Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l. 6,
N
o
.
3
,
Ju
n
e
201
6,
p
p
.
9
7
4
~
979
I
S
SN
: 208
8-8
7
0
8
,
D
O
I
:
10.115
91
/ij
ece.v6
i
3.9
155
9
74
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
Machine Learning Techniques
on Multidimensional Curve
Fitting Data Based on
R- Square and Chi-Square
Meth
ods
Vidyullatha P
,
D
.
Ra
jesw
ar
a Ra
o
Departement
of Computer
Scien
ce & Engi
neerin
g, KL University Guntur, India
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Sep 7, 2015
Rev
i
sed
No
v
23
, 20
15
Accepted Dec 10, 2015
Curve fitting
is one of th
e pro
cedures
in dat
a
anal
ysis
and is
helpful
for
prediction analysis showi
ng gra
phically
how the data poin
t
s ar
e related to
one another whether it is in linear or
non-linear model. Usually
,
the curve fit
will find
th
e
co
ncentr
ates
along
the
cu
rv
e
or it
will
just
use to sm
ooth
the
data and upgr
ade the presen
ce of the plot
. Curve fitting
checks the
relationship b
e
tween ind
e
pend
ent var
i
ab
les and
dependen
t
variables with
the
objec
tive
of
char
act
eriz
ing
a goo
d fit
m
odel.
Cur
v
e fi
tting
finds
m
a
them
atic
al
equation th
at b
e
st fits giv
e
n info
rmati
on. In
this
paper, 150 unor
ganized data
points of enviro
nmental var
i
ables are
used to develop Lin
ear
an
d non-linear
data modelling
which are
evalu
a
ted b
y
u
tilizing
3 dimensional ‘
S
ftool’ and
‘Labfi
t’
m
ach
in
e learning t
echn
i
ques
.
In Line
ar
m
odel, the bes
t
es
tim
ations
of the
co
effic
i
en
ts are
re
ali
zed
b
y
the
estim
a
tion
of R- squar
e
tur
n
s in to
on
e
and in
Non-Lin
e
ar m
odels
wi
th l
eas
t Ch
i-s
quare
are
the
cri
t
er
ia
.
Keyword:
C
h
i
-
s
qua
re
Cu
rv
e fit
In
terpo
l
an
tlin
ear
Labfit
Surface fitting tool
Copyright ©
201
6 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Vid
y
u
llath
a P,
Depa
rt
em
ent
of C
o
m
put
er
Sci
e
nce a
n
d
E
ngi
neeri
n
g
,
K L Un
iv
ersity,
V
a
dd
esw
a
r
a
m
5
225
02
, Gun
t
ur
D
i
st
r
i
ct, Andh
r
a
Pr
ad
esh
,
Ind
i
a.
Em
a
il: lek
a
n
a
04
cu
ty@g
m
a
il.c
o
m
1.
INTRODUCTION
Th
e an
alysis o
f
d
a
ta m
a
in
l
y
fo
cu
ses
on th
e relation
s
h
i
p
o
f
g
i
v
e
n
v
a
riab
les. Stat
isticall
y
, th
e
rel
a
t
i
ons
hi
p i
s
m
easured
by
u
s
i
n
g
co
rrel
a
t
i
o
n.
The
st
an
dar
d
m
e
t
hod
usi
n
g i
n
t
h
e c
o
rrel
a
t
i
on m
odel
i
s
P
earso
n
m
e
t
hod i
n
w
h
i
c
h co
ef
fi
ci
ent
are l
i
m
it
ed b
e
t
w
een m
i
nus
o
n
e a
n
d
pl
us
o
n
e.
The
r
e
w
o
n
’
t
be a
n
y
re
l
a
t
i
o
n
bet
w
ee
n dat
a
a
nd y
i
el
d v
a
ri
ab
l
e
s i
f
co
efficient is zero.
The
r
e is an abs
o
l
u
te relatio
n
ex
istin
g
if co
effici
en
t is
o
n
e
. On
th
e
off ch
an
ce th
at
free v
a
riab
le(x
) i
n
creases d
e
p
e
nd
en
t
v
a
riab
le(y
) will also
in
cre
m
en
ts p
r
ecisely i
n
lin
ear relation
.
So
m
e
th
in
g
else, if x
d
i
min
i
sh
es y exp
a
nd
s
th
en
th
e
relatio
n
is called
n
e
g
a
tiv
e lin
ear an
d
t
h
e
coefficient is
m
i
nus
one
. C
u
rve
fitting is a
m
o
re elevat
ed am
ount
of num
erical stru
cture tha
n
relationshi
p.
Th
e u
s
efu
l
n
e
ss o
f
cu
rv
e fitting
m
a
in
ly ai
ms
at fo
rm
u
l
atio
n
o
f
a m
a
th
e
m
ati
cal fu
n
c
tion
b
y
u
s
in
g
typ
e
of in
pu
t
d
a
ta. Differen
t
so
rts of cu
rv
es su
ch
as p
a
rametric cu
rv
es,
i
m
p
licit
cu
rv
es an
d
subd
iv
isi
o
n
cu
rv
es are
u
tilized
for fittin
g
.
