CURVE FITTING AND MATLAB

Tanushree Sharma , M.E. ( Electronics & Instrumentation

Control)@ Thapar University, Patiala

CURVE FITTING

 Curve Fitting: The process of constructing a curve,

or mathematical function, that has the best fit to a series of data points, possibly subject to constraints

It can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function is constructed that approximately fits the data

PURPOSE OF CURVE FITTING

Curve fitting, also known as regression analysis, is used to find the "best fit" line or curve for a series of data points

Most of the time, the curve fit will produce an equation that can be used to find points anywhere along the curve

In some cases, you may not be concerned about finding an equation. Instead, you may just want to use a curve fit to smooth the data and improve the appearance of your plot

TYPES OF CURVE FITTING

Let's start with a first degree polynomial equation: y = ax+ b This is a line with slope a. We know that a line

will connect any two points. So, a first degree polynomial equation is an exact fit through any two points with distinct x coordinates.

If we increase the order of the equation to a second

degree polynomial, we get: y=ax² + bx +c

This will exactly fit a simple curve to three points.

If we increase the order of the equation to a third degree polynomial, we get:

y=ax³+bx²+cx+d

POLYNOMIAL CURVES FITTING POINTS GENERATED WITH A SINE

FUNCTION

TYPES OF CURVE FITS

Least square Smoothing

curve fits curve fits

Non Linear

curve fits

Least Square Curve Fits It minimizes

the square of the error between the original data and the values predicted by the equation

Non Linear Curve Fits The

relationship between measured values and measurement variables is nonlinear. It also seeks to find those parameter values that minimize the deviations between the observed values and the expected values

Smoothing Curve Fits A new mathematical method is developed for interpolation from a given set of data points in a plane and for fitting a smooth curve to the points.

This method is devised in such a way that the resultant curve will pass through the given points and will appear smooth and natural.

It is based on a piecewise function composed of a set of polynomials, each of degree three, at most, and applicable to successive intervals of the given points

LINEAR LEAST SQUARE ALGORITHM

Data points (xi, yi ), i= 1…n

Choice of fitting function (linear) y=f(x)= ax+b

Errors between function & datapoints ei = yi – (axi +b)

Sum of the squares of the errors z = e1

2 + e22 +… +en

2

In compact form z = ∑ [yi – (axi + b)]2

Our goal is to determine the values of a and b that will minimize z, the sum of the squares of the errors

To find the minimum value for z, we set the derivative of z to zero and solve for a and b

0 2 4 6 8 10 120

1

2

3

4

5

6 This linear fit uses every data point

Notice the verysmall slope; y = f(x) is almost independent of x

CHOOSING THE RIGHT POLYNOMIAL

The degree of the correct approximating function depends on the type of data being analyzed

When a certain behavior is expected, we know what type of function to use, and simply have to solve for its coefficients

When we don’t know what sort of response to expect, ensure your data sample size is large enough to clearly distinguish which degree is the best fit

e.g. Ohms law ( curve between voltage and current)

POLYNOMIAL REGRESSION We need to fit a polynomial to data

using polynomial regression.

A second-order polynomial or quadratic fit is

y = a0 + a1 x + a2 x2 +

The sum of squares of the residues:

Differentiate Sr wrt all parameters:

n

iiiir xaxaayS

1

22210

Set the partials equal to zero and arrange:

These equations are called the normal equations

They form a system of linear equations with 3 equations and 3 unknowns

In general, an mth order polynomial requires solving a system of m+1 linear equations

EXPONENTIAL FIT Not all experimental data can be

approximated with polynomial functions.

Exponential data can be fit using the least squares method by first converting the data to a linear form

An exponential function, y = aebx

can be rewritten as a linear polynomial by taking the natural logarithm of each side, ln y = ln a + bx

By finding ln yi for each point in a data set, we can solve for a and b using the least squares method

CURVE FITTING TOOLBOX IN MATLAB

The Curve Fitting Toolbox is a collection of graphical user interfaces (GUIs) and M-file functions built on the MATLAB technical computing environment.

