CHAPTER 8 DESIGN O F OPTIMAL EXPERIMENTS
A PPENDIX
A ___________________
8.11 A plate which is subjected to a large instantaneous pulse of energy Q at x = 0
a nd is insulated at x = L has the solution for the temperature of
IDENTIFIABILITY CONDITION
where t + = 1/' 2a t iL 2, c is the densityspecific heat product, and Q has units
of energy (Btu or J) per unit area. For x = 0 the temperature is infinity at
time zero and decays to To+ Q I c L for large time. At X " L the temperature
starts at To a nd increased to To + Q I cL.
(a) Find an ellpression for the a sensitivity at x l L = I .
(b) Evaluate using a. c omputer the ellpression found in ( a) for 0 < t + < 3.
F or a filled value of Q (and no restriction on the range of T ) show that
the optimum time to take a single measurement is t + = 1.38. Also show
that this time corresponds to the time that the temperature at x = L has
reached one haIr of the m uimum temperature rise. This "onehaIr" time
is the basis of finding a in pulse or nash ellperiments. See the paper by
Parker, Jenkins, Butler, and Abbott (27).
( c) Also using a computer find the optimum ellperiment duration for many
equally spaced measurements at x I L = I.
8.13 ( a) A large number of measurements uniformly spaced in time have been
made at x = 0 a nd x = L in the heat conducting body discussed in
Section 8.5.2.2. For mo and m l sensors at x = 0 and L , respectively, show
that ~ + given by (8.3.7) can be written as
~+
...
[ zCIt.o+(Iz)Cltl][ z c2to+(IZ)C2t.]
 [zC I
i.o+(IZ)C ltlt
where z = m ol m and 1  z = m Il m and where
The third subscript in Cij~O o r Cij~1 refers to x =O or L , respectively. The
standard statistical assumptions are valid.
( b) Derive an ellpression for z at which ~ + is a m uimum, assuming that z
can assume any value in the interval 0 to I.
( c) T he following values are for the heat conducting body discussed in
Section 8.5.2.2:
c lto=0.07609
c lto=0.1062
c 2to=0.1552
c 2tl=O.126
C I t I =  0.0422
c ltl=0.0l48
T he values correspond to the dimensionless time =0.65. The first two
subscripts correspond to k (a I subscript) or c (a 2 subscript). Using the
ellpression derived in part (b), find a value for z.
( d) What conclusions can you draw from the results of this problem?
t:
A.I
I NTRODUcnON
T he problem of investigating the conditions under which parameters can be
uniquely estimated is called the identifiability problem. A convenient means of
anticipating slow convergence or even nonconvergence in estimating parameters
can save unnecessary time and expense. Also if easytoapply identifiability conditions are known, many times insight can be provided to avoid the problem of
nonidentifiability, through either the use of a different experiment o r a smaller set
of parameters that are identifiable.
The purpose of this appendix is to derive the identifiability criterion that the
sensitivity coefficients in the neighborhood o f the minimum sum of squares function
must be linearly independent over the range of the measurements. This criterion
applies for linear and nonlinear estimation. This criterion is derived only for a
weighted sum of squares function which includes least squares, weighted least
squares, and ML estimation with normal errors, in each case with no constraints on
the parameters. f or MAP estimation with prior parameter information it might be
possible to estimate the parameters even if the sensitivity coefficients are linearly
dependent.
This condition of independence of the sensitivity coefficients is particularly
convenient if the number of the parameters is not large, say, less than six. Even if
the number is larger, linear dependence between two o r three o f the parameters can
sometimes be readily detected from graphs of the sensitivity coefficients. The
p lolling of the coefficients is extremely important and should be done for each new
problem before attempting to estimate the parameter.
481
.1
~.
_

APPENDIX A 1DEN11F1ABILm C ONomON
.4.1 m EOIY
Define the determinant D, a s
1...1 THEORY
C onsider a general sum of squares function for measurements given hy
N
""
S = ~ ~ ( Y.  'I.)w.,,( Y"  'I.,)
c.n
SI,P,
S~ZIJI
( A.I)
where w"" is a n element of W. a square. symmetric. positivedefinite malr.i": lett~e
funclion S possess conlinuous derivatives in the neighborhood of ils minImum I n
the parameter space which occurs when 11 is evaluated al p•.
p
~
S (P)=S(p·)+
Sft,(P,fJP
II
p
+
,.
