Dust flux, Vostok ice core

Dust flux, Vostok ice core
Two dimensional phase space reconstruction of dust flux from the Vostok core over the period 186-4 ka using the time derivative method. Dust flux on the x-axis, rate of change is on the y-axis. From Gipp (2001).

Thursday, December 29, 2011


Perhaps you haven't noticed, but at The World Complex, we like gold. A lot. Not too long ago we ran an article about historical gold production, with some estimates of future production for gold. Today we will take a closer look.

It turns out that my main professional activity is exploring for gold, and yet that isn't why I spend so much time thinking about it.

Gold in a rubber pan, from an artisanal mining operation on a ridge in central Ghana.
The photo is about 4 cm across IIRC.

I began this blog to investigate application of mathematical methodology to geologic problems. Over the last year in particular, I have put increasing effort into using the same tools to look at economic problems. Yet ask geologists why they spend so much time looking for gold, and you will get many answers, none of them true.

In terms of exploration effort, gold is the most important mineral on the planet. Approximately 50% of all money spent on non-fuel mineral exploration during the last fifteen years was spent on gold exploration.

Sources here, here, and here (unfortunately the last two require a paid subscription).

I think people are mystified by the intense effort to find gold. Comments on gold at popular economic sites are polarized between those who find gold to be the most important commodity in the world and those who think it useless.

What does the USGS say about it?
Gold is by far the most explored mineral commodity target among those analyzed and the principal target for about 580 sites in 1995 and 1,800 sites in 2004. Gold’s popularity can be attributed to its demand in aesthetic and technological applications, its profitability (in terms of revenue minus costs), its widespread geological occurrence in relatively small deposits, some of which can deliver a high rate of economic return on investment, and its high price per unit weight.   (Wilburn, 2005)
To paraphrase--it's beautiful; profitable; and it scattered in small deposits worldwide. Rather than delve into the economic importance of gold, for now let us have faith that the market allocates investment as it does for a logical reason, whatever it happens to be. Based on exploration effort, gold is the most important non-fuel mineral in the world.

Copper is the second most important mineral--and at about 20% of global exploration expenditure, a distant second at that. Silver doesn't even appear on the scale.

Now let us look at the size-distribution of some known gold deposits. I will use the same approach used here to look for evidence of self-organization in the size of gold deposits. This article suggests that gold deposits exhibit scale-free behaviour. Let's investigate.

NRH Research has published a list of 296 deposits consisting of one million or more ounces of gold here. Although I haven't checked all the numbers, the ones I did check show some minor differences--mainly due to additional work carried out on the project since the date of the report. The only substantial issue is that for some of the deposits checked it appears that NRH has summed reserves, measured and indicated resources, and inferred resources; but not always. Rather than go through and update all 296 deposits, I have decided to use the data as is. Caveat emptor.

Size distribution of global gold deposits. Data from NRH report. Lines from my imagination.

The Chinese article cited above suggests that gold deposits are scale-free, meaning that if we plot them as we did the wealth distribution last week, we should see a straight line: however, we won't observe this because we have failed to discover all gold deposits. As a result, the graph will be concave downwards, reflecting possibly large numbers of yet-undiscovered gold deposits, particularly small ones.

On our graph is indeed concave downwards. The blue and yellow lines represent potential scenarios by which we may estimate how much more gold there remains to be found. Both suggested lines cover approximately one order of magnitude, however I believe the yellow line to be unrealistic, as it would suggest that there remain many very large deposits to be found--and projecting it all the way to the right would suggest one deposit larger than a billion ounces remains to be found!

The blue line would not permit much more in the way of large deposits, but would suggest that a great many smaller deposits are yet to be found. For instance, in the NRH report, there are 100 deposits larger than about 4 million ounces. From the blue line we would infer about 300 deposits larger than 4 million ounces. Two hundred more to go!

It may seem fantastic, but one thing to remember is that virtually all gold deposits ever discovered outcrop at surface. We are only just learning how to discover deposits that don't outcrop at surface. There's a lot of underground that hasn't been explored yet, so it's be fair to say the planet Earth is still an immature exploration district.

Friday, December 23, 2011

Innovation in complex systems

Innovation has been on my mind a lot lately. Unfortunately, not the kind that results in iPhones and the like.

We normally think of innovation as a good thing. But not all innovations are good ones. As counterexamples, let's consider recent political innovations in the US that allow indefinite detention without trial of anyone accused of terror-related activities; or the use of Predator drones to target American citizens.

My interest has been innovation in the Earth system--particularly in the behaviour of the climate system over the past two million years. The problem with recognizing innovation is that we tend to interpret any activities in light of what we already know--consequently it is difficult to discover anything new. Our first tendency would be to explain our new observations as a special case of what we already know. We resist the idea that something new is occurring.

