Gold

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.

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.

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.

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.

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.

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.

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.

Geological Hazards n+1: Caldera eruptions

Sorry about the title, bandwidth is too slow to go back and check what number I'm on.

Spent a few days at Axim, on a hotel on a hill where several years ago another geologist and I were digging a hole through a pyroclastic deposit full of doubly terminated quartz. The hotel was just being built at the time--only a few of the huts were in place. The workers pointed out a man dressed in rags just dismounting from a hammock slung between two coconut trees. "There's Rastaman," they said. A certain distinctive odor commonly associated with Jamaica (and Canada for that matter) wafted up from the shore. I thought the man a vagrant, but it turned out that he was the hotel owner.

Our view as we toiled in the sun.

We panned through the pyroclastics looking for a hint of diamonds. Diamonds in Ghana can be associated with pyroclastic flows, even though this model is somewhat at odds with the more typical kimberlite model.

Anyway, this association with volcanic flows is a good starting place for today's topic.

At the beginning of last year I gave a brief geological lesson to the grade four classes at my daughter's school. The first topic I decided to discuss was the age of the Earth. So I asked the class how old they thought the Earth was.

Most of them knew it was very old, but had no idea of the number. But to these kids, even ten thousand years seemed tremendously old. To be fair, most adults can't really conceive the concept of a billion years except as a one followed by nine zeroes.

One girl boldly stated, "According to my calculations, the Earth is exactly 699,998 years old." Well, I thought that was a fascinating answer, both for its length and its precision. Naturally I was curious as to the nature of her calculations.

She said that she knew the Earth was destroyed every 700 thousand years, and since it was going to end in two years (2012), it was 699,998 years old.

Well, that wasn't as involved a calculation as I had hoped, but that business of the Earth being destroyed every 700 thousand years was very interesting. Very interesting indeed.

Seven hundred thousand years is roughly the recurrence interval of supereruptions of the Yellowstone Caldera.

A caldera is a massive volcano, larger than anything seen in our lifetimes. Beneath the suface is a magma chamber that is so large that after it empties itself, the ground over the chamber collapses into it, leaving an enormous crater. Krakatau and Tambora (1815) are recent examples of calderas.

Caldera eruptions are so large they have the potential to destroy civilizations. The eruption at Santorini at about 1628 BCE devastated Minoan civilization. It also left a very picturesque crater which I hope to visit before the next one.

The Tambora eruption was even larger than that at Santorini, and had a devastating effect on local agriculture. It also affected global climate. Crops failed throughout the northern hemsiphere, leading to the worst famine in the 19th century.

While Santorini, Krakatau, and Tambora were all impressive, Yellowstone is on a completely different scale. The supereruptions spewed out about ten times as much magma as the Tambora eruption.

 Yellowstone is a particularly big one, which erupts leaving a crater tens of km in diameter, spewing out thousands of cubic kilometers of debris.

USGS map of Yellowstone caldera craters (click here to enlarge).

Ash from these supereruptions has covered up to half of North America to a depth of a metre. Imagine an event of that magnitude happening now. It would be a civilization-ending event.

Not to alarm you, but there has been a lot of recent activity, suggesting that the magma chamber is once again being filled. But one should remember that the majority of eruptions from the Yellowstone hotspot are smaller than the supereruptions, so the recent activity may be a precursor for an eruption but not necessarily a supereruption.

OWS should become PWS

It was interesting seeing the Occupy Wall Street movement trying to occupy the stock exchange.

It would have been a lot more interesting if it had happened a couple of decades ago, in the days of floor trading. Imagine the last part of "Trading Places" with protestors on the floor screaming out fake orders.

What they should do now if they want to shut down the market is borrow as much money as they can (from a major bank), stick it in a stock account (at the same bank), and short everything. When the margin call comes, declare bankruptcy. A few million people doing that would have a much bigger effect than occupying the place.

Government by fiat

I see Italy has a new government (maybe). The news is a little sketchy here--the newscasters seem very happy to tell us that this will be a government of technocrats (when did they stop being bureaucrats?)

It doesn't seem very democratic. Is this a glimpse of the future? A string of financial crises precipitated by the banks which in turn are used to justify placing the bankers in charge?

A day in Ghana

Internet is very slow and intermittent, so posting will be unpredictable until sometime next month.

