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# What next? - after NIPS

I will post a blog article about what I learned at NIPS when the videos of talks appeared. The best ideas I got were these:

1) To what extent have people systematically checked that Niño 3.4 is the best quantity for predicting other El Niño-related quantities, e.g. ones that actually matter to farmers and other people who need weather forecasts? There are a number of El Niño indices, but maybe we could seek "optimal" ones.

2) I should attend the workshop on Climate Informatics during September 25-26 in Boulder, Colorado.

3) In a 2013 paper in J. Climate, Jia DelSole and Tippett apparently discovered a low intrinsic dimensionality for the behavior of world climate models, due to El Niño and a few other features! Looking for a low-dimensional attractor in a high-dimensional phase space sounds fun.

In a similar vein, Kawale et al, SDM 2011 and Steinback et al, KDD 2003 did automated discovery of pressure dipoles like ENSO, NAO, etc. Also related to this is Ebert-Uphoff et al's paper A new type of climate network based on probabilistic graphical models: results of boreal winter versus summer J. Clim. 2012, which used Bayesian network ideas to infer causal relationships between the 4 biggest teleconnections.

Maybe there's a way to combine these ideas and start from climate data and build up a simplified but interesting model of the world's main variable climate elements.

4) The idea of rating the main existing El Niño forecasts, and/or developing a better one, still sounds very interesting to me. We've developed a certain amount of expertise in this subject and we haven't fully exploited it yet. For example: maybe we could try to predict Niño 3.4 using the entire matrix of link strengths, not just the average link strength. How much better is this?

5) I saw a rather shocking graph of world temperatures as predicted/retrodicted by the main climate models used by the IPCC. What was shocking is that these models stay close together in the best but then spread out wildly in the future. This suggests something like overfitting - the models seem to be fitting the known data "too well" compared to how much they agree about the future. Is there a way to quantify how bad this "overfitting effect" is, or prove that something bad is going on? This could have a lot of impact if it could be done well.

Needless to say, we won't have the energy to do all 5 of these things! I would love it if people could talk strategically about what we want to do. People tend to be more interested in discussing their latest favorite idea, but I'd like this thread to be about sketching a plan. It's possible that doing more of what we've just been doing is the best plan... but I plan to take a break from the energetic work I've been doing, and think a bit about what's good to do next.

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1.

People tend to be more interested in discussing their latest favorite idea

Yes. A big thing missing from what has been done here so far is a study of previous work, like those in the graph below for El Niño prediction. Nobody here seems interested in NCEP CFSv2, ..., UNB/CWC. We are not likely to make a useful contribution until we understand what has already been done by others.

The idea of rating the main existing El Niño forecasts, and/or developing a better one, still sounds very interesting to me.

A possible route to a quick publication would be to take one of the existing statistical approaches, add some sort of link strengths to its input, and hope that this improves the existing method. It is easier to demonstrate the superiority of one method over another if the two methods make similar mistakes (and I think these would).

Personally I don't think I'll be contributing a lot. I find biology more interesting than physics, and in particular phylogenetic analysis more interesting than climate science. Having said that, I have a bad track record of predicting what I will be interested in, so who knows.

Comment Source:> People tend to be more interested in discussing their latest favorite idea Yes. A big thing missing from what has been done here so far is a study of previous work, like those in the graph below for El Niño prediction. Nobody here seems interested in NCEP CFSv2, ..., UNB/CWC. We are not likely to make a useful contribution until we understand what has already been done by others. <img src = "http://math.ucr.edu/home/baez/climate_networks/2014-11-20-Nino34-predictions.jpg" alt = ""/> > The idea of rating the main existing El Niño forecasts, and/or developing a better one, still sounds very interesting to me. A possible route to a quick publication would be to take one of the existing statistical approaches, add some sort of link strengths to its input, and hope that this improves the existing method. It is easier to demonstrate the superiority of one method over another if the two methods make similar mistakes (and I think these would). Personally I don't think I'll be contributing a lot. I find biology more interesting than physics, and in particular phylogenetic analysis more interesting than climate science. Having said that, I have a bad track record of predicting what I will be interested in, so who knows.
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2.

** Is there a way to quantify how bad this “overfitting effect” is**

Backtesting as I showed how it was used to compute actual hard error for a forecast algorithm moving from past to present time, at each point of time re-compute the forecast function. At each location in time, only the previous data prior to that time to be used.

Otherwise to run one or two forecast algorithms, at times confused by the curve-fitting, issues great results for the data, and lousy forecasts.

Comment Source:** Is there a way to quantify how bad this “overfitting effect” is** Backtesting as I showed how it was used to compute actual hard error for a forecast algorithm moving from past to present time, at each point of time re-compute the forecast function. At each location in time, only the previous data prior to that time to be used. Otherwise to run one or two forecast algorithms, at times confused by the curve-fitting, issues great results for the data, and lousy forecasts.
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3.

Some would say that applying a GCM (general circulation model) to observations is over-fitting. The initial configuration of a GCM is based on physics but the minute that someone tries to fit it to actual data, it runs the risk of over-fitting as the number of parameters can be tweaked endlessly.

