Tables *of Bias Functions, and 82 for Variances Based on ... · TABLES OF BIAS FUNCTIONS, B1 AND...
42
Tables *of Bias Functions, B1 and 82 , for Variances Based on Finite Samples of Processes with Power law Spectral Densities JA. Barnes Time and Frequency Division Institute for Basic Standards National Bureau of Standards Boulder, Colorado 80302 Second printing; original issued January 1969 U.S. DEPARTMENT OF COMMERCE, Frederick 6. Dent, Secretary NATIONAL BUREAU OF STANDARDS. Richard W Roberts. Dlrector Issued June 1974 , ...
Transcript of Tables *of Bias Functions, and 82 for Variances Based on ... · TABLES OF BIAS FUNCTIONS, B1 AND...
Tables * o f Bias Functions, B1 and 82 , for Variances Based on Finite Samples o f Processes with Power l a w Spectral Densities
JA. Barnes
Time and Frequency Division Institute for Basic Standards National Bureau of Standards Boulder, Colorado 80302
Second printing; original issued January 1969
U.S. DEPARTMENT OF COMMERCE, Frederick 6. Dent, Secretary
NATIONAL BUREAU OF STANDARDS. Richard W Roberts. Dlrector
Issued June 1974
,
...
National Bureau of Standards Technical Note 375
Nat. Bur. Stand. (US.), Tecb Note 375, 40 pqen (June 1974)
CODER: RBTNAE
For nak by the Superintendent of Documents, U.S. Government Printing O G e , Uashington, D.C. 20402
(Order by SD G t d o g No. (33.46375.) S.50
CONTENTS
Page - ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
BODY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
The Bias Functions, B and B2 . . . . . . . . . . . . . . . . 2 1
Figure 1 , p - cy Mapping . . . . . . . . . . . . . . . . . . . . 4
Examples of the U s e of the Bias Functions . . . . . . . . . . 7
A CKNOW LEDGMENT . . . . . . . . . . . . . . . . . . . . . . . . 8
R E F E R E N C E S . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
TABLES OF B1 AND B2 . . . . . . . . . . . . . . . . . . . . . . 10
Figure 2, The Bias Function, B2(r,p) . . . . . . . . . . . . . 34
1
f
i i i
T A B L E S OF BIAS FUNCTIONS, B1 AND BZ, F O R VARIANCES
BASED ON FINITE SAMPLES O F PROCESSES WITH POWER
L A W S P E C T R A L DENS1 TIES
J . A. Barnes
ABSTRACT
D. W . Allan showed that if y(t) is a sample function of a random U
noise process with a power law spectral density (i. e . , S (f) = h I f I ),
then there is generally bias to the estimated variance of y, defined as Y
N 2 Y N- 1 0 (N, T , T ) = -
n= 1
where N is the number of samples, 7 is the average value of y(t) over
the n-th interval of duration T , T is the time between the beginnings of
any two successive sample intervals, and
n
N
<y> z 1 N c yn . n= 1
Allan also showed that, under these conditions, the expectation value
of the estimated variance i s proportional to T’ where p is a constant
related to a, the exponent in the spectral density; i. e . ,
P 7 .
Based on this work one may define the two bias functions
iV
and
where r T/T and the B ' s a r e functions of p through their dependence
on Y ( t ) . If one has a sample variance, oY 2 (N1, T l , T ~ ) , the bias functions
allow one to give an unbiased estimate for a2 ( N
spectral type is known (i. e . , p is known).
T T ) provided the y 2' 2' 2
The tables give values of B (N, r , p ) and B (r , p ) accurate to four 1 2
significant f igures for the following values of N, r , p :
p = -2 .0 to 2 .0 in steps of 0 .2;
N = 4, 8, 16, 32, 64, 128, 256, 512, 1024, co ;
r = 0.001,0.003,0.01,0.03,0.1,0.2,0.4,0.8,1,1.01,1.1,
2, 4, 8, 16, 32, 64, 128,256, 512, 1024, 2048,a .
