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.

References

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)

Recognizing change in complex systems part 3: Unemployment and real interest rates

This post is for those who still think that lower interest rates will lead to lower unemployment.

Information comes from Bureau of Labour Statistics and the Fred. Strangely, the historical data from the BLS does not match the data downloaded from the same site some months ago for previous posts on this topic--the differences are about 0.7% (i.e., the recent correction reduced the unemployment rate by 0.7% for December 2010).

The scatterplot of real interest rates (which is calculated by subtracting the official inflation rate calculated from CPI data for all urban consumers including all items--annualized and smoothed through a 3-pt MA--from the 3-month treasury yield) against unemployment rate shows two distinct areas of Lyapunov stability in phase space. These are separated by a brief (four month) excursion into relatively high real interest rates. The lower-unemployment region of phase space is occupied from January 2001 until August 2008.

Notice that there is no discernable correlation between unemployment rate and interest rate. I recognize that this observation based on possibly manipulated data sets is at odds with the axioms of Keynesian economics and therefore should not be discussed.

The system experienced a bifurcation in late 2008. When real interest rates fell in late 2008, unemployment unexpectedly rose and the system settled into a new area of stability, where it has remained since.

The policy of frantically lowering interest rates has failed to bring down unemployment because of a fundamental change within the economic system. Continuing to hold interest rates low will not undo the irreversible change that occurred in 2008. It might be a good thing to spend some effort on understanding the dynamics of the economic system rather than continuing with actions based on axioms that are clearly at odds with the actual universe.

I recognize that the idea of the economic system undergoing fundamental changes to its dynamics is at odds with the axioms of Keynesian economics and therefore should not be discussed.

Feedback is a common feature of dynamic systems. In certain dynamic systems, there are areas of phase space where the system is dominated by negative feedback. Perturbations to the system are resisted. If the perturbation is large enough, however, the system may enter a state wherein positive feedbacks are dominant, in which case the system evolves rapidly through phase space until it arrives in (usually) a new area of phase space, where once again negative feedbacks dominate and the system regains some form of stability. These areas of stability are sometimes described as attractors, but for reasons discussed previously, we prefer to describe them as areas of Lyapunov stability.

A similar change is observed in the plot of unemployment duration vs real interest rates, once again covering the period from 2001 to present. Notice that the average duration of unemployment actually shows no correlation with real interest rates.

Observing the change is easy (if we disregard Keynesian axioms). Deducing the nature of the change is more difficult.

One observation that leaps out at me is this. Real interest rates fell to an extreme low in August 2005, followed by an extreme high in October 2006. They fill to an extreme low in June 2008, and rose to an extreme high in November 2008. In the first case, there were no dire effects on unemployment. But the second time around, we got a bifurcation.

Is the answer here?

House prices were still rising in late 2005. They were falling in late 2008. Perhaps a fluctuation in interest rates when people believe they are becoming more wealthy is not harmful, but one that occurs during a time when our perception of wealth is falling led to a massive loss in confidence. Or at least a sudden realization that we couldn't afford all this debt.

If the change in economic dynamics is caused by a sudden negative perception of debt, then manipulating the interest rates downward will not and cannot bring us back to a paradise of low unemployment. Particularly if it is accompanied by declines in the Case-Schiller index and the stock market.

Irreversible damage to the economy from central bank interventions (example no. 92175)

The internet is a wonderful source of data. For instance, we can find estimates for M2 (a form of money supply) here.

I present for your edification or amusement charts of non-M1 M2 data since November 1980.

There are two charts--the upper chart simply shows the growth of non M1 M2 money in the US economy going back to November 1980.

Historical data exists prior to this date, but there was a change in the way the number was calculated--apparently new forms of money were included after November 1980 which were not counted before, making comparison before and after this date problematic.

The lower graph is a chart of the data, but with the exponential trend removed. The exact method I used after tabulating the data was to take the natural logarithm of the reported number, use a linear detrend on the logarithms, and then calculated e^(detrended ln) to obtain the logarithmic detrended data. The resultant graph shows the periods where monetary growth was faster than average over the past 30 years (upward sloping sections) and periods where monetary growth was slower than average over that period (downward sloping sections).

This is not to suggest that the average rate of monetary growth over the past 30 years is the correct one. To my knowledge, there is no "correct" rate of monetary growth (perhaps we could try 0?).

In the larger scheme we see faster than average monetary growth from '81 to '85, average monetary growth from '85 to '91, and slower growth (and even actual shrinkage) from '91 to '95 (the era of the "strong dollar policy"?), more rapid growth to about '03, followed by roughly average growth into the spike of '09, followed by slower than average growth (the data I have used ends in mid-January 2011).

Working from memory, there was a bit of a housing bubble collapse in the early '90s. There was something of a recession after '01 perhaps into '03. It's possible that there are connections between asset values and monetary growth. This will be investigated at a later date.

Now we will consider the impact of monetary growth on unemployment. One common argument of the Keynesians is that increasing the amounts of money in circulation is a requirement of maintaining acceptably low unemployment. Let us test this notion. We use the monthly unemployment figures from the BLS website which we have previously discussed here and here.

Below is a scatter plot showing official unemployment rate plotted against the detrended non-M1 M2 data discussed above. I have only unemployment data going back to January 2000, so here is unemployment vs money from Jan-00 to Feb-08 (before the amazing change in state).

I always supposed that more money was supposed to decrease unemployment. However this chart shows just the opposite. More money = more unemployment. So--who really benefits from money creation?

But the story gets better. Let's add the part where the unemployment data drop into the black hole (up to Dec 2010).

Yow! So now reversing the money growth machine doesn't reverse the rise in unemployment! Talk about a Keynesian nightmare!

This type of behaviour in a natural system would be described as "irreversible". Climate scientists frequently fret about the possiblity of irreversible changes in climate due to human or natural activities. What we observe above may be an actual observation of an irreversible change in a dynamic system brought about by human activity. How exciting! Bernanke et al. are making history! If any of the eight or so of you who read this know him, why not drop him a line to tell him how he's doing?

In the natural world, the behaviour is not completely irreversible, because if the driving force in reversed long enough, eventually the system does return to its previous state. But the system usually displays hysteresis, meaning the way back to the previous state may be long and arduous. I fear the same may be true for the unemployed in America.