Apparently I'm everything that's wrong with the world

Good news: the source of everything that's wrong with the world has been located.
Bad news: it's me

For a Christmas eve activity, my neighborhood decided to have an outing in the park directly in front of my house. My contribution was a bonfire. As tends to happen at such gatherings, you meet some of your neighbors for the first time. One neighbor discovers that I'm the owner / builder of the interesting Gaudi-esque house we're standing in front of and proceeds to tell me how much he loves the house and how great it is that I tried something different, etc. After a little while the conversation changes to the historic housing tax exemptions that are common in my neighborhood (for which I, of course, am not eligible since my house is new). They were discussing how much money they would save on taxes by declaring their expensive beautiful houses to be historic. I casually pointed out that while I understoodd their desire for a tax break, and indeed would probably do the same thing if I was in their position, that surely they'd have to understand that I'd be against such tax breaks for obvious reasons. "Oh but what about the character of the neighborhood?" they replied. "Which character would that be?" I replied, "Faux American Colonial? Faux Craftsmen? Faux Gaudi? It's not that I don't like these houses, I do, that's one of the reasons I live here, but let's not pretend that they represent some great monument to human achievement. Again, that said, I totally understand why you guys would want to use the law in your interest and get a tax break. But if it came up for a vote, I'd vote against it."

This was not greeted well. An aggressive: "You don't think we should save historic things?" is quipped back. "I dunno," I mused, "Life includes a lot of change and renewal, sometimes we should embrace it."

And for this bit of shocking opinion on the meaning of life I am met with my neighbor angrily saying: "I can't talk to you anymore." and storming away. At first I thought he was joking because his reaction seems so out of proportion to the topic. Someone else quietly said that they agreed with me and this brought him back angrily shouting: "He said that he wants to destroy all historic things!" I replied calmly, "actually that's not what I said...." but before I could explain my position, he yelled at me while wagging his finger: "You're everything that's wrong with the world. Money! Money! FUCK YOU!!". As he stormed away I quietly replied: "Merry Christmas to you too."

For the record, my full position is that, of course we should save historic things -- I just think that we should do it collectively by purchasing landmarks with public funds and then leasing them out or preserving them. The current method that permits some people to receive a tax break for living in old, beautiful houses strikes me as an extremely inequitable tax reward exclusively benefiting a small cadre of rich people because "historic" homes are almost always in old, nice neighborhoods where the property values are high. Furthermore, the extremely lax standards of what makes something "historic" are capricious. For example, a house around the corner from me has been declared historic because a man who lived in it once was said to be a "prominent doctor" when they found a 1 inch newspaper column about him from the Statesman decades ago. (He's actually still alive so technically he's not quite "historic" yet). By these low standards, every house that has been occupied by a professional (i.e. practically every house in every well-off neighborhood) will become "historic" eventually.

The absurdity of this to me is that after I die my unique house will probably become "historic" and the owner after me will get a big fat tax break. But I, who actually did all the work to create this unique house, who literally hurt his back laying the bricks, who spent a great deal of money and effort to create something interesting that might be appreciated into the future, I get nothing. Meanwhile, my neighbor who hypocritically likes my new house yet has forgotten that something had to be torn down to build it will get a tax break for living in and maintaining a house that is supposedly in the "character of the neighborhood" despite the fact that his house is only one of maybe two or three adobe-style houses in this supposedly "historic" neighborhood.

Absurd NY Times headline...

"Obama Tells Bankers That Lending Can Spur Economy"

In other news,

Cheney tells plumbers that pipes help keep people hydrated.
Clinton tells IT departments that working computers increase productivity.
Bush tells football players that scoring points increases odds of winning games.
Gore tells doctors that curing disease makes people healthier.
Pelosi tells real estate agents that houses may keep roof over people's heads.
McCain tells rodeo clowns that deflecting bulls' attention may improve rodeo riders' safety.

Talk for EAST tour tomorrow (Thurs)

6pm at Clayworks, 1209 E. Sixth St. It's a bit of a repeat of the genetic art design talk I gave a few months back, so if you saw that you're not missing anything. Otherwise, please join us.

Mexican wave quiver plots

A lesson about interactivity. Having realized that I should be using quiver plots to understand the dynamics of 2-variable differential systems, I went back to the Mexican wave and made a quiver plot. In Matlab it is a slow task to tweak a parameter and re-plot over and over again so I made a zlab (C++) version of it that allowed me to twiddle the parameters in real-time.

