Theory

Theory

  1. Initialize Population: 
population <- c(1, 3, 0)
  1. Evaluate Fitness:

In R, we write:

a = 1
b = -4
c = 4
f = function(x) {
  a*x^2 + b*x + c
}

Plotting the quadratic function f(x) First, we have to choose a domain over which we want to plot f(x).

Let’s try 0 ≤ x ≤ 3:

# Define the domain over which we want to plot f(x)
x <- seq(from=0, to=4, length.out=100)

# Define the domain over which we want to plot f(x) and create df data frame
df <- x |>
  data.frame(x =_) |>
  dplyr::mutate(y = f(x))

# Define the space over which we want to population to be
possible_xvalues <- seq(from=0, to=3, length.out=4)

# Create space data frame
space <- possible_xvalues |>
  data.frame(x = _) |>
  dplyr::mutate(y = f(x))

# Calculate fitness inline
fitness <- population^2 - 4 * population + 4

# Selected the surviving parents
num_parents <- 2
selected_parents <- population |>
  order(fitness, decreasing = FALSE) |>
  head(num_parents)

# Plot f(x) using ggplot
ggplot2::ggplot(df, aes(x = x, y = y)) +
  geom_line(color = "black") +  # Plot the function as a line
  geom_point(data = subset(space, x %in% c(1,3)), color = "coral1", size = 3, shape = 8) +  # Plot points at x=1 and x=3
  geom_point(data = subset(space, (x ==0 )), color = "blue", size = 3, shape = 8) +  # Plot a point at x=0
  geom_hline(yintercept = 0, linetype = "dashed", color = "blue") +  # Add horizontal line at y=0
  geom_vline(xintercept = 0, linetype = "dashed", color = "blue") +  # Add vertical line at x=0
  theme_minimal()

  1. Selection: 
# Plot f(x) using ggplot
ggplot(df, aes(x = x, y = y)) +
  geom_line(color = "black") +  # Plot the function as a line
  geom_point(data = subset(space, x %in% c(1, 3)), color = "coral1", size = 6, shape = 8) +  # Plot points at x=1 and x=3
  geom_hline(yintercept = 0, linetype = "dashed", color = "blue") +  # Add horizontal line at y=0
  geom_vline(xintercept = 0, linetype = "dashed", color = "blue") +  # Add vertical line at x=0
  theme_minimal()  # Use a minimal theme

  1. Crossover and Mutation:
  2. Replacement:
  3. Repeat Steps 2-5 for multiple generations until a termination condition is met.

The optimal/fitting individuals F of a quadratic equation, in this case the lowest point on the graph of f(x), is:

\[ F\left(\frac{-b}{2a}, f\left(\frac{-b}{2a}\right)\right) \]

find.fitting = function(a, b, c) {
  x_fitting = -b/(2 * a)
  y_fitting = f(x_fitting)
  c(x_fitting, y_fitting)
}
F = find.fitting(a, b, c)

Adding the Fitting to the plot:

# Plot f(x) using ggplot
ggplot(df, aes(x = x, y = y)) +
  geom_line(color = "black") +  # Plot the function as a line
  geom_hline(yintercept = 0, linetype = "dashed") +  # Add horizontal line at y=0
  geom_vline(xintercept = 0, linetype = "dashed") +  # Add vertical line at x=0
  geom_point(x = F[1], y = F[2], shape = 18, size = 6, color = "red") +  # Plot the vertex
  geom_text(x = F[1], y = F[2], label = "Fitting", vjust = -1, color = "red", size = 5) +  # Add label next to the vertex
  theme_minimal()  # Use a minimal theme

Existing alternative solution

Finding the x-intercepts of f(x)

The x-intercepts are the solutions of the quadratic equation f(x) = 0; they can be found by using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

The quantity \(b2–4ac\) is called the discriminant:

# find the x-intercepts of f(x)
find.roots = function(a, b, c) {
  discriminant = b^2 - 4 * a * c
  if (discriminant > 0) {
    c((-b - sqrt(discriminant))/(2 * a), (-b + sqrt(discriminant))/(2 * a))
  }
  else if (discriminant == 0) {
    -b / (2 * a)
  }
  else {
    NaN
  }
}
solutions = find.roots(a, b, c)

Adding the x-intercepts to the plot:

  # Plot f(x) using ggplot
ggplot(df, aes(x = x, y = y)) +
  geom_line(color = "black") +  # Plot the function as a line
  geom_hline(yintercept = 0, linetype = "dashed") +  # Add horizontal line at y=0
  geom_vline(xintercept = 0, linetype = "dashed") +  # Add vertical line at x=0
  geom_point(data = data.frame(x = solutions, y = rep(0, length(solutions))), shape = 18, size = 6, color = "red") +  # Plot x-intercepts
  geom_text(data = data.frame(x = solutions, y = rep(0, length(solutions)), label = "Fitting(x-intercept)"), aes(label = label), vjust = -1, color = "red", size = 5) +  # Add labels next to x-intercepts
  theme_minimal()  # Use a minimal theme