fill = "transparent", colour = NA))
cbPalette <- c("#000000", "#E69F00", "#56B4E9", "#009E73", 
               "#F0E442", "#0072B2", "#D55E00", "#CC79A7")
```
Load necessary libraries,

```{r libraries, cache=FALSE, message=FALSE, warning=FALSE}
library(nonparametricbayes) # loads the rest as dependencies
```



### Model and parameters

Uses the model derived in `citet("10.1080/10236190412331335373")`, of a Ricker-like growth curve with an allee effect, defined in the pdgControl package,


```{r}
f <- RickerAllee
p <- c(2, 8, 5)
K <- p[2]
allee <- p[3]
```


Various parameters defining noise dynamics, grid, and policy costs.  

```{r sdp-pars, dependson="stateeq"}
sigma_g <- 0.05
sigma_m <- 0.0
z_g <- function() rlnorm(1, 0, sigma_g)
z_m <- function() 1+(2*runif(1, 0,  1)-1) * sigma_m
x_grid <- seq(0, 1.5 * K, length=50)
h_grid <- x_grid
profit <- function(x,h) pmin(x, h)
delta <- 0.01
OptTime <- 50  # stationarity with unstable models is tricky thing
reward <- 0
xT <- 0
Xo <- K # observations start from
x0 <- Xo # simulation under policy starts from
Tobs <- 40
```

### Sample Data

```{r obs, dependson="sdp-pars"}
  set.seed(1234)
  #harvest <- sort(rep(seq(0, .5, length=7), 5))
  x <- numeric(Tobs)
  x[1] <- 6
  nz <- 1
  for(t in 1:(Tobs-1))
    x[t+1] = z_g() * f(x[t], h=0, p=p)
  obs <- data.frame(x = c(rep(0,nz), 
                          pmax(rep(0,Tobs-1), x[1:(Tobs-1)])), 
                    y = c(rep(0,nz), 
                          x[2:Tobs]))
```


## Maximum Likelihood

```{r mle, dependson="obs"}
alt <- par_est(obs,  init = c(r = p[1], 
                              K = mean(obs$x[obs$x>0]), 
                              s = sigma_g))
est <- par_est_allee(obs, f, p,  
                     init = c(r = p[1] + 1, 
                              K = p[2] + 2, 
                              C = p[3] + 2, 
                              s = sigma_g))
```


## Non-parametric Bayes


```{r gp-priors}
#inv gamma has mean b / (a - 1) (assuming a>1) and variance b ^ 2 / ((a - 2) * (a - 1) ^ 2) (assuming a>2)
s2.p <- c(5,5)  
d.p = c(10, 1/0.1)
```


Estimate the Gaussian Process (nonparametric Bayesian fit)

```{r}
gp <- gp_mcmc(obs$x, y=obs$y, n=1e5, s2.p = s2.p, d.p = d.p)
gp_dat <- gp_predict(gp, x_grid, burnin=1e4, thin=300)
```


Show traces and posteriors against priors

```{r gp_traces_densities}
plots <- summary_gp_mcmc(gp)
```


```{r}
# Summarize the GP model
tgp_dat <- 
    data.frame(  x = x_grid, 
                 y = gp_dat$E_Ef, 
                 ymin = gp_dat$E_Ef - 2 * sqrt(gp_dat$E_Vf), 
                 ymax = gp_dat$E_Ef + 2 * sqrt(gp_dat$E_Vf) )
```

### Parametric Bayes

We initiate the MCMC chain (`init_p`) using the true values of the parameters `p` from the simulation.  While impossible in real data, this gives the parametric Bayesian approach the best chance at succeeding.  `y` is the timeseries (recall `obs` has the $x_t$, $x_{t+1}$ pairs)

```{r}
# a bit unfair to start with the correct values, but anyhow...
init_p = p
names(init_p) = c("r0", "K", "theta")
y <- obs$x[-1] 
N=length(y);
```


We'll be using the JAGS Gibbs sampler, a recent open source BUGS implementation with an R interface that works on most platforms.  We initialize the usual MCMC parameters; see `?jags` for details.  


