# fit

Fit curve or surface to data

## Description

fitobject = fit(x,y,fitType) creates the fit to the data in x and y with the model specified by fitType.

example

fitobject = fit([x,y],z,fitType) creates a surface fit to the data in vectors x, y, and z.

example

fitobject = fit(x,y,fitType,fitOptions) creates a fit to the data using the algorithm options specified by the fitOptions object.

example

fitobject = fit(x,y,fitType,Name=Value) creates a fit to the data using the library model fitType with additional options specified by one or more Name=Value pair arguments. Use fitoptions to display available property names and default values for the specific library model.

example

[fitobject,gof] = fit(x,y,fitType) returns goodness-of-fit statistics in the structure gof.

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[fitobject,gof,output] = fit(x,y,fitType) returns fitting algorithm information in the structure output.

example

## Examples

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Load the census sample data set.

The vectors pop and cdate contain data for the population size and the year the census was taken, respectively.

Fit a quadratic curve to the population data.

f=fit(cdate,pop,'poly2')
f =
Linear model Poly2:
f(x) = p1*x^2 + p2*x + p3
Coefficients (with 95% confidence bounds):
p1 =    0.006541  (0.006124, 0.006958)
p2 =      -23.51  (-25.09, -21.93)
p3 =   2.113e+04  (1.964e+04, 2.262e+04)

f contains the results of the fit, including coefficient estimates with 95% confidence bounds.

Plot the fit in f together with a scatter plot of the data.

plot(f,cdate,pop)

The plot shows that the fitted curve closely follows the population data.

Load the franke sample data set.

The vectors x, y, and z contain data generated from Franke's bivariate test function, with added noise and scaling.

Fit a polynomial surface to the data. Specify a degree of two for the x terms and degree of three for the y terms.

sf = fit([x, y],z,'poly23')
sf =
Linear model Poly23:
sf(x,y) = p00 + p10*x + p01*y + p20*x^2 + p11*x*y + p02*y^2 + p21*x^2*y
+ p12*x*y^2 + p03*y^3
Coefficients (with 95% confidence bounds):
p00 =       1.118  (0.9149, 1.321)
p10 =  -0.0002941  (-0.000502, -8.623e-05)
p01 =       1.533  (0.7032, 2.364)
p20 =  -1.966e-08  (-7.084e-08, 3.152e-08)
p11 =   0.0003427  (-0.0001009, 0.0007863)
p02 =      -6.951  (-8.421, -5.481)
p21 =   9.563e-08  (6.276e-09, 1.85e-07)
p12 =  -0.0004401  (-0.0007082, -0.0001721)
p03 =       4.999  (4.082, 5.917)

sf contains the results of the fit, including coefficient estimates with 95% confidence bounds.

Plot the fit in sf together with a scatterplot of the data.

plot(sf,[x,y],z)

Load the franke data and convert it to a MATLAB® table.

T = table(x,y,z);

Specify the variables in the table as inputs to the fit function, and plot the fit.

f = fit([T.x, T.y],T.z,'linearinterp');
plot( f, [T.x, T.y], T.z )

Load and plot the data, create fit options and fit type using the fittype and fitoptions functions, then create and plot the fit.

Load and plot the data in census.mat.

plot(cdate,pop,'o')

Create a fit options object and a fit type for the custom nonlinear model $y=a\left(x-b{\right)}^{n}$, where a and b are coefficients and n is a problem-dependent parameter.

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

Fit the data using the fit options and a value of n = 2.

[curve2,gof2] = fit(cdate,pop,ft,'problem',2)
curve2 =
General model:
curve2(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 = struct with fields:
sse: 246.1543
rsquare: 0.9980
dfe: 19
rmse: 3.5994

Fit the data using the fit options and a value of n = 3.

[curve3,gof3] = fit(cdate,pop,ft,'problem',3)
curve3 =
General model:
curve3(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 = struct with fields:
sse: 232.0058
rsquare: 0.9981
dfe: 19
rmse: 3.4944

Plot the fit results with the data.

hold on
plot(curve2,'m')
plot(curve3,'c')
legend('Data','n=2','n=3')
hold off

Load the carbon12alpha nuclear reaction sample data set.

angle is a vector of emission angles in radians. counts is a vector of raw alpha particle counts that correspond to the angles in angle.