Much more defensible is to take a simple physics model and then one can tweak that.

Comment Source:Some would say that applying a GCM (general circulation model) to observations is over-fitting. The initial configuration of a GCM is based on physics but the minute that someone tries to fit it to actual data, it runs the risk of over-fitting as the number of parameters can be tweaked endlessly. Much more defensible is to take a simple physics model and then one can tweak that.
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4.

Graham, what are your research interests and directions in phylogenetic analysis? It is a really interesting area.

It would be nice to make a "subject map" that talks about various fields of science and how they fit into the context of the Azimuth project, and who is pursuing what. If you take as the starting point that Azimuth is a group of scientists etc. working together to understand environmental problems -- and the prospects for survival of life as we know it -- then it's clear that climate science and biology are core subjects. Since biology has many diverse aspects, it could be good to break out the "tableau" for biology into sections on the relevant sub-fields -- biodiversity, evolution (including its response to the environmental crises), bio-regeneration, ...

Today, my main interest at Azimuth is on the foundations of applied mathematics -- with the focus oriented towards applications to environmental science. For example, the study of Markov processes, and their applications to reaction networks and evolution mechanisms.

Comment Source:Graham, what are your research interests and directions in phylogenetic analysis? It is a really interesting area. It would be nice to make a "subject map" that talks about various fields of science and how they fit into the context of the Azimuth project, and who is pursuing what. If you take as the starting point that Azimuth is a group of scientists etc. working together to understand environmental problems -- and the prospects for survival of life as we know it -- then it's clear that climate science and biology are core subjects. Since biology has many diverse aspects, it could be good to break out the "tableau" for biology into sections on the relevant sub-fields -- biodiversity, evolution (including its response to the environmental crises), bio-regeneration, ... Today, my main interest at Azimuth is on the foundations of applied mathematics -- with the focus oriented towards applications to environmental science. For example, the study of Markov processes, and their applications to reaction networks and evolution mechanisms.
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5.
edited December 2014

At the moment I'm working on species delimitation. Basic problem: given genetic data from individuals eg a,b,c,d,e, infer how they cluster into species. Is it ab+cde or a+be+c+d or...? The approach is Bayesian, and it samples from the posterior using the Monte Carlo Markon chain technique. Right now I'm working on a package called STACEY for BEAST2. More info at my website.

Comment Source:At the moment I'm working on species delimitation. Basic problem: given genetic data from individuals eg a,b,c,d,e, infer how they cluster into species. Is it ab+cde or a+be+c+d or...? The approach is Bayesian, and it samples from the posterior using the Monte Carlo Markon chain technique. Right now I'm working on a package called [STACEY](http://biorxiv.org/content/early/2014/12/11/010199) for [BEAST2](http://beast2.org/). More info at [my website](http://www.indriid.com).
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6.

Let me make up an example to show how bad are all the approaches to what we are proposing to do.

When the dimensions for the data are reduced e.g. dropping coordinates or using averages or for that matter any other method of the kind, there might be a 'fit' for the data that is magnificent but shall produce no good forecasts!

Example1: Randomly spray a bunch of points into a plane in R^3 , then find a projection of that plane into R^2 which is basically a line, so we have a nice fitted line to the data, but in reality there is little help from this projection to forecast or classify the next set of points.

Example 2: Randomly spray a bunch of point into R3 with a globular range, then find a projection of that into R^2 which is basically a disk, so we have a nice way of constraining the points via a nice closed form inequality but that tells us very little about the next set of points.

El Nino indices are like Example 1, nice curve of some kind, we could find a nice fit to them, but forecasts are inaccurate, because somehow the dimensionality is reduced and a curve is obtained which is not of much help to forecast.

The taking of the averages to reduce the dimensions are the worst possible approach for what we are trying to do. Clearly the climate dimensionality is at least of degree 4 for a given point in time or 5 counting time as an additional dimensions.

Dara

Comment Source:Let me make up an example to show how bad are all the approaches to what we are proposing to do. When the dimensions for the data are reduced e.g. dropping coordinates or using averages or for that matter any other method of the kind, there might be a 'fit' for the data that is magnificent but shall produce no good forecasts! Example1: Randomly spray a bunch of points into a plane in R^3 , then find a projection of that plane into R^2 which is basically a line, so we have a nice fitted line to the data, but in reality there is little help from this projection to forecast or classify the next set of points. Example 2: Randomly spray a bunch of point into R3 with a globular range, then find a projection of that into R^2 which is basically a disk, so we have a nice way of constraining the points via a nice closed form inequality but that tells us very little about the next set of points. El Nino indices are like Example 1, nice curve of some kind, we could find a nice fit to them, but forecasts are inaccurate, because somehow the dimensionality is reduced and a curve is obtained which is not of much help to forecast. The taking of the averages to reduce the dimensions are the worst possible approach for what we are trying to do. Clearly the climate dimensionality is at least of degree 4 for a given point in time or 5 counting time as an additional dimensions. Dara
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7.

But then again, it may be possible that ENSO is about the scale of complexity as predicting tides. We just don't know quite the combination of forcing factors yet as the data is limited with respect to the length of a cycle.

