4 ms·

Bad news for this model: it's not stationary, due to the 11-year solar cycle, and to other behaviors at longer time scales superimposed on the cycle. But if yo

by mturmon 12y ago

Bad news for this model: it's not stationary, due to the 11-year solar cycle, and to other behaviors at longer time scales superimposed on the cycle. But if you average out over at least the solar cycle, you could come up with a rate of superstorms per-cycle. There have been about 15 cycles since the 1859 superstorm, which would mean that perhaps you expect ~ 1 chance in 15 of such a storm in the coming cycle.

graycat 12y ago
Of course what I mentioned is also called the 'axiomatic' derivation of the Poisson process. In part it's nice because it starts with 'qualitative' assumptions and ends up with the full algebra of the process and its distribution and independence of times between arrivals. Details are in E. Cinlar, 'Introduction to Stochastic Processes' in his chapter on the Poisson process. There he also touches on the non-stationary case, and what he has may (I'm in hurry this morning) be equivalent to your "average out over ... the cycle". Since a finite sum of independent Poisson processes is a Poisson process, your 1 in 15 over the next cycle may be equivalent to 1 in 2014 - 1859 years over the next year. I do the same for asteroids: We have a good ballpark estimate of the rate of killer asteroids, and the rate is low enough to f'get about it.
- mturmon 12y ago
  Yes, you described a case ("homogeneous P.P.") where the intensity, lambda, is constant over time. In the inhomogeneous P.P., there is a time-varying rate, lambda(t). The derivation of the properties in the inhomogeneous case is almost the same because the key property is independent increments. When you add up the number of catastrophic events over an 11-year cycle, you get a sequence of "numbers of catastrophic CMEs" N1, N2, ..., Nk each of which is Poisson with intensity \int_{cycle} lambda(t) dt [*] If you ignore the larger-than-a-cycle variations in intensity, this integral is the same for each cycle. So the N's above are iid Poisson with some common parameter lambda_0, say. Observing the N's for many cycles would allow you to estimate lambda_0. That's pretty much where the 1 in 15 comes from (indeed, lambda_0 is so low that there are only 0 or 1 events per cycle, so "estimation" is trivial). The "easy" problem is that, even averaging over a cycle, the activity ([*] above) is not constant, e.g. this graph of sunspot numbers vs. time: http://blogs.nicholas.duke.edu/thegreengrok/files/2012/07/sunspotnumbers.jpg http://blogs.nicholas.duke.edu/thegreengrok/files/2012/07/su... The "hard" problem is that the counts are not, in real life, independent, even across weeks-to-months time scales. This would invalidate the Poisson model at the ground level (see the talk referenced by @gone35 above). There are reasons (e.g., Maunder minimum, see the plot above) to believe that the counts are also not independent at the decadal time scale. Lack of independence would be a huge problem for any Poisson model. You'd have to go up to a doubly stochastic process, for example, in which lambda(t) is itself a random process.
  - graycat 12y ago
    You are guilty of letting a lot of really messy data ruin some really nice theory! So, with assumptions of stationary and independent increments, the Maunder minimum shown in the graph would have probability less than zip and zilch of happening in which case we would reject the assumptions. You also are analyzing sun spots while the OP is talking about dangerous CMEs. It's been a while since I studied sun spots: You are saying that a sun spot is just a small case of a CME? Or is a dangerous CME not just a big sun spot but a different animal?
    - mturmon 12y ago
      CMEs are associated with large or magnetically-complex sunspots. Simple sunspots are bipolar, they have a N and a S region with field lines arching up connecting one to the other. Complex sunspots are multipolar, and the field lines therefore cross in weird ways (in the 3D solar plasma). They can change configurations suddenly, releasing a lot of energy as the magnetic field lines re-arrange to a lower-energy configuration. This magnetic reconnection is what drives CMEs (roughly speaking).

by mturmon 12y ago

graycat 12y ago

Of course what I mentioned is also called the 'axiomatic' derivation of the Poisson process. In part it's nice because it starts with 'qualitative' assumptions and ends up with the full algebra of the process and its distribution and independence of times between arrivals. Details are in E. Cinlar, 'Introduction to Stochastic Processes' in his chapter on the Poisson process. There he also touches on the non-stationary case, and what he has may (I'm in hurry this morning) be equivalent to your "average out over ... the cycle". Since a finite sum of independent Poisson processes is a Poisson process, your 1 in 15 over the next cycle may be equivalent to 1 in 2014 - 1859 years over the next year. I do the same for asteroids: We have a good ballpark estimate of the rate of killer asteroids, and the rate is low enough to f'get about it.

mturmon 12y ago
Yes, you described a case ("homogeneous P.P.") where the intensity, lambda, is constant over time. In the inhomogeneous P.P., there is a time-varying rate, lambda(t). The derivation of the properties in the inhomogeneous case is almost the same because the key property is independent increments. When you add up the number of catastrophic events over an 11-year cycle, you get a sequence of "numbers of catastrophic CMEs" N1, N2, ..., Nk each of which is Poisson with intensity \int_{cycle} lambda(t) dt [*] If you ignore the larger-than-a-cycle variations in intensity, this integral is the same for each cycle. So the N's above are iid Poisson with some common parameter lambda_0, say. Observing the N's for many cycles would allow you to estimate lambda_0. That's pretty much where the 1 in 15 comes from (indeed, lambda_0 is so low that there are only 0 or 1 events per cycle, so "estimation" is trivial). The "easy" problem is that, even averaging over a cycle, the activity ([*] above) is not constant, e.g. this graph of sunspot numbers vs. time: http://blogs.nicholas.duke.edu/thegreengrok/files/2012/07/sunspotnumbers.jpg http://blogs.nicholas.duke.edu/thegreengrok/files/2012/07/su... The "hard" problem is that the counts are not, in real life, independent, even across weeks-to-months time scales. This would invalidate the Poisson model at the ground level (see the talk referenced by @gone35 above). There are reasons (e.g., Maunder minimum, see the plot above) to believe that the counts are also not independent at the decadal time scale. Lack of independence would be a huge problem for any Poisson model. You'd have to go up to a doubly stochastic process, for example, in which lambda(t) is itself a random process.
- graycat 12y ago
  You are guilty of letting a lot of really messy data ruin some really nice theory! So, with assumptions of stationary and independent increments, the Maunder minimum shown in the graph would have probability less than zip and zilch of happening in which case we would reject the assumptions. You also are analyzing sun spots while the OP is talking about dangerous CMEs. It's been a while since I studied sun spots: You are saying that a sun spot is just a small case of a CME? Or is a dangerous CME not just a big sun spot but a different animal?
  - mturmon 12y ago
    CMEs are associated with large or magnetically-complex sunspots. Simple sunspots are bipolar, they have a N and a S region with field lines arching up connecting one to the other. Complex sunspots are multipolar, and the field lines therefore cross in weird ways (in the 3D solar plasma). They can change configurations suddenly, releasing a lot of energy as the magnetic field lines re-arrange to a lower-energy configuration. This magnetic reconnection is what drives CMEs (roughly speaking).