The Earth system is driven by a few global parameters which interact with myriads of local agents; yet contrary to expectations instead of dissolving into noise, highly ordered global-scale structure arises. We may call such structures emergent properties, and the means by which they arise is termed emergence.

The problem of how these global structures arise from multitudes of interacting local agents is, shall we say, a non-trivial problem. They are in no way predictable from our knowledge of the local interactions; nevertheless we agree that emergence is in accordance with physical laws.

In earth systems, such emergent properties include plate tectonics, glaciations, superplume events, and some mass extinction events.

The emergent properties of a system may change. These changes may or may not be related to specific change(s) on the local level. For the purpose of this essay, I am referring to such changes as innovation.

Possible examples of innovation in Earth systems include the (somewhat controversial) proposed change in mode of tectonics in Archaean time; (very controversial) Neoproterozoic glaciation (i.e., "snowball Earth"); and magnetic pole reversals.

I have been considering change in operation of the climate system during the Mid-Pleistocene (from about 1 million years ago to about 500 thousand years ago).

I present the following probability density plots of the 2-d phase space reconstructions of the ice volume proxy, produced using the time delay method with a delay of 6 thousand years. Each of the figures below is calculated from 150 thousand years of data.

Starting from the Early Pleistocene . . .

Limit cycles (green dashed ellipses) are common in the Early Pleistocene, less so later.

Areas of Lyapunov stability, labelled A1 and A2, represent relatively ice-free conditions. Current global ice volume is comparable to A2, and A1 represents even less ice than at present. Limit cycles in the Early Pleistocene (representing slow, steady growth and decay of ice sheets) start from either the A1 or A2 condition.

The Late Pleistocene is characterized by discrete areas of high probability, suggesting rapid transitions between longer periods of stability. A2 represents an interglacial condition, and A3 to A6 represent separate metastable ice configurations of greater volume respectively. A6 represents a glacial maximum condition, as we experienced about 18,000 years ago.

Climate dynamics as inferred from global ice volume seems to have changed during the Pleistocene epoch. Was it innovation?

Opinions about what happened during the Mid-Pleistocene include changes in atmospheric CO2 leading to greater glaciations, cumulative cooling in the deep ocean changing the nature of the glacial-interglacial transition, erosive uncovering of crystalline bedrock leading to greater thickness of ice sheets, and spontaneous (chaotic) change. There is general agreement that there is no obvious external forcing or any fundamental change in the low-level dynamics leading to the change in climate behaviour, so it is at least possible to argue that the climate system began to act in an "innovative" fashion (provided we state that we do not view this innovation as having been directed in any way).

Let's look at another system instead--one represented by the share price of Century Casinos.

The chart of the daily closing price looks a little like my portfolio--up to a high in April, and all downhill from there.

The two-dimensional reconstructed phase space doesn't look much different from those of other stocks I've looked at in the past.

Actually, this has been smoothed a little, using a 3-point moving average.

There appears to be nothing interesting in the share price activity over the past year--unless we look at daily high prices instead of closing prices.

And here we see something unexpected--a singular spike in share price on June 21, where the share price bounced between about $3 and $8 several times over the day, on first a one-minute timescale, and around mid-day at a one-second timescale.

To investigate dynamics on this timescale, we have to construct our time-delay phase space with a small lag.

In two seconds of trading we have numerous fluctuations between $3 and $7. Lots of money to be made here! (or there would have been had the exchanges not cancelled all the trades).

A few minutes later we get this over one second.

This is orders of magnitude different from what we see in the annual behaviour of the stock, and even considerably different from the bowl of spaghetti above. This figure actually represents a phase space portrait of a random walk. Yes, you can trade randomly if you are quick enough.

So what is the difference between the trading in CNTY on June 21 and every other day this year? Another innovation--high-frequency trading, but in a form which creates the illusion of liquidity by placing lots of orders and then cancelling them as they begin to be filled. The resulting moves in a stock can be dramatic.

Suppose an institutional investor needs to buy a million shares of CNTY (perhaps part of some proprietary arbitrage position). The buyer looks at the depth chart and sees that there are a million shares being offered at $3, so the buyer attempts to fill the order--only to discover that he gets perhaps a thousand shares, the rest of the offer is cancelled, and there are now a million shares offered at $3.05. The tug-of-war may continue, but if the buyer is motivated, the share price may rise considerably in a remarkably short period of time.

Remember that the original intent of having a bid and ask price is that the various offerings were intended to be sold. The idea that these offerings would be used only as bait and not represent real liquidity is indeed innovative, but unhelpful.

Unlike the change in climate dynamics in the mid-Pleistocene, the change in dynamics in share price of CNTY is symptomatic of a fundamental change in the operation of the market, and this change is detrimental to the majority of its participants.

Thursday, December 22, 2011

Things I learned this month

It's hard to open a pull door when in a wheelchair.