Paleo data of ENSO does indeed suffer from averaging and the granular nature of sampling.

Comment Source:But then again, it may be possible that ENSO is about the scale of complexity as predicting tides. We just don't know quite the combination of forcing factors yet as the data is limited with respect to the length of a cycle. Paleo data of ENSO does indeed suffer from averaging and the granular nature of sampling.
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8.

This reminds me of the old times when people rubbed minerals and spices over copper surface with the hope of transforming it into gold! I cannot fathom how a few thousand numbers in a 1-D time-series could possibly predict such complex and vast weather system.

I am resigned to the fact that these El Nino man made simplified indices will not help in any serious forecasts.

Comment Source:This reminds me of the old times when people rubbed minerals and spices over copper surface with the hope of transforming it into gold! I cannot fathom how a few thousand numbers in a 1-D time-series could possibly predict such complex and vast weather system. I am resigned to the fact that these El Nino man made simplified indices will not help in any serious forecasts.
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9.

By this in #9 I was not referring to Paul, I was saying in general what is being done with the little data for El Nino

Comment Source:By **this** in #9 I was not referring to Paul, I was saying in general what is being done with the little data for El Nino
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10.
edited December 2014

Graham wrote:

Nobody here seems interested in NCEP CFSv2, …, UNB/CWC.

I'm interested in them; I just haven't had time to dig into them yet.

We are not likely to make a useful contribution until we understand what has already been done by others.

I agree. But this general issue is one reason for project 4): rating El Niño prediction methods. This would give us a good excuse to learn these methods, while not feeling the need to quickly create better methods. Instead of joining the competition, we'd be referees... at least for a while.

Since at least some of us here like statistics and "big ideas" more than the nitty-gritty details of simulating the Pacific Ocean, this project would also give us a good excuse to ponder the general question of how to rate predictions. What counts as a good prediction? This is clearly a tricky and interesting question, with lots of applications. I'm not completely sure, but it's possible that the Azimuth Project will do better at highly abstract issues like this. I know that I, at least, am drawn tot them.

Comment Source:Graham wrote: > Nobody here seems interested in NCEP CFSv2, …, UNB/CWC. I'm interested in them; I just haven't had time to dig into them yet. > We are not likely to make a useful contribution until we understand what has already been done by others. I agree. But this general issue is one reason for project 4): rating El Ni&ntilde;o prediction methods. This would give us a good excuse to learn these methods, while not feeling the need to quickly create _better_ methods. Instead of joining the competition, we'd be referees... at least for a while. Since at least some of us here like statistics and "big ideas" more than the nitty-gritty details of simulating the Pacific Ocean, this project would also give us a good excuse to ponder the general question of _how to rate predictions_. What counts as a good prediction? This is clearly a tricky and interesting question, with lots of applications. I'm not completely sure, but it's possible that the Azimuth Project will do better at highly abstract issues like this. I know that I, at least, am drawn tot them.
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11.

David Tanzer wrote:

Today, my main interest at Azimuth is on the foundations of applied mathematics – with the focus oriented towards applications to environmental science. For example, the study of Markov processes, and their applications to reaction networks and evolution mechanisms.

This is one of my favorite areas too. I'm working with Blake Pollard on "open" Markov processes. You may have seen his paper A second law for open Markov processes; this is the first of a series. We've created a category where the morphisms are open Markov processes; composing these lets you build big complicated Markov processes out of "pieces", and our goal is to understand big complicated Markov processes in terms of their pieces.

Markov processes are the same as reaction networks where each reaction has just one input and one output. So, everything we do with Markov processes, we want to generalize to reaction networks. The second law of thermodynamics for reaction networks is more subtle than for Markov processes, but it's very interesting. Manoj Gopalkrishnan blogged about the proof of this law a while ago, and Marc Harper hinted at its applications to evolutionary game theory. We want to generalize it to "open" chemical reaction networks.

All this comes much more naturally to me than working with software and/or studying El Niños. In a sense it's crazy for me to get involved in those other issues! But there's also something nice about getting my hands dirty with really concrete questions. For one thing, it seems to make it easier to collaborate with all the people here! We had a nice teamwork thing going on during the "crunch time" before my NIPS talk. It doesn't seem to work so well for me to talk about the theorems I want to prove...

Comment Source:David Tanzer wrote: > Today, my main interest at Azimuth is on the foundations of applied mathematics – with the focus oriented towards applications to environmental science. For example, the study of Markov processes, and their applications to reaction networks and evolution mechanisms. This is one of my favorite areas too. I'm working with Blake Pollard on "open" Markov processes. You may have seen his paper [A second law for open Markov processes](http://johncarlosbaez.wordpress.com/2014/11/15/a-second-law-for-open-markov-processes/); this is the first of a series. We've created a category where the morphisms are open Markov processes; composing these lets you build big complicated Markov processes out of "pieces", and our goal is to understand big complicated Markov processes in terms of their pieces. Markov processes are the same as reaction networks where each reaction has just one input and one output. So, everything we do with Markov processes, we want to generalize to reaction networks. The second law of thermodynamics for reaction networks is more subtle than for Markov processes, but it's very interesting. Manoj Gopalkrishnan [blogged about the proof of this law](http://johncarlosbaez.wordpress.com/2014/01/07/lyapunov-functions-for-complex-balanced-systems/) a while ago, and Marc Harper hinted at its [applications to evolutionary game theory](http://johncarlosbaez.wordpress.com/2014/01/22/relative-entropy-in-evolutionary-dynamics/). We want to generalize it to "open" chemical reaction networks. All this comes much more naturally to me than working with software and/or studying El Ni&ntilde;os. In a sense it's crazy for me to get involved in those other issues! But there's also something nice about getting my hands dirty with really concrete questions. For one thing, it seems to make it easier to collaborate with all the people here! We had a nice teamwork thing going on during the "crunch time" before my NIPS talk. It doesn't seem to work so well for me to talk about the theorems I want to prove...
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12.

