Weight control

[From Bruce Abbott (960329.0745 EST)]

Bill Powers (960329.0100 MST) --

Bruce Abbott (960328.1730 EST) --

    But the charts for 96006, 96008, and 96010 are the ones I find most
    interesting, from a weight-control standpoint. What do you think?
    Bill P., how about an interpretation?

They are indeed interesting, if the weight continues to be constant at
the same level as the final few entries. They look like a slow control
system recovering from a transient disturbance that took it considerably
outside the normal range of control. This interpretation would become
more defensible if all the rats eventually reach a constant weight. It
would be hard to explain why some rats control their weight and others

Yes, of course. It's still rather too early to tell whether the apparent
leveling off shown in these rats hold up in the long run, and we still have
several rats gaining weight steadily at rates varying from about 0.8-1.5
grams per day. But the pattern shown by these three is an absolutely
classic example of the response of a control system with a fair amount of
lag to a large transient disturbance. These observations haven't proven
anything, but finding what appears to be evidence control-system dynamics in
these weight curves is certainly encouraging. I used to see curves like
this on occasion in my former life as a research technician for
Owens-Illinios Glass Co. Some of our laboratory furnaces were hooked to
proportional controllers, but these had to be adjusted properly or you got
overshoots (and undershoots) looking much like these weight curves when the
furnace temperature set-point was changed to a much greater or lesser value.
The reason was that the heat being generated by the heating rods had to soak
into the large thermal mass of furnace's interior, creating a big lag
between rod temperature and the temperature being picked up by the
thermocouple at the furnace's center. If the gain were set too high the
controller would just keep pouring on the power at relatively small error
values. During an increase, by the time the thermocouple reached set-point
the deep interior of the furnace was already well above, leading to overshoot.

However, they also look like a high-gain control system with a reference
level that is varying in the patterns shown, or like a control system
with a constant reference level, a modest gain, and a slowly varying
disturbance, or a control system with a constant reference level, a
constant disturbance, and a slowly varying loop gain. Since we have no
independent measure of disturbances that affect weight, we can't really
distinguish these possible models.

Yeah, we'll have to investigate further. Remember, I wasn't really
conducting a study on these rats; in this sense the data are fortuitous. If
weight had already reached set-point by the time I started weighing the
animals, there would have been nothing interesting to see in the graphs
except perhaps a small periodic fluctuation owing to the estrous cycle.

If we could measure the daily food intake of the rats, we would have a
measure of an input variable that influences weight and that is one
primary means of controlling weight. When the slope of the weight curve
is positive, the food intake should be found to be greater than when the
slope is negative. We might be able to derive a transfer function
showing how weight depends dynamically on food intake rate.

A running-wheel, if used by the rat, should introduce a negative
disturbance of weight, and a corresponding increase in food intake.

I'd like to find that transfer function. For now, I'll see if I can get at
least a gross measure of food consumption on those rats still showing rapid
weight grain.

All measures of weight and food intake should be raw data, not smoothed.
If there is any smoothing, it should be done on the parameters of the
transfer function -- at the very end of the running calculations, not
before they are done.

I offered the smoothed data here to emphasize the long-term dynamics. The
running averages act like a low-pass filter, removing high-frequency
dynamics from the curves. That sounds odd, doesn't it -- "high" frequency
is periodic variation on the order of five _days_ or less in this system!
Any model we create would have to be fit to the raw data, as you say.

A clearer picture of the meaning of the data would be seen if the y
dimension of the plots started at zero. The Dow-Jones type of plot is
best for showing magnified changes, but the quality of control is best
seen when the data are presented relative to zero. The plots as shown
depict changes of about 6% below and above the average weight. Under
appropriate assumptions this could be taken as evidence of an equivalent
proportional loop gain of no less than 16.

The running averages I presented were created by Minitab using the MAPLOT
(moving average plot) command, which creates what is known in industry as a
control chart. Your point is well taken that for our purposes the plot
should originate at zero, but I had to take what Minitab provides. Of
course, we can write our own plot routine to do it the way we would like to
see it.