Tuesday, November 11, 2014

The Power Law of Slavery

This post may not be about what you think. The words "power", "law", and "slavery" are accurate but polyvalent. Here I discuss a type of mathematical distribution that appears in slave ownership in the antebellum southern United States. The form the distribution takes is called a power law.

A few weeks ago I was watching the Antiques Roadshow (American edition) on public television, and they showed a pre-Civil War historic map--I think it was of a county in the state of Virginia--, and in the legend I saw a table that listed the number of slave owners who owned given numbers slaves. I only got a glimpse, but I noticed that there seemed to be a lot of people who owned a few slaves and a tiny number of people who owned large numbers of slaves.

I am interested in the distribution of wealth, in which the right tail--representing the wealthy--is a power law, or Pareto, distribution, while the left part of the curve seems to fit an exponential distribution, although other mathematical functions have been proposed. Of course, I'm particularly interested in historical and especially archaeological examples. Archaeologically, the emergence--presumably through self-organization--of the power-law tail in the distribution of wealth appears related to the evolution of economic complexity.

Of course, in the Old South, slaves were economically significant assets, and I thought that slave ownership might be an interesting proxy for wealth, ghoulish as that certainly is.

The table on the map suggested a very skewed distribution, which could be a power law. So, I did some Googling, looking for data, and found the 1860 census, which listed number of slave owners by number of slaves owned. In the following table, the first and fourth columns come from the 1860 Census (Vol. III, Agriculture, p. 247).


No. of Slaves  Midpoint of Bin Width of Bin No. of Slaveholders Normalized frequency Log(Midpoint) Log(Normalized frequency)
1 1 1 77333 0.2025251 0 -0.693521113
2 2 1 43105 0.1128864 0.301029996 -0.947358321
3 3 1 34859 0.0912912 0.477121255 -1.039571046
4 4 1 28979 0.0758922 0.602059991 -1.119802576
5 5 1 24278 0.0635809 0.698970004 -1.196673064
6 6 1 20632 0.0540325 0.77815125 -1.267344642
7 7 1 17280 0.0452541 0.84509804 -1.344342233
8 8 1 14864 0.0389269 0.903089987 -1.409750274
9 9 1 12522 0.0327935 0.954242509 -1.484212271
10-14 12 5 40388 0.0211542 1.079181246 -1.674603628
15-19 17 5 21322 0.0111679 1.230448921 -1.952028036
20-29 24.5 10 20796 0.0054462 1.389166084 -2.263906162
30-39 34.5 10 9648 0.0025267 1.537819095 -2.597448676
40-49 44.5 10 5179 0.0013563 1.648360011 -2.86764006
50-69 59.5 20 5218 0.0006833 1.774516966 -3.165411892
70-99 79.5 20 3149 0.0004123 1.900367129 -3.384743306
100-199 149.5 100 1980 5.185E-05 2.174641193 -4.285220781
200-299 249.5 100 224 5.866E-06 2.39707055 -5.231637952
300-499 399.5 200 74 9.69E-07 2.601516784 -6.013684247
500-999 749.5 500 13 6.809E-08 2.874771637 -7.166912623
>1000* 1499.5 1000 1 2.619E-09 3.17594647 -8.581885971

*Actually, the largest slaveholder said to be the Estate of JOSHUA J. WARD, at SC, Georgetown, roll 1235 page 212, holding 1,130 slaves (http://freepages.genealogy.rootsweb.ancestry.com/~ajac/biggest16.htm).

The data are in a mixed format. If you look at the first column, you will see that some of the "intervals" are simple integers (1, 2, 3...) while the higher categories are intervals or ranges (called "bins" by statisticians). To adjust for the effects of the different bin widths I normalized the data (See Brown and Liebovitch 2010: 11-14). Graphing the last two columns on a double logarithmic graph (in which both axes are logarithmic) is a simple way of establishing whether the distribution is a power law: if the data form a straight line on a double log graph, the relationship is a power law. (Actually, I logged the data rather than logging the axes of the graph.)

So, here is a graph of the last two columns (which were already logged in the table above).