John could Category theory model non-associative structures (algebras), itself being associative?

Comment Source:John could Category theory model non-associative structures (algebras), itself being associative?
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13.
edited December 2014

Dara - this thread is about what the goals of the Azimuth Project. A better place to talk about nonassociative algebras is in my multi-part series on integer octonions on the $n$-Category Café.

Comment Source:Dara - this thread is about what the goals of the Azimuth Project. A better place to talk about nonassociative algebras is in my multi-part series on [integer octonions](http://math.ucr.edu/home/baez/octonions/integers/index.html) on the $n$-Category Caf&eacute;.
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14.

Paul wrote:

But then again, it may be possible that ENSO is about the scale of complexity as predicting tides. We just don’t know quite the combination of forcing factors yet as the data is limited with respect to the length of a cycle.

There are ways to estimate the dimension of the attractor in a chaotic system, starting from time series data. Roughly speaking, this lets us see how many numbers we need to know now to predict the future as well as we can. We could do this for El Niño! Susanne Still is doing something a bit similar with some help from my student Blake Pollard.

I would be happy to pursue this sort of issue: instead of merely trying to predict El Niños, try to scientifically study how hard they are to predict.

Comment Source:Paul wrote: > But then again, it may be possible that ENSO is about the scale of complexity as predicting tides. We just don’t know quite the combination of forcing factors yet as the data is limited with respect to the length of a cycle. There are ways to estimate the dimension of the attractor in a chaotic system, starting from time series data. Roughly speaking, this lets us see how many numbers we need to know now to predict the future as well as we can. We could do this for El Ni&ntilde;o! Susanne Still is doing something a bit similar with some help from my student Blake Pollard. I would be happy to pursue this sort of issue: instead of merely trying to predict El Ni&ntilde;os, try to scientifically study how hard they are to predict.
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15.

A recent paper

Stein, Karl, Axel Timmermann, Niklas Schneider, Fei-Fei Jin, and Malte F. Stuecker. "ENSO seasonal synchronization theory." Journal of Climate 2014 (2014).

based on a PhD thesis: Stein, Karl J. ENSO seasonal synchronization theory. Diss. UNIVERSITY OF HAWAI'I AT MANOA, 2013. http://www.soest.hawaii.edu/oceanography/PhD/2013-Stein.pdf

One equation that Stein is evaluating is the 2nd-order DiffEq

$\frac{d^2T}{d\tau^2} - \lambda (1-T^2) \frac{dT}{d\tau} + T = \omega F \cos(\omega \tau)$

This differs from the formulation that I have been using in that I am setting the $\lambda$ damping term to zero, and incorporating Mathieu terms on T.

What the damping term allows is to create a locking on the forcing frequency, which is curve D in the figure below.

"Further increasing the nonlinearity results in the oscillator becoming frequency locked to a rational multiple of the driving frequency (Figure 6.1, D)."

Comment Source:A recent paper Stein, Karl, Axel Timmermann, Niklas Schneider, Fei-Fei Jin, and Malte F. Stuecker. "ENSO seasonal synchronization theory." Journal of Climate 2014 (2014). based on a PhD thesis: Stein, Karl J. ENSO seasonal synchronization theory. Diss. UNIVERSITY OF HAWAI'I AT MANOA, 2013. <http://www.soest.hawaii.edu/oceanography/PhD/2013-Stein.pdf> One equation that Stein is evaluating is the 2nd-order DiffEq $\frac{d^2T}{d\tau^2} - \lambda (1-T^2) \frac{dT}{d\tau} + T = \omega F \cos(\omega \tau)$ This differs from the formulation that I have been using in that I am setting the $\lambda$ damping term to zero, and incorporating Mathieu terms on T. What the damping term allows is to create a locking on the forcing frequency, which is curve D in the figure below. > "Further increasing the nonlinearity results in the oscillator becoming frequency locked to a rational multiple of the driving frequency (Figure 6.1, D)." ![stein](http://imageshack.com/a/img903/2512/HE3e1X.gif)
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16.

Is wF smooth function? or it is a noisy non-smooth function?

Comment Source:Is wF smooth function? or it is a noisy non-smooth function?
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17.

F is just a scalar parameter. $\omega$ is the forcing frequency.

Comment Source:F is just a scalar parameter. $\omega$ is the forcing frequency.
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18.

Ok in that case the solutions are smooth functions, best I could tell, am I correct?

