Sines and cosines are great, but we might want to look for other patterns in a signal. A ‘continuous wavelet transform’ lets us scan a signal for appearances of a given pattern at different times and also at different time scales: a pattern could go by quickly, or in a stretched out slow way.

]]>I increase the derivative precision of the derivative (with iteration for higher degree):

with these definition, and with an optimization of the program velocity (more step in the same time) I obtain (this is not the absolute minimum, but the first minimum: there is a little difference):

I verify, with Mathematica, a solution that grow exponentially in the range of the time negative (for insolation).

This happen because the differential equation is linear.

The geometric solution of the differential equation is limited in derivative, and value (the warming and cooling of the Earth cannot be infinite); this can happen if the solution is a closed surface in the derivativ space ; so that the trajectory is limited trajectory on the surface.

If this is true, then must happen three connected surface: normal temperature surface, glacial period and global warming.

I think that the transition from normal temperature to glaciation can be studied in old data, to understand the geometry of the transition.

Saluti

DOmenico

]]>Hi Dan,

I mean 300 time units. Sorry I took so long to reply, I had to dig up the C functions that the R library, Rwave, actually calls to make sure.

]]>Florifulgurator,

in my experience such missing peaks are often one of the telltale signs of unmodeled nonlinear behavior. You should have tried perhaps with an Extended or Ensemble KF, but at the end of the day they are only as good as the (nonlinear) model of the system you have. If your model is not that great, they cannot make miracles.

Regarding your last comment, well, not really. A KF is a filter that you attach to a system in order to observe (reconstruct and filter out the noise) the state of the system at that particular time. It needs to have an internal model of the system to work (as well as access the system’s inputs and outputs).

An ARMAX model is just a model of the system, with no particular purposes rather than trying to reproduce the system’s output given its inputs.

The armax system that i have identified is this (i’ll copy and paste more info here below):

amx2221 =

Discrete-time ARMAX model: A(z)y(t) = B(z)u(t) + C(z)e(t)

A(z) = 1 – 1.873 z^-1 + 0.8778 z^-2

B1(z) = 19.87 z^-1 – 19.91 z^-2

B2(z) = 21.26 z^-1 – 21.68 z^-2

C(z) = 1 – 0.6272 z^-1 – 0.2794 z^-2

Name: amx2221

Sample time: 1 seconds

Parameterization:

Polynomial orders: na=2 nb=[2 2] nc=2 nk=[1 1]

Number of free coefficients: 8

Use “polydata”, “getpvec”, “getcov” for parameters and their uncertainties.

Status:

Estimated using POLYEST on time domain data “0-400-12”.

Fit to estimation data: 79% (prediction focus)

FPE: 0.3427, MSE: 0.3291

It is very simple indeed, just a few parameters, but often simple things work better and are anyway more useful than complicated ones. Indeed this outperforms more complicated systems.

Don’t try to read into these equations too much, stuff like “A(z)y(t)” is really a shorthand notation for convolution not a multiplication. And the sample time is not really one second (which is the default one) but one Kyear (this would matter when converting the model to a continuous-time one).

]]>Giampiero,

the missed peaks in your graph remind me of a Kalman filter I used at work to predict electricity load: It was very bad at predicting the daily peak.

Alas that job is gone and I haven’t gone any deep into time series stochastics stuff (some gargantuan Excel wizardry turned out sufficient to significantly improve prognosis). And before the job I never felt interested – which I regret now. Any book recommendation beyond A.N. Shiryaev’s Probability book?

Autoregressive models smell a bit oversimplifying to me – but that’s probably because I didn’t get the knack yet. My intuition (possibly wrong) is: ARMAX is a sort of “small” Kalman filter.

]]>By the way the structure of the ARMAX model is here:

where is the temperature, is a vector containing obliquity and eccentricity, is white noise.

]]>John wrote:

In physics, the state of a system is a complete description of the way it is at some given time.

Yes, assuming at least that the system is autonomous (no input), I am very comfortable with this definition, and perhaps it’s the best way to define the state even when the system has inputs.

Traditionally in control engineering too the cardinality of the set of possible states is infinity (yes, i know, technically aleph … something), and so we informally (and incorrectly) refer to the “number of states” as the number of variables that are needed to define the state of a system. I guess that “degrees of freedom” could be confusing when talking to mechanical engineers since a mechanical system that is said to have, for example, 2 DOF, needs a vector of 4 numbers (2 positions and 2 velocities) to represent its state.

I really liked your ‘best linear prediction’ here and I would like to know exactly how it’s defined.

That was a model identified with an ARMAX structure (see this and this). Very roughly speaking this is a discrete-time transfer function with some filtered additive noise. The transfer function (TF) has 2 poles, 2 zeros, and another zero that serves as a pure delay (this TF can be converted into a state space discrete-time model with 1 output, 2 inputs and 2 “degrees of freedom”). The noise filter also has 2 zeros (and the same 2 poles of the TF).

I am sorry if this is incomprehensible, but right now i can’t say much more, i’ll try to say more later if you are interested.

I think we could have fun quantitatively measuring the amount of nonlinearity of the Earth’s climate system, and I have some ideas for how to do that.

This looks really interesting, I wonder how you would do it, maybe just subtracting the outputs of the linear and nonlinear model and have a look at its energy.

Perhaps I can try to do something just once in a while. I think that however the first step would be to agree on which input to use (just insolation, obliquity, eccentricity, or maybe all of the above?) and which output to measure (temperature or ice volume). Otherwise I don’t think we can make an apple to apple comparison.

]]>Giampiero wrote:

I would not say that the qualitative model has “3 states” (e.g. as in 3 different dimensions like position and velocity), but just one state and 3 different allowed values (regimes or modes) along that dimension (that is i, g and G).

I should give you a more substantial reply, but I’m in a rush, and trivial issues of language are fun to argue about, so I’ll only do that.

In physics, the **state** of a system is a complete description of the way it is at some given time. For example, if we have a particle on a line with position 3 and velocity -7, its state is (3,-7). In traditional physics there’s usually an infinite set of states, since the set of real numbers is infinite. But there are also systems with finitely many states, and computer science gives many examples, called ‘finite-state machines’.

In physics, if we describe the state of a system by a list of *n* numbers, we say the system has *n* **degrees of freedom**.

So, I was trying to say that Paillard’s model has 3 different states: i, g, and G, together with certain rules for hopping between these states.

This is perhaps oversimplified since these rules involve *how long* the system has been in the g (mild glacial) state. So, one can argue that the *amount of time spent in the mild glacial state* should also be counted as part of the description of that state. Doing this makes sense because he essentially assumes that ice builds up linearly with time in this state, if the insolation stays low enough. So, the amount of time is a proxy for the amount of ice.

If you have the time and desire to do any more calculations or modelling, please let me know, and I’ll be glad to help out: I like to think and I don’t like to program!

I really liked your ‘best linear prediction’ here:

and I would like to know exactly how it’s defined. I think we could have fun *quantitatively measuring the amount of nonlinearity of the Earth’s climate system*, and I have some ideas for how to do that.