A day in Ghana

The heat sets in early. Kwame informed me that the shipment was ready for pickup, so we waited until the worst of the morning traffic was finished and began the journey to Tema. Kwame drove, and Kabi came along to help load the truck. There were a few administrative details to take care of--primarily the licence sticker on the truck had expired and had to be renewed. We drove the broken road up to Barrier, onto Winneba Road (a six-lane highway), headed west one junction to SCC and drove along a decent road to the licencing office. We were only mistaken for a trotro once, near the police barricade in SCC.

The licencing office was a series of simple buildings around a rough parking lot. After a few minutes we were on our way. Back on the Winneba Road, east in to Accra.

The main chokepoint is Malam Junction, where Winneba Road, which continues on toward the centre of the road, meets Kwame Nkrumah expressway, which leads off towards the airport. When the road was first built, it skirted the outside of the city, but since then the city has grown across the roadway and exploded into the virgin ground beyond. So now the expressway is wholly inadequate for the weight of traffic that tries to pass each day. Furthermore, as you approach the junction, Winneba Road (itself three lanes wide plus a dedicated lane for trotros separated by a concrete barrier) narrows to two lanes.

Making matters worse is the massive construction project whereby the junction will be made into an elevated interchange. Once completed this will greatly improve the flow of traffic through the junction (although without improving the rest of the road network, the jams will simply move elsewhere).

Improbably, we passed smoothly through the junction, being stopped only for a few moments by police to allow work vehicles to cross the road. We stopped for gas and discovered the truck was leaking oil. Next door was a fitting shop, but the mechanic advised us he would have to wait some hours for the engine to cool enough to open it up. So Kabi and I caught the trotro back to Barrier ("Kasoa! Kasoa direct!") and from there a shared cab back to the compound. The trotro cost 90 p (about 55c) and the taxi another 85 p for the last part of the trip. And with that, the day was done.

Ghana and Cote d'Ivoire and other marine boundary disputes

There's nothing like the discovery of a new marine resource to bring a marine boundary dispute to a head.

The recent discovery of oil off the coast of Ghana has naturally lead to a boundary dispute with their neighbour, Cote d'Ivoire. The confusing thing was that the dispute centred around not the discoveries that were closest to the official border--it seemed that Cote d'Ivoire was claiming jurisdiction over a portion of the seafloor well within Ghanaian maritime space, and clearly directly offshore from Ghana. The details are sketchy, especially here in Ghana where internet connectivity is painfully slow, but supposedly the two governments are to meet early in the New Year.

Maritime border issues tend not to be as clear as land disputes.

Oil has been the cause of disputes in the Aegean Sea between Turkey and Greece, so this is by no means an unusual development.

From memory, I can recall a significant dispute between Canada and the United States over Georges Bank, which was a significant scallop resource at the time. The Canadian position was that the border would be equally distant from both countries, which would give Canada the seaward portion of Georges Bank (which had the best scallop fisheries). The American position was that Georges Bank was connected to the US continental shelf, and separated from the Canadian continental shelf by a deep trough, and so should be entirely American.

Now this discussion is entirely from memory, so any inaccuracies are mine. There were precedents for both positions. But when the case came up for arbitration at the International Court of Justice, the US presented a proposal in which the maritime boundary was extended seaward in a straight line from the last segment of the land boundary. The Canadian proposal was as described above. The Americans presented a map showing only the continental US, and the position of their proposed boundary. The Canadians projected the  American's proposed boundary onto a world map, revealing that the proposed boundary cut through Nova Scotia. Since the case was up for arbitration, the court had no choice but to accept the Canadian boundary, which is how we ended up with the good part of Georges Bank. At least this is how the story was related to me.

Inference of dynamics for complex systems: Examples, part 2

A few more examples, as I have been under the weather and am also in last minutes of preparing to return to Ghana.

This chart, composed from monthly closing prices, covers the last fifteen years of the gold-silver ratio. As feared, the gold-silver ratio has reverted to its long-term (~10 year) area of stability, so regrettably we can only characterize the past year's action in silver as an excursion.

Once again, from The recent drop in the price of copper shows up well here at the right of the graph. In contrast to the first figure, this plot suggests that the fun with silver isn't over yet, and that silver is still doing great things in phase space. We may finally be seeing a long-term change in the relationship between silver and other commodities. We will have to wait and see how it unfolds over the next year or so.

