ESTIMATING SECOND ORDER PARAMETERS OF VOLCANICITY
FROM HISTORICAL DATA
by
Peter Guttorp
Mary Lou Thompson
TECHNICAL REPORT No. 144
October 1988
Department of Statistics, GN-22
University of Washington
Seattle, Washington 98195
from historical data
Peter Guttorp
Department of Statistics
University of Washington
Seattle, WA 98185
USA
Mary Lou Thompson
Institute for Biostatistics
Medical Research Council
P. O. Box 70, Tygerberg, 7505 Cape
South Africa
ABSTRACT
In order to study patterns of volcanicity it is important to use historical data. However, these data are
frequently incomplete. We develop a nonparametric procedure for estimating second order parameters of
the point process of eruption starts from a catalog. This method requires an assumption of underlying
stationarity, and of smoothness of the probability of recording an eruption as a function of time. We
illustrate the procedure on the Smithsonian catalog, using global data from the last 480 years or less, and
also specializing to data from Japan and Iceland from the last 200 years. The global and the Japanese data
sets show little structure, while the Icelandic data exhibits a tendency towards periodic behavior, with a
period of about 40 years. We do not see any regularity corresponding to tidal cycles.
Simkin et al. (1981). in their list of recorded volcanic eruptions from 8300 Be to 1980 AD. discuss
the possibility of finding trends or cycles in the historical data. In a plot of 3-year running average number
of active volcanos reported the last 120 years. they note that, although an increasing trend due to
reporting is dominating. some dips in the series correspond to events such as the World Wars and the
Depression. whereas some peaks follow particularly spectacular eruptions. such as.Krakatua and Pelee.
At any point in time. we assume that there is a corresponding probability that. should an eruption
have occurred. it would have been recorded. The available catalog hence represents only a partial listing
of the eruptions thathave occurred. There are. of course. many factors which lead to there being fewer
and fewer recorded eruptions as the record goes back' into the past. There has been a continuing increase
in the number of known volcanos throughout the history of the catalog. although the rate of increase
.
appears to have diminished in this century. About 30% of the known volcanos are erupting at anyone
time.
In this paper we have elected to look only at the yearly numbers of starts of volcanic eruptions from
1500 through 1980. As the earlier data are very uncertain. we win mostly concentrate on the last 200
years. The analysis presented here adapts and extends the methodology developed by Brillinger (1979. cf.
also Lee and Brillinger, 1979) in order to study Chinese earthquake data. to the volcanic data. We win be
estimating the (conditional) probability (or Conditional intensity) of an eruption k: years after a given
eruption. and assess the standard error of this estimate. with an aim at detecting cyclic behavior in the
historic records. We will look at worldwide data. in order to be able to make comparisons with Hamilton
(1973). who found evidence of a relation with tidal cycles. In addition. we will study the eruptions in the
Japanese region and on Iceland separately.
section 2 we present the general methodological problem and introduce an estimator for the
recoramg probabilities are
recording prebabilities is
section 3.
section 4 we
J\.lUJWlll.
metaods to
Simkin et al.
2. Point process analysis
By a point process we mean a sequence 'tl>'tz, . .. of occurrences of events of interest. In our present
context these would be times of volcanic eruptions. A general introduction to point process analysis can
be found, e. g., in Cox and Lewis (1966). LetMt denote the number of points (not all of which have been
observed) in the time interval (O,t], so Mt=Lj l('t(::;;t), where l(A) is a function taking on the value 1 if A
happens and zero otherwise. For each occurrence 'tj in the M, -process we associate a probability x('tj) of
it being recorded. The deletion process is assumed to act independently at different points. In other
words, the occurrence or non-occurrence of eruptions is assumed not to influence the probability of
recording another eruption. The discussion in Simkin et al. (1981) casts some doubt over this assumption,
but we will return to this problem in section 4. In this section we will furthermore assume that the values
of x(t) are known for all t within the range of the data record. The resulting process of observed events
(the thinned process) is denoted Nt. Properties of the thinned process are discussed in Brillinger (1979).
