Parametric Fitting
Parametric Fitting with Library Models
Parametric fitting involves finding coefficients (parameters) for one or more models that you fit to data. The data is assumed to be statistical in nature and is divided into two components:
data = deterministic component + random component
The deterministic component is given by a parametric model and the random component is often described as error associated with the data:
data = parametric model + error
The model is a function of the independent (predictor) variable and one or more coefficients. The error represents random variations in the data that follow a specific probability distribution (usually Gaussian). The variations can come from many different sources, but are always present at some level when you are dealing with measured data. Systematic variations can also exist, but they can lead to a fitted model that does not represent the data well.
The model coefficients often have physical significance. For example, suppose you collected data that corresponds to a single decay mode of a radioactive nuclide, and you want to estimate the halflife (T_{1/2}) of the decay. The law of radioactive decay states that the activity of a radioactive substance decays exponentially in time. Therefore, the model to use in the fit is given by
$$y={y}_{0}{e}^{\lambda t}$$
where y_{0} is the number of nuclei at time t = 0, and λ is the decay constant. The data can be described by
$$\text{data}={y}_{0}{e}^{\lambda t}+\text{error}$$
Both y_{0} and λ are coefficients that are estimated by the fit. Because T_{1/2 }= ln(2)/λ, the fitted value of the decay constant yields the fitted halflife. However, because the data contains some error, the deterministic component of the equation cannot be determined exactly from the data. Therefore, the coefficients and halflife calculation will have some uncertainty associated with them. If the uncertainty is acceptable, then you are done fitting the data. If the uncertainty is not acceptable, then you might have to take steps to reduce it either by collecting more data or by reducing measurement error and collecting new data and repeating the model fit.
With other problems where there is no theory to dictate a model, you might also modify the model by adding or removing terms, or substitute an entirely different model.
The Curve Fitting Toolbox™ parametric library models are described in the following sections.
Select Model Type
Select Model Type Interactively
Open the Curve Fitter app by entering curveFitter
at the MATLAB^{®} command line. Alternatively, on the Apps tab, in the Math, Statistics and Optimization group, click
Curve Fitter.
In the Curve Fitter app, go to the Fit Type section of the Curve Fitter tab. You can select a model type from the fit gallery. Click the arrow to open the gallery.
This table describes the models that you can fit for curves and surfaces.
Fit Group  Fit Category  Curves  Surfaces 

Regression Models  Polynomial  Yes (up to degree 9)  Yes (up to degree 5) 
Exponential  Yes  No  
Fourier  Yes  No  
Gaussian  Yes  No  
Power  Yes  No  
Rational  Yes  No  
Sum of Sine  Yes  No  
Weibull  Yes  No  
Interpolation  Interpolant  Yes, with methods:
 Yes, with methods:

