## Is y = f(x) really a Transfer Function?

Six Sigma practitioners often refer to a simple regression y= f(x) as a transfer function, but this equation does not represent a transfer function because it does not transform anything.

As transfer functions have important roles in engineering, for example in control circuit tuning^{1} and in filter design^{1}; as well as defining Acoustic Quality and Optical Image Quality, it seems only appropriate to try and identify a transfer function for a simple process, such as one described by y = f(x) the simple regression.

Another motivation for wanting to identify the transfer function of the simple process comes from our interest in causal inference – a subject of considerable neglect in Quality Engineering – in our opinion.

While researching the latter, we encountered an article entitled: “A Causal Law and Simple Regression,” in which the author Dr. Chendroyaperumal^{2} invokes Hume’s Principle of Causality, quoting it as: “Cause always necessarily precedes effect,” which he interprets as: “Cause always takes a non-zero and positive amount of time before yielding an effect.” He then applies this principle to our simple process to formulate a simple causal relationship having a single cause and one effect, of the form:

y = f (x, t) (1)

Given our recent concern about the good standing of Quality, particularly in the Financial Sector, we felt we ought to investigate the causal aspect further by adopting a form of equation (1) more amenable to transformation, such as equation (2) below:

y (x, t) = h (t) ⊗ x (t) (2)

Where we use the following notation in equation (2): ⊗ represents convolution, x (t) a test input, and h (t) represents the transfer function of interest. (For those unfamiliar with the mechanics of convolution there are some excellent descriptions and simulations on the internet^{3}).

It is worth noting equation (2) has a mathematical structure similar to the Optical Transfer Function used by optical engineers to assess image quality, and because optical illumination comprises pure energy, which has the same physical dimensions as work, we have inadvertently stumbled upon a transfer function suitable for describing any kind of work process, provided it is repetitive. Consequently, we can associate the work Quality transfer function h (t) with process efficiency.

### The frequency domain

Once we transform equation (2) into the frequency domain after substituting ω for the inverse of the repetition period, 1/τ, given in equation (3)

ω = 1/τ (3)

We can write:

Y (ω) = H (ω) X (ω) (4)

We should note that the repetition period, τ, in equation (3) is the time in which we run a sample through our process and replicate it with a second sample having the same process conditions of time, and temperature. (Relative to the Toyota Production System, our repetition period,τ corresponds to twice the takt time.)

After transformation, equation (3) becomes equation (4) where Y (ω) represents the output frequency response, and the input frequency response is represented by X (ω) and H (ω) is the Process transfer function, which we claim represents a member of the set of Quality transforms of the process.

In practice, we determine H (ω) by applying a periodic signal to X having a frequency, ω = 1 /τ, while at the same time measuring the output as a function of 1/τ. We can then calculate the modulation of each frequency, as described by equation (5):

Modulation = (max – min) / (max + min) (5)

We then plot the ratio of the output modulation to the input modulation as a function of ω. The resulting curve is the Fourier Spectrum, Y (ω) which we can use to determine the Quality Bandwidth at the -3 dB value if we wish.

We should note that any distortion of the test signal, as a function of frequency, is symptomatic of system noise. In our simple process, the system noise is due to gauging errors caused by scale inaccuracies and imprecision.

Of course, processes that are more complicated suffer more sources of error, which we will discuss in a later section. For now, we will limit ourselves to our simple process as illustrated in FIG. 1, where we have taken the average of the set points, but retained the effect of the extremes of tolerance N1 and N2.

Referring to FIG. 1, we illustrate our method for determining the process transfer function of our simple process by setting our input to a high condition, before running our first sample. We then wait a short time, τ, before replicating the same condition using a second sample – our replicate. (The repetition time and the number of replicates will have to be determined heuristically and usually depends on the process in question.)

We now set the process to a low condition, and run another sample, once again waiting a short time, τ, before replicating it.

To start the next cycle, we set the process back to the high condition and run another sample, but this time, we wait slightly longer, 2 τ before running another sample, also waiting slightly longer, 2 τ, before replicating with another sample.

By now, it should be apparent how we can set the process to a high or low condition while running samples and replicates to determine our process frequency response.

In terms of a Quality Transfer function, the presence of a delay, or time lag, represents a Phase shift and an aberration, or true defect. By true defect, we imply the defect has a one to one relationship to the true cause, and does present merely a symptom. (We discuss Phase shifts and delay in a later section; suffice to say we consider any delay an aberration or defect.)

### Sources of Noise

As most systems suffer disturbances from statistical noise, we ought to consider their sources:

- The first source is due to variation around the process operating point, or set point, which in the case of an amplifier corresponds to the bias point, which we typically choose to set the biasing condition of the amplifier, such as Class A, B, or C, or D.
- The second source of noise is due to tolerance variations from one process system to another. In the case of amplifiers, it corresponds to the tolerances of component used in the construction of the amplifier PCB.

In the case of a gauge previously mentioned, the accuracy of the set point on the scale corresponds to the first source type, and the precision of the set point corresponds to the second source type.

### Phase shifts, aberrations, etc.

When the process has an inherent delay, such as that imposed by hysteresis, or by residual gases, or conditions imposed by a previous process run, the transfer function takes a similar form to the Optical Transfer Function, which comprises two parts. The first part corresponds to the diffraction-limited case – the best we can achieve with a finite aperture – and the second part corresponds to a Phase shift due to an optical aberration, such as a deviation from spherical form, or shape.