Display a scatter plot of the counts plotted against the angles.

scatter(angle,counts)

The scatter plot shows that the counts oscillate as the angle increases between 0 and 4.5. To fit a polynomial model to the data, specify the fitType input argument as "poly#" where # is an integer from one to nine. You can fit models of up to nine degrees. See List of Library Models for Curve and Surface Fitting for more information.

Fit a fifth-degree, seventh-degree, and ninth-degree polynomial to the nuclear reaction data. Return the goodness-of-fit statistics for each fit.

[f5,gof5] = fit(angle,counts,"poly5");
[f7,gof7] = fit(angle,counts,"poly7");
[f9,gof9] = fit(angle,counts,"poly9");

Generate a vector of query points between 0 and 4.5 by using the linspace function. Evaluate the polynomial fits at the query points, and then plot them together with the nuclear reaction data.

xq = linspace(0,4.5,1000);

figure
hold on
scatter(angle,counts,"k")
plot(xq,f5(xq))
plot(xq,f7(xq))
plot(xq,f9(xq))
ylim([-100,550])
legend("original data","fifth-degree polynomial","seventh-degree polynomial","ninth-degree polynomial")

The plot indicates that the ninth-degree polynomial follows the data most closely.

Display the goodness-of-fit statistics for each fit by using the struct2table function.

gof = struct2table([gof5 gof7 gof9],RowNames=["f5" "f7" "f9"])
gof=3×5 table
__________    _______    ___    __________    ______

f5    1.0901e+05    0.54614    18      0.42007       77.82
f7         32695    0.86387    16      0.80431      45.204
f9        3660.2    0.98476    14      0.97496      16.169

The sum-of-squares error (SSE) for the ninth-degree polynomial fit is smaller than the SSEs for the fifth-degree and seventh-degree fits. This result confirms that the ninth-degree polynomial follows the data most closely.

Load the census sample data set. Fit a cubic polynomial and specify the Normalize (center and scale) and Robust fitting options.

f = fit(cdate,pop,'poly3','Normalize','on','Robust','Bisquare')
f =
Linear model Poly3:
f(x) = p1*x^3 + p2*x^2 + p3*x + p4
where x is normalized by mean 1890 and std 62.05
Coefficients (with 95% confidence bounds):
p1 =     -0.4619  (-1.895, 0.9707)
p2 =       25.01  (23.79, 26.22)
p3 =       77.03  (74.37, 79.7)
p4 =       62.81  (61.26, 64.37)

Plot the fit.

plot(f,cdate,pop)

Define a function in a file and use it to create a fit type and fit a curve.

Define a function in a MATLAB® file.

type piecewiseLine.m
function y = piecewiseLine(x,a,b,c,k)
% PIECEWISELINE   A line made of two pieces

y = zeros(size(x));

% This example includes a for-loop and if statement
% purely for example purposes.
for i = 1:length(x)
if x(i) < k
y(i) = a + b.*x(i);
else
y(i) = a + b*k + c.*(x(i)-k);
end
end
end

Save the file.

Define some data and create a fit type specifying the function piecewiseLine.

x = [0.81;0.91;0.13;0.91;0.63;0.098;0.28;0.55;...
0.96;0.96;0.16;0.97;0.96];
y = [0.17;0.12;0.16;0.0035;0.37;0.082;0.34;0.56;...
0.15;-0.046;0.17;-0.091;-0.071];
ft = fittype('piecewiseLine( x, a, b, c, k )')
ft =
General model:
ft(a,b,c,k,x) = piecewiseLine( x, a, b, c, k )

The inputs to ft are the coefficients in alphabetical order, followed by the independent variables. See Input Order for Anonymous Functions to learn more.

If you want to control the order of coefficients, then use an anonymous function input. For example, to change the order of the a and b coefficients:

ft = fittype(@(b,a,c,k,x) piecewiseLine(x,a,b,c,k))

You must specify the independent variable x last.

