This is awesome! I am so stocked about how they have found a 15D linear system with integer sparse off-diagonal matrix approximating Lorenz attractor with such an accuracy. It is a pure joy contemplating this mathematical elegance
I can't believe how much I have learnt in just the first 20 minutes. I have written down research ideas left and right. You are truely amazing, thank you so much for sharing.
This is truly fascinating. I have been doing Koopman analysis for some time now and every time I watch one of your videos I learn some new remarkable thing. Thank you a lot :)
Dr. Brunton - It has been some time since you uploaded this. Nevertheless, I thoroughly enjoyed your talk, and I greatly appreciate how well you connect so many seemingly different topics! Beautiful.
Hi Steve, I have the following 2 questions: 1. How to generalize HAVOK for input-output systems (Multiple Input and Single Output)? 2. After fitting linear regression model on time-delay coordinates how to reconstruct X'? Any leads on this would be really helpful.
If there was a way to communicate the profundity of this to the average person, it might have a big affect. You are talking about doing control in the region where the system is about to switch to a different attractor. So controlling climate change, virus propogation, etc at the right time could affect whether a system switches to a different state (e.g., ice age). Pretty amazing Steve, nicely explained for a guy like me. Thanks!
I think that ALL apparently chaotic systems can become linear in systems if you can get enough data and view it in the correct perspectives with a properly hypothesized analysis, but some systems simply do not have enough data, or an alternative viewing plane of of it. Many things in real numbers have asymptotes, poles, step discontinuity and the like which interfere in observation with current methods and that by changing methods of measurement, dimensions in complex planes, all systems can be displayed to show data if a person can define the correct origin point, the correct measurement or calculation, and most importantly the choice of data to construct a plane for application specific uses in a specifically designed and transformed koopman system. ALL choatics systems have patterns, we just need to learn how to measure, collect, and determine novel approaches to the problems. That is really the only thing that fits the patterns of math and the physical universe from my perspective, find a new way to look at it and figure out how you can transform the system to accommodate any particularity or non analytical aspect. Just need to focus on it long enough to find potential solutions and test them until you find one. All chaotic systems have a solution, I firmly believe that.
I applied HAVOK to high throughput network data for Anomaly Detection purposes but my first modes U1 and V1 are almost close to 0 in all cases and somewhat sinusoidal but shifted and noisy, but the other modes appear as clear sinusoids. Is this expected?
This is blowing my mind. Why only one forcing dimension? Could it be possible to have systems with multiple forcing dimensions? Or can all dynamical systems be represented with just one forcing dimension? Maybe there's a theorem there?
Looks to me that you have setup a linear system of nodes/modes that it can operate in but the other modes are connected by the intermittent force and acts as an indicator/switch like function that switches the linear system. E.g., it would be like having a linear system with a real value parameter. This parameter can cause the matrix to switch to another matrix which is the linear dynamics of another mode. The forcing term is tracked which allows for a bi-linear system and which the forcing term effectively tracks the mode/state switching in a reasonable/convincing way. It may be a good idea to expand out this idea of capturing multiple non-linear "parts" of a non-linear system analogous to capturing eigenfunctions of a linear system. Take the top N "non-linear subcomponents", apply some linearization theory such as HAVOK) then connect them up using "forcing terms" such that the forcing terms provide the correct dynamics. The combinations of multiple forcing terms should capture tremendous amount of non-linear dynamics. The only issue is determining a non-biased set of forcing functions. Also, I imagine that the forcing term is generally representing a mixture the modes. So spikes in forcing are to trigger a near complete mode change. It may be better to color the forcing terms in a smooth way based on the amplitude of the force. The larger the force(technically the closer it is to completely representing the mode) the more of a different color it is. This then will show when the system is selecting a mix of modes. One could, say, color the red component using threshold and the green component using scaled amplitude and the blue component using some other data. Such a system might be good at capturing high degree's of complex non-linear systems. If one can then apply the same optimizations to the forcing terms one can reduce their complexity too.
Thank you for your great videos! I have a question about the application of chaotic dynamical systems in turbulence. I know the chaos theory has raised from a turbulence-related phenomenon (i.e., the butterfly effect). But I rarely see articles in turbulence and turbulence modeling engaging chaos theory to analyze the turbulence nature. Do you have any idea about that?
@UCm5mt-A4w61lknZ9lCsZtBw I am trying to apply HAVOK + SINDy to a processor throughput prediction problem where my input is "Instruction" and output is "Time" to execute this instruction. This system is highly non-linear but quasi-periodic. I applied time delay embedding on "Time" and got very nice linear V embeddings which shows almost 15 dominant modes. Suppose I now find vdot using SINDy, how does it help me modeling output i.e "time" w.r.t. to input i.e. "instruction"? While prediction, I will only have instructions trace.
How did you work out that the forcing term as v15 for the lorentz system ? It's different for diff dataset so how do you know to stop at 15 and not 17 ? Thx.
![Hankel Alternative View of Koopman (HAVOK) Analysis [SHORT]](http://i.ytimg.com/vi/Q8VzAtGGlDQ/mqdefault.jpg)
![Hankel Alternative View of Koopman (HAVOK) Analysis [SHORT]](/img/tr.png)
This is awesome! I am so stocked about how they have found a 15D linear system with integer sparse off-diagonal matrix approximating Lorenz attractor with such an accuracy. It is a pure joy contemplating this mathematical elegance
I can't believe how much I have learnt in just the first 20 minutes. I have written down research ideas left and right. You are truely amazing, thank you so much for sharing.