We assume further that the observed data record arises fr()m sampling N, at discrete.sequispaced times,
i.e, observations are of the form Yj, i=I,2, ... ,T, where Yj=Nu1-N(j-l)l!., where d is the length of the
observation interval.
In attempting to establish the properties of the M t -process we are confronted with two problems:
that not all points have been recorded, and that the locations of the events that have been observed are
only known to be within an interval of length d. Various approaches to the problem of estimating point
process parameters in this context are developed and addressed in Guttorp and Thompson (1988). A
parameter of interest is the conditional intensity function
) is stationary,
conrntrcnar mtensity function measures the extent to
not depend
an
totally random (Poisson) process, this function is constant, and equal to the rate hu of the process. If the
process exhibits cyclic behavior, this shows up as periodic peaks (usually of decreasing magnitude and
increasing width) at integer multiples of the period.
We will estimate huu (u) from the data by
huu(u) = Y.t
(1)
[
where
/'1
T-u-1 Y/Y/
L
+u+1
1=1 1t11tl+u+1
r-«
+ 1;2 L
Y/Y/
+u
1=1 1t/ 1tl+ u
+ Y.t
T-u+1 Y/Y/
1]
L
+u- 1(1:i2(T -u )hu)
1=1 1t11tl+u-1
"T
1t/={I_l)A1t(t)dtll:i, h u
= Lj=l(Yjl1tj)ITI:i estimates the rate of the process, and u=m/1 for non-
negative integers m . If the conditional intensity is locally fairly smooth, it makes sense to use an
estimator which averages over contributions from points that are nearly u time units apart. The estimator
(1) reconstructs the unobserved process by reinflating the observed counts by the corresponding
.
observance probability. The numerator is roughly the average number of reinflated points that are
between u -1 and u +1 time intervals apart. In this respect it is related to the time series idea of hanning
(e.g., Bloomfield, 1976, p.83).
The properties of this estimator are discussed in Guttorp and Thompson (1988), and expressions for
the mean of the estimator and its variance in the case of an underlying Poisson process (i.e., a completely
random process) are given in the Appendix. While itis possible to derive an expression for the variance
of the estimator in more general cases than for the Poisson process, this expression is.a complicated and
not particularly illuminating function of higher order moments. These higher order moments are difficult
to estimate accurately from incomplete data. In many of the cases considered in Guttorp and Thompson
(1988), the Poisson variances appear to be conservative and should at least provide a guideline to the
precision of the estimator. Furthermore, Poisson variances enable us to test
random process.
hypothesis
completely
It is not usually very realistic to assume that 1t is known. However, one may be able to assess a
reasonable model for the recording probability. For example, Lee and Brillinger (1979) modelled the
recording probability as proportional to the population in the area they were studying, with a
proportionality factor that was a piecewise linear function of time. If the true recording probability is
"A
1t(t), a simple calculation shows that EY,,=hM
J 1t(s)ds. If 1t is smooth (over intervals of length several
("-I)A
.6.) we get that EY,,::::MM1t((k-Yl).6.). Consequently, we may estimate 1t by smoothing the observed
counts. The degree of smoothing used is determined by how smooth one feels the underlyingtunction 1t
is.
The problem of determining the smoothness of the probability function 1t is one that also arises in
nonparametric estimation of the trend function in time series. It may frequently be necessary to consult
.
with experts in the particular field of application, in order to decide to what extent local variation may be
regarded as being part of a long-term trend (in the present case, of the observance probability function) as
opposed to/variation in the underlying stationary process. In. some cases. the degree ofsnl()ot:J:m.~ss chosen
for 1t may have considerable impact on the resulting estimate of the conditional intensity. For an example
where this is the case, see Guttorp and Thompson (1988)~As arule, one wants to smooth the series rather
severely in order to avoid incorporating spurious local variation in the probability function.
