THE METHOD OF LEAST SQUARES
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
The probability that x is less than or equal to x is the value of
(x ) and is represented by the total area under the curve from - 00
shown as the shaded area in Figure 2-2. Note
Frequency distributions have two important characteristics ca
Eliminating K1 from Equation 925
Eliminating K2 from Equation 9.26
( 9. 27)
From the first Equation of 9.26
From the first Equation of 9.25
From the first Equation of 9.23
( 9. 30)
From the first Equation of
this estimator is the maximum likelihood estimator of X if V has a
the least squares unbiased estimator V of V is
vii) the least squares unbiased estimator cr2 of the variance factor cr 2
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
parameters)(.These deviations are represented as the difference between
the two uxl vectors,
The quadratic form (see Sections 3.1.6 and 4.10).
(X-X) + x2 ( u)
Note that the variance factor o 2 is assumed known since
find a. sequential expression for the eovariance matrix of the estimated
solution vector, we have from equations 10-8 and 10-9
+ Ak Nk-1
By the covariance law
LEAST SQUARES POINT ESTIMATORS:
LINEAR MATHEMATICAL MODELS
In this Chapter we follow the approaches of Schwarz  and
Hamilton 11964] in discussing the linear.mathematical model
where nLl is called the observation vector and is the
taken from Natrella, M.G. (1966). "Experimental Statistics", U.S.
National Bureau of Standards Handbook 91.
CUMULATIVE NOR.MAL DIS'TIUBUTION - VALUES OF Pr
Values of Pr corresponding to c for the normal curve.
and by letting y
x = by + b t + a
M( t) = f exp[ t (by + b 2t + a)]
[at + - ] f
exp [ _(by+b t) ] bdy
b(21T) 1 / 2
(x1 +x 2+ .
M(t) = E ( exp [
1 and for x.
independent M( t )= E[en Xl] E ( en X2) . . E [en
For x. C.i.istributed as n ( 11,
the m.g. f. is ( 3 -II.)
M(t) = E[e tx] = exp [11t + - - ]
thus in the pres
< w <
< V' <
Next consider the definition of third variable t as
The p.d.f. corresponding to the. two original variables w and
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
< lJ < x. + c
The bounds of the interval are evaluated from:
the measurement value x.
the known value :for the variance a 2
the tabulated (Table B-1 Appendix) value of c corresponding