Inference of dynamics for complex systems: Examples, part 1

This article continues from the theoretical discussions here, here, and here.

Today we begin looking at some reconstructed phase space portraits (in two dimensions). These are all figures that have been shown here.

All three of today's examples show multistable behaviour. As discussed last time, the implication of multistability is that there are two (or more) equilibrium states in the system, as opposed to just one (the most common assumption).

Stability arises from negative feedback. The instability results from positive feedback. Complex adaptive systems with many participants commonly exhibit both and are thus prone to multistability.

The Case-Shiller index is an inflation-adjusted measure of house prices (for houses of constant quality) in the United States. The reconstructed phase space (above) shows two areas of Lyapunov stability.

The tick marks on the trajectory mark the states at one-year intervals. As the lag is four years, the first point on the graph is the plot of the 1890 value against the 1894 value. The point is labelled as representing the state in 1894--consequently the 1894 state is the first one that can be plotted despite available observations going back to 1890.

The larger of the two areas of stability is occupied over two long stretches totalling nearly 70 years. The smaller of the two areas is occupied for 30 years. Thus of the 116 states (at one-year intervals), 100 of them occur in one of these two areas of stability. There are two short transitions, in about 1915, where inflation-adjusted housing prices suddenly fell, and again at about 1945, when they rose.

The principal era of instability began in the year 2000, whereupon the system embarked on an impressive excursion through phase space. Had this excursion wound up in another area of stability (this may yet be the hope of Greenspan, Bernanke, et al.) we would not be discussing a housing bubble now, but rather a new paradigm of high house prices. Unfortunately, there is no evidence of any stability--and for topological reasons, it is impossible for the current position in phase space to be an area of stability.

If housing prices were to remain at today's levels (adjusted for inflation), the trajectory of the curve would evolve directly towards a point just to the NE of the tip of the large area of stability--at about (130, 130). It would arrive there in four years.

If prices continue to fall, then the trajectory may fall into either the larger area of stability, or perhaps even the smaller one. Either outcome is more likely than developing a new area of stability at prices equal to or higher than today's prices. (FYI this does not constitute real-estate investment advice).

As for the drop in housing prices after 1915--there are a few possible explanations for that, but the easy one might be the introduction of income tax (about the same time as the Federal Reserve), which would have reduced the money most people had available for such a purpose. Our normal expectation when less money is available for discretionary purchases is that prices will fall. There followed a long period where for various reasons there just wasn't much money--the Depression, and WWII.

Interestingly, one reason there wasn't a lot of money available for buying houses despite scads of it being printed and distributed during WWII was the sale of War Bonds, which helped to draw excess money out of the economy and so prevent inflation. Curtailing this program at the end of WWII allowed inflation of house prices after 1945.

Two areas of stability over the past ten years--one of low unemployment, and more recently, a stable area of high unemployment.

The plot of unemployment vs interest rate also shows two distinct areas of stability in phase space. The existence of (at least) two areas of stability points to (at least) two equilibria in the system of unemployment and interest rates. This is at odds with the assumption of single equilibrium in the system which has informed the Central Bank's policy of lowering interest rates in order to stimulate employment.

A major problem with the relationship between interest rates and unemployment--like economic theory in general, this relationship is simply asserted. A great amount of effort has gone into justifying these assertions--and to be fair, the assertion doesn't seem unreasonable. However economic theory seems to assume that the preferences of the various participants in the system will never change in a way that has not been foreseen by an economist.

For instance, suppose that interest rates are high, around 10%. At such high rates, you had better be borrowing money for some productive purpose or the interest will kill you. Such high rates will make it difficult for a business founded on debt to succeed as interest on the entire debt has to be paid out of the profits. It is easy to see here that any marginal decline in interest rates will increase the likelihood of success of any business. More successful businesses mean more jobs. So lower those rates!

But economists don't consider that if interest rates fall below some level, some participants will see that is easier to try to make a living on speculation rather than productive industry. If real interest rates fall to zero, and you have an unlimited ability to borrow, then why not speculate on, say, the stock market. You just keep borrowing and gambling until you win big, and as interest rates are so low you can easily carry the debt until you win. It's a lot easier than building a factory to make refrigerators. The lower the interest rate falls, the greater the impetus to speculate rather than produce, as the costs of carrying the debt are minimal.

