by Lale Yurttas, Texas A&M UniversityChapter 4*Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.Chapter 4CURVE FITTING

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by Lale Yurttas, Texas A&M UniversityChapter 4*CURVE FITTINGDescribes techniques to fit curves (curve fitting) to discrete data to obtain intermediate estimates.

There are two general approaches for curve fitting:Data exhibit a significant degree of scatter. The strategy is to derive a single curve that represents the general trend of the data.Data is very precise. The strategy is to pass a curve or a series of curves through each of the points.

In engineering two types of applications are encountered:Trend analysis. Predicting values of dependent variable, may include extrapolation beyond data points or interpolation between data points.Hypothesis testing. Comparing existing mathematical model with measured data.

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by Lale Yurttas, Texas A&M UniversityChapter 4*Arithmetic mean. The sum of the individual data points (yi) divided by the number of points (n).

Standard deviation. The most common measure of a spread for a sample.

orMATHEMATICAL BACKGROUND

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by Lale Yurttas, Texas A&M UniversityChapter 4*Variance. Representation of spread by the square of the standard deviation.

Coefficient of variation. Has the utility to quantify the spread of data.

Degrees of freedom

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by Lale Yurttas, Texas A&M UniversityChapter 4*LINEAR REGRESSIONFitting a straight line to a set of paired observations: (x1, y1), (x2, y2),,(xn, yn).where: a1- slopea0- intercepte- error, or residual, between the model and the observationsMean values

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by Lale Yurttas, Texas A&M UniversityChapter 4*Criteria for a Best FitMinimize the sum of the residual errors for all available data:

n = total number of pointsHowever, this is an inadequate criterion.Minimize the sum of the absolute values:

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by Lale Yurttas, Texas A&M UniversityChapter 4*Minimizes the sum of residuals

Minimizes the sum of absolute values of residuals

Minimizes the max error of any individual point

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by Lale Yurttas, Texas A&M UniversityChapter 4*Best strategy is to minimize the sum of the squares of the residuals between the measured y and the y calculated with the linear model:

Yields a unique line for a given set of data.

Standard error of the estimate, Sy/x:

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by Lale Yurttas, Texas A&M UniversityChapter 4*Figure 17.3

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by Lale Yurttas, Texas A&M UniversityChapter 4*Linear regression with small residual errors

Linear regression with large residual errors

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by Lale Yurttas, Texas A&M University*Estimation of ErrorsSt-Sr quantifies the improvement or error reduction due to describing data in terms of a straight line rather than as an average value.

For a perfect fit:Sr=0 and r=r2=1, signifying that the line explains 100 percent of the variability of the data.For r=r2=0, Sr=St, the fit represents no improvement.r2 -coefficient of determinationr correlation coefficientChapter 4

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Example 1Fit a straight line to the x and y values in table 1. Then compute the total standard deviation, standard error of the estimate and the correlation coefficient for the databy Lale Yurttas, Texas A&M UniversityChapter 4*Table 1

xiyi10.522.532.044.053.566.075.5

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by Lale Yurttas, Texas A&M UniversityChapter 4*POLYNOMIAL REGRESSIONSome engineering data is poorly represented by a straight line. For these cases a curve is better suited to fit the data. The least squares method can readily be extended to fit the data to higher order polynomials.For example: 2nd order polynomial

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Set of normal equations:

In matrix form:

by Lale Yurttas, Texas A&M UniversityChapter 4*

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Standard error:

Where: m = m-th order polynomialby Lale Yurttas, Texas A&M UniversityChapter 4*

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Example 2Fit a second-order polynomial to the x and y values in table 2. Then compute the total standard deviation, standard error of the estimate and the correlation coefficient for the data.by Lale Yurttas, Texas A&M UniversityChapter 4*Table 2

xiyi02.117.7213.6327.2440.9561.1

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by Lale Yurttas, Texas A&M UniversityChapter 4*MULTIPLE LINEAR REGRESSIONUseful extension of linear regression in case where y is a linear function of 2 or more independent variables.For example:

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Set of normal equations:

In matrix form:

by Lale Yurttas, Texas A&M UniversityChapter 4*

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Standard error:

Where: m = dimension

by Lale Yurttas, Texas A&M UniversityChapter 4*

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Example 3The data from table 3 was calculated from the equationy = 5 + 4x1 3x2. Use multiple linear regression to fit this data.by Lale Yurttas, Texas A&M UniversityChapter 4*Table 3

x1x2y00521102.5291304637227

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by Lale Yurttas, Texas A&M UniversityChapter 4*GENERAL LINEAR LEAST SQUARESGeneral model:

Express in matrix form:

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Outcome / Solution to this model:

Equivalent to the normal equation developed previously3 solution technique: (i) Gauss Elimination; (ii) Choleskys ; (iii) matrix inverse

Sum squares of the residuals:by Lale Yurttas, Texas A&M UniversityChapter 4*Minimized by taking its partial derivative with respect to each of the coefficients and setting the resulting equation equal to zero

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NONLINEAR REGRESSIONIn engineering many cases where nonlinear model must be fit to data.General equation:

For example:

Gauss Newton method algorithm for minimizing the sum of squares of residual between data and nonlinear equations.by Lale Yurttas, Texas A&M UniversityChapter 4*

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ProceduresLinearized the original model with respect to the parameter (e.g. : 2 parameter)

In matrix form:by Lale Yurttas, Texas A&M UniversityChapter 4*

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Where:by Lale Yurttas, Texas A&M UniversityChapter 4*

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Applying linear least square theory; solve {A}

Improve values of parameters:

Repeated until the solution converges falls below an acceptable stopping criterion.

by Lale Yurttas, Texas A&M UniversityChapter 4*

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Sum of squares of the residualsby Lale Yurttas, Texas A&M UniversityChapter 4*

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Example 4Fit the function

to data in table 4. Use initial guesses of a0 = 1.0 and a1 = 1.0 for the parameters.by Lale Yurttas, Texas A&M UniversityChapter 4*Table 4

xiyi0.250.280.750.571.250.681.750.742.250.79

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