# Discover Ezyfit: a sample session

## Contents

- Simple fit: exponential decay
- Initial guesses
- Fitting in linear or in log scale
- Using the fit structure f
- Weigthed fit

## Simple fit: exponential decay

First plot some data, say, an exponential decay

    plotsample exp nodisp

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

    showfit exp

    Equation: y(x) = a*exp(b*x)
         a = 1.835
         b = -0.18352
         R = 0.99878  (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 = 1.835
         tau = 5.4484
         R = 0.99878  (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 t and tau into Matlab, you need to creates these variables into the base workspace:

    makevarfit
    a
    tau

    a =

        1.8350

    tau =

        5.4484

## 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))');

    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 = 31.316
         a_2 = -70.893
         s_1 = 5.1413
         s_2 = 6.1581
         x_1 = 13.45
         x_2 = -17.695
         R = 0.39514  (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 = 112.83
         a_2 = 65.573
         m_1 = 6.9957
         m_2 = 14.817
         s_1 = 0.40839
         s_2 = 1.4242
         R = 0.98866  (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 = 0.37324
         n = 0.29379
         R = 0.9359  (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 = 0.38668
         n = 0.29087
         R = 0.91515  (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 = 0.38668
         n = 0.29087
         R = 0.91515  (lin)
    Equation: y(x) = a*x^n
         a = 0.37324
         n = 0.29379
         R = 0.9359  (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.