In our case, since we have already equated optical illumination with energy and work, we can associate Q (ω) in equation (6) with useful work, and D (ω) with wasted work.

Y (ω) = Q (ω) D (ω) X (ω) (6)

In other words, D (ω) is due to defects because following Taguchi’s admonition we ascribe defects to wasted 6 work – either to too much work or too little work.

In general, we write the process transfer function as follows:

Y (ω) = Γ(ω) X (ω) (7)

In using equation (7) we like to refer to the term Γ(ω) as the Quality Transform Function, but in truth it is only one member of a set of transfer functions. If we encounter a process exhibiting hysteresis, without an appreciation of causal inference, we might well conclude the effect has been due to a ‘hidden variable,’ when no such variable exists.

### Interpretation

Now we have identified a transfer function for our simple process, we must ask does it have any utility. We think it does, for the following reasons. Consider the following example: Let us say the reader has just had a new tap (faucet) installed, and would like to check the fitting while the plumber writes an invoice. How can we go about it? Should we grab a cup and calculate the time taken to fill a cup, and then plot a histogram of the times, and later determine the output process capability against tolerances of our choice? (Under the Six Sigma concept, customers can define any tolerance they like and regard anything outside the tolerance as a defect.)

Alternatively, we could turn the tap on and off with a variety of durations to check the seal, and leave it on and off for a little longer to check for any air in the system.

While we do not dispute the importance of determining the output performance of a process in the form of process capability studies, we do not believe performance bears any relationship to Quality, which to our mind is a function of both input and output, and not just output alone as dictated by Six Sigma. Therefore, we conclude Six Sigma Performance, Sigma Levels, or Process Sigma only relate to business process performance, and not to Quality. This distinction is important because we would not want to see any further Quality neglect in the financial industry.

### Targets

There is considerable concern at present about the role of performance targets in the NHS because there have been some cases where staff have been pressured to cheat or mismanage Patient Care. How they achieved this is anyone’s guess – perhaps the BMA also believe it is possible to assess Quality based on outputs alone!

To our mind, any layperson (in Greek: idiotès) can set an arbitrary target, but it takes skill to be reasonable. Under the Toyota Production System, engineers practice the principle of ‘no-unreasonableness’, or ‘Muri,’ by using standard work times. Not to make people work harder – but to ensure they have a reasonable amount of time to complete their jobs properly, and without having undue pressure to take short cuts, lie, or cheat, to meet the production manager’s end of month output targets.

Therefore, we conclude Muri is a member of the set of Quality transfer functions since it takes both input and output into account.

### Using Policies as input

Sometimes, we do not even have to run physical tests and all we have to do is to interpret policies, such as a Bank’s withdrawal policy. Banks, which act as a repository for other people’s money, have an implied Quality … a duty of care … and any unnecessary exposure of their customer’s deposits to liquidity risk represents a serious diminishment of that responsibility. Even a slight delay between a notice of withdrawal and payment, which we represent as a Phase Shift, represents an aberration, or Quality defect.

If we plot the Fourier Spectrum of the Return on our investment, as a function of the Bank’s policy for short, medium, and long-term deposits, we obtain one of the Bank’s Quality Transfer Functions. Of course, the bank may well reject this claim by countering they have to sacrifice liquidity to offer higher rates of return. If this is true, then banks ought to offer depositors a variety of accounts with levels of capitalization commensurate with the risk.

### Quality as a Transformation

Previously, we wrote:

Y (ω) = Γ(ω) X (ω) (8)

Where Γ(ω) signifies a Quality Transfer function having a mathematical form similar to the Optical Transfer Function used by Optical Engineers to characterize Image Quality, and we were able to associate Γ(ω) with efficiency. Accordingly, equation (8) provides formal justification for considering Quality as a transformation, as identified by Harvey and Green^{5} when they considered qualitative change in Education. (In another article, we will illustrate how we can use efficiency for non-judgmental learning, making it more natural and enjoyable.)

## Conclusions

In conclusion, we believe one cannot assess Quality based on output alone; and, we must reject Sigma levels, Process Sigma, Process Capability, and DPMO as measures of Quality. Rather, we should consider Quality as a transformation – one that is the result of Care^{4}.

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### Acknowledgements:

Although the author takes full responsibility for any conceptual flaws or errors in this article, he would like to thank Giuseppe (Peppe) Calcara and Matt Moore for proof reading the article.

### References:

Ref. 1: http://lorien.ncl.ac.uk/ming/robust/freqapps.pdf

Ref. 2: “A Causal Law and Simple Regression Models,” by Dr. C. Chendroyaperumal, http://ssm.com/abstract = 1334690

Ref. 3: http://mathworld.wolfram.com/Convolution.html

Ref. 4: “Zen and the Art of Motorcycle Maintenance, “ by Robert M. Pirsig

Ref. 5: ‘Defining Quality,’ Harvey&Green, 1993

Ref. 6: Saying attributed to Dr. Taguchi: “Wasted energy – either too much or too little – are the true causes of defects”.

© Andy Urquhart 2009. All rights reserved. Not be reprinted without permission.

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