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).

Monday, January 30, 2012

Confidence and ruin amongst the PIIGS

For today's installment we'll take a look at the debt:gold ratio for the PIIGS countries to see who puts the IG in PIIGS (perhaps you've already guessed).

I am using the same methodology used in these postings. The data come from the World Gold Council, the World Bank (albeit indirectly, for instance here), and here.

We'll start off with a look at the biggest of them--Italy.

And just for comparison's sake, here is the American plot, over a somewhat longer timeframe.

Once again--the ratio represents the multiple by which the country's debt exceeds its gold holdings. To an optimist, a high ratio means that the rest of the world has great confidence in the economy of the country in question. To a pessimist, a high ratio means the country is ruined.

At a quick glance, it appears that Italy is no worse off than America--assuming that both countries actually have the gold the World Gold Council claims they have. Italy may have trouble getting theirs from New York, if that is where it is.

Notice the decline in the ratio over the past decade--that is a reflection of the rising price of gold, not a decline in these nations' debts. Debt has increased over the past decade. The price of gold has apparently risen more. So does this mean these countries are becoming solvent? Can a rising price of gold solve our economic woes?

Historically, a decline in this ratio can been used by governments to justify monetary expansion, particularly if it happened during an episode of such expansion. Why not? The improvement of the ratio suggests that the government isn't printing enough. The destruction of the value of the currency (and the country's debt) begins to occur faster than the rate of monetary creation (thus the label in the US graph "Ben proposes, the Market disposes"). The government counters this by printing faster, but the destruction of the currency's value is faster still.

Here we see Spain. They aren't as well off as the Italians, but I've seen worse. Canada, for instance. Or Japan. And we'll see some others later.

We notice that the confidence ration hasn't declined that much for Spain in the last decade. A good part of the reason is that the Spanish sold off over half of their gold since 1998. However the Spanish were a little more shrewd than Gordon Brown--their biggest sale was in 2007, so it wasn't quite at the bottom. Still regrettable, though.

Now this is a county! I know where I'm relocating. It'll be convenient for getting to my various African projects too.

Why, why, why, oh why did Portugal let the escudo die? It looks like it might have been the greatest currency in the world in the 70's and 80's. Oh sure, the economy was small. But why do we think this is a country with serious economic troubles? Their only real problem is the fact that their gold is probably all in New York.

It's hard to believe, but there are PIIGS countries worse off than the US.

Here are our winners--the Irish and the Greeks. They are the IG in PIIGS.

The Greeks have been getting the bad press, but look at the Irish! Their problem is that they have never really had any gold. But who needs gold when you've got booming real estate?

Bad as things look in Europe, there are worse.

Stupendous! And it kind of makes Harper's recent boasting about Canada's solvency darkly humorous.

Friday, January 27, 2012

Phase space portrait for Detour Gold . . .

The last time we looked we saw that the reconstructed two-dimensional phase space was characterized by a long period of relative stability in the $30 area, from at least January until early September. It moved rapidly to the $37 area, seemed to complete a cycle, then the price fell in late September, along with practically everything else in the sector.

Since the end of September, the phase space portrait for price has generally remained in what we identified as LSA 30--an area of Lyapunov stability centred at the $30 region, with one excursion above (in November) and below (in December).

Our best hope in the intermediate term is for continued strength in the sector and the price to move back to what we proposed to be LSA 37. But there is a lot of inertia in LSA 30.

Disclosure: Long DGC-T.

Monday, January 23, 2012

A critical year ahead for silver

Below I have the two-dimensional reconstructed phase space portrait for the gold-silver ratio with a one-year lag. I have actually calculated the three-point moving average of the ratio to smooth out some of the noise.

The last point labelled here is 1111 (November 2011), because the last available month-end close is December; in the convention I have been using, I plot the three-point moving average in the middle of the last three months, which is November. I appreciate that this is not the way everyone does it, particularly TA specialists, but this convention is a reflection of my background.

Recall that this plot is generated by plotting the current value of the 3-pt mav against a lagged value--in this graph, I have used a lag of one year, so the 3 pt mav is plotted against that from one year previously. This means that we already know something of how this graph is going to unfold over the next year--the values on the ordinate are going to fall to about 38 in the next six months, and then return to approximately the present value near 52 within 12 months. We don't know whether that will be a move straight down and straight back up (which would happen if the gold-silver ratio stayed near its present value over that period), or deflect towards either the left or the right in that time frame.

