What is Nonlinear Regression?
Nonlinear regression is a type of regression analysis that fits data to a model and then expresses it as a mathematical function. While simple linear regression relates two variables (X and Y) with a straight line (y = mx + b or y = ax + c), nonlinear regression shows the relationship between two variables in a nonlinear (curved) manner.
The primary goal of the model is to minimize the sum of the squares. The sum of squares measures how far the Y observations deviate from the nonlinear function that predicts Y.
Understanding the Sum of Squares
It starts by calculating the difference between the fitted nonlinear function and each Y point in the dataset. Each difference is squared, and these squared differences are summed up. The closer the sum of these values to zero, the better the function fits the data. Nonlinear regression employs logarithmic, trigonometric, exponential, power functions, Lorenz curves, Gaussian functions, among other methods.
Key Highlights
- Prediction: Both linear and nonlinear regression predict Y responses from X variables.
- Function Form: Nonlinear regression is a curved function of X variables predicting a Y variable.
- Applications: Ideal for modeling population growth over time.
Nonlinear regression modeling follows a concept similar to linear regression. Both seek to graphically track a particular response from a set of variables. Nonlinear models are more intricate, built on a series of approximations (iterations) which might involve trial-and-error. Many methods exist for this, including the Gauss-Newton method and the Levenberg-Marquardt method.
While some models may initially appear nonlinear, they might be linear. Curve estimation helps identify functional relationships in your data, allowing you to choose the accurate model. Linear regression, though typically forming straight lines, can also create curves based on its equation. Similarly, nonlinear equations can be transformed to mimic linear equations, termed as “intrinsically linear.”
Case Study: Predicting Population Growth
Example: Predicting population growth using nonlinear regression. A scatterplot of changing population data over time might suggest a nonlinear relationship between time and population growth. Employing a logistic population growth model estimates the population over unmeasured periods and predicts future growth.
For accurate nonlinear regression results, variables should be quantitative. Categorical variables (e.g., region or religion) should be converted into binary or other quantitative forms.
Correctly describing the relationship between variables and ensuring good initial values are crucial. Incorrect starting values may lead to a non-converging model or a local rather than a global solution, even when the model’s form is correctly specified.
Related Terms: linear regression, sum of squares, logistic growth model, Gauss-Newton method, Levenberg-Marquardt method
References
- IBM. “Nonlinear Regression”.