Key Words: S tatist ics, variance, spectral density, unbiased estimate.
V
. .
TABLES OF BIAS FUNCTIONS, B1 AND B2, FOR VARIANCES
BASED ON FINITE SAMPLES OF PROCESSES WITH POWER
LAW SPECTRAL DENSITIES
Consider a random variable y with mean m. .The variance of y is
defined as the expectation value of (y - m) 2 , that is
V a r y E E[(y - m) 2 1 .
This is defined as an average over the entire ensemble but, for an
ergodic process, y(t),it can alternatively be defined as an average over
all time, t .
Typically, the variance of y is estimated from a finite set of
experimental values according to the relation
N
i= 1
where <y>= - 5 yi is the mean of the y.. The factor - 1 is used in N- 1 1
i= 1 order that the estimate have no bias for non-correlated y: that is, one
typically wants to obtain the true variance which would be obtained as
N - m . For finite N, the estimated variance has some expected value,
If the y. a r e independent (actually, non-correlated is sufficient) random
variables, then the expected value of this estimated variance for finite 1
, 2
i N is exactly equal to the true (infinite N) variance.
related, however, the estimate based on (1) may indeed be biased.
This fact has been recognized and discussed in some detail for the case
of power law spectral densities by Allan [ 11.
If the y. a r e cor - 1
2 The Bias Functions, B and B 1
Following Allan [l], consider a random process y(t) with con--
tinuous sample functions.
S (f), which obeys the law
We assume that y(t) has a spectral density,
Y .
where h is a constant, the limit frequencies fe and f satisfy the relations U
O i f < < f < a 0 , e u
and any intervals of time, At, of any significance satisfy the relations
In short , y(t) has a power law spectral density over the entire range of
significance.
Consider a measurement process which determines an average
value of y(t) over the interval t to t t 7 . That is,
T(t) = - y(t ')dt' .
One, now, may determine an estimated variance from
measurements spaced every T units of time; that i s ,
2
c
_- .. -
roup
(3)
f N such
. . . _ _ .. ..
N N
k= 1 I
which i s called the "Allan variance" [ 13 . Allan [ 13 has shown that under these conditions,
E 0 (N, T, T ) a?', N and T/T constant, 1; J * where p is related to a according to the mapping shown in Figure 1
(see references 1 and 2 ). The relation between p and a may be given as
v = 1 -a -1 i f -3c a 5 1
-2 i f a 1 1
not defined otherwise.
This mapping involves a simple extension of Allan's work [ 11 to the
range O < p < 2 . This extension was also mentioned in [SI.
Allan [ 13 considered in some detail the case where T = T . This
i s the case of exactly adjacent sample averages--no "dead time" between
measurements. Allan defined a function, X (N, p), as follows
where i t i s again assumed that
* the spectrum of phase fluctuations while variances a re taken over average frequency fluctuations. exponent, a, in [l] plus two. a r e confined to one variable, y(t) (analogous to frequency in [l]) and the spectral density of y, S (f). spectrum of the integral of y(t) .
It should be noted that in reference 1 the exponent, a, corresponds to
In the present paper, a is equal to the Thus, in this paper, all considerations
This paper does not consider the Y
3
i
FIG. I p-a MAPPING
4
E u (N, T,T) a T’, N constant . [: 1 Allan shows [l] that experimental evaluations of X (N, p ) may be used to
infer p and hence the spectral type by use of the mapping of Figure 1.
Since many experiments actually have-dead time present, i t is of
value to make two different extensions of this function, X (N, p). Firs t ,
define B ( N , r , p ) by the relations 1
E[02 (N, T, 7 1 I B1(N, r , t . )
where r E T/T and
E (N, T, 7 ) a T’, N and r constant. K 1 The second function, B ( r , p), is defined according to the relation
2
I
!
where r = T / T .
N samples to the expected variance for 2 samples (everything else fixed);
while B
no dead time (with N = 2 and T held constant). The B’s , then, reflect
bias relative to N = 2 rather than N = co .
