## Compare Fits in Curve Fitter App

### Interactive Curve Fitter Workflow

The next topics fit census data using polynomial equations up to the sixth degree, and a single-term exponential equation. The steps demonstrate how to:

Load data and explore various fits using different library models.

Search for the best fit by:

Comparing graphical fit results

Comparing numerical fit results including the fitted coefficients and goodness-of-fit statistics

Export your best fit results to the MATLAB

^{®}workspace to analyze the model at the command line.Save the session and generate MATLAB code for all fits and plots.

### Loading Data and Creating Fits

You must load the data variables into the MATLAB workspace before you can fit data using the Curve Fitter app. For this
example, the data is stored in the MATLAB file `census.mat`

.

Load the data.

`load census`

The workspace contains two new variables.

`cdate`

is a column vector containing the years 1790 to 1990 in 10-year increments.`pop`

is a column vector with the US population figures that correspond to the years in`cdate`

.

Open the Curve Fitter app.

curveFitter

On the

**Curve Fitter**tab, in the**Data**section, click**Select Data**. In the Select Fitting Data dialog box, select the variable names`cdate`

and`pop`

from the**X data**and**Y data**lists, respectively.The Curve Fitter app creates and plots a default fit to the X input (or predictor) data and the Y output (or response) data. The default fit is a linear polynomial fit type. Observe the fit settings display in the

**Fit Options**pane. The fit is a first-degree polynomial.In the

**Fit Options**pane, change the fit to a second-degree polynomial by selecting`2`

from the**Degree**list.The Curve Fitter app plots the new fit. The Curve Fitter app calculates a new fit when you change fit settings because

**Auto**is selected by default. If refitting is time consuming, as is sometimes the case for large data sets, you can turn off the automatic behavior. On the**Curve Fitter**tab, in the**Fit**section, select**Manual**.The Curve Fitter app displays results of fitting the census data with a quadratic polynomial in the

**Results**pane, where you can view the library model, fitted coefficients, and goodness-of-fit statistics.Change the name of the fit. In the

**Table Of Fits**pane, double-click`untitled fit 1`

in the**Fit name**column and type`poly2`

.Display the residuals. On the

**Curve Fitter**tab, in the**Visualization**section, click**Residuals Plot**.The residuals indicate that a better fit might be possible. Therefore, continue exploring various fits to the census data set.

Add new fits to try the other library equations.

Right-click the fit in the

**Table Of Fits**pane, and select**Duplicate "poly2"**. Alternatively, on the**Curve Fitter**tab, in the**File**section, click**Duplicate**.**Tip**For fits of a given type (for example, polynomials), duplicate a fit instead of creating a new fit because copying a fit requires fewer steps. The duplicated fit contains the same data selections and fit settings.

Change the polynomial

**Degree**to`3`

and rename the fit`poly3`

.When you fit higher degree polynomials, the

**Results**pane displays this warning.Equation is badly conditioned. Remove repeated data points or try centering and scaling.

Normalize the data by selecting the

**Center and scale**check box in the**Fit Options**pane.Repeat steps a and b to add polynomial fits up to the sixth degree. Then add an exponential fit. On the

**Curve Fitter**tab, in the**File**section, click**New**and select**New Fit**. In the**Fit Type**section, click the arrow to open the gallery, and click**Exponential**in the**Regression Models**section.For each new fit, look at the

**Results**pane information, and the residuals plot in the app.The residuals from a good fit should look random with no apparent pattern. A pattern, such as a tendency for consecutive residuals to have the same sign, can be an indication that a better model exists.

#### About Scaling

The warning about scaling arises because the fitting procedure uses the
`cdate`

values as the basis for a matrix with very large values. The
spread of the `cdate`

values results in a scaling problem. To address
this problem, you can normalize the `cdate`

data. Normalization scales
the predictor data to improve the accuracy of the subsequent numeric computations. For
example, you can normalize `cdate`

by centering and scaling the data to
have zero mean and unit standard deviation.

