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Today we will look at one of the software models from the Azimuth code project, which aims to produce educational software that is relevant to the study of climate. The program, by Allan Erskine and Glyn Adgie, demonstrates the concept of “stochastic resonance,” which is a widely studied phenomenon that has an application to the theory of ice-age cycles. The Azimuth models are programmed as interactive web pages, which run right in the browser, as javascript.
Start by opening the stochastic resonance model web page. It displays a sine wave, called the forcing signal, alongside an intricate, time-varying function, called the output signal. There are four sliders, labelled A, B, C and D.
One slider controls the frequency of the sine wave, and another controls its amplitude. Try them.
The output signal depends, in a complex way, through the a “mechanism” of stochastic resonance, on the sine wave. Observe how the amplitude and frequency sliders affect the output signal.
The process involves a random component, whose magnitude is controlled by the noise slider. Set it to zero, and see that the output becomes completely smooth. As you increase the noise, verity that the output becomes increasingly chaotic.
Changing the Sample-Path parameter gives different instances of the process.
The program runs a discrete simulation for a stochastic differential equation (SDE), which is a specification for the derivative of a function in terms of time, its current value, and a noise process.
Here are the functional components of the program:
Interactive controls to set parameters
Plot of the forcing signal (the sine curve)
Plot of the output signal
A function which defines a particular SDE. The stochastic resonance is a property of the solutions to this equation.
A general simulator for SDE’s, based on the Euler method
I would like everyone now to locate the source code, by the following method.
Open the web page for the model.
The code is now downloaded and running in your browser.
While on the page, run your browser’s view-source function. For Firefox on the Mac, it’s Apple-U, for Firefox on the PC it’s … (TODO: fill in)
You should see the html for the web page itself.
Observe The these following lines at the top of the file: html file load javascript programs, from various locations on the web, into the browser’s internal javascript interpreter:
<script src='http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=default'></script> <script src='http://cdnjs.cloudflare.com/ajax/libs/jsxgraph/0.93/jsxgraphcore.js'></script> <script src='./StochasticResonanceEuler.js'></script> <script src='./normals.js'></script>
These Here’s lines what load each javascript line program does. files, from various locations on the web, into the broswer’s internal javascript interpreter. The first two lines load platform support files:MathJax is an open-source formula rendering engine, engine. andJSXGraph is a cross-platform library for interactive graphics, function plotting, and data visualization. The StochchasticResonanceEuler.js next line references StochchasticResonanceEuler.js, which is the main code for the model. Finally, And normals.js contains a table of random numbers, used in the main program.
Now, click on the link for StochasticResonance.js – and you’re there!
Now that you’ve reached the source, StochasticResonanceEuler.js, your next challenge is to scan through it and look for associations with the program inventory listed in the preceding section. Try to put a blur lens over any items that look obscure, since the goal here is only to form rough hypotheses about what might be going on.
Now let’s analyze the particular SDE that used is in being simulated. In this equation, the model. This equation sets deterministic part of the derivative is set to a sinusoidal function of time plus a bistable function of the current value:
DerivNonrandom(t, x) = SineWave(t, amplitude, frequency) + Bistable(x),
where Bistable(x) = x * (1 - x^2).
Let’a analyze the effects of each of these terms, both separately and in together. combination.
Alone, the sine wave would cause the output signal to vary sinusoidally.
Now let’s consider the bistable polynomial, which has roots at -1, 0 and 1. The root at zero is an unstable equilibrium, and -1 and 1 are points of stable equilibrium. The basin of attraction for -1 is all the negative numbers, the basin for 1 is the positive numbers, and the point of unstable equilibrium separates the basins.
We will view each basin as one of the states of a bistable system.
Now let’s put the sine wave and the bistable polynomial together. If the wave amplitude is not too large, the system will gravitate towards one of the attractors, and then continue to oscillate around it thereafter – it will stays never in leave the basin. But if it is large enough, the system will be pulled back and forth between the two basins – it will resonate. resonate with the driving signal.