22
SJ.fI, ( P, 
fJ,· )( P  fJ;. ) + . . .
j
(A.3a)
iI,I
(A.6)
T hen S. a pproximated by the terms explicitly given by (A.3a), has a unique local
minimum if in addition to (A.S) being true it is also true Ihat
(A.2)
A T aylor series expansion o f S in the neighhorhood of its minimum is
S lzlJl
D,=
.,1,1
10
SI,PI
D ,>O
for , 1.2 . .. ..p
(A.1)
which is Ihe condition that D, b e positivedefinite; see reference I . A minimum can
exist with weaker conditions. however. F or example. if D ,  0 a minimum may exist
b ut it may n ot b e unique; that is. the minimum could b e a long a line rather than a t
a point. T he conditions given by (A.S) a nd (A.1) are necessary a nd sufficient
conditions for a unique local minimum.
We wish t o relate the conditions of D, > 0 a nd D,  0 10 the sensitivity
coefficien Is.
Let us define
where
(A.Sa)
I IS( W)
(A.3h)
S ;'=ap;·
(A.Sb)
Using S defined by (A .I) in (A.3bl gives
""
S ;'",2 ~ ~ H·••. (Y"'I.~)X:.;
. 1, 1
n
S;'~=2 ~
n
2
H,., _ [X,~X.:(Y.'I.~)X:'I]
Then using
(A.4a)
SA  0 for all i. (A.3a) c an be written
p
(A.4h)
where (A.Sa) a nd (A.Sb) a re employed in
S ;/  2
( A.4c)
The expression X ui in (A.4a) is called a sensitivity co~rric!ent..
.
F or a model linear in Ihe parameters the crossdenvatlve X"V I n (A.4b) IS equal
to zero as are also Ihe third and higher order derivatives of S. Note that the
condition of continuous derivalives of S with respect to P will be satisfied if 11 a nd
ils derivatives are continuous functions of p . A necessary condition for S to possess a minimum a lp· is thai
f or i = 1.2 . .... p
(A.9)
S ;/flIJ/llP/
i I j  I
u I t 11
Sft,=O
p
S(lI>S("·),",,~ ~ ~
( A.S)
S;/.
""
2 2 w",,[ X ..i X '; 
(Yo  'I:)X..lj
]
( A. 1 0)
1 11.,1
N ow (A.9) is a q uadratic form a nd if a unique minimum is to exist it is necessary
that all the determinants
D:
, 1.2. ....p
( A.l1)
S ,j
b e greater than zero. Suppose that a minimum ellists a t II· b ut t hat the minimum is
APPENDIX A IDENTIFIABILITY C ONDITION
n ot unique. for example. it ellists along a line o r in a plane. This results in D,+ =0
for some r.
Suppose first that the term in (A. IO) involving
is negligible in its c ontribution to S / . Notice that this term becomes negligible as the residuals. Yv  ,,:.
become small but this is n ot true for the Xu: Xcj terms. ( For linearintheparameters cases.
is always zero.) Furthermore. assume that
A.J
C OMMENTS
Suppose (A. 16) is written in the form
,
xut
Xuu
= ou 2
"'",,=0
" 'w
a nd then
S,/
U
= v a nd a;"eO
~ 0 Xuj = 0.
u = 1.2 • ...• n:
j I
~here a t least o ne
( A.II)
0
r = 1.2. .. .. p
(A.17
is n ot equal to zero. Also form the summation involving a r m
In
n
,
u "ev
r
~
( A . 12)
S./0= ~ ~ ~
j I
j I u I
n
~_I
!
"'uv[Xu XJ(Yv 11:)Xu
ij]S
given by (A. IO) becomes
S'/ =
n
~ ( ouIXu!
HauIXu; )
(A.18 .
( A.I3)
u I
Differentiating (A. 17) with respect to
_[a~ IXI; 1
8; 
.
