I’m reading “Chaos Theory Tamed” by Garnett P. Williams, copyrighted in 1997. It’s a very good book. The necessary math is explained without calculus. I’m not going to go into chaos. No reason to call Maxwell Smart. The book is definitely not a mud thick, quicksand, mind resistant book, aka a textbook. I recommend this book. My public library always surprises.
Williams does something that most researchers do, he talks about the shape of his functional culture. He probably does this more than other authors I’ve read lately. So I wanted to show how he does this, and give requirements elicitors a few hints to help them make a functional culture and its subcultures more visible.
So off I go to look up the quotes, the quotes I didn’t highlight when I dug into this book. How did I know I’d be writing about this? I didn’t. My requirements forked. The original requirements didn’t change. Not at all. I wanted a fast read. But, it turned into a slow know this stuff by the time I put the book down project.
So right here in the Preface, Williams stories the discipline of chaos right here in his goals for this book. Storying the data is the act of contextualizing a thesis. Like so and so, but differ from …. Every PhD thesis starts this way. So go back to 1997 and take a look at chaos theoreticians. Williams uses another term for them, but after looking up that term up in Google 14 years later, I’ll not use that term in this post. Yes, in 14 years, chaos has changed, but it’s still chaos. As Williams puts it, chaos has an insider, a surface, and outsider population (in a bullet):
Most books on chaos … use a high level of math. Those books have been written by specialists for other specialists, even though the authors often label them “introductory.” Amato (1992) refers to a “cultural chasm” between a small group of mathematically inclined initiates who have been touting chaos theory, on the one hand, and scientists (and, I might add, “everyone else, on the other….
So we have three populations divided by two cultural divides. Cultural divides, because meaning partitions populations into those that don’t know, those that know a little, those that know it cold.
I play this old game called Snake. It looked like an Ito process to me. So did this Chaos stuff, so I tweeted someone who would be able to tell me if I was correct. Well, he felt is was Markov. I was a bit stunned. Markov is a process with no memory, Ito some memory (finite), Gaussian/Bayesian all memory. His answer reflects my now refined definition of Ito, Zero is finite, so all Markovian processes are Ito processes. I had to dig. I finally figured out what my peep was saying. His answer was a perfectly clear statement. It didn’t mean what he said. Yes, a nuclear submarine could have squeezed through there. As experts in human ambiguity, you’d think our ambiguity sensors would have gone off. Nope. And, like every good requirements elicitor, we’d claim that the requirements changed. I’m on the surface. Peep was an insider. Not faulting my peep. In fact, thank you very much. Understanding is always gold. Your elicited won’t anticipate your misunderstanding either unless they are writing a multiple choice math test. I hate those. Grr.
Another argument that Williams is taking up here is the pedagogical pathway, the sequence through which one teaches, or was taught. Where you anchor your introduction selects your readers, students, and if your media is software your users. How much must they know? How much of an insider must they be? That introduction exists in a space of constraints and enablers (affordances), which pretty much accounts for why there are so many of them for a given subject, and why you can always write another one.
Pedagogical pathways turned out to be important in Argentine tango. I learned “going to the cross” a certain way. I teach it that way. Other instructors don’t teach it, or teach it differently. I went on to deconstruct my tango and relearn it. One day I learned to go to the cross without all the conventions we used. Instead of those conventions, you just extend the two men in a donkey metaphor, and the leader’s right shoulder puts the follower, her left foot to be more exact where it belongs, while you give a little down lead on her back. Done in the distance of a single step, a single beat of music, rather than step outside, step inside if, and finally the right shoulder left foot thing. Why did we teach it the longer way? We did it so the learner would believe in themselves. Learning is a matter of moving from the unaware, unknowing to aware, unknowing to aware, knowing, to unaware knowing. With that aware unknowing comes fear, an obstacle to learning, and in a voluntary situation, it is libel to cause the student to give up and quit. We needed more dancers, not less. But, why learn it the donkey way? Because the conventions are shared and where the conventions are not known, conventions become problematic. When you walk into a new venue and ask a non-dancer, rapt watcher in the audience, if she wants to dance, she’ll say no, because she never had a lesson–she is afraid. Don’t get her wrong. She wants to. And, guess what, you have to make that happen, and you can if you don’t have to rely on conventions.
Those pedagogical pathways are important to us elicitors, because they reveal the conceptual underpinnings of doing, the meaning at the core of the ritual, the ritual we hope to encode and encase in GUI pixels. If all of the people elicited went to different schools and read different textbooks, expect that they have taken very different pedagogical pathways through the given subject. Even an Aha! is arrived at via a sequential cognitive pathway of this idea before that one. In chaotics, these sequential pathways become phase space pathways and dimensions.
