Probabailistic techniques in practice

Is a model needed for probabilistic analysis and optimization?

Most who have been exposed to the concept of robustness in product development have been introduced to the experimental approaches. However, few are aware of the analytical approach, and many hold the belief that the experimental approach to robustification is the sole approach. Why is this the case and is there any advantage to the analytical approach, which is rarely considered?

I believe that the focus on the experimental approach comes from the dominance of Taguchi and his approach in this field. While Taguchi’s approach has often provided remarkable improvements in quality, it should be noted that it as been heavily criticized by a number of statisticians. The key criticism revolves around the extra experimental effort that the outer array demands. By refocusing this effort upon the parameters in the inner array, it is argued, one can develop an empirical model that can then be combined with various probabilistic methods.

I personally believe that each approach (using the outer array or focusing solely on the inner array) has its advantages, and the engineer needs to consider each case as it is presented. Therefore, I can’t say that one of these approaches is better than the other; however, the notion of developing an empirical model raises something worth considering further: the model itself.

This brings us to the advantages of the analytical approach. If we have a model, we are able to apply probabilistic methods (Monte Carlo, FORM/SORM, error propagation and such). This allows us to predict the effects of randomness and to optimize against the negative effects. Therefore, it is worth developing a model that can be used for optimization.

One of the major problems that I see many have with the development of a model is the implicit assumption that it must be developed empirically. I believe that this is a carry over from the dominance of the Taguchi approach and the assumption that quality is a statistical phenomenon; statisticians seem to have a natural tendency toward the empirical approach. The result is that few people consider developing an analytical model for robustification.

An analytical model has a number of advantages. It improves your understanding of the system being considered, it typically requires less effort/resources to develop, and it can be kept simple for initial considerations, but increased in complexity later on when more accuracy is required. With these advantages, it is clearly worth developing an analytical model for robustification. There are however two common problems: many are reluctant to make a model and the full power of the analytical model is not appreciated.

• Many of us were taught the basics of scientific theory, but few of us were ever taught to use these theories to create a model. This in turn seems to result in a lack of confidence. I can’t tell you too much about developing a model here, but I can tell you that it is more of an art than a science, and it requires practice. Also, you don’t need to worry about getting it wrong; you can always try again, especially given that the development requires relatively little effort. Don’t be surprised if up to 5 attempts are required.

• When you have a model you are no longer restricted to empirical methods, and the faster analytical or numerical optimization methods can be considered. However, I have seen cases where an analytical model was developed and then combined with Monte Carlo and DOE for optimization. Not only would this take longer, but DOE would only offer a limited improvement. This reiterates the current focus on the empirical approach, and the extra consideration you might want to give to the alternative methods optimization methods.

If you are prepared to have a go at creating a model and then combine it with probabilistic methods and various optimization techniques, you can robustify a design at the early stages of development. With the extra insight this offers, you can anticipate greater quality once the product is developed. However, don’t forget that you might get it wrong first time around, and another attempt will be needed. Also, even though an analytical model is ideal, there will still be times when the experimental approach is the only option.

Statistics Vs Probability

To many, probability and statistics are synonymous. While they have much in common, they do have some significant differences. When these differences are fully appreciated, you will develop a perspective that allows you to fully appreciate how you can make the most of a probabilistic approach.

While there are a number of differences, the key difference focuses upon how we acquire systemic information:

• In a statistical approach, which is probably the most common approach, we would use experimentation and the collected data to improve our understanding of the quality we would expect in the system that we are considering. The key problem with this approach is that the actual system is required for experimentation. This means we need to have built a prototype or perhaps have even started production before we can perform these experiments. In the first case, much of the design works needs to have been completed, and in the second case, we would have already finalised the design and stared production. In either case, we would have made a considerable investment before we would know if we were going to be able to meet our quality target.

• In a probabilistic approach, we start with an analytical model and use this for probabilistic analysis. By taking this approach the quality can be predicted, and even optimized, before we commit to a final design or manufacturing. This can save considerable cost and time. In particular, we can determine if a design will even be capable of providing the quality we want, and rejecting it if it can’t.

The advantage of the probabilistic approach over the statistical is clear. However, it does require an analytical model and this can be overwhelming for some. Still, you should always try developing an analytical model and analyzing the model for quality before committing to take a design further. Paradoxically, there will be times when you will need to perform an experiment(s) to develop the model so it should not be thought that the probabilistic approach is a complete replacement of the statistical. This is especially the case when dealing with quality control as opposed to product development.

 In summary, a probabilistic approach allows us to be more predictive than does the statistical. Therefore, the probabilistic approach should be used whenever possible, but it should be noted that the statistical approach is still essential. Also, we need to bring our focus onto the development of an analytical model so that we can apply probabilistic methods.

Choosing the best distribution

One of the key tasks in probabilistic analysis is the selection of the appropriate distribution. While this is of key importance in many cases it is not something that many are sufficiently familiar with. If you have a large data sample, it is fairly easy to use a number of tests, such as the Chi-Square or Anderson Darling, to find the best-fitted distribution to the data. However, there are times when you won’t have data. Still, there are a few principles that can be used to guide you in selecting the most appropriate distribution if you don’t have any data to guide you.

The most significant of these is the central limit theorem. The central limit theorem tells us that if a number of distributions are added to each other, the resultant distribution will be very close to a Normal distribution. In fact, if the relationship between the inputs and the outputs is close to linear over the region of random variability, then the output distribution will also be close to a Normal distribution. This has lead to the Normal distribution being a default assumption for any input distribution. James Siddall, a researcher who put a lot of focus on the selection of input distributions, argued that because of the frequent use of the Normal distribution many think that it is indeed normal to use the Normal distribution. This is not to say that the Normal distribution should not be used or that it should only be used on special occasion, but it is probably used too often.

There is another element of the central limit theorem that is less commonly known. If a number of distributions are multiplied (or divided by each other) then the outcome is close to a Lognormal distribution. The Lognormal distribution can look a lot like a Normal distribution, but it can also be very different. Therefore, given how common multiplicative phenomena are, there are likely to be many cases where a Lognormal distribution is the best choice. This was most evident to me when I met an analyst who frequently dealt with biological processes. In his experience the Lognormal distribution was very much the safest choice.

If you can work out what kind of a process produces the variable you are interested in, you can potentially determine a suitable distribution for that variable. For example, casting operations are essentially multiplicative: the dimension of a cast feature is equal to the respective dimension of the tool, multiplied by the thermal expansion to the tool temperature, multiplied by the thermal contraction as the casting cools. Therefore, we would be safest to assume that the distribution for the dimension of a cast feature is likely to be well representative of a Lognormal distribution. Another example is composite materials. Some composites are made by gluing sheets together in different orientations so that the composite effectively has no grain, and the material’s strength is more uniform. The thickness of such a material is equal to the summation of the thickness of the material that makes it up. Therefore, the distribution of the thickness of the composite is likely to be well represented by a Normal distribution. 

This only covers two processes: multiplicative and additive. However, this covers many real life processes. Therefore, when you are choosing a distribution for an input variable try to think about what kind of an operation produces the respective feature: is it additive or multiplicative? It is not always easy to determine this, but multiplicative is more common. Still, the operation might be something different or there might be more to it. Consider the length of a hypotenuse of a 3D triangle. Three edges are added together after being squared, this will give us a Normal distribution. However, it is then square rooted; therefore, the final distribution will be the square root of a Normal distribution.