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.pap.,,",,-"''-
Implications of the Exact
Bradford Distribution
by
Philip M. Morse
OR 098-80
January 1980
-
I
-
IMPLICATIONS OF THE EXACT BRADFORD DISTRIBUTION.
by
Philip M. Morse
Operations Research Center
Massachusetts Institute
of Technology.
Abstract
The exact Bradford distribution 6 fits remarkably well
data on the scatter of articles in a given specialty among
journals, perhaps better than the approximate form of the
distribution.
This implies certain tendencies of authors in
the specialty to distribute their future articles among the
journals, on the basis of the past productivity (the number of
articles previously published) of the journals.
In particular
it is shown that if this tendency follows a Bradford law for
each group of journals with a given past productivity, then
the distribution of papers among all the journals active in
the specialty must also follow the exact Bradford distribution.
Vice versa, since this distribution does fit the data, it
implies that the author tendencies also are Bradford.
The
consequences, when the statistical situation is fairly constant
with time, are explored. -Applications to citation scattering also
are discussed, in Appendix A.
The Bradford distribution1 turns up in the discussion of
several aspects of the analysis of published information.
For
example, in the case of articles in a given specialty appearing
in a number of journals, the scatter among the journals in the
number of articles per year fits the Bradford distribution
fairly well4, in many cases.
Usually the productivity n (the
number of articles in the specialty published per year),of a
journal in the collection, is considered to be a continuous
variable and the usual formulas2 '5 for the Bradford distribution
make this approximation.
It is a reasonably satisfactory
approximation as long as n is greater than about 20; but for
productivities of 10 or less it is not satisfactory, and there
have been many discussions of this fact2 '7'8.
The appropriate response, of course, is to utilize mathematical techniques to obtain an exact solution, complying
exactly with the Bradford requirement relating cumulative numbers
of journals and cumulative production, keeping in mind that n is
a discrete, not a continuous variable.
This has recently been
done6 and the results show that the data in many cases fit the
exact solution better than they do the approximate one; in fact
the fit continues clear down to the periodicals publishing only
one article in the specialty per year.
Since the fit is closer and more general than would be
expected from a chance resemblance between the data and some
arbitrarily chosen mathematical function, it might be profitable
to see whether there are any properties of the exact Bradford
distribution that could make it particularly suitable to describe
these data. ) The present paper discusses some of these properties
and suggests a few reasons why the fit may not be fortuitous.
A few of the mathematical properties of the exact Bradford
distribution are given in Appendix A.
Details of their deriv-
ation have been given in anearlier paper6.
Only a few of these
properties are needed for our discussion here.
The fraction of
all the elements in the collection (all the active journals,
publishing in a given specialty, for example) that have productivity n (have n articles published in the specialty in a year,
for instance) will be denoted f'
The Bradford distribution
differs from most others by not including elements with n
0
.(inactive elements) and by having an upper limit for n as well.
Above a certain value N of productivity the elements, in practice,
become sparsely distributed; mathematically the distribution must
be cut off because the total production, the sum of nfn would
diverge if it were extended to infinity.
In fact the Bradford
distribution starts at this upper limit N of n, lumps the few
higher productivity elements together into what is denoted the
core, and
to n
1.
then extends
down through smaller values of n
All that are specified about the core elements are
the fraction F
of the elements in the core and their mean
productivity, which we shall denote by qN (which cannot be
smaller than N).
The distribution, in its exact form, then
specifies the fraction f of elements with productivity n, from
n N-l to n =l. Note tat one has a fairly wide range of choice
of the value of N, for the characteristics of the distribution
are rather weakly dependent on N, as will be seen in Appendix A.
More important for our present discussion, the exact
Bradford distribution differs from most others because the
quantities fn are proportional to the same sequence of quantities y
(as Appendix A shows), no matter what set of elements
is considered,
f'n
(
(1<
n< N)
where the values of yn are given in Table III.
