In boettiger-lab/pomdpplus: Planning and Learning in Uncertain Systems

Setup

library("mdplearning")
library("appl")
library("pomdpplus")
library("tidyverse")
library("seewave") ## For KL divergence only

knitr::opts_chunk$set(cache = FALSE, message=FALSE)
theme_set(theme_bw())

Identify available solutions in the log that match the desired parameters

log_dir <- "https://raw.githubusercontent.com/cboettig/pomdp-solutions-library/master/tuna-100"
meta <- appl::meta_from_log(data.frame(model ="ricker", sigma_m = 0.3), log_dir) %>% arrange(r)

# drop repeated values
meta <- meta[-c(4,9),]
meta

Read in the POMDP problem specification from the log

true_i <- 2

setup <- meta[1,]
states <- seq(0, setup$max_state, length=setup$n_states) # Vector of all possible states
actions <- states
obs <- states
sigma_g <- setup$sigma_g
sigma_m <- setup$sigma_m
reward_fn <- function(x,h) pmin(x,h)
discount <- setup$discount 
models <- models_from_log(meta, reward_fn)  ## Not valid for non-uniform

names(models) <- meta$r

alphas <- alphas_from_log(meta, log_dir)

reformat model solutions for use by the MDP functions as well:

transitions <- lapply(models, `[[`, "transition")
reward <- models[[1]]$reward
observation <- models[[1]]$observation
names(transitions) <- names(models)

Verfication & Validation

Compute the deterministic optimum solution:

f <- f_from_log(meta)[[1]]
S_star <- optimize(function(x) x / discount - f(x,0), c(min(states),max(states)))$minimum
h <- pmax(states - S_star,  0)
policy <- sapply(h, function(h) which.min((abs(h - actions))))
det <- data.frame(policy, value = 1:length(states), state = 1:length(states))

Convergence testing

(only evaluate if log has intermediate policies)

inter <- appl:::intermediates_from_log(meta, log_dir = log_dir)

if(length(inter[[1]]) > 0){
df1 <- 
purrr::map_df(1:length(models), function(j){
  alphas <- inter[[j]]
  m <- models[[j]]
  purrr::map_df(1:length(alphas), function(i)
    compute_policy(alphas[[i]], m$transition, m$observation, m$reward),
    .id = "intermediate") 
}, .id = "model_id")

df1 %>% 
  ggplot(aes(states[state], states[state] - actions[policy], col=intermediate)) + 
  geom_line() + 
  facet_wrap(~model_id, scales = "free") + 
  coord_cartesian(ylim = c(0,0.5))
}

Examine MDP

unif <- mdp_compute_policy(transitions, reward, discount)
prior <- numeric(length(models))
prior[1] <- 1
low <- mdp_compute_policy(transitions, reward, discount, prior)
prior <- numeric(length(models))
prior[true_i] <- 1
true <- mdp_compute_policy(transitions, reward, discount, prior)

bind_rows(unif = unif, low = low, true = true, det = det, .id = "model") %>%
  ggplot(aes(states[state], states[state] - actions[policy], col = model)) + geom_line()

Examine the policies from POMDP/PLUS solutions

Compare a uniform prior to individial cases:

prior <- numeric(length(models))
prior[1] <- 1
low <-  compute_plus_policy(alphas, models, prior)
prior <- numeric(length(models))
prior[true_i] <- 1
true <-  compute_plus_policy(alphas, models, prior)
unif <- compute_plus_policy(alphas, models) # e.g. 'planning only'
prior <- numeric(length(models))
prior[length(prior)] <- 1
high <-  compute_plus_policy(alphas, models, prior)
df <- dplyr::bind_rows(low = low, true=true, unif = unif, high = high, det = det, .id = "prior")

ggplot(df, aes(states[state], states[state] - actions[policy], col = prior, pch = prior)) + 
#  geom_point(alpha = 0.5, size = 3) + 
  geom_line()

Analysis

Hindcast

Historical catch and stock

set.seed(123)
data("scaled_data")
y <- sapply(scaled_data$y, function(y) which.min(abs(states - y)))
a <- sapply(scaled_data$a, function(a) which.min(abs(actions - a)))
Tmax <- length(y)

data("bluefin_tuna")
to_mt <- max(bluefin_tuna$total) # 1178363 # scaling factor for data
states_mt <- to_mt * states
actions_mt <- to_mt * actions
year <- 1952:2009
future <- 2009:2067

plus_hindcast <- compare_plus(models = models, discount = discount,
                    obs = y, action = a, alphas = alphas)

mdp_hindcast <- mdp_historical(transitions, reward, discount, state = y, action = a)
names(mdp_hindcast$posterior) = names(plus_hindcast$model_posterior)

left_join(rename(plus_hindcast$df, plus = optimal, state = obs),  
          rename(mdp_hindcast$df, mdp = optimal)) %>%
  mutate("actual catch" = actions_mt[action], "estimated stock" = states_mt[state], 
         plus = actions_mt[plus], mdp = actions_mt[mdp], time = year[time]) %>%
       select(-state, -action) %>%
gather(variable, stock, -time) -> 
  hindcast

hindcast %>% 
ggplot(aes(time, stock, color = variable)) + geom_line(lwd=1) #  + geom_point()

