In many papers, we can find the statements similar to the one below:

Because linear models can hardly capture the nonlinear relationship between load and temperature, we use Artificial Neural Networks (or other black-box models) in this paper.The major conceptual error of the above statement is due to a common misunderstanding that linear models

*cannot*capture nonlinear relationship.

I'm showing a nonlinear curve in the figure below, which is from a 3rd order polynomial function.

Is this a linear model? Yes! Polynomial regression models in general are linear models.

Why?

The "linear" in linear models refers to the equations we use to solve the parameters. By definition, a regression model is linear if it can be written as

*Y*=

*XB*+

*E*, where

*Y*is a vector of values of the response variable;

*B*is a vector of parameters to be solved;

*X*is a matrix of values of explanatory variables;

*E*is a vector of independent normally distributed errors.

Maybe the above description is too abstract. Let's check out an example of parameter estimation for a 3rd order polynomial regression model:

*y*=

*b0*+

*b1x1*+

*b2x2*+

*b3x3*+

*e*,

where

*x2*is the square of*x1*, and*x3*is the cube of*x1*. There are 4 parameters to be estimated. Now let's say we have 6 observations as shown in the table below.Then we can come up with 6 linear equations:

3 =

*b0*+*b1*+*b2*+*b3*;
1 =

*b0*+ 2*b1*+ 4*b2*+ 8*b3*;
4 =

*b0*+ 3*b1*+ 9*b2*+ 27*b3*;
9 =

*b0*+ 4*b1*+ 16*b2*+ 64*b3*;
5 =

*b0*+*b1*+*b2*+*b3*;
2 =

*b0*+ 3*b1*+ 9*b2*+ 27*b3*;These equations can be written in the following form:

Therefore, a 3rd order polynomial regression mode is a linear model. By solving the above equations, we can obtain the values of

*b0*,

*b1*,

*b2*and

*b3*:

To further clarify the concept of linear vs. nonlinear, here are a few examples of nonlinear regression models:

*y*=

*b0*+

*b1x1*/(

*b2x2*+

*b3*) +

*e*, ...... (1)

*y*=

*b0*(

*exp*(

*b1x*))

*U*, ...... (2)

Not to make it too complicated, some nonlinear regression problems can be solved in a linear domain. For instance, eq(2) can be transformed to a linear model by taking the logarithm on both side:

Ln(

*y*) = Ln(*b0*) +*b1x*+*u*, .... (3)Back to Load Forecasting Terminology.

## No comments:

## Post a Comment

Note that you may link to your LinkedIn profile if you choose Name/URL option.