---
title: "nonparametric distribution"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{nonparametric distribution}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

```{r setup}
library(denim)
```

## Non-parametric vs parametric

In denim, the users have 2 options to define dwell-time distribution:

-   As a parametric distribution: using `d_*()` functions.

-   As a non-parametric distribution: using `nonparametric()` function and the user must provide the histogram of distribution where bin-width matches time step.

### Example 1

To demonstrate the difference between 2 approaches, we can try modeling an SIR model with Weibull distributed infectious period using `nonparametric()` and `d_weibull()`.

***Model definition using `d_weibull()`***

```{r}
sir_parametric <- denim_dsl({
  S -> I = beta * (I/N) * S
  I -> R = d_weibull(scale = r_scale, shape = r_shape)
})
```

Model parameters that must be defined are: `beta`, `N`, `r_scale`, `r_shape`

***Model definition using*** ***`nonparametric()`***

```{r}
sir_nonparametric <- denim_dsl({
  S -> I = beta * (I/N) * S
  I -> R = nonparametric(dwelltime_dist)
})
```

Model parameters that must be defined are: `beta`, `N`, `dwelltime_dist` (the discrete dwell time distribution)

***Run model***

We will run both models under the following model settings

```{r}
# parameters
mod_params <- list(
  beta = 0.4,
  N = 1000,
  r_scale = 4,
  r_shape = 3
)
# initial population
init_vals <- c(S = 950, I = 50, R = 0)
# simulation duration and timestep
sim_duration <- 30
timestep <- 0.05
```

Running the model with `d_weibull()` is straight forward

```{r}
parametric_mod <- sim(sir_parametric,
    initialValues = init_vals,
    parameters = mod_params,
    simulationDuration = sim_duration,
    timeStep = timestep) 

plot(parametric_mod, ylim = c(0, 1000))
```

However, to run the model using `nonparametric()`, we first need to compute the discrete dwell time distribution (`dwelltime_dist`).

Since all parametric distributions are asymptotic to 1, we will set the maximal dwell time as the time point where the cumulative probability is sufficiently close to 1 (i.e. above the threshold `1 - error_tolerance`).

A helper function to compute discrete dwell time distribution from a distribution function in R is provided below.

<details>

<summary>Helper function</summary>

```{r}
# Compute discrete distribution of dwell-tinme
# dist_func - R distribution function for dwell time (pexp, pgamma, etc.)
# ... - parameters for dist_func
compute_dist <- function(dist_func,..., timestep=0.05, error_tolerance=0.0001){
  maxtime <- timestep
  prev_prob <- 0
  prob_dist <- numeric()
  
  while(TRUE){
     # get current cumulative prob and check whether it is sufficiently close to 1
     temp_prob <-  ifelse(
       dist_func(maxtime, ...) < (1 - error_tolerance), 
       dist_func(maxtime, ...), 
       1);

     # get f(t)
     curr_prob <- temp_prob - prev_prob
     prob_dist <- c(prob_dist, curr_prob)
     
     prev_prob <- temp_prob
     maxtime <- maxtime + timestep
     
     if(temp_prob == 1){
       break
     }
  }
  
  prob_dist
}
```

</details>

We can then run the model as followed

```{r}
# Compute the discrete distribution
dwelltime_dist <- compute_dist(pweibull, 
                               scale = mod_params$r_scale, shape = mod_params$r_shape,
                               timestep = timestep)

# Compute the discrete distribution
nonparametric_mod <- sim(sir_nonparametric,
    initialValues = init_vals,
    parameters = list(
      beta = mod_params$beta,
      N = mod_params$N,
      dwelltime_dist = dwelltime_dist
    ),
    simulationDuration = sim_duration,
    timeStep = timestep) 
plot(nonparametric_mod, ylim = c(0, 1000))
```

### Example 2

By using `nonparametric()`, we can run the model with any dwell time distribution shape

Consider the following multimodal distribution.

```{r echo=FALSE}
first_dist <- compute_dist(pweibull, 
                   scale = 1.5, shape = 4,
                   timestep = timestep)
second_dist <- compute_dist(pweibull, 
                   scale = 3, shape = 3.5,
                   timestep = timestep)
first_dist <- c(rep(0, length(second_dist) - length(first_dist)), first_dist)
multimodal_dist <- first_dist + second_dist
```

```{r}
timestep <- 0.05
plot(seq(0, by = 0.05, length.out = length(multimodal_dist)), 
     multimodal_dist, 
     type = "l", col = "#374F77", lty = 1, lwd = 3,
     xlab = "Length of stay (days)", ylab = "", yaxt = 'n')
```

We can also run the `sir_nonparametric` model from last example with this dwell time distribution

```{r}
# model parameter
parameters <- list(beta = 0.4, N = 1000,
                   dwelltime_dist = multimodal_dist)
# initial population
init_vals <- c(S = 950, I = 50, R = 0)
# simulation duration and timestep
sim_duration <- 30
timestep <- 0.05

# Run the model with multimodel distribution
nonparametric_mod <- sim(
  sir_nonparametric,
  initialValues = init_vals,
  parameters = parameters,
  simulationDuration = sim_duration,
  timeStep = timestep) 
plot(nonparametric_mod, ylim = c(0, 1000))
```
