462 THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA
DECOMPOSING THE LONG WAVE
THROUGH .A NON-PARAMETRIC
STOCHASTIC MEASUREMENT MODEL
By
Ronald Schoenberg
National Institute of Mental Health
and
Albert J. Bergersen
University of Arizona
March, 1986 ~
THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY.- CHINA 463
ABSTRACT
Earlier work demonstrated the presence of two long waves of
colonial administration of different lengths (1490 - 1825
and 1826 - 1969). Whether these were separate episodes or
examples of deeper underlying cyclical dynamics has implica-
tions for the existence of a common
long wave. To further inquire into
common cyclical rhythm these larger
through the use of a non-parametric
model.
system dynamic over the
the existence of a
waves were decomposed
stochastic measurement
To do this a 490 year time series of colonial administra-
tions is divided into ten episodes.
A conditional Normal-
Poisson model is proposed based on the assumptions of a
stochastic process. . The mean number of colonial administra-
tions established and terminated over each episode are
estimated, controlling for a quadratic time trend which
would be induced if the system was not constant throughout
an episode as assumed. Two sub-cycles are observed withing
each of the two long waves of colonial administrations
previously reported (Bergesen and Schoenberg, 1980).
The presence of these matching sub-cycles provides strong
evidence supporting a common system
economic but political aspects of international life.
dynamic in not only
464 THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA
In 1415 Portugal established the first European colony on
the coast of North Africa. This launched the first of two
long waves of European colonization that eventually. swept
the globe. The first wave began very slowly, a second
colony was established 65 years after the first in.1480,
another in 1482 and 1485, and two more in 1496. From there
the European empire steadily mounted, increasing to the apex
of the first wave at the end of the 18th century. The
colonial system then collapsed with the dissollution of the
French and Spanish empires, and shortly afterwards a new,
second wave of colonization began.
An interesting feature of the European colonization is that
it is borne out of the competition of independent states.
Eligible states, members of the "core," vie with one another
in the subjegation of alien territories.- And this process
is not cooperative. The colonized territories become ele-
ments in a centuries long dispute among the core countries
that was (and perhaps will.continue to be) more devastating
to human life than anything. that the planet has ever seen.
Figure 1 contains a diagram of the 490 history of European
colonization from 1480 to 1969 (the 65 years from 1415 to
1480 are omitted since there were no colonies established in
those years and for analytical purposes we shall begin the
colonial system with 1480). The curve in Figure 1 traces
the net number of colonial governors established by European
core states in existence in each year from 1480 to 1969 as
reported in Henige (1970). This curve is the joint outcome
of a process of the establishment of colonies and the ter-
mination of colonies. This in any given year the net number
of colonies is the number of colonies that have been estab-
lished minus the number of colonies terminated or transfered
to other ‘states.
THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA 465
PIGURE 1.
Net number of colonies of all European states from 1480 to 1969,
t
466 THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA
We have used the establishment of a colonial governor as a
measure of colonies, first, because this is the only data
available for the entire history of colonialism, and second,
because a political, as opposed to some geographical,
definition is more consistent with our intended purposes.
In our earlier report (Bergesen and Schoenberg, 1980), we
did not propose any kind of systematic model to account for
variation in the number of colonies. a sophisticated theory
sufficient for the specification of a math model did not
then exist, nor does one seem to exist at this time. An
attempt to apply series analysis of the Box-Jenkins type
(Box and Jenkins, 1970) was made by McGowan (1985). In our
opinion, however, the application of such a methodology was
inappropriate (Bergesen, 1985, and Schoenberg, 1985b).
First, the curve of net colonies is clearly nonlinear in
both frequency and amplitude. Box-Jenkins time series
analysis requires an assumption of stationarity which is
clearly violated. These defects may be corrected by the
appropriate selection of a "filter" or transformation of the
data, but this would remove from the data precisely what we
want to explain.
In fact it is not the net curve. of colonies that we want to
explain anyway. First, the underlying process is revealed
in both the establishment and the termination of colonies.