Create a fit using the fit type ft and plot the results.

f = fit(x, y, ft, 'StartPoint', [1, 0, 1, 0.5]);
plot(f, x, y)

Load some data and fit a custom equation specifying points to exclude. Plot the results.

Load data and define a custom equation and some start points.

[x, y] = titanium;

gaussEqn = 'a*exp(-((x-b)/c)^2)+d'
gaussEqn =
'a*exp(-((x-b)/c)^2)+d'
startPoints = [1.5 900 10 0.6]
startPoints = 1×4

1.5000  900.0000   10.0000    0.6000

Create two fits using the custom equation and start points, and define two different sets of excluded points, using an index vector and an expression. Use Exclude to remove outliers from your fit.

f1 = fit(x',y',gaussEqn,'Start', startPoints, 'Exclude', [1 10 25])
f1 =
General model:
f1(x) = a*exp(-((x-b)/c)^2)+d
Coefficients (with 95% confidence bounds):
a =       1.493  (1.432, 1.554)
b =       897.4  (896.5, 898.3)
c =        27.9  (26.55, 29.25)
d =      0.6519  (0.6367, 0.6672)
f2 = fit(x',y',gaussEqn,'Start', startPoints, 'Exclude', x < 800)
f2 =
General model:
f2(x) = a*exp(-((x-b)/c)^2)+d
Coefficients (with 95% confidence bounds):
a =       1.494  (1.41, 1.578)
b =       897.4  (896.2, 898.7)
c =       28.15  (26.22, 30.09)
d =      0.6466  (0.6169, 0.6764)

Plot both fits.

plot(f1,x,y)
title('Fit with data points 1, 10, and 25 excluded')

figure
plot(f2,x,y)
title('Fit with data points excluded such that x < 800')

You can define the excluded points as variables before supplying them as inputs to the fit function. The following steps recreate the fits in the previous example and allow you to plot the excluded points as well as the data and the fit.

Load data and define a custom equation and some start points.

[x, y] = titanium;

gaussEqn = 'a*exp(-((x-b)/c)^2)+d'
gaussEqn =
'a*exp(-((x-b)/c)^2)+d'
startPoints = [1.5 900 10 0.6]
startPoints = 1×4

1.5000  900.0000   10.0000    0.6000

Define two sets of points to exclude, using an index vector and an expression.

exclude1 = [1 10 25];
exclude2 = x < 800;

Create two fits using the custom equation, startpoints, and the two different excluded points.

f1 = fit(x',y',gaussEqn,'Start', startPoints, 'Exclude', exclude1);
f2 = fit(x',y',gaussEqn,'Start', startPoints, 'Exclude', exclude2);

Plot both fits and highlight the excluded data.

plot(f1,x,y,exclude1)
title('Fit with data points 1, 10, and 25 excluded')

figure;
plot(f2,x,y,exclude2)
title('Fit with data points excluded such that x < 800')

For a surface fitting example with excluded points, load some surface data and create and plot fits specifying excluded data.

f1 = fit([x y],z,'poly23', 'Exclude', [1 10 25]);
f2 = fit([x y],z,'poly23', 'Exclude', z > 1);

figure
plot(f1, [x y], z, 'Exclude', [1 10 25]);
title('Fit with data points 1, 10, and 25 excluded')

figure
plot(f2, [x y], z, 'Exclude', z > 1);
title('Fit with data points excluded such that z > 1')

Generate some noisy data using the membrane and randn functions.

n = 41;
M = membrane(1,20)+0.02*randn(n);
[X,Y] = meshgrid(1:n);

The matrix M contains data for the L-shaped membrane with added noise. The matrices X and Y contain the row and column index values, respectively, for the corresponding elements in M.

Display a surface plot of the data.

figure(1)
surf(X,Y,M)

The plot shows a wrinkled L-shaped membrane. The wrinkles in the membrane are caused by the noise in the data.