This is truly fascinating. I have been doing Koopman analysis for some time now and every time I watch one of your videos I learn some new remarkable thing. Thank you a lot :)
CAN YOU PROVIDE ME YOUR EMAIL ID
I NEED SOME INFORMATION
Dr. Brunton - It has been some time since you uploaded this. Nevertheless, I thoroughly enjoyed your talk, and I greatly appreciate how well you connect so many seemingly different topics! Beautiful.
Hi Steve, I have the following 2 questions:
1. How to generalize HAVOK for input-output systems (Multiple Input and Single Output)?
2. After fitting linear regression model on time-delay coordinates how to reconstruct X'?
Any leads on this would be really helpful.
If there was a way to communicate the profundity of this to the average person, it might have a big affect. You are talking about doing control in the region where the system is about to switch to a different attractor. So controlling climate change, virus propogation, etc at the right time could affect whether a system switches to a different state (e.g., ice age). Pretty amazing Steve, nicely explained for a guy like me. Thanks!
An important work related to this is the Matrix Pencil Method of 1995.
I think that ALL apparently chaotic systems can become linear in systems if you can get enough data and view it in the correct perspectives with a properly hypothesized analysis, but some systems simply do not have enough data, or an alternative viewing plane of of it. Many things in real numbers have asymptotes, poles, step discontinuity and the like which interfere in observation with current methods and that by changing methods of measurement, dimensions in complex planes, all systems can be displayed to show data if a person can define the correct origin point, the correct measurement or calculation, and most importantly the choice of data to construct a plane for application specific uses in a specifically designed and transformed koopman system. ALL choatics systems have patterns, we just need to learn how to measure, collect, and determine novel approaches to the problems. That is really the only thing that fits the patterns of math and the physical universe from my perspective, find a new way to look at it and figure out how you can transform the system to accommodate any particularity or non analytical aspect. Just need to focus on it long enough to find potential solutions and test them until you find one. All chaotic systems have a solution, I firmly believe that.
no, lol
I applied HAVOK to high throughput network data for Anomaly Detection purposes but my first modes U1 and V1 are almost close to 0 in all cases and somewhat sinusoidal but shifted and noisy, but the other modes appear as clear sinusoids. Is this expected?
This is blowing my mind. Why only one forcing dimension? Could it be possible to have systems with multiple forcing dimensions? Or can all dynamical systems be represented with just one forcing dimension? Maybe there's a theorem there?
Ahhh you explained this about 3/4ths of the way through and it's blowing my mind! Amazing!
Hats off! Amazing job!
Higher respect to applied maths!
Can we just use Vr to predict the rare event in original variable x(t)?
Looks to me that you have setup a linear system of nodes/modes that it can operate in but the other modes are connected by the intermittent force and acts as an indicator/switch like function that switches the linear system. E.g., it would be like having a linear system with a real value parameter. This parameter can cause the matrix to switch to another matrix which is the linear dynamics of another mode. The forcing term is tracked which allows for a bi-linear system and which the forcing term effectively tracks the mode/state switching in a reasonable/convincing way.
It may be a good idea to expand out this idea of capturing multiple non-linear "parts" of a non-linear system analogous to capturing eigenfunctions of a linear system. Take the top N "non-linear subcomponents", apply some linearization theory such as HAVOK) then connect them up using "forcing terms" such that the forcing terms provide the correct dynamics. The combinations of multiple forcing terms should capture tremendous amount of non-linear dynamics. The only issue is determining a non-biased set of forcing functions.
Also, I imagine that the forcing term is generally representing a mixture the modes. So spikes in forcing are to trigger a near complete mode change. It may be better to color the forcing terms in a smooth way based on the amplitude of the force. The larger the force(technically the closer it is to completely representing the mode) the more of a different color it is. This then will show when the system is selecting a mix of modes. One could, say, color the red component using threshold and the green component using scaled amplitude and the blue component using some other data.
Such a system might be good at capturing high degree's of complex non-linear systems. If one can then apply the same optimizations to the forcing terms one can reduce their complexity too.
Thanks
If we want to turn back to original units to predict in relevant physical variable, donwe simply multiply by U*sigma ?
Thank you for your great videos! I have a question about the application of chaotic dynamical systems in turbulence. I know the chaos theory has raised from a turbulence-related phenomenon (i.e., the butterfly effect). But I rarely see articles in turbulence and turbulence modeling engaging chaos theory to analyze the turbulence nature. Do you have any idea about that?
@UCm5mt-A4w61lknZ9lCsZtBw I am trying to apply HAVOK + SINDy to a processor throughput prediction problem where my input is "Instruction" and output is "Time" to execute this instruction. This system is highly non-linear but quasi-periodic. I applied time delay embedding on "Time" and got very nice linear V embeddings which shows almost 15 dominant modes. Suppose I now find vdot using SINDy, how does it help me modeling output i.e "time" w.r.t. to input i.e. "instruction"? While prediction, I will only have instructions trace.
How did you work out that the forcing term as v15 for the lorentz system ? It's different for diff dataset so how do you know to stop at 15 and not 17 ? Thx.
Discussed in supplemental note 5 on p.15
static-content.springer.com/esm/art%3A10.1038%2Fs41467-017-00030-8/MediaObjects/41467_2017_30_MOESM1_ESM.pdf
Nice. Thanks for sharing that last part.
This is amazing
Very exciting results! Thanks, professor! I am wondering how you choose the threshold as it would have an effect on the analysis?
I love you Steve you are the best !! Like really.