What a difference! What took me a week before trying to understand the parameter space of this system now took me only 3 minutes. It is proof again of the power of interactive twiddling. When twiddling the variables the fast response and ability to "scratch" back and forth allows you to quickly intuit both the action and derivative of each parameter. When twiddling I was saying things like: "Oh... this is causing all the arrows on the left to go up" and "this is moving the steady-state point." I understand this system a lot better now.

In the following plots, I show two time traces at different places in space as indicated on the northwest space-time plot.

Plot 1: the stable Mexican wave. The third quadrant of the phase plot means "sitting & excitable (not tired)". Note that there is a stable equilibrium in that quadrant so if the system ends up anywhere in the 3rd quadrant then it falls into that basin and stays there until disrupted by, for example, a neighbor who pulls it towards the standing side.


Plot 2: two stable points. In this configuration, the 3rd and 4th quadrants have stable points so it simply transitions from the starting point in q3 until it gets knocked into q4. I guess if I were to hit it with a pulse of "tired" then I could get it to transition back again so this system is akin to a 1 bit memory. Haven't decided how to take advantage of this yet, but I'm sure there's something cool to be done with it.


Plot 3: oscillator. Now the equilibirum point has been removed from Q3 and so the system just perpetually oscialltes. Note that the oscillating attractor is stable -- the green and blue traces converge into the same limit cycle. I'm not positive what gives it this property, I thought that diffusion might be helping to stabalize it, but that's not the case as shown...

... in the following with no diffusion.

Plot 4: a system that has the Q4 equilibrium right on the boundary with Q1 so that it momentarily seems to be commit but then gets sucked into a small oscillator around the origin. This oscillator, however, does not appear to have a stable limit cycle so it dampens out.

Plot 5: a pattern forming system. Here Q1 and Q2 have a huge shearing force compared to Q3 & Q4. Somehow this seems to make it into a chaotic attractor but I don't really understand how yet.

Stability analysis

I haven't been posting for a few months because I've been suffering from another bout of depression (as I have many times before) which leaves me mostly bed-ridden and unable to concentrate. Thanks to Welbutrin and an SSRI I'm starting to get better although I still tire easily and spend many days partially in bed. That said, at least I can now concentrate long enough to get a little work done and use my brain for a few hours a day. This one blog entry has taken me about two weeks to write. Thanks to everyone who's been helpful and understanding while I've been sick.


To help with the amorphous computing projects, I've been trying to get my head around concepts of stability analysis in differential equations. In most of the textbooks I've read, stability is discussed with an analogy to balls rolling on curved surfaces such as the one below showing the ball (representing the state of some system) near an unstable equilibrium at the the top of a hill and thus tending to roll down into a stable equilibrium at the bottom (these analogies ignore the momentum -- its as if the ball, i.e. the system, is moving through viscous honey).




While these "ball rolling" analogies are helpful, they become detrimental to understanding when the functions have complex-valued curvature. This confused me for a long time until I realized that I should think about systems as fields using quiver plots like the following.

In a quiver plot, an arrow is drawn to indicate the solution to the function at each point and this permits viewing both the magnitude and direction of the field. In the above case, we see a 2D section of a counter-clockwise rotational field from the Lotka Volterra equation which is a simplistic model of a predator / prey relationship; the details of this system do not matter, suffice to say the model creates oscillating behavior as a rise in population of the prey creates a corresponding rise in the population of the predator which in turn reduces the prey which reduces the predator, etc.

If we were to start the system at some arbitrary point in this space, say 0.5 rabbits/acre and 0.4 foxes/acre and then we can integrate the system forward to get a trajectory representing the evolution of the system. In the following example, the system starts at the red star and then the blue arc is ticked off in equal intervals of time. As the rabbit population grows, so too does the fox population until there's a rapid turn around as the foxes over-hunt the rabbits causing the fox population to plummet. If we let it continue we see that it would get back to where we started and it would keep on cycling in this trajectory forever.

The idea of stability analysis is to examine such plots for patterns around the equilibria. The above case entails a fairly complex analysis, so let's start simpler and return to this case later.

Let's start instead with a simple linear system with a single stable equilibrium at the origin. No matter where we start the system, it will evolve into this equilibrium at the origin. This is a stable equilibrium as all the arrows point into the same place.

