# fit

Fit curve or surface to data

## Syntax

## Description

creates a fit to the data using the algorithm options specified by the `fitobject`

= fit(`x`

,`y`

,`fitType`

,`fitOptions`

)`fitOptions`

object.

creates a fit to the data using the library model `fitobject`

= fit(`x`

,`y`

,`fitType`

,`Name=Value`

)`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.

## Examples

### Fit a Quadratic Curve

Load the `census`

sample data set.

`load census;`

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.

### Fit a Polynomial Surface

Load the `franke`

sample data set.

`load franke`

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)

### Fit a Surface Using Variables in a MATLAB Table

Load the `franke`

data and convert it to a MATLAB® table.

```
load franke
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 )
```

### Create Fit Options and Fit Type Before Fitting

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`

.

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

Create a fit options object and a fit type for the custom nonlinear model $$y=a(x-b{)}^{n}$$, 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
adjrsquare: 0.9979
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
adjrsquare: 0.9980
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

### Fit Multiple Polynomials

Load the `carbon12alpha`

nuclear reaction sample data set.

`load carbon12alpha`

`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*
sse rsquare dfe adjrsquare rmse
__________ _______ ___ __________ ______
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.

### Specify Normalize and Robust Options

Load the `census`

sample data set. Fit a cubic polynomial and specify the `Normalize`

(center and scale) and `Robust`

fitting options.

load census; 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)

**Fit a Curve Defined by a File**

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)
```

### Exclude Points from Fit

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')
```

### Exclude Points and Plot Fit Showing Excluded Data

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.

load franke 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')

### Compare Extrapolation Methods

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.

### Return Goodness-of-Fit Statistics and Fitting Algorithm Information

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.

`load enso;`

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")

### Fit a Single-Term Exponential

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)
```

### Fit a Custom Model Using an Anonymous Function

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.

### Find Coefficient Order to Set Start Points and Bounds

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.

load enso 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

`x`

— Data to fit

matrix

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`

`y`

— Data to fit

vector

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`

`z`

— Data to fit

vector

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`

`fitType`

— Model type to fit

character vector | string scalar | string array | cell array of character vectors | anonymous function | `fittype`

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"`

`fitOptions`

— Algorithm options

`fitoptions`

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**

`Normalize`

— Option to center and scale data

`'off'`

(default) | `'on'`

Option to center and scale the data, specified as the comma-separated pair consisting of `'Normalize'`

and `'on'`

or `'off'`

.

**Data Types: **`char`

`Exclude`

— Points to exclude from fit

expression | index vector | logical vector | empty

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`

`problem`

— Values to assign to problem-dependent constants

cell array | 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**

`SmoothingParam`

— Smoothing parameter

scalar value in the range (0,1)

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`

`Span`

— Proportion of data points to use in local regressions

0.25 (default) | scalar value in the range (0,1)

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**

`ExtrapolationMethod`

— Extrapolation method

`"auto"`

(default) | `"none"`

| `"linear"`

| `"nearest"`

| `"thinplate"`

| `"biharmonic"`

| `"pchip"`

| `"cubic"`

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

Value | Description | Supported Fits |
---|---|---|

`"auto"` | Default value for all interpolant fit types. Set | All interpolant fit types and |

`"none"` | No extrapolation. Query points outside of the convex hull of the fitting data evaluate to | Curve fits — Surface fits — Curve and surface fits — |

`"linear"` | Linear extrapolation based on boundary gradients. | Surface fits — Curve and surface fits — |

`"nearest"` | Nearest neighbor extrapolation. This method evaluates to the value of the nearest point on the boundary of the fitting data's convex hull.
| Curve fits — Surface fits — Curve and surface fits — |

`"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 | Surface fits — |

`"biharmonic"` | Biharmonic spline extrapolation. This method extends the biharmonic interpolating spline outside of the fitting data's convex hull. | Surface fits — |

`"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 | Curve fits — |

`"cubic"` | Cubic spline extrapolation. This method extends a cubic interpolating spline outside of the fitting data's convex hull.
| Curve fits — |

**Data Types: **`char`

| `string`

**Linear and Nonlinear Least-Squares Options**

`Robust`

— Robust linear least-squares fitting method

`'off'`

(default) | `LAR`

| `Bisquare`

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`

— Lower bounds on coefficients to be fitted

[ ] (default) | vector

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`

— Upper bounds on coefficients to be fitted

[ ] (default) | vector

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**

`StartPoint`

— Initial values for the coefficients

[ ] (default) | vector

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`

— Algorithm to use for fitting procedure

'Trust-Region' (default) | 'Levenberg-Marquardt'

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`

`DiffMaxChange`

— Maximum change in coefficients for finite difference gradients

0.1 (default)

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`

`DiffMinChange`

— Minimum change in coefficients for finite difference gradients

10^{–8} (default)

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`

— Display option in Command Window

`'notify'`

(default) | `'final'`

| `'iter'`

| `'off'`

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`

`MaxFunEvals`

— Maximum number of evaluations of model allowed

`600`

(default)

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`

`MaxIter`

— Maximum number of iterations allowed for fit

`400`

(default)

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`

`TolFun`

— Termination tolerance on model value

10^{–6} (default)

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`

`TolX`

— Termination tolerance on coefficient values

10^{–6} (default)

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

`fitobject`

— Fit result

`cfit`

| `sfit`

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.

`gof`

— Goodness-of-fit statistics

`gof`

structure

Goodness-of-fit statistics, returned as the `gof`

structure including the fields in this table.

Field | Value |
---|---|

| Sum of squares due to error |

| R-squared (coefficient of determination) |

| Degrees of freedom in the error |

| Degree-of-freedom adjusted coefficient of determination |

| Root mean squared error (standard error) |

**Example: **`gof.rmse`

`output`

— Fitting algorithm information

`output`

structure

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 |
---|---|

| Number of observations (response values) |

| Number of unknown parameters (coefficients) to fit |

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

| Jacobian matrix |

| 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. |

| Number of iterations |

| Number of function evaluations |

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

| Fitting algorithm employed |

| Exit message |

**Example: **`output.Jacobian`

## Version History

**Introduced before R2006a**

### R2024a: Specify natural neighbor interpolant surface fits

Starting in R2024a, you can create natural neighbor interpolant surface fits. For more information, see List of Library Models for Curve and Surface Fitting.

### R2023b: Specify extrapolation method for curve interpolant fits

Starting in R2023b, you can specify additional extrapolation methods for interpolant curve
fits by using the `ExtrapolationMethod`

name-value argument. For
more information, see Extrapolation for Interpolant Fit Types.

### R2023b: Specify sigmoidal and logarithmic fit types

Starting in R2023b, you can specify sigmoidal and logarithmic fit types for curve fits. For more information, see List of Library Models for Curve and Surface Fitting.

### R2023a: Specify extrapolation method for surface interpolant fits

Starting in R2023a, you can specify the extrapolation method for interpolant fits by using the
`ExtrapolationMethod`

name-value argument. For curve fits,
Curve Fitting Toolbox™ supports only the default extrapolation methods available in previous
releases.

## See Also

### Apps

### Functions

`fittype`

|`fitoptions`

|`prepareCurveData`

|`prepareSurfaceData`

|`feval`

|`plot`

|`confint`

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