Figure 1. Double logarithmic graph showing the distribution of slave ownership.


The left tail of the graph has a gentle curve while the right tail is a straight line. Let's look at that right tail more closely by isolating just the last few rows of the table.

Figure 2. Double logarithmic graph showing the power-law right tail of the distribution.
The coefficient of determination for the regression line (the R-squared) is quite high. The maximum is 1, so our fit is nearly perfect. This statistic provides a measure of the goodness of fit between the straight line and the data.

Curiously, though, the right tail is not a good fit to an exponential. In Figure 3, below, the right tail would be a straight line on this semi-logarithmic graph if it were an exponential. (A semi-logarithmic graph is one in which only a single axis is logarithmic, while the other is linear.)

Figure 3. Semi-logarithmic graph of the right tail of the slave ownership distribution, which exhibits a poor fit to an exponential distribution.



In fact, there may be another power-law segment in the middle reach of the data. If we take just the middle few points of the data set, above (left) of the ones in Figure 2, and plot them on a double logarithmic graph, we see another straight line, but with a different slope than the first.

Figure 4. Double logarithmic graph showing a power law relation for the middle portion of the slave ownership distribution.

Now if we take just the points to the left (above) these, we do get an exponential distribution for the left tail provided we drop the first point, which is weird for some reason that I haven't figured out.

Figure 5. Semi-logarithmic plot showing an exponential distribution for the left tail of the slave ownership distribution excluding the first (leftmost) point.


The fit to an exponential in Figure 5 is really quite excellent.

Perhaps if we were to include the number of non-slaveholders, that is, those with 0 slaves, we would be able to extend the distribution father to the left and understand that part of the process better.

Why should we care about the mathematical form of the distribution? The form of the distribution is a clue to the type of process that created it. There is a pretty substantial literature on the economic processes that create distributions of wealth. Unfortunately, there is no consensus on either the best mathematical forms of the distributions or the dynamical processes that have created them. They do, however, seem to share a general shape much like the one discussed here. I find that interesting. 


There is a large literature on the economics of slavery, both in the Old South and elsewhere in the world. Perhaps someone who is immersed in that literature will understand the historical significance on these results. I, unfortunately, don't have the time to delve into the literature and figure it out.


Please let me know, by commenting on this post if nothing else, if you have any thoughts.

Tuesday, October 28, 2014

Panel Discussion on Urbanism, Patrimony, Ownership, and Culture in Florida

I received an announcement for a panel discussion on "Urbanism, Patrimony, Ownership, and Culture in Florida" to be held at:

Wolson Campus of Miami Dade College
300 NE 2nd Ave., Room 7128, 
First Floor of Building 7

on Thursday, November 6th
at 6:30 pm

Free and Open to the Public

Here's the poster.


The announcement I received says it is a "panel on the current archaeological excavations at Met Square (which include a prehistoric village, Ft. Dallas, and the historic Royal Palm Hotel). This panel will include the County Archaeologist Jeff Ransom, the UM Archaeologist who has an ongoing  legal battle with the City in an attempt to save the site, Dr. Will Pestle, Mary Lou Pfeiffer from FIU who wrote a book about the legal aspects of the handling of the famous Miami Circle, and an Urbanism Professor from UM, Dr. Allan Shulman.


"This panel is supported by a grant from the state of Florida and news alerts have been sent to all local South Florida museums, universities, and news organizations including the Miami Herald and NPR."

Sounds interesting. I urge you attend.

Friday, October 24, 2014

Native American DNA in Easter Islanders

An article has come out in Current Biology in which the authors report the sequencing of  over 650,000 SNPs (single nucleotide polymorphisms) from 27 Rapa Nui individuals. I don't have access to the full article, but I've read the abstract and a discussion of it from today's issue of Science (Lawler, Andrew, 2014, Epic pre-Columbian Voyage Suggested by Genes, Science Vol. 346, No. 6208, p. 406). Native American DNA composed about 8% of the Rapanui genome. They explain this by invoking pre-Columbian trans-Pacific journeys from Easter Island to the Americas and back. They argue that the introgression took place in pre-Columbian times, rather than in recent or historical times, because the Native American DNA segments are fragmented and scattered. The degree of fragmentation and dislocation allow the authors to infer a date of AD 1280–1495 for the contact.