Wheelchair-accessible washrooms in hospitals aren't, really.

Canada's new $100 bills melt in the microwave. But the bank still takes them.

Sunday, December 18, 2011

Self-organization and wealth distribution

The question of wealth inequality has been making headlines, in everything from the Occupy Wall Street movement and their decrying the wealth of the 1%, to discussion in the Republican Presidential-Candidate Popularity Contest currently ongoing in the US.

There have always been voices clamouring for equal wealth for everyone, but the real world doesn't work like that. Wealth inequality doesn't seem particularly unfair given the inequalities in natural abilities and access to capital or resources. Intuitively, it seems that the distribution of wealth in society will follow a power-law distribution. A power-law distribution is one in which the observations show a 1/f distribution, as described in this article.

Recent modeling studies suggest a 1/f distribution over most of the population, but wealth distribution becomes exponential near the tails. The model distribution is described as Pareto-like, with a relatively few super-wealthy floating over an ever-changing middle class.

So wealth inequality should be expected in any society, no matter how even the playing field. The skills necessary to navigate through the economy are not evenly distributed. Some individuals play better than others. Therefore, some individuals will be wealthier than others. Let's take a look through some public data and see if we can recognize a power-law distribution.

According to Wolff (2010), the breakdown of wealth among different quintiles (and finer groups) is:

Fraction of                        Fraction of 
population                         wealth

Lowest 40%                      0.2%
40 - 60%                            4.0%
60 - 80%                          10.9%
80 - 90%                          12.0%
90 - 95%                          11.2%
95 - 99%                          27.1%
99 - 100%                        34.6%

Given that the wealth of Americans in 2007 was reported by the Fed to be $79.482 trillion, and the population of the US at that time was 299,398,400 (roughly), we can plot a logarithmic graph of individual wealth vs population to check for self-organization in wealth distribution.

In order to do this, I have estimated that the wealth of the individual in the middle of each group to have the average wealth of the group. Based on past experience, this estimate will tend to be biased--however given the number of orders of magnitudes on the resulting graph, the errors are so small as to be unnoticed.

To interpret this graph, consider the first two points--they suggest that roughly 80 million people have less than about $2,500, and about 130 million people have less than about $75,000. Most of the data appear to lie along a line of fit, but there are a few exceptionally rich individuals, including some on the Forbes 400 list, who plot far above the line. 

Also note that "the 99%" includes people that have about $8 million in assets.

The observed distribution agrees somewhat with the models described above--a few super-wealthy lording it over the rest. However, there is a significant difference between our observed slope and the slope of the models--the models suggest a slope for the straight-line of about 2. On our graph, the slope of the straight line is over 4 (meaning four orders of magnitude in wealth over one order of magnitude of population).

On our graph, roughly 290,000,000 people have less than $1 million, and 29,000,000 have less than $100. Seems a tad steep. With a slope of 2, the 29,000,000 would have less than $10,000.

If the wealth of the entire population were described by a 1/f distribution, then the richest American would have a wealth of only about $1.5 million. We here at the World Complex think it would be difficult to manage that summer home in the Hamptons with such a paltry sum.

The Ebert and Paul (2009) paper linked to above attempts to explain the semi-permanent nature of the super-rich. The super-rich have benefited from leverage in the system, and remain at the top due to the ongoing access to greater leverage than is possible for the average citizen. 

A poor geologist like me can only wonder--what happens when leverage becomes wealth-destroying rather than wealth-enhancing? Unfortunately, the answer we are seeing is that the super-rich get bailed out of their losing positions by everyone else.

And here we come to the question of fairness in the system. A fair system with an even playing-field will always result in inequalities--but even extreme inequalities will be tolerated to the extent that the system is perceived as being fair. In the past, during times when the system was fair(er), people tended to respect that someone had earned money and was able to enjoy the fruits of success. Under the present system, there is a widespread and growing skepticism that unusually wealth individuals have obtained their wealth not through production of wealth but through gaming the system and even stealing wealth from those lower down the socio-economic ladder.

Lastly we see the same plot as above, but with the estimated and "ideal" wealth distributions as determined from a series of nationwide interviews with over 5500 respondents reported in Norton and Ariely (2010).

Clearly most Americans thought the system was more equitable that was actually the case, and interestingly, they seemed to wish the system were more equitable still. I would like to point out that the "ideal" distribution is actually mathematically impossible (the third and fourth quintiles had equal wealth), which seems fitting. 

In an ideal world, according to the survey, only 10 million Americans would have less than $100,000 in assets, and no one would have as much as a million.

Unfortunately the survey neglected to ask respondents what they felt the wealth of Mssrs Gates and Snyder (no. 1 and 400 on the Forbes 400 list) should be in an ideal world, which might have been very interesting.

