We define linearity of a model in terms of the coefficient, not the variable therefore we treat Y=βX as a linear regression model [here coefficient and variable both are linear]
If the coefficient is linear but the variable is non-linear Y=βX^2 then why do some texts call this a non-linear regression model.[here variable is non-linear but the coefficient is still Linear]
What is meant with "linear" vs. "non-linear" depends on the context.
A linear model is "linear" because of the coefficients. In this context, Y=βX² can be written as Y = Y=βZ, where Z = X² (or any other transformation fo the variable). So it's linear in Z.
If the response or a predictor variable in a linear model is transformed, then the model fits a non-linear relationship between the original, untransformed predictor and the respone. Some authors call this a non-linear model, but this is confusing and should be discouraged.
Sometimes the term "curvilinear" is used for this kind of model, since in the bivariate case, we're looking for a curved fit.... Or "polynomial" regression.... As opposed to the case of simple linear regression.... Authors should probably separate these from "nonlinear" regression, even if only because the fitting methods are usually different.
Hi
I agree with the answer given to your question by Jochen Wilhelm above, and the linear relationship between the variables means that they are linear based on coefficients and not based on the quadratic or tertiary power of the variables.
That is the reason why "it's often said that" curves or exponential expressions or some polynomials, etc. can be fitted by linear model, i.e., through re-expression or log-transformation or anyother transformation. Y=ẞX + ẞX^2 is linear in its parameters but fitting a quadratic (non linear) function or model, hence a non linear model.
In the case of linearity, as x change by one unit, y must be changed by a constant amount (i'e y=BX), but for y=BX^2 as X changes by one unit y can not change by a constant amount. This is why y= BX^2 is considered as non-linear
I think one of the reasons why this question comes up is one of context. The explanation given by Teshita Uke Chikako is one that is common in introductory analytical geometry classes, and related to the discussion of nonlinear systems in general in math and engineering (https://en.wikipedia.org/wiki/Nonlinear_system). In the context of fitting models to data, a quadratic function is linear, as explained by Jochen Wilhelm , and can be thought of as a example of a general linear model. Practically, when fitting a general linear model, we might use OLS regression (although there are other alternatives). Whereas if we have a nonlinear model ---- consider a segmented model like a linear plateau model or a general exponential model ( https://online.stat.psu.edu/stat501/lesson/15/15.7 ) ---- we would use an iterative approach like nls() in R or PROC NLIN in SAS. Because of this practical point, when we talk about fitting models or in an analysis of experiments context, it's a good idea to keep the linear / non-linear distinction in tact.
As others have noted, I think we can help students by emphasizing the distinction between the model, which is linear in the coefficients, and the functional relationship between X and Y, which can be linear or non-linear. To emphasize this distinction, I like to show a scatter-plot with a clear quadratic relationship between X and Y, and then show that I can generate a model that fits very well* using a command for estimating linear regression models. (Most of the previous posters have said more or less the same thing in somewhat different words.) HTH.
* i.e., Y = B0 + B1X + B2X2 + error
PS- I like Jochen's suggestion to set Z = X2 and substitute Z in the equation above. That makes it even clearer that the model is linear in the coefficients.
HI. A function y=f(x) is linear on a scalar field if it satisfies two properties defined by vector spaces, where x is the independent variable, y is the dependent variable, x1 and x2 are two values of the independent variable and k is any scalar
f(x1+x2) = f(x1) + f(x2)
f(kx) = k f(x)
As a consequence of the second property, f(0)=0, necessary but non sufficient.
For instance y= 3x satisfies both properties
3(x1+x2) = 3(x1)+3(x2)
3(kx) = k (3x)
but y=3x^2 does not satisfy the properties:
3(x1+x2)^2 = 3 (x1^2 + 2 x1x2 + x2^2) 3 x1^2 + 3 x2^2
3(kx)^2 = 3 k^2 x^2 k (3x^2)
by the way y=3x+1 is non linear, because
3(x1+x2)+1 [3x1+1] + [3x2 +1] = 3(x1+x2)+2
3(kx)+1 k( 3x + 1) = k3x + k
A shorter demonstration that it is non linear is:
3(0)+1 = 1 0
These kind of functions that are similar to a linear one but does not pass by the origin define an affine space.
Nevertheless it is very common to think that any straight line is "linear" and any polynomial or other kind of functions are non linear.
Hope this helps
What matters is the relationship between X and Y. If Y depends on X, it's said to be linear, if Y depends on X^2, it's said to be non-linear.
For example, you can say that the relation between speed and acceleration is linear, but the relation between space and acceleration is non-linear.
Dear Khursheed Hussain , the non-linear function could be transformed to be a linear function. Kindly visit the link.
https://www.youtube.com/watch?v=2t9f0fHKTLw
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