Modification of probability density for stocks rising steeply in price

Today's study is based on an 18-month record of daily closing prices for Silver Wheaton Corp. (disclosure--long position)

The chart shows a pretty nice move up to the $45 range, and has since pulled back.

The double bumps near the end of the record provide enough information to define a time delay at the first minimum of the average mutual information of the time series against sequential lags. A reasonable approximation of this lag is sixteen trading days, and this lag is used in creating the phase space portrait below.

Share price phase space and turbulent flow

The turbulent eddies diffuse outward as share prices rise. The simplest reason is there is no inherent scale for price changes.

The graph above shows the daily change in closing price for Silver Wheaton, expressed as a percentage, over the past 18 months. There is no real trend, nor are the percentage moves larger or smaller when the price was higher. The effect of this is to produce a diffuse phase space portrait for the higher price area of the graph. Consequently, when the probability density is plotted, the areas in phase space representing higher prices will have lower probabilities than might otherwise be the case.

The simplest way to correct for this effect is to plot the reconstructed phase space on a logarithmic scale rather than a normal scale.

The bin size remains constant, which really means that the bins become larger as we move into the high-price
area of the reconstructed state space. The effect is to increase somewhat the probability density in the high-price area.

The only compelling attractors for price here are in the 17-$21 range. Like the Pelangio chart last time, this is a reflection of the amount of time spent at these price levels. The only cure is to stay at higher prices (SLW is near $35 at this writing).

Also recall that like most other technical approaches, resistance and support are due to psychological factors. For mining companies the drill can defeat technical analysis.

Probability density in phase space--Pelangio Exploration Inc.

Today we'll take a look at some charts for Pelangio Exploration Inc. (disclosure--have formerly done some contract work for this company, currently have a long position). They are currently active in Ghana at two sites, and a little less active in northern Ontario.

I remember back in December and January hoping this was forming a cup with handle. But the handle keeps going down. Will it bottom soon?

A two dimensional phase space projection (no trajectory, only the data) is presented below. As for the similar charts for Detour Gold, I have used a four-day time-delay.

The data are dense enough to create a probability density graph using a box $0.05 on a side.

Contouring the probability density shows us that from our present price in the 50-55c range, a further fall will bring the price under the influence of the massive attractor in the 15-20c range. There are attractors in the 35c and 45c areas, but these are weak.

If the price can hold or rise slightly, it will remain under the influence of the attractors in the 60c to 80c range.

Drill, drill!

BTW, this is not investment advice. DYODD.

IMF gets ransom note

After the recent arrest of IMF Head Dominique Strauss-Kahn, the following note showed up at IMF offices.

No word yet on whether the IMF will comply.

Gadaffi to face ICC

As related here, Gadaffi and some of his assistants are being charged with crimes against humanity by the International Criminal Court.

How long before we see similar charges against Ouattara? Or does having liberal, democratic values make genocide okay?

(Scaling) laws of life and death

Another blast from the past*

One of the enduring problems in historical geology is the relationship between speciation and extinction. Geological history is punctuated by episodes of mass extinction, when in response to tectonic, climatic, or even astronomic events, large numbers of species become extinct in a short period of time. What happens in the aftermath?

One idea is that with the Earth denuded of many lifeforms, there are a large number of ecological niches "up for grabs" by the first applicant. Natural selection may select for those organisms which have undergone morphological and/or behavioural changes that more efficiently exploit the opportunities of the vacant niche. The logical consequence is that speciation should increase rapidly after a mass extinction.

One problem with this idea is that natural selection is a mechanism for culling away the unfit--it is not a mechanism by which innovation occurs. Innovation has been linked to random mutation, although occasionally papers appear claiming that the mutations are not random. The Lamarckian idea of directed mutation in nature has been discredited.

We might say this is an existential problem, for just as one may contemplate one's own mortality, one may contemplate the mortality of the human race. Are there predictable laws of extinction, and are we governed by them?

In pondering this issue we are not helped by a belief in our own exceptionalism. So for the duration of this essay, let us consider merely the idea of extinction of any number of species, and only afterwards ponder its relevance for ourselves.

Extinction (for our purposes) represents termination of a species.

What can we say about the processes of speciation and extinction? Are they governed by the same dynamics?