If the gold-silver ratio rises in the next year; or if it stays near its present value, then the state will return to the region of phase space that has dominated the system for at least fifteen years. That would seem to imply a long wait (years, perhaps?) before seeing a major improvement in the silver price with respect to gold.

If, on the other hand the gold-silver ratio declines to at least 40 (or less) over the next six months and holds that through the year, then the phase space will trace out a large arc to the left of the area occupied over the last fifteen years, which I would then view as strong evidence that the dynamics of the gold-silver ratio had changed in silver's favour.

If silver is to outperform gold, that is the scenario we have to see. For that ratio to hold, if gold stays in the $1600-$1900 range over the next year, then silver has to rise to nearly $50. Failing that, it could be a long wait for silver. 

Thursday, January 19, 2012

Why geology (also economics) cannot be a branch of physics

I have been working my way through a new paper by Ellison et al. (2012), from the same group that brought us epsilon machines.

In this paper the authors investigate systems which lie between those governed by classical mechanics (which are reversible, and both their past and future are easily divined from present observations assuming we know the equations of motion) and thermodynamic systems (the past of which, once in equilibrium, are fundamentally unknowable); which could be to say, systems which lie between physics (undergraduate, at any rate) and chemistry.

Systems for which our observations are incomplete, and which are characterized by extreme sensitivity to initial conditions are the focus of the paper. These systems are stochastic.

At this my ears (or maybe my eyes) perk up--we have been looking at just these sort of systems in the course of this blog.

The key insight in the paper is the potential for irreversibility in such systems. Irreversibility, in this case, means that the computational effort of prediction (forecasting the future) and retrodiction (modelling the past) are not equivalent. For these systems, it is insufficient to characterize the forward time-evolution of the system--one must also characterize the backward time-evolution as well; otherwise its description is incomplete.

These characterizations are built through Markov chain analysis, leading to an epsilon-machine construction. A reversible process is one in which both the forward-evolving and backward-evolving epsilon machines are identical.

In an earlier post, I constructed epsilon machines for early Quaternary paleoclimates using as predictive states the high-probability ice volumes A1, A2, and A3, from probability density plots like the one below:

Probability density for reconstructed phase space portrait of global ice 
volume proxy from 1700 to 1550 ka.

In the above diagram, A1 represents an interglacial condition (much less ice than at present), A2 was an intermediate glacial condition (comparable to what we call an interglacial state today), and the maximal glacial state A3, which here in the late Quaternary would only be considered a mild glacial state.

These states were defined from the probability density plots of reconstructed phase spaces way back when I was still using a window of length 270 ky. More recently I have reconstructed phase space for global ice volume with a window length of only 150 ky, but haven't transcribed all the state changes yet.

In retrospect, I could have placed the border between α1 and α2  in a different place than I did in the figure above. As a demonstration, I shall leave things as they are.

Of the three epsilon machines depicted, α2 and   α3 are reversible, whereas α1 is not. The sequence of states in α1 was as follows: A2-A3-A1-A3-A2-A3-A2-A3-A2-A1. The probability computed for the arrow from A2 to A3 is the probability that A3 occurs given A2; which we find by observing all occurrences of A2 (4) and counting the number of them that are immediately followed by A3 (3), for a probability of 0.75. The other probabilities are calculated in a similar fashion.

To time-reversed epsilon machine is constructed the same way, but with the states in reverse order: A1-A2-A3-A2-A3-A2-A3-A1-A3-A2. The structure will appear to be the same, but the probabilities of each transition will differ.

The tilde (~) over the α1 tells us that this is the time-reversed epsilon machine. We note that it is not the same as α1.

The Mid-Quaternary epsilon machine appears to be irreversible.

The general structure of the forward and time-reversed version of α4 are similar. The Markov chain was short, beginning with A1 and ending with A4. The reversed version, therefore, has one more A4 and one less A1 than the forward version, and it is this difference that explains the changes in probabilities for the different transitions. Hence, α4 may actually be reversible.

The late Quaternary epsilon machine appears to be irreversible.

We recognize these as being irreversible because the backward evolving epsilon machine is different.