In this scenario, lowering interest rates no longer creates employment, as it simply encourages more speculation. No doubt, there will be a few hardy fools out there trying to start a business, but they are a distinct minority.

The empirical evidence suggests the policy of lowering interest rates to stimulate employment has failed. Unfortunately, because our observations are at odds with classical economic theory, it is unlikely we will see any change in Central Bank policy.

I think the only option is higher interest rates, but this will only be possible after the debt that is currently choking the system is somehow purged.

Canadian Mint launches new gold "ETR"

This was actually announced on Friday but only hit the papers today. The Royal Canadian Mint is proposing to sell a fraudulent product they call an "Exchange Traded Receipt" (ETR), each one of which represents a fixed amount of gold. The initial proposed price is $20 per unit, and the amount of gold represented by each receipt will be determined by the gold price on the Closing Date (not stated if this is the closing price, maximum price, London PM fix?).

The objective is to sell $250 million worth of units, through the usual culprits. The Closing Date is expected to be in late November.

You should note that this is not a good method for holding gold in the long term, as there is a management fee of 35 basis points per year, which is deducted daily from the gold represented by each receipt. So it's a little like buying a bag of gold with a very small hole--each day, a little will leak out.

According to the Mint's announcement, there will be a procedure for taking delivery of the gold represented by each certificate. No word yet on how onerous or time-consuming this procedure will be.

And if by chance the gold represented by your receipt goes missing at the Mint for whatever reason, there is this:

ETR holders will have no recourse to the Mint or the Government of Canada for any loss on their investment.

Of course if you never try to take delivery, you never have to find out if there is any gold backing this instrument.

Congrats to the Toronto Star for spicing up the otherwise bland coverage of this momentous achievement with the following quote from Dr. Moshe Milevsky, a professor at the Schulich School of Business:

“What I worry about is if people somehow think the government is somehow telling them gold is a good investment,” said Milevsky. 

 Oh no, we'd never want people to think that! 

Inference of dynamics for complex systems, part 3

Equilibria in linear systems

It is common for linear systems to evolve towards a single fixed state.

The assumption that most dynamic systems have a single preferred fixed state for a given set of circumstances appears to be one of the drivers behind Central Bank policies. It is debatable whether the Central Banks recognize nonlinearities in socio-economic systems, or whether they do but are unable to express it it the typical 10-second soundbite that the average consumer of their "products" can absorb.

Certainly, the common example of using lower interest rates to combat unemployment sounds like the kind of thinking one expects about a system with a single equilibrium state. The notion that low interest rates ultimately lead to higher inflation reflects similar thinking.

Here are some examples in phase space of single equilibria in a simple linear system.

When all trajectories within a given region of phase space gradually and continually converge towards a single point, then the equilibrium can be described as being asymptotically stable, and is commonly referred to as a point attractor. Note that even though trajectories 1 and 3 appear to cross, they actually do not as one passes over the other in a three- (or higher-) dimensional projection.

Line crossings in a properly "unfolded" phase space portrait cannot cross, as each uniquely defined state represents a unique value and a unique sequence of values of the higher time derivatives of the time series from which they are projected. If two trajectories converged onto a single point, it would be impossible for them to diverge again--hence no crossings. If there appear to be crossings in a reconstructed phase space, then the phase space needs to be projected in more dimensions. Unfortunately, there are logistical problems with presenting even three (let alone more) dimensional figures, which is why I have limited the figures presented to two dimensions, with a caveat that apparent crossings mean we should really be looking at a projection in at least three dimensions.

Often we find that rather than continuous convergence, two states which originate close to one another as well as to a particular point in phase space tend to stay close together and near to the point. Such stability is called Lyapunov stability (sometimes a Lyapunov-stable area, or LSA). More formally, we might state that all states with a region of phase space (a) will tend to remain in another (possibly smaller) region of phase space (b). Once again, note that the apparent crossing of trajectories only suggests that we should construct this phase space in at least three dimensions.

Another form of equilibrium is the limit cycle. It represents a form of asymptotic stability whereby the final equilibrium is not a point but a continuous cycle. There is no reason that the cycle needs to form an ellipse--any closed shape is possible. In higher-dimensional projections, a limit cycle may be in the form of a shell, or a torus.

