Constrained Polynomial Regression - Fixed Maximum

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In this article, we will explore the concept of constrained polynomial regression and how it can be applied to real-world problems. We’ll delve into the details of fixed maximum constraint and provide a step-by-step guide on how to implement this in R.

What is Constrained Polynomial Regression?

Constrained polynomial regression is a type of regression analysis that involves fitting a polynomial curve to a dataset while satisfying certain constraints. In our case, we want to fit a quadratic function (performance vs output power) and ensure that the derivative of the function has a maximum value within a specific range.

The Problem with Unconstrained Polynomial Regression

When performing unconstrained polynomial regression, the goal is to minimize the sum of the squared errors between the observed responses and the predicted values. However, in our case, we need to ensure that the derivative of the function obtained through polynomial regression has a maximum value within a specific range.

The Derivative Constraint

The derivative constraint can be mathematically represented as:

∂f(x) / ∂x = 0

where f(x) is the fitted quadratic function. For a quadratic function, the derivative is given by:

∂f(x) / ∂x = 2ax + b

To satisfy the constraint, we need to find values of a and b such that the maximum value of 2ax + b occurs within the specified range.

Fixed Maximum Constraint

The fixed maximum constraint can be expressed mathematically as:

(max_value - min_value) / (2 * std_dev) <= α

where max_value is the upper bound, min_value is the lower bound, std_dev is the standard deviation of the derivative values, and α is a constant that determines the range of acceptable values.

R Implementation

To implement constrained polynomial regression in R, we can use the following steps:

Step 1: Define the Fitted Quadratic Function

f(x) = ax^2 + bx + c

We’ll use the polyfit function from the stats package to fit the quadratic curve.

a <- polyfit(x, y, 3)[1]
b <- polyfit(x, y, 3)[2]
c <- polyfit(x, y, 3)[3]

f_x <- function(x) {
  a * x^2 + b * x + c
}

Step 2: Compute the Derivative of the Fitted Function

df_dx <- function(x) {
  2 * a * x + b
}

We’ll use this function to compute the derivative values.

Step 3: Apply the Fixed Maximum Constraint

To apply the fixed maximum constraint, we need to calculate the standard deviation of the derivative values.

std_dev_dx <- sd(df_dx(x))

Next, we’ll define the constant α that determines the range of acceptable values.

alpha <- 1.96  # for 95% confidence interval

Finally, we can check if the maximum value of the derivative falls within the specified range.

max_value_dx <- max(df_dx(x))
min_value_dx <- min(df_dx(x))

if (abs(max_value_dx - min_value_dx) / (2 * std_dev_dx) <= alpha) {
  # The constraint is satisfied
} else {
  # The constraint is not satisfied
}

Step 4: Check the Derivative Values

To ensure that the derivative values are within the specified range, we can plot a histogram of the derivative values.

hist(df_dx(x), main = "Histogram of Derivative Values",
     xlab = "Derivative Values", ylab = "Frequency")

This will give us an idea of whether our fitted quadratic function satisfies the fixed maximum constraint.

Example Use Case

Suppose we have a dataset x and corresponding response variable y. We want to fit a quadratic curve to this data while ensuring that the derivative of the fitted function has a maximum value within a specific range (e.g., -10 < df/dx(x) < 10). We can use the steps outlined above to implement constrained polynomial regression in R.

# Load necessary libraries
library(stats)

# Define the dataset
x <- c(1, 2, 3, 4, 5)
y <- c(-14e-05*1^2 + 0.009*1 + 0.31545,
       -14e-05*2^2 + 0.009*2 + 0.31545,
       -14e-05*3^2 + 0.009*3 + 0.31545,
       -14e-05*4^2 + 0.009*4 + 0.31545,
       -14e-05*5^2 + 0.009*5 + 0.31545)

# Fit the quadratic curve
a <- polyfit(x, y, 3)[1]
b <- polyfit(x, y, 3)[2]
c <- polyfit(x, y, 3)[3]

f_x <- function(x) {
  a * x^2 + b * x + c
}

df_dx <- function(x) {
  2 * a * x + b
}

std_dev_dx <- sd(df_dx(x))
alpha <- 1.96

max_value_dx <- max(df_dx(x))
min_value_dx <- min(df_dx(x))

if (abs(max_value_dx - min_value_dx) / (2 * std_dev_dx) <= alpha) {
  cat("The constraint is satisfied.\n")
} else {
  cat("The constraint is not satisfied.\n")
}

hist(df_dx(x), main = "Histogram of Derivative Values",
     xlab = "Derivative Values", ylab = "Frequency")

This example demonstrates how to implement constrained polynomial regression in R using the steps outlined above.

Last modified on 2023-11-26