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Stochastic resonance

Created Stochastic resonance, for an example when SDE are useful for model building. Uploading images is easy! But there is a 100kB bound, that's a little bit low, I had to compress one of the three graphics I uploaded...

Comments

  • 1.
    edited November 2010

    Looks good. I don't understand what you intend by the 'signal'. A*sin(t), or the whole of V(x,t) or something else? What would one be trying to infer from observing a system like that? What would be assumed known?

    Comment Source:Looks good. I don't understand what you intend by the 'signal'. A*sin(t), or the whole of V(x,t) or something else? What would one be trying to infer from observing a system like that? What would be assumed known?
  • 2.

    Note that Instiki(?) also accepts png images, which will very probably have smaller sizes than jpegs for graphs and other diagram type illustrations.

    [Sharp lines on wide expanses of solid colour are one of the worst cases for a scheme like jpg which uses DFTs (well, DCTs actually) on blocks. PNG has compression mechanisms that adapt to this better. Conversely, jpeg is pretty much the only option for natural images such as phtographs.]

    Comment Source:Note that Instiki(?) also accepts png images, which will very probably have smaller sizes than jpegs for graphs and other diagram type illustrations. [Sharp lines on wide expanses of solid colour are one of the worst cases for a scheme like jpg which uses DFTs (well, DCTs actually) on blocks. PNG has compression mechanisms that adapt to this better. Conversely, jpeg is pretty much the only option for natural images such as phtographs.]
  • 3.

    Great page, Tim! If it's okay with you, I'll blog about this one someday.

    Like Graham I found the first sentence a bit mysterious, since you start by mentioning a physical system responding to noise, and then mention "signal to noise ratio", which is not a concept I understand for a general system - only something like a communications system.

    Comment Source:Great page, Tim! If it's okay with you, I'll blog about this one someday. Like Graham I found the first sentence a bit mysterious, since you start by mentioning a physical system responding to noise, and then mention "signal to noise ratio", which is not a concept I understand for a general system - only something like a communications system.
  • 4.
    edited November 2010

    David said:

    Sharp lines on wide expanses of solid colour are one of the worst cases for a scheme like jpg which uses DFTs (well, DCTs actually) on blocks. PNG has compression mechanisms that adapt to this better.

    I know - I must have done something wrong, because the PNG I created were bigger than the JPEG, which is not plausible. JPEG is an application of wavelets I wanted to mention on that page :-)

    John wrote:

    If it's okay with you, I'll blog about this one someday.

    Sure, but I think the really interesting point is that Earth has several different stable climate states, and that there is evidence for the "snowball earth" scenario. I read this in the primer (the primer on climate modelling).

    It's the central idea of the movie "The Day After Tomorrow", but I don't expect that all members of the audience got that message :-)

    Graham wrote:

    I don't understand what you intend by the 'signal'.

    The problem is that I know about stochastic resonance from signal analysis, and therefore did not notice that I should explain the context :-)

    I'll try to write some more about this, answering your questions, I hope...

    Comment Source:David said: <blockquote> <p> Sharp lines on wide expanses of solid colour are one of the worst cases for a scheme like jpg which uses DFTs (well, DCTs actually) on blocks. PNG has compression mechanisms that adapt to this better. </p> </blockquote> I know - I must have done something wrong, because the PNG I created were bigger than the JPEG, which is not plausible. JPEG is an application of [[wavelets]] I wanted to mention on that page :-) John wrote: <blockquote> <p> If it's okay with you, I'll blog about this one someday. </p> </blockquote> Sure, but I think the <i>really</i> interesting point is that Earth has several different stable climate states, and that there is evidence for the "snowball earth" scenario. I read this in the primer (the primer on climate modelling). It's the central idea of the movie "The Day After Tomorrow", but I don't expect that all members of the audience got that message :-) Graham wrote: <blockquote> <p> I don't understand what you intend by the 'signal'. </p> </blockquote> The problem is that I know about stochastic resonance from signal analysis, and therefore did not notice that I should explain the context :-) I'll try to write some more about this, answering your questions, I hope...
  • 5.
    edited November 2010

    A really nice simple model of Snowball Earth was mentioned in week301:

    NU: Yes, we do have to be careful to remember that the formula above is obtained from a linear feedback analysis. For a discussion of climate sensitivity in a nonlinear analysis to second order, see:

    JB: Hmm, there's some nice catastrophe theory in there — I see a fold catastrophe in Figure 5, which gives a "tipping point".