(1)
The value of
Li
the constant parameter ,
for a particular example (a set of
active journals, for instance, and their publication of articles
in a given specialty) is determined by the mean productivity qN
of the core and the mean productivity of all elements ql (total
production divided by total number of elements), according to
the formula
-
(qN - vN)/(q - q)
(2)
where UN and VN are defined in Appendix A and tabulated in
Table III.
Thus the distribution is completely determined, for
a particular case, by the core productivity q
and the mean
total productivity ql and, of course, N, the upper limit of the
distribution, beyond which lies the core (in fact the value of
B is not very sensitive to the choice of the exact value for N,
as long as the data is fairly continuous for values of n less
than N).
The distribution of Eq.(l) is completed, for n
fN
1 - (UN/i)
(n = N)
N,by
(3)
as detailed in Appendix A.
Table I shows the cumulative distribution
Fn = fn + fn+l +
fn+2
+
+
*
fN
=
1-(U/)
(4)
and cumulative productive weight
G n = nf + (n+l)fn+l + . · .
=
+ (N-l)fN-l + Nf N
l
(Vn/0)
(5)
(if A is the total number of elements in the collection, then
AFn is the number of elements and AGtheir total production for
all elements with productivity equal to n or greater).
given for two examples from the literatures,
Data is
and compared the
exact Bradford distribution for the two appropriate values of
TABL
I.
(see Reference 4)
Articles on
N= 9
F
0.111
Data 0.094
Operations Research Published in Various Journals.
n =8
7
6
5
4
3
2
1
0.121 0.134 0.151 0.177 0.217 0.287 0.440 1.000
0.116 0.138 0.154 0.181 0.227 0.287 0.451 1.000
G
3.123 3.203
Data -2.949 3.122
3.295
3.273
3.402
3.370
3.530
3.505
= 1.5493 ;
3.689
3.689
3.898
3.924
4.205
4.216
4.765
4.765
4.765
Articles on Statistical Methodology Published in Various Journals.
n=8
7
N=9
6
5
4
3
2
1
F
0.156 0.166 0.178 0.195 0.219 0.257 0.323 0.469 1.000
Data 0.159 0.165 0.171 0.189 0.201- 0.232 0.305 0.451 1.000
G
Data
7.372
7.488
7.448 7.535 7.636
7.537 7.579 7.689
= 1.6332 ;
7.758
7.750
ql
7.909 8.108
7.872 8.091
8.933
8.399
8.384
8.933
8.933
Ql and B. We see the closeness of fit, clear down to the lower
values of n. Another example, this time of the scatter among
journals of the citations of articles in a given journal, is
discussed in Appendix A and displayed in Table IV.
tM
e exac tBradfo dr
-,
Because of the form of Eqs.-l)
distribution has a unique property, illustrated by the following
example.
The various articles on a given subject, published
in the collection of journals active in the field, are contributed by
-a collection of authors, who have submitted
the articles.
Their tendency to contribute papers to a given
journal is influenced by the number of papers in the specialty
previously published by that journal.
The apportionment of the
newly submitted articles among the journals, by these authors,
will of course affect next year's distribution of articles
among journals.
The interesting fact is that if this apportionment by
authors of articles among journals follows the exact Bradford
pattern, for each journal's past production n, then next year's
distribution of articles among
pattern.
ournals must follow the Bradford
This may appear to be a tautology, but actually the
result depends crucially on the form of the Bradford distribution, as displayed by Eqs.(l) and (3).
If the allocation of
authors' articles among journals did not follow this form, then
next year's distribution of articles among journals would not
have this form.
To make this statement more explicit, let us call p(j if n)
the probability an active journal will publish j articles in
the specialty next year if
it
published n such articles last year.
Also let the fraction of active journals that published n articles
in the specialty last year be fn.
Then the fraction f
of active
journals that will publish j articles in the specialty next year
is -the sum of the products, fraction that published n last year
times-the chance that, having published n last year, j will be
published next year, summed over n,
f
. p(j if 1)f1 + p(j if 2)f2 +
·
(6)
There may be some borderline journals which were active last
year but are not this year, or vice versa.