```{r}
jags.data <- list("N","y")
n.chains = 1
n.iter = 40000
n.burnin = floor(10000)
n.thin = max(1, floor(n.chains * (n.iter - n.burnin)/1000))
```


The actual model is defined in a `model.file` that contains an R function that is automatically translated into BUGS code by *R2WinBUGS*.  The file defines the priors and the model, as seen when read in here


```{r}
cat(readLines(con="bugmodel-UPrior.txt"), sep="\n")
``` 



We define which parameters to keep track of, and set the initial values of parameters in the transformed space used by the MCMC.  We use logarithms to maintain strictly positive values of parameters where appropriate. 


```{r}
# Uniform priors on standard deviation terms
jags.params=c("K","logr0","logtheta","stdQ", "stdR")
jags.inits <- function(){
  list("K"=init_p["K"],"logr0"=log(init_p["r0"]),"logtheta"=log(init_p["theta"]), "stdQ"=sqrt(0.05),"stdR"=sqrt(0.1),"x"=y,.RNG.name="base::Wichmann-Hill", .RNG.seed=123)
}

set.seed(12345)

time_jags <- system.time(       
  jagsfit <- jags(data=jags.data, inits=jags.inits, jags.params, n.chains=n.chains, 
                  n.iter=n.iter, n.thin=n.thin, n.burnin=n.burnin,model.file="bugmodel-UPrior.txt")
)         
time_jags <- unname(time_jags["elapsed"]);
```


#### Convergence diagnostics for parametric bayes

```{r}
jags_matrix <- as.data.frame(as.mcmc.bugs(jagsfit$BUGSoutput))
par_posteriors <- melt(cbind(index = 1:dim(jags_matrix)[1], jags_matrix), id = "index")
```

```{r parametric_bayes_traces}
# Traces
ggplot(par_posteriors) + geom_line(aes(index, value)) + facet_wrap(~ variable, scale="free", ncol=1)
```


```{r parametric_bayes_posteriors}
## priors (untransformed variables)
K_prior <- function(x) dunif(x, 0.01, 40)
logr_prior <- function(x) dunif(x, -6, 6)
logtheta_prior <- function(x) dunif(x, -6, 6)
stdQ_prior <- function(x) dunif(x, 0.001, 100)
stdR_prior <- function(x) dunif(x, 0.001, 100)

par_priors <- list(K = K_prior, deviance = function(x) 0 * x, 
                   logr0 = logr_prior, logtheta = logtheta_prior, 
                   stdQ = stdQ_prior, stdR = stdR_prior)

par_prior_curves <- ddply(par_posteriors, "variable", function(dd){
    grid <- seq(min(dd$value), max(dd$value), length = 100) 
    data.frame(value = grid, density = par_priors[[dd$variable[1]]](grid))
})

# posterior distributions
ggplot(par_posteriors, aes(value)) + 
  stat_density(geom="path", position="identity", alpha=0.7) +
  geom_line(data=par_prior_curves, aes(x=value, y=density), col="red") + 
  facet_wrap(~ variable, scale="free", ncol=2)

```


```{r}
# um, cleaner if we were just be using the long form, par_posterior
mcmc <- as.mcmc(jagsfit)
mcmcall <- mcmc[,-2]
who <- colnames(mcmcall)
who 
mcmcall <- cbind(mcmcall[,1],mcmcall[,2],mcmcall[,3],mcmcall[,4],mcmcall[,5])
colnames(mcmcall) <- who


pardist <- mcmcall
pardist[,2] = exp(pardist[,2]) # transform model parameters back first
pardist[,3] = exp(pardist[,3])


bayes_coef <- apply(pardist,2,mean)
bayes_pars <- unname(c(bayes_coef[2], bayes_coef[1], bayes_coef[3]))
bayes_pars
```



