NBS TECHNICAL NOTE

375

Tables of Bias Functions, Bj and B2, for

Variances Based on Finite Samples of

Processes with Power Law

Spectral Densities

J. A. BARNES

Q

,/

TK^T or

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Q

*<"*au of *

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NBS TECHNICAL NOTE 375

ISSUED JANUARY 1969

TABLES OF BIAS FUNCTIONS, Bi AND B2, FOR VARIANCES

BASED ON FINITE SAMPLES OF PROCESSES WITH POWER

LAW SPECTRAL DENSITIES

J. A. BARNES

Time and Frequency Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

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Price 50 cents.

TABLE OF CONTENTS

Page

ABSTRACT v

BODY 1

The Bias Functions, B and B 2

1 ^ Figure 1, fi - a Mapping 4

Examples of the Use of the Bias Functions 7

ACKNOWLEDGEMENT . 8

REFERENCES 9

TABLES OF B AND B 10

Figure 2, The Bias Function, B (r,u) 34

iii

TABLES OF BIAS FUNCTIONS, B AND B FOR VARIANCES

BASED ON FINITE SAMPLES OF PROCESSES WITH POWER

LAW SPECTRAL DENSITIES

J. A. Barnes

ABSTRACT

D. W. Allan showed that if y(t) is a sample function of a random

CL noise process with a power law spectral density (i. e . , S (f) = h | f | ), then there is generally bias to the estimated variance of y, defined as N

n= 1

where N is the number of samples, y 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= 1

Allan also showed that, under these conditions, the expectation value

of the estimated variance is proportional to T where [i is a constant

related to a, the exponent in the spectral density; i.e.,

E[a (N, T, t)] « t^ . y

Based on this work one may define the two bias functions
E[o (N, T,t)]

B (N, r, n ) = f

E[a (2, T, t)]

y

and

B (r, p.) = E[a

2

(2, T,t)]

y i

2 E[a 2

(2, t,t)]

y

where r = T/t and the B's are functions of u through their dependence

on y(t).

2

If one has a sample variance, a (N , T,,T,), the bias functions

y 1 1 1 allow one to give an unbiased estimate for o^ (N , T , t ) provided the

spectral type is known (i.e. , \i is known).

The tables give values of B (N, r,u) and B (r, \x ) accurate to four

significant figures for the following values of N, r, jjl :

u = -2. to 2. 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, & .

Key Words: statistics, variance, spectral density, unbiased estimate

VI
TABLES OF BIAS FUNCTIONS, B AND B , FOR VARIANCES

BASED ON FINITE SAMPLES OF PROCESSES WITH POWER

LAW SPECTRAL DENSITIES

J. A. Barnes

Consider a random variable y with mean m. The variance of y is

defined as the expectation value of (y - m) , that is

Var y = E[(y - m) ] .

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

(Var y) es t. = TTT Z2 (Y

i <y»' (i)

i= 1 N

1 - 1 where <^y^> = — \ y. is the mean of the y.. The factor ——— is used in

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 —* co . For finite N, the estimated variance has some expected value }

N —N-l l^i V(y, -<y>)'

n= 1

If the y are independent (actually, non-correlated is sufficient) random

i variables, then the expected value of this estimated variance for finite
BfN, r,u) = E[a 2

(N, T,t)]

I

E[a 2

(2, T,t)]

y

and

B (r,M.) = E[g2

(2,T,t)]

i i

2 E[g 2

(2, t,t)]

y

where r = T/t and the B's are functions of [i through their dependence

on y(t).

Z

If one has a sample variance, o (N , T,t ), the bias functions

y 1 1 1 allow one to give an unbiased estimate for Q^ (N , T , t ) provided the

spectral type is known (i.e. , u is known).

The tables give values of B (N, r, u) and B (r, u) accurate to four

significant figures for the following values of N, r, u :

|jl = -2. to 2. 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, . 2, 0. 4, 0. 8, 1, 1.01, 1.1,

2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, co .