There are several nonparametric smoothers available. We use the supersmoether (Friedman and
Stuetzle, 1982). This uses a locally linear fit, modified to reduce the effect of outlying values. In essence,
to compute the smoothed value x(t) at time t , one fits a least squares line to J points (k.6., Y,,) around
time
s,
The span J is chosen locally (i.e., for each value of t) using the data by a method called
crossvalidation (e.g., Stone, 1977). Large values of the span produces very smooth
bias), whereas a small span yields a highly variable (noisy) fit,
(and possibly a
user to
a
purposes.
4. Application to volcano data
The Simkin et al. catalog groups nearby vents by the major volcanic edifice on which they are
found. This tends to separate distinct volcanos by at least 20 km. The arrival of volcanic products at the
earth's surface is termed an eruption. Small steam clouds are ignored, although they may have entered
into earlier compilations as eruptions. Large steam columns are included in the catalog. An eruption
following its predecessor at the same volcano by less than threemonthseis deemed to be a part of the
previous eruption, unless careful study has shown it to be distinct. Although the catalog contains exact
starting dates for many eruptions, these dates are only known for 44% of the eruptions. For example, in
1980, two eruptions in Kamchatka and one in the Aleutian Islands are only known to the nearest month,
and a Chilean eruption is only known to have started before June 20. 62% of the eruptioas are known to
the nearest month. We will work with the number of recorded eruptions in each year. The number of
actual eruptions is very uncertain for the older parts of
tl1~data
sets, and consequently we will often
concentrate on the last 200 years. We have used all the entries in the catalog from the year 1500 onward.
The size of an eruption has not been taken into account, although this certainly affects the reliability of a
record.
Before
any estimate of the conditional intensity can be evaluated, it is necessary to estimate the
observance probability function. Using the supersmoother described in the previous section, we arrived at
the probability function shown in Figure 1. We have arbitrarily set 1t(1980)=L This is clearly not a
reasonable assumption, judging from the shape of the estimated 1t, but it just introduces a scaling factor
into
reconstructed counts. As we are mainly interested
of the parameter function
reconnng a
"h~,rnl:u
hMM
the shape
particular peaks and troughs)
(a ), this scaling will not be important. The
probability of
event is seen to increase
increase at a constant rate
our
This indicates
are
events
discussed in Section 1. the former explanation is not implausible. The latter explanation could mean a
genuine increase in volcanicity. There is no way to use the data to statistically test a hypothesis of nonstationarity. This also relates back to the question of smoothness of the probability function discussed
earlier.
Figure 2 shows the reconstructed data. obtained by dividing the observed counts by the estimated
probability of recording an event. The variability in the early parts of the plot is considerable. since we
are multiplying small numbers by large inverse probabilities. In analyses of this sort. there is a trade-off
in terms of variability when extending the length of the record further into the past. The advantages of
increased series length will at some point be outweighed by the associated low observance probabilities.
Judging from Figure 2. there is considerable uncertainty in the part of the record that is over 200 years
old. We have therefore performed several analyses; on the entire data set. the last 200 years. and this
century.
Figure 3 shows the reconstructed data from the 20th century only. The discussion in Simkin et al.
(1981) explaining peaks and troughs in the'observediseries can also be applied to this reconstructed series.
especially taking into account the smooth form of the estimated observance probabilities. There is a clear
decrease during the two World Wars. although this could well be part of the natural fluctuations of the
series. It is difficult to explain the peaks. In particular. the eruption of Pelee in 1902. the birth of Surtsey
in 1963. and the Heimaey eruption in 1973. do not seem to have affected the estimated number of
eruptions. Thus the assumption that recording probability is unrelated to the series of eruption starts does
not seem unreasonable.
Figures 4 through 6 show the estimates of the conditional intensity function and associated
Poisson-based 95% confidence bands based on data from the Simkin
for
period 1500 through
on
at
may well
structure, with a trough around 11 years.The data from this century also has a trough at 32 years. These
estimates provide little evidence that the process of volcanic eruptions, viewed globally, is different from
a Poisson (or totally random) process, since the estimates mostly are well within the Poisson standard
errors. It should be noted that the confidence bands shown in Figures 4-6 are pointwise, not simultaneous,
so one would expect about 5% of the bands to not cover the estimated Poisson mean
hM , even
if the
process is Poisson.