The above simple linear system is represented by:


where Z(t) is the 2-vector that is the state of the system and and M = [ -0.1 0; 0 -0.1 ]. Note that M in this case is a uniform negative scaling matrix.

Now consider the opposite: a uniform positive scaling matrix, we get the graph where all the arrows point outwards:

In this case, even if we start extremely close to the equilibrium, say at [0.01 0.01], the system diverges rapidly. This is an unstable equilibrium.

The following is the in-between case: stable in one direction, unstable in another. This is called a "saddle point" because of the shape of a saddle: curving up (from horse's nose to rear) and curving down (from side to side). This is M = [ +0.1 0; 0 -0.1 ] . In this case we have a change in convexity around the equilibrium so an approach from any direction towards the equilibrium causes a deflection. We can therefore see that only in the case that all directions point inwards will an equilibrium be stable.

So far, these systems are like the valley and the mountain analogies typically used. But now consider this linear system which isn't like those:

Here, M is a rotation matrix = [ 0 0.1; -0.1 0 ]. Here if we start anywhere we get a circular orbit. It's for this kind of linear system that the "ball-rolling" analogies fail. There is no "surface" shape that would make a ball roll around in an orbit maintaining whatever radii it started at. The "ball-rolling" analogies just don't make any sense but the quiver plot does.

Finally, for illustration, here's a case where there's rotation and a saddle point at the same time.


So, how can characterize these cases mathematically? Eigen decomposition.

First, we solve the system for the places where the d/dt is zero, i.e. the equilibria. For the above linear cases, this is extremely simple: the equilibrium is at the origin. Now, we look at the curvature of the system around that equilibrium. Again, for these linear cases, this is very simple: the curvature is the same everywhere. That is to say, no matter where you look, the percentage change of the arrows is the same.

Eigen decomposition tells us the components of curvature. Consider the following system which is unstable all directions away from the equilibrium but it is more curved (ie "more unstable") in one direction than another. In green are plotted the eigen vectors scaled by the eigen values. In this case, the eigen values are 0.1 and 0.3, both positive, telling is that this system is unstable in all directions. The eigen vector (ie. the direction of instability) doesn't really matter if we're just asking the simple question "is this stable or not?" but I've chosen to plot it anyways just for illustration -- as you can see, the eigen vectors point in the directions of maximum and minimum curvature.

The sign of the eigen values tells us about the local curvature. A system is only stable if all the eigen values are negative: i.e. all the arrows point inwards. If the system has rotation, then the eigen values will be complex valued. That's what complex value means: that there's rotation in the scale factor. For example, the following has eigenvalues 0.17 + 0.09i and 0.17 - 0.09i and you can see the rotation in the field. Note that the two eigen values are complex conjugates of each other. This is just saying that you can look at this as either clockwise or counter-clockwise rotation by just changing the sign. Complex eigen values will always come in conjugate pairs.


Another way to intuit the complex-valued eigen values is to consider the definition of eigen direction as those directions that don't change under the given transformation. Consider a rotation matrix. What direction is invariant under a rotation? Nothing in the plane of the rotation! Whatever vector is invariant must somehow be perpendicular to the plane -- and what do we call the direction perpendicular to any system? -- the (stupidly named) "imaginary" direction.



You can also see from this why the complex valued eigen values come in conjugate pairs -- both the out-of-paper and into-paper directions are valid rotational axes.


Now that we've seen positive, negative, and complex-valued eigenvalues for linear system, we're ready to look at a more complicated system. In a non-linear system there may be more than one equilibrium and the curvature may vary from place to place. However, the eigen-decomposition tool can only be used on linear systems so we look at the approximate linear system right next to the equilibrium. This is called "linearization".

For example, consider this plot for a dampening pendulum. There's two equilibria: one at the bottom (angle zero) and one at the top (angle pi). The one at the top is a shearing unstable point -- a tiny displacement either way will knock the pendulum away. By the way, the graph is periodic: read x axis 2*pi the same as zero. You can see that starting the pendulum higher near the top causes it to spin around a lot faster and more times as you'd expect.

It's taken me 2 weeks to write this part so I'll post it now and another entry later on linearization of complicated systems like this one and the stadium wave we're working on for the amorphous computing demo.

ScienceNews

A few points from my talk yesterday made it into the web edition of ScienceNews today.

Talk at Council for the Advancement of Science Writing

Just gave a talk for a conference of science writers. They shot some (not so high quality) video of it viewable here.

The slide deck is here.