The goal is to then combine the forward and reverse models into a single bidirectional model (Crutchfield et al., 2009; Ellison et al., 2009). I haven't figured out how to do that yet.

Consider for a moment the reconstructed phase space portrait for the Case-Shiller index of house prices.

Defining causal states by a similar process as for the global ice volume proxy, we would see something like "BAB?"--with the question mark referring to the ongoing excursion generated by our friends at the Federal Reserve. It isn't really a long enough string to do any interesting Markov chain analysis or to construct an epsilon machine. We just haven't seen enough Fed intervention* to devise a predictive model.

Well, my homework now is to figure out the business of constructing the bidirectional model, and how to calculate the difference in entropy between the forward and backward process.

*On the other hand, I think we have already seen quite enough.


Crutchfield, J. P., C. J. Ellison, and J. R. Mahoney. Time's barbed arrow: Irreversibility, crypticity, and
stored information. Phys. Rev. Lett., 103(9):094-101, 2009.

Ellison, C. J., J. R. Mahoney, and J. P. Crutchfield. Prediction, retrodiction, and the amount of information
stored in the present. J. Stat. Phys., 136(6):1005-1034, 2009.

Ellison, C. J., J. R. Mahoney, R. G. James, et al., 2012. Information symmetries in irreversible processes. on arxiv (waiting for official publication)

Sunday, January 15, 2012

Wednesday, January 11, 2012

Scale invariance and the scaling laws of Zipf and Benford

Scaling laws have been empirically observed in the size-distributions of parameters of complex systems, including (but not limited to): 1) incomes; 2) personal wealth; 3) cities (both population and area); 4) earthquakes, both locally and globally; 5) avalanches; 6) forest fires; 7) mineral deposits; and 8) market returns. Several years ago one of my students showed that various measures for the magnitude of terrorist attacks also observed scaling laws.

The general prevalence of scale invariance in geological phenomena is the reason for one of the first rules taught to all geology students--every picture must have a scale. The reason for this is that there is no characteristic scale for many geological phenomena--so one cannot tell without some sort of visual cue whether that photo of folded rocks is a satellite photo or one taken through a microscope--whereas one can make such a distinction about a picture of, say, a moose.

Numerous empirical laws (by which I mean equations) have been developed to describe the size-distribution of scale invariant phenomena. Most of these empirical laws were developed before the idea of scale invariance was well understood. One famous example is the Gutenberg-Richter law describing the size distribution of earthquakes.

Another statistical law, Zipf's Law, describes the relationship between size and rank. For cities, for instance, the largest city in a country will tend to have twice the population of the second-largest city and three times the population of the third. More formally, the relationship is stated as follows:

for a distribution where C is the magnitude of the largest individual in the population, y is the magnitude of an individual with rank r, and k is a constant which characterizes the system--but is commonly about 1.

If we plot rank vs size on a log-log plot, the graph should approximate a straight line with a slope of -1/k.

For instance, a plot of city size vs rank for US cities appears as follows:

Data sourced here.

From the same data source we find a similar relationship when city size is determined from area rather than population:

In the first plot we obtain a value for k very close to 1. The plot where cities are ranked by area is not as clear, but this may be due to the arbitrary nature of city limits. To characterize either of the above plots by Zipf's law is fairly straightforward--draw the straight line from the top-ranked city that best follows the line of observations.

A recent article published in Economic Geology argues that mines in Australia follow Zipf's Law. In summary, not only do the known deposits in Australian greenstone belts follow Zipf's law fairly closesly, but the early estimates of as yet undiscovered gold projected from early Zipf's law characterizations compared favourably with the amount of gold eventually discovered.

The weakness that I see with this approach is that it is all rather strongly dependent on the estimates of the size of the largest deposit. In any given area, it will be true that the largest known deposit will be well studied, but history has shown us that mines can be "mined out" only to be rejuvenated by a new geological or mining idea.

I am unable to reconcile the size-distribution data from the Nevada mineral properties presented recently with Zipf's Law, although they do seem to follow some sort of power law.

Using the straightforward approach to a Zipf Law characterization gives us the red line, which appears to show that there have been far too many gold deposits of > 1/2 million ounces for the largest mine. To reconcile the known gold discoveries with Zipf's Law (green line), someone would need to find a 100-million-ounce deposit (if that doesn't get explorers interested in Nevada, I don't know what will)!

