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Fit type for curve and surface fitting

`aFittype = fittype(libraryModelName)`

`aFittype = fittype(expression)`

`aFittype = fittype(expression,Name,Value)`

`aFittype = fittype(linearModelTerms)`

`aFittype = fittype(linearModelTerms,Name,Value)`

`aFittype = fittype(anonymousFunction)`

`aFittype = fittype(anonymousFunction,Name,Value)`

creates the `aFittype`

= fittype(`libraryModelName`

)`fittype`

object `aFittype`

for the model specified by `libraryModelName`

.

creates a fit type for the model specified by the MATLAB`aFittype`

= fittype(`expression`

)^{®} expression.

constructs the fit type with additional options specified by one or more
`aFittype`

= fittype(`expression`

,`Name,Value`

)`Name,Value`

pair arguments.

creates a fit type for a custom linear model with terms specified by the
cell array of character vector expressions in
`aFittype`

= fittype(`linearModelTerms`

)`linearModelTerms`

.

constructs the fit type with additional options specified by one or more
`aFittype`

= fittype(`linearModelTerms`

,`Name,Value`

)`Name,Value`

pair arguments.

creates a fit type for the model specified by
`aFittype`

= fittype(`anonymousFunction`

)`anonymousFunction`

.

constructs the fit type with additional options specified by one or more
`aFittype`

= fittype(`anonymousFunction`

,`Name,Value`

)`Name,Value`

pair arguments.

Construct fit types by specifying library model names.

Construct a `fittype`

object for the cubic polynomial library model.

`f = fittype('poly3')`

f = Linear model Poly3: f(p1,p2,p3,p4,x) = p1*x^3 + p2*x^2 + p3*x + p4

Construct a fit type for the library model `rat33`

(a rational model of the third degree for both the numerator and denominator).

`f = fittype('rat33')`

f = General model Rat33: f(p1,p2,p3,p4,q1,q2,q3,x) = (p1*x^3 + p2*x^2 + p3*x + p4) / (x^3 + q1*x^2 + q2*x + q3)

For a list of library model names, see `libraryModelName`

.

To use a linear fitting algorithm, specify a cell array of terms.

Identify the linear model terms you need to input to `fittype`

: `a*x + b*sin(x) + c`

. The model is linear in `a`

, `b`

and `c`

. It has three terms `x`

, `sin(x)`

and `1`

(because `c=c*1`

). To specify this model you use this cell array of terms: `LinearModelTerms = {'x','sin(x)','1'}`

.

Use the cell array of linear model terms as the input to `fittype`

.

ft = fittype({'x','sin(x)','1'})

ft = Linear model: ft(a,b,c,x) = a*x + b*sin(x) + c

Create a linear model fit type for `a*cos(x) + b`

.

ft2 = fittype({'cos(x)','1'})

ft2 = Linear model: ft2(a,b,x) = a*cos(x) + b

Create the fit type again and specify coefficient names.

ft3 = fittype({'cos(x)','1'},'coefficients',{'a1','a2'})

ft3 = Linear model: ft3(a1,a2,x) = a1*cos(x) + a2

Construct fit types for custom nonlinear models, designating problem-dependent parameters and independent variables.

Construct a fit type for a custom nonlinear model, designating `n`

as a problem-dependent parameter and `u`

as the independent variable.

g = fittype('a*u+b*exp(n*u)',... 'problem','n',... 'independent','u')

g = General model: g(a,b,n,u) = a*u+b*exp(n*u)

Construct a fit type for a custom nonlinear model, designating `time`

as the independent variable.

g = fittype('a*time^2+b*time+c','independent','time','dependent','height')

g = General model: g(a,b,c,time) = a*time^2+b*time+c

Construct a fit type for a logarithmic fit to some data, use the fit type to create a fit, and plot the fit.

x = linspace(1,100); y = 5 + 7*log(x); myfittype = fittype('a + b*log(x)',... 'dependent',{'y'},'independent',{'x'},... 'coefficients',{'a','b'})

myfittype = General model: myfittype(a,b,x) = a + b*log(x)

myfit = fit(x',y',myfittype)

Warning: Start point not provided, choosing random start point.

myfit = General model: myfit(x) = a + b*log(x) Coefficients (with 95% confidence bounds): a = 5 (5, 5) b = 7 (7, 7)

plot(myfit,x,y)

You can specify any MATLAB command and therefore any `.m`

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.

function y = piecewiseLine(x,a,b,c,d,k) % PIECEWISELINE A line made of two pieces % that is not continuous. 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) = c + d.* x(i); end end end

Save the file.