Friday, December 16, 2011

Inference of dynamics for complex systems, part 4: long records

Today we look at phase space reconstructions of long climate proxies (which are records of some geological parameter which is believed to be related to some climatic parameter--used because we have no way of directly measuring temperatures or global ice volumes of the distant past).

The proxy I will be looking at today is the ca. 2-million-year-long record of deep-sea d18O (difference in concentration of O-18 from some standard)  from ODP 677.

The record is actually inverted, as it is a proxy for ice volume. In the figure above, the curve is near the top of the graph at times of low ice volume (i.e., interglacials) and near the bottom during glacial maxima.

Reconstructing the phase space over the past 585,000 years (since 585 ka in the figure below), using a delay of 6000 years (noted as 6 ky below), gives us the following.

Now we need to decide what sort of system this graph describes. Is it like this?

Or more like this one?

There are a lot of loops in our reconstructed phase space portrait. Are there any areas of Lyapunov stability? It is not too easy to see directly from the portrait.

To simplify it, we can divide up the data into bins and contour the density of data in each bin. I have called these "probability density plots" in previous posts. With sufficient data, you may be able to use a Gaussian kernel estimator--as many commonly used mathematical software packages contain such a feature (you may have to create your own subroutines to work in two or more dimensions).

The probability density plot (modified from the Paleoceanography paper) is a lot easier to interpret than the original phase space portrait (second figure from top). The peaks in probability density (labelled A2 through A5) are interpreted as areas of Lyapunov stability. Global ice volume over the past 750 thousand years (and much more) is characterized by multistability--there are multiple equilibria. At any given interval, only one equilibrium is "in play"; but the equilibrium point is subject to abrupt change from, say, A5 to A2, over very short intervals.

The above image was constructed from a "window" (a shorter section of the record) of 750 thousand years. The entire record might be studied in a series of three such windows. Windows of lesser duration offer a higher-resolution characterization of the system dynamics.

Same data set, shorter window.

Creating a probability density plot is a robust method of limited computational difficulty which can simplify your interpretations of the dynamics from long, complex data sets.

Wednesday, December 7, 2011

Two war criminals; but only one faces trial

As Gbagbo faces trial at the Hague, Ouattara runs Cote d'Ivoire.

Our housekeeper was outraged. Like many Ghanaians, she recognized that massacres were committed by both sides during the recent Ivoiran Civil War, but it is only Gbagbo going to the ICC. I told her it was a historical truth that the winners get to try the losers.

A few days ago we were having trouble with our truck and dropped in to a service station. Where in North America you would expect to find a Pirelli calendar, at this station they had the poster below depicting some of the crimes in question.

(Warning--graphic and disturbing imagery below).

How QE helped Main Street, example 2: Collectors of fine wines

Following earlier discussion posted here, we investigate another way in which typical Main Street individuals benefited from quantitative easing.

Below we show the graph of the Liv-Ex Wine 100 Index over the past four years. The Wine 100 Index is an index based on the auction prices of the 100 wines selected as most representative of the secondary market. It represents an indicator of the price of the typical wine in your neighbourhood liquor outlet, such as Lafite Rothschild 2006.

For a blue-collar worker who's just lost his job and had his mortgage go underwater, the 20% decline in value of his wine collection in late 2008 would have been a devastating blow. But fortunately the Fed stepped in, first with TARP and then with quantitative easing. The resulting 80% rise in the price of fine wines has certainly helped our friend. It's true his mortgage is still underwater and he has no jobs, but as he must feel enriched as he drinks to his misfortune.

For myself, I prefer cocoa.

Your correspondent in Africa.

Friday, December 2, 2011

A day in Ghana redux

Today was spent cruising up the Ankobra River, in our mighty fishing vessel, which we will be taking out to sea tomorrow.

We set out from our dock just north of the highway bridge crossing the Ankobra (in the background), near the coast, and headed upriver. About 2 km upstream was a nice artisanal gold mining operation.

If you look closely you can see the man working under the tarp. The operation draws water from the river to run through the sluices. All that bare rock represents forest cleared for the mining operation.

If my memory serves me correctly, this land is on a concession held by Adamus Resources Ltd. (excuse me, that's now Endeavour Mining Corp., with which I have no relationship, and of which I own no shares). So this is most likely an illegal operation.

Speaking of questionable mining operations, last week on the Ankobra we stumbled across these.

The dredge is rigged for running up the river, not for operation. The locals here tell of a very large contingent of Chinese miners working with artisanal miners near Prestea. The dredges were not on this part of the river today.

The high price of gold has caused an explosion in artisanal mining all over Ghana. New operations were reported last week on the beach at Elmina, in the Central Region, but local authorities have shut these down. We saw a very large operation along the south side of the coastal highway about twenty minutes west of Agona Nkwanta, which still appears to be active.