If the number A of
active journals remains roughly the same each year, so that as
many enter the active collection as leave it, then a journal
which leaves the collection can be paired, statistically, with
one that enters.
We show in Appendix B that Eq.(6) is valid, if
A is constant, even though it does not include such terms as
p(O if n) or p(j if
).
Our earlier statement means that if the transition probability distributions p(j if n) are each of them exact Bradford
distributions,
p(j if
for ln.
)
=
Yj/n
(14 j< N);
- (UN/n) (j N)
N, then (as shown in Appendix B) distribution f
(7)
of
next year's articles must be Bradford, no matter what kind of
a distribution f
last year's articles had.
The value of f
for fj is the reciprocal of the mean of the (1/
i)'s,
averaged
over fG.
The process specified by Eq.(6) is called a Markov chain 3 .
If editorial and auctorial policies have not changed from year
to year, so that neither A nor the transition probabilities
7
p(J if n) change appreciably, then the distribution of articles
among active journals will reach a statistical steady state.
When this occurs, fn will equal f
if the p(j if n)'s are Bradford.
and both will be Bradford
Thus the known fact that
articles -in a specialty are scattered over the active journals
according to a Bradford distribution implies that authors submit
their articles to journals according to a p(j if n) which is
Bradford.
A pattern which differs from that of Eqs.(7) will
not produce a steady-state distribution of articles in accord
with Eqs.(l) and (3).
It is not immediately apparent why authors should tend to
accord with Eq.(7)insubmitting articles to those journals that
published n papers in the specialty last year (call them the
n-group of journals).
For each value of n (for each n-group)
there will be the cumulative functions
Fj(if
) = p(j if
Gj(if n)
) + p(j+l if n) * · · · p(N if n)
jp(j if n) + (j+l)p(j+l if n)
+ (-l)p(N-l
if
) +
(8)
p(t if n)
The average productivity of the n-group will be ql(if n) =
Gl(if n) and p(N if n) of them will have N or more articles in
the specialty next year, with a mean productivity q(if n)
GN(if n)/FN(if n) for this portion.
It is reasonable to
suppose that these mean productivities for the n-group should
increase as n increases, i.e., that authors would tend to send
more papers to those journals that had published more papers
in their specialty in the past.
If Eq.(7) holds, as seems to
be the case, then not all authors will send their articles to
to the most productive journals.
Some authors might prefer the
less productive journals for various reasons; some may wish to
--call attention to their work to others outside their specialty
or- to bring out connections between their work and other specialties.
Whatever the reason, among the group of journals that published
n, less than N, articles in the specialtylast year, there will
be a :core group" that will publish a mean number qN(if n) of
articles, less than q(if N), and a diminishing number that will
publish less than N, the average productivity of the n-group
being ql(if n), less than ql(if N).
For each n-group, the
implication is that the distribution of articles published in
the specialty is Bradford, in accord with Eqs.(7).
This is not to say that the different n-groups remain
distinct from year to year, even in steady state,
If a journal
publishes n articles in the specialty this year, it need not
publish n next year.
All that is required for steady state is
that the same fraction of the active journals publish n articles
next year as last year; this fraction will contain some different
journals, but the size of the fraction will remain the same.
Indeed, as mentioned earlier, some journals previously inactive
may become active and a similar number will become inactive.
Thus we should not expect ql(if n) to be equal to n, all we
should expect is that it increases as n increases.
All that is
required by the steady-state relationships is that the parameter
is and the mean productivity qls,of the distribution f
active journals, be related to all the individual
for all
in's and ql(if n)'s
for the different n-groups, in accord with Eqs.(5B) and (6B) of
Appendix B.
Therefore, once we have verified that the scatter of
articles in a given specialty over a collectioa of active journals
is in accord with the exact Bradford distribution,
have
determined that it remains more or less the same from year to
year, and have determined the values of its parameters 0s and qls,we -can infer that the distribution of articles within n-groups
is also Bradford.
