THE METHOD OF LEAST SQUARES
1.
INTRODUCTION
The method of least squares is the standard method used to obtain
unique values for physical parameters from redundant measurements of
those parameters, or parameters related to them by a known
model.
The first
15
The probability that x is less than or equal to x is the value of
,o
(x ) and is represented by the total area under the curve from  00
0
to x
0
shown as the shaded area in Figure 22. Note
Frequency distributions have two important characteristics ca
141
Eliminating K1 from Equation 925
=0
.
(9.26)
Eliminating K2 from Equation 9.26
or
( 9. 27)
From the first Equation of 9.26
(9.28)
From the first Equation of 9.25
(9. 29)
From the first Equation of 9.23
( 9. 30)
From the first Equation of
(9. 31)
142
T
100
P
v)
= cr2
o
1
this estimator is the maximum likelihood estimator of X if V has a
normal distribution
vi)
the least squares unbiased estimator V of V is
V=AXL
"
vii) the least squares unbiased estimator cr2 of the variance factor cr 2
0
0
is
where
p
113
coefficient matrix consists in large part of zero elements, the storage
of which in a computer is unnecessarily wasteful.
Therefore in the next
section we will derive more explicit expressions for the solutions for
X, K and V.
7.4 DERIVATION OF EXPLIC
127
parameters)(.These deviations are represented as the difference between
the two uxl vectors,
XX.
(XX)
The quadratic form (see Sections 3.1.6 and 4.10).
T
1
d
A
(XX) + x2 ( u)
Note that the variance factor o 2 is assumed known since
0
where
is the
155
'Eo
find a. sequential expression for the eovariance matrix of the estimated
solution vector, we have from equations 108 and 109
where
C2
=
1
T
Ak
1
1
+ Ak Nk1
T 1
Ak)
.
By the covariance law
1
Nk
1
Nk1
=
Xk
= [Cl
0
CT
0
pk
1
CT
2
C2]
1
wher
86
6.
LEAST SQUARES POINT ESTIMATORS:
LINEAR MATHEMATICAL MODELS
In this Chapter we follow the approaches of Schwarz [1967] and
Hamilton 11964] in discussing the linear.mathematical model
(6.1)
AXL=V
where nLl is called the observation vector and is the
APPENDIX B
STATISTICAL TABLES
taken from Natrella, M.G. (1966). "Experimental Statistics", U.S.
National Bureau of Standards Handbook 91.
170
TABLE B1.
CUMULATIVE NOR.MAL DIS'TIUBUTION  VALUES OF Pr
Values of Pr corresponding to c for the normal curve.
29
00
M(t)
=f
e
tx
dx
00
00
=f
e
tx
co
and by letting y
= xa
b
 bt
(xa)
1
exp [
b(21T)l/2
2b 2
2
] dx
310
we have
2
x = by + b t + a
00
M( t) = f exp[ t (by + b 2t + a)]
 00
= exp
b2t2
00
[at +  ] f
2
_oo
2 2
1
exp [ _(by+b t) ] bdy
b(21T) 1 / 2
2
58
(x1 +x 2+ .
M(t) = E ( exp [
t
+xn)
1 and for x.
t
1
t
independent M( t )= E[en Xl] E ( en X2) . . E [en
g),
For x. C.i.istributed as n ( 11,
1
statistically
X :"1
n_,
the m.g. f. is ( 3 II.)
o2t2
M(t) = E[e tx] = exp [11t +   ]
2
x
thus in the pres
44
=o
r
00
< w <
0
< V' <
:]
elsewhere.
Next consider the definition of third variable t as
339
The p.d.f. corresponding to the. two original variables w and
V'
can be transformed into a new p.d.f., e.g. in terms of the new
variables t and u through the
while the confidence interval for
a
lJ
is
< lJ < x. + c


l
a] .
The bounds of the interval are evaluated from:
( l)
the measurement value x.
(2)
the known value :for the variance a 2
( 3)
the tabulated (Table B1 Appendix) value of c corresponding
l
to