Table 1 gives the estimated values and Poisson standard errors for different series length and lags.
The final row, labeled lag co, has the estimated limiting value of hMM(U), namely the rate per year hM.
***Table 1 about here***
The uncertainty of the estimates based on all the data is about the same as that based on the last 200
years. In the former case this depends on the small observance probabilities, whereas in the latter case it is
a result of a smaller number of data points.
While volcanic eruptions may represent a purely random processona global
scal~,one
may expect
more structure in a series of eruptions confined to a restricted locality or region. With this in mind, we
applied the methodology described in the previous sections to the record of Japanese eruptions for the last
200 years. We defined a Japanese eruption as any eruption in the Simkin catalog with geographic
description including the word Japan, i.e., the subregions Kyushu, Honshu, Izu and Bonin Islands, and
Hokkaido. This region contains 77 known volcanos, and is one of the most active regions in the catalog.
The observance probability estimates and the conditional intensity estimate for the Japanese data are
...
u",'n~
in Figures 7
average 1
8. The
for the Japanese eruptions (which compose on the
of the global record in anyone year, and have over the last
is similar to
stracmre,
the
estimate at
had at most 36% of the
one
are
is an
is minimal, Table 2 contains estimated values and standard errors.
***Table 2 about here***
In some areas of the world we would expect the record of eruptions to be fairly complete over the
last few centuries. We therefore looked separately at the Icelandic eruptions for the last 200 years. Indeed,
the observance probability estimate, as shown in Figure 9, is quite flat Assuming that the observance
probability is constant, the estimate in Figure 10 resulted. There is substantially more structure in this
estimate than in any of the other ones, inpanicular,a clear indication of a periodicity of about 40 years.
The-same feature showed up in a spectral estimate. The peak at 39 years exceeds the rate significantly, by
about eight Poisson standard errors. From the data, it appears that Icelandic eruptions indeed come in
waves, with about 20 years of relative quiet, followed by a decade of increasing activity, and a decade of
high activity. This is similar to the observation of Tcmasson, reponed in Gudmundsson and Saemundsson
(1980), of 25 years of quiescence alternating with 25 years of tectonic unrest. The results of
Gudmundsson and Saemundsson, on the other hand, indicated no significant deviation from a Poisson
process. We believe that our estimation procedure is more sensitive to regularity.
5. Discussion
We have investigated reeordsof volcanic eruptioIlS with
a view
towards identifying any temporal
structure that may be present in these series. Our procedure has been to estimate the conditional intensity
of eruption starts for different lengths of the global record. We also have studied the subsets of Japanese
and Icelandic volcanic eruptions within the last 200 years.
Both globally
record. The global
for
Japanese data there appears to
a very little structure in the eruption
seemed to indicate a relatively smaller chance of eruptions a decade after an
appeared to
a
one.
some coupling, navme a
feamres were not
average pro,bat'iltl:y
sufficient mlagrritulie to
non-overlapping time intervals are statistically independent random variables, and hence not predictable.
For the Icelandic data there was a statistically significant increase in the probability of an event about 40
years from an other event.
These findings are relevant with respect to the results in Hamilton (1971), who demonstrated a
relation between various tidal cycles and the frequency of eruptions (see also the study of Stromboli data
in Mauk and Johnston, 1973). He used the data of Lamb (1970), who was interested in volcanic dust
clouds in the atmosphere. In particular, there was a rather high eruption frequency in 1821, a Yl;larwhen
perigee,maxhnallunar declination,
new moon, and winter solstice allcOiricided. In Lamb's data the five
year interval fl"(}m 1820-1825 stands out clearly. There is no elearindication of an increase in reportings
over the twentieth century, although there is a paucity of data before 1800, similar to the Simkin et al.
catalog. Lamb argues that large dust douds are less likely to be missed than lava eruptions in remote
areas, but it is not clear from his paper what were the selection criteria used for inclusion in his catalog.