But knowledge of Us and qls alone does not
determine the parameters
n and ql(if n) for the various n-groups
beyond the requirements of Eqs. (5B) and (6B).
Data for the
n-group distributions themselves would have to be obtained, to
fix the values of the «D's and ql(if n)'s and to verify that
Eqs. (B)
/
and (6B) are indeed satisfied.
Nevertheless Table II shows what kind of distributions
p(j if n) might have in order to result in the first of the
distributions of Table I.
Here we tabulate Fj(if n) and G(if n)
for each n-group, assuming a set of Bn's and ql(if n)'s which
satisfy Eqs.(5B) and (6B), just to show the possible scatter of
articles within each n-group that would result in the distribution
of Table I.
For example, for those journals that published one
article in the specialty last year, only one twenty-fifth of
them would be expected to publish 9 or more next year and nearly
two-thirds of them are likely again to publish only one article.
In contrast, two-fifths of the journals that published 9 or
more articles last year would again publish 9 or more articles
and less than two fifths of them would publish only one.
As mentioned earlier, this is just an illustrative set of
values for the F's and G's that will yield the first distribution
_of Table I.
Other sets of values for
and qlO can be obtained
from Eqs.(5B) and (6B) that would display greater author
--
preference for the higher-n journals.
Further data is needed
-to decide which set is in accord with actuality.
With the data
at hand (Table I) all we can say is the the p(j if n)'s must be
Bradford, with the
and (6B).
n 's and ql(if n)'s satisfying Eqs.(5B)
TABLE II.
-
Illustrative values of expected cumulative journal count Pj(if n)
and cumulative article count Gj(if ) for next year, derived from
P(j if n),for those journals in the first example of Table I that.
published n operations research articles last year. See Appendix B.
,-1.55;
ls 4.77"
-
=4/90 ;
;
- 14/ 9 ;
1o 0.582
I
. 1-375 ;
-~---~-''
Pvt_.
........ .....
n
- J9
1G F
0.043
030
2
j
6
5
3
__
2
1
____I·^IIC·__
G 0.370
0.087
0.670
0.114
0.808
0.156
0.979
0.232
1.205
0.378
1.535
1.000
2.138
F
G
0.097
2.090
0.111
2.184
0.129
2.293
0.155
2.425
0.196
2.588
0.268
2.804
0.426
3.119
1.000
3.693
0.154
3.814
0.172
3.918
0.200
4.043
0.235
4.198
0.304
4.403
0.454
4.703
1.000
5.249
0.087
2.008
--
0.132
0.141
G 3.645
3.724
_
4
· ______I_
0.176
0.185
5.367
__________
_I_
___ _____ ____·^___ _ ___I
________
I_____
5.442
0.214
5.541
0.238
5.660
0.275
5.807
0.339
6.002
0.482
6.286
1.000
6.804
G 6.921
0.230
6.991
0.241
7.072
0.25?
7.165
0.279
7.277
0.314
7.417
0.375
7.601
0.510
7.870
1.000
8.360
F 0.265
G 8.559
0.274
8.635
0.284
8.701
0.299
8.789
0.320
8.895
0.353
9.026
0.411
9.200
0.538
9.453
1.000
9.916
G
0 .10
10.20
0.318
10.26
0.328
10.33
0.342
10.41
0.361
10.51
0.392
10.64
0.447
10.80
0.566
11,04
1.000
11.47
F
G
0.354
11.83
0.362 0.371
11.89 ~~ 11.96
0.399 0.406 0.415
13.47
13.53 13.59
-------
0.384
12.04~
0.403
12.13
~
0.432
12.25
0.482
12.40
0.594
12.62
1.000
13.03
G 5.283
F
0.221
_
7
8
8
_
0.067
0.555
F
6
7
0.053
0.456
-
___
8
____1_
_-
F
G
-
--
0.198
__
_-
-
-
`-
--
I
.
--
.
-
-
0.427 0.444
13.66
-----I13.75
"'- -'I'-`
_-
.`
.~
.