I refer to John's earlier foray into F being a non-smooth noise like function, that is a much better model for diff eq governing the climate.

What basically these models you quoting do, are modelling the TREND of the signal, where the most important part of the signal i.e. the high frequency noise-like is ignored. The latter is what causing the most concern.I believe and need to check my ideas that these noisy F functions could show the instability of the climate we see these days. F being non-smooth should be a requirement for these diff eq.

Comment Source:Ok in that case the solutions are smooth functions, best I could tell, am I correct? I refer to John's earlier foray into F being a non-smooth noise like function, that is a much better model for diff eq governing the climate. What basically these models you quoting do, are modelling the TREND of the signal, where the most important part of the signal i.e. the high frequency noise-like is ignored. The latter is what causing the most concern.I believe and need to check my ideas that these noisy F functions could show the instability of the climate we see these days. F being non-smooth should be a requirement for these diff eq.
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19.

Here some code I wrote learning from earlier notes made by John:

Noisy Hopf Bifurcation

If you scroll down the CDF file, you see noisy solutions, and these are models that we need to pay attentions too.

My problem is, where does the noise come from?

Comment Source:Here some code I wrote learning from earlier notes made by John: [Noisy Hopf Bifurcation](http://mathematica.lossofgenerality.com/2014/06/24/noisy-hopf-bifurcation/) If you scroll down the CDF file, you see noisy solutions, and these are models that we need to pay attentions too. My problem is, where does the noise come from?
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20.

Yet the QBO shows relatively little noise and this is an important forcing factor for ENSO.

Comment Source:Yet the QBO shows relatively little noise and this is an important forcing factor for ENSO.
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21.

Yet the QBO shows relatively little noise and this is an important forcing factor for ENSO.

I go the same thing from Wavelets, I just checked from some stuff I posted earlier. We get 4 decompositions of magnitude 0.026% contribution to the original signal, and one major contribution of 0.88%.

So the major Trend ratio to the sub-trends is about 40x. So this 40x factor as opposed to 4000x is what you mentioned as relatively little noise.

Now my next question: Do you know the QBO data published had already been smoothed or denoised? Or do you know where the specs are for how the data was post-processed.

Dara

Comment Source:>Yet the QBO shows relatively little noise and this is an important forcing factor for ENSO. I go the same thing from Wavelets, I just checked from some stuff I posted earlier. We get 4 decompositions of magnitude 0.026% contribution to the original signal, and one major contribution of 0.88%. So the major Trend ratio to the sub-trends is about 40x. So this 40x factor as opposed to 4000x is what you mentioned as **relatively little noise**. Now my next question: Do you know the QBO data published had already been smoothed or denoised? Or do you know where the specs are for how the data was post-processed. Dara
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22.

Here is some ideas about the processing of mean monthly values for QBO, correct me if wrong:

The Quasi-Biennial-Oscillation (QBO) Data Series

Basically the values we are using are once averaged over the month (nearly equivalent to moving average window of 30 days) and again averaged for zonal winds (which I have no idea what it means):

30mb zonal wind at the equator, zonal average

Each of these averages serve as a smooth or denoising which explains why the data has low noise, not because in nature it is as such so smooth, but because of the moving average on large windows

Dara

Comment Source:Here is some ideas about the processing of mean monthly values for QBO, correct me if wrong: [The Quasi-Biennial-Oscillation (QBO) Data Series](http://www.geo.fu-berlin.de/en/met/ag/strat/produkte/qbo/) Basically the values we are using are once averaged over the month (nearly equivalent to moving average window of 30 days) and again averaged for zonal winds (which I have no idea what it means): [ 30mb zonal wind at the equator, zonal average](http://www.esrl.noaa.gov/psd/data/correlation/qbo.data) Each of these averages serve as a smooth or denoising which explains why the data has low noise, not because in nature it is as such so smooth, but because of the moving average on large windows Dara
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23.

I guess my point is that there is a significant periodicity in the QBO time series. Some research discusses this as a factor in ENSO but others ignore it.

I don't have much knowledge on how the data is processed but it is collected by radiosonde measurements which were first made in the early 1950's. So there is a fairly long uninterrupted streak whereby the underlying 2.33 year period is quite visible, albeit containing a jitter about this value.

Comment Source:I guess my point is that there is a significant periodicity in the QBO time series. Some research discusses this as a factor in ENSO but others ignore it. I don't have much knowledge on how the data is processed but it is collected by radiosonde measurements which were first made in the early 1950's. So there is a fairly long uninterrupted streak whereby the underlying 2.33 year period is quite visible, albeit containing a jitter about this value.
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24.

So there is a fairly long uninterrupted streak whereby the underlying 2.33 year period is quite visible

My point exactly, the diff eq right hand side being constant/smooth means you are modelling only the Trend.

Comment Source:>So there is a fairly long uninterrupted streak whereby the underlying 2.33 year period is quite visible My point exactly, the diff eq right hand side being constant/smooth means you are modelling only the Trend.
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25.

However the jitter in the signal is frequency demodulated by the differential equation and will create fine details in the result.

That is the eye-opener and what contributes to the erratic nature of ENSO.

