Previously, i have written about when to select nonlinear regression and also how to design curvature with both linear and nonlinear regression. Since then, I’ve obtained several comments expressing confusion around what differentiates nonlinear equations from direct equations. This man is understandable since both types can design curves.

You are watching: Difference between linear and nonlinear equations

So, if it’s not the ability to version a curve, what is the difference between a linear and nonlinear regression equation?

Linear Regression Equations

Linear regression needs a linear model. No surprise, right? but what does that really mean?

A model is straight when every term is one of two people a constant or the product of a parameter and also a predictor variable. A straight equation is constructed by including the outcomes for each term. This constrains the equation to simply one an easy form:

Response = constant + parameter * predictor + ... + parameter * predictor

Y = b o + b1X1 + b2X2 + ... + bkXk

In statistics, a regression equation (or function) is straight when the is straight in the parameters. While the equation need to be direct in the parameters, you have the right to transform the predictor variables in methods that develop curvature. For instance, friend can encompass a squared change to develop a U-shaped curve.

Y = b o + b1X1 + b2X12

This design is still straight in the parameters even though the predictor variable is squared. Friend can likewise use log and also inverse functional forms that are linear in the parameters to create different species of curves.

Here is an example of a direct regression model that uses a squared term come fit the bent relationship in between BMI and also body fat percentage.


Nonlinear Regression Equations

While a direct equation has actually one an easy form, nonlinear equations deserve to take many different forms. The easiest method to recognize whether one equation is nonlinear is to focus on the term “nonlinear” itself. Literally, it’s not linear. If the equation doesn’t meet the criteria over for a straight equation, that nonlinear.

That covers plenty of different forms, i beg your pardon is why nonlinear regression provides the most flexible curve-fitting functionality. Here are several instances from’s nonlinear function catalog. Thetas stand for the parameters and X represents the predictor in the nonlinear functions. Unlike straight regression, these attributes can have more than one parameter every predictor variable.

Nonlinear functionOne feasible shape
Power (convex): Theta1 * X^Theta2
Weibull growth: Theta1 + (Theta2 - Theta1) * exp(-Theta3 * X^Theta4)
Fourier: Theta1 * cos(X + Theta4) + (Theta2 * cos(2*X + Theta4) + Theta3

Here is an example of a nonlinear regression design of the relationship between density and electron mobility.

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The nonlinear equation is so lengthy it the it doesn"t right on the graph:

Mobility = (1288.14 + 1491.08 * thickness Ln + 583.238 * density Ln^2 + 75.4167 * density Ln^3) / (1 + 0.966295 * density Ln + 0.397973 * density Ln^2 + 0.0497273 * thickness Ln^3)

Linear and nonlinear regression space actually named after the functional form of the models that each evaluation accepts. Ns hope the distinction between linear and also nonlinear equations is clearer and that you understand just how it’s possible for straight regression to version curves! It also explains why you’ll watch R-squared displayed for some curvilinear models also though it’s impossible to calculate R-squared for nonlinear regression.