Tables *of Bias Functions, and 82 for Variances Based on ... · TABLES OF BIAS FUNCTIONS, B1 AND...

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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

'3

N

m 0

I

L + 3 0 4

P

W .AI 0

a. 3 W

0

c. . L

N + 0 0 *

I 0

'u 3

0

w 1 u 43

3 3 3

D

3 cp s

. 5- - -

R

* 4 P

c

. - L

4

3 3 P

3 3 4

N

W -4 4

a 3 N

.

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