# Matlab regressions

This program is under development.
is complete, remove the {{develop}} tag.

Many people use MATLAB to perform regressions. Let's give some examples.

## Linear regression

Linear least squares regressions can be calculated using the \ operator:

```B = x \ y
```

It is also possible to use the regress function from the MATLAB Statistics Toolbox:

```[B, Bint, R, Rint, Stats] = regress(y, x)
```

## Apparently nonlinear regressions

### Simple example

#### Setting the question

Consider a question asked on the MATLAB newsgroup (Help to curve fitting):

```I have some data (x,y)and one equation like y=(a*x)/(1+ax) and I need to plot the data and the curve [...]
```
 When you have such a parametrized equation: you can sometimes rewrite it in a: so if you have some (x,y) distribution, you can plot them and use (with a define as in (2)). result of the trivial estimation script

First generate a test set:

• f is defined as an anonymous function
• the real value of a is arbitrary chosen
• a noise coefficient s is chosen
• the domain of x is [0,10], the chosen resolution for x is 0.05
• y is the theoretical value of y
• ny the noisy (observed) values for y
```<<generate test set>>=
f  = @(a,x)(a*x./(1+a*x))
a  = sqrt(2)
s  = .05;
x  = (0:.1:10)';
y  = f(a,x);
ny = y + randn(size(y))*s;
figure('Color',[0.9412 0.9412 0.9412 ]);
h1 = subplot(1,2,1);
plot(x,y,'-b',x,ny,'.k','linewidth',2);
```

Now we can get the implied values for a:

```<<a implied values>>=
g  = @(x,y)(y./(x-x.*y))
ai = g(x,ny);
h2 = subplot(1,2,2);
boxplot(ai);
```

And a trivial estimator for a is the empirical expectation (the mean) of the implied a:

```<<trivial estimation>>=
ah = mean(ai(~isinf(ai)));
title(sprintf('\$\\hat a\$ = %5.5f ; real a = %5.5f', ah, a), 'interpreter', 'latex');
axes(h1);
hold on
plot(x, f(ah, x), ':r', 'linewidth', 2);
hold off
legend({'theoretical', 'observed', 'estimation'});
```

Let's produce the whole code:

```<<test_trivial_estimation.m>>=
generate test set
a implied values
trivial estimation
```

(see the upper figure for the result).

But it's a not so simple answer, because from a statistical point of view, you have to master some of the properties of the estimator:

• is it efficient?
• is it biaised?
• etc