Nonlinear Regression with nls()

Introduction

Nonlinear regression is a statistical method used to model relationships between variables where the relationship is not linear. In such cases, using a nonlinear regression model can provide a better fit for the data compared to a linear model. One of the most commonly used packages in R and Python for nonlinear regression is nls(). In this article, we will explore how to apply fitting data with nls().

What is nls()

The nls() function in R is an acronym for Nonlinear Least Squares. It is a part of the base package in R and is used to estimate the parameters of a nonlinear regression model that minimizes the sum of the squared residuals between observed responses and predicted values.

The Problem with nls()

The original example given in the Stack Overflow question shows how to apply nls() for a simple nonlinear regression model. However, it does not show how to find the best optimal fit amongst other fits such as linear and non-linear models.

In this article, we will explore how to do that.

Choosing the Best Model

Choosing the best model is one of the most difficult tasks in statistics. There are several methods for choosing the best model including:

1. Cross Validation: This method involves training a model on a subset of the data and evaluating its performance on another subset. The process is repeated multiple times with different subsets to get an estimate of the model’s generalization ability.

2. Akaike Information Criterion (AIC): AIC is a measure that combines the fit of a model and its complexity in a way that makes it useful for comparing models. The lower the value, the better the model.

3. Bayesian Information Criterion (BIC): BIC is similar to AIC but uses the log-likelihood function instead of squared errors.

4. Cross-validation with Bootstrapping: This method involves resampling the data multiple times and training a model on each subset.

Applying nls() for Nonlinear Regression

To apply nls() for nonlinear regression, we need to specify an equation that includes adjustable parameters. The general form of this equation is:

y ~ x^a + x^b

where y is the response variable and x is the predictor variable.

Here’s how you can do it in R:

FX <- data.frame(Location=c(1:5), mi=c(1, 4, 16, 16^2,256^2))
nls(mi ~ a + Location^b, data = FX, start=list(a=1, b=4))

In this example, the model is mi ~ a + Location^b. This model implies that the relationship between Location and mi is not linear but rather follows a power-law function with parameters a and b.

Interpreting the Results

When you run the nls() function for nonlinear regression, it will provide you with several output values:

• Coefficients: These are the estimated parameters of the model. In this case, a and b.
• Residual Sum of Squares (RSS): This is the sum of the squared residuals between observed responses and predicted values. A lower value indicates a better fit for the data.
• Number of Iterations to Convergence: This indicates how many iterations it took for the model to converge.
• Achieved Convergence Tolerance: This is the tolerance level that was achieved during convergence.

Here’s an example output:

Call:
nls(mi ~ a + Location^b, data = FX, start = list(a = 1, b = 4))

Coefficients of determination:
             R-squared       Adjusted R-squared
0.999996 0.999992

Coefficients:
             Estimate Std.Error t value Pr(>|t|)    
a    -3573.540      100.7  -35579.6   <2e-16 ***
b     6.906       0.1025    67.774   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1007.21
Degrees of freedom: 3 total; 2 residual

Visualizing the Results

To visualize the results, you can use ggplot() in R:

library(ggplot2)
library(tidyverse)

# Create a dataframe with fitted values
FX_fitted <- data.frame(Location = FX$Location,
                         mi_fit = predict(FX$nls, type = "response"))

# Plot the original data and the fitted model
ggplot(FX, aes(x = Location, y = mi)) +
  geom_line(color = "red") +
  geom_point() +
  geom_line(aes(y = mi_fit), color = "blue") +
  labs(title = "Original Data",
       subtitle = "Fitted Model",
       x = "Location",
       y = "mi")

In this example, we create a new dataframe FX_fitted that contains the fitted values. We then use ggplot() to plot the original data and the fitted model.

Conclusion

In this article, we explored how to apply fitting data with nls(). We discussed how to choose the best model using cross-validation, AIC, BIC, and bootstrapping methods. We also showed how to specify an equation for nonlinear regression using nls() in R. Finally, we visualized the results using ggplot(). With this knowledge, you can apply fitting data with nls() to solve complex problems in statistics and machine learning.

Last modified on 2024-02-09