The Chemistry Maths Book, Second Edition

(Grace) #1

620 Chapter 21Probability and statistics


Simple least squares fitting


The simplest and one of the most popular methods of fitting a set of data points to a


model function (21.49) is to minimize the quantity


(21.50)


with respect to the parametersa


1

, a


2

, =, a


k

:


(21.51)


As shown in Figure 21.11, each term of the sum is the


square of the difference


ε


i

1 = 1 y


i

1 − 1 y(x


i

; a)


between the actual and predicted values of yfor each


value ofx 1 = 1 x


i

.


This method of least squares can be regarded either


as an empirical curve-fitting procedure that is justified


on the grounds that it often gives sensible results, or it


can be derived from the theory of random errors for


the special case that theσ


i

values are all equal; that is, when all they


i

have the same


precision. It is also the method to be used when no information about precision is


available.


5

The straight-line fit


The quantity to be minimized for a fit to the straight liney 1 = 1 mx 1 + 1 cis


(21.52)


Then




=− − − = − −


===

∑∑∑


D


c


ymxc ym xc


i

N

ii

i

N

i

i

N

i

i

20


111

()


==


=


1

10


N



=− − − = −


===

∑∑


D


m


xy mx c xy m


i

N

ii i

i

N

ii

i

N

20


111

()or


∑∑∑


−=


=

xcx


i

i

N

i

2

1

0


Dymxc


i

N

ii

=−+








=


1

2

()




=,




=, ,




=


D


a


D


a


D


a


12 k

00 0...


Dyyx


i

N

ii

i

N

i

=−;








=


==

∑∑


1

2

1

2

()a ε


5

The method of least squares (la méthode des moindres quarrés) was developed by Legendre in 1805 in connection


with the determination of the orbits of comets, and quickly became a standard method for solving problems in


astronomy and geodesy. The method also appeared in 1809 in Gauss’ Theory of motion of the heavenly bodies.


Gauss did not quote Legendre, and claimed that he had been using the method since 1795. Legendre argued that


precedence in scientific discoveries could only be established by publication, and was still accusing Gauss in 1827


of appropriating the discoveries of others.








x


i

y


i

y(x


i

;a)


y(x;a)


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Figure 21.11

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