Equilibria in chaotic systems

A different form of equilibrium was discovered by Edward Lorenz in 1963. A system defined by three nonlinear differential equations reached a complex equilibrium in which any state evolved continually towards the "attractor"; however any two states starting arbitrarily close would diverge exponentially, even though at the same time they would both evolve in similar fashion through phase space.

This form of attractor was called a "strange attractor". Although it occupies a finite volume of phase space, the trajectory from any arbitrary point would evolve in unique fashion, so that all evolving trajectories from any arbitrary starting point would appear similar, yet would occupy subtly different regions of phase space.

Multiple equilibria (multistability)

In some complex systems, we observe any number of disjoint Lyapunov-stable areas (LSA), each separated in phase space by a separatrix [Kauffman, 1993]. At any given time, the state of the system occupies only one such LSA, so that their number therefore constitutes the total number of alternative long-term behaviors, or equilibrium states, of the system. Since an LSA is likely to be smaller than the total allowable range of states, the system tends to become boxed into an LSA unless it is subjected to external forcing. When the state approaches a separatrix, small perturbations can trigger a change to a nearby state (a bifurcation), resulting in chaotic changes in the evolution of the system [Parker and Chua, 1989]. Thus very complex behavior can arise in multistable systems.

Multistable behaviour generally arises from systems in which feedbacks, both negative and positive, impact a system which is perturbed by some sort of external forcing. Negative feedback tends to resist external forcing, resulting in stability in some regions of phase space. If the external forcing is sufficient to overcome the feedback, then positive feedback may actually accelerate the changes, causing the state to evolve rapidly towards another area of stability.

The smooth variation of one or more parameters in the system may result in a change in the type of or the number of attractors in the system, or even in the order in which the attractors are visited. Such a response is called a bifurcation. Bifurcations can represent a sudden transition within a system characterized by purely chaotic attractors to one with one or more LSA; between one LSA and multiple LSA; or between different configurations of LSA.

Bifurcations represent changes in the organization of the system, and their existence has been suggested by models [e.g., Ghil, 1994; Rahmstorf, 1995], and more rarely from observations [Livina et al., 2010; discussed here]  in future installments we will demonstrate such behaviour in natural systems. Initially, however, we will concentrate on interpretation of some of the phase space portraits presented last time.

References


Ghil, M. (1994), Cryothermodynamics: the chaotic dynamics of paleoclimate, Physica, 77D: 130-159.

Kauffman, S. (1993), The Origins of Order: Self-Organization and Selection in Evolution, Oxford Univ. Press, New York, 734 p.

Livina, V. N., Kwasniok, F., and Lenton, T. M. (2010), Potential analysis reveals changing number of climate states during the last 60 kyr. Climate of the Past, 6, 77-82, doi: 10.5194/cp-6-77-2010.

Parker, T. S. and L. O. Chua, (1989), Practical Numerical Algorithms for Chaotic Systems, Springer-Verlag, New York.

Rahmstorf, S. (1995), Bifurcation of the Atlantic thermohaline circulation in response to changes in the hydrologic cycle. Nature, 378, 145-149.

Snapshots of economic crises in phase space

Part 3 of inference of dynamics is delayed as for some reason I can't open the post and edit.

Today we look at the monthly change in net foreign purchases of US long-term securities. Data comes from the US Treasury site. By net foreign purchases they mean the difference between foreign purchases of US long-dated securities and US purchases of foreign securities.

What I found most surprising is the negative bias. In this chart, a negative number means US purchases of foreign long-dated securities exceeds foreign purchases of US long-dated securities. We note the negative bias becomes quite pronounced beginning in the early '90s (wasn't this the era of the US strong dollar policy?)

The plot below is a reconstructed phase space in two dimensions, using the monthly change in net foreign purchases of US securities, expressed as a percentage, and smoothed by a three point moving average.

What we are looking at is a measure of policy responses to economic crises of the last 30 years. Two things really leap out at me: 1) the size of the response to the Russian crisis in early 1998, which arguably led to the collapse of Long Term Capital Management; and 2) where is the great 2008/2009 crisis?

The net change of about 13000% in a single month reflects foreigners dumping US bonds and the US buying foreign bonds. The actual peak value is higher because I have plotted the phase space using a 3 pt moving average instead of the actual values to reduce noise. Of course the super-spike is so large that any noise has faded into the background.