Fittin
g
a su
itab
l
e cu
rv
e or
m
o
d
e
l to
a p
r
og
ression
o
f
d
a
ta p
o
i
n
t
s is a
maj
o
r n
e
cessity in
n
u
merou
s
fi
el
ds,
f
o
r
i
n
st
ance,
com
put
e
r
gra
p
hi
cs, im
age pr
ocessi
ng
and dat
a
m
i
ni
ng. T
h
ere are
di
ffere
nt
t
y
pes o
f
m
o
d
e
ls to
fit t
h
e curv
e. Th
e d
i
fferen
t
m
o
d
e
ls
are lin
ea
r,
p
o
l
y
nom
i
a
l
of vari
ous
deg
r
ees,
p
o
we
r fi
t
,
l
o
ga
ri
t
h
m
i
c
cu
rv
e
fit and
no
n-lin
ear curve fits. Th
e cu
rv
e fitting
n
o
t
ju
st fits the uno
rg
an
ized
d
a
ta in
to
v
a
riou
s
m
o
d
e
ls
ad
d
ition
a
lly p
e
rfo
r
m
s
v
a
ri
o
u
s
task
s su
ch
as to
redu
ce th
e no
ise,
find
t
h
e m
a
th
e
m
a
tica
l
relatio
n
s
h
i
p
a
m
o
n
g
v
a
riab
les and
assessm
en
t th
e
q
u
a
lities b
e
tween
d
a
ta sam
p
les. Fo
r th
e cu
rv
e fittin
g
m
o
d
e
llin
g
pro
cesses,
Wenn
i Zh
eng
etal [1
] p
r
op
osed
a B-sp
lin
e cu
rv
e fitting
m
e
th
od
in
v
i
ew
of th
e L-BFGS
o
p
tim
izat
io
n
strateg
y
on
di
s
o
r
d
e
r
l
y
i
n
f
o
rm
at
i
on p
u
r
pos
es
of
f
oot
p
o
i
n
t
pr
o
j
ec
tio
n
an
d d
e
m
o
n
s
trates th
at it is th
e sp
eed
i
est techn
i
qu
e
in
ev
ery cycle con
t
rasted
with
trad
itio
n
a
l
strateg
i
es.
Ya
ng etal [2]
desc
ribes s
p
line
re
presentation in c
u
rve
fitting m
e
thod.
W
eenie Z
h
e
ng [3] pr
opos
ed optim
i
zatio
n m
e
thod
for fast
fitting of B-spline c
u
rves to
u
norg
a
n
i
zed data p
o
i
n
t
s Francis etal [4
] ex
amin
ed
3D
p
a
rameter yield
cu
rv
e fitting
m
e
th
o
d
fo
r
pred
ictin
g t
h
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Ma
ch
in
e
Lea
r
nin
g
Techn
i
qu
es on
Mu
ltid
imensio
n
a
l
Curve Fittin
g
Da
ta
Ba
sed
o
n
.... (Vid
yu
lla
tha
P)
9
75
sho
r
t
an
d l
o
ng
t
e
rm
s st
ruct
ure
of G
o
ver
n
m
e
nt
securi
t
y
y
i
eld
s
. Paul Norm
a
n
etal [5] evaluated the unit values
fro
m
co
llected
in
fo
rm
atio
n
lik
e ag
e and
salary g
r
oup
s to
m
a
k
e
non
-lin
ear reg
r
ession
strat
e
g
i
es of curv
e
fitting
m
o
d
e
ls u
s
in
g
th
e SPSS software. Fox
j
an
d
Weisb
e
rg
[6
] p
o
rtrayed
fittin
g
o
f
n
on-p
a
ram
e
tric regression
m
o
d
e
ls u
tilizin
g
R prog
rammin
g
langu
ag
e
wh
ich
lik
ewise in
corpo
r
ated s
m
o
o
t
h
i
ng
sy
ste
m
s o
n
scatter p
l
ot
str
a
teg
i
es. Lian
Fang
etal [
7
]
p
r
esen
ted
a meth
od
fo
r
p
r
o
d
u
c
ing
a sm
o
o
t
h
e
n
p
a
r
a
m
e
tr
ic cu
rv
es on
unor
d
e
r
e
d
dat
a
p
o
i
n
t
s
f
o
c
u
s
on
2
D
dat
a
.
Ana
n
dt
hi
rt
ha e
t
al
[8]
w
o
r
k
e
d
on
speec
h
dat
a
,
ap
pl
i
e
d c
o
r
r
el
at
i
on c
o
ef
fi
ci
ent
an
d
curve fitting
m
e
thod to foc
u
s the va
rying degrees of
e
x
trem
e speech disabilities to
com
p
are with norm
al
child. In
this pape
r,
the
surface fitting t
ool
, which is a
3D represe
n
ting data tool is
utilized for fitting the
u
norg
a
n
i
zed
en
v
i
ron
m
en
tal
d
a
ta po
in
ts and
p
l
o
ttin
g
th
e
g
r
aph
s
.
