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Today let’s look into some of the software that has been created in the Azimuth code project, what it accomplishes, and how it works. There are a number of models there, which are implemented as interactive web pages. The code runs right in the browser, so they have a very responsive behavior.
Here we’ll focus on the stochastic resonance model, by Allan Erskine and Jim Stuttard. This will also prepare you for looking at the other models, which use the same programming language and application support libraries (javascript language, MathJax for formula rendering, and JSXGraph for interactive plotting).
Let’s explore the page. There is a graph showing two functions over time: a green sine wave, and a randomized, funky curve. There are four sliders that control the generated curves. The amplitude and frequency of the green sine wave are controlled by two of the sliders. These sliders also affect the funky curve. Verify this by experimenting with the sliders.
At the heart of this model is a random process, which generates the second curve. This process is determined by the combination of a deterministic component – known as the forcing function, which here is the sine wave – and a noise component. Experiment with the noise slider, to get the output to range from perfectly smooth to completely chaotic. Try the amplitude and frequency controls, to see how they affect the output signal. Changing the “seed” parameter gives you different instances of the random process.
This program performs a discrete simulation for a “stochastic differential equation” (SDE), which is an equation that specifies the derivative of a function in terms of time, the current value of the function, and a noise process. If the noise amplitude is zero, you get an ordinary differential equation, where the derivative depends only on time and the current value of the function.
This program consists of:
A general SDE simulator
The formula for the specific SDE used in this model
Auxiliary functions used by this formula
Interactive controls used to set parameters
Plot of the intermediate time series (the sine curve)
Plot of the final output time series
I would like everyone now to try to open up the source code, and at least skim through it. Since the code is running in your browser, you have already downloaded it! Here are the steps that I went through to view it:
Open the web page for the model.
Run your browser’s view-source function. This is browser specific. At home I’m running Firefox on the Mac, and for some reason this function is not appearing in the menu. But a Google search told me to type Apple-U, which worked. (TODO: how to do this with other browsers / environments)
Then the window opens up, with some html stuff that clearly contains the text of the web page.
But where’s the code? The only possible culprits could be these lines at the head:
<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 lines tell the browser to the following javascript program files:
MathJax.js is an open source engine for rendering math formulas.
JSXGraph is a cross-platform library for interactive graphics, function plotting, and data visualization.
StochasticResonanceEuler.js is the main application code. Click on it!
normals.js, also clickable, contains an array of random numbers, used in the model
If you had trouble getting to the file, you can get it here.
Now that we’re there, at the source file StochasticResonanceEuler.js, I would like to ask one more thing of you, before we return to our regularly scheduled programming. Please review the succinct, six-point (precis) structural description of the code, from the previous section. Then skim through this text, and see how many of the elements that I wrote about there you can see being mentioned. This will involve putting a blur lens over all kinds of non-obvious implementation-dependent statements, to sniff out what might be going on there, and form some rough hypotheses.
Before proceeding, to avoid any possible mathematical liability, I must express the following disclaimer. We said that the simulation is driven by a function for the “derivative” of the random variable X. Now, for ordinary differential equations derivative this of course means instantaneous time rate of change, but for stochastic equations with Brownian-motion-like components, the definition involves subtleties that are outside of the scope here. But, almost miraculously, for the “Euler-based” discrete numerical approximation that we are using, we are permitted to finagle out of these complications, because the simulator is just a “stepper” that takes an ostensibly instantaneous time rate of change, and extrapolates over the interval between sample points.
Now, as we said, the derivative of the main random variable X is specified as a function of time t, the current value x, and a noise term. For stochastic resonance, this Deriv(t,x) is given as the sum of a sinusoidal function of t, called the forcing function, a “bistability” function of x, and a random noise variable:
Deriv(t, x) = SineWave(sine-amplitude, sine-frequency, t) + BiStablility(x) + NoiseSample(noise-amplitude)
Considered alone, the effect of the sinusoidal forcing function would be to cause the value of X(t) itself to oscillate sinusoidally. What makes the model interesting is the bi-stability term:
BiStability(x) = x * (1 - x^2).
Let’s see what would happen if this were the sole term that defined Deriv(t,x).
First, BiStability has zeros at -1, 0 and 1, so these are the equilibrium points of x. The derivative of BiStability is negative at -1 and 1, and positive at zero, so -1 and 1 are stable equilibria, and 0 is unstable. All the points from -infinity to 0 are in the basin of attraction for -1, and all the points from 0 to infinity are in the basin for 1.
