>>> John Boone on Biofuzz 
 WE> Given a function f measured at discrete times t1, t2, ..., 
 WE> ti, ... with ti < ti' where i' = i+1, plot the pairs <  
 WE> f(ti), f(ti') >.  So what's the spread function?  A measure  
 JB> The spread function is the thousand dollar question and 
 JB> one I am still pondering.  Something, more mathematical, 
 JB> "what is the minimum domain of the representation function 
 JB> (f(ti),f(ti')) such that the heart shows sufficient chaotic 
 JB> behavior?" 
I would think the spread function would be a measure of distribution such as 
standard deviation of the distances between all pairs of points.
 
 WE> of what?  The function of your example is 60/(Pulse 
 WE> Interval).  Why the reciprocal? 
 JB> In response to "Why the reciprocal?"  I assume by this 
 JB> why plot (f(ti),f(ti'))?  It was arbitary and gleamed by 
 JB> the work of some in "Scientific American" four years ago. 
But why plot 60/(Pulse Interval)?  The 60 is most likely because of 1 minuet 
= 60 seconds.  But why not (Pulse Interval) directly or (Pulse Interval)/60?
 JB> There are many other ways to represent the data. 
 JB> However, it a reasonable way to represent the data. 
 JB> If you look at a -simple- system of degree 2, pendulum 
 JB> with moving pivot point, the representation of the 
 JB> movement of pendulum as a function of time could be 
 JB> done by (P(ti),P(ti')). 
(P(ti),P(ti')) and then (P(ti'),P(ti'')).  This switch of ti to ti' defies my 
ability to visualize just what is happening.  This glib switching the x and y 
axis indicates some conceptual symmetry I'm not grasping.  Why the two 
dimensional data display?  Why not just calculate delta-f(t) = f(t') - f(t) 
and take the standard deviation of the delta-f's?  
 JB> It seems reasonable to continue the method of 
 JB> representation with heart rate.   
Don't get the pendulum example, too complex.  How about a drummer, thump, 
thump, thump or a typist, click, click, click.
 
 JB> Ah, bingo, in healthy hearts, the pulse interval doesn't 
 JB> hold steady, but in fact is "choatic." Reason, the pulse 
 JB> interval is a representation of a multiple of excitable 
 JB> tissue or cells, in this case, the heart. 
 JB> IOW, the less "choatic" the heart the more unhealthy 
 JB> it is.  
Hm, what about fibrillation and other erratic heart mummers?  Aren't they 
chaotic?  A healthy person movements has a large plot domain while a dying 
person movements has a shrinking plot domain.
 
 WE> will all fall at about the same point.  A -small- spread 
 WE> but a likelihood of death?   
 JB> We find that as the plot domain diminishes, the less spread, 
 JB> the more unhealthy the heart.  
Plot domain?  The smallest rectangle with sides parallel to the axis that 
contains all the plotted points.
 WE> Any fuzz in any of these magazines? 
 JB> Nope, the one I am currently reading, reentrant supraventricular 
 JB> tachycardia, fast heart beat. 
 JB> There doesn't seem to be much research in this area, perhaps, 
 JB> there doesn't seem to be an awareness or understanding of "fuzzy" 
 JB> or fractals. 
I've heard it rumored some years ago that medical students are mathematical 
shy.  Is that so?  Are you in medicine or biology?  Are you considering a 
fuzzy thesis or research proposal.  Wish I could help you, but my fuzzy 
background is scant.  What you're considering would require understanding 
application examples.  BTW, do you have access to the SCIENCE fidonet echo?
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