Comment Source:However the jitter in the signal is frequency demodulated by the differential equation and will create fine details in the result. That is the eye-opener and what contributes to the erratic nature of ENSO.
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26.
edited January 2015

Paul and Dara,

I don't know if you came across this excellent thread on climate network theory Nathan Urban described what I regard as a pretty big challenge::

as [far as] the control theory perspective is concerned, what I’m most interested in is systems of coupled PDEs, where the goal is to understand how coupling one PDE to another changes the behavior of both. (Or rather, coupling one system of PDEs to another system of PDEs.) i.e., coupling together atmospheric circulation, ocean circulation, atmospheric chemistry, terrestrial biogeochemistry, etc. Or even understanding how different terms in a single PDE contribute to the global dynamics. I realize you’re far from doing this, but it’s ultimately where my interest would lie. (And also in the idea of model reduction: what are the “essential” dynamics of these PDEs, and can we replace them by low-order ODE or SDE systems?).

He also pointed out this paper:

Paillard and Parrenin, Earth and Planetary Science Letters, 227, 263-271, 2004) Saedeleer and Crucifix, Is astronomical forcing a reliable and unique pacemaker for climate? A conceptual model study (2011) http://goo.gl/OyKZTh

I thought that Chua circuits might be more interesting under the mistaken assumption that Van der Pol oscillators showed only deterministic and periodic but not chaotic behaviour but I found out from these slides by Rial that the Van der Pol oscillator can also exhibit chaotic behaviour. I don't have the foggiest about how to evaluate one against the other except on some time series?

Miko Kiviranti immediately posted a Chua circuit emulation in LTSpice.

http://forum.azimuthproject.org/discussion/1100/hello-from-mikko-kiviranta/?Focus=7924#Comment_7924

This is the so-called Chua’s circuit, which is considered one of the simplest continuous systems showing chaotic behaviour. At least it seems minimal in the sense of Poincare-Bendixson theorem, because it has just three degrees of freedom (charges on the two capacitors and the flux of the inductor) and just one nonlinear element (“B1”, so-called Chua’s diode). The concept of chaos pops up regularly in discussions about weather and climate.

Here’s how it simulates. Looks a bit like the average earth temperature over the past 1E6 yrs, don’t you think?

When I tried to construct my own Chua’s circuit from scratch, I was surprised about how carefully one must tune the parameters in order to make the circuit chaotic. When following the climate discussions by non-professionals one gets the idea that whenever a non-linear dynamic system has three degrees of freedom or more, it is doomed to become chaotic. Judging by my N=1 statistics (a brief look at one simple system) this may be a misconception.

I was definitely under this misconception :).

John B. posted the last comment on the thread:

how various kinds of network theories are related. Many of these amount to ways to describe the interaction between systems modeled by multivariable ODE’s. When we get to PDE’s things get higher-dimensional and we may need to replace categories by n-categories; in highly theoretical physics this is almost “fashionable”, but I’m not sure the rest of the world is ready for it (or needs it).

If n-categories are needed to deal with PDEs then the world needs n-categories.

How a PDE can be conceptualised as an n-category? Any implementation of a solver obviously follows a recipe and compiles a graph but I suppose that would be a decategorified description?

http://forum.azimuthproject.org/discussion/1300/relation-of-climate-models-to-network-theory/?Focus=10030#Comment_10030

Ian Ross made a 3-ring oscillator model which I take as a great demo of how to do FRP in Haskell, html5 canvas and javascript. Nathan pointed out that a 2-oscillator model has very different behaviour when fully coupled.

http://www.geo.brown.edu/research/Fox-Kemper/classes/CU/ATOC6020_09/notes/07-Newman.pdf

There are a couple of methodological distinctions which seem germane: stochastic/deterministic and Markov/memory models. To sort out which approaches are used in which models I guess would be a considerable thrutch.

Many authors keep pointing to the deep ocean as a, if not the, site of most needed research wrt. a host of phenomena. Unfortunately the data record is so far too sparse, even if reliable (I guess nad might be able tell us about the quality). Nathan also pointed to theories involving high salinity in the deep ocean as even being possibly related to Milankovitch cycles. This also seems to be worth learning about.

The climate network theory thread also has some comments on detrending and smoothing issues which is another thing I haven't properly come to grips with.