Finally, let’s complete the picture by adding in the noise. Suppose the sine wave were large enough to periodically pull the system close to zero, zero. but never to actually reach it, or cross over to the other side. If we add in some noise, then a well-timed random event could push the system over to the other side. Therefore, So the noise may trigger transitions state between changes, states, and this will occur with higher probability at certain phases of the sine wave. Increasing More the noise will lead to transitions on more and more cycles of the sine cycles wave will lead to transitions – we may expect to see, in the flip-flopping between states, states will contain a “stochastic reflection” of the frequency of the driving sine wave. Even Further more noise will lead cause to transitions occurring across during a wider range of phases phases, of the sine wave, and a enough sufficient amount of noise will lead drown to out transitions at all phases – then the noise overtakes the signal, and turning the output becomes to noise.
Moral: under the right conditions, the noise may amplify the effect of the input signal.
The theory of the timing of the “ice ice ages” ages is a fascinating, challenging and unsolved open problem in science.
Note: what is colloquial colloquially called an ice “ice age age” – a frozen period – is technically known as a glacial maximum, maximum. and The the technical meaning of Ice Age is a huge period of time that spans thousands of glacial minima and maxima. There have only been four Ice Ages in the history of the Earth, Earth. each of which is characterized by a different pattern in the cycling of the ice ages that it contains.
Here One we talk about one of the current hypotheses, hypotheses which uses a stochastic resonance model. model, In where this account, the climate is modeled as a multistable system, and the forcing is results a from result of certain cyclical, slowly varying changes in astronomical parameters variables such as the tilt of the Earth’s axis. These are known as Milankovitch cycles, and their durations can be measured in units of tens of thousands of years, which at least puts them in the right ballpark as the intervals between “ice ages.”
the The forcing purpose of this section is produced to by sketch astronomical out variations the known hypothesis as – Milankovitch not cycles, to make a claim, but rather to suggest how programs like this can play a role in the scientific enterprise.
In the very simplest model, the climate has two stable states: a cold, “snowball” Earth, and a hot, iceless Earth. Each state is self-reinforcing. A stochastic frozen resonance Earth simulator is program white, can so therefore it play doesn’t absorb much solar energy, which keeps it cold and frozen. A hot Earth is dark, so it absorbs a critical lot role of solar energy, and which keeps it hot and melted. The model also covers the fact that the glaciers are concentrated in the testing northern of latitudes such – a that hypothesis. thenorthern temperatures are capable of triggering a change in the state of the climate.
A There major hypothesis for the timing of the ice ages uses a stochastic resonance model, where the climate is modeled as a multistable system, and the forcing function is produced by astronomical variations known as Milankovich cycles, which periodically vary in the amount of solar energy received at the northern latitudes – where the glaciers are concentrated. three The astronomical duration of these cycles is that on contribute to the order forcing of function, all with cycle times measured in tens of thousands of years, years: which is at least in the ballpark for the intervals between ice ages.
In the very simplest model, the climate has two stable – self-reinforcing – states: a cold, “snowball” Earth, and a hot, iceless Earth. Each of these states is self-reinforcing. A frozen Earth is white, so it doesn’t absorb much solar energy, and this keeps it cold and frozen. A hot Earth is dark, so it absorbs a lot of solar energy, and this keeps it hot and melted. The model also includes the fact that the glaciers are concentrated in the northern latitudes, so the northern temperatures are capable of triggering a change in the state of the climate.
There are three astronomical cycles that contribute to the forcing function, all with cycle times measured in in tens of thousands of years:
Changing of the eccentricity (ovalness) of the Earth’s orbit, with a period of 100 kiloyears
Changing of the obliquity (tilt) of the Earth’s axis, with a period of 41 kiloyears
Precession (rotary wobbling) of the Earth’s axis, with a period of 23 kiloyears
Together, these produce a multi-frequency variation in the amount of solar energy received in the northern latitudes. But the induced temperature changes are not large enough to trigger a state change. According to the stochastic resonance hypothesis, it is other, random variations in the heat received up north that may trigger the climate to change states. One such source of variation is changes in the amount of heat-trapping gases in the atmosphere.
Note also the following interesting interchange that took place on the Azimuth blog:
For further information, see:
As Our we’ve seen, the present program demonstrate serves an educational function, which is to show the concept of stochastic resonance, and allows allow you to interactively explore its behavior. But this kind type of software also has a functions function within research, research. both First, for they exploring can be used to empirically explore theoretical questions, questions. Suppose we asked how the effectiveness of a forcing function depends on its frequency. This can be explored, with the current program, by manually varying the frequency parameter, and for observing contributing the generated results. On a more systematic basis, we could write meta-program that varies the parameters and applies measures to the testing output of signal. empirical theories.
