## Interactive Fitting

### Basic Fitting UI

The MATLAB^{®} Basic Fitting UI allows you to interactively:

Model data using a spline interpolant, a shape-preserving interpolant, or a polynomial up to the tenth degree

Plot one or more fits together with data

Plot the residuals of the fits

Compute model coefficients

Compute the norm of the residuals (a statistic you can use to analyze how well a model fits your data)

Use the model to interpolate or extrapolate outside of the data

Save coefficients and computed values to the MATLAB workspace for use outside of the dialog box

Generate MATLAB code to recompute fits and reproduce plots with new data

**Note**

The Basic Fitting UI is only available for 2-D plots. For more advanced fitting and regression analysis, see the Curve Fitting Toolbox™ documentation and the Statistics and Machine Learning Toolbox™ documentation.

### Preparing for Basic Fitting

The Basic Fitting UI sorts your data in ascending order before fitting. If your data set is large and the values are not sorted in ascending order, it will take longer for the Basic Fitting UI to preprocess your data before fitting.

You can speed up the Basic Fitting UI by first sorting your data. To create sorted
vectors `x_sorted`

and `y_sorted`

from data
vectors `x`

and `y`

, use the MATLAB
`sort`

function:

[x_sorted, i] = sort(x); y_sorted = y(i);

### Opening the Basic Fitting UI

To use the Basic Fitting UI, you must first plot your data in a figure window,
using any MATLAB plotting command that produces (only) *x* and
*y* data.

To open the Basic Fitting UI, select **Tools > Basic
Fitting** from the menus at the top of the figure window.

### Example: Using Basic Fitting UI

This example shows how to use the Basic Fitting UI to fit, visualize, analyze, save, and generate code for polynomial regressions.

#### Load and Plot Census Data

The file `census.mat`

contains U.S. population data for the
years 1790 through 1990 at 10 year intervals.

To load and plot the data, type the following commands at the MATLAB prompt:

load census plot(cdate,pop,'ro')

The `load`

command adds the following variables to the
MATLAB workspace:

`cdate`

— A column vector containing the years from 1790 to 1990 in increments of 10. It is the predictor variable.`pop`

— A column vector with U.S. population for each year in`cdate`

. It is the response variable.

The data vectors are sorted in ascending order, by year. The plot shows the population as a function of year.

Now you are ready to fit an equation the data to model population growth over time.

#### Predict the Census Data with a Cubic Polynomial Fit

Open the Basic Fitting dialog box by selecting

**Tools > Basic Fitting**in the Figure window.In the

**TYPES OF FIT**area of the Basic Fitting dialog box, select the**Cubic**check box to fit a cubic polynomial to the data.MATLAB uses your selection to fit the data, and adds the cubic regression line to the graph as follows.

In computing the fit, MATLAB encounters problems and issues the following warning:

This warning indicates that the computed coefficients for the model are sensitive to random errors in the response (the measured population). It also suggests some things you can do to get a better fit.

Continue to use a cubic fit. As you cannot add new observations to the census data, improve the fit by transforming the values you have to

*z-scores*before recomputing a fit. Select the**Center and scale x-axis data**check box in the top right of the dialog box to make the Basic Fitting tool perform the transformation.To learn how centering and scaling data works, see Learn How the Basic Fitting Tool Computes Fits.

Under

**ERROR ESTIMATION (RESIDUALS)**, select the**Norm of residuals**check box. Select**Bar**as the**Plot Style**.

Selecting these options creates a subplot of residuals as a bar graph.

The cubic fit is a poor predictor before the year 1790, where it indicates a
decreasing population. The model seems to approximate the data reasonably well
after 1790. However, a pattern in the residuals shows that the model does not
meet the assumption of normal error, which is a basis for the least-squares
fitting. The **data 1** line identified in the legend are the
observed *x* (`cdate`

) and
*y* (`pop`

) data values. The
**Cubic** regression line presents the fit after centering
and scaling data values. Notice that the figure shows the original data units,
even though the tool computes the fit using transformed z-scores.

For comparison, try fitting another polynomial equation to the census data by
selecting it in the **TYPES OF FIT** area.

#### View and Save the Cubic Fit Parameters

In the Basic Fitting dialog box, click the **Expand Results**
button to display the estimated coefficients and the
norm of residuals.

Save the fit data to the MATLAB workspace by clicking the **Export to
Workspace** button on the Numerical results panel. The Save Fit to
Workspace dialog box opens.

With all check boxes selected, click **OK** to save the fit
parameters as a MATLAB structure `fit`

:

fit

fit = struct with fields: type: 'polynomial degree 3' coeff: [0.9210 25.1834 73.8598 61.7444]

Now, you can use the fit results in MATLAB programming, outside of the Basic Fitting UI.

#### R^{2}, the Coefficient of Determination

You can get an indication of how well a polynomial regression predicts your
observed data by computing the *coefficient of
determination,* or *R-square* (written as
R^{2}). The R^{2} statistic,
which ranges from 0 to 1, measures how useful the independent variable is in
predicting values of the dependent variable:

An R

^{2}value near 0 indicates that the fit is not much better than the model`y = constant`

.An R

^{2}value near 1 indicates that the independent variable explains most of the variability in the dependent variable.