,
.
0.471 0.518
13.85
14.00
-`--`---- I------
.
.
.
0.622
14.20
--^-
.
.
.
.
..
_
_.
1.000
14.58
' --- -- '
References.
1-
S.C.Bradford, Documentation, Crosby Lockwood, London, 1948.
2 ---B.C.Bookes,"The Derivation and Application of the Bradford-Zipf
Distribution", J.Documentation 24, 247-265 (1968).
3
Wm.Feller, Introduction to Probability Theory and its
Applications, Chapter IV, John Wiley and Sons, New York, 1957.
4
M.G.Kendall,"The Bibiography of Operational
Research",Opnl.Res.Quart. 11,
5
F.P.Leimkuhler,"The Bradford Distribution",
J.
6
31-36 (1960).
Documentation, 2,
197-207 (1967).
P.M.Morse and F.F.Leimkulhler,"Exact Solution for the Bradford
Distribution and its Use in Modelling Informational
Data", Ops.Res. 27, 187-198 (1979).
7
H.A.Simon,"On a Class of Skew Distribution Functions",
Biometrika, 42, 425-440 (1955).
8
B.C.Vickery,"Bradford's Law of Scattering",
J.Documentation, 4,
198-203 (1948).
APPENDIX A.
The Exact Bradford Function (see Reference 6).
General Definitions:
A
=Total number of productive items.
Afn - Number of items with productivity n (l
n< N).
-All items with productivity N or greater are lumped
together in the core.
ffN
Fraction of all items that have productivity n.
FN
Fraction of all items in the core.
+
M = ,fmt
Total
T
production of items with productivity n or greater.
AGn
Gn
qn
FN = Fraction of items of productivity n or greater.
=
L
mf
Gn/Fn
m
+ GN
;
AGN - Total production of core.
mean productivity of items with prod. n or greater.
ql = Mean productivity of all items ;
F1
1
qN - Mean productivity of core.
For the distribution to satisfy the Bradford requirement,
Fn
= C exp
F
(G n-GN)
- 1
these quantities must be related to those in Table III by the
following equations;
fn w yn/
(,
Ya-
Gn= -
n<)
Z
;
;
1
- (/0)
-n
where
Tn = Ylexp(-Vn)
and the parameter
F.i (
nnY
(<
) ;
1
Yn-
FAnZ S
n< N) ;
1 -
(/B)
m
qF
(1A)
(2A)
(n= N)
Em;m
(3A)
(4A)
(Bradford requirement)
, for a given collection of items, is related
to the mean productivity qN of its core and the mean productivity
Il__·_IIL_11__1__1_11___11_1-11_111
I_
APPENDIX A
Continued.
ql of the whole collection, by the equation
-
(qNUN - VN)/(qN
(5A)
- q)
and V
20, Table III gives values of yn' U
For n
For no 20
.
the following equations,
yn±' (1/n
-
n
(1/ 4 nL)
(1/n) + (1/2 2 ) + (1/12n 3 ) - (1/8n4)
AU
= 1.4954639 - Yn
0.4024484 +
vn
-
(6A)
n(n-1) + (1/2n) + (1/2n 2 )
+ (1.76/n3 )
(Il/n4)
are accurate to at least six places after the decimal point.
To determine the parameters
and q
for some collection of
items, list the number A. having productivity n,from n= 1 to as
high a value of n as there are items.
Choose N such that for n
less than N there are few (or no) An's equal to zero and for n
greater than N there are many An's equal to zero.
sn =
and
Qn
-mAm
Next tabulate
for n from 1 to N
(7A)
where the sums include the highest n for which A. differs from
zero.
S1
of course equals the total number of active items;
Sn/S1 should equal Fn of Eq. (lA) and Qn/S1 should equal Gn of
Eq.(3A).
Mean productivity ql should equal Q/S
1
and q
Qn/Sn.
Therefore if the distribution is exact Bradford, the fraction
Bn= [(Qn/Sn)Un - V7]/ (Q./S.