We would want, then, to study these outcomes of the process
separately. To analyze them combined in a. single number
would be throwing away information. Second, the true de-
pendent variable is the rate of colonization. As in the
study of population dynamics in which the birth rate (that
is, the number of children born per unit time) is the essen-
tial structural measure of productivity rather than the
sheer number of people in existence, so it will be with
colonization: a constant level of the underlying generating
process over, say, 10 years, will produce a constant prob-
ability or "propensity" to generate a colony in each of
those years, and thus though the net numbér of colonies
changes each year (from one in the first year to ten in the
tenth year) the rate of change of the number of colonies,
one per year, is constant as is the propensity.
The fundamental dependent variables, then, will be the
THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA 467
propensities to establish and to terminate colonies. And as
we shall see, these propensities may be interpreted as
instantaneous rates of change of the number of colonies,
That is, the number of colonies established and terminated
per year.
THE TIME SERIES MODEL
We shall assume that the time series of colonial governors
is an outcome of a stochastic process.. Denote the propen-
sity for the establishment of a colonial governor by \(t).
The propensity for an event to occur is.the instantaneous
rate of change of probability of that event occuring:
(1) A(t) = lim Pr{N(t+dt)-N(t)=1}/dt
dtyo
where N(t) is the number of colonial governors established
by time t. From the definition in equation (1) and the
assumption that no more than one event can occur in dt, we
may conclude that the probability of k colonial governors
being established during time intervals is
Pr(N(t+s)-N(t)=k) = Q(t,s)*exp{-Q(t,s)}/k!
k = 0,1,2,...
where
Q(t,s) = 16°Sa(tjyar = sba(t yar
Over the 490 year time series of the establishment of
colonial governors there appears to be wide variation in the
propensity, and \(t) is therefore a complicated function of
time. For example, the first 75 years of the 17th century
seems to be a period of very high propensity to establish
colonies and is followed by another 75 year period of a
relative lower propensity. It seems reasonable to suppose,
however, that the change in propensity over short periods of
time is quite small relative to larger periods of time,
decades, say, and that we may therefore assume that the
propensity is approximately constant over the range of a
single year. It follows that in any given year \(t) = 4,
and
= peel t. =
Q(t,1) = Sg “Adt = Jokdt = 2
The distribution of the observed number of established
colonial governors within any one year, then, is a Poisson
distribution with expected value equal to its propensity.
Let x, = N(t+1)-N(t) be the number of colonial governors
468 THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA
established in year t, then
Pr(X,=k) = \fexp(-d,)/! k = 0,1,2,...
where XE is the propensity of year t.
At this point a model that specified the behavior of Xe for
t = 1480, 1481,...1969 would be advantageous. However, a
theory that specifies the behavior of the propensity across
years does not exist at this time. This behavior is clearly
nonstationary and nonlinear and thus the application of any
of several well-known linear models of time series would be
inappropriate. We shall here attempt, then, to specify a
model that will entail very limited claims while providing
enough description of the process to permit some conclusions
about what is happening.
An inspection of the total number of colonial governors
(established minus terminated colonial governors) suggests
that the two global cycles may be crudely characterized by
the following sequence: (a) an initial low level of
colonization (the 16th century and middle 19th century), (b)
rapid colonization (17th century and later 19th century),
({c) a tapering off of colonization (18th Century and and
1920 to 1940), (d) fluctuating colonization (late 18th
century and 1940's, actually this is a period of primarily
the exchange of colonies as a product of armed conflict),
and (e) the collapse of colonial empires (early 19th century
and middle 20th century).
Let us postpone for a moment the discussion of the deter-
mination of the precise boundaries of these ten time seg-
ments. Instead we shall at this time develop the time
series model. First, let us distinquish between
the establishment and the termination of colonial governors.
The total number of colonial governors existing at time t is
the net outcome of the number of established colonial gover-
nors minus the number of terminated governors. Since it is
likely that the propensity to establish colonial governors
is subject to a different set of determinants than the
termination of colonial governors, we shall model them
separately. Therefore, let Xitj denote the number of
colonial governors established in the t-th year of the j-th
time segment, and X2e5 the number of colonial governors
terminated in the t-th year of the j-th segment.
ss e k =
Pr(Xyyetkl a5) = (1/k!) Mage ‘exp ( Maye)
THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA 469
Je 0/152) sae
where a = 1,2, t= 1,2,...T 5, j = 1,2,...10, and T; is the
number of years in the j-th time segment.
