# Discover Ezyfit: A free curve fitting toolbox for Matlab

F. Moisy, 19 nov 2008.

Laboratory FAST, University Paris Sud.

## Contents

## About the Ezyfit Toolbox

The EzyFit toolbox for Matlab enables you to perform simple curve fitting of one-dimensional data using arbitrary fitting functions. It provides command-line functions and a basic graphical user interface for interactive selection of the data.

## Simple fit: exponential decay

First plot some data, say, an exponential decay

plotsample exp nodisp

A predefined fit called 'exp' allows you to fit your data:

```
showfit exp
```

Equation: y(x) = a*exp(b*x) a = 4.3408 b = -0.22743 R = 0.99697 (lin)

Suppose now you want to use your own variable and function names. Let's fit this data with the function f(t)=a*exp(-t/tau), and show the fit with a bold red line:

undofit % deletes the previous fit showfit('f(t)=a*exp(-t/tau)','fitlinewidth',2,'fitcolor','red');

Equation: f(t) = a*exp(-t/tau) a = 4.3409 tau = 4.397 R = 0.99697 (lin)

Note that showfit recognizes that t is the variable, and the coefficients of the fit are named a and tau.

If you want to use the values of the coefficients a and tau into Matlab, you need to create these variables into the base workspace:

makevarfit a tau

a = 4.3409 tau = 4.3970

## Initial guesses

Now suppose you want to fit more complex data, like a distribution showing two peaks. Let's try to fit these peaks with two gaussians, each of height a, mean m and width s.

plotsample hist2 nodisp showfit('a_1*exp(-(x-x_1)^2/(2*s_1^2)) + a_2*exp(-(x-x_2)^2/(2*s_2^2))');

Exiting: Maximum number of function evaluations has been exceeded - increase MaxFunEvals option. Current function value: 369920.454653 Equation: y(x) = a_1*exp(-(x-x_1)^2/(2*s_1^2))+a_2*exp(-(x-x_2)^2/(2*s_2^2)) a_1 = 41.052 a_2 = -2492.7 s_1 = 1269.4 s_2 = 13.262 x_1 = 1062.3 x_2 = -38.289 R = 0.32602 (lin)

The solver obviously get lost in our 6-dimensional space. Let's help it, by providing initial guesses

undofit showfit('a_1*exp(-(x-m_1)^2/(2*s_1^2)) + a_2*exp(-(x-m_2)^2/(2*s_2^2)); a_1=120; m_1=7; a_2 = 100; m_2=15', 'fitcolor','blue','fitlinewidth',2);

Equation: y(x) = a_1*exp(-(x-m_1)^2/(2*s_1^2))+a_2*exp(-(x-m_2)^2/(2*s_2^2)) a_1 = 128.41 a_2 = 77.126 m_1 = 6.9929 m_2 = 14.783 s_1 = 0.42396 s_2 = 1.4307 R = 0.98977 (lin)

The result seems to be correct now. Note that only 4 initial guesses are given here; the two other ones, s_1 and s_2, are taken as 1 -- which is close to the expected solution.

## Fitting in linear or in log scale

Suppose you want to fit a power law in logarithmic scale:

plotsample power nodisp showfit power

Equation: y(x) = a*x^n a = 0.4784 n = 2.494 R = 0.99934 (log)

would you have obtained the same result in linear scale? No:

swy % this shortcut turns the Y-axis to linear scale showfit('power','fitcolor','red');

Equation: y(x) = a*x^n a = 2.9016 n = 2.2564 R = 0.99608 (lin)

The value of the coefficients have changed. In the first case, LOG(Y) was fitted, whereas in the second case Y was fitted, because the Y-axis has been changed.

You may however force showfit to fit LOG(Y) or Y whatever the Y axis, by specifying 'lin' or 'log' in the first input argument:

rmfit % this removes all the fits showfit('power; lin','fitcolor','red'); showfit('power; log','fitcolor','blue');

Equation: y(x) = a*x^n a = 2.9016 n = 2.2564 R = 0.99608 (lin) Equation: y(x) = a*x^n a = 0.4784 n = 2.494 R = 0.99934 (log)

In the equation information, it is specified (lin) or (log) after the R coefficient.

## Using the fit structure f

You can fit your the data without displaying it:

```
x=1:10;
y=[15 14.2 13.6 13.2 12.9 12.7 12.5 12.4 12.4 12.2];
f = ezfit(x,y,'beta(rho) = beta_0 + Delta * exp(-rho * mu); beta_0 = 12');
```

f is a structure that contains all the informations about the fit:

f

f = name: 'beta(rho)=beta_0+Delta*exp(-rho*mu)' yvar: 'beta' xvar: 'rho' fitmode: 'lin' eq: 'beta_0+Delta*exp(-rho*mu)' r: 0.9992 param: {'Delta' 'beta_0' 'mu'} m: [3.9949 12.1058 0.3237] m0: [1 12 1] x: [1 2 3 4 5 6 7 8 9 10] y: [1x10 double]

From this structure, you can plot the data and the fit:

```
clf
plot(x,y,'r*');
showfit(f)
```

Equation: beta(rho) = beta_0+Delta*exp(-rho*mu) Delta = 3.9949 beta_0 = 12.106 mu = 0.32368 R = 0.99925 (lin)

you can also display the result of the fit

dispeqfit(f)

Equation: beta(rho) = beta_0+Delta*exp(-rho*mu) Delta = 3.9949 beta_0 = 12.106 mu = 0.32368 R = 0.99925 (lin)

or create the variables in the base workspace

makevarfit(f) beta_0 mu Delta

beta_0 = 12.1058 mu = 0.3237 Delta = 3.9949

## Weigthed fit

Suppose now we want to fit data with unequal weights, shown here as error bars of different lengths:

```
x = 1:10;
y = [1.56 1.20 1.10 0.74 0.57 0.55 0.31 0.27 0.28 0.11];
dy = [0.02 0.02 0.20 0.03 0.03 0.10 0.05 0.02 0.10 0.05];
clf, errorbar(x,y,dy,'o');
```

In order to perform a weighted fit on this data, the vectors y and dy have to be merged into a 2-by-N matrix and given as the second input argument to ezfit. Compare the results for the usual and weighted fits:

fw = ezfit(x, [y;dy], 'exp'); showfit(fw,'fitcolor','red'); f = ezfit(x, y, 'exp'); showfit(f,'fitcolor','blue');

Equation: y(x) = a*exp(b*x) a = 2.0017 b = -0.2519 R = 0.98832 (lin) Equation: y(x) = a*exp(b*x) a = 2.0071 b = -0.24013 R = 0.99067 (lin)

The red curve (weighted fit) tends to go through the data with smaller error bars.