Fitting a Custom Function to Data Using R’s nls2

In this post, we’ll explore the process of fitting a custom function to data using R’s nls2 package. We’ll start by examining an example problem where a custom function fails to fit to the data due to a mathematical issue.

The Problem: Fitting Custom Function to Data

The problem involves fitting a custom function, defined as $A_par(x)$, which is derived from another function, $LEV_par(x)$. The goal is to use nls2 to fit this custom function to a dataset, fit_df, containing two variables: y and x. The nls2 package is used for non-linear regression in R.

Understanding the Custom Function

The custom function $A_par(x)$ is defined as:

LEV_par &lt;- function(x, alpha,lambda){
  (lambda - lambda^alpha*(lambda+x)^(1-alpha)) / (alpha - 1)
}

A_par &lt;- function(x,base,alpha,lambda){
  
  LEV_par(x=x,alpha=alpha,lambda=lambda) / LEV_par(x=base,alpha=alpha,lambda=lambda)
  
}

The Error: Missing Value or Infinity Produced

When trying to fit the custom function using nls2, we encounter an error message indicating that a missing value or infinity is produced. This error occurs when the function $LEV_par(x)$ returns NaN (Not a Number) due to the mathematical term $lambda^alpha$.

The Issue with LEV_par

The issue arises because $LEV_par(x)$ contains the term $lambda^alpha$, which may result in NaN if $lambda$ is negative and $alpha$ is rational. This is because any non-zero number raised to a fractional power can result in an infinite value, causing the expression to produce NaN.

A Solution: Handling Negative Lambda Values

To resolve this issue, we need to handle cases where $lambda$ is negative or introduces NaN values into our calculations. We will implement checks within the LEV_par function to deal with these potential problems and ensure that it produces a valid result even when encountering NaNs due to the term $lambda^alpha$.

Implementing Checks in LEV_par

We can rewrite the LEV_par function as follows:

LEV_par &lt;- function(x, alpha, lambda){
  
  # Check if lambda is negative
  if (lambda < 0) {
    stop("Lambda cannot be negative.")
  }
  
  result &lt;-
    (lambda - exp(lambda * alpha) *
     (lambda + x) ^ (-alpha)) / (alpha - 1)
  
  # Ensure that the result doesn't exceed infinity
  if (is.infinite(result)) {
    stop("Result exceeds infinity.")
  }
  
  return(result)
}

In this version, we added checks to prevent $lambda$ from being negative and handle cases where the expression would otherwise produce NaN due to the term $lambda^alpha$. We use R’s built-in exp() function to calculate e to the power of $lambda * alpha$, instead of directly using $lambda^alpha$. This way, even if $lambda$ is negative, we avoid dealing with complex numbers in our calculations.

Implementing the Solution

With these changes, we can now implement a solution for fitting our custom function $A_par(x)$ to the data. We’ll use nls2 and modify its parameters to account for potential issues like NaN values from the mathematical terms in the function $LEV_par$.

Last modified on 2024-01-11