(cdate - mean(cdate))./std(cdate)

**Note**

Because the predictor data changes after normalizing, the values of the fitted coefficients also change when compared to the original data. However, the functional form of the data and the resulting goodness-of-fit statistics do not change. Additionally, the data is displayed in the Curve Fitter app plots using the original scale.

### Determining the Best Fit

To determine the best fit, you should examine both the graphical and numerical fit results.

#### Examine the Graphical Fit Results

Determine the best fit by examining the graphs of the fits and residuals. To view plots for each fit in turn, click the fit in the

**Table Of Fits**pane. The graphical fit results indicate that:The fits and residuals for the polynomial equations are all similar, making it difficult to choose the best one.

The fit and residuals for the single-term exponential equation indicate it is a poor fit overall. Therefore, it is a poor choice and you can remove the exponential fit from the candidates for best fit.

Examine the behavior of the fits after the year 2000. The goal of fitting the census data is to extrapolate the best fit to predict future population values.

Click the sixth-degree polynomial fit in the

**Table Of Fits**pane to view the plots for this fit.In the fit plot, click the Pan button in the axes toolbar and pan until the fit is visible for several years after the year 2000. The axes limits of the residuals plot adjust accordingly.

Examine the fit plot. The behavior of the sixth-degree polynomial fit beyond the data range makes it a poor choice for extrapolation and you can reject this fit.

#### Evaluate the Numerical Fit Results

When you can no longer eliminate fits by examining them graphically, you should examine the numerical fit results. The Curve Fitter app displays two types of numerical fit results:

Goodness-of-fit statistics

Confidence bounds on the fitted coefficients

The goodness-of-fit statistics help you determine how well the curve fits the data. The confidence bounds on the coefficients determine their accuracy.

Examine the numerical fit results.

For each fit, view the goodness-of-fit statistics in the

**Results**pane.Compare all fits simultaneously in the

**Table Of Fits**pane. Click the column headings to sort by statistics results.Examine the sum of squares due to error (SSE) and the adjusted

*R*-square statistics to help determine the best fit. The SSE statistic is the least-squares error of the fit, with a value closer to zero indicating a better fit. The adjusted*R*-square statistic is generally the best indicator of the fit quality when you add additional coefficients to your model.The largest SSE for

`exp`

indicates it is a poor fit, which you already determined by examining the fit and residuals. The lowest SSE value is associated with`poly6`

. However, the behavior of this fit beyond the data range makes it a poor choice for extrapolation, so you already rejected this fit by examining the plots with new axes limits.The next best SSE value is associated with the fifth-degree polynomial fit,

`poly5`

, suggesting it might be the best fit. However, the SSE and adjusted*R*-square values for the remaining polynomial fits are all very close to each other.Resolve the best fit issue by examining the confidence bounds for the remaining fits in the

**Results**pane. Click a fit in the**Table Of Fits**pane to open the fit figure (or select it, if the figure is already open), and view the**Results**pane. Each fit figure displays the plots for a single fit.Display the fifth-degree polynomial

`poly5`

and the second-degree polynomial`poly2`

fit figures side by side. Examining results side by side can help you assess fits.To show two fit figures simultaneously, you can drag and drop the fit figure tabs in the

**Fits**pane. Alternatively, you can click the Document Actions button located to the far right of the fit figure tabs. Select the`Tile All`

option and specify a 1-by-2 or 2-by-1 layout.Compare the coefficients and bounds (

`p1`

,`p2`

, and so on) in the**Results**pane for both fits,`poly5`

and`poly2`

. The toolbox calculates 95% confidence bounds on coefficients. The confidence bounds on the coefficients determine their accuracy. Check the equations in the**Results**pane (`f(x)=p1*x+p2*x`

...) to see the model terms for each coefficient. Note that`p2`

refers to the`p2*x`

term in`Poly2`

and the`p2*x^4`

term in`Poly5`

. Do not compare normalized coefficients directly with non-normalized coefficients.**Tip**If you want more space to view and compare plots and results, as shown next, drag down the

**Table Of Fits**pane. You can also hide the**Results**pane to show only plots.The bounds cross zero on the

`p1`

,`p2`

, and`p3`

coefficients for the fifth-degree polynomial. This means you cannot be sure that these coefficients differ from zero. If the higher order model terms might have coefficients of zero, they are not helping with the fit, which suggests that this model overfits the census data.However, the small confidence bounds do not cross zero on

`p1`

,`p2`

, and`p3`

for the quadratic fit`poly2`

, indicating that the fitted coefficients are known fairly accurately.Therefore, after examining both the graphical and numerical fit results, you should select

`poly2`

as the best fit to extrapolate the census data.

