Optimization Example of Lexical Scoping in R: Exploring optim, optimize, and nlm Functions

The Beginner’s Guide to Optimization Example of Lexical Scoping in R:

When it comes to optimization in R, lexical scoping can be a useful tool for optimizing complex functions that involve multiple variables. In this blog post, we will explore how lexical scoping can be used to optimize a function using the NLL (negative log-likelihood) function, and how the optim, optimize, and nlm functions can be used to perform optimization in R.

Optimizing the NLL Function using Lexical Scoping

The NLL function is a common function used in optimization problems. It is defined as the negative log of the likelihood function, which is used to estimate the parameters of a statistical model. In R, the NLL function can be defined using lexical scoping, which allows us to pass arguments to the function and access variables from within the function.

Here is an example of how to define the NLL function using lexical scoping in R:

nll <- function(data, parameters) {

  # Define local variables

  x <- data$x

  y <- data$y

  a <- parameters[1]

  b <- parameters[2]

  c <- parameters[3]

  

  # Define the likelihood function

  likelihood <- sum((y - (a + b * x + c * x^2))^2)

  

  # Return the negative log-likelihood

  return(-likelihood)

}

In this example, the NLL function takes two arguments: data, which is a data frame containing the variables x and y, and parameters, which is a vector containing the parameters a, b, and c. The function then calculates the likelihood function and returns the negative log-likelihood.

Using the Optim, Optimize, and Nlm Functions for Optimization

Once the NLL function is defined, we can use the optim, optimize, or nlm functions to optimize the function and estimate the values of a, b, and c. These functions take different approaches to optimization, so it's important to choose the right one for your problem.

The optim function is a general-purpose optimization function that can be used to optimize any function. It works by minimizing the function using a variety of algorithms, such as Nelder-Mead or BFGS. Here is an example of how to use the optim function to optimize the NLL function:

# Define the data and starting values for parameters

data <- data.frame(x = c(1, 2, 3, 4, 5), y = c(1.1, 2.1, 3.3, 3.9, 5.2))

start <- c(a = 1, b = 1, c = 1)

# Use optim to optimize the function

result <- optim(par = start, fn = nll, data = data)

In this example, we first define the data and starting values for the parameters. We then use the optim function to optimize the function, passing in the starting values and the data as arguments.

The optimize() function in R is used to optimize a one-dimensional function. It takes a function and a range of values to search within, and returns the minimum or maximum value of the function and the argument that produces that value.

Here's an example of how to use the optimize() function in R:

# Define a function to be optimized

myfunc <- function(x) {

  return(x^2 - 4*x + 3)

}

# Use optimize() to find the minimum value of the function

result <- optimize(myfunc, c(0, 5))

In this example, we define a function myfunc that takes a single argument x and returns the value of x^2 - 4*x + 3. We want to find the minimum value of this function within the range of values from 0 to 5.

We use the optimize() function to find the minimum value of the function by passing the function and the range of values as arguments. The optimize() function returns an object that contains the minimum value of the function ($minimum) and the argument that produces that value ($objective). We can access these values like this:

cat("Minimum value of function:", result$minimum, "\n")

cat("Argument that produces minimum value:", result$objective, "\n")

This will print the following output:

Minimum value of function: 1 

Argument that produces minimum value: 2 

So, we can see that the minimum value of the myfunc function within the range of values from 0 to 5 is 1, and the argument that produces that value is 2.

The nlm() function in R is used to minimize a multivariate function using a Newton-type algorithm. It takes a function to be minimized, an initial guess for the minimum, and various other arguments that control the algorithm, and returns the minimum value of the function and the argument that produces that value.

Here's an example of how to use the nlm() function in R:

# Define a function to be minimized

myfunc <- function(x) {

  return(x[1]^2 + x[2]^2)

}

# Use nlm() to find the minimum value of the function

result <- nlm(myfunc, c(1, 2))

# Print the results

cat("Minimum value of function:", result$minimum, "\n")

cat("Argument that produces minimum value:", result$estimate, "\n")

In this example, we define a function myfunc that takes a two-element vector x and returns the value of x[1]^2 + x[2]^2. We want to find the minimum value of this function.

We use the nlm() function to find the minimum value of the function by passing the function and an initial guess for the minimum as arguments. The nlm() function returns an object that contains the minimum value of the function ($minimum) and the argument that produces that value ($estimate). We can access these values like this:

cat("Minimum value of function:", result$minimum, "\n")

cat("Argument that produces minimum value:", result$estimate, "\n")

This will print the following output:

Minimum value of function: 0 

Argument that produces minimum value: 0 0 

So, we can see that the minimum value of the myfunc function is 0, and the argument that produces that value is a two-element vector with both elements set to 0.

Practice Material:

Here are some practice problems to help you get started with optimization in R using optimize(), optim(), nlm(), and the NLL function:

• Use the optimize() function to find the minimum of the function f(x) = x^3 - 6x^2 + 11x - 6 in the interval [1, 3].

• Use the optim() function to minimize the Rosenbrock function:
  rosenbrock <- function(x) {
     sum(100 * (x[-length(x)]^2 - x[-1])^2 + (1 - x[-length(x)])^2)
  }
  Set the initial values to c(-1, -1) and set the control argument to list(fnscale = -1).

• Use the nlm() function to minimize the function f(x) = x^2 - 5x + 6 with an initial guess of x = 0.

• Write a negative log-likelihood (NLL) function for a linear regression model with intercept b0 and slope b1:
  nll <- function(b, x, y) {
    y_hat <- b[1] + b[2] * x
    -sum(dnorm(y, mean = y_hat, sd = 1, log = TRUE))
  }
  Use the optim() function to find the maximum likelihood estimates of b0 and b1 for the following data:
  x <- 1:10
  y <- c(2.1, 4.0, 5.8, 9.2, 12.4, 15.5, 18.7, 22.1, 25.0, 28.5)
  Set the initial values to c(0, 0) and set the control argument to list(fnscale = -1).

• Use the nlm() function to minimize the function f(x, y) = x^2 - 4xy + 3y^2 with an initial guess of x = 1 and y = 2.

For more practice you should start swirl's lessons in R Programming. Complete download process of swirl and R Programming is here, click on the link!

You can look in to the practice and reading material that is provided in the text book, click here to download the textbook.

Lecture slides can be downloaded from here. You'll find the code that we used in the lecture for more practice. It would be great if you go through them too.

These practice problems should help you get started with optimization in R using optimize(), optim(), nlm(), and the NLL function. And I hope that you'll find this material useful going in to your career as a Data Scientist. Good luck!