Define some data, create a fit type specifying the function
`piecewiseLine`

, create a fit using the fit type
`ft`

, and plot the results.

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

To use a linear fitting algorithm, specify a cell array of terms.

Identify the linear model terms you need to input to `fittype`

: `a*x + b*sin(x) + c`

. The model is linear in `a`

, `b`

and `c`

. It has three terms `x`

, `sin(x)`

and `1`

(because `c=c*1`

). To specify this model you use this cell array of terms: `LinearModelTerms = {'x','sin(x)','1'}`

.

Use the cell array of linear model terms as the input to `fittype`

.

ft = fittype({'x','sin(x)','1'})

ft = Linear model: ft(a,b,c,x) = a*x + b*sin(x) + c

Create a linear model fit type for `a*cos(x) + b`

.

ft2 = fittype({'cos(x)','1'})

ft2 = Linear model: ft2(a,b,x) = a*cos(x) + b

Create the fit type again and specify coefficient names.

ft3 = fittype({'cos(x)','1'},'coefficients',{'a1','a2'})

ft3 = Linear model: ft3(a1,a2,x) = a1*cos(x) + a2

Create a fit type using an anonymous function.

g = fittype( @(a, b, c, x) a*x.^2+b*x+c )

Create a fit type using an anonymous function and specify independent and dependent parameters.

g = fittype( @(a, b, c, d, x, y) a*x.^2+b*x+c*exp(... -(y-d).^2 ), 'independent', {'x', 'y'},... 'dependent', 'z' );

Create a fit type for a surface using an anonymous function and
specify independent and dependent parameters, and problem parameters
that you will specify later when you call `fit`

.

g = fittype( @(a,b,c,d,x,y) a*x.^2+b*x+c*exp( -(y-d).^2 ), ... 'problem', {'c','d'}, 'independent', {'x', 'y'}, ... 'dependent', 'z' );

Use an anonymous function to pass workspace data into the
`fittype`

and `fit`

functions.

Create and plot an S-shaped curve. In later steps, you stretch and move this curve to fit to some data.

% Breakpoints. xs = (0:0.1:1).'; % Height of curve at breakpoints. ys = [0; 0; 0.04; 0.1; 0.2; 0.5; 0.8; 0.9; 0.96; 1; 1]; % Plot S-shaped curve. xi = linspace( 0, 1, 241 ); plot( xi, interp1( xs, ys, xi, 'pchip' ), 'LineWidth', 2 ) hold on plot( xs, ys, 'o', 'MarkerFaceColor', 'r' ) hold off title S-curve

Create a fit type using an anonymous function, taking the values from
the workspace for the curve breakpoints (`xs`

) and the
height of the curve at the breakpoints (`ys`

).
Coefficients are `b`

(base) and `h`

(height).

`ft = fittype( @(b, h, x) interp1( xs, b+h*ys, x, 'pchip' ) )`

Plot the `fittype`

specifying example coefficients of
base `b=1.1`

and height `h=-0.8`

.

plot( xi, ft( 1.1, -0.8, xi ), 'LineWidth', 2 ) title 'Fittype with b=1.1 and h=-0.8'

Load and fit some data, using the fit type `ft`

created using workspace values.

% 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] ) % Plot fit plot( f, xdata, ydata ) title 'Fitted S-curve'

This example shows the differences between using anonymous functions with problem parameters and workspace variable values.

Load data, create a fit type for a curve using an anonymous function
with problem parameters, and call `fit`

specifying the
problem parameters.

