Matlab通过自定义方程实现数据拟合

>> help fittype
 fittype   Fittype for curve and surface fitting
 
    A fittype encapsulates information describing a model. To create a
    fit, you need data, a fittype, and (optionally) fit options and an
    exclusion rule.
 
 
    LIBRARY MODELS
 
    fittype(LIBNAME) constructs a fittype for the library model
    specified by LIBNAME.
 
    Choices for LIBNAME include:
 
        LIBNAME           DESCRIPTION
        'poly1'           Linear polynomial curve
        'poly11'          Linear polynomial surface
        'poly2'           Quadratic polynomial curve
        'linearinterp'    Piecewise linear interpolation
        'cubicinterp'     Piecewise cubic interpolation
        'smoothingspline' Smoothing spline (curve)
        'lowess'          Local linear regression (surface)
 
    or any of the names of library models described in "List of Library Models
    for Curve and Surface Fitting" in the documentation.
 
 
    CUSTOM MODELS
 
    fittype(EXPR) constructs a fittype from the MATLAB expression
    contained in the string, cell array or anonymous function EXPR.
 
    The fittype automatically determines input arguments by searching
    EXPR for variable names (see SYMVAR). In this case, the fittype
    assumes 'x' is the independent variable, 'y' is the dependent
    variable, and all other variables are coefficients of the model. If
    no variable exists, 'x' is used.
 
    All coefficients must be scalars. You cannot use the following
    coefficient names in the expression string EXPR: i, j, pi, inf,
    nan, eps.
 
    If EXPR is a string or anonymous function, then the toolbox uses a
    nonlinear fitting algorithm to fit the model to data. To use a
    linear fitting algorithm, use a cell array of terms.
 
 
    ANONYMOUS FUNCTIONS
 
    If EXPR is an anonymous function, then the order of inputs must be
    correct. The input order enables the fittype class to determine
    which inputs are coefficients to estimate, problem-dependent
    parameters and independent variables. The order of the input
    arguments to the anonymous function must be:
 
    EXPR = @(<coefficients>, <problem parameters>, <x>, <y>) expression
 
    There must be at least one coefficient. The problem parameters and
    y are optional. The last arguments, x and y, represent the
    independent variables: just x for curves, but x and y for surfaces.
    If you don't want to use x and/or y as the names of the independent
    variables, then you can specify different names by using the
    'independent' property name/value pair. However, whatever name or
    names you choose, these arguments must be the last arguments to the
    anonymous function.
 
    Anonymous functions make it easier to pass other data into the
    fittype and fit functions. For example, to create a fittype using
    an anonymous function and variables xs and ys from the workspace:
 
        % Variables in workspace
        xs = (0:0.1:1).';
        ys = [0; 0; 0.04; 0.1; 0.2; 0.5; 0.8; 0.9; 0.96; 1; 1];
        % Create fittype
        ft = fittype( @(b, h, x) interp1( xs, b+h*ys, x, 'pchip' ) )
        % Load some data
        xdata = [0.012;0.054;0.13;0.16;0.31;0.34;0.47;0.53;0.53;...
            0.57;0.78;0.79;0.93];
        ydata = [0.78;0.87;1;1.1;0.96;0.88;0.56;0.5;0.5;0.5;0.63;...
            0.62;0.39];
        % Fit the curve to the data
        f = fit( xdata, ydata, ft, 'Start', [0, 1] )
 
 
    LINEAR MODELS
 
    To use a linear fitting algorithm specify EXPR as a cell array of
    terms. That is, to specify a linear model of the following form:
 
        coeff1 * term1 + coeff2 * term2 + coeff3 * term3 + ...
 
    (where no coefficient appears within any of term1, term2, etc) use
    a cell array where each term, without coefficients, is specified in
    a cell of EXPR, as follows:
 
        EXPR = {'term1', 'term2', 'term3', ... }
 
    For example, the model
 
        a*x + b*sin(x) + c
 
    is linear in 'a', 'b' and 'c'. It has three terms 'x', 'sin(x)' and
    '1' (since c=c*1) and so EXPR is
 
        EXPR = {'x','sin(x)','1'}
 
 
    ADDITIONAL PROPERTIES
 
    fittype(EXPR,PROP1,VALUE1,PROP2,VALUE2,....) uses the property
    name/value pairs PROP1-VALUE1, PROP2-VALUE2 to specify property
    values other than the default values.
 
    PROPERTY         DESCRIPTION
    'independent'    Specifies the independent variable name
    'dependent'      Specifies the dependent variable name
    'coefficients'   Specifies the coefficient names (in a cell array
                     if there are two or more). Note excluded names
                     above.
    'problem'        Specifies the problem-dependent (constant) names
                     (in a cell array if there are two or more)
    'options'        Specifies the default 'FITOPTIONS' for this
                     equation
 
    Defaults: The independent variable is x.
              The dependent variable is y.
              There are no problem dependent variables.
              Everything else is a coefficient name.
 
    Multi-character symbol names may be used.
 
 
    EXAMPLES
 
       g = fittype('a*x^2+b*x+c')
       g = fittype('a*x^2+b*x+c','coeff',{'a','b','c'})
       g = fittype('a*time^2+b*time+c','indep','time')
       g = fittype('a*time^2+b*time+c','indep','time','depen','height')
       g = fittype('a*x+n*b','problem','n')
       g = fittype({'cos(x)','1'})                            % linear
       g = fittype({'cos(x)','1'}, 'coefficients', {'a','b'}) % linear
       g = fittype( @(a,b,c,x) a*x.^2+b*x+c )
       g = fittype( @(a,b,c,d,x,y) a*x.^2+b*x+c*exp(-(y-d).^2), ...
            'independent', {'x', 'y'}, ...
            'dependent', 'z' ); % for fitting surfaces

load census
plot(cdate,pop,'o')
hold on

s = fitoptions('Method','NonlinearLeastSquares',...
               'Lower',[0,0],...
               'Upper',[Inf,max(cdate)],...
               'Startpoint',[1 1]);
f = fittype('a*(x-b)^n','problem','n','options',s);

[c2,gof2] = fit(cdate,pop,f,'problem',2)

c2 = 

     General model:
     c2(x) = a*(x-b)^n
     Coefficients (with 95% confidence bounds):
       a =    0.006092  (0.005743, 0.006441)
       b =        1789  (1784, 1793)
     Problem parameters:
       n =           2

gof2 = 

           sse: 246.1543
       rsquare: 0.9980
           dfe: 19
    adjrsquare: 0.9979
          rmse: 3.5994

[c3,gof3] = fit(cdate,pop,f,'problem',3)

c3 = 

     General model:
     c3(x) = a*(x-b)^n
     Coefficients (with 95% confidence bounds):
       a =   1.359e-05  (1.245e-05, 1.474e-05)
       b =        1725  (1718, 1731)
     Problem parameters:
       n =           3

gof3 = 

           sse: 232.0058
       rsquare: 0.9981
           dfe: 19
    adjrsquare: 0.9980
          rmse: 3.4944

plot(c2,'m')
plot(c3,'c')
legend('data','fit with n=2','fit with n=3')