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This page is a blog article in progress, written by David Tanzer. To see discussions of this article while it was being written, visit the Azimuth Forum. Please remember that blog articles need HTML, not Markdown.
guest post by David Tanzer</i>
Today we’ll 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. Our subject is a program, contributed by Allan Erskine and Glyn Adgie, that demonstrates the concept of “stochastic resonance.” This is a widely studied phenomenon, which has an application to the theory of the timing of the ice ages. The models are implemented as interactive web pages. They have a responsive behavior because they are written in javascript, which runs right in the browser.
The model page shows a green sine wave, a funky randomized curve, and four sliders.
The sine wave is input to a random process, which produces the output signal.
One of the sliders controls the frequency of the sine wave, and another controls its amplitude. Try them.
Observe how these sliders also affect, in a complex way, the generated output signal.
The noise slider controls the randomness of the process. Set it to zero, and verify that the output is completely smooth. Max it out, and see the output turn to noise.
The Sample-Path parameter gives different instances of the process.
The program runs a discrete simulation for a “stochastic differential equation” (SDE), which specifies the derivative of a function in terms of time, the current value of the function, and a noise process.
The program contains the following elements:
A general simulator for SDE’s, based on the Euler method
The specific derivative function, for the stochastic resonance model
Auxiliary functions used by this derivative function
Interactive controls to set parameters
Plot of the intermediate time series (the sine curve)
Plot of the output time series
I would like everyone now to locate the source code, by the following method.
Open the web page for the model.
Since the code is running in your browser, you have already downloaded it!
Now, 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)
A window should open up with some html stuff, in which you should clearly see the text of the web page.
The code is loaded into the browser by these lines at the head of the file, which reference javascript files at different locations on the web:
<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>
The line loads the MathJax formula rendering engine. JSXGraph* is a cross-platform library for interactive graphics, function plotting, and data visualization. Then we see the main program, StochasticResonanceEuler.js. Finally, normals.js defines a table of random numbers that is used by the calculation.
Now click on the link for StochasticResonance.js – and you’re there!
Now that you’ve reached the code file StochasticResonanceEuler.js, your next challenge is to scan through it and attempt to make loose associations between code fragments and the points in the program synopsis given above. Try to put a blur lens over any items that look obscure; the goal here is only to form rough hypotheses about what might be going on.
In the stochastic resonance model, the deterministic component of the derivative is equated with the sum of a sinusoidal function of t, called the forcing function, and a function of x that has two stable points of equilibrium:
DerivDeterministic(t, x) = SineWave(t, amplitude, frequency) + Bistable(x),
where Bistable(x) = x * (1 - x^2).
On its own, the sinusoidal forcing function will cause the value of X(t) to oscillate sinusoidally.
Consider the effect of Bistable(x) in isolation. Bistable(x) has roots at -1, 0 and 1, which are the equilibrium points for X(t): -1 and 1 are stable equilibria, and 0 is unstable. The basin of attraction for -1 is all the negative numbers, the basin for +1 is the positive numbers, and the unstable equilibrium is on the fence between the basins.
We regard each basin as one of the states of a bistable system.
Now consider the combination of the sine wave and the bistable polynomial. If the wave amplitude is below some threshold, then the system will remain in one basin, forever oscillating around its attractor. But if the amplitude is large enough, it will pull the system back and forth between the two basins, i.e., the system will oscillate between the two stable states, in resonance with the forcing sine wave.
Now let’s complete the picture by adding noise… Now, suppose the sine wave were large enough to periodically pull the system close to zero, but not enough to cross over to the other side – so it remains in a single state. If we add in some noise, then a well-timed random event may push the system over to the other side. So we will see random transitions between states, with higher probabilities at certain phases of the sine wave. As the noise amplitude increases, more and more cycles of the sine wave will lead to transitions, and the frequency of the flip-flopping will approach that of the sine wave. The noise here has amplified the effect of the input signal. As noise increases further, it will dominate the state transitions, which will themselves become noise.
One of the leading theories for the timing of the ice ages is based on a stochastic resonance model. In Didier Paillard’s model, the climate of the Earth is modeled as a tristable system, and the forcing function is produced by certain astronomical variations called Milankovich cycles, which produce periodic variation in the amount of solar energy received at the northern latitudes.
In the most simplified model, the climate has two stable states: a cold, “snowball Earth,” and a hot Earth with no ice. The snowball Earth is stable: because it is white, it doesn’t absorb much solar energy, which keeps it cold and frozen. The hot Earth is stable: because it is dark, it absorbs a lot of solar energy, which keeps it hot and melted.
Now in order to bring the Milankovitch cycles into the picture, we bring into the picture the fact that the glaciers are concentrated up north, and it is the temperatures there that control the state of the system. If it gets warm enough up north, the glaciers melt, and the state goes to hot. If it gets cold enough, the glaciers form, and the state goes to cold.
There are three Milankovitch cycles that have been identified:
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 of the Earth’s axis (rotation of the direction of the axis, like a wobbling top), with a period of 23 kiloyears
Now the amplitude of these temperature changes in not enough to move the climate from one state to another. But random temperature events may cause crossings if they occur at the right phase of a Milankovitch cycle.
Researchers have found correspondences (complex correlations) between the actual spacing of the ice ages and the timings of these three Milankovitch cycles.
For further information, see:
The program for the stochastic resonance model consists of seven functions. The top level function is initCharts, which dispatches to two other functions, initControls and initSrBoard. (Key: sr = stochastic resonance, board = graphical gizmo.) The job of initControls is to build the sliders.
The main logic of the application is contained in this second function, initSrBoard. Two curve objects are constructed there, positionCurve and forcingCurve. The forcing curve is given by the locally defined forcingFunction, which defines the sine wave. This function reads its values from the amplitude and frequency sliders.
Now a crucial function “mkSrPlot” gets attached to the “update—” method of the positionCurve object. This function is responsible for redrawing the curve, whenever its defining parameters get changed. It returns an object with a list of time values, and a corresponding list of values for X(t).
The main algorithm is spelled out by the function mkSrPlot. It first constructs a function that gives the deterministic part of the derivative:
deriv = Deriv(t,x) = SineCurve + BiStability,
Then a “stepper” function is constructed, by the call to Euler(deriv, tStep). A stepper function takes inputs as input the current point (t,x) and a noise sample, and returns outputs the next point (t’,x’). The Euler stepper maps (t,x) to (t + tStep, x + tStep * Deriv(t,x) + noiseSample).
A The stepper top function level is loop all of that is needed for the general simulator toplevel loop, which is implemented by the function sdeLoop, to which generate the full time series for the output. The main loop of the simulator is passed: passed the following arguments: (1) the stepper function, (2) the noise amplitude amplitude, (“dither”), or “dither”, (3) the initial point (t0,x0), a (4) seed value for the randomization, randomization seed, and (5) the number of points to be generated. The loop initializes a currentPoint to (t0,x0), and then repeatedly applies the stepper function to the current point and the next noise sample; the output returned is just this sequence of (t,x) values.
The noise samples are taken from a block of values from the (large) array normals(i), and scaling them by the dither value. The “seed” Sample-path variable controls which section of the array gets used.
How to post it.
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.