For Such a software case can of also exploring be a used theoretical to query generate in stochastic resonance, suppose we asked how the effectiveness predictions of a forcing theory function and depends compare on them its to frequency. actual This measurement can data. be One explored could through imagine, for example, a computer program experiment, that either implements where a you model manually of vary the frequency Milankovitch parameter, astronomical and cycles, observe then outputs this signal into the generated state results, changing or, model more of systematically, through a meta-program particular which theory varies of the climate, parameters, and then applies finally calculated compares measures to the output signal. signal of the climate model with observed (or inferred) data.
The This same is modelling (scientific) software programming can in also be modified to apply empirical tests of theory predictions. For starters, we could replace the sinusoidal service forcing function with a function that used computed astronomical data about the Milankovitch cycles. Another step would be to replace the bistable polynomial with a more accurate modelling of the our multistable understanding process of the climate. Earth. Output could then be “data-mined” for correlations with the actual timings of the ice-ages. The model could be used to falsify the theory as well as to validate it.
This is a case of scientific programming at work, and it is an application of “science that really matters,” which is the motivating principle of the Azimuth Project.
Our scientific program consists of seven functions. The top-level function is initCharts, which dispatches to initControls and initSrBoard (board means a container for graphical widgets). The job of initControls is to build the sliders.
The application logic is encompassed within the scope of initSrBoard, which constructs one curve object for the forcing curve, and another for the position curve. Next, the update methods on these objects get set to the appropriate functions. These methods, which are responsible for redrawing the curves whenever its defining input data gets changed, return an object that contains a list of time values and a corresponding list of values for the curve. The update method on the forcing curve is set to a function that computes the sine wave. Note that this function (locally defined) is defined to read the amplitude and frequency values from the sliders. The function “mkSrPlot” gets set as the updater for the position curve object.
The simulation is performed by the function mkSrPlot. It first defines the function for the deterministic part of the derivative:
deriv = Deriv(t,x) = SineCurve(t) + BiStable(x),
Then a “stepper” function is constructed, by the call to Euler(deriv, tStep). The In idea general, of a stepper function is maps to map the current point (t,x) and a noise sample to the next point (t’,x’). The Euler stepper maps (t,x) ((t,x), noiseSample) to (t + tStep, x + tStep * Deriv(t,x) + noiseSample).
The simulator loop is implemented performed by the function sdeLoop, which is passed: given:
The stepper function
The noise amplitude (“dither”)
The initial point (t0,x0)
The A randomization offset
Count The of the number of points to generate
The current point is initialized to (t0,x0), and then the stepper is repeatedly applies applied to the current point and the current noise sample. The output returned is the sequence of (t,x) values.
The noise samples are read from an array, array called normals[i] normals, and scaled by the noise amplitude. The array is large, and contains many more data points than are needed by the calculation. The randomization offset controls the starting point in the array, which leads to different instances of the random process.
Now that we’ve tried out the program, downloaded its source code, and understood how it works, it’s time to roll up our sleeves and start tweaking it to do new things!
We’re We’ll going to proceed by a series of “baby steps.” The First first let’s step get is to make a local copy of the program to run on your machine, machine. and Copy get that to run. So, copy thehtml file and the main java script to a folder on your local machine. I’ll suppose that you’ve stored them into the following folder on your machine: c:\pkg\webmodels.
Now check that the html file is active, by
Effect of frequency
Design a study of the effectiveness of signal transmission, as a function of noise amplitude and signal frequency. How you define the effectiveness measure?
How would you restructure the code for general, statistical studies of the output time series?
When the sliders are moved, an event must be fired, which causes the recalculation to take place. How is this mechanism implemented in the javascript / JSXGraph application library?
Modify to add an exponent slider
Modify to show graph of expected value (add slider for nTrials) (Not enough random numbers.)
Add a standard deviation plot
If you are a climate scientist, let us know of next steps
Begin to study this book —-, and think of how to write programs for some of the models. Simplify! The hierarchy of models. All models that you post here will be considered as candidates for the Azimuth Code Project page. This may be a way for programmers, ultimately, to give back to the Earth.