---
title: "Logistic growth model with bssm"
author: 
  - Jouni Helske^[Linköping University, Department of Science and Technology, Sweden, University of Jyväskylä, Department of Mathematics and Statistics, Finland]
date: "21 November 2017"
link-citations: true
output: 
  pdf_document:
    fig_caption: yes
    fig_crop: no
    fig_height: 6
    fig_width: 8
vignette: |
  %\VignetteIndexEntry{Logistic growth model with bssm}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
  %\VignetteDepends{ggplot2}
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

## Logistic growth model

This vignette shows how to model general non-linear state space models with `bssm`. The general non-linear Gaussian model in `bssm` has following form:

$$
y_t = Z(t, \alpha_t, \theta) + H(t, \alpha_t, \theta)\epsilon_t,\\
\alpha_{t+1} = T(t, \alpha_t, \theta) + R(t, \alpha_t, \theta)\eta_t,\\
\alpha_1 \sim N(a_1(\theta), P_1(\theta)),
$$
with $t=1,\ldots, n$, $\epsilon_t ~ N(0,\textrm{I}_p)$, and $\eta ~ N(0,\textrm{I}_k)$. Here vector $\theta$ contains the unknown model parameters. Functions $T(\cdot)$, $H(\cdot)$, $T(\cdot)$, $R(\cdot)$,$a_1(\cdot)$, $P_1(\cdot)$, as well as functions defining the Jacobians of $Z(\cdot)$ and $T(\cdot)$ and the prior distribution for $\theta$ must be defined by user as a external pointers to `C++` functions, which can sound intimidating at first, but is actually pretty simple, as this vignette illustrates. All of these functions can also depend on some known parameters, defined as `known_params` (vector) and `known_tv_params` (matrix) arguments to `nlg_ssm` functions.

As an example, consider a noisy logistic growth model
$$
\begin{aligned}
y_t &= p_t +  \sigma_y\epsilon, &\quad \epsilon_t \sim N(0, 1),\\
r_{t+1} &= r_t + \sigma_r\eta, &\quad \eta_t \sim N(0, 1),\\
p_{t+1} &= K p_0 \exp(t r_t) / (K + p_0 (\exp(t r_t) - 1)) + \sigma_p\xi, &\quad \xi_t \sim N(0, 1),
\end{aligned}
$$

Let's first simulate some data, with $\sigma_r=\sigma_p=0$:

```{r data}
set.seed(1)

#parameters
K <- 100 # carrying capacity
p0 <- 10 # population size at t = 0
r <- .2 # growth rate

#sample time
dT <- .1

#observation times
t <- seq(0.1, 25, dT)

# simulate true population size (=p) at the observation times
p <- K * p0 * exp(r * t) / (K + p0 * (exp(r * t) - 1))

# observations
y <- ts(p + rnorm(length(t), 0, 5))

```


## Model in bssm

The functions determining the model functions are given in file `model_functions.cpp`. Here are few pieces from the file. The first one defines the state transition function $T(\cdot)$:

```{Rcpp, eval = FALSE}
arma::vec T_fn(const unsigned int t, const arma::vec& alpha, 
  const arma::vec& theta, const arma::vec& known_params, 
  const arma::mat& known_tv_params) {
  
  double dT = known_params(0);
  double k = known_params(1);
  
  arma::vec alpha_new(2);
  alpha_new(0) = alpha(0);
  alpha_new(1) = k * alpha(1) * exp(alpha(0) * dT) / 
    (k + alpha(1) * (exp(alpha(0) * dT) - 1));
  
  return alpha_new;
}
```

The name of this function does not matter, but it should always return Armadillo vector (`arma::vec`), and have same signature (ie. the order and types of the function's parameters) should always be like above, even though some of the parameters were not used in the body of the function. For details of using Armadillo, see [Armadillo documentation](http://arma.sourceforge.net/docs.html). After defining the appropriate model functions, the `cpp` file should also contain a function for creating external pointers for the aforementioned functions. Why this is needed is more technical issue, but fortunately you can just copy the function from the example file without any modifications. 