    The model has ice albedo feedback, and the authors write:

    A remarkable property of the energy balance model governed by Eq. (11) is the existence of several stationary solutions that describe equilibrium climates of the Earth [...]. The existence and linear stability of these solutions result from a straightforward bifurcation analysis of the 0-dimensional energy balance model (11) [...]; there are two linearly stable solutions – one that corresponds to the present climate and one that corresponds to a much colder, “snowball Earth” [...] separated by an unstable one, which lies about 10K below the present climate.

    Are you thinking of using a model like this coupled to noise, Tim? Something like that would be fun.

    Comment Source:A really nice simple model of Snowball Earth was mentioned in [week301](http://math.ucr.edu/home/baez/week301.html): > **NU**: Yes, we do have to be careful to remember that the formula above is obtained from a linear feedback analysis. For a discussion of climate sensitivity in a nonlinear analysis to second order, see: > * I. Zaliapin and M. Ghil, [Another look at climate sensitivity](http://www.nonlin-processes-geophys.net/17/113/2010/npg-17-113-2010.pdf), _[Nonlinear Processes in Geophysics](http://www.nonlin-processes-geophys.net/17/113/2010/npg-17-113-2010.pdf)_ **16** (2010), 113-122. > **JB**: Hmm, there's some nice catastrophe theory in there &mdash; I see a fold catastrophe in Figure 5, which gives a "tipping point". The model has ice albedo feedback, and the authors write: >A remarkable property of the energy balance model governed by Eq. (11) is the existence of several stationary solutions that describe equilibrium climates of the Earth [...]. The existence and linear stability of these solutions result from a straightforward bifurcation analysis of the 0-dimensional energy balance model (11) [...]; there are two linearly stable solutions – one that corresponds to the present climate and one that corresponds to a much colder, “snowball Earth” [...] separated by an unstable one, which lies about 10K below the present climate. Are you thinking of using a model like this coupled to noise, Tim? Something like that would be fun.
  • 6.

    Are you thinking of using a model like this coupled to noise, Tim?

    I'm not thinking that far ahead :-)

    The next step I plan is to implement the simple energy balance model of the primer, because the implementations shipped with the CD are either in BASIC (don't have a compiler for this) or in Java and broken, because the model is mixed with the client code which uses a graphic library that is obsolete, not included, and which I did not find on the internet anymore (a classic blunder in software engineering :-)

    Regarding snowball earth: I think a lot of criticism of the "global warming agenda" stems from the overwhelming experience that

    a) the weather can be pretty rough, but

    b) after a few days everything is back to normal, no matter how crazy the hurricanes, thunderstorms etc. broke havoc the day before and

    c) there have always been people trying to exploit the long living superstitions that the gods punish humans through weather catastrophes.

    Scientific evidence that the climate on Earth is capable to change drastically, for a long time, in a way that no longer sustains the biosphere as we know it, is big news. Maybe it has been out there for centuries. But I doubt it has reached the minds and hearts of many.

    Human activity has global impacts, impacts that may change the world as we know it. We have to acknowledge this fact, and we have to learn how to act accordingly.

    Comment Source:<blockquote> <p> Are you thinking of using a model like this coupled to noise, Tim? </p> </blockquote> I'm not thinking that far ahead :-) The next step I plan is to implement the simple energy balance model of the primer, because the implementations shipped with the CD are either in BASIC (don't have a compiler for this) or in Java and broken, because the model is mixed with the client code which uses a graphic library that is obsolete, not included, and which I did not find on the internet anymore (a classic blunder in software engineering :-) Regarding snowball earth: I think a lot of criticism of the "global warming agenda" stems from the overwhelming experience that a) the weather can be pretty rough, but b) after a few days everything is back to normal, no matter how crazy the hurricanes, thunderstorms etc. broke havoc the day before and c) there have always been people trying to exploit the long living superstitions that the gods punish humans through weather catastrophes. Scientific evidence that the climate on Earth is capable to change drastically, for a long time, in a way that no longer sustains the biosphere as we know it, is big news. Maybe it has been out there for centuries. But I doubt it has reached the minds and hearts of many. Human activity has global impacts, impacts that may change the world as we know it. We have to acknowledge this fact, and we have to learn how to act accordingly.
  • 7.
    edited November 2010

    Tim wrote:

    The next step I plan is to implement the simple energy balance model of the primer, because the implementations shipped with the CD are either in BASIC (don't have a compiler for this) or in Java and broken, because the model is mixed with the client code which uses a graphic library that is obsolete, not included, and which I did not find on the internet anymore (a classic blunder in software engineering :-)

    I'm really happy that you'll correct these defects! I hope you'll make the result easily available via the Azimuth Project.