```{r}
par_bayes_means <- sapply(x_grid, f, 0, bayes_pars)
```


## Parametric Bayes based on the structurally wrong model

We initiate the MCMC chain (`init_p`) using the true values of the parameters `p` from the simulation.  While impossible in real data, this gives the parametric Bayesian approach the best chance at succeeding.  `y` is the timeseries (recall `obs` has the $x_t$, $x_{t+1}$ pairs)

```{r}
init_p = p
names(init_p) = c("r0", "K")
y <- obs$x[-1] 
N=length(y);
```


We'll be using the JAGS Gibbs sampler, a recent open source BUGS implementation with an R interface that works on most platforms.  We initialize the usual MCMC parameters; see `?jags` for details.  


```{r}
jags.data <- list("N","y")
n.chains = 1
n.iter = 400000
n.burnin = floor(10000)
n.thin = max(1, floor(n.chains * (n.iter - n.burnin)/1000))
```


The actual model is defined in a `model.file` that contains an R function that is automatically translated into BUGS code by *R2WinBUGS*.  The file defines the priors and the model, as seen when read in here


```{r}
cat(readLines(con="ricker-UPrior.txt"), sep="\n")
``` 



We define which parameters to keep track of, and set the initial values of parameters in the transformed space used by the MCMC.  We use logarithms to maintain strictly positive values of parameters where appropriate. 


```{r}
# Uniform priors on standard deviation terms
jags.params=c("K","logr0", "stdQ", "stdR")
jags.inits <- function(){
  list("K"=init_p["K"],"logr0"=log(init_p["r0"]), "stdQ"=sqrt(0.05),"stdR"=sqrt(0.1),"x"=y,.RNG.name="base::Wichmann-Hill", .RNG.seed=123)
}

set.seed(12345)

time_jags <- system.time(       
  jagsfit <- jags(data=jags.data, inits=jags.inits, jags.params, n.chains=n.chains, 
                  n.iter=n.iter, n.thin=n.thin, n.burnin=n.burnin,model.file="ricker-UPrior.txt")
)         
time_jags <- unname(time_jags["elapsed"]);
```


#### Convergence diagnostics for parametric bayes Ricker model

```{r ricker_traces}
jags_matrix <- as.data.frame(as.mcmc.bugs(jagsfit$BUGSoutput))
par_posteriors <- melt(cbind(index = 1:dim(jags_matrix)[1], jags_matrix), id = "index")

# Traces
ggplot(par_posteriors) + geom_line(aes(index, value)) + 
  facet_wrap(~ variable, scale="free", ncol=1)
```

```{r ricker_posteriors}
## priors (untransformed variables)
K_prior <- function(x) dunif(x, 0.01, 40)
logr_prior <- function(x) dunif(x, -6, 6)
stdQ_prior <- function(x) dunif(x, 0.001, 100)
stdR_prior <- function(x) dunif(x, 0.001, 100)

par_priors <- list(K = K_prior, deviance = function(x) 0 * x, logr0 = logr_prior, stdQ = stdQ_prior, stdR = stdR_prior)

par_prior_curves <- ddply(par_posteriors, "variable", function(dd){
    grid <- seq(min(dd$value), max(dd$value), length = 100) 
    data.frame(value = grid, density = par_priors[[dd$variable[1]]](grid))
})

# posterior distributions
ggplot(par_posteriors, aes(value)) + 
  stat_density(geom="path", position="identity", alpha=0.7) +
  geom_line(data=par_prior_curves, aes(x=value, y=density), col="red") + 
  facet_wrap(~ variable, scale="free", ncol=2)
```