After creating the file for `C++` functions, you need to compile the file using `Rcpp`:

```{r pointers}
Rcpp::sourceCpp("nlg_ssm_template.cpp")
pntrs <- create_xptrs()
```

This takes a few seconds. et's make less than optimal initial guess for $\theta$, the standard deviation of observational level noise, the standard deviations of the process noises (which were zero but let's pretend that we do not know that), and define the prior distribution for $\alpha_1$:

```{r theta}
initial_theta <- c(3, 0.5, 0.5)

# dT, K, a1 and the prior variances
psi <- c(dT, 100, 0.3, 5, 4, 10)
```

If you have used line `// [[Rcpp::export]]` before the model functions, you can now test that the functions work as intended:

```{r test}
T_fn(0, c(100, 200), initial_theta, psi, matrix(1))
```

Now the actual model object using `nlg_ssm`:

```{r model}
library("bssm")
model <- nlg_ssm(y = y, a1=pntrs$a1, P1 = pntrs$P1, 
  Z = pntrs$Z_fn, H = pntrs$H_fn, T = pntrs$T_fn, R = pntrs$R_fn, 
  Z_gn = pntrs$Z_gn, T_gn = pntrs$T_gn,
  theta = initial_theta, log_prior_pdf = pntrs$log_prior_pdf,
  known_params = psi, known_tv_params = matrix(1),
  n_states = 2, n_etas = 2)
```

Let's first run Extended Kalman filter and smoother using our initial guess for $\theta$:
```{r ekf}
out_filter <- ekf(model)
out_smoother <- ekf_smoother(model)
ts.plot(cbind(y, out_filter$att[, 2], out_smoother$alphahat[, 2]), col = 1:3)
ts.plot(cbind(out_filter$att[, 1], out_smoother$alphahat[, 1]), col = 1:2)
```

## Markov chain Monte Carlo

For parameter inference, we can perform full Bayesian inference with \texttt{bssm}. There are multiple choices for the MCMC algorithm in the package, and here we will use $\psi$-PF based pseudo-marginal MCMC with delayed acceptance (note that in case of multimodal state density, it is typically better to use bootstrap filter). Let us compare this approach with EKF-based approximate MCMC (note that the number of MCMC iterations is rather small):

```{r mcmc}
out_mcmc_pm <- run_mcmc(model, n_iter = 5000, nsim_states = 10, method = "da", 
  simulation_method = "psi")
out_mcmc_ekf <- run_mcmc(model, n_iter = 5000, method = "ekf")
summary_pm <- summary(out_mcmc_pm)
summary_ekf <- summary(out_mcmc_ekf)
ts.plot(cbind(summary_pm$states$Mean[, 1], summary_ekf$states$Mean[, 1]), col = 1:3)
ts.plot(cbind(y, summary_pm$states$Mean[, 2], summary_ekf$states$Mean[, 2]), col = 1:3)
```

Let's make some predictions based on our MCMC run. We first build a model for future observations, and then call `predict` method. Note that for EKF based prediction (`nsim = 0`) the forecasting is based on the summary statistics obtained with EKF, which produces smoother but potentially biased results.
```{r pred}
future_model <- model
future_model$y <- ts(rep(NA, 100), start = end(model$y) + c(1, 0))
out_pred_pm <- predict(out_mcmc_pm, future_model, type = "response", nsim = 50)
out_pred_ekf <- predict(out_mcmc_ekf, future_model, type = "response", nsim = 0)
library("ggplot2")
autoplot(out_pred_pm, y = model$y, plot_median = FALSE)
autoplot(out_pred_ekf, y = model$y, plot_median = FALSE)
```

## Appendix

This is the full `nlg_ssm_template.cpp` file:

```{Rcpp nlg_ssm_template, code=readLines('nlg_ssm_template.cpp'), eval = FALSE, echo = TRUE}
```