Of course, these issues--Easter Island and trans-Pacific contact--are popular and therefore controversial questions. So, naturally some Norwegians have to defend Thor Heyerdahl by saying that this means that the South Americans sailed to Easter Island. Right. And just happened to get there at the same time as the Polynesians. Sure. That's more likely than the Polynesians, who'd been sailing around the Pacific for thousands of years and who had developed unique maritime technology and navigational knowledge, sailing to South America. Nothing against the South Americans, but dude, this was the Polynesians' thing. In fact, it's hard to imagine that the Polynesians, who, over a period of two or three thousand years, had found and settled virtually all the tiny Polynesian islands, would suddenly stop at Hawaii and Easter Island. Right. You sail all over the Pacific for millennia and find all the little islands, but miss the giant continent in front of you. Yeah, that's really likely.

Of course, the South Americans did have boats and sailed around some, but there's little evidence that they crossed the open ocean to Polynesia. Sure, a boat might have drifted there, as they occasionally do today, but that's not really in the same league.

Added to the pre-Columbian chicken bones from Chile with (possibly) Polynesian chicken DNA in them and the evidence that the yam diffused to Polynesia from South America, and the evidence for pre-Columbian trans-Pacific contact is getting stronger.


The citation for the article is:


Moreno-Mayar, J. Víctor, Simon Rasmussen, Andaine Seguin-Orlando, Morten Rasmussen, Mason Liang, Siri Tennebø Flåm, Benedicte Alexandra Lie, Gregor Duncan Gilfillan, Rasmus Nielsen, Erik Thorsby, Eske Willerslev, and Anna-Sapfo Malaspinas. (2014). Genome-wide Ancestry Patterns in Rapanui Suggest Pre-European Admixture with Native Americans. Current Biology In press. DOI:  http://dx.doi.org/10.1016/j.cub.2014.09.057

Beware the words in your Custom Dictionary file!

I accidentally added a misspelled word to my word processor's custom dictionary, so I had to edit it, which required a long, convoluted, painful web search for instructions. But once I had it open, I reviewed it and found all kinds of weird stuff in it. Here are the interesting ones:

alluviation, bioarchaeology, bioarchaeological, calendrically, ceiba, cenote, Chaak, chert, Chichigalpa, Chontal, Chorotega, Chorrera, chroma, conchoidal, Cosigüina, Cosmapa, Cusirisna, debitage, eastings, ecofacts, econophysics, ethnoarchaeological, Ethnoarchaeology, ethnohistoric, ethnohistorical, ethnohistorically, ethnohistory, extrajudicially, fechamiento, flintknapper, flintknapping, fractality, francosa, francoso, geoarchaeological, georeference, georeferenced, georeferencing, hammerstone, hammerstones, handaxes, Ixtepeque, Izamal, Jōmon, knapper, knappers, lacunarity, lithics, Manteño, Maribio, Maribios, matings, Mayapán, Mazatega, megafauna, micaceous, midden, middens, Middens, Nahua, Ocós, osteological, paleobotánicos, Paleodemography, perimortem, Physiographically, Postclassic, Preclassic, Puuc, replicability, Research, schadenfreude, sedentism, seriation, sherds, Somotillo, Speleothem, Springer, Stela, stelae, suprayacente, swidden, symphysis, tafonomía, tafonómica, tafonómicos, taphonomy, Tecoh, Telchaquillo, Términos, toponyms, uncountably, Undergraduate, undiagnostic, unslipped, vigesimal, Yucatec

Why should you care?

It's very revealing, practically a list of keywords for your life.

If someone steals your custom dictionary file, perhaps with malware, they can figure out a lot about you!

Beware! What's in your file?

