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).

Wednesday, October 19, 2011

Inference of dynamics for complex systems, part 1

Today I will start over with the analysis of dynamic systems, describing a methodology and some of the rationale behind the interpretations from previous postings, as it occurs to me that all of this stuff, though discussed before, is buried in the archives and is not easy to pull together.

This will also be good for me as I have to put together some kind of paper on the topic for one or more conferences in the first half of next year. GAC, in St. John's next year, will be a given as it is my old alma mater, but I am giving thought to presenting at the upcoming 3rd Multiconference on Complexity, Informatics and Cybernetics.

You are studying an interesting system, with many components. You know that many of the components interact, but you don't know the details of their interaction. If the interactions vary with changing conditions  within the system (feedback) it may be described as a complex adaptive system. Examples of such systems include, but are not limited to, ecosystems and other biological systems, the stock market and other economic systems, the climate system, and some would argue, the entire earth system.

The behaviour of such systems is typically nonlinear, and typically characterized by self-organization and emergent phenomena. The presence of negative feedback gives the system a form of resilience, allowing it to resist perturbations; and the presence of positive feedback causes the system to experience episodes of rapid change, usually resulting in a shift from one equilibrium condition to another. Multistability (the presence of more than one equilibrium condition) is a common feature of such systems.

The system has input signals, which may be time-dependent, however it may be that you are only able to observe some of these signals; furthermore there may be input signals of which you are unaware. There are output signals, which you observe, and compile into one or more time series; however there is no way to know if your output signal is important in terms of developing a global understanding of the system of interest.

There are conditions within the system which influence the manner in which the input signals feed through to the output signals. You may have an inkling of some of these rules (commonly expressed as differential equations) but normally your understanding of these rules is incomplete. You hope to understand your system by deducing these equations on the basis of your observations.

Here are some examples of systems we may wish to study.

Daily closing prices for Detour Gold Corp. (DGC-T), from late November 2009 to October 2011.

Gold-silver ratio.

Case-Shiller index. Data from Robert Shiller data page.

Unemployment rate (from US BLS site).

Trading activity in Century Casinos, June 21, 2011. From Nanex.

Paleoclimate proxy records over the past two million years. Magnetic susceptibility of loess (proxy for Himalayan monsoon strength) at top. Deep water 18-O record (proxy for global glacial ice volume) at bottom.

At first glance, the problem seems insurmountable. How do you study a system when you can't even be sure that your observations are meaningful? What if you have failed to observe the most important observable parameters?

It is especially bad for the geological time series, for in addition to the above problem, there are both errors in measurement and errors in the date (or time) of each observation.

In future installments, we will work through the data sets shown above; but we will start with some thoughts on equilibrium.

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