The model at this point assumes constant propensities within
time segments. This is a rather strong assumption which can
be relaxed. To account for variation in the propensity
within a segment we may assume that the propensity has some
distribution within a segment. Propensity is a positive
real variable and therefore let us propose that it has a
log-normal distribution:
= = 2g2-_y7l/2 = sy 2 jog2
PrO ait A) (20h?o aj) exp[-(logé Yaz) 26 aj!
where Yay is the mean of the log of the propensity of the
a,j-th segment, and Oey is the variance of the log of the
propensity of the a,j-th segment.
The assumption that the means of the propensities are con-
stant throughout the time segments is also unrealistic since
the segments are cut out of a time series which clearly
changes smoothly with time. Thus we might expect that a
segment cut from the rising part of the curve would have a
higher mean at the end of the segment than at the beginning.
Segments cut from the top parts of the curve might have
higher means at the middle than at either end. For these
reasons we shall also specify a quadratic time trend for the
propensities within segments:
= 2
(2) Wait = Fajo * Tajit * Tajat * Faget
where fay is IN(1,97 5), andt=t- aye where taj is the
mean of t within the a,j-th segment.
This model requires 60 parameters to describe 980 observa-
tions (= 2 X 490 years). The results reported here were
produced from the maximization of the marginal, likelihood
through the EM method. The marginal likelihood function is
2 10 +o"
(3) log L — gett FFF Rage Magee Ooze) Mage
In the EM method (a more complete description of which, as
applied to the type of problem presented here, may be found
in Schoenberg, 1985a), equation (3) is divided into two
parts, the first part containing sufficient statistics,
treating the log-propensities as latent or unmeasured vari-
ables, as parameters in a log-likelihood conditional on the
observations and the unknown parameters, that is, the 60
regression coefficients, 20 residual variances, and 20
470 THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA
auto-correlation coefficients; and the second part contain-
ing the unknown parameters conditional on the sufficient
statistics. The first step in the EM method, then, computes
the sufficient statistics of the latent log-propensity given
the observations and either initial guesses of the
parameters or estimates from previous iterations, and the
second step computes maximum likelihood estimates of the
parameters given the sufficient statistics.
Finally, we must consider the determination of the bound-
aries of the time segments. Such a determination amounts to
the estimation of the elements of Ty The simplest proce-
dure, and the one applied here, is to vary the size of the
time segments at the boundaries beginning with a best guess
for the T;- The likelihood function in (3) is then maximized
for each variation of the Tj. The values for T; are those
chosen that are associated with the largest maximum.
RESULTS
Table 1 contains the estimates of the mean propensities to
establish and terminate colonies for the five time segments
with each cycle of colonization. These estimates reveal a
set of two sub-cycles within each global cycle that seem to
be comparable across the global cycles. An important
obstacle to such a comparison, though, is a nonlinearity in
the global cycles, that is, an acceleration in the frequency
of the global cycles and the sub-cycles. To improve com-
parability of the sub-cycles across the global cycles, an
exponential transformation was performed on the time axis
the parameters of which were chosen to equalize the length
of the two main global cycles. Figure 2 contains a plot of
the mean propensities to establish and terminate colonial
governors, as well as the net propensity, on the transformed
time axis.*
The sub-cycles appear to have a period of .44 time units
(i.e., in units of global cycles: 1 = 1480, 2 = 1825, 3 =
1970, etc.) that is roughly constant. A constant period or
frequency in the transformed time scale would seem to be
justified if we are willing to concede that the first sub-
cycle in the second global cycle finished a little late--the
sub-cycle actually ended in 1921, but should have ended
THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA 471
time
1j0 250
segment
1 1480-1606 +476 0.000
2 1607-1674 1.186 282
Cycle 3 . 1675-1742 % +347 148
1 4 1743-1811 696 638
5 1812-1825 -341 2.108
1 1826-1874 -9168 314
2 1875-1921 2.014 707
Cycle 3 1922-1932 091 371
2 4 1933-1962 +512 1.931
5 1963-1969 ~142 2.833
TABLE 1: Mean propensities to establish (150) and
to terminate (S259) colonial governors for 10 time segments
spanning 490 years from 1480 to 1969.