45,000-Year-Old Modern Human Genome from Western Siberia is the Earliest Yet

Svante Pääbo's team has produced the earliest Homo sapiens genome so far, extracted from a directly radiocarbon-dated femur found in western Siberia. Two AMS dates on the collagen combined to produce a calibrated date of 46,880-43,210 cal BP. Stable C and N isotope analyses indicate a diet based on C3 plants and animals that ate them. The bone is from a male whose mtDNA falls near the root of the R haplogroup, which is widespread in modern Europe and Asia. The Y chromosome is similarly ancestral a widespread Eurasian haplogroup. They called the specimen  "Ust'-Ishim" after the location in which it was recovered.

The authors find that the individual was probably related to or derived from an early population involved in the dispersal of modern humans out of Africa. Principal components analyses (PCAs) of the autosomal DNA suggest this guy was more closely related to non-Africans than Africans. In fact, when comparing the autosomal DNA only to modern non-Africans the authors observe that the specimen falls near the origin of the graph (of the first two components) and interpret that to indicate that he was equally-closely related to all modern non-Africans.

There may, however, be a better interpretation. In both PCA plots the Ust'-Ishim genome fell closest to modern Central and South Asians, which would make sense too, given the location of the find. I haven't read the Supplementary materials yet (they're 115 pages long), but I did review Section 10, which addresses this issue. To my mind, the results of the admixture analysis shown in Supplementary Figure S10.3 do indeed suggest that the Ust'-Ishim genome is closest to Central Asians, such as the Pathan, Sindhi, Burusho, Hazara, and Uygur. (The ethnic labels come from the article.) These are all central Asian peoples from Pakistan, Afghanistan, and western China. The match between the bars representing those peoples and the Ust'-Ishim genome is not perfect (the former are missing a purple patch and a bit of dark blue that appear in the ancient genome), but to me that match it does seem markedly more similar than the others. Unfortunately, the subsequent phylogenetic analyses do not address this question as they do not include central Asian genomes. So, I'm not sure I agree with the authors' interpretation that this new specimen is equally related to all Eurasians. If the Ust'-Ishim individual is more closely related to central Asians than to others, it implies a remarkable degree of geographic stability. This is an obvious hypothesis that probably should have been addressed. I hope it is investigated in the future.

They also found a 2.3 +/- 0.3% Neanderthal admixture, showing that their inter-species cuddling had already taken place by 45,000 BP. Curiously, given the date and location, there was no Denisovan admixture. The percentage of Neanderthal admixture is a bit higher than in modern humans, but not by much. As one would expect, the characteristic length of the Neanderthal DNA segments is substantially greater (~1.8-4.2 x) in the Ust'-Ishim genome than in modern ones, because the fragments were subsequently broken up further by recombination. Actually, this fragmentation process can provide an estimate of the length of time since the admixture took place, which the authors estimate as 232-430 generations earlier, or 50,000-60,000 BP, using a 29-year generation length. This corresponds approximately to some estimates of the time of expansion of modern humans out of Africa, but post-dates the putatively early modern humans from Skhul and Qafzeh. Some longer Neanderthal segments in the Ust'-Ishim genome imply that later admixture events also occurred. Interestingly, Figure 5 in the article illustrates homozygous versus heterozygous Neanderthal-derived alleles in the Ust'-Ishim genome compared to several modern ones. The Neanderthal alleles in the ancient genome are all heterozygous whereas some of the modern ones are homozygous. This makes sense, as one would expect by chance alone (panmixia) that heterozygous derived alleles would be more common earlier in the gene flow process. Probabilistically, it would take time for homozygosity to develop. Figure 5 only shows Chromosome 12, but it is suggestive.

Because the specimen is directly dated, it could be used (along with other specimens) to help calibrate the "genetic clock" that calculates age of genetic divergence from mutation rates, and the authors did so. The new estimates of mutation rates in autosomal DNA from this study are lower (slower) than some, possibly implying a longer time interval since the split from archaic human relatives.They also provided new Y Chromosome and mtDNA mutation rate estimates.

Finally, it is noteworthy that the authors find that the Ust'Ishim individual was not more closely related to the Andaman Islanders than he was to modern East Asians or Native Americans. I say this is noteworthy because some documentaries have made, in my opinion, far too much of the Andaman Islanders' genes as evidence for a coastal, southern migration route for early modern humans.