**Note**

The fitted coefficients associated with the constant, linear, and quadratic terms are nearly identical for each normalized polynomial equation. However, as the polynomial degree increases, the coefficient bounds associated with the higher degree terms cross zero, which suggests overfitting.

### Analyzing Best Fit in the Workspace

You can export the selected fit and the associated fit results to the MATLAB workspace. On the **Curve Fitter** tab, in the
**Export** section, click **Export** and select
**Export to Workspace**. The fit is saved as a MATLAB object and the associated fit results are saved as structures.

Right-click the

`poly2`

fit in the**Table Of Fits**pane, and select**Save "poly2" to Workspace**. Alternatively, click**Export**and select**Export to Workspace**. The app opens a dialog box.Click

**OK**to save the fit with the default names.`fittedmodel`

is saved as a Curve Fitting Toolbox™`cfit`

object.`whos fittedmodel`

Name Size Bytes Class Attributes fittedmodel 1x1 925 cfit

Examine the `cfit`

object `fittedmodel`

to display the
model, the fitted coefficients, and the confidence bounds for the fitted
coefficients.

fittedmodel

Linear model Poly2: fittedmodel(x) = p1*x^2 + p2*x + p3 Coefficients (with 95% confidence bounds): p1 = 0.006541 (0.006124, 0.006958) p2 = -23.51 (-25.09, -21.93) p3 = 2.113e+04 (1.964e+04, 2.262e+04)

Examine the `goodness`

structure to display goodness-of-fit
results.

goodness

goodness = struct with fields: sse: 159.0293 rsquare: 0.9987 dfe: 18 adjrsquare: 0.9986 rmse: 2.9724

Examine the `output`

structure to display additional information
associated with the fit, such as the residuals.

output

output = struct with fields: numobs: 21 numparam: 3 residuals: [21×1 double] Jacobian: [21×3 double] exitflag: 1 algorithm: 'QR factorization and solve' iterations: 1

You can evaluate (interpolate or extrapolate), differentiate, or integrate a fit over a specified data range with various postprocessing functions.

For example, evaluate `fittedmodel`

at a vector of values to
extrapolate to the year
2050.

x = 2000:10:2050; y = fittedmodel(x)

y = 274.6221 301.8240 330.3341 360.1524 391.2790 423.7137

plot(fittedmodel,cdate,pop) hold on plot(fittedmodel,x,y,"k+") hold off legend(["data","","extrapolated data","fitted curve"], ... "Location","northwest")

For more examples and instructions for interactive and command-line fit analysis, and a list of all postprocessing functions, see Fit Postprocessing.

For an example reproducing this interactive census data analysis using the command line, see Polynomial Curve Fitting.

### Saving Your Work

The Curve Fitter app provides several options for saving your work. You can save one or more fits and the associated fit results as variables to the MATLAB workspace. You can then use this saved information for documentation purposes, or to extend your data exploration and analysis. In addition to saving your work to MATLAB workspace variables, you can:

Save the current curve fitting session. On the

**Curve Fitter**tab, in the**File**section, click**Save**and select**Save Session**. The session file contains all the fits and variables in your session and remembers your layout. See Save and Reopen Sessions.Generate MATLAB code to recreate a fit and its associated plots. In the

**Export**section, click**Export**and select**Generate Code**. The Curve Fitter app generates code for the currently selected fit and displays the file in the MATLAB Editor.You can recreate your fit and plots by calling the file at the command line with your original data as input arguments. You can also call the file with new data, and automate the process of fitting multiple data sets. For more information, see Generating Code from the Curve Fitter App.