```{r}
ricker_pardist <- jags_matrix[! names(jags_matrix) == "deviance" ]
ricker_pardist[,"logr0"] = exp(ricker_pardist[,"logr0"]) # transform model parameters back first

posterior.mode <- function(x) {
  ux <- unique(x)
  ux[which.max(tabulate(match(x, ux)))]
}

apply(ricker_pardist,2,mean)
bayes_coef <- apply(ricker_pardist,2, posterior.mode) # much better estimates
ricker_bayes_pars <- unname(c(bayes_coef[2], bayes_coef[1]))
ricker_bayes_pars
```





## Write the external bugs file

```{r}
logr0_prior_p <- c(-6.0, 6.0)
logtheta_prior_p <- c(-6.0, 6.0)
logK_prior_p <- c(-6.0, 6.0)
stdQ_prior_p <- c(0.0001, 100)
stdR_prior_p <- c(0.0001, 100)

bugs.model <- 
paste(sprintf(
"model{
  logr0    ~ dunif(%s, %s)
  logtheta    ~ dunif(%s, %s)
  logK    ~ dunif(%s, %s)
  stdQ ~ dunif(%s, %s)
  stdR ~ dunif(%s, %s)", 
  logr0_prior_p[1], logr0_prior_p[2],
  logtheta_prior_p[1], logtheta_prior_p[2],
  logK_prior_p[1], logK_prior_p[2],
  stdQ_prior_p[1], stdQ_prior_p[2],
  stdR_prior_p[1], stdR_prior_p[2]), 
  "
  
  iQ <- 1 / (stdQ * stdQ);
  iR <- 1 / (stdR * stdR);

  r0 <- exp(logr0)
  theta <- exp(logtheta)
  K <- exp(logK)

  x[1] ~ dunif(0, 10)

  for(t in 1:(N-1)){
    mu[t] <- r0 * pow(abs(x[t]), theta) / (1 + pow(abs(x[t]), theta) / K)
    x[t+1] ~ dnorm(mu[t], iQ) 
  }

  for(t in 1:(N)){
    y[t] ~ dnorm(x[t], iR)
  }
}")

writeLines(bugs.model, "myers.bugs")
```



```{r}

## priors (untransformed variables)
logK_prior     <- function(x) dunif(x, logK_prior_p[1], logK_prior_p[2])
logr_prior     <- function(x) dunif(x, logr0_prior_p[1], logr0_prior_p[2])
logtheta_prior <- function(x) dunif(x, logtheta_prior_p[1], logtheta_prior_p[2])
stdQ_prior     <- function(x) dunif(x, stdQ_prior_p[1], stdQ_prior_p[2])
stdR_prior     <- function(x) dunif(x, stdR_prior_p[1], stdR_prior_p[2])

```



### Parametric Bayes based on myers model 

Contains the 


```{r}

init_p = c(1, 1, 1)
names(init_p) = c("r0", "theta", "K")
y <- obs$x[-1] 
N=length(y);
```


We'll be using the JAGS Gibbs sampler, a recent open source BUGS implementation with an R interface that works on most platforms.  We initialize the usual MCMC parameters; see `?jags` for details.  


```{r}
jags.data <- list("N","y")
n.chains = 1
n.iter = 40000
n.burnin = floor(10000)
n.thin = max(1, floor(n.chains * (n.iter - n.burnin)/1000))
```


```{r}
jags.params=c("logr0", "logtheta", "logK", "stdQ", "stdR")
jags.inits <- function(){
  list("logr0"=log(init_p["r0"]), "logtheta" = log(init_p["theta"]),  
       "logK"=log(init_p["K"]), "stdQ"=sqrt(0.05), 
       "stdR"=sqrt(0.1),"x"=y,.RNG.name="base::Wichmann-Hill", .RNG.seed=123)
}
```


```{r}
set.seed(12345)
time_jags <- system.time(       
  jagsfit <- jags(data=jags.data, inits=jags.inits, jags.params, 
                  n.chains=n.chains, n.iter=n.iter, n.thin=n.thin,
                  n.burnin=n.burnin, model.file="myers.bugs")
)         
time_jags <- unname(time_jags["elapsed"]);
```

Convergence diagnostics for parametric bayes