In words, B is the ratio of the expected variance for 1
i s the ratio of the expected variance with dead time to that of 2
It is apparent that B (N, r = 1, p) 1 E X (N, t.).
For the conditions given above and with reference to Allan [l],
one may write expressions for both B and B 1 2’ as follows:
5
in particular for r = 1,
and
except that by definition, B2(1, p ) E 1.
on the r -1 te rm when r e 1, and, indeed, proper.
involved with r L 1 the magnitude ba r s were dropped in reference 1.
The magnitude bars a r e essential
Since Allan [l] was
For p = 0, equations (8), (9), and (10) a r e indeterminate of form
0/0 and must be evaluated by L’Hospital’s rule.
also be given when expressions of the form 0
Special attention must 0
a r i se .
One may obtain the following results:
N(Nt 1)
N B1 (N, 1, 1) = - 2
1
B1(N,r , 2) = 6
B1 (N, r, -1) = 1 i f r 2 1
B1 (N, r , -2) = 1 i f r # 1 or 0
B 2 ( 0 , p ) = 0
B ( 1 , p ) I 1. 2
6
2 B2 (1, 2) = r
1 B2 (r , 1) = - (3r -1) i f r I 1 2 r i f O<rSl
B~ (r, - I ) = { 1 i f r > l
O i f r = O
2/3 otherwise I
Values of the functions B (N, T, p ) and B2(r, p ) a r e tabulated on 1 the following pages for values of N, r , p as shown below:
p = - 2 . 0 to 2.0 in steps of 0.2;
N = 4, 8, 16, 32, 64, 128, 256, 512, 1024, (0 ;
r = 0.001, 0.003, 0.01, 0.03, 0. 1, 0. 2, 0 .4 , 0.8,
1, 1.01, 1. 1, 2,4, 8, 16, 32, 64, 128, 256, 512, 1024,
2048, Q) . Figure 2 i s a graphical representation of B (r, p ) for O l r i 2 and 2 - 2 s p 5 2.
Examdes of the use of the bias functions
The spectral type, that is , the value of p, may be inferred by
varying T, the sample time [I, 21. Another useful way, however, of
determining the value of p i s by using B (N, r, p ) as follows: calculate
an estimate of E 0 and of E D 2 (2, T, T ) and hence B (N, r , p);
then by use of the tables the value of p may be inferred.
1 (N, T, T ) [ f 1 [ Y 1 1
2 Suppose one has an experimental value of o (N T T ) and its
11 1, I spectral type is known--that is , p i s known.
wishes to know the variance at some other set of measurement para-
Suppose also that one
2 meters , N2, TZ, T ~ . An unbiased estimate of o (N T T ) may be 2, 2, 2 calculated by the equation:
7
where r = T / T and r2 = T / T 1 1 1 2 2 '
Obviously one might be interested in N = a . the expected value of 0 (a, T , T ) is also infinite.
In this case if p 1 0,
This is true because, - 2 2
Y 2 2
Lim B (N r , p ) = a , N2+ a 1 2' 2
for p 2 0.
Also, it should be noted that, for p = 2, E (N, T, T) i s a 1 function of f for any N 2 2 , T, T, even though B (N, r J 2) and B (r, 2)
as determined from (8), (9), and (10) a r e finite and well behaved [3].
In this region, p
1 1 2
2, the low frequency behavior is critically important.
ACKNOWLEDGMENT
The author wishes to acknowledge the capable assistance of
Mrs . Bernice Bender who wrote the computer programs for evaluating
the functions and Messrs . D. W. Allan and D. J. Glaze for many helpful
discussions.
8
f REFERENCES
1 D. W. Allan, "Statistics of Atomic Frequency Standards, I' Proc . IEEE, Vol. 54, No. 2, February 1966, pp. 221-230.
2 R. F. C. Vessot, L. Mueller, and J . Vanier, "The Specification of Oscillator Characteristics from Measurements made in the Time Domain," Proc. IEEE, Vol. 54, No. 2, pp. 199-207, February 1966.
