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Curve Fitting and Optimization

Introduction

Least Square Regression

Linear

Non-Linear

Curve Fitting and Optimization

Introduction

Data is often given for discrete values along a continuum. However, you may require estimates at points between the discrete values

Describes techniques to fit curves to such data to obtain intermediate estimatesestimates

General approaches for curve fitting : where the data exhibits a significant degree of error or noise, the strategy is to

derive a single curve that represents the general trend of the datathis nature is called least-squares regression

where the data is known to be very precise, the basic approach is to fit a curveseries of curves that pass directly through each of the pointsvalues between well-known discrete points is called

Data is often given for discrete values along a continuum. However, estimates at points between the discrete values

techniques to fit curves to such data to obtain intermediate

eneral approaches for curve fitting :where the data exhibits a significant degree of error or noise, the strategy is to derive a single curve that represents the general trend of the data. One approach of

squares regressionwhere the data is known to be very precise, the basic approach is to fit a curve or a series of curves that pass directly through each of the points. The estimation of

points is called interpolation

Three attempts to fit a best curve through five data points.

(a) Least-squares regression, (b) linear interpolation,

Three attempts to fit a best curve through five data

b) linear interpolation, (c) curvilinear interpolation

Two types of applications are generally encountered when fittingexperimental data: trend analysis and hypothesis testing.

Trend Analysis :represents the process of using the patternUsed to predict or forecast values of the dependent variableUsed to predict or forecast values of the dependent variable

Hypothesis Testing :an existing mathematical model is compared with measured datamodel coefficients are unknown, it may be necessary to determine values that best fit the observed data. On the other hand, if estimates of the model coefficients are already available, it mayvalues of the model with observed values to test theOften, alternative models are compared and the best one is selectedbasis of empirical observations.

Two types of applications are generally encountered when fittingexperimental data: trend analysis and hypothesis testing.

pattern of the data to make predictions. to predict or forecast values of the dependent variableto predict or forecast values of the dependent variable

is compared with measured data. If the are unknown, it may be necessary to determine values that

data. On the other hand, if estimates of the model coefficients are already available, it may be appropriate to compare predicted values of the model with observed values to test the adequacy of the model. Often, alternative models are compared and the best one is selected on the

Least Square Regression

(a) Data exhibiting significant

error,

(b) Polynomial fit

oscillating beyond the range of

the data

(b) Polynomial fit

oscillating beyond the range of(c) More satisfactory

result using the least-squares fit.

Linear Regression

The simplest example of a least-square approximation is fitting a straight line paired observation : (x1, y1), (x2, y2), . . . , (

y = a0 + a1x + e a0 = coefficients representing the intercept

a coefficients representing the slope a1 = coefficients representing the slope

e = error, or residual between the model and the observations

e = y - a0 - a1x

The error, or residual, is the discrepancy between the true value of y and the approximate value, a0 + a1x , predicted by the linear equation

square approximation is fitting a straight line paired ), . . . , (xn, yn).

coefficients representing the intercept

coefficients representing the slopecoefficients representing the slope

error, or residual between the model and the observations

he error, or residual, is the discrepancy between the true value of y and the predicted by the linear equation

Linear Regression

Criteria for a "Best" Fit

minimize the sum of the residual errors for all the available data

(a) minimizes the sum of the residuals,

minimize the sum of the residual errors for all the available data

of the residuals,

Linear Regression

Criteria for a "Best" Fit

(b) minimizes the sum of the absolute

values of the residuals,

b) minimizes the sum of the absolute

values of the residuals,

Linear Regression

Criteria for a "Best" Fit A third strategy for fitting a best line is the

technique, the line is chosen that minimizes the maximum distance that an individual point falls from the line

It should be noted that the minimax principle is sometimes wellfitting a simple function to a complicatedWilkes, 1969).

(c) Minimizes the maximum error of any individual point

third strategy for fitting a best line is the minimax criterion. In this is chosen that minimizes the maximum distance that an

principle is sometimes well-suited for fitting a simple function to a complicated function (Carnahan, Luther, and

Linear Regression

Minimize the sum of the squares of the residuals between the measured y and the y calculated with the linear model

the sum of the squares of the residuals between the with the linear model

Linear Regression

Least-Squares Fit of a Straight Line

To determine values for a0 + a1

Setting these derivatives equal to zero will result

in a minimum Sr

Squares Fit of a Straight Line

Setting these derivatives equal to zero will result

realizing that a0 = na0

where y and x are the means of y and x, respectively.y and x are the means of y and x, respectively.

Example

Fit a straight line to the x and y values in the first two columns

SOLUTION

Xi Yi Xi^2

1 0.5 1

2 2.5 4

3 2 9

4 4 164 4 16

5 3.5 25

6 6 36

7 5.5 49

jml X 28 jml Y 24 jml Xi^2 140

rata X 4 rata Y 3.428571

Quantification of Error of Linear Regression

the square of the residual represented the square of the discrepancy between the data and a single estimatebetween the data and a single estimatetendencythe mean

the square of the residual represents the square of the vertical distance between the data and another measure ofstraight line

Quantification of Error of Linear Regression

of the residual represented the square of the discrepancy between the data and a single estimate of the measure of central between the data and a single estimate of the measure of central

represents the square of the vertical distance between the data and another measure of central tendencythe

Regression data showing (a) the spread of the data around the mean of the dependent variable

data around the best-fit line. The reduction in the spread in going from

the right, represents the improvement due to linear regression

a) the spread of the data around the mean of the dependent variable and (b) the spread of the

fit line. The reduction in the spread in going from (a) to (b), as indicated by the bell-shaped curves at

linear regression

Polynomial Regression

For example, suppose that we fit a secondquadraticFor example, suppose that we fit a second-order polynomial or

where all summations are from i = 1 through n. Note that the above three equations are linear and have three unknowns: The coefficients of the unknowns can be calculatedobserved data.

The two-dimensional case can be easily extended to an polynomial as

the standard error is formulated asmth

= 1 through n. Note that the above and have three unknowns: a0, a1, and a2.

The coefficients of the unknowns can be calculated directly from the

dimensional case can be easily extended to an mth-order

mth-order polynomial

EXAMPLE

Fit a second-order polynomial to the data in the first

two columns

order polynomial to the data in the first

SOLUTION

Solving these equations through a technique such as Gauss

elimination gives

a1 = 2.35929, and a2 = 1.86071. Therefore, the least

quadratic equation for this

case is y = 2.47857 + 2.35929x + 1.86071x2

The standard error of the estimate based on the regression

polynomial is

Solving these equations through a technique such as Gauss

elimination gives a0 = 2.47857,

a1 = 2.35929, and a2 = 1.86071. Therefore, the least-squares

quadratic equation for this

y = 2.47857 + 2.35929x + 1.86071x2

The standard error of the estimate based on the regression