# Explain “Linear regression is very extensible and can be used to capture nonlinear effects”

I encountered this statement from Statistical Learning by Stanford University. Can anyone give me an example?

• What do you understand by non-linear effeccts ? Logically, a variable could affect another directly or through a third variable. It may be useful to define variables from different angles - mathematics, statistics, computing etc. It seems to depend on design of experiment or statistical model, construction of variables and so on. To me, given the basic formulations and formulas; it sounds good. – Subhash C. Davar Oct 3 '17 at 3:11
• You can get some kinds of nonlinear relationships via transformed $x$'s and multiple linear regression e.g. 1.$\:$See the discussion here for a specific example. 2 See the nonlinear relationship between $y$ and $t$ here and the several periodic models: How to find a good fit for semi-sinusoidal model in R? fitted with linear regression by using sin and cos terms.3. Polynomial regression 4. regression spline models. etc ... ctd – Glen_b -Reinstate Monica Oct 3 '17 at 4:55
• ctd... there's some elementary discussion of 3&4 here (also try searches of our site for lots of examples of 3 and 4), and 5. a particular example here: regression where response variable is a function. 6. For another example some of the fits here were done using multiple linear regression. ...ctd – Glen_b -Reinstate Monica Oct 3 '17 at 4:59
• ctd... The host of possible examples aren't necessarily the explanation you're seeking though; it may help if you could be more specific about what particular kinds of information you want. – Glen_b -Reinstate Monica Oct 3 '17 at 5:19

In linear regression, the word "linear" applies to the coefficients: the dependence between $Y$ and the coefficients is linear. This does not mean the dependence between $Y$ and $X$ is linear.

Assume $X$ is a one dimensional variable. Basic linear regression is (I omit the noise and intercept for simplicity): $$Y=\beta X$$

But this is still linear regression:

$$Y=\beta_1 X+\beta_2X^2+\beta_3\log(X)$$

The latter is the same as basic linear regression with feature vector $(X,X^2,\log(X))$ instead of $X$.