3 J. A. Barnes and D. W. Allan, "An Approach to the Prediction of Coordinated Universal Time, " Frequency, Vol. 5, No. 6, November/December 1967, pp. 15-20.
,
9
TABLES OF B1 AND B 2
The tables a re photographic reproductions of the computer output.
Each entry for the value of the functions, B and B consists of a
decimal number followed by an integer which is the exponent of 10.
ra ised to this power should multiply the decimal number.
table entry 2 . 752 t 003 could be written 2 . 7 5 2 X10
Similarly, "9.869 - 001" = 9.869 X 10-1 = 0.9869.
1 2 Ten
Thus the 3 or , simply 2752.
10
. , - .. - . . . , . - . . . . . - - ____ -. .
r(
0 1 0
0 0
Y a
8
J N 0 d
N In r.
9 in N
a nI -
4 9
N m
5 4
a
d
- - z
a
ul c 0 + U d b 0
d
J 0 0 + CD I- m
J
J c 0 + f- 0 0 0
m c, 0 + n r U
N
N 0 c m (P 0'
.a
N C c * C 5. r- . 4
.-4
c = + m m LT.
5
4
=' C + 0 0 N r.
c C c f?
F, m (I:
c c N
a
In c c + 0 0. 9 b #-I
J 0 0 + 9 N m 4
J c 0 + N * c d
m C 0 + Q J r- (c
N 0 C + .-4
c P u: .
N c 0 + 0 5: b
.-4
d 0
m m In t
- -
d c c * 0 0 N . c
c c 0 + m r, m m .
C
9 .-a
8
In c t 6 6 h' 9 b d
d 0 c + M 9 N
.i* 0
J c 0 + r- 0
r.
m 0 C + m J I-
N .
N 0 c 9
Q
5
n
N c C + 0 ;n IC
r(
d
2 (1 m m J
4
C C
0 C N
e
C C 0
m c; m m . c 9
.-a
a
In 0 0 + 0 9 In
0 4
4 0 a + f- Q
4
d 8
d C T
0 + F c m . d
m 0 0 + J F: .c- N
N e 0 * 9
Q
9
c
N C 0
a IJ? r- d
4 c C
N m In t .
4 3 c 0 0 N r.
c c C
m r m
r, . C #
d
8
ul c c, + d a # a r.
# 0 0 + d (0 0
4 8
J 0 0 + 9 9 c . .4
(1 c c * 5
r- N
c
N c 0 + U U
< :
N c t
4 u- r- 4
4
3 c
m N ul J
-4 0 CI
0 C N c
t c c
m 0 m c,
c h 9
d
8
In t c N Q m a d
4 0 0 + d (1
m 0:
3 c 0 + J 0
d
8 d
0 C c + N U s N
0
N 0 0
t- 9 Q.
5
N c c
C f . .-4
- c c + v % In 5
d c C + 0 0. r.
c
c c c
n m m m
c c 4
8
In c c + 9 (c (u
a r.
4 0 0 + 9 0 * m
0
J 0 0 + d 0 0
.-I
m 0 0 + F c
N
N 0 0
c. U r- e
(\i c 0
r m b
4
- - - - - + f- s d
f
-4 c c lr 0 .-4 b - c 0 c + U N m m
C CL
0 .
8
VI C 0 + d N d 0
d
4 0 0 + m b m m
0
m c 0 + 4 J I-
*
0 0 .= I
Q U
N
N c c + U
U
c
c
% c
U OI 9 - - - - c + m l-7 4
4
* c c c z 4 . c
C G c I-
(n
0
c
C .K
c.
8
J c c r- d R
(F .
J 0 0 + r- r- CD
N
m 0 0 + m 0 no ID
m 0 0
0 LT
&
n
N t C
U ;I c,
c
n! c C
3 a
?
4
3 C
r3 U .A.