Spectral power graphs for extinctions and originations of marine 

families over the last 500 million years. From Kirchner (2002).

The Fourier power spectra for rates of extinction and origination of fossil marine families provides insight into the dynamics of extinction and evolution. On the graph above, the extinction power spectrum is relatively constant, with undulations. The highest frequency undulation (the last wiggle on the left) is Raup and Sepkoski's (1984) "death star" peak--the extinction peak at 28 million year intervals attributed to the periodic approach of a neutron star to our solar system.

The graph for originations shows a steady decline with increasing frequency, so that at higher frequencies (shorter timescales) the rate of originations falls below the rate of extinction. Kirchner (2002) used this behaviour to infer that the rate of originations, especially at higher frequencies (periodicities less than 25 million years) was lower than the rates of extinctions; implying that there was a limit to the rate at which innovations could appear in the fossil record.

Although a correct observation, I believe the explanation for it is incorrect. The significance of the decline in rate of origination with frequency (note the nearly straight line of best fit on a log-scale) is that originations have a scale-invariant character, whereas extinctions (horizontal line of best fit) are random.

Over geologic time, rates of originations have to be at least as high as rates of extinction, or all life would be extinguished. We see that on long time intervals, rates of originations are higher than rates of extinction.

Scale invariance or random chance?

Scale invariance is a common characteristic of geological systems (Turcotte, 1997), and has been observed in such diverse phenomena as earthquakes, volcanic eruptions, and climate change. Such behaviour can be clearly demonstrated from the power spectra of some geological record. When projected on a log-log plot, the best fit of the power spectrum is a straight line of constant slope. The slope of this line is called the scaling exponent, and can be related to the fractal dimension which characterizes the size-time distribution of events.

Discrete scale invariance is a weaker form of invariance, where the scaling is not apparent at all frequencies, but only for a certain range of frequencies (Sornette, 1998). Discrete scale invariance is only described by a complex fractal dimension, the imaginary part of which is a simple function of the discrete scaling exponent (Saleur et al., 1996). Such behaviour is shown to exist in a wide variety of conditions, including diffusion, fracture propogation, fault rupture (in time), hydrodynamic cascades, turbulence, the Titus-Bode law and gravitational collapse of black holes (Sornette, 1998).

Thus, spectral power (P) varies as a function of frequency (f) so that for an arbitrary change f → λf there exists a number μ such that P(f) → μP(λf). This is a homogeneous function encountered in the theory of critical phenomena (Bak et al., 1988), and is solved by a power law P(f) = Af**(-α), where α = -log(μ)/log(λ). Power laws are the “fingerprint” of scale invariance, as the ratio of P(f):P(λf) is independent of f. Thus the relative spectral density is a function of the ratio of the two frequencies, and this property is the fundamental one which associates power laws to scale invariance, self-similarity, and self-organized criticality (Bak et al., 1988).

Fractal analyses of the scaling behaviour can be used to provide more information about the dynamics of speciation and diversification. The scaling exponent is the slope of the best fit line fitted to the log-log power spectrum. There is a correlation between the size-time distribution of events and the scaling exponent. If this slope is ~0, then the the distribution of events through time resembles white (i.e., random) noise.

Importantly, there is no trend observed in the power spectrum for extinctions. The implication to be drawn from this observation is that extinctions are randomly distributed, both in size and in time. They are not governed by processes at all like those that control diversification. The apparently random nature of extinctions is something of a mystery, as it would suggest that extinctions are not related to originations (new critters outcompeting the old). Nor does it fit with extinctions being related to large tectonic, climatic, or external events (bolides), all of which are believed to be systems at a state of self-organized criticality (Bak et al., 1988), and show a measured increase in power with decreasing frequency. Randomness in extinctions may imply that they are dominated by gambler's ruin.

By contrast, the power spectrum for the originations undulates about a line with a constant negative slope on the log-log graph. The slope of the eyeballed line of best fit (don't have access to the real data) is about -0.9 (i.e. a = 0.9). This is consistent with a system at a state of self-organized criticality.

Natural systems displaying self-organized criticality (SOC) are known throughout the geological realm. Tectonic and volcanic activity shows such a distribution, as does the distribution of large climatic disturbances. From the data analysis of Kirchner (2002), it is unclear whether the fingerprint of SOC arises from external influences or is an internal character of the evolutionary process.

From a geological perspective, it is natural to assume that SOC is imprinted on evolution by environmental processes. But scaling laws are observed at the level of proteins (Unger et al., 2003) as well as at the gene- and species level (Harrada et al., 2011), suggesting that SOC is inherent in life itself.

Bak, P., Tang, C., and Wiensenfeld, K., 1988. Self-organized criticality. Physical Review A, 38: 364-374.

Bonnet, E., Bour, O., Odling, N. E., Davy, P., Main, I., Cowie, P., and Berkowitz, B., 2001. Scaling of fracture systems in geological media. Reviews of Geophysics, 39: 347-383.

Erwin, D. H., 1998. The end and the beginning: recoveries from mass extinctions. Trends in Ecology and Evolution, 13: 344-349.

Herrada, E. A., et al., 2011. Scaling laws of protein family phylogenies. http://arxiv.org/PS_cache/arxiv/pdf/1102/1102.4540v2.pdf

Kirchner, J. W., 2002. Evolutionary speed limits inferred from the fossil record. Nature, 415: 65-68.

Kirchner, J. W. and Weil, A., 1997. No fractals in fossil extinction statistics. Nature, 395: 337-338.

Raup, D. M., and Sepkoski, J. Jr., 1984. Periodicity of extinctions in the geological past. Proceedings of the National Academy of Sciences 81 (3): 801–805.

Saleur, H., Sammis, C. G., and Sornette, D., 1996. Discrete scale invariance, complex fractal dimensions, and log-periodic fluctuations in seismicity. Journal of Geophysical Research, B101: 17,661-17,677.

Sepkoski, J. J. Jr., 1993. Ten years in the library: new data confirm paleontological patterns. Paleobiology, 19: 43-51.

Sornette, D., 1998. Discrete-scale invariance and complex dimensions. Physics Reports, 297: 239-270.

Unger, R., Ulierl, S., and Havlin, S., 2003. Scaling law in sizes of protein sequence families: from super-families to orphan genes. Proteins, 51 (4): 567-576. doi:10.1002/prot.10347.



* I wrote this article nearly ten years ago, which is why the references are so dated.Updated recently just for you.

Geopolitics and Economics: Cloaking evil in a mantle of scientific respectability

I have written before about Economics and how it differs from the hard sciences.

The hard sciences are predominantly inductive, premised on the repeated testing of hypotheses, which are discarded if they are disproven. The social sciences, especially Economics and Geopolitics are largely axiomatic--based on logical development of premises.

It doesn't have to be this way. It's just the way it is practiced.

The political geographers argue that there is a fundamental difference between, say, geopolitics and chemistry. Geopolitics, they argue, is not reducible to a test tube. You cannot rerun human history from the beginning while changing events (having the Axis win WWII, for instance) to see how things turn out. It is argued that as geopolitical theory has failed to advance our understanding of international relations, and even to generate testable hypotheses, it should be relegated to the periphery of political thought. Nevertheless, people like George Friedman continue to hype geopolitical theory as a guide for US foreign policy.

The notion of testable hypotheses is so fundamental to science that it is impossible to argue you are practicing science if you aren't using them.


Is it that testable hypotheses don't work for these type of sciences? Or are testable hypotheses difficult to formulate, and frustratingly enough, rarely have anything to do with any of the problems that need solving?

It's science. Move along.

Can testable hypotheses exist in the geopolitics?

I would argue that they can, and do. There are hard sciences which are essentially historical as well--fields like paleoclimatology, and evolutionary biology, which are predominantly much like the social sciences--you can't rewind history and run it again with some controlled changes. These sciences have been built upon a foundation of numerous testable hypotheses, admittedly some are a little limited and there is no question that there is a great deal of scientific frustration at our limited understanding of the Earth systems--however, progress does occur. It is true that grand hypotheses concerning the fundamental operation of the Earth's climate system are lacking--but there are plenty of testable hypotheses. Even so, we are constantly beset by unpredictable disasters.

In Economics, I would argue a similar situation exists. There are numerous testable hypotheses concerning the connections between variables in the economic system, but grand hypotheses about the overall workings of the economy are hard to formulate. This is no justification, however, for geologists have the same problem--nevertheless they proceed in a scientific fashion.

Is there no way to generate testable hypotheses in historical sciences?

I agree that generating testable hypotheses in a complex system is challenging.

Two of the prominent social sciences which purport to understand the way the world works are Geopolitics and Economics. These are of import due to their role in setting (or at least appearing to set) both the economic policies and the foreign policies of major governments.

Geopolitical theory is an attempt to explain the impact of physical geography on political systems.It is tied to the notion of economic and military strength being controlled by size of land mass, available natural resources, and means of approaching and traversing the territory. Economic theory is currently dominated by Keynesian policies, which are pitched by governments around the world as the only approach to operating a modern economy.

The "science" of Geopolitics, according to G. Friedman in his book, "The Next Hundred Years" stipulates that one of the world's great powers must arise in Eurasia because of the size of the landmass and its resources. This is purported to be the reasoning behind the "Great Game" and much of the history of meddling in Afghanistan. The internal logic of geopolitics has driven such acts as the invasion of Afghanistan by the British, Russians, and Americans; the invasion and destruction of Iraq; possible invasion or destruction of Iran; and carving up the Ottoman Empire into divided, squabbling countries. The old Soviet Union was driven to search for suitable locations for launching nuclear attacks against its enemies in North America; just as the United States has done for its enemies in China.

Fettweis (2006) writes:

Social scientists look skeptically upon research programs that use conjecture as the primary support for their conclusions. Evidence from neither present nor past state behavior could help demonstrate the validity of the future-oriented theories of geopolitics . . . [G]eopolitical scholarship has often been merely thinly disguised expressions of parochial national interests or strategic recommendations for individual states masquerading as science.

If we allow ourselves to be seduced by the idea that geopolitics is a purely explicative science, the explanations of which cannot be judged on moral grounds any more than could be the boiling point of water; then we do not allow ourselves to perceive that these actions are wholly evil, predicated as they are on the necessity of destroying lives and capital.

Similarly, the conclusions drawn by economic "science" are that we should keep the current economic system as is, regardless of the damage done to the real economy (involved with the making of real things). Keynesian policies have resulted in the creation of enormous amounts of credit, all to fund government programs which for political reasons could not be funded through taxation. Keynesian theory prescribes some desirable (non-zero) rate of inflation combined with a zero-interest-rate policy. The end result is the theft of purchasing power from the savers of society. In the absence of the economic "science", we would recognize this for a grand act of immorality--a crime committed against savers. But under the cover of economic "science", it becomes a necessary act, as free from moral judgement as the boiling temperature of water.

The immoral acts of government are presented as being scientifically necessary. Science, after all, is not governed by morality. However, there is a difference between economic or foreign policy and chemistry--economic policies are policies of choice. We can choose to raise interest rates. We can choose not to attack Iran. We can't choose the boiling temperature of water. Do not become confused over what is and is not a science, or you will awake to find Genghis Khan your Minister of Peace, and Kwame Sikani the governor of your central bank.

Reference

Fettweis, C. J., 2006. Examining the Chessboard: A Disappointing Century of International Political Geography. US Naval War College December 2006, 43 pp.

Gold is protection against depredations of the State

I return from Africa to be greeted by David Olive's latest hatchet job against gold in the Toronto Star.

Investment advisors have a tough time with gold. It doesn't pay performance fees. It doesn't pay you a little bonus when you offload a thousand shares into a client account. It just sits there looking back at you.

Olive argues that investing in gold is a drain on the economy.

MarketWatch contributor David Weidner put it best, I think, in this recent Swiftian passage:
“Gold makes a statement. It says, ‘I think the American people work as hard and are as resourceful as anyone on the planet,’ and then adds, ‘but for now, I’m just going to hide out in this bunker and eat Spam until someone else has the guts to create a new business with a great idea and gives me a job.’”
Weidner implies that gold investing is unpatriotic. I wouldn’t go that far. But as with taking possession of a Rembrandt at auction or scoring a $384,000 half-litre of Bordeaux, a goldbug is not doing his or her bit for fellow citizens.

In real terms, the cost of commodities has fallen relentlessly through time as a consequence of the progressive application of human ingenuity. Buying and holding gold bullion is, in effect, a bet against this process. So why do I (and other individuals who are similarly optimistic about human ingenuity) hold gold as opposed to, say, dollars?

It is simply because as ingenious as humans are, that ingenuity is failing before the economic influence of the State. Those of us who hold gold do not object in principle to investing in sound businesses, or even buying government bonds, provided that those bonds are denominated in a currency that doesn't lose purchasing power over time.

The real drag on the economy is the interference of government through taxation, inflation and artificially fixing interest rates so low that savers are forced into the great global casino when their preference would have been to keep it in the bank. In the bank, the money would be available for investment and capital formation. The experience of the majority in the great casino is heartache and loss of capital.

Even those of us who have been lucky enough to invest in the casino with gains that match the real (as opposed to the official) rate of inflation, lose a significant portion of those gains to taxation. Under the current system, the vast majority of us face nothing but loss of capital. Our only means to preserve that capital is to remove it from the system, which means gold, silver, productive land, preserved food, gasoline, and other similar investments too numerous to belabour here.

To say that we are "not doing our bit for our fellow citizens" is exactly the opposite of the truth. We are doing our bit for our fellow citizens by withdrawing our support for an inherently corrupt system in a way that directly damages that same system.

Keep your paper if it makes you happy, David. Perhaps you can sew it into a quilt to keep you warm at night.

Bifurcation in gold-silver ratio! Where are we headed?

Is the recent exciting rise in the price of silver telling us something? Let's investigate.

The World Complex presents a two-dimensional reconstructed phase space portrait (using the time delay method) for monthly average gold and silver prices from January 1996 to April 2011. The time delay is twelve months, meaning that this plot shows the gold/silver ratio plotted against the ratio one year earlier.

Silver pricing data comes from here. Gold price monthly data from the World Gold Council website.

Don't get lost in the details. The important thing isn't every loop and swirl in the trajectory, nor is it even the trajectory itself. It is the space. For almost the entire data set, the system has cycled within a large area, which may have been an area of Lyapunov stability. The only exceptions are the excursion to the lower right area of the graph (an unusually high ratio coupled with an unusually high rate of increase in the gold/silver ratio, which happened in late 2008 to early 2009--I'm sure we all remember that one); and two excursions to the far left of the graph (in which the ratio is low and had also declined rapidly in the previous year)--these being the "Warren Buffett event" of 1998, and our present excursion.

Notice that the Warren Buffet excursion was short-lived. Indeed, the sudden reversal of the trajectory looks a little unnatural. We eagerly await to see whether our current excursion will be similarly short-lived, and marked by a similar reversal. Unfortunately, although we are eager, the time taken for such an event to play out is months to years, so patience is the order of the day.

The next several months will be crucial. If there are truly problems in silver supplies, it would be logical to expect to see the trajectory of our dynamic system leap into a new area of phase space--probably one beyond the boundaries of the graph.

One idea currently circulating is that bifurcations are preceded by a period of extreme stability (what we might term low volatility). It is not clear if the period of low volatility is dynamically necessary (clearing the road, as it were), or is merely an empirical observation. The volatility of the gold/silver ratio over the past couple of years has been breathtaking, so if a bifurcation is occurring, it would provide a counterargument to the extreme stability hypothesis.

Odds and ends

The news cycle is slow here in Ghana, especially when travelling. We had a gruelling all-day trip yesterday in which the truck suffered some sort of significant, yet hidden injury, requiring us to dump in a couple of litres of oil   every 20 km or so. As we approached one of the few bright spots on the bleak road from Mankessim Circle to Winneba Circle, my adventurous driver decided he wanted to see if the oil on hand would be enough to reach Winneba. I convinced him otherwise, arguing that the expense of a few litres of oil was cheap compared to the costs of blowing up the engine on a long, deserted stretch of highway in the middle of the night.

So I make it back to the office and what do I find? The TV has been fixed in my absence! And what's this? Osama Bin Laden is dead?!? And what's this? Harper wins a majority government, while the Liberals retreat to Isengard to lick their wounds? Perhaps Saruman will be thrown from the tower.

My last observation was in driving through Buduburam, a small town about 40 km west of Accra. The UN High Commission on Refugees (UNHCR) has a very snazzy building that has just gone up in this town, in acute counterpoint to the grinding poverty around it. Pretty blue banners wave in the breeze. The building has something to do with the Ivoiran refugees in Ghana, have no doubt, but does look a little too nice for actually housing them. I suspect it is for storing bureaucrats.