Numerical Tech for Interpolation & Curve Fitting

7/27/2019 Numerical Tech for Interpolation & Curve Fitting

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Advanced Mathematical Methods

for Civil Engineering Applications

Wonsiri Punurai, PhDWonsiri Punurai, PhDDepartment of Civil EngineeringRoom 6391, EG Building 3Faculty of Engineering, Mahidol UniversityClass Web: www.egmu.net/~civil/wonsiri

Numerical techniques for interpolation & curve fitting


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Basic Problems in relation to CE work

Dial Gauge

0

75

25

50

Experimental Data


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Curve Fitting

{

Linear Regression{ Polynomial Regression

{ Multiple Linear Regression


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Linear Regression

Errors do exist between model and observation.

Find the best line which minimizes the sum of error for all data

( )


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Least-square Fit of a Straight Line

{ Minimize sum of the square of the errors

{ Differentiate with respect to each coefficient:


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How good is our fit?


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Confidence Interval (CI)


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Linear Regression with MATLAB

{ Matlab function regress addresses the

multiple linear regression problem based onthe least squares approach.

{ Matlab function regress estimates the

model parameters and performing theregression model automatically.

{ Theregress function requires that thematrix of independent variables include a

column of ones so that the model containsa constant term.


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Example use of functionregress

{ Given the following independent

observations for variable Q and t:

19.919151412.5127.95.502Q

10987654321t


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1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

16

18

20

t

Q

19.919151412.5127.95.502Q

10987654321t

Figure 1: Plot of variable Q vs. t


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Now one wants to check if there exists a linear

relationship between Q and t (e.g. one wants to

predict Q when t is 11 and 12

>> Use the regress function to estimate the model

parameter and perform regression analysis.

>>The r e g r e s s function requires that the matrix of

independent variables include a column of ones so that

the model contains a constant term.

Create x in matlab


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Now lets apply theregress function to our data (note that Q and t have

to be transposed according to the function requirements):

Vector of regressionQ=b1+tb2+

Confidence intervalsDefault is 95% CIIf contains zero, theconstant shall be excluded.

95% CI for b195% CI for b2


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Discussion

As a result, our model that can be used for prediction Q for given t look as follow

Q=2.0081t

Q


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1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

t

Q

Data

Model

Figure 2 Original data and estimated regression model


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Q

Qt

Qt-1


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Q

Q

Qt-1Qt


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Curve Fitting

{ Linear Regression

{ Polynomial Regression

{ Multiple Linear Regression


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Polynomial Regression

0 =

Model equation

Sum sq. error


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Normal Equations


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Polynomial Regression with MATLAB


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Solving previous example using Matlab

Fit a 2nd orderpolynomial


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Matlab Polyval function


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Polyval Example


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Error Bounds


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Multiple Linear Regression


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Multivariate Fit in Matlab


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Example

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Numerical Tech for Interpolation & Curve Fitting