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Fit type for curve and surface fitting

- example
`aFittype = fittype(libraryModelName)`

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

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

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. Oberve 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 single 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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