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

Setup

library("mdplearning")
library("appl")
library("pomdpplus")

library("ggplot2")
library("tidyr")
library("purrr")
library("dplyr")

library("seewave") ## For KL divergence only

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

(Scaled data created from RAM data via script in mdplearning/data-raw/scaled_data.R)

library("nimble")
data("scaled_data")
ricker_code <- nimbleCode({
  x[1] <- x0
  for(t in 1:(N-1)){
    s[t] <-  x[t] - min(x[t], a[t])
    mu[t] <- log(s[t])  + r * (1 - s[t] / K) 
    x[t+1] ~ dlnorm(mu[t], sd = sigma) 
  }
})

ricker_model <- nimbleModel(code = ricker_code, 
                            constants = list(N = length(scaled_data$a), a = scaled_data$a),
                            inits =  list(K = 1, r = 0.1, sigma = 0.1, x0 = scaled_data$y[1]), 
                            data = data.frame(x = scaled_data$y))

mloglik <- function(pars){
    if(any(pars < 0 )) return(1e12)
    ricker_model$r <- pars[1]
    ricker_model$K <- pars[2]
    ricker_model$sigma <- pars[3]
    - calculate(ricker_model)
}

ricker_fit <- optim(c(r = .1, K = 1, sigma = 0.1), mloglik, control = list(maxit = 20000))
ricker_fit$par

Problem definition

(Code to create the logged alpha vectors; cached & not evaluated, very slow)

mc.cores <- round(parallel::detectCores() / 2)

log_dir <- "tuna"
states <- seq(0, 1.2, len=150) # Vector of all possible states
actions <- states  # Vector of actions: harvest
obs <- states
K = 0.9903371
r = 0.05699246
sigma_g = 0.01720091
discount = 0.99

vars <- expand.grid(r = rev(seq(0.025, 0.2, by =0.025)), sigma_m = c(0.1, 0.2, 0.3, 0.4, 0.5))

# At 150 length, initialization time ~ 100,000 seconds, exceeds runtime

## Detect available memory (linux servers only)
memory <- round(0.95 * as.numeric(gsub(".* (\\d+) .*", "\\1", system("cat /proc/meminfo", intern=TRUE)[1])))
## Bind this to a data.frame listing each of the fixed parameters across all runs
fixed <- data.frame( K = K, C = NA, sigma_g = sigma_g, discount = discount, model = "ricker", 
                     precision = 0.0000001, memory = memory / mc.cores, timeout = 250000, timeInterval = 10000,
                     max_state = max(states), max_obs = max(obs), max_action = max(actions), 
                     min_state = min(states), min_obs = min(obs), min_action = min(actions))
pars <- data.frame(vars, fixed)
## Usual assumption at the moment for reward fn
reward_fn <- function(x,h) pmin(x,h)
## Compute alphas for the above examples
models <- lapply(1:dim(pars)[1], function(i){
  ## Select the model
  f <- switch(as.character(pars[i, "model"]), 
              allen = appl:::allen(pars[i, "r"], pars[i, "K"], pars[i, "C"]),
              ricker = appl:::ricker(pars[i, "r"], pars[i, "K"])
  )
  ## Compute matrices
  fisheries_matrices(states, actions, obs, 
                     reward_fn, f = f, 
                     sigma_g = pars[i, "sigma_g"], 
                     sigma_m  = pars[i, "sigma_m"])
})

alphas <- sarsop_plus(models, 
                      discount = pars[1, "discount"], 
                      precision = pars[1, "precision"], 
                      timeout = pars[1, "timeout"],
                      timeInterval = pars[1, "timeInterval"],
                      log_dir = log_dir, 
                      log_data = pars,
                      mc.cores = mc.cores)

Identify available solutions in the log that match the desired parameters

log_dir <- "https://raw.githubusercontent.com/cboettig/pomdp-solutions-library/master/tuna"
meta <- appl::meta_from_log(data.frame(model ="ricker", sigma_m = 0.4), log_dir) %>% arrange(r)

meta

Read in the POMDP problem specification from the log

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

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

---

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

# log dir must be local
log_dir = "tuna/"
inter <- appl:::intermediates_from_log(meta, log_dir = log_dir)

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[2] <- 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[2] <- 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)

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) %>% 
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)], rep = 1) %>%
gather(model, value, -time, -rep, -method) %>%
group_by(time, rep, method) %>% 
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,]))

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)

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

plus_forecast <- 
plus_replicate(40, 
               sim_plus(models = models, discount = discount,
                        model_prior = as.numeric(plus_hindcast$model_posterior[length(y), ]),
                        state_prior = as.numeric(plus_hindcast$state_posterior[length(y), ]),
                        x0 = x0, Tmax = Tmax, true_model = models[[1]], alphas = alphas), 
               mc.cores = parallel::detectCores())

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

set.seed(123)
mdp_forecast <- 
plus_replicate(40, 
               mdp_learning(transition = transitions, reward = models[[1]]$reward, 
                            model_prior = as.numeric(mdp_hindcast$posterior[length(y),]),
                            discount = discount, x0 = x0,  Tmax = Tmax,
                            true_transition = transitions[[1]], 
                            observation = models[[1]]$observation),
               mc.cores = parallel::detectCores())

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$df, 
          mdp = mdp_forecast$df,
          .id = "method")  %>% 
select(-value, -obs) %>% 
mutate(state = states_mt[state], action = actions_mt[action], time = future[time]) %>% 
bind_rows(historical) %>%
rename("catch (MT)" = action, "stock (MT)" = state) %>%  
gather(variable, stock, -time, -rep, -method) %>%
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) + 
  facet_wrap(~variable, ncol = 1, scales = "free_y")

boettiger-lab/pomdpplus documentation built on May 24, 2019, 3:05 a.m.