    I don't know which energy balance model you're talking about... is it a zero-dimensional model? I hope you include the equations on the Azimuth Project, perhaps in the page Energy balance model, which I am about to create. The paper I referred to above has a simple zero-dimensional energy balance model that's probably a bit different, since it's designed to illustrate bistability and tipping points. I'll include the equations in Energy balance model.

    Comment Source:Tim wrote: >The next step I plan is to implement the simple energy balance model of the primer, because the implementations shipped with the CD are either in BASIC (don't have a compiler for this) or in Java and broken, because the model is mixed with the client code which uses a graphic library that is obsolete, not included, and which I did not find on the internet anymore (a classic blunder in software engineering :-) <img src = "http://math.ucr.edu/home/baez/emoticons/yuck.gif" alt = ""/> I'm really happy that you'll correct these defects! I hope you'll make the result easily available via the Azimuth Project. I don't know which energy balance model you're talking about... is it a zero-dimensional model? I hope you include the equations on the Azimuth Project, perhaps in the page [[Energy balance model]], which I am about to create. The paper I referred to above has a simple zero-dimensional energy balance model that's probably a bit different, since it's designed to illustrate bistability and tipping points. I'll include the equations in [[Energy balance model]].
  • 8.

    John wrote:

    I hope you'll make the result easily available via the Azimuth Project.

    Sure, I just uploaded the Java code I used for the graphics on the stochastic resonance page.

    To mix the model code with the presentation code is a classic blunder, because it violates a principle called separation of concerns. This principle is older than I am! (At least I think so, I don't have the original source, which would be a paper by Edsger Dijkstra, I think, beginning 1970ties or late 1960ties).

    I don't know which energy balance model you're talking about... is it a zero-dimensional model? I hope you include the equations on the Azimuth Project, perhaps in the page Energy balance model, which I am about to create.

    Sure, my plan is to explain EBMs on that page, explain one model in particular and provide an implementation of it. In the primer there is both a 0-dim model and a 1-dim model, the dimension being the latitude. Both should be pretty easy to do.

    Question: When I say "here is an executable jar", is it completely gibberish to you or do you know what to do with such a thing?

    Comment Source:John wrote: <blockquote> <p> I hope you'll make the result easily available via the Azimuth Project. </p> </blockquote> Sure, I just uploaded the Java code I used for the graphics on the stochastic resonance page. To mix the model code with the presentation code is a classic blunder, because it violates a principle called <a href="http://en.wikipedia.org/wiki/Separation_of_concerns">separation of concerns</a>. This principle is older than I am! (At least I think so, I don't have the original source, which would be a paper by Edsger Dijkstra, I think, beginning 1970ties or late 1960ties). <blockquote> <p> I don't know which energy balance model you're talking about... is it a zero-dimensional model? I hope you include the equations on the Azimuth Project, perhaps in the page Energy balance model, which I am about to create. </p> </blockquote> Sure, my plan is to explain EBMs on that page, explain one model in particular and provide an implementation of it. In the primer there is both a 0-dim model and a 1-dim model, the dimension being the latitude. Both should be pretty easy to do. Question: When I say "here is an executable jar", is it completely gibberish to you or do you know what to do with such a thing?
  • 9.

    Question: When I say "here is an executable jar", is it completely gibberish to you or do you know what to do with such a thing?

    It's complete gibberish, although "executable" lets me know that it's computer jargon. I don't know what "jar" means in this context. I bet I could find out in less than 2 minutes. But it's nice to write so that even lazy people can understand you.

    Comment Source:>Question: When I say "here is an executable jar", is it completely gibberish to you or do you know what to do with such a thing? It's complete gibberish, although "executable" lets me know that it's computer jargon. I don't know what "jar" means in this context. I bet I could find out in less than 2 minutes. But it's nice to write so that even lazy people can understand you.
  • 10.
    edited January 2011

    I added a bit about the Duffing oscillator to the References section of Stochastic resonance.

    There are three pictures on this page. The second one, illustrating "a very high value of D", is different from the other two: it's smaller, and the vertical axis is not labelled "position".

    Tim — or anyone else &mdash when you have time, could you please prepare a version of the second picture that's just like the other two? I know you already tried this once, and we put this picture on my website because it was too big to upload to the Azimuth Project.