472 THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA
ste of change
Sf Golonies ~
1 @iesolved
0 sus -oba n'y
ee ee ee
net rate of change
of colonies
FIGURE J: Estimated rates of change of colonial governors from
1680 to 1969 corrected for quadratic tine trend.
THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA 473
about 1905. In the untransformed time scale, the one
measured in calendar years, the period would be nonlinear,
.44 units being equal to about 150 years in the 16th century
and 70 years in the 19th century. The equivalent period for
1986 is 27 years; thus a complete subcycle today would
complete its course in 27 years. If we are undergoing a
third global cycle, and assuming that it started about 1975,
then the first sub-cycle would be over by about the turn of
the century.
An inspection of the net propensities, i.e., established
minus terminated propensities, we see that the second wave
of colonization peaked in the 19th century with a rather
larger propensity (1.336) than the respective peak in the
first wave of colonization in the 17th century (.903).
However, the second wave clearly deteriorated more rapidly
as well. Without more waves we cannot say whether there is
a gradual change in the waves towards greater increased
intensity of colonization accompanied by a greater
deterioration of the system, or whether there is some com-
pensating mechanism which penalizes waves with greater
increasing intensity with later increasing deterioration.
Since we have proposed no model (other than the simple
stochastic Poisson measurement model) our findings are
primarily qualitative. And the most striking result is that
the overall smooth curve of net colonization conceals oscil-
latory motion, that is, the sub-cycles, in its components.
This suggests some kind of synchonization in the causal
system. In future work on the global cycles we shall inves-
tigate models that will attempt to explain first the oscil-
latory motion of the colonization, and second the coordina-
tion of these components into the smooth curve of net
colonies that. we observe over the 490 years of European
colonization.
474 THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA
FOOTNOTES
lane transformation of the time axis has only heuris-
tic purposes in this paper. The formula is t = 2072.173 -
1418.7183*exp(-.8737*t.), where t is calendar time in years
and ts is measured in global cycle units, i.e., 1 = 1480,
the beginning of the first global cycle, 2 = 1825, the
beginning of the second global cycle, and 3 = 1969, the
beginning of the third global cycle. Taking this transfor-
mation seriously one could conclude that, (i) the system
started in 653 (ty = 0), (ii) the third global cycle will be
over by 2029 (t. = 4), and (iii) the system will have to
re-calibrate itself by 2072 (t, =)
On the basis of this formula it is also tempting to
conclude that a single year in the 1980's is equal to about
6.8 years in the 15th century, and about 2.8 years in the
early 19th century.
THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY, CHINA 475
REFERENCES
Bergesen, A.J.
1985 "How to Model the Cyclical Dynamics of the World-System:
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Bergesen, A.J., and Schoenberg, R.J.
1980 "Long Waves of Colonial Expansion and Contraction,
1415-1969," in Albert Bergesen (ed.) Studies of the
Modern World System, Academic Press, 1980.
Box, G.E.P., and G.M. Jenkins
1970 Time Series Analysis, Forecasting and Control.
San Francisco: Holden Day.
Henge, D.
1970 Colonial Governors. Madison: University of
Wisconsin Press.
McGowan, P.
1985 "Pitfalls and Promise in the Quantitative Study of
the World-System: A Reanalysis of Bergesen and
Schoenberg's ‘Long Waves' of Colonialism," Review,
8(4),477-500.
Schoenberg, R.J.
1985a "Latent Variables in the Analysis of Limited Dependent
Variables," in N.B. Tuma (ed.) Sociological
Methodology 1985, Jossey-Bass, Inc.
1985b-"Statistical Models Must Be Appropriate: A Reply to
Pat McGowan," Review, 8(4),513-516.