Now, let’s consider the effect of the sine function, plus the BiStability polynomial, but still without noise. If the amplitude of the forcing function is low enough, then, depending on initial conditions, the solution will converge towards on of the attractors -1 and 1, and oscillate around it according to the forcing function. If the amplitude of the forcing function is large enough, then the forcing function may have enough activity, at the right moment, to pull the function from one basin of attraction to another.
Next, let’s consider the bistability polynomial, plus the forcing function, plus the noise – the whole magilla. Suppose that the forcing function is weak, and the signal is oscillating around an attractor. Then, depending on the noise amplitude, a random noise event may pull the signal from one attractor to another. The transitions between basins, then, will contain both a periodic element, and a randomized element.
This is interesting / applied in the study of the Milkanovich cycles of the ice ages.
Now the program logic is packaged up in seven functions, and spans a mere 3.5 pages of printed text – it’s impressive that this platform can produce such nice, interactive applications, with this economy of coding.
The top level entry point in the function initCharts (hint: find it and look at it). This is a short little function, which does its work by making two function calls: initControls, and initSrBoard. Please at least scan through initControls, which, as you can see, is building the objects that represent the sliders, and returning them as a dictionary of slider objects.
All the real, logical content of the application is encapsulated in this second function, initSrBoard, which is at the end of the file. There, we see that two curve objects are constructed, one called positionCurve, and the other called forcingCurve. The forcing curve is constructed from the locally defined forcingFunction, which defines the sine wave. Note that in this function, the values of the amplitude and frequency sliders are used. (Note that 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?)
(How to make your own version of the app.)
(Brownian motion, for noise process)
Next, and most centrally to the application, a function is attached to the updateDataArray member of the positionCurve object, which makes the call to the stochastic resonance model proper: this is the call to the “mkSrPlot” (i.e. MakeStochasticResonancePlot) function, which takes as arguments the values for all of the sliders: forcing amplitude, forcing frequency, noise amplitude, and seed value. This call returns a “plot” object that contains a list of time values, and a corresponding list of values for the main variable X.
So now we must drill down into the logic of the mkSrPlot function. The first step here is to construct a function object, and store in in the variable deriv, which represents the derivative computation:
Deriv(t,x) = SineCurve + BiStability,
where represents the deterministic component of the derivative.
Then a “stepper” function is constructed, by the call to the function Euler(deriv, tStep), which returns a function that performs “stochastic Euler extrapolation,” with a step size of tStep, using the derivative function deriv that was just constructed. The idea of a stepper function is very simple: it takes as input the current point (t,x), and a noise sample, and returns the next point (t’,x’).
A stepper function of this form is all that is needed for the general toplevel loop, which is implemented by the function sdeLoop, to generate the full time series for the output. This sdeLoop function takes as arguments the stepper function, the “dither value” (amount of noise), the initial point (t0,x0), a seed value for the randomization, and a number of points to be generated. The loop simply 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 the sequence of (t,x) values that are generated (by this iteration).
The noise samples are generated by taking a contiguous subsequence of the array normals[i], and scaling them by the dither value. The “seeding” of the randomization is implemented by using the seed variable as a control value, which controls which subsequence of the (large) array of normal random variates gets used.
One last point here: How did the Euler function compute a stepper based on what I labeled “stochastic Euler extrapolation?” Well, it’s quite simple: given the deterministic derivative function Deriv(t,x), and step size tStep, it returns the following function:
((t,x), noiseSample) –> (t + tStep, x + tStep * Deriv(t,x) + noiseSample).
(Transitions may resonate with the forcing function, in a stochastic manner)
The time series are generated as sequences of (t,x) pairs. Note that x here is the dependent variable, which is the representation in code of a random variable X.
This program could be naturally evolved into a simulator for another SDE, by changing the formula, the intermediate functions, and the associated slider parameters (hint: homework problem to come). Here, the specific components are the SDE derivative formula, the sine wave, which used in that formula, and the sliders for the sine wave parameters.
Note also, for a given amplitude of the sine wave, a longer period will have a stronger effect on pulling the state from one pole to another – i.e., the oscillations are more sensitive to low frequencies. One could examine this more systematically by analyzing – or measuring through computational examples – the expected values of the frequency components in the spectrum of the output signal, as a function of forcing amplitude, frequency, and noise amplitude.
Effect of frequency
How would you study this more systematically, using Fourier analysis in the setting of experimental, stochastic programming
How would you restructure the code for general, statistical studies of the output time series?
Modify the code to compute —-
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.