The toolbox provides you with these main features:

1. Data preprocessing such as sectioning and smoothing

2. Parametric and nonparametric data fitting

3. Standard linear least squares, nonlinear least squares, weighted least squares, constrained least squares, and robust fitting procedures

A graphical environment that allows you to:

1. Explore and analyze data sets and fits visually and numerically

2. Save your work in various formats including M-files, binary files, and workspace variables

The Curve Fitting Toolbox consists of two different environments:

1. The Curve Fitting Tool, which is a graphical user interface (GUI) environment

2. The MATLAB command line environment

OPENING THE CURVE FITTING TOOL

The Curve Fitting Tool (GUI) allows you to : Visually explore one or more data sets and

fits as scatter plots Graphically evaluate the goodness of fit

using residuals and prediction bounds Access additional interfaces for • Importing, viewing, and smoothing data • Fitting data, and comparing fits and data sets• Marking data points to be excluded from a fit• Selecting which fits and data sets are

displayed in the tool• Interpolating, extrapolating, differentiating,

or integrating fits

STEPS TO CURVE FITTING You can explore the Curve Fitting Tool by typing

cftool

Before you can import data into the Curve Fitting Tool, the data variables must exist in the MATLAB workspace.

For this example, the data is stored in the file census.mat, which is provided with MATLAB.

Hence first we call ‘load census’

The workspace now contains two new variables, cdate and pop:

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

2. pop is a column vector with the US population figures that correspond to the years in cdate.

You can import data into the Curve Fitting Tool with the Data GUI.

You open this GUI by clicking the Data button on the Curve Fitting Tool.

The Data GUI consists of two panes: Data sets and Smooth. The Data Sets pane allows you to Import predictor (X) data, response (Y) data, and weights.

If you do not import weights, then they are assumed to be 1 for all data points. Specify the name of the data set.

To load cdate and pop into the Curve Fitting Tool, select the appropriate variable names from the X Data and Y Data lists. The data is then displayed in the Preview window. Click the Create data set button to complete the data import process.

DATA FITTING PROCEDURE

For this example, begin by fitting the census data with a second degree polynomial. Then continue fitting the data using polynomial equations up to sixth degree, and a single-term exponential equation.

The data fitting procedure follows these general steps:

From the Fit Editor, click New Fit. Note that this action always defaults to a linear polynomial fit type.

You use New Fit at the beginning of your curve fitting session, and when you are exploring different fit types for a given data set.

Because the initial fit uses a second degree polynomial, select quadratic polynomial from the Polynomial list. Name the fit poly2.

Click the Apply button or select the Immediate apply check box . The library model, fitted coefficients, and goodness of fit statistics are displayed in the Results area.

Fit the additional library equations. For fits of a given type (for example, polynomials), you should use Copy Fit instead of New Fit because copying a fit retains the current fit type state thereby requiring fewer steps than creating a new fit each time

You display the residuals as a line plot by selecting the menu item View->Residuals->Line plot from the Curve Fitting Tool.

The residuals indicate that a better fit may be possible. Therefore, you should continue fitting the census data following the procedure outlined in the beginning of this section.

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.

When you fit higher degree polynomials, the Results area displays this warning:

‘Equation is badly conditioned. Remove repeated data points or try centering and scaling’

DETERMINING THE BEST FIT

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

Your initial approach in determining the best fit should be a graphical examination of the fits and residuals.

The graphical fit results shown here 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 for extrapolation.

Exponential fit residues

Use the Plotting GUI to remove exp1 from the scatter plot display.

The census data and fits are shown above for an upper abscissa limit of 2050. The behavior of the sixth degree polynomial fit beyond the data range makes it a poor choice for extrapolation. Hence you should exercise caution when extrapolating with polynomial fits because they can diverge wildly outside the data range.

EXAMINING THE NUMERICAL FIT RESULTS

There are two types of numerical fit results displayed in the Fitting GUI:

1. Goodness of fit statistics 2. Confidence intervals on the fitted

coefficients

The goodness of fit statistics help you determine how well the curve fits the data.

The confidence intervals on the coefficients determine their accuracy.

In this example, the sum of squares due to error (SSE) and the adjusted R-square statistics are used to help determine the best fit.

As described in Goodness of Fit Statistics, 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 p1, p2, and p3 coefficients for the fifth degree polynomial suggest that it overfits the census data.

However, the confidence bounds for the quadratic fit, poly2, indicate that the fitted coefficients are known fairly accurately.

Therefore, after examining both the graphical and numerical fit results, it appears that you should use poly2 to extrapolate the census data

SAVING THE FIT RESULT

By clicking the Save to workspace button, you can save the selected fit and the associated fit results to the MATLAB workspace.

The fit is saved as a MATLAB object and the associated fit results are saved as structures.

This example saves all the fit results for the best fit, poly2.

CONCLUSION

Working with MATLAB not only makes our job easier but also adds fun to otherwise tedious problems.

It saves time and is very accurate.

The different commands, toolbox and GUI’s offer a range of operations on the click of a mouse.

THANK YOU