3
c C
&
9
- c c
.j
- -
a t-? n'
t f
0
8
4 C c 9 m r . 9
J 0 0 + 0 0 N
N 8
m c 0 + m 9 LT
9 . m c c U F a 4
N C 0 + G
h
Lp
n
ru C c + U F, -3
4
5
Tc, U o m
- -
4 - - c
C 0
4
4
0 c
c C w m
- -
c n, 0
8
e 0 + 0
e - nl . 0
4 0 0 + m N 3
c(
m c 0 + d * d
J .
m 0 c + u LT c .-4
N 0 C
in 0.
yr,
a
N E 0 + t- w - - e - - 0
0 !I- (I:
m
- - - c
IC r- a3
0
C c C
r- N .= Fi
C 0
0 I
m 0 0 + 9 9 0
m 0
d c 0 + 0 N H
N 0
(Ti 0 0 + N (u 9
b 8
m c 0 + al m U
N 0
N C CI
P n: 3r
c
8
N c c
U R .c N
- C c + Fs fi
CC
c
.-4 0 c LI: T UI
V
3 3 C + U 51 N
U
c c c - 4. r- &
t N
0 I
0
0 0 + r- 0 m
0 9
0 c c + 0 c (0
8 W
m 0 0 + m * m m
m c 0 + M m ru d
N 0 c + In 5 IT
J
N 0 c 0
In n
d C c
C 0 h.
m
r. 0
+ r- 0 C.
- -
0
- d - - c
0 t u-l
4
c c c
C In d
rk
c U
0 I
.
4 0 c + N U IC
0 r.
m c 0 + 9 N nJ 8
m
m 0 0 + In r ] m . CI
k c 0 + m N I-
In
N C C + Q. (c c N
c(
0 0
P- a I
U
c c 0 + n IC - ?.,
- 3 C
a dl h
d
C 0
m J) 0
U
- -
*
0 c c + c If,
N
- c 8 3 I
m 0 0 + m m m c)
0
m c c + 9 r- 0 . d
N 0 0 + In 9 a J .
N C 0 + N 9
N
- . c. 0 0 + rp 9 u 0
L.
0 0
4 c n4
U
- c 0
U si S
e
c 0 c + r- D rl
I
- - - c + X c
m
f?
0 C c + 0 a2 I(
c
c 5
0 I
.
0 0 0 + 0 0 0 0 c1
N c C + r- c(
4 m
N 0 0 + 0
IC -r.
d
r. c 0 + r- 9 Y
ID 0
0-l
C c + c c F
t 9
- c 0
b c
n,
d c 0 + t 0 r(
H
0 3 3 * I- 9 4
lJ--
3 '3 C
0 C c CI
C e C
I- rc 1c
c
C c c. I
N 0 0 + N
VI N
.-I
0
N c 0 + (c In 0 8
.-a
r. 0 0 + 9 m (F
VI 8
H c 3 + 9 0 (r,
m
- c 0 + In U 0 . c.
d 0 c.
3 Rf .-I
CI
0 c C
t- J
rc y:
0 0 C
(0 .p CD
r? . 3 c c + .d 4 l 7
N
c C t
0.
4 a
c K
d I
d 0
+ 0
IC)
9
a
d
0
d c c + 9 m ? m
d 0 0 + 9 9 0
(u 8
d c 0 + m In m
0
6
0 c c + d (L Q
0
a,
0 0 0 + c nu 0.
U 8
0 c 0
0 m OI
m
0 cs C
c 5 rc
N
c 0 0
t IF 9
4
c 0 c + r- N m c
c 4
I
0
r.
d 0 0 + In m VI . M
0 C 3 + 9 e 6
(r
0 0 0 + (c
0
(c
- 0
0 0 0 + d
c(
b u m
0 0 0 + Q I-
J
C
0 c 0 + In 0 4
m
0 c 0
0 J
N
n
0 0 c
0 P Uf
M
0 C. 0
aD 4
r-
c, c, C + a Q c . r(
c 9
I
. .-l
0 0 c + Q d
0; (1
0 C 0 + 9 0 r(
8
P)
0 0 0 + 9 In 9
N 0
0 0 0 + 9 0 (1
N 0
0 c 0 + 0
C
(u
c(
0 C 0 + IC U r- e
0 c c + 4 kn
c
c
G 0 c 0 s rl
d
0
C
0
(u
- -
- 0
c
c c C
4
0 0 - c Q!