Comment Source:Paul and Dara, I don't know if you came across this excellent thread on [climate network theory](http://forum.azimuthproject.org/discussion/1300/relation-of-climate-models-to-network-theory/?Focus=10030#Comment_10030) Nathan Urban described what I regard as a pretty big challenge:: > as [far as] the control theory perspective is concerned, what I’m most interested in is systems of coupled PDEs, where the goal is to understand how coupling one PDE to another changes the behavior of both. (Or rather, coupling one system of PDEs to another system of PDEs.) i.e., coupling together atmospheric circulation, ocean circulation, atmospheric chemistry, terrestrial biogeochemistry, etc. Or even understanding how different terms in a single PDE contribute to the global dynamics. I realize you’re far from doing this, but it’s ultimately where my interest would lie. (And also in the idea of model reduction: what are the “essential” dynamics of these PDEs, and can we replace them by low-order ODE or SDE systems?). He also pointed out this paper: Paillard and Parrenin, Earth and Planetary Science Letters, 227, 263-271, 2004) Saedeleer and Crucifix, [Is astronomical forcing a reliable and unique pacemaker for climate? A conceptual model study (2011)](http://arxiv.org/abs/1109.6214) http://goo.gl/OyKZTh I thought that Chua circuits might be more interesting under the mistaken assumption that Van der Pol oscillators showed only deterministic and periodic but not chaotic behaviour but I found out from these [slides](http://goo.gl/OyKZTh) by Rial that the Van der Pol oscillator can also exhibit chaotic behaviour. I don't have the foggiest about how to evaluate one against the other except on some time series? Miko Kiviranti immediately posted a Chua circuit emulation in LTSpice. http://forum.azimuthproject.org/discussion/1100/hello-from-mikko-kiviranta/?Focus=7924#Comment_7924 > This is the so-called Chua’s circuit, which is considered one of the simplest continuous systems showing chaotic behaviour. At least it seems minimal in the sense of Poincare-Bendixson theorem, because it has just three degrees of freedom (charges on the two capacitors and the flux of the inductor) and just one nonlinear element (“B1”, so-called Chua’s diode). The concept of chaos pops up regularly in discussions about weather and climate. > Here’s how it simulates. Looks a bit like the average earth temperature over the past 1E6 yrs, don’t you think? * [Chua93](http://www.eecs.berkeley.edu/~chua/papers/Chua93.pdf) > When I tried to construct my own Chua’s circuit from scratch, I was surprised about how carefully one must tune the parameters in order to make the circuit chaotic. When following the climate discussions by non-professionals one gets the idea that whenever a non-linear dynamic system has three degrees of freedom or more, it is doomed to become chaotic. Judging by my N=1 statistics (a brief look at one simple system) this may be a misconception. I was definitely under this misconception :). John B. posted the last comment on the thread: > how various kinds of network theories are related. Many of these amount to ways to describe the interaction between systems modeled by multivariable ODE’s. When we get to PDE’s things get higher-dimensional and we may need to replace categories by n-categories; in highly theoretical physics this is almost “fashionable”, but I’m not sure the rest of the world is ready for it (or needs it). If n-categories are needed to deal with PDEs then the world needs n-categories. How a PDE can be conceptualised as an n-category? Any implementation of a solver obviously follows a recipe and compiles a graph but I suppose that would be a decategorified description? http://forum.azimuthproject.org/discussion/1300/relation-of-climate-models-to-network-theory/?Focus=10030#Comment_10030 Ian Ross made a [3-ring oscillator model](http://www.skybluetrades.net/blog/posts/2012/11/13/fay-ring-oscillator/) which I take as a great demo of how to do FRP in Haskell, html5 canvas and javascript. Nathan pointed out that a 2-oscillator model has very different behaviour when fully coupled. http://www.geo.brown.edu/research/Fox-Kemper/classes/CU/ATOC6020_09/notes/07-Newman.pdf There are a couple of methodological distinctions which seem germane: stochastic/deterministic and Markov/memory models. To sort out which approaches are used in which models I guess would be a considerable thrutch. Many authors keep pointing to the deep ocean as a, if not the, site of most needed research wrt. a host of phenomena. Unfortunately the data record is so far too sparse, even if reliable (I guess nad might be able tell us about the quality). Nathan also pointed to theories involving high salinity in the deep ocean as even being possibly related to Milankovitch cycles. This also seems to be worth learning about. The climate network theory thread also has some comments on detrending and smoothing issues which is another thing I haven't properly come to grips with.
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edited January 2015

Thanx Jim, for these really important notes you posted.

The damping function (F on the right hand side of diff eq) need not be smooth, we assume it smooth, then we only have modeled the Trend, or some sub-trend. This a huge issue for me to even understand what these theories are all about.

I give you an example: imagine we listen to the hums from an engine, if no gear shifted, the sound's period is quite computable and we could gain meaningful insight as to what is happening. The driver reving the engine (without gear change) also could be easily be discerned and understand from the varying rpm (reving change smooth). Then suddenly there is a gear shift, which is non-smooth function so to say, this is what I have difficulty to understand how to study, then after the gear shift we are back to smooth variations in rpm, all back to normal understanding. The gear shift changes the energetics of the car or direction of the movement, the smooth rpm of the engine cannot explain any of that. We need to understand the non-smooth gear shift in order to understand if the car is going backwards, or up the hill or pulling a larger load.

When we get to PDE’s things get higher-dimensional and we may need to replace categories by n-categories; in highly theoretical physics this is almost “fashionable”, but I’m not sure the rest of the world is ready for it (or needs it).

Generally speaking without John's full cooperation I would not even think of dabbling into such areas.