% Load some data. xdata = [0.098;0.13;0.16;0.28;0.55;0.63;0.81;0.91;0.91;... 0.96;0.96;0.96;0.97]; ydata = [0.52;0.53;0.53;0.48;0.33;0.36;0.39;0.28;0.28;... 0.21;0.21;0.21;0.2]; % Create a fittype that has a problem parameter. g = fittype( @(a,b,c,x) a*x.^2+b*x+c, 'problem', 'c' ) % Examine coefficients. Observe c is not a coefficient. coeffnames( g ) % Examine arguments. Observe that c is an argument. argnames( g ) % Call fit and specify the value of c. f1 = fit( xdata, ydata, g, 'problem', 0, 'StartPoint', [1, 2] ) % Note: Specify start points in the calls to fit to % avoid warning messages about random start points % and to ensure repeatability of results. % Call fit again and specify a different value of c, % to get a new fit. f2 = fit( xdata, ydata, g, 'problem', 1, 'start', [1, 2] ) % Plot results. Observe the specified c constants % do not make a good fit. plot( f1, xdata, ydata ) hold on plot( f2, 'b' ) hold off

Modify the previous example to create the same fits using workspace
values for variables, instead of using problem parameters. Using the
same data, create a fit type for a curve using an anonymous function
with a workspace value for variable `c`

:

% Remove c from the argument list. try g = fittype( @(a,b,x) a*x.^2+b*x+c ) catch e disp( e.message ) end % Observe error because now c is undefined. % Define c and create fittype: c = 0; g1 = fittype( @(a,b,x) a*x.^2+b*x+c ) % Call fit (now no need to specify problem parameter). f1 = fit( xdata, ydata, g1, 'StartPoint', [1, 2] ) % Note that this f1 is the same as the f1 above. % To change the value of c, recreate the fittype. c = 1; g2 = fittype( @(a,b,x) a*x.^2+b*x+c ) % uses c = 1 f2 = fit( xdata, ydata, g2, 'StartPoint', [1, 2] ) % Note that this f2 is the same as the f2 above. % Plot results plot( f1, xdata, ydata ) hold on plot( f2, 'b' ) hold off

`libraryModelName`

— Library model to fitcharacter vector

Library model to fit, specified as a character vector. This table shows some common examples.

Library Model Name | Description |
---|---|

| Linear polynomial curve |

| Linear polynomial surface |

| Quadratic polynomial curve |

| Piecewise linear interpolation |

| Piecewise cubic interpolation |

| Smoothing spline (curve) |

| Local linear regression (surface) |

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

**Example: **`'poly2'`

**Data Types: **`char`

`expression`

— Model to fitcharacter vector

Model to fit, specified as a character vector. You can specify any
MATLAB command and therefore any `.m`

file. See
Fit a Curve Defined by a File.

**Data Types: **`char`

`linearModelTerms`

— Model to fitcell array of character vectors

Model to fit, specified as a cell array of character vectors. Specify the model terms by the expressions in the character vectors. Do not include coefficients in the expressions for the terms. See Linear Model Terms.

**Data Types: **`cell`

`anonymousFunction`

— Model to fitanonymous function

Model to fit, specified as an anonymous function. For details, see Input Order for Anonymous Functions.

**Data Types: **`char`

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside quotes. You can specify several name and value
pair arguments in any order as `Name1,Value1,...,NameN,ValueN`

.

`'coefficients',{'a1','a2'}`

`'coefficients'`

— Coefficient namescharacter vector | cell array of character vectors

Coefficient names, specified as the comma-separated pair
consisting of `'coefficients'`

and a character
vector, or a cell array of character vectors for multiple names. You
can use multicharacter symbol names. You cannot use these names:
`i`

, `j`

,
`pi`

, `inf`

,
`nan`

, `eps`

.

**Data Types: **`char`

| `cell`

`'dependent'`

— Dependent (response) variable name`y`

(default) | character vectorDependent (response) variable name, specified as the
comma-separated pair consisting of `'dependent'`

and a character vector. If you do not specify the dependent
variable, the function assumes `y`

is the dependent
variable.

**Data Types: **`char`

`'independent'`

— Independent (response) variable names`x`

(default) | character vector | cell array of character vectorsIndependent (response) variable names, specified as the
comma-separated pair consisting of `'independent'`

and a character vector or cell array of character vectors. If you do
not specify the independent variable, the function assumes
`x`

is the independent variable.