Back to Williams:
In this book, I assume no prior knowledge of chaos, on your part. …, I try to discuss only the most important ones [topics]… I try to present a plain vanilla treatment, ….
So we will visit the peaks, not the valleys of the cultural terrain of chaos. And, really Williams has a bit of attitude with this vanilla thing. It comes up again and again. We won’t be joining Williams up there on the summit. That’s OK. This is the 101 class, the surface, feet wet, freeze at the knees spring.
Williams is himself a geologist/hydrologist that came to chaos much like us, non-mathematicians, from somewhere else. He mentions his writing style and its goal of reducing the distance from the subject and the reader, a UX goal.
In defining nonlinear, he says:
… An alternate and sort of “cop-out” definition is that nonlinear refers at to anything that isn’t linear,…
I still remember in linear algebra class, the fill-in professor, who was taking the place of THE linear algebra professor, who died the day before the semester started, writing an equation on the chalkboard and saying, This is an eigenvector” So I had to ask, “What is an eigenvector?,” “This,” he answered. That was the end of my attention to the subject for the rest of the semester.
“Us against them (you),” a cultural boundary. No not a stupid question. And, not a wrong answer. Just an answer with no batteries for the flashlight. Williams went on to say that we didn’t really have to know what nonlinearity was beyond this cop-out definition. We were not going to become one of them. Let’s face it, we are happy to be BAs and PMs. We wouldn’t take the pay cut to become one of them. Although another day like yesterday, and we’ll be feeling around wondering where is that towel?
That cop-out definition was William’s proof that he is one of us, not them.
You’ll often see the term “flow” with differential equations. To some authors … a flow is a system of differential equations. To others … a flow is the solution of differential equations.
So where so we stand? I’m sure some of us can handle differential equations. I can, but only lightly right now. It’s an active goal. At any rate, start with the universal set U, draw a subset inside that for the differential equation crowd, and divide that crowd into two mutually exclusive populations (subsets). The line between those two differential equation crowds is where those pedagogical pathways would be to take someone in the unknowing crowd into the knowing crowd. “Belay on!”
Not all differential equations can be solved, so you might have to rappel down and climb up again. This thin line in the diagram is really a deep box canyon, and it’s an easier climb if you learned your integral calculus cold way back there on your pedagogical pathway. There will be a moment or two on the line when someone will yell out, “To the right, slightly above you, a crack!” You’ll reach out. You can’t see it. You have to feel it. There it is. Fear and relief, learning again. On the climb again, get off that plateau. Ratcheting up. You’d think that the people you are eliciting would remember the guy that tipped them off, or that they would forever recall that they didn’t know the tip. When you elicit them, it was smooth, they were brilliant, nobody helped them, the teacher/professor sucked, and education failed them. Really?
Don’t worry, that hint will turn into a bug. Why didn’t you …? Well, you didn’t …, and … (not the time to drag out the tape and prove it).
That last step in learning, unaware knowing, is in knowledge management terminology a reimplication, the explicit disappearing back into the implicit/tacit, as if it was so obvious why would anyone bother to explicate it at all. It’s just the way all of us humans are. Culture is so transparent that you can’t see it even if it obscures things.
Some authors like to label the first observation as corresponding to “time zero” rather than time 1.
As programmers, how many times to we make this error, the off by one error? You probably see it as random event, rather than the behavior of a population. Well, for authors, it’s not an error. It’s a way of seeing the world. Nobody is going to get in a fight over A0 vs. Ax. They look at it and think, oh, one of them. Likewise, you can run into a differential equation book written in Newton’s notation, rather than the usual Leibniz notation. Still, a frontier checkpoint where the customs guy has gone home for lunch leaving the gate up.
When you’re looking at a conceptualization, just remember it was constructed to sell it’s constituent concepts. And, every concept is a producer to some population’s consumers. “Are you ready to consume this idea yet?”
Vectors are points. I only recently came to that. They look like lines between two points with an arrow at one end. But, normalized, a point is all you need to specify a vector. If you know that, congrats, you’ve been ratcheted up. Everything is easier where you are. If you didn’t, so everything was tougher until the day I joined you on the other side. The point again, a conceptualization dividing two populations. Tougher and more relevant is that BAs and PMs will end up eliciting requirements from both of these populations, because you didn’t sort them out. You can’t do it both ways can you? So you do it one way, half way between both. Sure, with vectors, no way, but with distant stuff, subtle stuff, how do you know? And, again, the customer didn’t know what they wanted, so the requirements changed. No, no the requirements didn’t.
