Chapter 8Chapter 8

Curve FittingCurve Fitting

ContentContent

•Introduction•Linear Regression•Multidimensional unconstrained•Example

Introduction (1)Fit curves (curve fitting) to discrete data to obtain

intermediate estimates.

Two general approaches :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.

Two types of engineering applications:Trend analysis (Extrapolation or Interpolation).Hypothesis testing.

Introduction (2)

Introduction (3)Mathematic Background: Simple StatisticsThese descriptive statistics are most often selected to

representThe location of the center of the distribution of the data

(mean)

The degree of spread of the data. (standard deviation)

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Introduction (4)Mathematic Background: Simple Statistics (cont’d)Variance. Representation of spread by the square of the

standard deviation.

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

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Introduction (5)

Linear Regression (1)Fitting a straight line to a set of paired observations:

(x1, y1), (x2, y2),…,(xn, yn).

y=a0+a1x+e

a1- slope

a0- intercept e- error, or residual, between the model and the observations

Linear Regression (2)Criteria for a “Best” Fit/Minimize the sum of the residual errors

for all available data:

n = total number of pointsHowever, this is an inadequate criterion,

so is the sum of the absolute values

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Linear Regression (3)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.

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Linear Regression (4)

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Normal equations, can be solved simultaneously

Mean values

Derivation of a0 and a1

Linear Regression (5)

Linear Regression (6)“Goodness” of our fit/ IfTotal sum of the squares around the mean for the

dependent variable, y, is St

Sum of the squares of residuals around the regression line is Sr

St-Sr quantifies the improvement or error reduction due to describing data in terms of a straight line rather than as an average value.

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r2-coefficient of determination

Sqrt(r2) – correlation coefficient

Linear Regression (7)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.

Interpolation (1)Estimation of intermediate values between precise data

points. The most common method is:

Although there is one and only one nth-order polynomial that fits n+1 points, there are a variety of mathematical formats in which this polynomial can be expressed:The Newton polynomialThe Lagrange polynomial

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Interpolation (2)Newton’s Divided-Difference Interpolating Polynomials

Linear Interpolation/The simplest form of interpolation, connecting two data

points with a straight line.

f1(x) designates that this is a first-order interpolating polynomial.

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Linear-interpolation formula

Slope and a finite divided difference approximation to 1st derivative

Interpolation (3)

Interpolation (4)Quadratic InterpolationIf three data points are available, the estimate is

improved by introducing some curvature into the line connecting the points.

A simple procedure can be used to determine the values of the coefficients.

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Interpolation (5)

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Bracketed function evaluations are finite divided differences

General Form of Newton’s Interpolating Polynomials

Interpolation (6)Lagrange Interpolating PolynomialsThe Lagrange interpolating polynomial is simply a

reformulation of the Newton’s polynomial that avoids the computation of divided differences:

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Interpolation (7)Lagrange Interpolating Polynomials(cont’d)

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Interpolation (8)Coefficients of an Interpolating PolynomialAlthough both the Newton and Lagrange polynomials are

well suited for determining intermediate values between points, they do not provide a polynomial in conventional form:

Since n+1 data points are required to determine n+1 coefficients, simultaneous linear systems of equations can be used to calculate “a”s.

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Interpolation (9)Coefficients of an Interpolating Polynomial

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where “x”s are the knowns and “a”s are the unknowns.