)
-(/S)j
See Eq.(5A)
(SA)
(with Un, V n given in Table III) should have the same value for
all n's from 2 to N inclusive.
nearly the case, with B
If this is the case or if it is
showing no secular change with n, then the
TABLE III.
The Bradford Functions.
nY
Yn
1
2
0.8671469
.2378476
3
.1084065
4
.0615988
5
.0396210
6
7
.0275921
.0203070
8
.0155653
9
.0123083
10
.0099753
11
12
13
14
15
16
.0082476
.0069325
.0059085
.0050956
.0044396
.0039024
17
18
19
20
21
22
23
24.
25
26
27
28
29
30
V]
Un
1.4954639
0.6283170
.3904694
.2820629
.2204641
0
0.8671469
1.3428421
1.6680616
1.9144567
0
0.8671469
1.1049945
1.2134010
1.2749998
.1808432
.1532511
.1329441
.1173789
.1050706
2.1125615
2.2781139
2.4202630
2. 5447851
2.6555595
1.3146208
1.3422128
1.3625198
1.35780851
1.3903934
.0034573
.0030840
.0027682
.0024984
.0950953
.0868476
.0799152
.0740066
.0689111
.0644715
.0605691
.0571118
.0540278
.0512596
2.7553128
2.8460366
2.9292264
3.0060370
3.0773748
3.1439682
3.20q4072
3.2651803
3.3206929
3.3732885
1.4003687
1.4086163
1.4155488
1.4214573
1.4265529
1.4309924
1.4548949
1.4385521
1.4414361
1.l444 2043
.0022663
.0020650
.0018895
.0017354
.0015994
.0014787
.0013713
.0012751
.0011887
.0011108
.0487612
.0464949
.0444298
.0425404
.0408050
.0392057
.0377269
.0363556
.0350805
.0338918
3.4232571
3.4708495
3.5162804
3.5597384
3.6013868
3.6413710
3.6798182
3.7168427
3.7525456
3.7870180
1.4467028
1.4489691
1.4510341
1.4529236
1.4546589
1.4562583
*1.4577370
1.4591083
1.4603334
1.4615721
APPENDIX A
Continued
distribution is exact Bradford and an appropriate value of f
is the average value of these ratios from n 2
to n
N.
The
appropriate value of ql is (q1/S1 ) and that of qN is then
[(Vql
VN)/(
A more accurate value for
UN)
-
again from Eq.(5A).
(9A)
, the least squares solution, is
obtained by calculating the sums
2
C. 2L(Qn/Sn ()<1/S1)
n
D=
and
n'
t(Qn/S)Un
Vn1(Qzn/Sn)
-
(Q,/S 1 )
and setting
- (D/C)
(10A)
-
ql is
again Q1/S1 and qN is obtained from this
and q
by
using Eq.(9A).
Note that when Bn is more or less independent of n (i.e.,
when the distribution is Bradford) then a small change in the
chosen value of N will make little change in the value of
calculated from Eq. (10A).
as
In other words the value of N can be
chosen arbitrarily within the borderline range where few An's
are zero for n
N and few An's are not zero for n> N.
Occasionally data for the few smallest values of n are missing or
Appendii A, continued
not complete.
J -i
=
In this case one computes the sums
(/S)Un
11
L
n,2
-
_~nSnU
4'
V);(Q
Z)(Q/Sn)
p£(/n/S)U-n
K =
;
~
2
vnU
(1lA)
a
(/Sn) [(Qn/Sn)Un- Vn1
X.
#.rlM
where
m, the lower limit of the sums, is the lowest value of
n for which there is complete data.
Then the two parameters
for the distribution, the least-squares values, are
K(N-n +1) M("
rMf(-
2
n+)++l) -- JL
JL.
nm
KL - MJ
,
K(N -n+l)
- J?
( 12A ) --
Since the data is incomplete for the lower values of n, the
value of A (the total number of active items) also is
undetermined.
Its mean-square value is given by the formula
Appendix
A
tsjL
where the
continued.
