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 half-life (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

$y = y_{0} e^{- λ t}$

wherey₀is the number of nuclei at timet= 0, and λ is the decay constant. The data can be described by

$data = y_{0} e^{- λ t} + error$

Bothy₀and λ are coefficients that are estimated by the fit. BecauseT_1/2= ln(2)/λ, the fitted value of the decay constant yields the fitted half-life. However, because the data contains some error, the deterministic component of the equation cannot be determined exactly from the data. Therefore, the coefficients and half-life 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 enteringcurveFitterat the MATLAB^®command line. Alternatively, on theAppstab, in theMath, Statistics and Optimizationgroup, clickCurve Fitter.

In the Curve Fitter app, go to theFit Typesection of theCurve Fittertab. You can select a model type from the fit gallery. Click the arrow to open the gallery.

Fit Type model gallery

This table describes the models that you can fit for curves and surfaces.

Fit Group	合适的类别	曲线	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: Nearest neighbor Linear Cubic Shape-preserving (PCHIP)	Yes, with methods: Nearest neighbor Linear Cubic Biharmonic Thin-plate spline
Smoothing	Smoothing Spline	Yes	No
Smoothing	Lowess	No	Yes
Custom	Custom Equation	Yes	Yes
Custom	Custom Linear Fitting	Yes	No

TheResultspane displays the model specifications, coefficient values, and goodness-of-fit statistics.

Tip

If your fit has problems, messages in theResultspane help you identify better settings.

The Curve Fitter app provides a selection of fit types and settings in theFit Optionspane 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, seeSpecify 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, seeCreate 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 thefitfunction. For example, you can specify a quadraticpoly2model:

f = fit(x,y,"poly2")

To view all available library model names, seeList of Library Models for Curve and Surface Fittingto view all available library model names.

You can also use thefittypefunction to construct afittypeobject for a library model, and use thefittypeas an input to thefitfunction.

Use thefitoptionsfunction 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 inSelect Model Type Interactively. For details on all the functions for creating and analysing models, seeCurve and Surface Fitting.

Center and Scale Data

Most fits in the Curve Fitter app provide theCenter and scaleoption in theFit Optionspane. When you select this option, the app refits the model with the data centered and scaled. At the command line, use thefitoptionsfunction with theNormalizeoption set to'on'.

To alleviate numerical problems with variables of different scales, normalize the input data (also known aspredictor 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 scalegenerally 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), thenCenter and scaleis 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 theCenter and scalecheck box. The plots in the Curve Fitter app always use the original scale, regardless of theCenter and scalestatus.

At the command line, to center and scale the data before fitting, create theoptionsstructure by using thefitoptionsfunction withoptions.Normalspecified as'on'. Then, use thefitfunction with the specified options.

options = fitoptions; options.Normal ='on'; options options = Normalize:'on'Exclude: [1x0 double] Weights: [1x0 double] Method:'None'loadcensusf1 = 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 theFit Optionspane. All fits exceptInterpolantandSmoothing Splinehave 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.
LowerandUppercoefficient constraints are the only fit options available in theFit Optionspane forPolynomialfits.
Nonparametric fits (that is,Interpolant,Smoothing Spline, andLowessfits) do not haveAdvanced Options.

TheFit Optionspane for the single-termExponentialfit is shown here. TheCoefficient Constraintsvalues are for thecensusdata.

Fit Options pane showing Advanced Options for exponential fit

Fitting Method and Algorithm

Finite Differencing Parameters

Fit Convergence Criteria

Coefficient Parameters

For more information about these fit options, see thelsqcurvefit(Optimization Toolbox)function.

Optimized Starting Points and Default Constraints

The default coefficient starting points and constraints for fits in theFit Typepane 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 theFit Optionspane.

Fit	起点	Constraints
`Linear Fitting`	N/A	None
`Custom Equation`	Random	None
`Exponential`	Optimized	None
`Fourier`	Optimized	None
`Gaussian`	Optimized	c_i> 0
`Polynomial`	N/A	None
`Power`	Optimized	None
`Rational`	Random	None
`Sum of Sine`	Optimized	b_i> 0
`Weibull`	Random	a,b> 0

TheSum of SineandFourierfits 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 theNormalize,Exclude, orWeightsfields, and then fit your data using the same options with different fitting methods. For example:

loadcensusf1 = fit(cdate,pop,"poly3",options); f2 = fit(cdate,pop,"exp1",options); f3 = fit(cdate,pop,"cubicsp",options);

Data-dependent fit options are returned in the third output argument of thefitfunction. For example, the smoothing parameter for smoothing spline is data-dependent:

[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 thefitoptionsfunction.