Nonlinear model

TL;DR

Models that capture complex, non-straight-line relationships between inputs and outputs.
Often more accurate than linear models for nonlinear data but harder to fit and interpret.
Require careful model choice and can be sensitive to data size and parameter initialization.

Definition

Nonlinear models are mathematical models that exhibit nonlinear relationships between the input variables and the output variables. They cannot be represented using a straight line on a graph and, in contrast to linear models, have non-constant coefficients.

Explanation

Nonlinear models describe complex systems that linear models cannot accurately represent. Because their relationships are not linear, the model form and coefficients can vary with the inputs rather than remaining constant. This flexibility often yields more accurate representations of real-world relationships, but it also makes nonlinear models more difficult to fit and interpret. Fitting challenges include greater data requirements, sensitivity to initial parameter values, and the possibility of multiple local minima that complicate finding the global minimum and the best model fit.

Examples

Logistic regression

The logistic regression model is a nonlinear model used in classification problems where the output variable is binary (e.g. 0 or 1). It uses a logistic function to model the probability of an event occurring. The logistic function has a s-shaped curve, with the probability increasing as the input variables increase. Logistic regression can predict the probability that an individual will have a certain disease based on age, gender, and other risk factors.

Polynomial regression

Polynomial regression describes relationships that follow a curve rather than a straight line. It fits the data with a polynomial function; the degree of the polynomial determines the curve’s complexity. For example, a polynomial regression model with a degree of 2 (quadratic) will have a parabolic curve, while a degree of 3 (cubic) will have a cubic curve. Polynomial regression can be used to predict the cost of a product based on its production volume, with the cost increasing at a nonlinear rate as the production volume increases.

Use cases

Binary classification problems (e.g., predicting a 0 or 1 outcome) using logistic regression.
Modeling curved relationships between variables (e.g., predicting cost from production volume) using polynomial regression.
Situations where data exhibits nonlinear relationships and linear models are insufficient.

Notes or pitfalls

Nonlinear models often require more data to achieve an accurate fit.
They can be sensitive to the initial values of model parameters.
Multiple local minima can exist, making it difficult to determine the global minimum and the most accurate model fit.
Careful model selection and cautious interpretation of results are important to ensure accuracy.

Linear model
Logistic regression
Polynomial regression