PGE 310 - HW8 - Solution

# PGE 310 - HW8 - Solution - The University of Texas at...

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1 The University of Texas at Austin PGE 310: Formulation and Solution in Geosystems Engineering Homework #8: Curve Fitting and Interpolation By HosseinRoodi 1. Curve Fitting: (By HAND) A graduate student is looking for an analytical equation that can describe his lab data the best. In his experimental tests, he is changing parameter x, which can be something like the temperature of the test, and he is measuring parameter y, which can be something like the stress in the specimens. After investigation of different types of equations, he has come up with 4 final choices as follows: a) 1 1 b x a y ! " 1 1 b x a y ! " b) x b e a y 2 2 " x b a y 2 2 ) ln( ) ln( ! " c) 3 3 b x a y " ) log( ) log( ) log( 3 3 x b a y ! " No. x y X =x Y=y XY X 2 Y 2 y pred (y pred " y) 2 1 5 2.760 5 2.760 13.800 25 7.618 2.839 0.006 2 10 3.050 10 3.050 30.500 100 9.303 3.070 0.000 3 15 3.577 15 3.577 53.655 225 12.795 3.301 0.076 4 20 3.355 20 3.355 67.100 400 11.256 3.532 0.031 # 50 12.742 165.055 750 40.971 # 12.742 0.114 Slope =(a 1 )= 0.046 a 1 = 0.046 S r = 0.114 Interc =(b 1 )= 2.608 b 1 = 2.608 r 2 = 0.701 Error Calc y=a 1 x+b 1 Part a 1 1 b x a y ! " No. x y X =x Y=ln(y) XY X 2 Y 2 y pred (y pred " y) 2 1 5 2.760 5 1.015 5.076 25 1.031 2.835 0.006 2 10 3.050 10 1.115 11.151 100 1.244 3.054 0.000 3 15 3.577 15 1.275 19.118 225 1.624 3.291 0.082 4 20 3.355 20 1.210 24.209 400 1.465 3.545 0.036 # 50 4.615 59.554 750 5.364 # 12.725 0.124 Slope = (b 2 ) = 0.015 a 2 = 2.632 S r = 0.124 Interc = ln(a 2 ) = 0.968 b 2 = 0.015 r 2 = 0.721 Part b Error Calc ln(y)=b 2 x+ln(a 2 ) x b e a y 2 2 " No. x y X =log(x) Y=log(y) XY X 2 Y 2 y pred (y pred " y) 2 1 5 2.760 0.699 0.441 0.308 0.489 0.194 2.770 0.000 2 10 3.050 1.000 0.484 0.484 1 0.235 3.116 0.004 3 15 3.577 1.176 0.554 0.651 1.38319 0.306 3.338 0.057 4 20 3.355 1.301 0.526 0.684 1.69268 0.276 3.506 0.023 # 4.176 2.004 2.127 4.56443 1.012 # 12.730 0.084 Slope = (b 3 ) = 0.170 a 3 = 2.107 S r = 0.084 Interc = log(a 3 ) = 0.324 b 3 = 0.170 r 2 = 0.814 Part c log(y)=b 3 log(x)+log(a 3 ) Error Calc 3 3 b x a y " X Y interceptslope X Y interceptslope

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2 d) x b x a y ! " 4 4 4 4 4 1 1 1 a x a b y ! " Note that although it is possible to calculate the error, (S r ), in X-Y space (i.e 2 )) ( ) ( ( i Ypredict i Y \$ ), but as shown in the above tables, it would be better to obtain it in original space x-y (i.e. 2 )) ( ) ( ( i ypredict i y \$ ). Here is a good sample of his measured data: % & %& 355 . 3 577 . 3 050 . 3 760 . 2 20 15 10 5 " " y x Basedon this sample data help him to find which equation fits better to the data. Use linearization and find the coefficients of the fitted curve for each type of equations. Then compare these four types of equations based on the r S and 2 r values. Based on the obtained values for S r and r 2 it is obvious that Equation 4 can represents the best (among the four equations) fit to the data. Because it returns larger value for r 2 and lower value for S r . You should do this problem by HAND. You may use EXCEL to make your calculations easy and faster, but you are not supposed to use curve fitting feature of EXCEL.
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## This note was uploaded on 04/12/2011 for the course PGE 310 taught by Professor Klaus during the Spring '06 term at University of Texas at Austin.

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PGE 310 - HW8 - Solution - The University of Texas at...

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