Not Completely Linear Regression
In simple linear regression the computer learns a linear relationship between a single input and a single output by calculating two values and . These values define a line that best fits the training examples .
Multivariate linear regression is an extension that finds a relationship between multiple inputs and an output. We say that the input is “ dimensional.” The computer calculates values . These values define a plane (if ) or hyperplane (if ) that best fits the training examples.
To handle the multiplicity of variables and values, it is convenient to use matrices.
Let \( \vec{\theta} = \) `[[\theta_0], [\theta_1], [\vdots], [\theta_n]]`. Its transpose `[\theta_0, \theta_1, …, \theta_n]`
Let the inputs be represented as `[[x_0 = 1], [x_1], [\vdots], [x_n]]`. (Below you’ll see why we set .)
Instead of as for simple linear regression, we use the rules of matrix multiplication to get the model equation:
The cost function can also be expressed using matrix notation:
The partial derivatives are as follows:
The model equation, cost function, and its partial derivatives should look familiar. In the special case where they are exactly the equations for simple linear regression.