Key Words: statistics, variance, spectral density, unbiased estimate

VI
TABLES OF BIAS FUNCTIONS, B AND B FOR VARIANCES

BASED ON FINITE SAMPLES OF PROCESSES WITH POWER

LAW SPECTRAL DENSITIES

J . A. Barnes

Consider a random variable y with mean m. The variance of y is

2

defined as the expectation value of (y - m) , that is

Var y = E[(y - m) ] .

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

(Var y) es t. = ih z2 (y

i -<y»' (i)

i= 1 N

where <(y^> = — \ y. is the mean of the y.. The factor is used in

i=l

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 —* cd . For finite N, the estimated variance has some expected value t

N

N *-!>. <y>

n= 1

If the y are independent (actually, non -correlated is sufficient) random

i variables, then the expected value of this estimated variance for finite
N is exactly equal to the true (infinite N) variance. If the y. are correlated, 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 \_l]-

The Bias Functions, B and B

Following Allan (_lj, consider a random process y(t) with continuous sample functions. We assume that y(t) has a spectral density,

S (f), which obeys the law

y

S (f) = h|f| a

, f/ <|f|<f

u

(2)

where h is a constant, the limit frequencies in and f satisfy the relations

0<f/7 <<f<ao , It u

and any intervals of time, At, of any significance satisfy the relations

1 A 1 « At <<

f ^ -" ^ f

.

u I

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 + j . That is,

i r

t+T

y(t) = - / y(t')df . (3) T

t

One, now, may determine an estimated variance from a group of N such

measurements spaced every T units of time; that is,
N

y<

N' T' T>=NTil]Jy(t+nT) N^V^'i , (4)

n= 1

which is called the "Allan variance" [l] . Allan [l] has shown that under these conditions,

E a

2

(N, T, t) <x t , N and T/t constant,

where u is related to a according to the mapping shown in Figure 1

(see references 1 and 2 ). The relation between \x and a may be given as

-2 if a > 1

|jl= \ -a -1 if -3< a < 1 not defined otherwise.

This mapping involves a simple extension of Allan's work [ lj to the

range 0< (J. < 2. This extension was also mentioned in |_3J.

Allan L considered in some detail the case where T = t. This

is the case of exactly adjacent sample averages --no "dead time" between

measurements. Allan defined a function, X (N, \i), as follows

E|~C7

2

(N, T,T)1

X(N,u) = —1-| J , (5)

E[a (2,t,t)J

where it is again assumed that

It should be noted that in reference 1 the exponent, a, corresponds to

the spectrum of phase fluctuations while variances are taken over

average frequency fluctuations. In the present paper, a is equal to the

exponent, a, in [[ljplus two. Thus, in this paper, all considerations

are confined to one variable, y(t) (analogous to frequency in Llj) and

the spectral density of y, S (f). This paper does not consider the

spectrum of the integral of y(t).
FIG. I H--<* MAPPING

4
a (N, t,t)

y

a T , N constant

Allan shows [lj that experimental evaluations of X (N, |j. ) may be used to

infer \x and hence the spectral type by use of the mapping of Figure 1. Since many experiments actually have dead time present, it is of

value to make two different extensions of this function, X (N, \x). First,

define B (N, r, u) by the relations

EJV(N, T,t)]

B (N, r,u) = E (T(2, T.T)

L y J

where r = T/t and

a

2

(J y

;. t, t

)j

<x t , N and r constant

The second function, B (r, \x), is defined according to the relation

E[a2

(2, T,T)

B

2

<^> s Mr,—;- E a (2,t,t)

where r = T/T. In words, B is the ratio of the expected variance for

N samples to the expected variance for 2 samples (everything else fixed);

while B is the ratio of the expected variance with dead time to that of

no dead time (with N = 2 and T held constant). The B's, then, reflect

bias relative to N = 2 rather than N = co . It is apparent that B (N, r= 1, |jl)

sX(N, n).