L 0 Xuij=0.
i =I. . ..• P :
r =I. . . .. p
,
L S/0=0.
r I . ...•p ;
i = I . ....p
j I
a fa,
D +=
,
(A.19 )
Using (A. 17) a nd (A. 19) in ( A.l8) then produces
( A .14)
Use this interpretation in (A.J3) a nd i ntroduce (A. 13) into (A . II) to get
( A . IS)
a ;a l
u =I. . ..• n:
j I
a. IX.!
a ia l
Pi yields
,
T he summation in (A . I3) can be considered to form an inner product involving
vectors a,. i = I . 2. .... r :
8 ;a,
which can be considered a G ram d eterminant of al . a1 . ... . a,. I t is known that D,+
is equal to zero if a nd only if the vectors a, a re linearly dependent. which means
(A.20 1
~e have shown for linear dependence o f the sensitivity coerficients. (A. ' 7). that a
~Iven c olumn o f the square matrix D,+ given by C A.") c an b e considered to be a
I~near c ombi.nati.on o f the other columns. But if a ny c olumn o f a s quare matrix is a
h near c ombinatIon o f t he other columns. then the determinant o f t hat matrix is
zero. Consequently. the sum o f s quares function S does not have a unique
mi~imum in th~ P I,Pl. ' . . ,p, space a nd thus not all o f these parameters can be
umquely d etermined.
T~e results given ab~ve a pply for r equal to I to p p arameters. Note. however. if
the h near d ependence condition o f t he sensitivities given by (A .• 7) is satisfied for
r < p . then it is also satisfied for r + I r + 2 . .. p because C It C' +2• . .. ' C can be
set equal to zero I n (A.17).
p
•
•
t
t
,
. ..
(A .16a)
A.3
or
(A .16b)
for k  1.2 • .. .. n a nd for not all C; being equal to zero. In other words. i f the
(continuous) sensitivity coefficients are linearly dependent in the neighborhood o f t hf
minimum there is no unique minimum and all the r parameters cannot be simultaneously and uniquely estimated. This is the desired relation. Note. however. that this
result assumes that the term involving Xuij in (A. 10) c an be dropped: "'110 is given
by (A.12): there is n o prior information; a nd there are no parameter constraints.
C OMMENTS
( a) P arameters c annot all be uniquely estimated for 11 being linear o r n onlinear in
the parameters if the sensitivity coefficients a re linearly dependent over the range
o f the measurements. This is true if (i) the S function is formed by some weighted
least ~~uare f u?ction. (ii) the sensitivities a re c ontinuous functions o f the p arameters. (m) t~ere IS n o prior information regarding the parameters. a nd (iv) there are
no c onstraints o n the parameters.
. (b) I f t~e
termrin (A.IO~ is negligible o r is d ropped. the determinant o f D /
IS p roportIonal to IX WXI whIch must not be zero when attempting to estimate
xut
APPENDIX A IDEN11FIA8IUrv C ONDmON
7.4.
parameters using the Gauss method discussed in Section
Hence: .u~i~g the
Gauss method. n one of the parameters can b e estimated ,f the sens,t,v,ttes a re
linearly dependent. Other methods might permit one to obt~in certain .p.ara~eters,
b ut n ot all since there would b e no unique minimum of S ,f the cond,ttons t n ( a)
above are true.
T
(e) Regardless of the form of W if IXTXIO it is also true .that IX WXI:O.
Hence if the sensitivities are linearly dependent, there is no chOIce o f W poss,ble
that will cause IXTWXI to be n ot zero.
( d) I f IXTXI * 0. then IXTWXI m ayor may not be equal to zero. But if M L
estimation is used as mentioned in (b).IXTWXI would not b e zero if IXTXI*O.
REFERENCES
where
(A.2S)
which is also limited to between zero a nd one. F or small ( it c an b e demonstrated
that
(A.26)
( I
S ,t S2;'
Only for
does
T he d eterminant D + is numerically equal to the product of its eigenvalues
",.A 2." .,A.. N ow t h; matrix on the right side of (A. I I) is real and symmetric which
results in all the eigenvalues being real. Also since
(A.21)
D, + will b e then equal zero if a nd only if a t least one of the A, values is equal to
zero.
Since S. + is normalized so that the scale of the A, values (or the choice of their
units) is u~important. the relative magnitudes of the >./ values is significant. I f o ne
value is much smaller than the others (but not zero). the XTWX matrix is p robably
illconditioned and the minimum of S is not welldefined. This would occur when
there is " almost" linear dependence of the sensitivity coefficients. In such cases
there will b e relatively large inaccuracy (large variances) in the parameters.
Consider the case of two parameters a nd then the eigenvalues >., a nd A2 in
I
SI~
(A.22)
where f l is the determinant in (A.22) with>. set equal to zero. Let A, be the smaller
eigenvalue. The the ratio of the eigenvalues. A,/A 2• is always between zero a nd o ne;
A.lA2 is given by
1 _(l_()I/l
A2 = I + (I_()I/l
S,;  S2t  0 and
(A.27)
could b e used to see if there is n ear linear d ependence o f the sensitivity coefficients.
