CurveFitting
In manybranchesofappliedMathematics
Engineering and sciences we come across
i i a n be n sets
let Cai y
variables
numericalvalues
C
n
dependent
two
C
and
independent
Forexample it is
of
of
y
known that the speed n'of a shipvaries with the horsepower p ofan engineaccordingto
theformula p a tbr Here a and b is constant tobe determine For this purpose
we takeseveralsets ofreadings ofspeeds and the correspondinghorsepowers The
problemis tofind the best value for a'and b using the observedvaluesof n and p
Thus thegeneral problem is tofind a suitablerelation or law that mayexistbetweenth
twovariables xandy from agivenset of observedpairedvaluesCmi yD i 1,2 3 n
experiments and problems whichinvolvestwo variables
Sucha relationconnectingthetwovariables onand y is known as Empirical law
Theprocessoffindingtheequationofthecurveofbestfitwhichmaybemostsuitable forpredic
theunknownvalues isknownascurvefitting
Methodsofcurve Fitting
Following arestandardmethods forfitting a curve
Cis graphicalmethod
Crismethodof group averages
Ciii Methodofmoments Civ MethodofLeastsquares
Linearlaw
supposetherelationshipbetweenthevariable n and y is linear and of theform
y an tb Thu
thecurvetobefittedforthegivendatais astraightline In simplecases a straight linemay b
fittedbyeye But if thepointsare scatteredthen it becomesunreliable Hence we use some
mathematicalprinciplestodetermine thevalues of a'and b But it mustbenoted thatgiv
datamaynotfollow a linearlaw Insuchcases it maybepossibletofindsomeothernonlinear
lawwhich canbereduced to linear lawbysuitable transformation
Thefollowing aresomeLawswhicharereducibleto linearLaw
1
y
an't b
Taking x
Inthiscase wecannotapply commonlogarithm
logo asthereisn't lawfor log m n so we
and y y the
n
can resolve thisproblemdirectly
above law becomes
2
y arb
iiiii
TakingcommonlogarithmsClog
sides
iii weget
log y
logoy
Log
carb
logo a t log
substituting
ab
7
If nonlinear equationconta
power multiplication or exponet
on both
Cmn 109m logn then go for commonlogarithm
109
log 7 logm logn
Logmm
nlogm
b y ab x
loga x
y logoy x logionNote Suppose B isknownvalue in case2
A b B log a
Then theequation becomes
y AX B Whichis LinearLaw
we canfind
thevalueofa'directlyin calculator pressing shift1
i.e CForproblem B log a logoCB valueo
a
aebn
Taking commonlogarithmsClogo onbothsides
3 y
e
a can be
Inthiscasevalueof
foundbyapplying a log
EB
log y log Caeb
log Ca t logo ebm
log y
log Cy
log Ca balog e
bnlog e log Ca
log Cy
Putting Y log Cy x
n
we
A blogwe B
calculator
valueof e incalculator shiftInc
or Alpha in C271828.2
log Ca
4 my a
Applying commonlogarithmonbothsides weget
logo ayn
logoa
nlogio y
logoCy
logo n t logioa
logo n log yn
logo n t nlogio y
logoa
logioa
Inlogo n
Putting Y logo y
A th and B
Inlog
a
x logo n then
Inlog a
weget y AX B whichis linearequation
Exercise 1.1
convert thefollowing equationsintolinearform
a
Fbn
ath
y
É
y
If
a bn
y My
n
yn
X N
A b and B A
an
b
y yn X n Aca and b b
Here y Ax B is linearegn
c
d
ye at buy
I
g
g
Y
Here
g
Y
bn
A
b and B 1
Y AX TB is linear equation
nB b
is in aneg
go
Y
Applying logarithmonbothsideweget
Logiontylogioa
109iob
logion loglob
F F
ya
i X
N
A
J
B
a
Here Y AX TB is lineareqn
9N t b22
In
bn
a
but a
Y
g mad b
In
In
Y
ÉÉax
to itself
ylogioa
a
of
but 1
Xen
ca
A Ib
In
h
Xen
and Ba
y ab
applying logarithmon bothsid
Logy logcab
ligatalogiob
logy
Y logioy X N A logiob
B
log