Our reconstruction (Figure 2) shows no spectacular feature near the year 1821. In addition, Hamilton
reports evidence ·ofcycles of 8.85 .(the sum •of the .·lunatdeclinational· and anomalistic
ev(~les:).
(variation in the declination of the moon) and 19 (the sum of the synodic and solar annual cycles) years.
We see no evidence of any of these cycles in Figure 5. There is a-slight indication of a 55 year cycle
(three Saros cycles)
was also
by Hamilton. However, our analysis of the eruption
catalog has not found any evidence of dear tidal effects. We are, of course, unable to find high frequency
events, such as fortnightly, monthly, and yearly cycles, since our data are yearly counts.
Acknowledgements
This work: had partial support from the U.S. Environmental Protection Agency through a
cooperative agreement with SIMS. Part of
work was done while the authors were visiting the
- 11 -
Bloomfield, P. (1976): Fourier analysis oftime series: an introduction. New York: Wiley.
Brillinger, D. R. (1979): Analyzing point processes subject to random deletion.
Can. J. Statist. 7: 21-28.
Cox, D. R., and Lewis, P. A. W. (1966): The statistical analysis ofseries of events. London: Methuen.
Friedman, J. H., and Stuetzle, W. (1982): Smoothing of scatter plots. Project ORION Tech. Rep. 003,
Dept of Statistics, Stanford Univ.
Gudmundsson,
a.,
and Saemundsson, K. (1980): Statistical, analysis of damaging earthquakes and
volcaniceruptions'fnIceland from 1550-1978. J.Geophys. 47: 99-109.
Guttorp, P., and Thompson, M. L (1987): Nonparametric estimation of intensities for sampled counting
processes. Tech. Rep. 35 (rev.), Dept. of Statistics, Univ, of Washington.
Guttorp, P., and Thompson, M. L (1988): Estimating second order properties of partially observed point
processes. Manuscript.
Hamilton, W. L (1973): Tidal cycles of volcanic eruptions: fortnightly to 19 yearly periods.iJ. Geophys.
Res. 78: 3363-3375.
Lamb, H. H. (1970): Volcanic dust in the atmosphere with a chronology and assessment of its
meteorological significance. Phil. Trans.Roy. Soc. London, 226: 425..531.
Lee, W. H. K., and Brillinger, D. R. (1979): On Chinese earthquake history-an attempt to model an
incomplete data set by point process analysis. Pure Appl. Geophys. 117: 1229-1257.
Mank, F. J., and Johnston. M. J. S. (1973): On the triggering of volcanic eruptions by earthtides, J.
Geophys. Res. 78: 3356-3362.
Simkin, T., Siebert, L.. McClelland.
Bridge. D., Newhall, C., and Latter, J. H. (1981): Volcanos ofthe
Stone. M.
r10''7'"7\.
Statist. Soc. B 39: 44-47.
In this section we give a general expression for the mean and an expression for the variance under
the hypothesis of a Poisson process for the estimator hMM (u). Given the observance probability function
n:(s), the mean of the estimator PMM (u )=hMM (u )h"M is given by
EV51lJJ(u » = Jh (s ,u ,T)PMM (8)ds
where PMM (s )=h MM (8 )h M is the probability density of points 8 time units apart, and
1
T-ultJ.
ItJ.
TCI1TCi--';ultJ.
TC(W+t)TC(t)dt,u-J1Sw:S;u
tJ.2(T-'!!") 1=1
(1-1)tJ.+u-w
J
L
tJ.
1
T-ultJ.
h(w,u,T)=
tJ.2(T _.!!..)
o
ItJ.+u-w
TCI-1TCl"1:ultJ.
TC(W+t)TC(t)dt, u::';;w::';;u+tJ.
1=1
(1-1)tJ.