Gen
e
rally in
MATLAB, b
a
sic fitti
n
g
t
o
ol
,
curve fitting tool and surface fitting tool
are
available to fit the data. The
first two fitting tools are 2D
whe
r
e
the “S
ftool”, in Matlab R2010b a
n
d La
bfit te
chni
que
s are
3dim
ensional da
ta fitting m
e
thods
.
2.
R
E
SEARC
H M
ETHOD
2.
1.
Ch
oosi
n
g a C
u
rve Fi
t Mo
de
l
C
h
o
o
si
n
g
a ki
n
d
of
c
u
r
v
e fi
t
n
o
rm
al
l
y
depen
d
s up
o
n
t
h
e
t
y
pe of dat
a
. In
t
h
i
s
pa
pe
r, fo
r b
e
st
cur
v
e fi
t
,
sftoo
l
in
Matla
b
and
Labfit are u
tilized
. In lin
ear
m
o
d
e
l, for min
i
m
i
zin
g
th
e su
m
o
f
th
e sq
u
a
res, th
ere are
di
ffe
re
nt
m
e
t
hods
nam
e
l
y
Gaus
s-
Newt
on
m
e
t
hod,
G
r
a
d
i
e
nt
desce
n
t
m
e
t
hod an
d
Leven
b
er
g
-
M
a
rq
uar
d
t
Meth
od
. In
non
-lin
ear m
o
d
e
l, co
effi
cients a
r
e calculated a
t
minim
u
m
C
h
i
-
sq
uare val
u
e.
Data fittin
g
is th
e
p
r
o
c
ed
ure
o
f
fittin
g
m
o
d
e
ls to
info
rm
atio
n
an
d inv
e
stig
atin
g th
e exactn
e
ss
o
f
th
e fit. Sp
ecialists and
research
ers
use in
form
at
io
n
fitting
p
r
oced
ures, in
cl
u
d
i
n
g
scien
tific
m
a
th
e
m
ati
cal state
m
en
t
s
and
nonparam
etric techniques, t
o
m
odel gaine
d
inform
at
i
on.
M
A
TLAB
®
gi
ves y
o
u
a c
h
ance t
o
im
po
rt
and
p
i
ctu
r
e you
r i
n
form
at
io
n
,
and p
e
rfo
rm
essential fittin
g
m
e
t
h
od
s. Th
e cu
rve fit m
o
d
e
ls are no
t on
ly fit th
e
d
a
ta
but
al
s
o
red
u
ce
t
h
e
noi
se
an
d
sm
oot
hen t
h
e
d
a
t
a
[8]
.
2.
2.
LAB Fit
to
o
l
The La
bfit is programm
ing package
cu
st
o
m
ized
fo
r testing an
d
t
r
eatm
e
n
t
o
f
d
a
ta. In
LAB Fit, th
ere
are num
e
rous
favora
ble circum
s
t
ances like treating
of co
m
p
arab
le informatio
n
,
n
on-practically
id
en
tical d
a
ta,
d
i
scov
er pro
b
a
b
ilities fo
r a few m
o
v
e
m
e
n
t
s, fo
cu
s eng
e
n
d
e
red m
i
s
t
ak
es,
plo
t
2D an
d 3D
g
r
aph
s
, and
execu
te
a few estim
a
t
i
o
n
s
in
an
arran
g
e
m
e
n
t
of linear co
m
p
ar
ison
an
d Curv
e
Fittin
g
.
Th
e Labfit is m
a
in
ly in
ten
d
e
d
for cu
rv
e fittin
g
u
s
ing
no
n
l
in
ear reg
r
ession
. Th
e Lab
f
it
so
ftware can b
e
u
tilized
t
o
fit cu
rv
e up to
six
aut
o
nom
ous
va
ri
abl
e
s a
n
d
o
n
e
su
bo
r
d
i
n
at
e
v
a
ri
abl
e
. T
h
e
r
e
are ab
ou
t
2
8
0
fu
n
c
tion
s
with two free
v
a
riab
les in
Labfit lib
rary
.
The
u
s
ers
o
f
t
h
e s
o
ft
ware
ca
n als
o
c
o
m
p
o
s
e th
eir
own
p
a
rticu
l
ar
fit fu
nctio
n
i
n
Lab
f
it
. It is
h
e
lpfu
l to
fit cu
rv
es in
bo
th
2D and
3D cases. Th
e Lab
F
it
is u
s
ed
to
treat
d
i
stin
ctiv
e sorts o
f
d
a
ta su
ch
si
m
ila
r
d
a
ta, non
-p
ract
ically
id
en
tical d
a
ta
and error propa
g
ation.
The La
bfit has
g
i
v
e
n
10
scien
tific m
a
th
e
m
atical
eq
u
a
tion
s
wh
ich
were shown
in
tab
l
e 1
fo
r
b
e
tter fittin
g
of g
i
v
e
n
d
a
ta as in
d
i
cated
b
y
Ch
i-sq
u
a
re estimates.