Notice the straight lines from the tangle near the origin along the axes. It is rare to see straight segments in a phase space portrait--it tells us that the massive change happened over the course of a single observation. Imagine what must have happened to a heavily levered player in such an environment? Well, we hardly need to imagine. 

Looking at the last few years, we see the following.

I'm not sure what was going on in late 2005, but the last big crisis had a sudden (albeit small) response.

Since these are percentage changes, it could be that before the 2008-9 crisis really took off there was already a lot of net selling of US long treasuries, so the spike does not appear so impressive.

In conclusion--it looks like the Russian crisis caught monetary authorities by surprise, leading to an enormous response.

Inference of dynamics for complex systems part 2

Phase space portraits

As we left our last installment we had the problem of a series of observations from some interesting system, and we were seeking a means of understanding it. First of all, however, we had some doubts as to whether the measurements we have made will tell us anything about the system, or whether there will be other information needed in order to make any useful inferences.

Approaches to studying dynamic systems include both qualitative studies of the general trends of a system and quantitative studies in which invariant properties of the system are evaluated [Abarbanel, 1996]. System dynamics are evaluated by reconstructing the system’s phase space, which is a geometrical representation of the system projected in a “space” created of different variables [Packard et al., 1980; Abarbanel, 1996]. The climate system can be described by a phase space with coordinates x1, x2, x3, . . . xn, and the functions x1(t), x2(t), x3(t), . . ., xn(t) (the outputs of the system). As time (t) varies, the sequential plot of points of coordinate {x1(t), x2(t), x3(t), . . ., xn(t)} describes the time evolution of the system in phase space.

The number of output functions (n) is called the embedding dimension [Sauer et al., 1991]. The evolution of the system is marked by the trajectory traced out by sequential plots of individual states with coordinates defined by the values of the n functions at each observed time. Describing the trajectory of the system as it flows through phase space is a qualitative means of characterizing the dynamics of the system. The system may also be characterized quantitatively in terms of its invariant properties, such as the Lyapunov exponents and the correlation dimension of the system, which can be calculated from the phase space portrait [Abarbanel, 1996].

Phase space from multiple time series

How do we select the coordinates? One method is to create a phase space by plotting scatterplots of several different records which have been sampled at the same time intervals. For instance, Saltzman and Verbitsky [1994] created a phase space using, as variables, ice mass, ocean temperature, and atmospheric CO2. The state of the system is defined by its location in phase space at a particular time. The plot of successive states through time traces out the trajectory of the system. Traditionally the trajectory is constructed by drawing a curved line, rather than straight line segments through the states in sequence.

The drawback with the Saltzman and Verbitsky approach in paleoclimate is that is difficult to find many records that have been sampled at the same intervals. You are restricted to the portion of the geologic record covered by the shortest record. Additionally, there are errors in both magnitude and time.

Let's not worry about interpretation yet. Today is only about basic methodologies.

Economic systems can quite profitably be studied using this approach, mainly because there are so many of them, the errors tend to be small (except see here), and the timing is usually well constrained as well. So we can compare US unemployment rate to interest rates, for instance.

Data from BLS site.

Commonly we might look at observations like the one above, and not draw the trajectory (the curve that runs sequentially through the data). Instead, a traditional approach might have been to draw a line of best fit in the hopes of defining a correlation. In looking at the above figure, we see two clusters of observations. Past experience tells us it is risky to define a line of best fit using the traditional methods in this way, as the result is heavily weighted by the line between the centres of each cluster.

Similarly we can look at the average duration of unemployment vs unemployment rate.

Data from BLS site.

Or unemployment rate (vertical axis) vs monetary measures.

Data from BLS and St. Louis Fed site.

Or house prices vs real interest rates.

Data from Shiller [2005].

Defining a phase space from multiple variables requires multiple records. The state space can only be characterized over the duration of the shortest record. Dating errors will lead to various forms of distortion in the projected phase space. The economic time series tend to lend themselves well to this form of projection, because many of them exist to any arbitrary level of precision. If you choose month-end or year-end prices, there are normally no dating errors.

Phase space from a single time series

It is pretty uncommon to have more than one geological time series of sufficient length with good dating control. So geologists will normally have to work with a single time series. The method below can similarly be used in other types of time series as well.