Fit two surfaces through the wrinkled membrane using linear interpolation. For the first surface, specify the linear extrapolation method. For the second surface, specify the extrapolation method as nearest neighbor.

flinextrap = fit([X(:),Y(:)],M(:),"linearinterp",ExtrapolationMethod="linear");
fnearextrap = fit([X(:),Y(:)],M(:),"linearinterp",ExtrapolationMethod="nearest");

Investigate the differences between the extrapolation methods by using the meshgrid function to evaluate the fits at query points extending outside the convex hull of the X and Y data.

[Xq,Yq] = meshgrid(-10:50);

Zlin = flinextrap(Xq,Yq);
Znear = fnearextrap(Xq,Yq);

Plot the evaluated fits.

figure(2)
surf(Xq,Yq,Zlin)
title("Linear Extrapolation")
xlabel("X")
ylabel("Y")
zlabel("M")

figure(3)
surf(Xq,Yq,Znear)
title("Nearest Neighbor Extrapolation")
xlabel("X")
ylabel("Y")
zlabel("M")

The linear extrapolation method generates spikes outside of the convex hull. The plane segments that form the spikes follow the gradient at points on the convex hull's border. The nearest neighbor extrapolation method uses the data on the border to extend the surface in each direction. This method of extrapolation generates waves that mimic the border.

Fit a smoothing spline curve, and return goodness-of-fit statistics and information about the fitting algorithm.

Load the enso sample data set. The enso sample data set contains data for the monthly averaged atmospheric pressure differences between Easter Island and Darwin, Australia.

Fit a smoothing spline curve to the data in month and pressure, and return goodness-of-fit statistics and the output structure.

[curve,gof,output] = fit(month,pressure,"smoothingspline");

Plot the fitted curve with the data used to fit the curve.

plot(curve,month,pressure);
xlabel("Month");
ylabel("Pressure");

Plot the residuals against the x-data (month).

plot(curve,month,pressure,"residuals")
xlabel("Month")
ylabel("Residuals")

Use the residuals data in the output structure to plot the residuals against the y-data (pressure). To access the residuals field of output, use dot notation.

residuals = output.residuals;
plot( pressure,residuals,".")
xlabel("Pressure")
ylabel("Residuals")

Generate data with an exponential trend, and then fit the data using the first equation in the curve fitting library of exponential models (a single-term exponential). Plot the results.

x = (0:0.2:5)';
y = 2*exp(-0.2*x) + 0.5*randn(size(x));
f = fit(x,y,'exp1');
plot(f,x,y)

You can use anonymous functions to make it easier to pass other data into the fit function.

Load data and set Emax to 1 before defining your anonymous function:

data = importdata( 'OpioidHypnoticSynergy.txt' );
Propofol      = data.data(:,1);
Remifentanil  = data.data(:,2);
Algometry     = data.data(:,3);
Emax = 1;

Define the model equation as an anonymous function:

Effect = @(IC50A, IC50B, alpha, n, x, y) ...
Emax*( x/IC50A + y/IC50B + alpha*( x/IC50A )...
.* ( y/IC50B ) ).^n ./(( x/IC50A + y/IC50B + ...
alpha*( x/IC50A ) .* ( y/IC50B ) ).^n  + 1);

Use the anonymous function Effect as an input to the fit function, and plot the results:

AlgometryEffect = fit( [Propofol, Remifentanil], Algometry, Effect, ...
'StartPoint', [2, 10, 1, 0.8], ...
'Lower', [-Inf, -Inf, -5, -Inf], ...
'Robust', 'LAR' )
plot( AlgometryEffect, [Propofol, Remifentanil], Algometry )

For more examples using anonymous functions and other custom models for fitting, see the fittype function.

For the properties Upper, Lower, and StartPoint, you need to find the order of the entries for coefficients.

Create a fit type.

ft = fittype('b*x^2+c*x+a');

Get the coefficient names and order using the coeffnames function.

coeffnames(ft)
ans = 3x1 cell
{'a'}
{'b'}
{'c'}

Note that this is different from the order of the coefficients in the expression used to create ft with fittype.

Load data, create a fit and set the start points.

fit(month,pressure,ft,'StartPoint',[1,3,5])
ans =
General model:
ans(x) = b*x^2+c*x+a
Coefficients (with 95% confidence bounds):
a =       10.94  (9.362, 12.52)
b =   0.0001677  (-7.985e-05, 0.0004153)
c =     -0.0224  (-0.06559, 0.02079)

This assigns initial values to the coefficients as follows: a = 1, b = 3, c = 5.