```{r}
jags_matrix <- as.data.frame(as.mcmc.bugs(jagsfit$BUGSoutput))
par_posteriors <- melt(cbind(index = 1:dim(jags_matrix)[1], jags_matrix), id = "index")

## Traces
ggplot(par_posteriors) + geom_line(aes(index, value)) +
  facet_wrap(~ variable, scale="free", ncol=1)

```

```{r}
## prior curve functions, must be in order
par_priors <- list( deviance = function(x) 0 * x, logK = logK_prior,
                    logr0 = logr_prior, logtheta = logtheta_prior, 
                    stdQ = stdQ_prior, stdR = stdR_prior)

## calculate points along the prior curve
par_prior_curves <- ddply(par_posteriors, "variable", function(dd){
    grid <- seq(min(dd$value), max(dd$value), length = 100) 
    data.frame(value = grid, density = par_priors[[dd$variable[1]]](grid))
})
```

```{r}
## posterior distributions
ggplot(par_posteriors, aes(value)) + 
  stat_density(geom="path", position="identity", alpha=0.7) +
  geom_line(data=par_prior_curves, aes(x=value, y=density), col="red") + 
  facet_wrap(~ variable, scale="free", ncol=2)
```

```{r}
myers_pardist <- as.matrix(jags_matrix[2:6])
myers_pardist[,1] = exp(myers_pardist[,1]) # transform model parameters back first
myers_pardist[,2] = exp(myers_pardist[,2]) # transform model parameters back first
myers_pardist[,3] = exp(myers_pardist[,3]) # transform model parameters back first
colnames(myers_pardist) = c("K", "r0", "theta", "stdQ", "stdR")


apply(myers_pardist,2,mean)
bayes_coef <- apply(myers_pardist,2, posterior.mode) # much better estimates

myers_bayes_pars <- unname(c(bayes_coef[2], bayes_coef[3], bayes_coef[1]))
```















### Phase-space diagram of the expected dynamics

```{r}
true_means <- sapply(x_grid, f, 0, p)
alt_means <- sapply(x_grid, alt$f, 0, ricker_bayes_pars[c(1,2)])
est_means <- sapply(x_grid, est$f, 0, est$p)
par_bayes_means <- sapply(x_grid, f, 0, bayes_pars)

myers_means <- sapply(x_grid, Myer_harvest, 0, myers_bayes_pars)

models <- data.frame(x=x_grid, GP=tgp_dat$y, True=true_means, 
                     MLE=est_means, Ricker=alt_means, 
                     Parametric.Bayes = par_bayes_means,
                     Myers = myers_means)

models <- melt(models, id="x")
names(models) <- c("x", "method", "value")
```


```{r Figure1}
plot_gp <- ggplot(tgp_dat) + geom_ribbon(aes(x,y,ymin=ymin,ymax=ymax), fill="gray80") +
    geom_line(data=models, aes(x, value, col=method), lwd=1, alpha=0.8) + 
    geom_point(data=obs, aes(x,y), alpha=0.8) + 
    xlab(expression(X[t])) + ylab(expression(X[t+1])) +
    scale_colour_manual(values=cbPalette) 
print(plot_gp)
```




## Optimal policies by value iteration

Compute the optimal policy under each model using stochastic dynamic programming. We begin with the policy based on the GP model,


```{r}
MaxT = 1000
# uses expected values from GP, instead of integrating over posterior
#matrices_gp <- gp_transition_matrix(gp_dat$E_Ef, gp_dat$E_Vf, x_grid, h_grid)

# Integrate over posteriors 
matrices_gp <- gp_transition_matrix(gp_dat$Ef_posterior, gp_dat$Vf_posterior, x_grid, h_grid) 

