RE: Basis for quadratic Taguchi Loss Function

[jmsiegel@sbcglobal.net]

So far as I know, the principle basis for the Taguchi Loss Function is
that it usually seems to work. 

But it doesn't always work. Where one has data, one is always better off
predicting or estimating ones real losses rather than using a made-up
formula. 

The quadratic loss function does seem to work under quite a variety of
circumstances. When it works, an average provides a reasonable estimate
of overall behavior of a process. 

Deming illustrated a number of situations where the Taguchi loss
function isn't a very useful model. One example Deming used is missing a
train -- if one misses the train, it doesn't really matter whether it
was by five seconds or five hours.

Another situation that sometimes comes up in my own work is where one
has a distribution is highly skewed. The Taguchi loss function is
minimized by keeping a process at its or average. When the average value
of a process is not a useful control parameter, that is often an
indicator that the Taguchi loss function isn't a good model of ones real
losses. For example, if Bill Gates gains or loses a couple of tens of
billions of dollars, the "average" American's wealth appears to change
substantially. But here the average just isn't a very good indicator of
what's really going on. The difficulty is that Bill Gate's personal gain
or loss simply doesn't reflect as great a gain or loss for the country
as a whole as the Taguchi loss function would predict. The Taguchi loss
function here isn't a good model of actual economic losses. A flatter
loss function provides a more useful estimate of average income which is
why the median -- which is based on a linear loss function -- is often
used instead of the mean or average. Of course, the presence of outliers
and skew values may simply indicate that a process is out of control.
But not always. Statistical control under a linear or other
flatter-than-normal loss function is sometimes possible and may
sometimes better reflect ones economic situation. When one has data on
or a basis for predicting ones actual empirical losses, ones own actual
losses should be used. The Taguchi function often works, but not always.


Statistical control theory is designed so as to minimize ones economic
losses over the long term. One result of C.I Lewis' perspective is that
the aame process could be considered in control with respect to one
person's loss function, but out of control from the point of view of
another's.

Standard analysis of variance in experimental design use a least squares
approach, which is also based on the Taguchi quadratic loss function.
When one has a different empirical loss function, one can use a
different approach, not based on least squares. to optimize results. 

Sincerely,
 
Jonathan Siegel
jmsiegel@yahoo.com