There are three ways to show a set of sequential measurements….
- The time series….
- The pseudo phase space plot….
- Th wave-characteristic plot….
Because wave-characteristic graphs show a parameter’s array or spectrum of values…, people often attach the word “spectrum,” as in “”power spectrum,” or “frequency spectrum.” The terminology isn’t yet fully standardized. Commonly, a “variance spectrum” or “power spectrum” ….
Some authors reserve the term “spectrum” for the case of a continuous (as opposed to discrete) distribution of ….
Statisticians just love frequency-domain transformations of time series….
Different terminology sure. Different times. Different people. Different functional cultures. And, different rituals where Williams is telling us what statisticians love. Different states of adoption as well.
Williams brings up an interesting aside when he talks about Fourier series being limited to waves of finite periods:
… Mathematicians had to figure out a way around that problem….
Until this problem, this constraint, was solved, Fourier’s methods could only be used by a small group of people, the market. Once the mathematicians cracked this constraint on the use of Fourier’s methods, and people adopted this new technology, the market grew larger.
Most techniques and concepts in chaos analysis assume that raw data were measured at equally spaced time intervals. Only a few techniques  have sophisticated variants to account for irregularly spaced data. Hence the best approach by far is to measure data at equal time intervals.
Williams goes on to list three methods, and follows this list with
Our discussion will be based on equal time intervals, unless specified otherwise.
The park ranger is telling you to stay on the trail and not to create any new switchbacks. Easier on you that way. The trail is again the pedagogical pathway. He did this earlier when he covered DFTs [nevermind what they are]:
The mathematics behind the DFT are vast and complicated. Many books….
At least he hinted at where the adventurous could find what they were looking for. The people you elicit won’t do that, and somehow, you’ll have to be able to become one of them if you aim to capture ethnographic field notes and eliminate requirements volatility at its source: populations and their various meanings–the preludes to doing.
The way he handled the dangers of DFTs to learners is exactly what we did as river canoeing guides. Waterfalls are dangerous. It’s ok to portage them. They portage it, we run it safely after they are downstream. It was only four feet high and didn’t have a deep keeper. You have to ditch the life jacket and dive out under the keeper. No thanks. Yeah, now that we’re done with that, feel free to explore. Just don’t let them get behind you on the river.
Then, Williams dives into standardization and differencing. He walks us through standardization. Yuck! It reminds me of factor analysis and linear programming. Nothing difficult, just endless iterative mathematics, over and over again. And, when you are fed up with that differencing comes to the rescue. Easy. Done! So why did he take the pedagogical pathway that he took? Did he do it to give us something spreadsheetable? Did he do it to make us love differencing? Work is persuasion. Lazy wins. What about the people you elicit from? Did they walk you the long way to the destination, or the fastest way possible? Knowing that they were not going to get what they are asking for, where they punishing you, or did they think that the fastest way would get you out of their hair? How would you know? And, in the end, when they don’t get what they needed, who will pay the costs? Not you. Not the vendor if you are the vendor. Not the IT department after months of effort to an anti-heroic end, except that the functional unit had to hire a few more unemployed people. BAs and PMs can get people back to work by ensuring that meaning can only be recovered from an application by many more hours of staff labor than was previously slated in that units budget. Hire another staffer.
When discussing attractors Williams used the terminology of approaching the attractor asymptotically. Doesn’t he mean limit I wondered. Well, yes, eventually he got around to defining epsilon and walking down that geometry-based intuitive approach to limits we got hit with in calculus class. But, he went on to say that statisticians see it as a matter of hitting the noise and having the signal disappear, the confidence interval becomes epsilon. Then, there’s me and that line as a collection of convergent sequences using a set theory-based definition of a post-geometrically intuitive mathematical limit.
I brought outside stuff to my understanding. What about when we elicit? Sure we do. How do we not? Do we have practices that eliminate spill over. Would we stop asking questions once we thought we knew? Would we have ever gotten to the statisticians definition? Set theory doesn’t belong in the mix at all.
And, lastly, I’ll go back to the side point about constraints. It’s 1997, Williams is saying
… There’s no way to predict long-term evolution…
That was a constraint back in 1997. It might have been a collection of constraints. This is where value lives. Eliminate that constraint, and the dollars will flow into your pockets.
Chaos is deterministic, and non-stochastic. There is no reason whatsoever that you can’t predict long-term evolution, at least not from where I stand.
Williams was always saying “Other authors…” so comments please. Cross the cultural divide. (more…)