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
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
Freque
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
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 e
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
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 va
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 la
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
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
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 +
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
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.,
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 tabul