Alternatively, you can get the fit options and set start points and lower bounds, then refit using the new options.

options = fitoptions(ft)
options =
nlsqoptions with properties:

StartPoint: []
Lower: []
Upper: []
Algorithm: 'Trust-Region'
DiffMinChange: 1.0000e-08
DiffMaxChange: 0.1000
Display: 'Notify'
MaxFunEvals: 600
MaxIter: 400
TolFun: 1.0000e-06
TolX: 1.0000e-06
Robust: 'Off'
Normalize: 'off'
Exclude: []
Weights: []
Method: 'NonlinearLeastSquares'

options.StartPoint = [10 1 3];
options.Lower = [0 -Inf 0];
fit(month,pressure,ft,options)
ans =
General model:
ans(x) = b*x^2+c*x+a
Coefficients (with 95% confidence bounds):
a =       10.23  (9.448, 11.01)
b =   4.335e-05  (-1.82e-05, 0.0001049)
c =   5.523e-12  (fixed at bound)

## Input Arguments

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Data to fit, specified as a matrix with either one (curve fitting) or two (surface fitting) columns. You can specify variables in a MATLAB® table using tablename.varname. Cannot contain Inf or NaN. Only the real parts of complex data are used in the fit.

Example: x

Example: [x,y]

Data Types: double

Data to fit, specified as a column vector with the same number of rows as x. You can specify a variable in a MATLAB table using tablename.varname. Cannot contain Inf or NaN. Only the real parts of complex data are used in the fit.

Use prepareCurveData or prepareSurfaceData if your data is not in column vector form.

Data Types: double

Data to fit, specified as a column vector with the same number of rows as x. You can specify a variable in a MATLAB table using tablename.varname. Cannot contain Inf or NaN. Only the real parts of complex data are used in the fit.

Use prepareSurfaceData if your data is not in column vector form. For example, if you have 3 matrices, or if your data is in grid vector form, where length(X) = n, length(Y) = m and size(Z) = [m,n].

Data Types: double

Model type to fit, specified as a character vector or string scalar representing a library model name or MATLAB expression, a string array of linear model terms or a cell array of character vectors of such terms, an anonymous function, or a fittype created with the fittype function. You can use any of the valid first inputs to fittype as an input to fit.

For a list of library model names, see Model Names and Equations.

To a fit custom model, use a MATLAB expression, a cell array of linear model terms, or an anonymous function. You can also create a fittype using the fittype function, and then use it as the value of the fitType input argument. For an example, see Fit a Custom Model Using an Anonymous Function. For examples that use linear model terms, see the fittype function.

Example: "poly2"

Algorithm options constructed using the fitoptions function. This is an alternative to specifying name-value pair arguments for fit options.

### Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: Lower=[0,0],Upper=[Inf,max(x)],StartPoint=[1 1] specifies fitting method, bounds, and start points.

Options for All Fitting Methods

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Option to center and scale the data, specified as the comma-separated pair consisting of 'Normalize' and 'on' or 'off'.

Data Types: char

Points to exclude from the fit, specified as the comma-separated pair consisting of 'Exclude' and one of:

• An expression describing a logical vector, e.g., x > 10.

• A vector of integers indexing the points you want to exclude, e.g., [1 10 25].

• A logical vector for all data points where true represents an outlier, created by excludedata.

For an example, see Exclude Points from Fit.

Data Types: logical | double

Weights for the fit, specified as the comma-separated pair consisting of 'Weights' and a vector the same size as the response data y (curves) or z (surfaces).

Data Types: double

Values to assign to the problem-dependent constants, specified as the comma-separated pair consisting of 'problem' and a cell array with one element per problem dependent constant. For details, see fittype.

Data Types: cell | double

Smoothing Options

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Smoothing parameter, specified as the comma-separated pair consisting of 'SmoothingParam' and a scalar value between 0 and 1. The default value depends on the data set. Only available if the fit type is smoothingspline.

