Transcript Boyko Zlatev - Mathematical and Statistical Sciences

Statistical Approaches to
Length of Reign
Boyko Zlatev
University of Alberta
Canadian Young Researchers Conference in Mathematics
and Statistics. Edmonton, 18-20 May 2010
The Date of Founding of Rome
• Last Roman King
deposed c. 503 BC.
• Traditional date of
founding of Rome – c.
760 BC.
• Reigns of 7 Roman
Kings cover 257
years in total.
• Isn’t it a very large
value?
Sir Isaac Newton (1642-1727) and “The
Chronology of Ancient Kingdoms Amended” (1728)
• Newton was not much
interested in probability
(Sheynin, 1971). But he
was interested in
chronology!
• He reconstructed ancient
chronology and made it
“fit better with the course
of Nature”.
• For that he analyzed
some data, available in
his time.
Newton’s Data and Results
• Average lengths of
reign for different
states concentrated in
the interval 18÷20
years.
• In fact, this is roughly
a 65% confidence
interval, as shown by
Stigler (JASA, 1977).
François-Marie Arouet de Voltaire
(1694-1778)
• Confirmed Newton’s
results on the reigns on
the German Emperors
(elected!) – average is
18.4 years.
• Applied Newton’s rule to
Chinese Kings.
• Noted that in states
where revolutions are
frequent, Newton’s rule
shows too large a value.
Marie Jean Antoine Nicolas de Caritat, marquis
de Condorcet (1743-1794)
• Determined the chance
that 7 kings shall reigned
exactly 257 years.
• Applied “la probabilité
propre” – “appropriate
probability” (not clear
what is it) to find the
solution.
• Contributed to shortening
the reign of Louis XVI –
King of France.
Condorcet’s Solution
• Following De Moivre’s Law of Mortality with 90
years as limit of life, and the ages of (s)elected
kings between 30 and 60 years, the probability
seven kings reigning s years is the coefficient cs
of xs in the expansion of:
7
420
 31x  31x  x  x 
s

   cs x
2
45  31(1  x)
s 7


2
31
62
Condorcet’s Solution
(continued)
• Condorcet found P = c257 = 0.000792 (may be it is an
approximation only)
• Computation with R gives P = c257 = 0.000733, but it is
also not very accurate:
• Mathematica gives P = c257 = 0.000727
Condorcet’s Solution
(continued)
• “Appropriate probability” for 414 events possible:
Pappropriate
P
413P


 0.2466
1  P 1  412 P
P
413
• If P from R computation is used, the result is
0.2325
• To find P(s≥257), one has to compute:
256
P( s  257)  1   c j  0.0102
j 7
Karl Pearson
“Biometry and Chronology” (Biometrika, 1928)
• “The frequency distribution
of the reigns of
souvereigns is given.
What is the chance that a
sample of seven selected
at random will give a total
length of reigns of 257 or
more years?”
Karl Pearson
“Biometry and Chronology” (Biometrika, 1928)
(continued)
• The frequency distribution we have is not
for the same country at the same time
(further discussions are possible about
which is more important – time or space).
• In reality averages for different countries
and times are (according to Pearson) in
surprising agreement with Newton’s
results, i.e. concentrated between 18 and
20 years.
Karl Pearson
“Biometry and Chronology” (Biometrika, 1928)
(continued)
Histogram for 250 European Monarchs
Karl Pearson
“Biometry and Chronology” (Biometrika, 1928)
(continued)
Distribution of Means in Samples of Seven Reigns
• According to this
distribution, Pearson
found the probability
for seven kings
reigning at least 257
years to be 0.00496
• This is about ½ of the
result based on de
Moivre’s Law of
Mortality
Karl Pearson
“Biometry and Chronology” (Biometrika, 1928)
(continued)
• “Seven men are chosen at random
between the ages of 30 and 60. Find the
chance that a sample of seven selected at
random will give a total length of reigns of
257 or more years”.
• The life-table chosen will not be from
same time and same country.
• The choice of age interval is questionable.
Solution by Charles F. Trustam
(Biometrika, 1928)
• HM life table is used
(healthy males from
England and Scotland,
mid. XIX century).
• Linear approximation
for λ between 0 and 45.
• P = 0.00274.
1943
• E.Rubin. The Place of Statistical Methods in Modern
Historiography. American Journal of Economics and
Sociology.
• S.Ya.Lurie. Newton – Historian of the Antiquity. In
Proceedings of Symposium (USSR), dedicated to 300-th
anniversary from Newton’s birth
In both papers Newton’s approach to the date of founding of
Rome and similar problems is examined. As pointed by Lurie, in
some cases Newton’s method gives surprisingly accurate results,
confirmed by evidence found later.
According to Rubin: “Statistical methods may only be applied if
the historian knows something as to the reliability of the data. (…)
There must be a clear comprehension of these techniques, which
implies an understanding of their possibilities and limitations”.
Thomas R. Trautmann. Length of Generation and
Reign in Ancient India. Journal of the American
Oriental Society (1969)
• Hypotheses concerning ancient Indian chronology must
be tested by comparing with a sample of kings from
medieval India whose dates are better known.
• The problems of interregna and missing data are
considered.
• Estimations based on the number of generations are
more accurate then those based on the number of
reigns.
E. Khmaladze, R. Browning, J. Haywood
Brittle power: On Roman Emperors and exponential
lengths of rule. Statistics & Probability Letters (2007)
• Hypothesis tested: a sample T1, …, Tn of n independent r.vs. follows
exponential distribution with some a priori unspecified λ>0. One
could consider an empirical distribution function Fn (t ) of the sample
and compare it to F (t , ˆ) , with  estimated by ̂ from the same
data. Instead of taking the difference Fn (t )  F (t, ˆ) there is an
advantage to take Fn (t )  K (t, Fn ) where K is called the “compensator”
in the martingale theory of point processes. For the family of
exponential distributions its form is simple. If the hypothesis of
exponentiality is true, the version of the empirical process
wn (s)  n Fn (t )  K (t , Fn ), s  F (t , ˆ),
converges in distribution to the standard Brownian motion. So, the
classical Kolmogorov-Smirnov statistic can be used.
E. Khmaladze et al. (continued)
Results
• For the three chronological tables of Roman Emperors
(those of Kienast, Parkin and Gibbon) the p-values are
0.26, 0.17 and 0.20, respectively. Hence, the hypothesis
of exponentiality is not rejected.
• Exponentiality rejected for Chinese Emperors (p=0.028).
• Age of ascent dependence for British and Spanish
monarchs.
Why?
• “…their [Roman Emperors’] lives were
constantly exposed to turmoil and life
threatening challenges, no matter in peace or
war” (Khmaladze at al.)
• But the same must hold for other monarchs also!
• There are differences in the inheritance rules!
Topology of Data:
Family tree (graph!) mapped on a time interval
Idea: to apply FDA
• Considering non-homogeneous Poisson
process with intensity µ(t) = exp(W(t)),
representing W(t) in terms of a set of basis
functions (B-splines, Fourier, etc.) and
minimizing the following:
(Ramsay & Silverman, Functional Data
Analysis, 2006).
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