Signal to Noise Ratios

https://www.youtube.com/watch?v=zHU9xdU51FY

https://www.youtube.com/watch?v=Hy-T26gF3Ic

To reduce the M values of y at each design configuration d to a single number, Taguchi developed the concept of a statistical quality loss function. A loss function L(p) is any function over a performance variable p that represents quality lost in terms of the performance deviation , where T is a desired nominal output specification value on p.

Such deviations on output p are due to noise variations. The next question is what is L(p)? What is a function that can represent the quality lost by a product when it is not designed and manufactured and performing exactly on specification? To answer this, consider a design performance . The quality lost is a function of the performance p, call it L(p). Suppose our target value of performance p is a value τ. Expanding the loss L in a Taylor series:

Now there is no quality lost, by definition, when the design produced is on target. Thus

Further, the quality lost increases when moving away from target. That is, T is a local minimum for L. Therefore,

These results lead to the following quality loss function

Thus, the loss is proportional to the square of the deviation from specification target. Quality loss L is effectively represented as a quadratic loss function

Even with the loss function of Equation 19.4, we do not yet have a means of transforming M experimental points, for a design configuration d, reduced to a single number. To do this, we will make a thought extension to the quality loss function. Taguchi roughly considered quality (as observed on the output p) as the difference between the effects of the intended output as compared to the undesired variations. That is, quality is a measure of the intended output-the signal-relative to the noise. Thus, Taguchi calls a measure of quality incorporating the noise variations as a signal-to-noise ratio, or S/N ratio. An S/N ratio is, generally speaking, any function that takes, for any design d, the values of p across the M noise experiments and converts them into a single number. Of importance are the particular functions that also are quality loss functions.

There are many such S/N ratios. We develop here forms suitable for different types of problems, depending upon the nature of the specifications available on the performance variable p.

Target Problems

The first S/N ratio to consider is when we have a target value T for performance p. Then we define our loss as above, evaluated across our noise variations. Also, Taguchi always suggests transforming variations through a logarithmic transformation. The logarithmic transformation converts errors that are multiplicative into being additive, which Taguchi argues is appropriate for manufacturing. Thus, our first S/N ratio is

The M - 1 normalizes the sum, and the form is a quadratic loss function.

Date: 2016-03-03; view: 773