Well, I'll start now and probably need to get back to it tonight...
On 4/10/07, Stephen Guerin <[EMAIL PROTECTED]> wrote:
> Well, yea, you're onto a parallel design there. I'm > usually referring to the individual instances of physical > processes that correspond to the general models of 'basins of > attraction'. When you say individual instances of physical processes, I translate that to "a specific trajectory through the phase space of a system". <http://en.wikipedia.org/wiki/Phase_space> Am I correct?
If we were both talking about systems of definitions, then yes, because a set of related definitions considered as a universe of relations defines an individual system. The corollary for physical systems, has to be observed and represented differently (because physical things can be predicted, but not defined, and the difference is significant) When studying how an individual system evolves, say an air current, how all the possible air currents that satisfy the ideal laws of mechanics might evolve is not what the individual system is doing. The ideal laws of mechanics are also 'ideal' and don't take into account the actual physical mechanisms by which individual systems operate, just some general predictabilities of the expected range of behaviors. Since one of the main reasons for systems theory is that very very small differences can have large consequences, it is highly useful to study both systems models which are well described but rather inaccurate representations of real things and ones that are completely accurate, even if imperfectly described. When you use them together you find big substantive differences.
If so, wouldn't a trajectory moving through a phase transition be a growth curve? > As far as I can tell 'basins of attraction' > are hypothetical constructs designed to improve the accuracy > of statistical models. A basins of attraction is a way of characterizing points in phase space of a dynamical system, real or modeled. They may be an abstract description, but I don't think they're hypothetical. A basin of attraction is the set of states in a dynamical system with future trajectories that tend toward a common stable state (attractor). Or from MathWorld: "Basin of Attraction: The set of points in the space of system variables such that initial conditions chosen in this set dynamically evolve to a particular attractor."
The parallel between the two approaches, like two metaphors for the same thing, are on the surface. The two approaches are built in fundamentally different ways, but produce comparable descriptions of many of the same kinds of macro level phenomena. Physical systems are definitely 'dynamic' but because of the confusion of terms I tend to use 'animated' instead, because I use kinds of generalities that are different from equations. To me, modeling nature with a concept of 'phase space' is a sort of 'cheating'. It presupposes that the system contains all information about all it's future states. Since reaction to change is such a prevalent behaviour of real systems I've concluded that systems don't know their future, that system evolution is a discovery process. I've think this is a highly useful finding, even if it's a 'generalization' that can't be derived from a set of axioms. It may be the kind of generalization from which a set of axioms can be derived, however, and that's more or less what the 'bump on a curve' model is.
and are a very useful construct but > don't physically exist. Do growth curves physically exist? As in, "oh my god, did you see that growth curve crawl behind the couch?"
Separating physical things from their images is dicey, of course. To me an image is a projection of a set of rules, and physically exists only in the sense that the rules are recorded somewhere, and an 'agent' reads and applies them. If you mean by 'growth curve', as I usually do, a shape which *I imagine* in a set of data, then no, growth curves don't physically exist. Another thing which helps to tell images from things is that things generally have fairly evident complex hierarchical structures and history based changes that you can describe in more detail the more effort you make, but still always remain imperfectly defined. Because my approach to systems is aimed at finding things that are physically real, not just projections of my own, I pay a fair amount of attention to keeping the difference straight, and correctly using terms in natural language that seem as if they were invented for the purpose.
What I find, anyway, when looking > to see how patterns of organization develop in individual > instances of emerging systems is a lot different, and so my > language parallels the standard physics models but addresses > different phenomena. > > What I start from might be with identifying the boundary > between the inside and the outside of the feedback loop > network, finding the wiggly line separating the complex > interior network of relationships that is acting as a whole > and its environment. Lost me. Can you give an example of a wiggly line that separates the complex interior network of relationships that is acting as a whole and its environment.
Well, let's say a news shop in town lowers the price of their picture magazines, and that other shops notice their customers disappearing and then firs one then flurry of them discover why and match and exceed the price break, starting a little price war which ends with a stable lower price structure and a few old time vendors dropping out of the market. That's an example of the usual sort of thing one would call a 'single emergent systems event'. It's pretty easy to tell if the action is all in the one town, and that the 'S' shaped magazine price curve of the aggregate reflects the separate 'S' shaped price curves for each individual shop. As such you can draw a circle around the event in space and before/after markers in time, i.e. localize it. From there on it gets more complicated, because you might find that what triggered it was actually involved with a particular popular issue of a particular magazine... etc etc. That's where the line between the internal loops and the external connections gets harder to separate. gtg... thanks for good questions! phil
It's good > to mention that reading subtle changes in the evolution of > the system from subtle changes in the continuity of the curve > of the data does usually involve a projection of idealized > continuity from the actual dots of measurements. some kind of nonlinear regression with extra moxy? > I favor > things that are much less heavy handed than splines for that. > The useful assumption seems to be that there is a form > there and it'll be easier to see if the 'clothing' you drape > it with is loose fitting... Perhaps a easy-breathing cotton bézier would do the trick? ;-) > Where the parallels separate is that when studying individual > instances of anything there is no 'definition' of the system, > and no feature of the physical thing which 'describes the > state of the system' as an 'order parameter' might. > As > close to a 'state variable' as one might get is the hard to > explain origin of a growth system, its starting design. > Because growth structures are 'sticky' and accumulate around > and branch off from the original loops, the character of the > original loops remains to the end. Are you familiar with lindenmeyer systems (l-systems)? <http://en.wikipedia.org/wiki/L-system> "The recursive nature of the L-system rules leads to self-similarity and thereby fractal-like forms which are easy to describe with an L-system. Plant models and natural-looking organic forms are similarly easy to define, as by increasing the recursion level the form slowly 'grows' and becomes more complex. Lindenmayer systems are also popular in the generation of artificial life." Are you talking about something different? > What I've come to as a workable technical definition of a > 'growth curve' is a period of time in a measured behavior > when all the higher derivatives have the same sign. So, you take some measurements from a system over time and then do some kind of regression on those measurements? And then look at the derivatives of the resulting equation? As a brain-dead example, let's say I launch a rocket and continually increase the rate of fuel burn while escaping the gravitational field until I'm in orbit. During the launch, I record the height every 5 seconds. If I graph height on the y-axis and time on the x-axis and fit a polynomial to it, I would have positive 2nd and 3rd derivatives in velocity and acceleration, right? I realize that's probably not what you had in mind as a growth curve but it fits the definition... > If for > ising model a measure of it's behavior displays such curves > then I'd say so. I would explect the graph of the phase transition to be sigmoidal which would have positive first derivatives and mixed positive and negative second derivatives. Initial growth is exponential but slows in the end as most of the spins are locked in. BTW, the sigmoid function is the solution to the logistic equation < dx/dt = rx(1-x) > which is used to model population growth...Is that of interest? http://mathworld.wolfram.com/LogisticEquation.html > What may be difficult with the kind of lab > setup used for helping to refine prediction models is that > you'll have a hard time telling the difference between one > run of the system and another, I'm not sure. If you can, > and see eventfulness (presence of growth curves) in the trace > of the differences, then you're in a position to ask pointed > question about what made those system developments. You may > not find the answer, of course, but you often find new stuff > of some kind when you ask new questions. > > does that make any sense? Not really. But I can wait until you answer the other questions. -S ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