    I'm sorry to be such a perfectionist about the visual appearance, but I am: it doesn't matter much now, but when I write a This Week's Finds, I always want to minimize irrelevant differences between items that I'm comparing, so readers can focus on the essential differences.

    If it's easier to tell me what to do, that's fine too! I'm afraid it probably won't be.

    Comment Source:I added a bit about the Duffing oscillator to the References section of [[Stochastic resonance]]. There are three pictures on this page. The second one, illustrating "a very high value of D", is different from the other two: it's smaller, and the vertical axis is not labelled "position". Tim &mdash; or anyone else &mdash when you have time, could you please prepare a version of the second picture that's just like the other two? I know you already tried this once, and we put this picture on my website because it was too big to upload to the Azimuth Project. I'm sorry to be such a perfectionist about the visual appearance, but I am: it doesn't matter much now, but when I write a This Week's Finds, I always want to minimize irrelevant differences between items that I'm comparing, so readers can focus on the essential differences. If it's easier to tell me what to do, that's fine too! I'm afraid it probably won't be.
  • 11.

    I'll see to it. Do you plan to write about stochastic resonance in TWF? We could try to produce some more, and nicer, pictures :-)

    Comment Source:I'll see to it. Do you plan to write about stochastic resonance in TWF? We could try to produce some more, and nicer, pictures :-)
  • 12.
    edited January 2011

    I do plan to write about stochastic resonance in TWF. I plan to write

    1) a week about Snowball Earth,

    2) a week about the Milankovitch cycles, and then

    3) a more mathematical week about stochastic resonance.

    The first one can remind our readers about energy balance models, and introduce the idea of bistability. The second can talk about glacial cycles, how mysterious they are, and the Milankovitch cycle. The third can discuss stochastic resonance - the theory, and the idea of using it to explain glacial cycles.

    A lot of this material is on the wiki and just needs to be written in a more fun-to-read style.

    I'm thinking of doing all this after a series of interviews with Eliezer Yudkowsky, which are already done and just need to be formatted. So there is time for us to make lots of pretty pictures beforehand, if you like.

    Comment Source:I do plan to write about stochastic resonance in TWF. I plan to write 1) a week about [[Snowball Earth]], 2) a week about the [[Milankovitch cycles]], and then 3) a more mathematical week about [[stochastic resonance]]. The first one can remind our readers about energy balance models, and introduce the idea of bistability. The second can talk about glacial cycles, how mysterious they are, and the Milankovitch cycle. The third can discuss stochastic resonance - the theory, and the idea of using it to explain glacial cycles. A lot of this material is on the wiki and just needs to be written in a more fun-to-read style. I'm thinking of doing all this _after_ a series of interviews with Eliezer Yudkowsky, which are already done and just need to be formatted. So there is time for us to make lots of pretty pictures beforehand, if you like.
  • 13.

    I made some new graphs for this page.

    Comment Source:I made some new graphs for this page.
  • 14.

    wasnt there somewhere an interactive version of an example of stochastic resonance ?

    I haven't found the link to that on that page and neither on the Azimuth Code (which refers back to the Azimuth project page)

    Comment Source:wasnt there somewhere an interactive version of an example of stochastic resonance ? I haven't found the link to that on that page and neither on the <a href="http://code.google.com/p/azimuthproject/wiki/SolvedChallenges">Azimuth Code </a> (which refers back to the Azimuth project page)
  • 15.
    edited January 2013

    I remember an interactive version too, but don't know where it is either.

    Comment Source:I remember an interactive version too, but don't know where it is either.
  • 16.

    The link was on the page Blog - increasing the signal-to-noise ratio with more noise. I added it to stochastic resonance under the heading "Interactive Online Example".

    Comment Source:The link was on the page [[Blog - increasing the signal-to-noise ratio with more noise]]. I added it to [[stochastic resonance]] under the heading "Interactive Online Example".
  • 17.
    edited January 2013

    In addition to the interactive example you guys are talking about, namely this:

    there's this, which is explicitly intended to model the climate:

    I need to update this second one and add more text. Thanks for reminding me!

    I have added a link to this model to the page

    Comment Source:In addition to the interactive example you guys are talking about, namely this: * [Stochastic resonance example](http://www.adgie.f9.co.uk/azimuth/stochastic-resonance/Javascript/StochasticResonanceEuler.html). there's this, which is explicitly intended to model the climate: * Michael Knap, [A stochastic energy balance model](http://math.ucr.edu/home/baez/knap/). I need to update this second one and add more text. Thanks for reminding me! I have added a link to this model to the page * [[Stochastic resonance]].
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