Fro
m
th
e ab
ove tab
l
e, it
is cl
early ev
id
en
t th
at th
e eq
u
a
ti
on
(1) b
e
st fits th
e g
i
v
e
n
d
a
ta sin
ce it h
a
s
m
i
n
i
m
u
m
C
h
i
-
s
qua
re
va
l
u
e. S
u
bse
que
nt
t
o
g
u
ara
n
t
e
ei
ng t
h
e
be
st
fit nu
m
e
rical
m
a
th
em
a
tical state
m
ent, click on
"Resu
lts: d
i
agra
m
"
to
g
e
t a
3D ch
art as
indicated in
Figure
5.
2.
3.
Surf
a
ce Fi
ttin
g
T
ool in
M
a
tl
ab
For fitting t
h
e
curves a
n
d surfaces t
o
the
data poi
nts, surface Tool
box i
s
one of the a
pplications
pr
o
v
i
d
e
d
by
M
a
t
l
a
b [9]
.
T
h
e c
u
r
v
e fi
t
t
o
ol
b
o
x
gi
ves a c
h
a
n
ce t
o
pe
rf
orm
expl
orat
ory
i
n
fo
rm
ati
on anal
y
s
i
s
, fo
r
p
r
ep
ro
cessi
n
g
an
d
co
m
p
are can
d
i
d
a
te m
o
d
e
ls an
d
rem
o
v
e
o
u
tliers. It wi
ll p
r
ov
id
e th
e
lin
ear & no
n-l
i
n
ear
m
odel
s
and
a
l
so i
n
di
cat
e c
u
st
om
m
a
t
h
em
at
i
cal
equat
i
ons
.
It
al
so
sup
p
o
rt
s
n
o
n
-
param
e
t
r
i
c
m
odel
i
n
g
t
echni
q
u
es s
u
ch as spl
i
n
es
,
i
n
t
e
rp
ol
at
i
o
n
and sm
oot
hi
ng
of
dat
a
po
i
n
t
s
. M
a
ny
st
a
t
i
s
t
i
cal packag
es
suc
h
as R and
nu
m
e
r
i
cal
so
f
t
w
a
r
e
su
ch
as GNU
Scien
tif
ic
Lib
r
ar
y,
Map
l
e,
MA
TLA
B,
SciPy and
O
p
en
O
p
t ar
e
u
s
efu
l
for d
o
i
ng
cu
rv
e fittin
g
in
a v
a
ri
o
u
s
situ
atio
ns.
Th
ere
are ad
d
ition
a
l
p
r
og
ram
s
p
a
rticu
l
arly k
e
p
t
in
to
u
c
h
with
th
e curve fittin
g
.
The co
mman
d
Sftoo
l
o
p
e
n
s
curv
e fitting
applicatio
n
wh
ich g
i
v
e
s an
ad
ap
tab
l
e
interface a
nd c
u
rve can
be vi
ewed the
plots. Sftool [1
0] ha
s num
erous fa
vora
ble ci
rcumstances, for e
x
a
m
ple,
to
create, p
l
o
t
& co
m
p
are m
u
ltip
le fits and
also
au
to
m
a
tica
l
ly p
r
o
d
u
ces t
h
e m
a
th
e
m
atical
eq
u
a
tion
s
.
3.
R
E
SU
LTS AN
D ANA
LY
SIS
On
e m
u
st cho
o
se righ
t fittin
g
to
o
l
prior to
selectin
g
a d
a
ta set. By selectin
g
a wro
n
g
too
l
, th
e u
s
er i
s
g
o
i
n
g
to
g
e
t an
in
app
r
op
riat
e curv
e thu
s
, it is id
eal to p
i
ck
righ
t fittin
g m
o
d
e
l con
ting
e
n
t
up
on th
e
g
i
v
e
n
inform
ation. In gene
ral, each
m
odel
has its own
presum
ptions for c
o
m
puti
ng t
h
e fitting e
r
ror to
fit the curve
.
C
ont
i
n
ge
nt
o
n
t
h
e o
u
t
c
om
es, t
h
ree
para
m
e
ters are com
puted such
as sum
of
sq
uar
e
s
o
f
er
r
o
r (SSE
), ro
ot m
ean
sq
uar
e
d
er
ro
r
(RMSE)
an
d R-
squ
a
r
e
erro
r
(R
2
). T
h
ese t
h
ree statistical param
e
ters have
the ca
pacity to
give
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E
V
o
l
.