For the conditions given above and with reference to Allan [_lj,

one may write expressions for both B and B , as follows:
1 + B (N, r.n) =

N-l

n=

£1

N-n

N(N-l)

-.1 i M- + 2 , _ . u. + 2 . , , u + 2 2|nr| r - |nr+ir - |nr-l|^

1 + 2|r|

fl+2 -|r + ll^

+2

-lr-ll fl+2

; (8)

in particular for r = 1,

B^N, I, is.) = 2(N-1)(1-2M') (9)

and

B

2 (r,|i) =

1 1+ I

2lr'r

+2 -lr+lP +2

r-1

|JL +2

2(1 - 2^)

(10)

except that by definition, B (1, [j.) = 1. The magnitude bars are essential

on the r-1 term when r < 1, and, indeed, proper. Since Allan |_1J was

involved with r > 1 the magnitude bars were dropped in reference 1. For |i. = 0, equations (8), (9), and (10) are indeterminate of form

0/0 and must be evaluated by L'Hospital's rule. Special attention must

also be given when expressions of the form arise.

One may obtain the following results:

B (2,r,n) = 1

1 N

6

B^N, 1,1) =~

B (N, r, -1) = 1 if r > 1 B (N, r, -2) = 1 if r * 1 or

B

2

(0,u) = B

2

(l,u)= 1.
B

2

(r, 2) = r

2

B (r, 1) = j(3v -1) if r > 1

Jr if 0<r<l B

2

fr 0<r

(r '- 1) = ilifr>l

(0 if r = B

2

(r, -Z) = h if r = 1 12/3 otherwise

Values of the functions B (N, r, jjl ) and B (r, u ) are tabulated on

the following pages for values of N, r, \x as shown below:

jjl = -2. to 2. in steps of 0.2;

N = 4, 8, 16, 32, 64, 128, 256, 512, 1024, en

;

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, co .

Figure 2 is a graphical representation of B (r, u) for 0<r <Z and

-2<m-<2.

Examples of the use of the bias functions

The spectral type, that is, the value of \x , may be inferred by

varying T, the sample time |_1, 2J . Another useful way, however, of

determining the value of u is by using B (N, r, u ) as follows: calculate

an estimate of E a (N, T, t )1 and of E \a Z

(2, T, t )land hence B (N, r, (jl);

then by use of the tables the value of u may be inferred.

2

Suppose one has an experimental value of a (N , T , t ) and its

spectral type is known --that is, (J. is known. Suppose also that one

wishes to know the variance at some other set of measurement parameters, N , T , t . An ui

calculated by the equation:

2

meters, N_, T\, T_. An unbiased estimate of a (N_,, T\, t_) may be

2 2 2 y 2 2 2
a (NTt)

y Z Z Z W B

l (N2' r Z^ )B2

(r 2^>

B

1 (N1 ,r

1

, (x)B

2

(r

1 ,|a.) a (NT t ) y 1 1 1

where r = T

1 / T1 and r

? = T ?/ T

Obviously one might be interested in N = co . In this case if |jl > 0,

2

the expected value of a (co, T , t ) is also infinite. This is true because.

y Z Z Lim

N-co

Z

B(N.r, ,*)=..

is a

for |a > 0.

Also, it should be noted that, for [x - Z, E a (N, T, t)

function of f /; for any N>Z, T, t, even though B (N, r, Z) and B (r, 2) as determined from (8), (9), and (10) are finite and well behaved [3].

In this region, u ~ Z, the low frequency behavior is critically important.

ACKNOWLEDGEMENT

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.
REFERENCES

D. W. Allan, "Statistics of Atomic Frequency Standards, " Proc.

IEEE, Vol. 54, No. 2, February 1966, pp. 221-230.

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.

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.
TABLES OF B AND B

J. C*

The tables are 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. Ten

raised to this power should multiply the decimal number. Thus the

3

table entry 2. 752 + 003 could be written 2. 752 X10 or, simply 2752.

Similarly, "9.869 -001" = 9.869 X 10

_1

= 0.9869.

10
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