In earlier essays I discussed creating plots of the probability density of a 2-d reconstructed phase space portrait as a means for investigating the dynamics of complex systems. For examples I used proxy records representing global ice volume, Himalayan paleomonsoon strength, and oceanographic conditions.

Today's discussion centres around using entropy (in an information theoretic sense) as a tool for deciding whether the window selected for calculating the probability density is sufficient.

We've seen this before. The probability density plot of the trajectory of the 2-d phase space portrait for one 270-ky window of the paleomonsoon strength proxy data suggests that the phase space is characterized by five areas of high probability--five areas where the trajectory of the phase space tends to be confined in small areas for episodes of time before rapidly moving towards another such area. In earlier articles we have argued that these represent areas of Lyapunov stability (LSA).

I used a 270-ky window for all the plots. Why this number? Why any number?

Selection of the width of the window is important, and at the time I was creating these plots, I did not have a technique in place for prescribing this width. If the window is wide, resolving power is lower. If the window is narrow, resolving power is higher--consequently one would think to choose a narrow (i.e. short length of time) window. But if the window is too short, then the observed trajectory over the window is not representative of the phase space of the function.

If we constructed probability-density plots from the two trajectories in the diagram above, the plots would not be very interesting. It would be even worse if the windows were shorter--say 1 ky, for instance. Then each successive probability density plot would be a blob translated some short distance in phase space, tracing out the trajectory of the entire function, alternating between being stretched a little and "snapping back".

Using entropy to prescribe window length

Entropy will be inversely related to the size of the region of phase space that encompasses the trajectory within the window of time under investigation. If the phase space travels from one LSA to another, then the phase space is characterized by alternating episodes of confinement to a small area punctuated by sweeping trajectories as the system moves to a new state of metastability. A graph of the entropy of successive windows of probability data would appear very noisy, as the entropy varies from very low values for windows dominated by confinement to one LSA, to much higher values during the shifts from one LSA to another.

As the window is lengthened, the noise level of the graph of successive values of entropy declines.

The plot here seems to suggest that I can get by using a window of only 150 ky, rather than the 270 ky that I actually used.

The story for the trajectories in the Late Quaternary is the same--the minimum window width looks to be about 150 ky.

You'll have to excuse me. I have a lot of redrafting to do.

A more formal treatment of this method is given here.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Note: "ky" means thousand years, and refers to duration

"ka" means thousand years ago and refers to a specific time

Today's discussion centres around using entropy (in an information theoretic sense) as a tool for deciding whether the window selected for calculating the probability density is sufficient.

We've seen this before. The probability density plot of the trajectory of the 2-d phase space portrait for one 270-ky window of the paleomonsoon strength proxy data suggests that the phase space is characterized by five areas of high probability--five areas where the trajectory of the phase space tends to be confined in small areas for episodes of time before rapidly moving towards another such area. In earlier articles we have argued that these represent areas of Lyapunov stability (LSA).

I used a 270-ky window for all the plots. Why this number? Why any number?

Selection of the width of the window is important, and at the time I was creating these plots, I did not have a technique in place for prescribing this width. If the window is wide, resolving power is lower. If the window is narrow, resolving power is higher--consequently one would think to choose a narrow (i.e. short length of time) window. But if the window is too short, then the observed trajectory over the window is not representative of the phase space of the function.

*The trajectories from two neighbouring 30-ky windows in phase space.*

If we constructed probability-density plots from the two trajectories in the diagram above, the plots would not be very interesting. It would be even worse if the windows were shorter--say 1 ky, for instance. Then each successive probability density plot would be a blob translated some short distance in phase space, tracing out the trajectory of the entire function, alternating between being stretched a little and "snapping back".

Using entropy to prescribe window length

Entropy will be inversely related to the size of the region of phase space that encompasses the trajectory within the window of time under investigation. If the phase space travels from one LSA to another, then the phase space is characterized by alternating episodes of confinement to a small area punctuated by sweeping trajectories as the system moves to a new state of metastability. A graph of the entropy of successive windows of probability data would appear very noisy, as the entropy varies from very low values for windows dominated by confinement to one LSA, to much higher values during the shifts from one LSA to another.

As the window is lengthened, the noise level of the graph of successive values of entropy declines.

The diagram above is a series of graphs of successive values of entropy calculated over overlapping windows of length 30 ky (at top), 60 ky, 90 ky, 120 ky, and 150 ky (at bottom) using the calculated probability density of the 2-dimensional phase space reconstructed from the time series of the paleomonsoon strength proxy, during the early Quaternary. As the window lengthens, we see the plot become smoother until it reflects primarily secular changes rather than accidents of local trajectories.

The plot here seems to suggest that I can get by using a window of only 150 ky, rather than the 270 ky that I actually used.

You'll have to excuse me. I have a lot of redrafting to do.

A more formal treatment of this method is given here.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Note: "ky" means thousand years, and refers to duration

"ka" means thousand years ago and refers to a specific time

## No comments:

## Post a Comment