Re: Basis for quadratic Taguchi Loss Function

[Carl Betterton <Carl.Betterton@gmail.com>]

I wish to thank Jonathan Siegel, B. Murtjahjanto, and Myron Tribus for
their responses on my question about the basis for the quadratic form
of Taguchi's Loss Function. Let me also share with you some
information that came from three posts off one of the American Society
for Quality (ASQ) lists, where I had also posed this question. I went
to the DEN after seeing Deming in a video drawing a diagram of the
loss function. Then I thought of ASQ because the loss function has
obvious quality connections. As you will see, the responses from the
two lists are fairly similar.

On the ASQ list James Howe suggested a web link, Richard DeRoeck
suggested a text, and Timothy Folkerts suggested that the basis of the
quadratic form was found in its usefulness. The explanation at the
link James suggested

http://www.mv.com/ipusers/rm/loss.htm 

is similar to one explanation from the DEN list. At the above given
link it is stated, "The quadratic curve is the first term when the
first derivative of a Taylor Series expansion about the target is set
equal to zero." This is related to the more expanded description from
the DEN list's B. Murtjahjanto.

The text suggested by Richard DeRoeck (Advanced Topics in Statistical
Process Control by Don Wheeler, SPC Press) is unfortunately not in our
university library. The text B. Murtjahjanto recommends, Taguchi, G.,
E. A. Elsayed and T. C. Hsiang, Quality Engineering in Production
Systems, McGraw-Hill, 1989 is also not at our university library nor
have I been able to find it on line yet. Go figure!

>From the DEN list, Jonathan Siegel says the basis for the quadratic
function was that "it works" (usually). This parallels the
"usefulness" rationale given by Timothy Folkerts on ASQ.

>From the post of Myron Tribus it is comforting to me that Taguchi was
"perplexed" about what function to use for the general case. (I know
how that feels!) It is also interesting that Taguchi reportedly
settled on the quadratic form based on its simplicity and usefulness.
I still need to find the references suggested and take a look at the
math in that Taylor Series expansion, but even without doing so there
seems to be harmony between the mathematical basis and the practical
basis. The quadratic term is the simplest portion of the Taylor Series
expansion that gives useful results with little error. It also
satisfies one's intuitive notion that loss increases progressively as
deviation from the target grows.

Thanks again for your help.

Best regards,

Carl

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