nonparametric distribution

Non-parametric vs parametric

In denim, 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 user must provide the histogram of distribution where bin-width matches timeStep.

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()

sir_parametric <- denim_dsl({
  S -> I = beta * (I/N) * S * timeStep
  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()

sir_nonparametric <- denim_dsl({
  S -> I = beta * (I/N) * S * timeStep
  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

# 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

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.

Helper function

# 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
}

We can then run the model as followed

# 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.

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

# 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))