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ezfit
Fit data with arbitrary fitting function
Description `

    ezfit(FUN) fits the active curve with the function FUN. See below for
    the syntax of FUN. If FUN is not specified, 'linear' is used.

    ezfit(X,Y,FUN) or ezfit(Y,FUN) fits the data (X,Y) (or Y) using the
    function FUN. X and Y must be vectors of equal length. If X is not
    specified, X=[1, 2, 3...] is used.

    ezfit(X,Y,FUN), where X is a 1-by-N vector and Y is a 2-by-N matrix,
    also specifies the weights for Y(2,:) (error bars). By default, when
    Y is a 1-by-N vector, all the weights are 1.

    Note that ezfit only computes the coefficients, but does not display the
    fit. Use showfit to display the fit, or selectfit to fit only a part of
    the current curve.

    By default, the first curve in the active figure is fitted - see
    fitparam to change this default behavior. To fit another curve, select
    it before calling ezfit.

    The function string FUN can be:
       - the name of a predefined fitting function (see below).
       - the name of a user-defined fitting function (see editfit).
       - an equation, in the form 'y(x)=...', where 'x' represents the
         X-data, and all the other variables are parameters to be fitted
         ('a', 'x_0', 'tau', ...). Example: 'y(x)=a*sin(b*x)'. If the
         left-hand-side 'y(x)' is not specified, 'x' is taken for the
         X-Data. All the parameter names are accepted, except Matlab
         reserved strings ('sin', 'pi', ...)

    The predefined fitting functions are:
       - linear             y = m * x
       - affine or poly1    y = a*x + b
       - poly{n}            y = a0 + a1 * x + ... + an * x^n
       - power              y = c*x^n
       - sin                y = a * sin (b * x)
       - cos                y = a * cos (b * x)
       - exp                y = a * exp (b * x)
       - log                y = a * log (b * x)
       - cngauss            y = exp(-x^2/(2*s^2))/(2*pi*s^2)^(1/2)
       - cfgauss            y = a*exp(-x^2/(2*s^2))
       - ngauss             y = exp(-(x-x0)^2/(2*s^2))/(2*pi*s^2)^(1/2)
       - gauss              y = a*exp(-(x-x0)^2/(2*s^2))
    ('ngauss' is a 2-parameters normalized Gaussian, and 'gauss' is a
    3-parameters non-normalized (free) Gaussian. 'cngauss' and 'cfgauss'
    are centered normalized and centered free Gaussian, respectively.

    ezfit is based on Matlab's built-in FMINSEARCH function (Nelder-Mead
    method), which performs an unconstrained nonlinear minimization of
    the SSR (sum of squared residuals) with respect to the various
    parameters.

    Nonlinear minimization requires starting guesses (or starting estimates)
    for the fit parameters. By default, all the starting guesses are taken
    as 1, or, when using predefined fits (e.g., exp, gauss, power...), the
    starting guesses are determined depending on the range of the data to
    be fitted. However, in most cases, values closer to the expected
    result should be specified to "help" the convergence. It is sufficient
    to choose values that have the correct sign and correct order of
    magnitude, e.g. 0.01, 1, 100...

    The starting guesses for the parameters may be specified in two ways:
      - directly in the string FUN, after the fit definition:
           'c0 + a*sin(pi*x/lambda); c0=1; a=0.1; lambda=100'
           ('!' or '$' may also be used instead of ';').
      - by specifying them as an additional input argument for ezfit:
           ezfit(x,y,'c0 + a*sin(pi*x/lambda)',[0.1 1 100]);
        Note that in this case the parameters must be ordered alphabetically.
    Note that if both methods are used, only the starting guesses given in
    the string FUN are considered.

    By default, Y is fitted in linear mode. If you want to fit LOG(Y)
    instead, you must specify the option 'log' to the string FUN, separated
    by the symbol ';' or '$' or '!' (eg, FUN='a*x^n;log'). This is
    specially useful to fit power laws with equally weighted points in a
    log scale.  If nothing specified, the option 'lin' is used.

Example

         plotsample('power')
    and compare the results of:
         ezfit('power;lin')
         ezfit('power;log')

    F = ezfit(...) also returns a structure F having the following fields:
       - name       name of the fit
       - eq         equation of the fit
       - param      cell array of strings: names of the parameters
       - m          values of the coefficients
       - m0         initial guess for the coefficients
       - r          correlation coefficient R (Pearson's correlation)
       - fitmode    'lin' (y is fitted) or 'log' (log(y) is fitted) mode

    This structure F can be further used with showfit, selectfit, dispeqfit,
    showeqbox, makevarfit and editcoeff.

    From F, you can get the values of the fitted parameters. If you want to
    create in the current Matlab workspace the variables associated to
    these parameters, use makevarfit (or set the option 'automakevarfit'
    to 'on' in fitparam).

    The correlation coefficient R is defined as (SSR/(SSE+SSR))^(1/2), where
       SSR = sum ((y_fit - mean(y)).^2)   % sum of squared residuals
       SSE = sum ((y_fit - y).^2)         % sum of squared errors
    (see http://mathworld.wolfram.com/CorrelationCoefficient.html)

Examples

      plotsample damposc
      f = ezfit('u(t) = c + u_a * sin(2*pi*t/T) * exp(-t/tau); T=5; tau=20');
      showfit(f);

      plotsample poly2
      [x,y] = pickdata;
      f = ezfit(x, y, 'z(v) = poly3');
      editcoeff(f);

      plotsample poly2
      f = ezfit('beta(z) = poly2');
      showfit(f, 'fitcolor', 'red', 'fitlinewidth', 2);

See Also

    selectfit, showfit, plotsample, dispeqfit, editcoeff,
    FMINSEARCH, makevarfit, fitparam.

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