(Note t hat if the X"; term in (A.lO) can b e d ropped, the components of
(X+)TWX+ a re given b y S /.) W hen ~ goes t o zero, a t least o ne eigenvalue is
equal to zero; the maximum value o f ~ is unity.
Thus in addition t o plotting the sensitivity coefficients, o ne could examine ~ to
see if it is n ear zero. I f it is, the experiment is poorly designed a nd o ne o r more of
the parameters should not b e estimated, b ut r ather certain groups o f parameters. I f
possible, the experiment 'should b e redesigned so t hat ~ is n ot so small. However,
the recommended criterion for accomplishing this is n ot ~, b ut r ather the numerator o f (A.27), subject to certain constraints. See C hapter 8.
REFERENCES
I.
are
>.,
unity; ( equals one only when
T he above analysis suggests for more than two parameters that the criterion of
small
A.4 RELATION T O EIGENVALUES
s,t  >.
"'/"2 equal
(A.24)
G. S. G . Beveridge a nd R. S. Schechter. Optimizatio,,: Theory a u Practice. McGrawHili
Book Company. New York, 1970, p. 217.
tsz
>
."
."
f Il
2
C
>;;
eo
AppeacUx B
Estimators and Covariances" for Various Estimation Methods for the Linear M odel" = Xf1
Name
of
Estimator
Assumptions
Used
Estimator
Covariance
Matrix of b
cov(y) for Y Xlb
Usual Estimator for Unknown a l
F or'; alO, 0 Known
(XTX)IXTy
Ordinary
Least Squares
I III
Same as above
( Ots)
111111
Same as above
Maximum
111111
(XT.;IX)XT.; Iy
Likelihood
(ML)
111011
(XTOIX)  'XTOIy
G auuMarkov
11011
Maximum
a posteriori
111112
(XTX) 'XT';X(XTX)  , XI(XTX) 'XT';X(XTX)  IXr
a1(XTX)'
a 2XI (XTX)  'Xf
s 2_(y_y>T(YY>/(np) where V Xbu
( XT.;IX)'
a 2(X TO  IX)  ,
X ,(XT';IX)'Xf
a 2Xf<X TOIX)  'Xf
(a 2 assumed known)
s l_(Y _ V)TOI(Y  V)/(np)
where Y  XbML
(XTOIX)  'XTOIy
a 2(X TOIX)'
a 2X I(X TOI)X)IXf
s 2_(Y  y>TOI(y  Y>/(np)
where Y  XbOM
111012
c ov(bIl)P MAP
XIPMA,xf
"+~A,xTOI(Y_X,,)
c ov(bIl)
a2XI~A,xf
where~AP
(MAP)
P +PMA,xT';I(yX,,)
w herePMAP ( XT';IX+Vq'
a2~AP
( a 2 assumed known)
' ;2.(y_y> TOI(y  Y>/n
where Y  Xb MAP
[ XTOIX+';lVlr'
(Note iteration is required for ';2).
·';cov(~) for assumptions denoted 1 1. (See Section 6.1.5 o r inside rear cover for list of standard assumptions).