When you have one time series, you may wonder how much dynamic information it contains. Fortunately, ergodic theory suggests that dynamic information about the entire system is contained in each time series output from the system [Abarbanel, 1996]. Therefore, a phase space portrait reflecting the dynamics of the entire system may be reconstructed from a single time series.

Time-derivative method

Packard et al. [1980] propose a method in which the function is plotted along one axis, and its various time derivatives are plotted on the other axes. If we use the simplest two-dimensional case, the graph would consist of a scatterplot of the function against its first time derivative. (i.e. y vs. dy/dt). An example of such a plot appears on the masthead of the blog.

In the above figure, we see the ice volume proxy plotted on the horizontal axis (ice volume increases towards the right) plotted against its first derivative over an interval of time lasting about 120,000 years. The numbers on the graph represent the time in thousands of years before present (ka BP). The rate of change of ice volume is plotted with +ve on top, so that as global ice volume grows (near A, for example), the system will move towards the right through phase space.

Any equilibria in this type of figure must necessarily occur along the zero rate of change axis.

Note the error bars presented on some of the states. Similar error bars would be found at all other states in the figure as well. The error in estimating the rate of change is a consequence of the error in measurement being similar in size to the difference between successive measurements. The size of the error bars is large compared to the variability of some parts of the trajectory--consequently our confidence in this trajectory is not as great as it otherwise might be.

Time-delay method

We reduce these errors by reconstructing the phase space by the time delay method [Packard et al., 1980], in which the elements of a time series are plotted against n-1 lagged observations from the same series (figure 2B). Identifying the lags and the embedding dimension (n) are key decisions in the reconstruction. To simplify things in the following discussion we shall only use two dimensions. Thus we reconstruct our phase space portrait by a scatterplot of the data against a lagged copy of itself. The optimum lag is defined by the first minimum of the average mutual information function [Fraser and Swinney, 1986]; however for quasiperiodic data we find that this tends to be the first minimum of the autocorrelation function (about ¼ of the period of the dominant waveform).

Thus for ice volume:

Here we are looking at a two-dimensional phase space reconstructed from ice volume proxy data covering about 200,000 years. In this projection, lower glacial ice volume is at the lower left corner of the plot, with greater ice volume towards the upper right corner. We'll interpret these later. Moving on

Case-Shiller index

Official unemployment rate

Detour Gold Corp.

CNTY busted trades (1 s of trading activity each figure)

Gold-silver ratio in phase space

Dynamic systems, like climate, have historically been analyzed using power spectral methods, such as the Fourier transform and wavelet analysis [Hays et al., 1976; Imbrie et al., 1992]. This has been a reflection of the predominantly linear assumptions underlying early analytical methods.

The power spectrum is not an invariant property of a nonlinear time series [Abarbanel, 1996], meaning that significant changes may appear in the power spectrum despite the lack of changes in the dynamics of the system. Therefore, changes in power spectrum are insufficient evidence to infer changes in dynamics.

In our next installment we'll talk a bit about equilibrium and what any of the above plots have to say about it.

References

Abarbanel, H. D. I. (1996), Analysis of Observed Chaotic Data, Springer-Verlag, New York.

Fraser, A. M., and H. L. Swinney (1986), Independent coordinates for strange attractors from mutual information, Phys. Rev. A, 33, 1134-1140.

Hays, J. D., J. Imbrie, and N. J. Shackleton (1976), Variations in the Earth’s orbit: Pacemaker of the ice ages. Science, 194, 1121-1132.

Imbrie, J., et al. (1992), On the structure and origin of major glaciation cycles, 1, Linear responses to Milankovitch forcing, Paleoceanography, 7, 701-738, 1992.

Packard, N. H., J. P. Crutchfield, J. D. Farmer and R. S. Shaw (1980), Geometry from a time series, Phys. Rev. Lett., 45, 712-716.

Saltzman, B., and M. Verbitsky (1994), Late Pleistocene climatic trajectory in the phase space of global ice, ocean state, and CO2: observations and theory, Paleoceanography, 9, 767-779.

Sauer, T., J. A. Yorke and M. Casdagli (1991), Embedology. Journal of Statistical Physics, 65, 579-616.

Shiller, R. J. (2005), Irrational Exuberance, 2nd ed., Princeton University Press.