6,
No
. 3,
J
u
ne 2
0
1
6
:
97
4 – 9
7
9
97
6
d
e
fi
n
ite elu
c
idatio
n
with
resp
ect to
ri
g
h
t
n
e
ss o
f
curv
e
fit. If a m
o
d
e
l i
s
in
lin
ear, t
h
e co
nstrain
t
s su
ch
as
con
s
t
a
nt
, pa
ra
m
e
t
e
r and a p
r
edi
c
t
o
r
has a basi
c eq
uat
i
o
n
repr
esent
e
d a
s
y
= a0 + a1x1 + a
2
x
2
. F
o
r l
i
n
ear
equation, t
h
ere
is only one ba
sic form
whereas for nonlin
e
a
r there a
r
e many differe
n
t form
s such as
powe
r,
para
b
o
l
i
c
, exp
one
nt
i
a
l
et
c. In t
h
i
s
researc
h
wor
k
,
15
0 da
t
a
poi
nt
s of e
n
vi
r
onm
ent
a
l
vari
abl
e
s are t
a
ken t
o
represe
n
t a dat
a
m
odel usi
n
g surface
fitting tool in Matla
b [11] whic
h re
sults a linear data re
present
a
tion
m
o
d
e
l. In
th
is wo
rk
, two
dep
e
nd
en
t v
a
riab
les an
d
o
n
e
in
d
e
p
e
nd
en
t variab
le d
a
ta po
in
ts are tak
e
n
in
to
co
nsid
eration
fo
r fitting
th
e 3D grap
h
wh
ich are sho
w
n
in
fig
u
res 2-4
h
a
v
i
n
g
lin
ear in
fun
c
tio
n. Utilizin
g
the
surface fitting
tool, it is easy
to pl
ot an
d exa
m
ine the fits at the comm
and
line.
At fi
rst sight it is obvious that
a good fit s
h
ould m
i
nimize the so called
resi
duals
. T
h
e
R
2
measu
r
e is th
e
m
o
st g
e
n
e
rally u
tilized
and
rep
o
rted
m
easure
of e
r
r
o
r a
n
d g
o
o
d
n
e
ss o
f
fi
t
f
o
r l
i
n
ear m
odel
s
.
T
h
e est
i
m
at
i
on
of R
2
m
easures
go
o
dne
ss o
f
fi
t
whi
c
h
l
i
e
s
som
e
wher
e
aro
u
nd 0.
0
a
nd 1.
0.
Whe
n
R
2
reaches to
1, it is better to
stop
t
h
e c
u
rve
fit process a
n
d
the best
fi
t
res
u
l
t
s
t
o
m
a
ke
per
f
ect
pr
edi
c
t
i
ons
are
s
h
o
w
n i
n
fi
gu
re
s i
n
1 t
o
4. R
2
is d
e
term
in
ed b
y
u
s
ing
t
w
o
terms
nam
e
ly
SSE (s
um
of sq
ua
res
of e
r
r
o
rs
of
re
gressi
o
n
) a
n
d
SST
(sum
of s
qua
res
of
t
o
t
a
l
)
.
SSE i
s
c
o
m
put
e
d
fro
m
th
e su
m
o
f
th
e squ
a
res
o
f
th
e
ob
serv
ed
v
a
lu
e and
p
r
ed
icted
v
a
lu
e.
Th
is is also
cal
led
th
e
su
m
o
f
sq
uares
of
re
gres
si
o
n
.
SST i
s
cal
cul
a
t
e
d f
r
o
m
t
h
e sum
of t
h
e
s
q
u
a
res
of
t
h
e
o
b
s
e
rve
d
val
u
e
a
n
d m
ean val
u
e.
If t
h
e
cu
rv
e fit is goo
d, norm
a
l
l
y
SSE is sm
aller
th
an
SST. Then
, R
2
is calculated
u
s
ing
th
i
s
equ
a
tio
n
:
R
2
=
[1
-
(SSE/
S
ST)]
=1
.0
-4
1
65/
62
7
3
5
=
0.9
3
3
6
.
I
n
g
e
neral
,
M
a
t
l
a
b
gi
ves f
u
nct
i
o
ns
i
n
t
h
e fo
rm
of array
s
. It
i
s
di
f
f
i
c
ul
t
t
o
fi
n
d
out
fu
n
c
t
i
ons at
di
ffe
r
e
nt
p
o
i
n
t
s
whi
c
h are
n
o
t
co
v
e
red i
n
ar
ray
s
.
The
r
e are t
w
o
m
e
t
hods t
o
fi
nd
o
u
t
v
a
lu
es
b
e
tw
een
d
a
ta po
i
n
ts an
d
b
e
yond
d
a
t
a
p
o
i
n
t
s; in
terp
o
l
ation
and
ex
tr
ap
o
l
ation
r
e
sp
ectiv
ely. A
str
a
ig
h
t
l
i
n
e can
be
de
fi
ned
bet
w
een
t
w
o
dat
a
p
o
i
n
t
s
(x
1,
y
1
)
an
d
(
x
2, y
2
)
as
gi
ve
n
bel
o
w:
.
Thi
s
ba
si
c ge
o
m
et
ri
c form
ul
a i
s
used t
o
l
i
n
earl
y
i
n
t
e
rp
ol
a
t
e bet
w
ee
n t
w
o dat
a
poi
nt
s.
For
bet
t
e
r
un
de
rst
a
n
d
i
n
g,
y
= (y
1
+ y
2
)/
2 i
f
x i
s
m
i
dway
bet
w
ee
n x
1
and x
2
.