c I
0 0 0 n
0 t c 0 + ! 0 0 0 j
i i
0 f
. L L
0
+ 0 0
0 0 0
1 d
0 0 0 + 0 c 0
c I
I
1 i
0 C C + c 0 a 1 - I
I
I
0 0 c 0 3 0 . rl
c c . N I
1 1
I 0
f. 3
0
c
s m -4
3 3 3
W
'D rl r J 3 3
0
2
w N '3 '3 3
Y
c
D 3 -4
0 3 c
E
c T. 4
0 3 c
Q
'3 3 AI
3 >
. e
N 0
3 m VI
0 3 %I
P W h) 0
0 0 h)
0
P
0 W 9
3 3
0
N
c 0
0 0 0 + 0 0 W
I 0
Cm 3
iu .- E
+ 3 =a 3
n
JI 3 x E
. - - - -
c
h' c3 u 0 0 L
W c c V
3 3 c
-4
P D c + 0 3 c
c . 9 D N
3 0 N
P
D 4 0
0 3
0
2 1
I 0
3 3
v c a d
2 3 - 4
D
ul 3 iT. + 3
0
- - w . Y
4 4
2 c I-
e
5 3 r 0 3 N
w 51 P
3 3 N
. c
c . I
P w 3 3 W
W
P 9 0
0 3 W
0
9
N w P
0 3 i )
0
c 0
4 9 P
a P
8
I N 0
3 >
L
z '3 3
3 3 c
r. . 5 7 3
0 3 c
r 0
3 0 2
c 0
- -
c
3 0 3
0 3 J
0
0 3 0
0 0 0
L 0
0 a 3
0 0 0
c 0
3 0 c + 0 3
I c . 3- >
d . 4
0 0
3 2 7
d
0
E
4
3 C
m
- - L
,n c
0 3
- v
N
E P J
3 2 0
.
iJ
Y
0 c
0 3 0
P
u 3
3 3 0
.n
ul P N w 0 3 0
.
4
4 0
c c + 0 3 3
9
* P 4 3 3 0
L
L 0
0
c + 0
nJ
I
p. 3
c . Y
.A N D
3 3 - - Y
m ;n P
3 c -.
N r I c.
- d
c - 3
w e 3 E
3 3 0
m c E
3 7 3
e
9
3 a . r.
0 0 0
r b 4
(r
0 3 c
m
N
VI 9 + 0 0
0
c
c
h)
4 OI P + 3 0 c
0
W
N * e + 0 0
0
c
I c . ru - L
E 9 u .=a 3 - - v u) E h
3 3 s
W
P rl 3
0 0
-
F
'3 u1 r 3 3 0
L
L
u P 0 0 c
L 0
0 3 3
0 a e
W
P m . c
0 0 c
e 0 0 0 c + 3 0 c
m
P 0) + 3
c
0 c
a.
c
0 P W + 3 0 N
I c. . 1 3
3 7 4
7 3 1
W - v 0 0
3 - - - - ul
r i\ 4
0 c 0
c.
L
c > 3 3 r.
u - r 4
¶ 0 e
P
'3 3 3
0 3 c
.
m Jl 0 4
0 3
0
L
c 0
e W 4 + 0 3 V
N
'4 iP a, + 0 3
0
N
W
W W W + 0 0 N
0
I 0
E 3
. N
e w * 3 3 >
n
r P w 0
3 '= c
L
-4 .t s
3 3 - 3
3 v U
0 3 c
d
0
E u '3
3 C Y
W
0 JI UI 3 3 h,
c . J In m
a 0
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