s
A PPENDIX
c____ LIST O F S YMBOLS
ENGLISH SYMBOLS
j
k
' I.(P)
L
LS
m
MA
M AP
ML
n
P
P (·)
P
Pu
PML
PMAP
Q
R
E NGLISH S VMBOLS
a;
al
A j(i+l)
AR
b
C/j
C
c ov(',' )
V
e
e
E( )
f (·)
f(VIII)
f(IIIV)
F
G
h
H
Hi
Coefficients in S; see (7 .6.3)
Firstorder AR errors
Term used in sequential methods, (6.7.8a) a nd (7.8.23a)
Autoregressive
P arameter vector; estimated from estimation equation ; Ip x I)
C omponent of XTWX; C Ijl:.l:,w.,X.;XIi , (7.4.13)
= XTWX, (7.4.12); IpxpJ
Covariance, c ov(A,B) E([A  E(A)II B  E(B )I}
Matrix used for depenslent observations. e = Du; In x nJ
Residual vector  V  V
Eigenvector; Section 6.8
Expectation operator
Probability density
Probability density o f V given II
Probability density o f II given V
Modified observation vector, F  VIV where V comes from
e Du
F statistic associated with (J  a) 1 00% confidence region a nd
p a nd n  p degrees o f freedom
Related to the slope of S, (7.6.8a)
Acceleration factor (7 .6. 1)
= XTW(y,,), (7.4. 15); Ipx II
=Ii_II:'_lw,onXt;( Yon 71",), (7 .4.16)
Subscript or superscript
$
491
Subscript
Subscript o r superscript
Coefficient for confidence region, Section 6.8
Likelihood function, Sections 3.2.5 and 6. 1.6
Least squares
Number of observations at a given time
Moving average
Maximum a posteriori
Maximum likelihood
Number of observations o r number of observation times
Number of parameters
Probability
Covariance matrix of estimators; Ip x pl
= (XTX) tXT",IX(XTX)1
= (XT",IX)I
= (XT",IX+Viltl
Quadratic form. AT
(6.1.27)
Minimum S ; for LS, R = (Y  y) T( y  Y)
Estimated standard deviation of observation errors, $ =
2
($2)1/2 where $2= a ; for independent constant variance
errors, $2  R/(n  p)
Sum of squares function. scalar
Least squares sum of squares. ( y_,,)T(y_,,)
Maximum likelihood sum of squares; for standard
 I(y _ , ,)
assumptions, ( Y MAP loss function; for standard MAP assumptions.
S MAP = ( Y  I(y  ,,) + ( I'll  fJ)TVi I( I 'll  fJ)
Time
t statistic associated with ( I  a) 100% confidence region
and n  p degrees of freedom
Random component for correlated errors. Appendix 6A.
.A.
")T,,,
")T,,,
U.,/2
V( · )
Vb
V~
W lj
W
x
X
l  Du
1 00(1 a/2) percentage point of the normal distribution
Variance operator; V (A) E{ [ A  £(A) t }
Covariance matrix of b, ( p X pI
Covariance matrix of I'~, ( pxp] (I'~ is prior vector of fJ)
Component o f weighting matrix W
Weighting matrix; for ML, W =",I. ( nxn]
Coordinate o r independent variable
I f " is linear in the
Sensitivity matrix; X = (V~"
XfJ. (V ~"
reduces to X.
parameter as in
,,=
Tt
Ty
APPENDIX C L IST OF S YMBOLS
A PPENDIX
Y
O bservation v ector, ( nX II
Pr~dicted v ector o f o bservations, (n X I I; f or l inear c ase
Z
M odified s ensitivity m atrix, Z = D 1X;
Y
Y =Xb
In x pl
1:)_____________________________________
S OME ESTIMATION PROGRAMS
GREEK SYMBOLS
0/
Associated with confidence interval o r region percent .confidence; see
Section 6.8
See Section 7.6 for parameter related to reducing the interval for calculating S (·'
Parameter vector. [ p X I)
G amma function; see Section 6.8
Optimum experiment criterion (see Chapter 8); for standard assumptions
of Y =,,+I:, £ (1:)=0, I: with normal density. known independent variable values. tit known within a multiplicative constant, we have 11=
IXT",  IXI
Error vector, [n X I) ; usually Y = " + t
Expected value vector, regression vector, model vector, (n X I)
Moving average parameter
Eigenvalue. Section 6.8
Parameter vector known from prior information [p x I)
Autoregressive parameter
Correlation coefficient; (2 .6. 17)
Standard deviation of constant variance observation errors
Constant variance of observation errors
Variance of E, ; V (Ej )=O/
a;.
V arianceofu, ; V(Uj)=o~
a
a
P
f (·)
11
t
lJ
9
>.
1'/3
p
P
a
0
2
0,2
~
X2
tit
o
Vari~nce of E
j • used for the AR case designated a I; 0,2 = o~(I p2)  I. See
below (6 .9.9) .
~iagonal matrix; usually </>= £ (uu T ) for £ (u)=O a nd where £ (u;"1)=O for
i =t:j ; [ nxn)
Chisquared statistic
Covariance matrix of the observation errors; for £ (t)=O. " '= £ (u T )
Known part of tit . as in tit = 0 20 where 0 2 is unknown; [n x n)
In this appendix a few c omputer programs are referenced. Many others are
available. F or additional references see Himmelblau ( 01. pp. 170, 171,203), Bard
[ 02 , pp. 323, 324), and Kuester a nd Mize [D3).
