[From Bruce Abbott (960120.1905 EST)]
Bill Powers (960120.1530 MST) --
B
B' = ----------
B + delta*B
The "B" on the right does exactly nothing, because this expression is
the same as saying
B' = 1/(1 + delta),
Sorry, that was a copying error. The correct equation (which I gave in an
earlier post) is
B
(2) B' = -----------
1 + delta*B
That's the trouble with this "instant" publication: no copy editing! I do
try to proof my work but don't always catch everything. Dividing top and
bottom by B gives
B/B 1
B' = --------------- = -----------
1/B + delta*B/B 1/B + delta
Thus B' = 1/(IRT + delta), since 1/B = IRT. Inverting:
IRT' = IRT + delta, as I stated.
I've also caught another one, in my description of the program's operation:
The simulation begins by setting IRT' to some reasonable baseline level and
computing the IIT that would result from this given the schedule ratio (IIT
= IRT'*N). When each incentive is delivered, the time since last incentive
(IIT) is saved and used in equation 1a to compute a new IRT. This is then
plugged into equation 2a to give the new IRT'. A response is emitted as
each IRT' time-period elapses, and the response count is incremented. When
this count reaches N, the next incentive is delivered, the new IRT recorded,
^^^
IIT
Your conversions to inter-response times are really unnecessary. You can
define new variables as the reciprocals of old variables and make the
equations look different, but this doesn't change the basic
relationships. Sometimes it makes the algebra easier and sometimes it
makes it harder. But it doesn't introduce anything new. And it can keep
you from seeing the obvious.
Yes, if you do the algebra correctly, you are only changing appearances.
The reason I have done this is to try to _understand_ the relationships a
little better. That's all mathematics is: you manipulate some equations and
reveal more clearly what was already implicit in the equations in the first
place. But sometimes the equations _as presented_ can keep you from seeing
the obvious. Manipulating them a bit may reveal it.
Manipulating equations can also reveal simpler ways to display the inherent
relationships. For example,
B' = B/(1 + delta*B) is a complicated function to plot and fit data to.
IRT' = IRT + delta is not; you can fit it to data with a ruler.
In addition, my simulation was easier to construct around the times between
events as opposed to the rates.
How about that complete set of equations?
By now you should have received KILLEEN.PAS via private post. But here are
the basic equations Killeen presents:
Activation: B = a*R/delta
Ceiling constraints: B' = B(1 + delta*B)
Behavior: B' = (zeta*R/delta)/(R + 1/a)
lamda: lamda = lamda'*delta/rho
coupling: zeta = rho*[1 - exp(-lamda*N)]
FR behavior rate: B' = (lamda/delta) - (N/a)
The behavior equation is produced by substituting the right side of the
activation equation for B in the ceiling constraints equation and
multiplying the whole thing by zeta, the coupling coefficient. The coupling
equation is specific to fixed ratio schedules; Killeen derives others for
other schedules.
The FR behavior rate is derived by substituting B'/n for R in the behavior
equation and solving for B'. This is the system equation for asymptotic
output, in which B' is NOT a function of R.
Terms:
B predicted behavior rate excluding response duration resp/sec
a specific activation sec/incent
R incentive rate incent/sec
delta response duration; minimum interresponse time sec/resp
zeta coupling coefficient
lamda' intrinsic rate of decay from short-term memory
lamda observed rate of decay from short-term memory
N ratio requirement resp/incent
rho proportion of instrumental responses in response stream;
set to 1.0 for terminal performance on ratio schedule.
Regards,
Bruce