Smoothing  Smoothing Spline  Yes  No 
Lowess  No  Yes  
Custom  Custom Equation  Yes  Yes 
Custom Linear Fitting  Yes  No 
The Results pane displays the model specifications, coefficient values, and goodnessoffit statistics.
Tip
If your fit has problems, messages in the Results pane help you identify better settings.
The Curve Fitter app provides a selection of fit types and settings in the Fit Options pane that you can change to try to improve your fit. Try the defaults first, and then experiment with other settings. For more details on how to use the available fit options, see Specify Fit Options and Optimized Starting Points.
You can try a variety of settings for a single fit and you can create multiple fits to compare. When you create multiple fits in the Curve Fitter app, you can compare different fit types and settings side by side. For more information, see Create Multiple Fits in Curve Fitter App.
Select Model Type Programmatically
You can specify a library model name as a character vector or string
scalar when you call the fit
function. For
example, you can specify a quadratic poly2
model:
f = fit(x,y,"poly2")
To view all available library model names, see List of Library Models for Curve and Surface Fitting to view all available library model names.
You can also use the fittype
function
to construct a fittype
object for a library model,
and use the fittype
as an input to the
fit
function.
Use the fitoptions
function to find out what parameters you can set, for
example:
fitoptions(poly2)
For examples, see the sections for each model type, listed in the table in Select Model Type Interactively. For details on all the functions for creating and analysing models, see Curve and Surface Fitting.
Center and Scale Data
Most fits in the Curve Fitter app provide the Center and
scale option in the Fit Options pane.
When you select this option, the app refits the model with the data centered
and scaled. At the command line, use the fitoptions
function with
the Normalize
option set to
'on'
.
To alleviate numerical problems with variables of different scales, normalize the input data (also known as predictor data). For example, suppose your surface fit inputs are engine speed with a range of 500–4500 r/min and engine load percentage with a range of 0–1. Then, Center and scale generally improves the fit because of the great difference in scale between the two inputs. However, if your inputs are in the same units or similar scale (for example, eastings and northings for geographic data), then Center and scale is less useful. When you normalize inputs with this option, the values of the fitted coefficients change when compared to the original data.
If you are fitting a curve or surface to estimate coefficients, or the coefficients have physical significance, clear the Center and scale check box. The plots in the Curve Fitter app always use the original scale, regardless of the Center and scale status.
At the command line, to center and scale the data before fitting, create the
options
structure by using the
fitoptions
function with
options.Normal
specified as
'on'
. Then, use the fit
function
with the specified options.
options = fitoptions; options.Normal = 'on'; options options = Normalize: 'on' Exclude: [1x0 double] Weights: [1x0 double] Method: 'None' load census f1 = fit(cdate,pop,"poly3",options)
Specify Fit Options and Optimized Starting Points
Fit Options in Curve Fitter App
In the Curve Fitter app, you can specify fit options interactively in the Fit Options pane. All fits except Interpolant and Smoothing Spline have configurable fit options. The available options depend on the fit you select (that is, linear, nonlinear, or nonparametric fit).
The options described here are available for nonlinear models.
Lower and Upper coefficient constraints are the only fit options available in the Fit Options pane for Polynomial fits.
Nonparametric fits (that is, Interpolant, Smoothing Spline, and Lowess fits) do not have Advanced Options.
The Fit Options pane for the singleterm
Exponential fit is shown here. The
Coefficient Constraints values are for the
census
data.
Finite Differencing Parameters
For more information about these fit options, see the lsqcurvefit
(Optimization Toolbox)
function.
Optimized Starting Points and Default Constraints
The default coefficient starting points and constraints for fits in the Fit Type pane are shown in the following table. If the starting points are optimized, then they are calculated heuristically based on the current data set. Random starting points are defined on the interval [0 1] and linear models do not require starting points. If a model does not have constraints, the coefficients have neither a lower bound nor an upper bound. You can override the default starting points and constraints by providing your own values in the Fit Options pane.
Fit  Starting Points  Constraints 

 N/A  None 
 Random  None 
 Optimized  None 
 Optimized  None 
 Optimized  c_{i} > 0 
 N/A  None 
 Optimized  None 
 Random  None 
 Optimized  b_{i} > 0 
 Random  a, b > 0 
The Sum of Sine and Fourier fits are particularly sensitive to starting points, and the optimized values might be accurate for only a few terms in the associated equations.
Specify Fit Options at the Command Line
Create the default fit options structure and set the option to center and scale the data before fitting:
options = fitoptions; options.Normal = 'on'; options options = Normalize: 'on' Exclude: [1x0 double] Weights: [1x0 double] Method: 'None'
Modifying the default fit options structure is useful when you want to
set the Normalize
, Exclude
, or
Weights
fields, and then fit your data using
the same options with different fitting methods. For example:
load census f1 = fit(cdate,pop,"poly3",options); f2 = fit(cdate,pop,"exp1",options); f3 = fit(cdate,pop,"cubicsp",options);
Datadependent fit options are returned in the third output argument
of the fit
function. For
example, the smoothing parameter for smoothing spline is
datadependent:
[f,gof,out] = fit(cdate,pop,"smooth");
smoothparam = out.p
smoothparam =
0.0089
Use fit options to modify the default smoothing parameter for a new fit:
options = fitoptions("Method","Smooth","SmoothingParam",0.0098); [f,gof,out] = fit(cdate,pop,"smooth",options);
For more details on using fit options, see the fitoptions
function.