LINEAR E STIMATION P ROGRAMS
U NFIT A linear least squares program with optional constraints to make the
parameters nonnegative, a dd to a constant, etc. This is o ne o f eighteen statistical
routines written by 1. R . Miller ( 04).
U NREG A linear least squares program that is described in reference 0 3 where
an example a nd the listing are given ..
O MNIT AB A general purpose computer program for statistical and numerical
analysis [OS) .
NONLINEAR ESTIMATION P ROGRAMS
O THER S YMBOLS
. ,.
Vp ()
M atrix d erivative o perator. V /3=(3;3p , . . . 3;3Pp l
B ARD A nonlinear least squares program that uses the Gauss method [ 03, p .
218).
BSOLVE A nonlinear least squares program that uses Marquardt's method ( 03).
N UN IBM Share Program SO 3094 written by Marquardt and others. Written in
F ORTRAN IV for IBM 7040. Uses M arquardt's method with derivatives o r finite
difference approximations to solve weighted least squares problems.
493
APPENDIX D SOME E STIMAnON PROGRAMS
NLiNA This is a program written at Michigan State University by 1. V. Beck and
available from him. It uses the sequential and BoxKanemasu modifications of the
Gauss method.
SSQMIN This program uses the Powell procedure and is discussed in reference
OJ.
Index
N
CJ1
CD
REFERENCES
D I.
02.
03.
0 4.
OS.
Himmelblau. O. M .• P rocm Analysis by Sialisilcal MtlllodJ. John Wiley . t Sons. Inc .•
New York. 1970.
Bard. Y.• Nonlintar Parameltr ESlimolion. Academic Press. Inc .• New York. 1974.
Keuster. J. l . and Mize. J. H .• Oplimizalion T«IIniquts Willi Fortran. McGrawHili
Book Co .• New York. 1973.
Miller. J. R.• O nLint Analysis for Social Scientists. MACTR40. Project MAC.
Massachusetts Institute o r Technology. Cambridge. Mass .• 1967.
Hilsenrath. J.• Ziegler. G .• Messina. C. G .• Walsh. P. 1.. and Herbold. R .. O MNITAB. A
Compu/er Program for Sia/is/ical and Numtrical Analysis. Nal. Bur. or Std. Handbook
101. U. S. Government Printing Orrice. Washington. O. c.. 1966. Reissued Jan. 1968.
with corrections.
A bbott, G. L , 4 15, 4 80
Abramowitz, M., 1 1
AJArajl, 5 ., 263, 319
Analysis o f co~arlance, 131
A naly,l, o harlance, 1 30, 1 31, 115, 178
ArI" R., 4 14
Arkin, H., 78
Assumptions, GaussMarkov, 1 34, 232
,tandard, 134, 228, 229
violation of, 1 85204, 2 90319, 3 93,
4 00,401,459,460
Atkinson, A. C., 4 35, 4 38, 439, 4 14
A utocovulance, 5 9
Bacon, D. W., 3 19,414
Badaval, P. C., 4 32, 4 74
B ud, Y., 4 , 24, 335, 362, 3 64, 375, 386,
4 11,414,472,475,493,494
Bayesian e 'tlmation, 9 7101. & ttll,o
Maximum a po,teriori estimation
Baye,·. t heorem. 4 6, 4 1,160,164,210
Beale. E. M. L ,414
Beck. I . V., 263, 3 19,415.474.475.494
Beveridse. G. S. G ., 3 38,414,481
Bevington. P. R., 24
Beyer, W. H., 7 8,319
Bias. 8 9
Bia, error, 1 80
Bonaelna, C., 474
Booth. G. W.• 4 14
Box, G. E. P., 2 4,114,129.162,204,229,
2 32.319,359,363.364,369376,
3 80,386,414.415,419.432,438,
4 39.469,470,474,475
Box. M. 1 .,4
BoxKanemasu i nterpolation method,
3 62311,381,494
BoxMuller transformation, 126
Brownlee, K. A., 2 04, 2 26
Bryson. A. E., J r., 24
BUrington, R. 5 •• 78, 2 04,319,415
Butler, C. P., 4 75, 4 80
Cannon, J. R., 4 74
C uslaw, H. 5 .,474
C entral limit theorem, 6 4, 6 1,186
Cheby,hev's i nequaUty,62
Chlsquared t est, 2 68, 269
Cochran's theorem, 176
Coefficient o f multiple determination,
1 73115
Colored errors, , tt Errors, correlated
495