Thread Subject: taguchi method

is there a connection between/codes for taguchi methods in matlab?
thank you.

"ben harper" <controlusc@gmail.com> wrote in message <h9o7q5$mnl$1@fred.mathworks.com>...
> is there a connection between/codes for taguchi methods in matlab?
> thank you.

There are several things that are commonly meant
by the phrase "Taguchi method".

For example, in a statistical tolerancing context, one
can compute the (lower order) moments of a function
of a random variable, using a Taguchi method. That
is suppose that x is normally distributed, with mean
zero and standard deviation 0.5, what is the distribution
of f(x) = 1+sin(x)? When I ask for the distribution here,
I recognize that the actual distribution is not available
in any standard form. So just compute the approximate
mean and variance of this nonlinear transformation
of x.

The standard approach is to use a first order Taylor
series approximation. In that event, the estimated
mean of f(x) will be 1, and the standard deviation
0.5. (Since the derivative of f(x), at x = 0 is 1.)

How well does this do, using Monte Carlo as a point
of comparison?

   X = randn(1000000,1)*0.5 + 0;
   f = @(x) 1 + sin(x);
   mean(f(X))
ans =
            1

  std(f(X))
ans =
      0.44335

As you can see, the mean agrees with the Taylor
series estimate, but the standard deviation is off
by a bit. That difference is real too. It is not just
due to simulation error.

Can we do better than this, without resorting to
Monte Carlo methods? A Taguchi method is an
option here. Given the known normal distribution,
evaluate the function f(x) at only THREE specific
points.

   sigma = 0.5;
   mu = 0;
   xt = mu + [-sqrt(3/2), 0 sqrt(3/2)]*sigma;
   
Now, compute the mean and standard deviation
of the original function at those three points. Use
the population estimator for the standard deviation,
where it is divided by N, not by N-1.

   mean(f(xt))
ans =
     1

   std(f(xt),1)
ans =
      0.46933

As you can see, the mean is correct, and the standard
deviation is much better than what we got from the
Monte Carlo, at a cost of only three function evaluations
instead of 1000000 function evaluations.

In fact, we can do better yet, if we are willing to
compute weighted means and variances. It turns out
that these Taguchi methods are equivalent to numerical
integrations. Nick Zaino and I showed this some years
ago.

D’Errico J.R. and Zaino, Jr. N.A. (1988) “Statistical
Tolerancing Using a Modification of Taguchi’s Method.”
Technometrics, Vol. 30, No. 4, November 1988: 397-405.

In this paper we showed how to transform the problem
to a Gauss-Hermite integration form, so we can now
compute quite accurate estimate of the mean, variance,
as well as higher order moments yet. In fact, a higher
order approximation using these methods yields these
as the correct distribution parameters:

mean: 1
std: 0.44355
skewness: 0
kurtosis: 2.2905

Having said all of that, you may have meant an entirely
different thing when you said Taguchi method. There
are also various Taguchi methods used in modeling and
experimentation. I'll stop here though, as that is as far as
my interest ever went.

HTH,
John

in statistical toolbox there is "design of experiments" (DOE) part.
.
i want use DOE with Taguchi's parameter design methods.

"ben harper" <controlusc@gmail.com> wrote in message <h9oi7r$jvl$1@fred.mathworks.com>...
> in statistical toolbox there is "design of experiments" (DOE) part.
> .
> i want use DOE with Taguchi's parameter design methods.

I do little by design. ;-) Sorry.

John

> in statistical toolbox there is "design of experiments" (DOE) part.
> .
> i want use DOE with Taguchi's parameter design methods.

The Statistics Toolbox doesn't have anything like a GUI to step you through
the process of a DOE. Instead it's a collection of tools. You can use those
tools to do Taguchi methods. For example:

1. If you want inner and outer arrays, you can generate one design for the
controllable factors as an inner array, and another for the noise factors as
an outer array.

2. If you want orthogonal arrays, the two-level fractional factorial designs
are in that category. If you have factors with more than two levels, you can
generate a D-optimal design for them. There's no need to "jam" factors into
an orthogonal array of a pre-determined size.

3. If you want to measure the response multiple times at each factor
combination, just do that and store the results in a matrix y with one row
per factor combination and one column per measurement.

4. If you want to compute a signal-to-noise ratio, you can easily do that as
a MATLAB expression. I'm not sure if I have the formula right, but I think
the nominal-is-better SN ratio is something like:

y = rand(10,4)
m = mean(y,2)
v = var(y,0,2)
sn = -10 * log10(v./m)

Then you can use regression functions to analyze "sn" as a function of the
factors.

If this is not the type of information you're seeking, try asking a more
specific question and I'll see if I can answer it.

-- Tom