R^{2} is computed from the
*residuals*, the signed differences between an observed
dependent value and the value your fit predicts for it.

residuals = y_{observed} -
y_{fitted} | (1) |

The R^{2} number for the cubic fit in this example,
0.9988, is located under **FIT RESULTS** in the Basic Fitting
dialog.

To compare the R^{2} number for the cubic fit to a
linear least-squares fit, select **Linear** under
**TYPES OF FIT** and obtain the
R^{2} number, 0.921. This result indicates that a
linear least-squares fit of the population data explains 92.1% of its variance.
As the cubic fit of this data explains 99.9% of that variance, the latter seems
to be a better predictor. However, because a cubic fit predicts using three
variables (*x*,
*x ^{2}*, and

*x*), a basic R

^{3}^{2}value does not fully reflect how robust the fit is. A more appropriate measure for evaluating the goodness of multivariate fits is

*adjusted R*. For information about computing and using adjusted R

^{2}^{2}, see Residuals and Goodness of Fit.

#### Interpolate and Extrapolate Population Values

Suppose you want to use the cubic model to interpolate the U.S. population in 1965 (a date not provided in the original data).

In the Basic Fitting dialog box, under **INTERPOLATE / EXTRAPOLATE
DATA**, enter the **X** value 1965 and check the
**Plot evaluated data** box.

**Note**

Use unscaled and uncentered `X`

values. You do not need
to center and scale first, even though you selected to scale
`X`

values to obtain the coefficients in Predict the Census Data with a Cubic Polynomial Fit. The Basic Fitting tool makes the necessary
adjustments behind the scenes.

The `X`

values and the corresponding values for
`f(X)`

are computed from the fit and plotted as
follows:

#### Generate a Code File to Reproduce the Result

After completing a Basic Fitting session, you can generate MATLAB code that recomputes fits and reproduces plots with new data.

In the Figure window, select

**File > Generate Code**.This creates a function and displays it in the MATLAB Editor. The code shows you how to programmatically reproduce what you did interactively with the Basic Fitting dialog box.

Change the name of the function on the first line from

`createfigure`

to something more specific, like`censusplot`

. Save the code file to your current folder with the file name`censusplot.m`

The function begins with:function censusplot(X1, Y1, valuesToEvaluate1)

Generate some new, randomly perturbed census data:

`rng('default') randpop = pop + 10*randn(size(pop));`

Reproduce the plot with the new data and recompute the fit:

censusplot(cdate,randpop,1965)

You need three input arguments:

*x,y*values (`data 1`

) plotted in the original graph, plus an*x*-value for a marker.The following figure displays the plot that the generated code produces. The new plot matches the appearance of the figure from which you generated code except for the

*y*data values, the equation for the cubic fit, and the residual values in the bar graph, as expected.

#### Learn How the Basic Fitting Tool Computes Fits

The Basic Fitting tool calls the `polyfit`

function to compute
polynomial fits. It calls the `polyval`

function to evaluate
the fits. `polyfit`

analyzes its inputs to determine if the
data is well conditioned for the requested degree of fit.

When it finds badly conditioned data, `polyfit`

computes a
regression as well as it can, but it also returns a warning that the fit could
be improved. The Basic Fitting example section Predict the Census Data with a Cubic Polynomial Fit displays this
warning.

One way to improve model reliability is to add data points. However, adding observations to a data set is not always feasible. An alternative strategy is to transform the predictor variable to normalize its center and scale. (In the example, the predictor is the vector of census dates.)

The `polyfit`

function normalizes by computing
*z-scores*:

$$z=\frac{x-\mu}{\sigma}$$

where *x* is the predictor data, *μ* is the
mean of *x*, and *σ* is the standard deviation
of *x*. The *z*-scores give the data a mean of
0 and a standard deviation of 1. In the Basic Fitting UI, you transform the
predictor data to *z*-scores by selecting the **Center
and scale x-axis data** check box.

After centering and scaling, model coefficients are computed for the
*y* data as a function of *z*. These are
different (and more robust) than the coefficients computed for
*y* as a function of *x*. The form of the
model and the norm of the residuals do not change. The Basic Fitting UI
automatically rescales the *z*-scores so that the fit plots on
the same scale as the original *x* data.

To understand the way in which the centered and scaled data is used as an intermediary to create the final plot, run the following code in the Command Window:

close load census x = cdate; y = pop; z = (x-mean(x))/std(x); % Compute z-scores of x data plot(x,y,'ro') % Plot data as red markers hold on % Prepare axes to accept new graph on top zfit = linspace(z(1),z(end),100); pz = polyfit(z,y,3); % Compute conditioned fit yfit = polyval(pz,zfit); xfit = linspace(x(1),x(end),100); plot(xfit,yfit,'b-') % Plot conditioned fit vs. x data

The centered and scaled cubic polynomial plots as a blue line, as shown here:

In the code, computation of `z`

illustrates
how to normalize data. The `polyfit`

function performs the
transformation itself if you provide three return arguments when calling
it:

[p,S,mu] = polyfit(x,y,n)

`p`

, now are based on
normalized `x`

. The returned vector, `mu`

,
contains the mean and standard deviation of `x`

. For more
information, see the `polyfit`

reference page.