I, however, would prefer to use the interpretation of the above data developed in our last installment--that there is a power-law relationship between size and rank, but this relationship breaks down for the largest deposits because there is some sort of limit to the size of gold deposits (at least near the Earth's surface), although I do not know what the limiting factor(s) would be.

Another scaling law is Benford's Law, which is an empirical observation that the first digits of measurements of many kinds of phenomena are not random. In particular, the first digit is a '1' approximately 30% of the time; '2' 18% of the time, '3' 12% of the time and so on, with the probability descending as the number increases.

First       Probability of
digit        occurrence

1            0.30103
2            0.176091
3            0.124939
4            0.09691
5            0.0791812
6            0.0669468
7            0.0579919
8            0.0511525
9            0.0457575

So if you had a table of the lengths of every river in the world, for instance, you would find that approximately 30% of the first digits were '1'--rivers with lengths of 1 904 km, or 161, or 11 km would fall into this category.

Furthermore, it doesn't matter what units you used--if you had measured the river lengths inches, you would observe the same relationship. The reason for this is that if you were to double a number which begins with '1', you end up with a number which either begins with '2' or '3'. Hence, the probability that the first digit is either '2' or '3' must be the same as the probability of it being '1'. In the table above, we see this is the case.

It isn't only natural phenomena that are characterized by Benford's Law. It has also been used as a tool to identify fraud in forensic accounting.

The deposit-size data from Nevada seem to conform to Benford's Law.

And if I convert the deposit size from ounces to metric tonnes . . .

So although Zipf's Law doesn't describe Nevada gold deposits well (at least at present), Benford's Law does.

Saturday, January 7, 2012

Gold, part 2: Is there a maximum size for gold deposits?

In our last installment, I presented a graph showing the size distribution for global gold deposits of greater than one million ounces. In it I tried to estimate the slope of the relationship between the size of deposits and their ranking, in terms of size,  in the hopes that the slope had some predictive power for the deposits that are yet to be found.

Two suggested scaling laws for the size-distribution of gold deposits (global).

Once again, the interpretation of these graphs is the rank, (in size, less one) of any deposit is the abscissa, and size is the ordinate. The reason for subtracting one from the rank number is that the largest deposit shown on the graph is actually the second-largest deposit in the state--and there is one deposit larger.

In our last installment, we assumed that the blue line was the better representation of the scaling law for gold deposits. Today we explain why the yellow line may be the correct answer, and that it does not mean we can expect to find multi-billion ounce deposits of gold (at least nowhere near the Earth's surface).

- - - - - - - -

The Earth system consists of myriads of local interacting subsystems. Intuitively, we might not expect the overall effects of these to merge into a background of white noise, we find instead that highly ordered structure arises on a variety of scales ranging up to that of the globe.

A simple scaling law for the size-distribution of gold is an example of red noise (or pink noise, depending on the slope). The observed power-law is a characteristic of a system at a state of self-organized criticality (SOC), as is nicely outlined here. In essence, the scaling law we observe in the size-distribution of gold deposits due to self-organization in the geological processes which control the reservoir size of crustal fluids which contained the gold, and possibly also the fracturing process which preceded the emplacement of the gold in the rocks.

Today we look at the size-distribution of gold deposits in Nevada.

The above graph was plotted using the data from the Nevada Bureau of Mines and Geology review of its mineral industry for 2009. There were 191 (unambiguous) significant deposits of precious metals for which I have combined the most recent mineral resources (all categories) plus any pre-existing historical production. I only counted gold ounces--and freely acknowledge that some of the mines in the above chart were probably better described as copper or silver mines--and treated all categories (proven and probable reserves, measured and indicated resources, and inferred resources) equally. If you feel the methodology is flawed you are invited to use your own.

We can compare the current size-distribution of gold deposits to the size-distribution of gold deposits in the Carlin Trend in 1989 (Rendu and Guzman, 1991).

Remarkably, both sets of data appear to be described by a straight line of constant slope, at least between for deposits between about 100,000 ounces and 10 million ounces in size.

During Nevada's "maturation" as a gold province, the scaling law describing the size-distribution of gold deposits remained constant over two orders of magnitude in size. The slope of these lines is about 1.5, placing the scaling law exponent between pink noise and red noise.