Compare rates of learning

# delta function for true model distribution
h_star = array(0,dim = length(models)) 
h_star[2] = 1
## Fn for the base-2 KL divergence from true model, in a friendly format
kl2 <- function(value) seewave::kl.dist(value, h_star, base = 2)[[2]]

bind_rows(plus = plus_hindcast$model_posterior,
          mdp = mdp_hindcast$posterior, 
          .id = "method") %>%
mutate(time = year[rep(1:Tmax,2)]) %>%
gather(model, value, -time, -method) %>%
group_by(time, method) -> df

df  %>% 
summarise(kl = kl2(value)) %>%

ggplot(aes(time, kl, col = method)) + 
stat_summary(geom="line", fun.y = mean, lwd = 1)

Final beliefs

Show the final belief over models for pomdp and mdp:

barplot(as.numeric(plus_hindcast$model_posterior[Tmax,]))

barplot(as.numeric(mdp_hindcast$posterior[Tmax,]))

posteriors <-
bind_rows(plus = plus_hindcast$model_posterior,
          mdp = mdp_hindcast$posterior, 
          .id = "method")  %>%
## include a time column
mutate(time = rep(1:Tmax, 2)) %>%
## but into long orientation
gather(model, value, -time, -method)

posteriors %>%
  filter(time == Tmax) %>%
  mutate(model = as.numeric(model)) %>%
  ggplot(aes(model, value, fill=method)) + 
  geom_bar(stat="identity", position="dodge") + 
  geom_vline(xintercept = meta$r[true_i], lwd = 1) + 
  xlab("r value") + ylab("probability density") + ggtitle("Final belief state")

Forecast simulations under PLUS and MDP-learning

All forecasts start from final stock, go forward an equal length of time:

x0 <- y[length(y)] # Final stock, 
Tmax <- length(y)
set.seed(123)
reps <- 40
mc.cores <- parallel::detectCores()

Note also that forecasts start with the prior belief over states and prior belief over models that was determined from the historical data.

## Need to average over model
state_prior = rowSums(plus_hindcast$state_posterior[length(y),, ])

plus_forecast <- 
parallel::mclapply(1:reps, function(i) 
               sim_plus(models = models, discount = discount,
                        model_prior = as.numeric(plus_hindcast$model_posterior[length(y), ]),
                        #state_prior = as.numeric(),
                        x0 = x0, Tmax = Tmax, true_model = models[[true_i]], alphas = alphas), 
               mc.cores = mc.cores)

We simulate replicates under MDP learning (with observation uncertainty):

set.seed(123)
mdp_forecast <- 
parallel::mclapply(1:reps, function(i) 
               mdp_learning(transition = transitions, reward = models[[true_i]]$reward, 
                            model_prior = as.numeric(mdp_hindcast$posterior[length(y),]),
                            discount = discount, x0 = x0,  Tmax = Tmax,
                            true_transition = transitions[[true_i]], 
                            observation = models[[true_i]]$observation),
               mc.cores = mc.cores)

Compare forecasts

historical <- bluefin_tuna[c("tsyear", "total", "catch_landings")] %>% 
  rename(time = tsyear, state = total, action = catch_landings) %>% 
  mutate(method = "historical")

bind_rows(plus = plus_forecast %>% map_df(`[[`, "df", .id = "rep"), 
          mdp = mdp_forecast %>% map_df(`[[`, "df", .id = "rep"),
          .id = "method")  %>% 
select(-value, -obs) %>% 
mutate(state = states_mt[state], action = actions_mt[action], time = future[time]) %>% 
bind_rows(historical) %>%
rename("catch" = action, "stock" = state) %>%  
gather(variable, stock, -time, -rep, -method) ->
  forecast

forecast %>%  
ggplot(aes(time, stock)) + 
#  geom_line(aes(group = interaction(rep,method), color = method), alpha=0.2) +
  stat_summary(aes(color = method), geom="line", fun.y = mean, lwd=1) +
  stat_summary(aes(fill = method), geom="ribbon", fun.data = mean_sdl, fun.args = list(mult=1), alpha = 0.25) +   
  #geom_line(data = hindcast, mapping = aes(time, stock, color = variable)) +
  facet_wrap(~variable, ncol = 1, scales = "free_y")

hindcast %>% rename(method = variable) %>% filter(method %in% c("mdp", "plus")) %>% mutate(variable = "catch", rep = "NA") -> hindcast2

forecast %>% full_join(hindcast2) -> df


df %>%  
ggplot(aes(time, stock)) + 
  stat_summary(aes(color = method), geom="line", fun.y = mean, lwd=1) +
  stat_summary(aes(fill = method), geom="ribbon", fun.data = mean_sdl, fun.args = list(mult=1), alpha = 0.25) +   
  facet_wrap(~variable, ncol = 1, scales = "free_y")