1- (Un/i)12
4
ncomputed from Eq.(12A) is
(13A)
used, the values of Ur are
taken from Table III. and the values of Sn from the data, using
Eq.(7A).
Having obtained A, the values of AF n and AGn appropriate
for this case are then
AJ n
A - (A/n)U
AGn
n
Aql - (A/O)V n
which are to be compared with Sn and
,
(14A)
obtained from the data.
To test whether the exact Bradford distribution applies also
to the scatter of citations an example was picked at random from
the 1977 Citation Index (journal cited, Ann. Rheum. Dis., scatter
of citations among journals).
N was chosen to be 20; the cumu-
lative sums of citing journals Sn and Qu,
are given in Table IV.
estimate that S
1
the cumulative citations,
The data stops at n
325 and Q1
-
6, but there is an
1940, making ql
-
5.97.
Using
this in Eq.(8A) we find that Bn rises almost linearly from 1.58
to 1.75, perhaps indicating that some citing journals with few
citations were not included in the estimate.
Therefore Eqs.(llA)
and (12A) were used and the sums computed;
J=634.16 ; K 28620 ; L 479.8 ; M=21500 ; N-nam +l = 15
resulting in
a
IP
1
.
A
0 _T*% 7
_.
i
-
1
From Eq.(13A) it was found that A
S
CQn
. /7V
653.
From this one could
use Eq.(14A) to compute AFn and AGn, which are given in Table IV.
The excellence of fit may persuade one that the number of lowproductivity journals (n<6) should be nearer 580 rather than 252
and that the total number of citations by these journals should
be 940 rather than 540.
Otherwise the distribution is a truncated
Bradford one, cutting off at n = 6 rather than at n= 1.
J
TABLE IV.
Comparison between 1977 data on scattering among journals of
: citations to articles in Ann. Rheum. Dis. and the Exact Bradford
distribution for A
653, q- " 3.590 ,
Data
Sn
Dist.
20
22
23
24
20
21
22
24
25
25
15
32
32
34
27
30
32
36
38
35
39
9
42
49
43
49
8
53
55
7
67
6
73
64
76
13
12
11
10
5
94
4
121
168
273
653
3
2
1
Dist.
AGn
AF n
20
19
18
17
16
14
Data
- 1.489.
872
910
928
945
961
864
888
912
938
965
1066
1066
1092
1116
1138
994
1026
1059
1095
1136
1178
1241
1273
1371
1407
1179
1228
1283
1345
1417
1504
1612
1755
1963
2344
See Appendix A.
APPENDIX B.
Suppose the collection of A items is distributed with
respect to productivity for last year such that Afn is the
number of items with productivity n
(1
nN)
and
fn = 1.
We define p(J if n) as the transition probability that those
items in-the collection that had productivity n last year will
have productivity J next year.
And suppose that p(J if n) is
a Bradford distribution on j (fn can be any distribution),
p(j if n)
=
(l< <
(yj/)
) ;
=
1
(uN/n) (j= )
(1B)
with parameter Bn for each different n-group of items (that have
productivity n); P. may differ for different values of n.
Then the distribution in productivity for the whole collection
of A items next year will be
fr
=
p(j
if
n)f
n
= yj
if (1/')
Z
1 - (UK/fin)]
(fn/n
=
.=1-
(y/)
(y/jN)
(2B)
Z(fn/o)
(UN/p')
This is a Markov chain 3, with the p(j if n) being the transition
probabilities.
The point to be noted here is that if the p(j if n)'s are
Bradford distributions on j,
for each n, then f
distribution, no matter what distribution f
is
a Bradf ord
is.
If the same conditions continue for several years then the
system will settle down to a steady state, and f' will equal f;
the yearly distribution in productivity will remain roughly the
same and will be Bradford, if the p(j if n)'s are.
