CT5 Regress

7/27/2019 CT5 Regress

1/19

Computational TechniquesComputational TechniquesM d lM d l R r i n nd In r l i nR r i n nd In r l i n

Dr. Niket Kaisare

Indian Institute of Technology - Madras


2/19

Given the following data:

x 0.8 1.4 2.7 3.8 4.8 4.9

Regression:

tain a straig t ine

that best fits the data


3/19

Given the following data:

x 0.8 1.4 2.7 3.8 4.8 4.9

Interpolation:

oin t e ots an

find a curve passing

through the data.


4/19

..

x 0.8 1.4 2.7 3.8 4.8 4.9

y 0.69 1.00 2.02 2.39 2.34 2.83

,

chosen function to data

=

In inter olation iven finite amount of data we

. .

are interested in obtaining new data-points

within this range.

Atx = 2.0,y = 1.87


5/19

Example: Kinetic Rate ConstantsExample: Kinetic Rate Constants

egress onegress on Experiments with conversion measured at

various temperatures

6

acekrRT)(

0

2

4

)

)ln(

1

)ln()ln( 0 ac

E

kr

-2

0log(

Slope = E/R

-3.5 -3 -2.5 -2 -1.5

x 10-3

-4

-1/T

Intercept = ln(k0)


6/19


7/19

Letx be an independent variable

andy be a dependent variable

Given the data:

NN yxyxyx ,,,,,, 2211

Find parameters to get a bestfit curve

;x


8/19

..

Choose a function Various standard

For a given , obtain

The interpolatingthe values from

the model

function passes

through all the points

iy

The best minimizes

the error

Can be used to fill-

in the data at newyy

points


9/19

..

Curve Fittin oinin the dots

Obtain a functional

form to fit the data

Obtain the value ofy

at intermediate oint


10/19


11/19

Model: ;xfy

Actual Data: NN yxyxyx ,,,,,, 2211 re iction:

Errors:

;ii xy

iii yye

iii exfy

;

Mean / Variance:

yi

Nxx i )1( Nxxs

ix

xs


12/19

--

Letx1,x2, ,xn be n variables.

Let there be N data points for each:

xxx

n yxxx ,;,,, 222212

NnNNN yxxx ;,,, 21

Obtain for

x;fy


13/19


14/19

yawux 101111

yawux 212221

NNNNyawux

31

Least Squares

TT 1


15/19

Exam le: S ecific heat as a function of T

Methane: cp = 85.8 + 1.126e-2 T 2.1141e-6 T2

Example: Antoines vapor pressure relationship

cTapsat

ln


16/19

21

2111 xxx

32Model:

3

2

2

221 xxxX

321 NNN xxx

x

xx

xx

1

11

1

ln1

ln11

xaa

xaay 32

10 )ln(

o e :

X

2

x xN


17/19


18/19

Niiixxxxxxxxxx

P...... 1121

j

Niiiiiii

xx

xxxxxxxxxx ...... 1121

ij ji xx

The interpolating

polynomial becomes

NNPyPyPyxf ...)( 2211


19/19

yy

iiiiyiiy

xx

1

],1[]1,2[

,

ii

iiiyiiiyiiii

xx

2

],1,2[]1,2,3[123

,,

ii xx 3

...,1,2,3,4,1,2,3,1,2, 32110 ycycycyc

...))(()()( 212110 xxxxcxxccxf

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