When we look at the figure on the top of the page, the blue line has a slope < 1, whereas the yellow line has a slope of about 1.5. For this reason, I propose the yellow line to be a better representation of the scaling law for the global deposits. The reason I first leaned towards the blue line was due to insufficiency of observations.

For comparison, if I only looked at deposits in Nevada greater than 1 million ounces, I would not be as confident describing the size-distribution with the yellow line.

SOC theory would seem to tell us the entire distribution should be characterized by a power law. Why not gold deposits?

In nature, there are limits. Infinity is not an option. Earthquakes are recognized as SOC processes, yet they have a maximum size, as the capacity for earth materials to store and transmit strain is finite. Similarly, we would expect there to be an upper limit for the size of crustal reservoirs of gold-bearing fluids. The result is that the largest gold deposit we find is much less than we would predict on the basis of our observed power law.

This explanation does not explain why there also appears to be a deficit in small deposits. For this the reason is economic. Under the current reporting regime (NI 43-101), gold in the ground cannot be considered a "deposit" unless it is reasonable to expect it to be exploited profitably. The requirement for economic exploitability will exclude many small--well, since they are not deposits, let's call them "collections"--of gold. Additionally, many company geologists will ignore such collections as soon as it becomes clear they are unlikely to become a deposit.

So it's up to these guys! (sorry about the quality--this is a point-and-shoot photo scanned way back in the '90s). He's using a rubber cut-out from an inner tube as a pan. This site is a thrilling walk north of Asanta village, western Ghana, on land almost certainly on a concession held by Endeavour.


Hronsky, J. M. A., 2011. Self-organized critical systems and ore formation: The key to spatial targeting? SEG Newsletter, 84, 3p.

Nevada Bureau of Mines, 2010. The Nevada Mineral Industry 2009. Special Publication MI-2009. http://www.nbmg.unr.edu/dox/mi/09.pdf, accessed today.

Rendu, J. M. and Guzman, J., 1991. Study of the size distribution of the Carlin Trend gold deposits. Mining Engineering, 43: 139-140.

Sunday, January 1, 2012

Change in state for Arctic sea ice?

The recent Arctic Report Card concludes that  . . .
 the Arctic Ocean climate has reached a new state with characteristics different than those observed previously. The new ocean climate is characterized by less sea ice (both extent and thickness) and a warmer and fresher upper ocean than in 1979-2000. 
Well, this sort of thing just happens to be a specialty here. Let's look at the data to see what they mean.

The record of sea ice can be inferred here.

The National Snow and Ice Data Center has kindly drawn a regression line through the points.

We'll use the time delay method to reconstruct the phase space in two dimensions.

The choice of a two-year time delay is somewhat arbitrary, as one year also serves. The two reconstructions do not appear materially different.

We observe that the state tends to occupy a small area in phase space, centred at about 7 million sq-km (yellow) from the initiation of the data set (1981 in this projection) until about 2002, after which the system has evolved into a new area of phase space characterized by reduced ice cover. It doesn't yet trace out anything that looks stable in this new area, so I would not exclude the possibility that it is currently tracing out a transient excursion.

The extent of sea-ice cover in November has declined in a stepwise fashion since the early 1980s. Here we have tenatively labelled three possible areas of Lyapunov stability.

- - - - - - - - - - - - - - - -

We observe a decline in Arctic sea-ice cover over the past 30 years. Like other components of the climate system, it appears that sea-ice cover is prone to sudden changes in state. The direction of the change appears to be consistent with projections of the global warming hypothesis.

The caveat is that the records presented here are too short. We cannot be certain that the period of observations--in particular those from 1980 to 2000--were representative of "normal" climate. Consequently, the statement quoted at the beginning of this post seems premature.

The 1970s were at the end of a multi-decade period of global cooling. The Arctic sea-ice cover at the end of the 1970s may therefore have been unusually large, and the observed shift to reduced cover may be simply the natural variability in the system. We can't tell for sure because we would need to observe at least one more cycle of variability, and we do not have the records we need.

The only way to obtain longer records will be through some form of proxy measurement, either through micropaleontology (dinoflagellates seem to be a favourite), or concentration of aerosols in nearby glacial ice.