Documentation

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

where y₀ is the number of nuclei at time t = 0, and λ is the decay constant. The data can be described by

Both y₀ 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 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.

Selecting a Model Type Interactively

Select a model type to fit from the drop-down list in the Curve Fitting app.

What fit types can you use for curves or surfaces? Based on your selected data, the fit category list shows either curve or surface categories. The following table describes the options for curves and surfaces.

For all fit categories, look in the Results pane to see the model terms, the values of the coefficients, and the goodness-of-fit statistics.

Selecting Model Type Programmatically

You can specify a library model name as a string when you call the fit function. For example, to specify a quadratic poly2:

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

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 Selecting a Model Type Interactively. For details on all the functions for creating and analysing models, see Curve and Surface Fitting.

Using Normalize or Center and Scale

Most model types in the Curve Fitting app share the Center and scale option. When you select this option, the tool refits with the data centered and scaled, by applying the Normalize setting to the variables. At the command line, you can use Normalize as an input argument to the fitoptions function. See the fitoptions reference page.

Generally, it is a good idea to normalize inputs (also known as predictor data), which can alleviate numerical problems with variables of different scales. 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 (e.g., 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 Curve Fitting app plots use the original scale with or without the Center and scale option.

At the command line, to set the option to center and scale the data before fitting, create the default fit options structure, set Normalize to on, then fit with the 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)

Specifying Fit Options and Optimized Starting Points

• About Fit Options

• Fitting Method and Algorithm

• Finite Differencing Parameters

• Fit Convergence Criteria

• Coefficient Parameters

• Optimized Starting Points and Default Constraints

• Specifying Fit Options at the Command Line

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);
f2 = fit(cdate,pop,'exp1',options);
f3 = fit(cdate,pop,'cubicsp',options);

[f,gof,out] = fit(cdate,pop,'smooth');
smoothparam = out.p
smoothparam =
    0.0089

options = fitoptions('Method','Smooth','SmoothingParam',0.0098);
[f,gof,out] = fit(cdate,pop,'smooth',options);