[FRIAM] predictive models v. causal mechanisms

[Stephen Guerin]

> 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 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."

> 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?"


> 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.

> 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