Quite the interesting article. Here's the reference:

Fu, Qiaomei, Heng Li, Priya Moorjani, Flora Jay, Sergey M. Slepchenko, Aleksei A. Bondarev, Philip L. F. Johnson, Ayinuer Aximu-Petri, Kay Prüfer, Cesare de Filippo, Matthias Meyer, Nicolas Zwyns, Domingo C. Salazar-García, Yaroslav V. Kuzmin, Susan G. Keates, Pavel A. Kosintsev, Dmitry I. Razhev, Michael P. Richards, Nikolai V. Peristov, Michael Lachmann, Katerina Douka, Thomas F. G. Higham, Montgomery Slatkin, Jean-Jacques Hublin, David Reich, Janet Kelso, T. Bence Viola & Svante Pääbo (2014). Genome sequence of a 45,000-year-old modern human from western Siberia. Nature 514: 445-450. doi:10.1038/nature13810.

Tuesday, October 14, 2014

Icebergs in Florida

In today's weird science news, icebergs in Florida. Yup:

Hill, Jenna C. and Alan Condron (2014). Subtropical icebergs scours and meltwater routing in the deglacial western North Atlantic. Nature Geoscience PUBLISHED ONLINE: 12 OCTOBER 2014 | DOI: 10.1038/NGEO2267.

Okay, so they recorded high resolution bathymetry of the continental shelf along the Carolinas and Florida coast. They noticed these big furrows that they apparently can attribute to the keels of icebergs dragging through the sediments. Then they ran ocean circulation simulations that showed that big meltwater discharges from Hudson Bay or the St. Lawrence during the last deglaciation could have pushed icebergs as far south as the tip of Florida, although they would have been more common further north, along the Carolina coast. These kinds of events ought to have provided bursts of cold fresh water. I wonder if signals from those events could have been recorded in coral or shellfish.

Pretty cool work! (No pun intended.)


Wednesday, October 8, 2014

The Archaeologist Was a Spy by Charles H. Harris III and Louis R. Sadler

I recently read The Archaeologist Was a Spy by Charles H. Harris III and Louis R. Sadler (Albuquerque: University of New Mexico Press, 2003). It's a detailed exposition and discussion of Sylvanus Morley's espionage work during World War I. I think I knew previously that he was one of those denounced for spying by Franz Boas in his famous letter to The Nation after the war. But I did not know how central the role that he played was in not only in the spying itself but also in setting up the entire network of spies. I also knew nothing about the object of the espionage or its significance. As it turns out, the historians who wrote the book showed that Morley played a pivotal role in identifying and recruiting others, particularly archaeologists, into the network, which was run out the Office of Naval Intelligence. In those years, German U-boats threatened not only United States shipping and even our ports and coastal towns, but also the vital military links between the US and its European allies. The possibility that German U-boats could refuel and re-arm in nearby waters was a grave concern because it would have multiplied their effectiveness. Morley's primary task was to survey the Atlantic coast of Mexico and Central America to determine where it was physically--bathymetrically--possible for the U-boats to put in to ports, bays or inlets (preferably undetected) and resupply themselves. He was also on the lookout for German activities ashore that might indicate the willingness or ability to assist such a mission. I was surprised to learn that he was one of the dominant actors in such a key drama.

A couple of warnings. First, there is exceedingly little archaeology in the book. It's not a biography of Morley as much as it is a history of American espionage. Second, toward the end, the authors celebrate Boas's censure, and by implication the vindication of their protagonists, by the American Anthropological Association. But it's not clear to me what they are really saying. That American scientists ought to spy using their profession as a cover? That it's patriotic to do so? That we can do so with a clear conscience because we can assume that foreign scientists (or perhaps American ones) are spying on us? What about the argument that spying by some puts us all under suspicion? This not just theoretical for me. I have worked in countries that are not America's best friends, and who therefore would reasonably be on the lookout for American spies. Not only am I an archaeologist, but I worked for the Navy for a time. Worst of all I went to Yale, the great hothouse in which American has long propagated her budding spies. None of that means that I'm a spy, and in fact I'm not, but I don't want to have convince an interrogator of that.