O
n
e s
h
o
u
l
d
be ca
ut
i
ous
w
h
i
l
e
usi
n
g t
h
i
s
form
ula. The l
i
near a
p
proxi
mation to the
curve
d
functio
n represe
n
ted
by the
das
h
e
d
line “a” is
pret
ty poor
si
nce t
h
e p
o
i
n
t
s
x = 0 a
nd
x =
1 o
n
w
h
i
c
h t
h
i
s
l
i
n
e i
s
dra
w
n
are far a
p
art
.
B
y
Addi
ng a
p
o
i
n
t
bet
w
een
0
and
1
at
x =
0.
5, t
h
en
we get
t
w
o-se
gm
ent
app
r
o
x
i
m
ati
on “c
” wh
ich
is
quite so
m
e
wh
at b
e
tter.
It can
also
b
e
obs
erved t
h
at that line “b” is
a pretty good
approxi
m
a
t
i
on si
nce t
h
e f
u
nc
t
i
on d
o
es
n’t
c
u
rve m
u
ch. T
h
e
sam
e
fo
rm
ul
a
m
a
y
be use
d
f
o
r e
x
t
r
apol
at
i
o
n al
s
o
.
It
can
be
sh
o
w
n
t
h
at
t
h
e
f
o
rm
ul
a f
N+1
= 2
fN
−
f
N
−
1.
If the end
values in the a
rray are f
N
−
1
and
f
N
. Matlab
is h
a
v
iing
its in
terpo
l
atio
n
code as “in
t
erp1
”. Fo
r ex
am
p
l
e,
if we
are ha
vi
n
g
a set
of dat
a
p
o
i
n
t
s
{x, y
}
and
we
are havi
ng a
d
i
ffere
nt
set
of
x-
val
u
es {
x
i
} f
o
r
whic
h we w
a
nt to
fi
n
d
out
c
o
r
r
es
po
n
d
i
n
g {y
i
}
valu
es,
we can
easily u
tilize th
e fo
llo
wi
n
g
th
ree fo
rm
u
l
ae for
in
terp1
co
mm
a
n
d
:
y
i
=interp1(x,y,x
i
, ‘lin
ear’)
y
i
=interp1(x,y,x
i
,
‘
c
ub
ic’)
y
i
=interp1(x,y,x
i
, ‘sp
lin
e’)
An ex
am
p
l
e cod
e
is
written
b
e
lo
w
fo
r
d
a
ta set represen
tin
g th
e si
n
e
fun
c
tion
.
clear;
% m
a
k
e
s th
e data set with
dx
dx
=
pi
/
5
;
x=
0:
dx:
2*
pi
;
y
=
si
n
(
x
)
;
% f
o
r
a fi
ne
x-
gri
d
x
i
=0:
dx/
20:
2*
p
i
;
% interpolate
on c
o
arse
grid
% ob
tain
y
i
val
u
es
% lin
ear i
n
terpo
l
atio
n
y
i
=in
t
erp1
(x
,y,x
i, ‘lin
ear’);
% p
l
o
t
th
e
d
a
ta and
th
e in
terpo
l
atio
n
Plo
t
(x
, y,
‘b*
’
, x
i
, y
i
,
‘r
-‘
)
title (‘Lin
ear In
terp
o
l
ation
’)
% cubic inte
rpolation
y
i
=interp1(
x
, y
,
x
i
, ‘cub
ic’);
% p
l
o
t
th
e
d
a
ta and
th
e in
terpo
l
atio
n
fig
u
re
p
l
o
t
(
x
,
y, ‘
b*’
,x
i
, y
i
, ‘r
-‘
)
title (‘Cu
b
i
c
Interpo
l
atio
n’)
% sp
lin
e in
terpo
l
atio
n
y
i
=i
nt
erp1
(x
,y
,
x
i
, ‘s
pline’
);
% p
l
o
t
th
e
d
a
ta and
th
e in
terpo
l
atio
n
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Ma
ch
in
e
Lea
r
nin
g
Techn
i
qu
es on
Mu
ltid
imensio
n
a
l
Curve Fittin
g
Da
ta
Ba
sed
o
n
.... (Vid
yu
lla
tha
P)
9
77
fig
u
re
pl
ot
(
x
,
y
,
‘
b
*’
,
x
i
,
y
i
,
‘r
-‘
)
title (‘Sp
lin
e Interpo
l
atio
n’)
In
case of
3
-
D
in
terpo
l
atio
n on
a
d
a
ta set of
{x
, y, z}
to
g
e
t th
e approx
im
a
t
e v
a
lu
es
o
f
z(x
,
y)
at po
in
ts
{x
i
y
i
},
w
e
u
s
e
an
y o
f
t
h
e f
o
llow
i
ng
z
i
= in
ter
p2(
x,
y, z,
x
i
, y
i
, ‘lin
ear’)
z
i
= in
terp2
(
x
,
y, z,
x
i
, y
i
,
‘c
ub
ic’)
z
i
= in
terp2
(
x
,
y, z,
x
i
, y
i
, ‘sp
lin
e’)
In t
h
i
s
pape
r,
t
h
e u
n
o
r
g
a
n
i
zed e
nvi
r
o
nm
ent
a
l
dat
a
po
in
ts a
r
e
u
s
ed
to
p
l
o
t
th
e c
u
r
v
e.