# Solve the SDP using the GP-derived transition matrix
opt_gp <- value_iteration(matrices_gp, x_grid, h_grid, MaxT, xT, profit, delta, reward)
```


Determine the optimal policy based on the true and MLE models

```{r}
matrices_true <- f_transition_matrix(f, p, x_grid, h_grid, sigma_g)
opt_true <- value_iteration(matrices_true, x_grid, h_grid, OptTime=MaxT, xT, profit, delta=delta)

matrices_estimated <- f_transition_matrix(est$f, est$p, x_grid, h_grid, est$sigma_g)
opt_estimated <- value_iteration(matrices_estimated, x_grid, h_grid, OptTime=MaxT, xT, profit, delta=delta)

```

Determine the optimal policy based on parametric Bayesian model

```{r}
allen_f <- function(x,h,p) unname(f(x,h,p[c(2, 1, 3)]))
matrices_par_bayes <- parameter_uncertainty_SDP(allen_f, x_grid, h_grid, pardist, 4)
opt_par_bayes <- value_iteration(matrices_par_bayes, x_grid, h_grid, OptTime=MaxT, xT, profit, delta=delta)
```

Bayesian Ricker

```{r}
ricker_f <- function(x, h, p) alt$f(x, h, unname(p[c(2, 1)]))
matrices_alt <- parameter_uncertainty_SDP(ricker_f, x_grid, h_grid, as.matrix(ricker_pardist), 3)
opt_alt <- value_iteration(matrices_alt, x_grid, h_grid, OptTime=MaxT, xT, profit, delta=delta)
```


Bayesian Myers model

```{r}
myers_f <- function(x,h,p) Myer_harvest(x, h, p[c(2, 3, 1)])
matrices_myers <- parameter_uncertainty_SDP(myers_f, x_grid, h_grid, as.matrix(myers_pardist), 4)
myers_alt <- value_iteration(matrices_myers, x_grid, h_grid, OptTime=MaxT, xT, profit, delta=delta)
```


Assemble the data

```{r}
OPT = data.frame(GP = opt_gp$D, True = opt_true$D, MLE = opt_estimated$D, Ricker = opt_alt$D, Parametric.Bayes = opt_par_bayes$D, Myers = myers_alt$D)
colorkey=cbPalette
names(colorkey) = names(OPT) 
```



## Graph of the optimal policies

```{r Figure2}
policies <- melt(data.frame(stock=x_grid, sapply(OPT, function(x) x_grid[x])), id="stock")
names(policies) <- c("stock", "method", "value")

ggplot(policies, aes(stock, stock - value, color=method)) +
  geom_line(lwd=1.2, alpha=0.8) + xlab("stock size") + ylab("escapement")  +
  scale_colour_manual(values=colorkey)
```



## Simulate 100 realizations managed under each of the policies


```{r sims}
sims <- lapply(OPT, function(D){
  set.seed(1)
  lapply(1:100, function(i) 
    ForwardSimulate(f, p, x_grid, h_grid, x0, D, z_g, profit=profit, OptTime=OptTime)
  )
})

dat <- melt(sims, id=names(sims[[1]][[1]]))
dt <- data.table(dat)
setnames(dt, c("L1", "L2"), c("method", "reps")) 
# Legend in original ordering please, not alphabetical: 
dt$method = factor(dt$method, ordered=TRUE, levels=names(OPT))
```

```{r Figure3}
ggplot(dt) + 
  geom_line(aes(time, fishstock, group=interaction(reps,method), color=method), alpha=.1) +
  scale_colour_manual(values=colorkey, guide = guide_legend(override.aes = list(alpha = 1)))
```



```{r profits}
Profit <- dt[, sum(profit), by=c("reps", "method")]
Profit[, mean(V1), by="method"]
```

```{r totalprofits}
ggplot(Profit, aes(V1)) + geom_histogram() + 
  facet_wrap(~method, scales = "free_y") + guides(legend.position = "none")
```