Data Types: double

Proportion of data points to use in local regressions, specified as the comma-separated pair consisting of 'Span' and a scalar value between 0 and 1. Only available if the fit type is lowess or loess.

Data Types: double

Interpolation Options

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Extrapolation method for an interpolant fit, specified as one of the following values.

ValueDescriptionSupported Fits
"auto"

Default value for all interpolant fit types. Set ExtrapolationMethod to "auto" to automatically assign an extrapolation method during fitting.

All interpolant fit types and cubicspline curve fits

"none"

No extrapolation. Query points outside of the convex hull of the fitting data evaluate to NaN.

fit sets the extrapolation method to "none" when you set ExtrapolationMethod to "auto" for cubicinterp and linearinterp surface fits.

Curve fits — cubicspline and pchipinterp

Surface fits — naturalinterp

Curve and surface fits — cubicinterp, linearinterp, and nearestinterp

"linear"

Linear extrapolation based on boundary gradients.

fit sets the extrapolation method to "linear" when you set ExtrapolationMethod to "auto" for "linearinterp" curve fits.

Surface fits — cubicinterp, nearestinterp, and naturalinterp

Curve and surface fits — linearinterp

"nearest"

Nearest neighbor extrapolation. This method evaluates to the value of the nearest point on the boundary of the fitting data's convex hull.

fit sets the extrapolation method to "nearest" when you set ExtrapolationMethod to "auto" for "nearestinterp" curve and surface fits.

Curve fits — cubicspline and pchipinterp

Surface fits — naturalinterp

Curve and surface fits — cubicinterp, linearinterp, and nearestinterp

"thinplate"

Thin-plate spline extrapolation. This method extends the thin-plate interpolating spline outside of the fitting data's convex hull. For more information, see tpaps.

fit sets the extrapolation method to "thinplate" when you set ExtrapolationMethod to "auto" for "thinplateinterp" surface fits.

Surface fits — thinplateinterp

"biharmonic"

Biharmonic spline extrapolation. This method extends the biharmonic interpolating spline outside of the fitting data's convex hull.

fit sets the extrapolation method to "biharmonic" when you set ExtrapolationMethod to "auto" for "biharmonicinterp" surface fits.

Surface fits — biharmonicinterp

"pchip"

Piecewise cubic hermite interpolating polynomial (PCHIP) extrapolation. This method extends a shape-preserving PCHIP outside of the fitting data's convex hull. For more information, see pchip.

fit sets the extrapolation method to "pchip" when you set ExtrapolationMethod to "auto" for pchipinterp curve fits.

Curve fits — pchipinterp

"cubic"

Cubic spline extrapolation. This method extends a cubic interpolating spline outside of the fitting data's convex hull.

fit sets the extrapolation method to "cubic" when you set ExtrapolationMethod to "auto" for cubicinterp and cubicspline curve fits. For more information, see spline.

Curve fits — cubicinterp and cubicspline

Data Types: char | string

Linear and Nonlinear Least-Squares Options

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Robust linear least-squares fitting method, specified as the comma-separated pair consisting of 'Robust' and one of these values:

• 'LAR' specifies the least absolute residual method.

• 'Bisquare' specifies the bisquare weights method.

Available when the fit type Method is LinearLeastSquares or NonlinearLeastSquares.

Data Types: char

Lower bounds on the coefficients to be fitted, specified as the comma-separated pair consisting of 'Lower' and a vector. The default value is an empty vector, indicating that the fit is unconstrained by lower bounds. If bounds are specified, the vector length must equal the number of coefficients. Find the order of the entries for coefficients in the vector value by using the coeffnames function. For an example, see Find Coefficient Order to Set Start Points and Bounds. Individual unconstrained lower bounds can be specified by -Inf.

Available when the Method is LinearLeastSquares or NonlinearLeastSquares.

Data Types: double

Upper bounds on the coefficients to be fitted, specified as the comma-separated pair consisting of 'Upper' and a vector. The default value is an empty vector, indicating that the fit is unconstrained by upper bounds. If bounds are specified, the vector length must equal the number of coefficients. Find the order of the entries for coefficients in the vector value by using the coeffnames function. For an example, see Find Coefficient Order to Set Start Points and Bounds. Individual unconstrained upper bounds can be specified by +Inf.