T
h
er
e
are
vari
ous interpolant
fitting methods a
r
e a
v
a
ilable base
d
on the
type
of
curves
a
n
d
surfaces. For Non-linea
r
m
odel
,
alm
o
st
10
0 dat
a
p
o
i
n
t
s
are t
a
ken t
o
get
a
m
a
t
h
em
at
i
cal
form
ul
a and
3D
gra
p
h b
y
usi
ng L
A
B
F
i
t
.
Thi
s
soft
com
put
i
n
g
t
echni
q
u
e has
eval
uat
e
d best
10 m
a
t
h
em
ati
cal
equat
i
o
n
s
an
d t
h
e eq
uat
i
on
Y=A
*
X2
*
*
(B
*X
1)
has gi
ve
n best
fi
t
si
nce
i
t
has got
l
o
we
st
C
h
i
-
sq
ua
re val
u
e f
o
r
gi
ven
dat
a
s
h
o
w
n
i
n
t
h
e be
l
o
w Tabl
e 1.
Tabl
e
1. T
h
e
L
i
st
of M
a
t
h
em
at
i
cal
Funct
i
o
ns
i
n
La
bfi
t
M
e
t
h
od
S.No
Mathe
m
ati
c
al Co
m
p
u
t
ing F
unctio
n
s
Chi-square
1
Y=A*X2**(
B*X1)
,
wher
e A=50.
6,
B
=
0.
004
242.
3
2
Y=A*(
X
1*X2)
**B,
wher
e A=18.
1,
B=0.
21
244.
1
3
Y=A*X1**(
B*X2)
,
wher
e A=55.
2,
B=0.
002
271.
6
4
Y=A*X2**(
B/X1)
,
A=78.
2,
B
=
-
0
.65
332.
3
5
Y=A*X1**(
B/X2)
,
wher
e A=75.
7,
B
=
-
0
.
73
334.
4
6
Y=A*(
X
1/X2)
**B,
wher
e A=68.
5,
B
=
-
0
.
005
352.
2
7
Y=A*X1+B*X2**2,
wher
e
A=2.
8,
B=0.
009
575.
2
8
Y=A*X2+B*X1**2,
wher
e A=1.
7,
B=0.
03
583.
8
9
Y=X1/(
A+B*X2),
wher
e A=0.
31,
B
=
-
0
.
001
616.
4
10
Y=X2/(
A+B*X1*
*2)
,
w
her
e
A=13.
1,
B= -
0
.
03
4637.
8
Fig
u
r
e
1
.
A 3D G
r
aph
By Labf
it Meth
od
Fi
gu
re
2.
I
n
t
e
r
pol
a
n
t
Li
nea
r
R
e
l
a
t
i
onshi
p A
n
d
R
e
si
d
u
al
Pl
ot
s S
h
o
w
i
n
g
I
n
M
a
t
l
a
b
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E
V
o
l
.
6,
No
. 3,
J
u
ne 2
0
1
6
:
97
4 – 9
7
9
97
8
Fi
gu
re
3.
I
n
t
e
r
pol
a
n
t
Li
nea
r
R
e
l
a
t
i
onshi
p S
h
o
w
i
n
g R
-
S
q
u
a
re E
qual
s
To
One
Fi
gu
re
4.
The
Dat
a
P
o
i
n
t
s
Sh
owi
n
g
Li
nea
r
R
e
l
a
t
i
onshi
p i
n
3
-
D
Vi
ew
Fi
gu
re
5.
The
Dat
a
P
o
i
n
t
s
Sh
owi
n
g
Li
nea
r
R
e
l
a
t
i
onshi
p I
n
3
-
D
Vi
ew
U
s
i
n
g
M
a
t
l
a
b
4.
CO
NCL
USI
O
N
Thi
s
resea
r
ch
pape
r has
des
c
ri
be
d t
h
e dat
a
re
pr
esent
a
t
i
o
n m
e
t
hods usi
ng M
A
Tl
ab a
nd L
A
B
F
i
t
.
Curve fitting
gives
detailed account of
inter-relation
of de
pende
n
t variab
le with respect to inde
pende
n
t
v
a
riab
les.
In this p
a
p
e
r, on
e dep
e
nd
en
t an
d two ind
e
p
e
nd
en
t
vari
a
b
l
e
s are
con
s
i
d
ere
d
t
o
e
vol
ve
best
fi
t
m
odel
.
In
lin
ear m
o
d
e
l, Cu
rv
e fittin
g fin
d
s th
e v
a
l
u
es o
f
th
e
co
effi
cien
ts (p
aram
e
t
ers) wh
ich
m
a
k
e
a fu
n
c
tion
match
the
data as cl
osely as possi
ble. T
h
e
best
va
lues
of
the c
o
efficients a
r
e
known wh
en
th
e
v
a
lu
e of
R-s
q
ua
r
e
b
eco
m
e
s o
n
e
.
