Discover Ezyfit: a sample session

Contents

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.

Published with MATLAB® 7.5