**Data Types: **`char`

`'options'`

— Fit options`fitoptions`

Fit options, specified as the comma-separated pair consisting of
`'options'`

and the name of a
`fitoptions`

object.

`'problem'`

— Problem-dependent (fixed) parameter namescell array

Problem-dependent (fixed) parameter names, specified as the
comma-separated pair consisting of `'problem'`

and
a character vector, or cell array of character vectors with one
element per problem dependent constant.

**Data Types: **`char`

| `cell`

`aFittype`

— Model to fit`fittype`

objectModel to fit, returned as a `fittype`

. A
`fittype`

encapsulates information describing a
model. To create a fit, you need data, a `fittype`

, and
(optionally) `fitoptions`

and an
exclusion rule. You can use a `fittype`

as an input to
the `fit`

function.

How do I decide which variables are dependent and independent?

To determine dependent and independent variables and coefficients, consider this equation:

$$y=f(x)=a+(b*x)+(c*{x}^{2})$$.

*y*is the dependent variable.*x*is the independent variable.*a*,*b*, and*c*are the coefficients.

The `'independent'`

variable is what you control. The
`'dependent'`

variable is what you measure, i.e., it
depends on the independent variable. The `'coefficients'`

are
the parameters that the fitting algorithm estimates.

For example, if you have census data, then the year is the independent
variable because it does not depend on anything. Population is the dependent
variable, because its value depends on the year in which the census is taken. If
a parameter like growth rate is part of the model, so the fitting algorithm
estimates it, then the parameter is one of the
`'coefficients'`

.

The `fittype`

function determines input arguments by
searching the fit type expression input for variable names.
`fittype`

assumes `x`

is the independent
variable, `y`

is the dependent variable, and all other
variables are coefficients of the model. `x`

is used if no
variable exists.

If the fit type expression input is an anonymous function,
then the order of inputs must be correct. The input order enables the
`fittype`

function 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:

fcn = @(coefficients,problemparameters,x,y) expression

`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`

to name the
independent variables, then specify different names using the
`'independent'`

argument 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.

Create a fit type using an anonymous function and a variable value (

`c`

) from the workspace.c = 1; g = fittype( @(a, b, x) a*x.^2+b*x+c )

The

`fittype`

function can use the variable values in your workspace when you create the fit type. To pass in new data from the workspace, recreate the fit type, e.g.,`c = 5 % Change value of c. g = fittype( @(a, b, x) a*x.^2+b*x+c )`

Here, the value of

`c`

is fixed when you create the fit type. To specify the value of`c`

at the time you call`fit`

, you can use problem parameters. For example, make a fit with`c = 2`

and then a new fit with`c = 3`

.g = fittype( @(a,b,x) a*x.^2+b*x+c, 'problem', 'c' ) f1 = fit( xdata, ydata, g, 'problem', 2 ) f2 = fit( xdata, ydata, g, 'problem', 3 )

How do I define linear model terms?

To use a linear fitting algorithm, specify
`linearModelTerms`

as a cell array of terms.

afittype = fittype({expr1,...,exprn})

`expr2,...,exprn`

. Do not include coefficients in the
expressions for the terms. If there is a constant term, use
`'1'`

as the corresponding expression in the cell
array.To specify a linear model of the following form:

```
coeff1 * term1 + coeff2 * term2 + coeff3 * term3 + ...
```

`term1`

,
`term2`

, etc., use a cell array where each term, without
coefficients, is specified in a cell of `expr`

, as
follows:LinearModelTerms = {'term1', 'term2', 'term3', ... }

For example, the model

a*x + b*sin(x) + c

`a`

, `b`

, and `c`

.
It has three terms `x`

, `sin(x)`

and
`1`

(because c=c*1) and therefore `expr`

is:LinearModelTerms = {'x','sin(x)','1'}

In the Curve Fitting app, see the `Linear Fitting`

model type.

If the fit type expression input is a character vector or anonymous function, then the toolbox uses a nonlinear fitting algorithm to fit the model to data.

If the fit type expression input is a cell array of terms, then the toolbox uses a linear fitting algorithm to fit the model to data.

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