In
Non
-
lin
ear
m
o
d
e
l, Cu
rv
e
fittin
g
assesse
s an
d
d
i
scov
ers th
e b
e
st scien
tific
m
a
th
e
m
atical
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Ma
ch
in
e
Lea
r
nin
g
Techn
i
qu
es on
Mu
ltid
imensio
n
a
l
Curve Fittin
g
Da
ta
Ba
sed
o
n
.... (Vid
yu
lla
tha
P)
9
79
state
m
en
t wh
ose Ch
i-sq
u
a
re
estee
m
is
least
.
Th
e fittin
g
mo
d
e
ls an
d
m
e
t
h
od
s
u
s
ed
h
e
re d
e
p
e
nd
on
th
e in
pu
t
d
a
ta
set.
Th
is p
a
p
e
r p
o
rtrays
h
i
storical
environ
m
en
ta
l d
a
ta con
s
ists of l
i
n
ear
v
a
riab
les
wh
ich
is fitted
b
y
in
fo
rm
atio
n
u
tilizin
g
3
d
i
m
e
n
s
io
n
a
l. Fo
r data fittin
g
,
1
5
0
no
n
lin
ear d
a
ta p
o
i
n
t
s are co
llected
fro
m
o
n
e
o
f
th
e
coal
co
nsum
i
ng p
o
we
r ge
ne
r
a
t
i
on pl
a
n
t
and
are use
d
t
o
pl
o
t
t
h
e grap
h u
s
i
ng “s
ft
o
o
l
”
co
m
m
a
nd i
n
M
a
tl
ab. F
o
r
cu
rv
e fittin
g
,
i
n
th
e Matlab
,
th
e d
a
ta is p
l
o
tted
in
in
te
rpo
l
an
t lin
ear sho
w
i
n
g
th
at R-sq
u
a
re eq
uals to
o
n
e an
d
it
i
s
repres
ent
e
d i
n
3 di
m
e
nsi
o
nal
m
e
t
hods
f
o
r al
l
vari
a
b
l
e
s. The
gi
ve
n v
a
ri
abl
e
s dem
onst
r
at
ed t
h
at
t
h
ey
are
linear in struct
ure so
that
, for forecast exam
inati
on, the multi linear re
gression is
t
h
e better decision
whic
h
fol
l
o
ws t
h
e m
a
t
h
em
at
i
cal
equ
a
t
i
on Z=
a+
b1
x1
+
b2
x
2
+
b3
x3
. F
o
r
N
o
n-l
i
near
m
odel
,
al
m
o
st
100
dat
a
poi
nt
s
are t
a
ken t
o
ge
t
a
m
a
t
h
em
at
i
c
al
form
ul
a and
3D
gra
ph
by
usi
n
g L
A
B
Fi
t
.
Thi
s
so
ft
com
put
i
n
g t
ech
ni
q
u
e h
a
s
ev
alu
a
ted
b
e
st
1
0
m
a
th
e
m
atic
al eq
u
a
tion
s
and
th
e equ
a
tion
Y=A*X2**
(B
*
X
1
)
h
a
s g
i
v
e
n
b
e
st fit. The ab
ility
to do c
u
rve
fitting is an extre
m
ely
helpful e
xpe
rtise for for
ecasting
purposes. The
fut
u
re
scope is
reac
hed
out
to
g
a
t
h
er m
o
re
in
fo
rm
atio
n
focu
ses t
o
sp
eak
to
in
4D
p
e
rsp
e
ctiv
es.
REFERE
NC
ES
[1]
Z. Me
i,
et al.
, “
C
urve fi
tting
an
d optim
al
int
e
rp
olation
on CNC
m
achines based
on quadra
tic
B-splines,
”
SCIENC
E
CHINA Information Sciences
,
vol/issue: 54(7)
, pp
. 1407–1418, 201
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[2]
X. Yang, “Curve fitting
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f
a
irin
g using con
i
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Computer
-Aided
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l. 36
, pp
. 461–4
72, 2004
.
[3]
W. Zheng,
et al.
, “
F
as
t B-s
p
line
curve fit
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y
L-BF
GS
,”
Computer Aided Geo
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etric Design
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l. 29, pp. 448–
462, 2012
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[4]
F. X. Diebold an
d C. Li, “Forecasting the
term structure of govern
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y
i
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s
,”
Journal of Econometrics
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[5]
P.
Norma
n
,
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e
tailed distribu
tio
ns from grouped
so
ciodemograp
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r m
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198, 2012
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[6]
J. Fox, “Nonpar
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ousand Oaks, C
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L. Fang
and D.
C. Gossard, “Multidim
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organized data p
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y
nonlin
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i
nim
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zat
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ComPuter-Aid
ed
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58, 1995
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[8]
http://lab-f
it-
cur
v
e-fit
ting-software.soft112.com/
[9]
http://cda.ps
y
ch
.uiuc.e
du/matlab
_pdf/curvef
it.pdf
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http://www.grap
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i
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[11]
http://cn
.
mathworks.com/help/pdf
_doc/curv
e
fit/cu
rvef
Evaluation Warning : The document was created with Spire.PDF for Python.