Collier et. al. data and The Test

[From Bill Powers (950810.1710 MDT)]

Bruce Abbott (950810.1040 EST) --

Excellent application of the test. The estimations were creative and
logical, and the remainder of the test was correctly applied.

This is a good first approximation. I think we can get even closer by
considering the details a bit more. The basic relationships are (with a
tentative control model added)

                         Ref intake rate
                                >
      ----------->------------[COMP]------error---->-----
     > >
perception [overall gain factor]
     ^ |
     > -<-- meal -<-- [RATIO] -<- Press-- g1<--|
     > / rate Rate |
   Intake<---[x] |
    Rate \ |
                  -<-- meal |
                       size -<-- Amount eaten<------ g2<--
                                 per meal

Not shown is the sequencer that switches from one feedback path to the
other as appropriate.

We can observe the meal rate which is 1/ratio times the pressing rate,
times the amount eaten per meal. The pressing rate and the meal size are
the behaviors we would consider for the Test. With a constant pressing
rate and constant meal size the intake rate would vary inversely with
the ratio.

Following your strategy we can estimate a baseline pressing rate and
meal size for the lowest ratio, FR-1. The free-feeding rate is used to
estimate the reference signal; we would use it only in fitting a model.

The first computation involves simply leaving the pressing rate and meal
size at the values measured or estimated for FR1, and then computing the
intake rate for higher ratios without changing pressing rate or meal
size. This is the estimated change in intake rate that would occur as
the ratio changed, if neither behavior changed. Note that the ratio
affects only the one path involving pressing rate. If you want to add
effects of pauses this is where to do it.

This gives the overall evaluation as to whether there is control. The
result should be pretty clear.

Next we can try to assess the amount of control due to each path. First
we introduce explicit gain factors and try to estimate them (checking to
see if control is still found), and then we use a control-system model
to try to refine the estimates of the gain factors.

We can say that the behavior of the system is represented by the error
signal, with the pressing rate being g1*error/ratio and meal size being
g2*error. Starting with the estimated meal rate and meal size for FR1,
we can compute values of g1 and g2 such that meal rate * ratio / g1 =
error and meal size / g2 = error. It should be possible to solve for g1
and g2 at least approximately for an assumed error value. Then, with the
assumption that the error signal is constant at its inferred value for
FR1, the expected intake rate can be plotted for each value of the
ratio. This again should give a clear indication that there is control.

Finally we can complete the control model as shown above, putting in an
overall gain factor (perhaps an integrator or leaky integrator) between
the error signal and the two factors g1 and g2, to allow adjusting
overall gain.

Notice that the net gain in the pressing-rate loop will decrease with
ratio, while the gain in the other loop remains constant. If the gain in
the meal size loop g2 were larger than g1, obviously most of the control
would occur because of that loop, so the error would be kept small and
the pressing-rate loop would see very little increase in error over the
ratios. So the pressing-rate loop would produce a steady decline in meal
rate with increasing ratio, as observed, while the meal size loop would
produce a steady increase in meal size, also as observed. By adjusting
the ratio g1/g2, it should be possible to match the behavior of the
model to the data on both pressing rate and meal size. Note that there's
no need to use logs.

Bravo. Keep going.

···

-----------------------------------------------------------------------
Best,

Bill P.