Comment Source:Thanx Jim, for these really important notes you posted. The damping function (F on the right hand side of diff eq) need not be smooth, we assume it smooth, then we only have modeled the Trend, or some sub-trend. This a huge issue for me to even understand what these theories are all about. I give you an example: imagine we listen to the hums from an engine, if no gear shifted, the sound's period is quite computable and we could gain meaningful insight as to what is happening. The driver reving the engine (without gear change) also could be easily be discerned and understand from the varying rpm (reving change smooth). Then suddenly there is a gear shift, which is non-smooth function so to say, this is what I have difficulty to understand how to study, then after the gear shift we are back to smooth variations in rpm, all back to normal understanding. The gear shift changes the energetics of the car or direction of the movement, the smooth rpm of the engine cannot explain any of that. We need to understand the non-smooth gear shift in order to understand if the car is going backwards, or up the hill or pulling a larger load. >When we get to PDE’s things get higher-dimensional and we may need to replace categories by n-categories; in highly theoretical physics this is almost “fashionable”, but I’m not sure the rest of the world is ready for it (or needs it). Generally speaking without John's full cooperation I would not even think of dabbling into such areas.
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edited January 2015

One last thing, I must note that models producing exactly accurate and same periodicity for QBO might not produce the same results for other parameters. Just like the example I gave for the car, in two different gears we might find the same RPM, but two different energetics for the car motion or even reverse motion.

So models matching some periodicity observed in nature, could be quite diverse on other computations and forecasts.

Comment Source:One last thing, I must note that models producing exactly accurate and same periodicity for QBO might not produce the same results for other parameters. Just like the example I gave for the car, in two different gears we might find the same RPM, but two different energetics for the car motion or even reverse motion. So models matching some periodicity observed in nature, could be quite diverse on other computations and forecasts.
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Comment #16 is actually a van der Pol circuit. This place much emphasis on the damping term, which is the factor modifying the 1st derivative. But from what I have read, the ocean is inviscid (i.e. not viscous), so I ask myself why is the damping that important?

Instead of that, I suggest that the varying factor should really go on the direct term, which is a category of Mathieu or Hill equation. What this does is easily model a form of jitter, which in the case of QBO is a variation of the frequency without impacting the amplitude much. This is really a circuit with frequency modulation: http://contextearth.com/2014/10/19/demodulation-and-the-soim/

Then if one inputs a frequency-modulated signal into a 2nd-order DiffEq, the output becomes amplitude modulated. Which is how an FM radio works. Bingo -- that is one example of how two simple circuits can interact to create a complex waveform.

Comment Source:Comment #16 is actually a van der Pol circuit. This place much emphasis on the damping term, which is the factor modifying the 1st derivative. But from what I have read, the ocean is inviscid (i.e. not viscous), so I ask myself why is the damping that important? Instead of that, I suggest that the varying factor should really go on the direct term, which is a category of Mathieu or Hill equation. What this does is easily model a form of jitter, which in the case of QBO is a variation of the frequency without impacting the amplitude much. This is really a circuit with frequency modulation: <http://contextearth.com/2014/10/19/demodulation-and-the-soim/> ![fm](http://imagizer.imageshack.us/a/img661/8997/2yJxW1.gif) Then if one inputs a frequency-modulated signal into a 2nd-order DiffEq, the output becomes amplitude modulated. Which is how an FM radio works. Bingo -- that is one example of how two simple circuits can interact to create a complex waveform.
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From the thesis cited in comment #16

"As can be seen, the driven van der Pol oscillator favors frequency locked solutions at odd multiples of the driving frequency; frequency locked solutions at even multiples of the driving frequency are very unstable (Mettin et al., 1993). No chaotic solutions were found in the neighborhood of realistic parameter ranges, reflecting the rarity of chaotic solutions within more complex ENSO models (Jin et al., 1996). "

I really have not been able to understand the frequency-locking mode, either in the data or in the models. A recent pair of papers in Physics Letters A has some more info: http://www.pas.rochester.edu/~douglass/recent-publications.html

The Sun is the climate pacemaker I. Equatorial Pacific Ocean temperatures David H. Douglass & Robert S.Knox Physics Letters A; ©2014 Elsevier B.V.; doi:10.1016/j.physleta.2014.10.057

The Sun is the climate pacemaker II. Global ocean temperatures David H. Douglass & Robert S.Knox Physics Letters A; ©2014 Elsevier B.V.; doi:10.1016/j.physleta.2014.10.058

Comment Source:From the thesis cited in comment #16 > "As can be seen, the driven van der Pol oscillator favors frequency locked solutions at odd multiples of the driving frequency; frequency locked solutions at even multiples of the driving frequency are very unstable (Mettin et al., 1993). No chaotic solutions were found in the neighborhood of realistic parameter ranges, reflecting the rarity of chaotic solutions within more complex ENSO models (Jin et al., 1996). " I really have not been able to understand the frequency-locking mode, either in the data or in the models. A recent pair of papers in Physics Letters A has some more info: <http://www.pas.rochester.edu/~douglass/recent-publications.html> The Sun is the climate pacemaker I. Equatorial Pacific Ocean temperatures David H. Douglass & Robert S.Knox Physics Letters A; ©2014 Elsevier B.V.; doi:10.1016/j.physleta.2014.10.057 The Sun is the climate pacemaker II. Global ocean temperatures David H. Douglass & Robert S.Knox Physics Letters A; ©2014 Elsevier B.V.; doi:10.1016/j.physleta.2014.10.058