Linear model

A linear model consists of a matrix $A$ which contains all coefficients. We can then write our set of possible regression functions like this:

$f_{\gamma }(t)=A\cdot \gamma$

Of course there are also other models which can include non-linearities like:

• Exponential Growth
• Allometric Growth

Simple linear model

This is how a simple two dimensional linear model can look like: $f(x)=w_0\cdot x_0+w_1\cdot x_1=y$

The input is the x vector with $x_0$ and $x_1$ as values. The same applies to the weights. Finding the best weights to describe $y$ with our function is called Fitting.

Usually a residual error $ϵ$ is part of the equation. A lot of times it has a Gaussian distribution.

In vector notation the general linear model looks like this: $f(x)=w^{T}x+ϵ=y$

Now we have to optimize the model parameters: