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).

Tuesday, September 27, 2011

Recognizing change in complex systems: excursions vs. bifurcations

Continued from last time.

Once again, we have a fifteen-year plot of gold/copper ratio vs. silver/rough rice ratio. We are continuing our discussion of whether the event labelled C, which is still unfolding, is likely to be an excursion (which will then return to the region populated by most of the graph) or a bifurcation (which will lead us to a new area of Lyapunov stability somewhere new in phase space).

It appears to be at least a once-in-a-generation event. But how significant is it?

For this spike to represent a bifurcation as opposed to an excursion, what we would have to see the function settle in around a new area of phase space. Looking at the shape of the spike, the likely area for such a new area of stability will be centred near the "C", and could extend from the far right of the graph to the kink in the curve near (350, 1.5). For this to form with any degree of satisfaction is likely to take about two more years. If we don't see any sign of orbiting about the "C", then the most likely outcome is a return to the 15-year area of stability.

I found it interesting that on this graph, the silver spike of 1998 does not appear. It is lost in the middle of area of stability.

Is it possible that the late spike in silver is due to its comparison to a soft commodity, which perhaps haven't performed as well as metals? We can check this by changing our ratios slightly.

Same four commodities--but compared differently. This time we are looking at the silver($/oz) to copper ($/lb) ratio against gold to rough rice. There are three significant excursions, labelled A, B, and C.

Excursion A represents the spike in silver price in 1998 due to the Warren Buffet purchase. Excursion B is the rise in price in copper in 2006, which exceeded (in percentage terms) the gain in silver which occurred at the same time.

Excursion C has two phases. For the first six months or so, the excursion is dominated by increased gold prices (compared to rice), and for the last 13 months or so, the excursion records the outperformance of silver.

Once again, we won't be sure that this is a bifurcation as opposed to an excursion unless the phase space settles somewhere near or above the "C" for at least another two years.

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It is obvious gold has not behaved well lately. Some of the stocks have been hit as well. We haven't looked at Detour Gold Corp. for awhile.

The recent price action has mirrored that of gold, with a sharp rise from the end of July and a sharper decline in the past few trading days.

The two-dimensional reconstructed phase space of the last eight months of the price of DGC-T.

The phase space portrait occupies a Lyapunov-stable area at around $30 for about six months, and from the beginning of August, the system evolves towards the $37 area. The system completes an orbit in the $37 range before leaving the area. This behaviour suggests that there may be another LSA in the $37 range, but I would feel more comfortable with it after a few more orbits.

What happens next? The two most likely scenarios would appear to be a return to the $30 LSA, or a return to the incipient LSA in the $37 range. Being long Detour, I would be encouraged by a reinforcement of the $37 LSA. Unfortunately, this outcome is nearly impossible. The reason is in the nature of the construction of the graph. Recall that the phase space reconstruction is generated by the price at closing of a particular day (on the vertical axis) against the closing price four trading days earlier (horizontal axis).

The coordinates of the last point are ($37.60, $31.00). In four days, the value of the x-coordinate is going to be $31 (today's closing price). In three days, the value of the x-coordinate is going to be $31.17 (yesterday's closing price). I don't know what the closing price of Detour will be on those days--but if it is at about the current price, then the state will lie within LSA30. The only possible hope for the Detour price state to reach the orbit at $37 would be for the price to recover to about $37 within the next three trading days--by Friday. Even then the state would not lie within LSA37, but at least the trajectory would indicated that that would be the likely outcome.

In fact, if we don't see some reassuring action tomorrow--when x falls to $33.60--we will be right on the outer edge of LSA30.

There is another, less desirable outcome. The price may continue falling to the previous LSA near the $23 level. Doh!

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Time to liquidate more cocoa. Sadly, the good Ghanaian stuff is all gone. I am forced to take the South American stuff. But I can at least sweeten it a little with this.

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