J
L
tJ.
elsewhere,
Note that in the case of a Poisson process we just get the integral of the weight function over the line,
hJ,
multiplied by
so that the estimator
is unbiased in the Poisson case. The mean of
PMM
substantially harder to compute, but for a large number of events, the approximation
will be adequate.
In the Poisson case we can determine the variance by tedious algebra. We obtain
var (PMM (n 8» =
hJ13
[m+1 T-S
6(T tJ. - m
1 T-~+1 (-1
+ 2"
2
~
j=1
TCj+m-l
L L
L
L (
s=m
+ 1tj1) - 5
s=m j=l
-1)
+ 1tj
+ TCf!s-l) +
j=l
~
j=1
T-2m
L
(
-1 + -1
TCj
(TCj.!m-l
L
j=l
T-m-l
T-m+l
L
"
J=l
+2
)
+ TCj.!m+l)
j=1
2
L
TCj+m+1
T-2m
-10
T-m
j=1
1 T~-1
+ 2"
m+1T-~+1
+2
(1tj';s
j=1
(1tj.!m
+ 1tjl)
hMM is
-14 -
hali.
+ 2J)(Tli.-mli.)2
+
h~A
~Ii.-m~
2
T-m
j~ 1tj!m 1tjl
[T-m
L (1tjim + 1tTI) + 2 T-2m
L 1t;:tm]
~
j~
While we have not been able to obtain exact expressions for the Poisson variances of the estimator hMM , a
simple linearization argument suggests that a reasonable approximation in the Poisson case is
where
h2
COV(PA1M(U),hM ) =
r-«
T(T~U) j~(1tjl+1tT}u),
It should be noted that, particularly in the case of small probabilities, the last covariance term may result
in considerable reduction in the variance of h"1flt as compared to the simpler approximation VarG1i))lha.
In the case of the global data, this reduction is by a factor of two or three.
Table 1:
for
All data
Last 200 years
This century
Lag
Estimate
Poisson se
Estimate
Poisson se
Estimate
Poisson se
I
37.3
.432
39.0
.451
38.5
.695
5
34.0
.431
37.6
.455
37.5
.736
10
35.4
.431
37.0
.463
36.1
.783
25
35.0
.437
37.3
.491
37.2
.876
50
34.6
.460
37.4
.539
00
34.0
37.9
37.7
Table 2: Estimates for Japanese data
Japan
Lag
Iceland
Estimate
Poisson se
Estimate
Poisson se
1
5.47
.22
.34
.056
5
4.39
.23
.25
.057
10
4.67
.24
.20
.060
25
4.72
.27
.08
.061
50
4.43
.31
.33
.066
00
4.87
.25
Recorded counts
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o
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•
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Reconstructed counts
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o
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o
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Pelee
...L
(0
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o
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o
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Conditional intensity
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--,
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---'
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)
(
..r;::..
0
,...,
I
r
....
-----
---
--
<. ......
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(Q
... ,
---
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CD
'::::. ,
Q.)
"""'l
en
........-
, ....
....
0'>
0
..-- ---
....
_J
------
--,...
----\
.::-
--_.....
--
........ ......
'--
---_
........
..-.. _... - --...
co
o
.... ....
.......
.. - --_----.. "
....
.. ---
.....
o
o
,
....
--- -- ----- --............
...
....
_._- --..,..
.....
....
' - ------
---
..... ...........
.. ....
...
.... " "
---- ..
"-
-;;
---- ---- ---------::-:~~~~
--- ---
Conditional intensity
o
'"
'""'":::
",-
--- --.-...
ro
o
-
....
'" .... ....
.... ...
, '"
'"
I
...'"
-...
.-
....-
....
.... ....
-- ......
...
....
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I
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'"'"
- ... -
... ...
....
.-
"'-
...'"...
(
.-
. .... ........
..
..
.- '"
.. '"
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\
I
.- .-
-
(
" ... ...
.. >
........
~
--- --
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....'"
.-I
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... ...
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.... ....
...... ..
..........
--- --
.. ..
ll>
-.,
(
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'<
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ll>
'"
r'"
\
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