Tuesday, November 19, 2013

State Diagrams

  For the global temperature anomaly the correlation between one month and those following it disappears rapidly and aside from a slight annual effect is essentially nonexistent after a couple of months. The drift function, the changes from the mean value, appears to be random and memoryless so it can be represented by a Markov process. To simplify the analysis we had to divide up the range of the anomaly change into a number smaller sections and use frequencies instead of probabilities. No change in the anomaly is the most probable outcome but finite positive and negative changes are also possible with the distribution being symmetric about the mean. So to understand changes in the anomaly we need to know what to expect for the drift, the change in the mean, due to this Markov process in order to distinguish it from a steady and persistent change that we have to associate with long-term global warming.

  A Markov process can be represented by a transition matrix giving the probability of the various changes from one state to another which in turn can be represented by a state diagram. A simple example of a state space is the permutation of three objects which can be represented by the triple (a,b,c) which in turn can be represented by the Euclidean coordinates of a point such as (0,1,2). For three objects we can define three elementary permutations consisting of the exchange of just two of the objects. Let R represent the exchange of the first two objects, G that of the second two, and B that of the first and last. If we start with (0,1,2) and apply R we get (1,0,2), G gives (0,2,1) and B (2,1,0). Applying R, G and B again to these new states we find that each elementary permutation is its inverse, taking us back to (0,1,2) and applying G and B to (1,0,2) gives (1,2,0) and (2,0,1) respectively. So the 3 elementary exchanges give a total of 6=1+3+2 permutations. We can label the states by the order in which we found them calling them 0 through 5. A state diagram makes it easier to keep track of all the changes that can take place when we apply R, G, and B to these states.

In the state diagram above the elementary exchanges R, G, and B are represented by the colors red, green, and blue.

  One can trick Mathcad 11's 3D plotter into drawing the figure above using sets of matrices containing the data for the points and lines. It took ten separate plots altogether and the labeling numbers had to be added later with MS Paint. A "next generation" plotter would only need 4 plots, one for the points and one for each set of lines.

Moving along the axes the required steps one can see that the point at the top of the figure is (0,1,2). The set of points that represent the states turn out to be in the same plane.

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