Well, getting on to the 2nd half : On 4/10/07, Phil Henshaw <[EMAIL PROTECTED]> wrote:
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?
Well, maybe, but I'm perhaps being more verbose than needed there. The only critical part of the 'extra moxy' is approximating dots with a shape using some kind of local rules rather than a formula. A formula describes a fixed set of relationships from beginning to end, and local rules are adaptive and will display change. Where it gets a little interesting is with things like my 'hollow peak' smoothing kernel. It wasn't till much later that I realized the kernel of the rule I developed had an important dip in the middle, but it has the effect of strongly smoothing the higher derivatives of the implied flowing shape through a series of points without having much effect on the scale of the shape. Gaussian smoothing (recombining points with a smooth center weighting) has a well known tendency to flatten all bumps. My 'derivative reconstruction' rule doesn't. It may not be an earth shaking discovery, well, like, that emerging systems develop by growth..., but smoothing the higher derivatives without changing the scale of the behavior is extremely useful for exploring behaviors where the variation is not statistical but systemic. Then it does a great job of helping to reveal the shapes of the dynamic events in the system. > 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? ;-)
Just what I had in mind! Natural systems are the real ghosts of our world in a way, doing all sorts of work for us, and to us, and we really can't see them for beans because they don't have well defined boundaries in the usual sense. The idea of 'dressing them lightly' came sort of from those old movies where you see the dark rooms in the mansion full of furniture covered with sheets. What if when you took off the sheets the furniture turned out to be invisible? Well, you might put the sheets back on, and notice more about the difference between the shape under the sheet and the shapes of the sheet. That 'discrepancy' is what I'm mostly paying attention to learn about natural systems. Another example of this, not knowing what to call them, tricks for making hidden things show themselves, is the one where the 'invisible man' gets caught by making a 'clear spot' in a thick London fog. Anyway, growth curves, so long as you're looking 'through' them and not 'at' them, are great because you can very precisely locate the turning points where developmental feedbacks switch direction. Knowing when, and more or less what, from reading a loosely form fitting curve helps give you good ideas for finding how and where. > 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."
Surely one of the interesting modern discoveries are the various ways arithmetic that wouldn't seem naturalistic can readily produce naturalistic forms. Fractals seem to me to produce more of an aesthetic suggestion of nature but not very naturalistic forms really. L-curves, if that's what was being used in the demonstrations of animated plant morphology, seem much more interesting. I'm skeptical though. The images and animations of the plant branching forms of weeds I was so impressed with once, are maybe too naturalistic, so I think they may be CG images since the rest of the study on the Wiki link seems to be of rigid geometries. Nature may arrive at rigid geometries occasionally, but always seems to me to do so by another means. Are you talking about something different?
yea, focusing on the parts of nature that work with a backward direction of causation. Inside out instead of outside in. I find it in the local animation of events, and an 'all together' kind of causation that's really quite common too. There are enough uncertainties, to be glad to have some rigor and to not dwell on unanswerable questions. I've been basically doing this for 25 years, but just starting to find a useful language for it. > 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?
No, it's not an equation, but has all the derivatives. It's a data curve with rules for filling in dots where you need them in a way that effectively constructs a mathematical continuity. Say the rule is put a dot between dots, smooth a little, then do it again. That provides a procedure for obtaining a point at any position along the curve and determining as close an approximation to an instantaneous slope at that point as you like (describing the combination of the data and the rule, the real and artificial description). It's helpful for finding exactly when curvature reversals occur and such things. For example, it's interesting to know that very very early evidence of the 80's crack epidemic in the statistics of murder rates in NYC is the turning point indicating suppression of the scourge in Staten Island... That sort of thing is helpful in building an understanding of systemic events. 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...
Well, where you'd find the non-zero higher derivatives of the same sign (growth curves) is at the beginning and ending of the finite polynomial periods (transitions to and from). The periods having constant description have higher derivatives all zero. The transitions show their beginning and ending of the steady states are non-linear, 'non-steady' states. That's the big thing you find when you don't use regressions! > 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.
Then they're not simple growth curves. It's possible to have a complex sequence of derivative sign changes, say if you had an 'S' curve for which the derivative was a simple hump that completes in a finite period (making a pair of mirrored 's' curves), then the derivatives of those might conceivably make two humps,... etc. Say you asked where the pure growth curve was in that, the one with all higher derivatives positive. It would have to be at the very beginning of the first half of the first 's'. It would be interesting to know if any physical behaviors have that kind of complex layering of beginnings and endings of beginning and ending, but I think it's not something one can write a formula for. I really have not gone very far with this, except to see that IF you have an event with a beginning and end, then you'll most likely find a continuity with all derivatives positive at the beginning and all derivatives negative at the end. It's a direct corollary of conservation, just a two page proof, as well as what I observe. 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
not much. the problem with equations is that their whole past and future is completely implied by any finite segment. Because an equation 'knows' it's future before it gets there it's not a very good representation of systems that can't reflect their futures until they get there. If you happen to have a formula for the future you think should work fine, it's a very helpful indicator of when nature has discovered another way to work that the discrepancies between the formula and observation are diverging on a growth curve. > 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.
Well, any progress? Hope I'm not digressing too much. Sometimes It's hard to understand the question is being asked...and so I answer something else... Phil -S
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