Curve Fitting Using Least
Squares Method
Bio-Fluids
Bien 301
Mootaz Eldib
The Problem
• Given the value for the density of water over the
temperature range of 0-100°C, fit these values to
a least squares approximation of the from
  a  bT  cT
2
and estimate the accuracy. Calculate the density
of water at 45°C and compare it to the actual
value of 990.1 Kg/m3.
The Problem
• Given:
(0,1000),(10,1000),(20,998),(30,996),(40,992),(50,
988),(60,983),(70,978),(80,972),(90,965),(100,9
58)
Find
  a  bT  cT
2
Least Squares Approximation
If data given in the form
(T1 , 1 ), (T2 ,  2 ),....., (Tn ,  n )
We assume a best fit curve f(t), where
the error R for each point is
R  1  f (T1 ), R   2  f (T2 ),......., d n   n  f (Tn )
Square the error and sum over all the points
n
R 2  d12  d 22  ......  d n2   [  i  f (Ti )]2
i
Least Squares Approximation
• The total error
n
R 2  d12  d 22  ......  d n2   [  i  f (Ti )]2
i
is MINIMIZED for least squares method
where
f (Ti )  a  bTi  cTi
2
Least Squares Approximation
n
R 2
 2 [ i  (a  bTi  cTi 2 )]  0
a
i 1
n
R 2
 2 [ i  (a  bTi  cTi 2 )]  0
b
i 1
n
R 2
 2 [ i  (a  bTi  cTi 2 )]  0
c
i 1
n
n
n
n
i 1
i 1
i 1
i 1
2


a
1

b
T

c
T
 i  i i
n
n
n
n
i 1
i 1
i 1
i 1
2
3
T


a
T

b
T

c
T
i i i i i
n
n
n
n
i 1
i 1
i 1
i 1
2
2
3
4
T


a
T

b
T

c
T
i i i i i
n
 n
  n
  i   1  Ti
i 1
 i 1
  i 1
n
 n
  n
2
T


T
T
  i i   i  i
i 1
 in1
  i n1
n
3
 T 2   T 2
T
 i i    i  i
i 1
 i 1
  i 1

Ti 

i 1

n
3
Ti 

i 1

n
4 
T

i 
i 1

n
2
a 
 
 
b
 
 
c
Density Vs Temperature
1005
1000
y = -0.0036t2 - 0.0699t + 1000.6
R2 = 0.9993
995
Density(kg/m^3)
990
985
980
975
970
965
960
955
0
20
40
60
Temp(c)
80
100
120
Solution
Density Vs Temperature
1010
1000
y = -0.43t + 1006
R2 = 0.9474
Density(kg/m^3)
990
980
970
960
950
0
20
40
60
80
Temp(c)
First order least square approximation
Ρ=1006.05 -0.43 t
100
120
Density vs. Temperature
1005
1000
995
3
2
y = 1E-05t - 0.0054t - 0.001t + 1000.2
2
R = 0.9997
Density(kg/m^3)
990
985
980
975
970
965
960
955
0
20
40
60
80
100
Temp(c)
Third order approximation. Mathematica equation is
1000.21 0.000971251 t
2
0.00540793 t
3
0.0000120435 t
120
Density vs. Temperature
1005
1000
y = -2E-07t4 + 6E-05t3 - 0.0083t2 + 0.0573t + 1000
R2 = 0.9998
995
Density(kg/m^3)
990
985
980
975
970
965
960
955
0
20
40
60
Temp(c)
80
100
120
Summary
• Very powerful method
• Widely used
• Biomedical engineering application
References
• http://www.efunda.com/math/leastsquares/
lstsqr2dcurve.cfm
• http://mathworld.wolfram.com/LeastSquar
esFitting.html
• http://math.fullerton.edu/mathews/n2003/L
eastSqPolyMod.html
Questions?