Introduction to Parallelizing Simulations in R

Parallel computing is a technique used to speed up computation by using multiple processors or cores. In this article, we will explore how to parallelize simulations in R using various methods.

Background on the Wiener Process and Simulation

The Wiener process is a mathematical concept that models Brownian motion. It is defined as a continuous-time stochastic process whose paths are Gaussian processes with correlated increments. The Wiener process has many applications in finance, physics, engineering, and biology.

To simulate the Wiener process, we can use the rnorm function to generate random numbers and then calculate the cumulative sum of these numbers. This gives us a sequence of random values that represent the Wiener process.

Understanding the Take_expected_value Function

The Take_expected_value function is used to estimate the expected value of the maximum return over a given interval using Monte Carlo simulations. It takes three arguments:

• interval_end: The end point of the interval.
• points: The number of points in the interval.
• number_of_trajectories: The number of trajectories to simulate.

The function uses the Max_from_Wiener_on_interval function to calculate the maximum return for each trajectory and then takes the mean of these values.

Understanding the Max_from_Wiener_on_interval Function

The Max_from_Wiener_on_interval function calculates the maximum return on investment (ROI) over a given interval using the Wiener process. It takes two arguments:

• interval_end: The end point of the interval.
• points: The number of points in the interval.

The function first calculates the time increment (Delta) and then generates a sequence of random numbers representing the Wiener process. It then calculates the cumulative sum of these numbers and returns the maximum value minus the time moment.

Understanding the R Code

The provided R code uses the sapply function to apply the Take_expected_value function to each value in the vector 1:1000. However, this approach is not parallelizable as it uses a sequential approach.

To make the code parallelizable, we can use the snowfall package, which provides a way to distribute R code across multiple processors. In this example, we use the sfLapply function to apply the Take_expected_value function in parallel using multiple cores.

Parallelizing Simulations Using Snowfall

To parallelize the simulations using snowfall, we can follow these steps:

1. Install and Load the Snowfall Package: First, we need to install the snowfall package and load it into R.
2. Initialize Snowfall: We then initialize snowfall with the number of available processors.
3. Export All Functions: We export all functions using sfExportAll.
4. Run Parallelized Code: We use sfLapply to apply the Take_expected_value function in parallel using multiple cores.
5. Remove and Stop Snowfall: Finally, we remove snowfall and stop it.

Example Code

# Install and Load the Snowfall Package
install.packages("snowfall")
require(snowfall)

# Initialize Snowfall
cpucores = as.integer(Sys.getenv('NUMBER_OF_PROCESSORS'))
sfInit(parallel = T, cpus = cpucores)

# Export All Functions
sfExportAll()

# Define a Junk Function with 2 Arguments
Iter_vals = as.list(c(1:16)) # The values to iterate the function with

fx_parallel_run = function(Iter_val, multiplier){ 
    jnk = round(runif(1) * multiplier)
    jnk1 = runif(jnk)
    for (i in 1:length(jnk1)){
        jnk1[i] = (jnk1[i] * runif(1)) + Iter_val[[1]]
    }
    return(jnk1) 
}

# Run Parallelized Code
results = sfLapply(Iter_vals, fun = fx_parallel_run, multiplier = 800)

# Remove and Stop Snowfall
sfRemoveAll()
sfStop()

Alternative Approach Using Parallel

R has a built-in package called parallel that provides a way to parallelize code. We can use this package to achieve the same result as shown above.

# Load the Parallel Package
library(parallel)

# Define the Number of Cores
cores = detectCores()

# Create a Cluster with Multiple Cores
cluster <- MakeCluster(cores)
registerDoParallel(cluster)

# Define the Take_expected_value Function
Take_expected_value = function(interval_end, points, number_of_trajectories){
    return(
        mean(
            exp(
                replicate(
                    n = number_of_trajectories,
                    expr = Max_from_Wiener_on_interval(interval_end, points)
                )
            )
        )
    )

}

# Define the Max_from_Wiener_on_interval Function
Max_from_Wiener_on_interval = function(interval_end, points){
    # Time increment
    Delta <- interval_end / points

    # Time moments
    time <- seq( 0, interval_end, length = points + 1)

    # Wiener process
    W <- cumsum(sqrt(Delta) * rnorm(points + 1))
    # return max of "Wiener * sqrt(2) - time moment"
    return(max(sqrt(2) * W - time))
}

# Run Parallelized Code Using parallel Package
results = sapply(1:1000, function(i){
    Take_expected_value(interval_end = 1, points = 10^7, number_of_trajectories = 10^6)
})

# Stop the Cluster
stopCluster(cluster)

Conclusion

Parallelizing simulations can significantly improve computation time. In this article, we explored how to parallelize simulations in R using various methods.

We discussed the snowfall package and provided an example code snippet that demonstrates how to use it for parallel computing. We also showed how to achieve the same result using the built-in parallel package in R.

In addition to these packages, there are other methods for parallelizing computations, such as using a distributed computing framework like Apache Spark or a library specifically designed for parallel computing, such as foreach.

Last modified on 2024-08-21