Available when the Method is LinearLeastSquares or NonlinearLeastSquares.

Data Types: logical

Nonlinear Least-Squares Options

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Initial values for the coefficients, specified as the comma-separated pair consisting of 'StartPoint' and a vector. Find the order of the entries for coefficients in the vector value by using the coeffnames function. For an example, see Find Coefficient Order to Set Start Points and Bounds.

If no start points (the default value of an empty vector) are passed to the fit function, starting points for some library models are determined heuristically. For rational and Weibull models, and all custom nonlinear models, the toolbox selects default initial values for coefficients uniformly at random from the interval (0,1). As a result, multiple fits using the same data and model might lead to different fitted coefficients. To avoid this, specify initial values for coefficients with a fitoptions object or a vector value for the StartPoint value.

Available when the Method is NonlinearLeastSquares.

Data Types: double

Algorithm to use for the fitting procedure, specified as the comma-separated pair consisting of 'Algorithm' and either 'Levenberg-Marquardt' or 'Trust-Region'.

Available when the Method is NonlinearLeastSquares.

Data Types: char

Maximum change in coefficients for finite difference gradients, specified as the comma-separated pair consisting of 'DiffMaxChange' and a scalar.

Available when the Method is NonlinearLeastSquares.

Data Types: double

Minimum change in coefficients for finite difference gradients, specified as the comma-separated pair consisting of 'DiffMinChange' and a scalar.

Available when the Method is NonlinearLeastSquares.

Data Types: double

Display option in the command window, specified as the comma-separated pair consisting of 'Display' and one of these options:

• 'notify' displays output only if the fit does not converge.

• 'final' displays only the final output.

• 'iter' displays output at each iteration.

• 'off' displays no output.

Available when the Method is NonlinearLeastSquares.

Data Types: char

Maximum number of evaluations of the model allowed, specified as the comma-separated pair consisting of 'MaxFunEvals' and a scalar.

Available when the Method is NonlinearLeastSquares.

Data Types: double

Maximum number of iterations allowed for the fit, specified as the comma-separated pair consisting of 'MaxIter' and a scalar.

Available when the Method is NonlinearLeastSquares.

Data Types: double

Termination tolerance on the model value, specified as the comma-separated pair consisting of 'TolFun' and a scalar.

Available when the Method is NonlinearLeastSquares.

Data Types: double

Termination tolerance on the coefficient values, specified as the comma-separated pair consisting of 'TolX' and a scalar.

Available when the Method is NonlinearLeastSquares.

Data Types: double

## Output Arguments

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Fit result, returned as a cfit (for curves) or sfit (for surfaces) object. See Fit Postprocessing for functions for plotting, evaluating, calculating confidence intervals, integrating, differentiating, or modifying your fit object.

Goodness-of-fit statistics, returned as the gof structure including the fields in this table.

Field

Value

sse

Sum of squares due to error

rsquare

R-squared (coefficient of determination)

dfe

Degrees of freedom in the error

rmse

Root mean squared error (standard error)

Example: gof.rmse

Fitting algorithm information, returned as the output structure containing information associated with the fitting algorithm.

Fields depend on the algorithm. For example, the output structure for nonlinear least-squares algorithms includes the fields shown in this table.

Field

Value

numobs

Number of observations (response values)

numparam

Number of unknown parameters (coefficients) to fit

residuals

Vector of raw residuals (observed values minus the fitted values)

Jacobian

Jacobian matrix

exitflag

Describes the exit condition of the algorithm. Positive flags indicate convergence, within tolerances. Zero flags indicate that the maximum number of function evaluations or iterations was exceeded. Negative flags indicate that the algorithm did not converge to a solution.

iterations

Number of iterations

funcCount

Number of function evaluations

firstorderopt

Measure of first-order optimality (absolute maximum of gradient components)

algorithm

Fitting algorithm employed

message

Exit message

Example: output.Jacobian

## Version History

Introduced before R2006a

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