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, March 12, 2012

Constructing an epsilon machine--a worked example, part 1

Started feeling a little under the weather just as the madness of PDAC settled down. Really busy over the past few days.Fell asleep feverish and had this world-transforming idea of an entirely new kind of testable hypothesis. I was convinced that I had something about which papers--no, entire scientific volumes--would be published. Unfortunately when I awoke it was gone, and the parts that I remember (it seemed to involve measuring the girth of everyone who left a particular poster session at which I would present--that and a great deal of cotton stuffing) do not seem promising. But I'll work on it and let you know how it goes.

In the meantime I want to work in more detail on concepts that I have just glossed over--epsilon-machine (which may unfortunately be confused with machine-epsilon--I do wish these researchers would do a search before naming something) reconstruction from paleoclimatic data. I have earlier posted here and here on this topic, and have presented here at GAC a couple of years ago.

We will be working with the ocean condition probability density plots shown here, based on data presented in Raymo et al. (2004) and methodologies described in Gipp (2001) (also here).

Probability density plot. The probability density portrait was computed over the interval 1299 ka to 1029 ka. The two ellipses are preliminary interpretations based on the contours of probability density--I inferred limit cycles in addition to the single LSA labelled O4. The lower limit cycle I have called LC12, and the upper one is LC23.

Now let's add some of the reconstructed phase space portrait.

This is an enlargement of the above plot. The green curve is the reconstructed phase space over the interval 1119 ka to 1059 ka (so 60 000 years). At 1119 ka, the system is in the lower limit cycle--it then moves into the upper limit cycle where it stays until at least 1059 ka. The little yellow blobs represent direct observations (at 3 ky intervals). The green line is a curve fitted through the points, as suggested by Abarbanel (1997). We record this as a transition from LC12 to LC23. Over the entirety of the record--or over a distinct subset--we will see many transitions. These will be counted and enumerated to produce our epsilon machine.


Abarbanel, H. D. I., 1996. Analysis of Observed Chaotic Data, Springer-Verlag, New York.

Crutchfield, J. P., 1994. The calculi of emergence: Computation, dynamics, and induction. Physica D, v. 75, 11-54.

Gipp, M. R., 2001. Interpretation of climate dynamics from phase space portraits: Is the climate system strange or just different? Paleoceanography, v. 16, 335-351.

Raymo, M. E., Oppo, D. W., Flower, B. P., et al., 2004. Stability of North Atlantic water masses in face of pronounced climate variability during the Pleistocene. Paleoceanography, v. 19, 13p.
doi: 10.1029/2003PA000921.

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