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by Lale Yurttas, Texas A&M Universit

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CURVE FITTING

Part 5 Describes techniques to fit curves (curve fitting to !iscrete

!ata to obtain inter"e!iate esti"ates#

There are t$o %enera& a''roaches t$o curve fittin%

) Data exhibit a significant degree of scatter# The strate%* is to !erive asin%&e curve that re'resents the %enera& tren! of the !ata#

) Data is very precise.The strate%* is to 'ass a curve or a series ofcurves throu%h each of the 'oints#

In en%ineerin% t$o t*'es of a''&ications are encountere!) Tren! ana&*sis# Pre!ictin% va&ues of !e'en!ent variab&e+ "a* inc&u!e

e,tra'o&ation be*on! !ata 'oints or inter'o&ation bet$een !ata 'oints#

) -*'othesis testin%# Co"'arin% e,istin% "athe"atica& "o!e& $ith"easure! !ata#

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Fi%ure PT5#.

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/athe"atica& 0ac1%roun!

2i"'&e 2tatistics3

In course of en%ineerin% stu!*+ if severa&"easure"ents are "a!e of a 'articu&ar quantit*+

a!!itiona& insi%ht can be %aine! b* su""ari4in% the!ata in one or "ore $e&& chosen statistics that conve*as "uch infor"ation as 'ossib&e about s'ecificcharacteristics of the !ata set#

These !escri'tive statistics are "ost often se&ecte! tore'resent) The &ocation of the center of the !istribution of the !ata+

) The !e%ree of s'rea! of the !ata#

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Arithmetic mean# The su" of the in!ivi!ua& !ata

'oints (*i !ivi!e! b* the nu"ber of 'oints (n#

Standard deviation# The "ost co""on "easure of a

s'rea! for a sa"'&e#

or

ni

n

yy

i

++.=

=

==

(

.

yyS

nSS

it

ty ( )

.3

= n

nyySii

y

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Variance# Re'resentation of s'rea! b* the square of

the stan!ar! !eviation#

Coefficient of variation# -as the uti&it* to quantif* the

s'rea! of !ata#

.

( 55

=

n

yyS

i

y De%rees of free!o"

6.77##y

Svc y=

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Fi%ure PT5#

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Fi%ure PT5#8

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9east 2quares Re%ression

Cha'ter .:

9inear Re%ression

Fittin% a strai%ht &ine to a set of 'aire!observations (x1, y1), (x, y),!,(xn, yn).

y"a#$a1x$e

a1% s&o'e

a#% interce'te% error+ or resi!ua&+ bet$een the "o!e& an! theobservations

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Criteria for a ;0est< Fit3

/ini"i4e the su" of the resi!ua& errors for a&&

avai&ab&e !ata

n = tota& nu"ber of 'oints

-o$ever+ this is an ina!equate criterion+ so is the su"

of the abso&ute va&ues

==

=n

i

ioi

n

i

i xaaye.

.

.

(

==

=n

i

ii

n

i

i xaaye.

.7

.

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Fi%ure .:#

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0est strate%* is to "ini"i4e the su" of the squares of

the resi!ua&s bet$een the "easure! * an! the *

ca&cu&ate! $ith the &inear "o!e&

>ie&!s a unique &ine for a %iven set of !ata#

= ==

===n

i

n

i

iiii

n

i

ir xaayyyeS. .

.7

.

("o!e&+"easure!+(

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9east?2quares Fit of a 2trai%ht 9ine3

[ ]

=

=

==

==

.7

.7

.

.

.

7

7

7(

7(

iiii

ii

iioir

ioi

o

r

xaxaxy

xaay

xxaaya

S

xaay

a

S

( )

( )xaya

xxn

yxyxna

yaxna

naa

ii

iiii

ii

.7

.

.7

77

=

=

=+

=

Nor"a& equations+ can be

so&ve! si"u&taneous&*

/ean va&ues

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Fi%ure .:#8

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3igure 17)/

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3igure 17)0

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1!Copyright 2! The M"#ra$%ill Co'panies, (n") *er'ission re+uire -or reprou"tion or isplay)

;Goo!ness< of our fit3

If

Tota& su" of the squares aroun! the "ean for the!e'en!ent variab&e+ *+ is St

2u" of the squares of resi!ua&s aroun! the re%ression

&ine is Sr St%Srquantifies the i"'rove"ent or error re!uction

!ue to !escribin% !ata in ter"s of a strai%ht &ine rather

than as an avera%e va&ue#

t

rt

S

SSr =

r?coefficient of !eter"ination

2qrt(r ) corre&ation coefficient

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For a 'erfect fit

2r=7 an! r=r5=.+ si%nif*in% that the &inee,'&ains .77 'ercent of the variabi&it* of the

!ata#

For r=r5

=7+ 2r=2t+ the fit re'resents noi"'rove"ent#

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Po&*no"ia& Re%ression

2o"e en%ineerin% !ata is 'oor&* re'resente!

b* a strai%ht &ine# For these cases a curve is

better suite! to fit the !ata# The &east squares

"etho! can rea!i&* be e,ten!e! to fit the !ata

to hi%her or!er 'o&*no"ia&s (2ec# .:##

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Chapter 17 1

Genera& 9inear 9east 2quares

{ } [ ]{ } { }

[ ]

{ }

{ }

{ } resi!ua&sE

tscoefficienun1no$n@

variab&e!e'en!enttheofva&ue!observe!>t variab&ein!e'en!entheofva&ues"easure!at the

functionsbasistheofva&uesca&cu&ate!theof"atri,

functionsbasis.are.7

55..77

+=

+

+++++=

&

'A&(

m, ),, ))

e)a)a)a)ay

m

mm

. 7

= =

=

n

i

m

*

*i*ir ayS

/ini"i4e! b* ta1in% its 'artia&

!erivative $#r#t# each of the

coefficients an! settin% the

resu&tin% equation equa& to 4ero

chap_17

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