# Functions for empirical data fitting

## Idea

Various kinds of functions used in data fitting. TODO: an important issue is “redundant variables” which can confuse the fitting. These functions are chosen because they may be in some way revealing about some structure in the dataset. This is a different idea to general function approximation with with, e.g., non-parametric density estimation.

## Details

In general data variables are denote $x$ for “input” variables and $y$ for “output” variables, with $\sim$ denoting “basically fits with unmodelled noise term”. (This is deliberately loose.)

### Functions for 1-dimensional fitting

Linear, vertical errors:

(1)$y \sim a x +b$

Gaussian:

(2)$y \sim a \exp -\frac{(x-\mu)^2}{\sigma^2}$

Exponential:

(3)$y \sim a \exp \lambda x$

Combined Gaussian and exponential:

(4)$y \sim a \exp -(b x^2 + c x)$
(5)$y \sim a x^b + c$

Beta:

(6)$y \sim a x^b (1-x^c)$

Need to see how these break down in terms of (possibly null) affine transformations of basic variable combined with other functions.

### Functions for multidimensional fitting

In some cases, particularly if the choice of axis corresponds to meaningful variables, a multidimensional dataset may be well-fitted by a separable function composed from several of the 1-dimensional functions above. Otherwise an intrinsically multidimensional function like those below may be better.