In that case
-l
Appendix B, continued
n
where
_
=
f
=
Yn/ps
(1
1- (U/os) (n
n< N) ;
N)
(3B)
s is the steady-state parameter for the whole collection; so
j/
'' '
(J /n)(7/Ps)-
)1
+ (/
- (U/s
(l .v
N)
(4-B)
i
1
(UN/Ps)
-(=P/)
+ 11- (UK/O)jll-- (U/,
where
(aQ
N)
n is the parameter for those items that had productivity
n last year (the n-group).
all values of
These equations are satisfied for
, from 1 to N inclusive, if
VA-i
-Ps
OX +
which relates the
-N -
(5B)
NE,,(Yn/n)
n's for the n-groups to the steady-state
s
for the whole collection.
Similarly we must have the mean productivity qls the same
from year to year in steady state.
Therefore the mean productivities
ql(if n) for each n-group must be related to the mean productivity
qls of the whole collection by the equation
ql8s
-
q 1 (if
=
n)fn
q(if
)(Yz/
)
+ ql(if N)1
-
(UN/S)l
(6B)
If the two parameters Bn and ql(if n) are determined for each
n-group, then the transition probabilities p(j if n) are determined and the parameters
,s
and qls' for the whole collection,
are also determined.
It may be, of course, that some items with productivity n
last year will have no productivity next year, i.e., that
Appendix B continued
pt(O if n) is not zero.
But if the collection A' next year is
to be equal in size to the A of last year, there must be some
items with zero productivity last year that have productivity j
next year, i.e., Pt(j if O) is not zero and
A Pt(O
if
n)
Pt(j
if
O)f_
-
((7B)
-where ft represents the temporary members of the collection of
items, that move in and out of the collection each year, as many
moving in as out.
They need not be counted separately from the
fn's, for they are only present when they are productive and
are thus among those measured by fn or f j.
The collection of
active items remains the same in number, the components
change somewhat from year to year.
However the transition probabilities Pt(j if n) need to be
"renormalized" if Eqs.(lB) to (6B) are to remain valid.
For
example, we now have, for steady state,
M
fj
Pt(J if
j
-
from Eq.(7B).
Z|Pt(J
)ft
+
Pt(j
if
if n) + Pt(
n)fn
if O)Pt(O if n)]f
Thus Eqs.(lB) to (6B) will remain valid if
(8B)
we
define the transition probability p(j if n) as
p(j if n)
for all
Pt(j if n) + Pt(j if O)pt(O if n)
(9B)
values of j and n from 1 to N inclusive, and if we set
this p(j if n) to satisfy the Bradford distribution of Eq.(1B).
Since probability distributions do not specify which item
belongs where, but simply assign the number of items that have
productivity n, we can consider the items that move into the
collection to be the same as those that moved out, and allocate
them to the n's as though no items moved in or out.
_
__I_··1_···
_·II_
Appendix B continued
all we kWow about the collection of active items
Of course if
_
is that the steady-state distribution is Bradford, with known
.parameters--
and ql,
then there is not enough data to determine
the-individual transition probabilities p(J if
-- Sure that-they are Bradford.
If
ql(if n) and
n), beyond being
n happen to vary
-linearly with n (a not unlikely event), then -Eqs. (5B)
-can-be somewhat simplified.
(1 + n)/01
(1/3=)
and
If we define
and
ql(if n) - q1
jL
+
and (6B)
as follows,
n
(lOB)
and we utilize Eq.(4A) so that Table III can be used. to carry
out the calculations, we then find that
-
+s [N(-q UN)
+ V
(11B)
lO'
qls - (p/s)[N(s -UN)
+
]
Table II is constructed, as an illustrative example, for the
first example of Table I, by assuming that Eqs.(lOB) hold, with
X
L
4/90 and
= 14/9.
Without further data on the transition
probabilities, i.e., on how items change in productivity from
year to year, we are, of course, unable to determine the true
values of
or g, or, indeed, if (l/
linearly with n.
) and ql(if n) do vary
All that can be determined, given the excellent
fit with the data on overall productivity, as illustrated in
Table I, is that the transition probabilities p(j if n) must
also follow a Bradford distribution for each value of n.