Averaging settlements

[Mccoy]

BigH wrote in one of the latest threads:

for sands, a paper years ago in Ground Engineering implied to pick three or four of the available analyses (he reviewed something like 15), then take the average of them - arithmetic average? Nominal average; geometric average? - I'll let that be as per another thread in the main forums!

As a statistics buff, I'll bite Howard's bait and say whatever comes to my mind about such point.

First and foremost, we must assume that the sample is homogeneous, which is to say that all methods are equal, i.e. they have no bias, they have shown to be reliable, they are suited to all conditions...
We know well this is not true; so we should analyze the model and pick 3-4 methods which have been shown to be reliable in the specific case under examination.

We further assume that the results are random, i.e. there is no bias due to the source of measurements (lab tests, SPT,CPT....), to stratigraphy, presence of water...

We end up with a sample of small dimension, in a non -parametric context (we don't know if a specific distribution function suits
the model.

I've no real-case data at hand (I usually average values of bearing capacity, not settlements), but the rule usually applies that arithmetic mean yields higher values than geometric mean, since the latter filters the extreme values out, the former rules them in. Of course, outliers are suspicious values per se, so sometimes they are excluded a priori.

At the end, I believe it's a matter of personal judgment (the subjectivity which is such an important part of the geotechnical engineer's job) to decide wether to rule out the outlier(s), to rule'em in because we want to include that possibility, to rule'em out because we want an average which is little sensitive to extremes...
If data are clustered, arithmetic or geometric mean makes very little difference.

If we want to deepen the subject, I'll say that, in this specific case with very small samples, Bayesian analysis might do a very reasonable job, where the average value is "corrected" by our prior knowledge of the model. One example might be as follows: let's figure out all the possible values of settlements by all methods in our knowledge; the values will make up the range of all possible settlements in our model (the Bayesian prior); let's then calculate the settlements by our one ore two chosen methods; this will be what we think is the "actual value", or most likely value, or Bayesian likelyhood. Combining the two by Bayes' theorem we end up with a posterior value (or distribution) which is our final value to be used in calcs.
If the above isn't clear, please don't bother, it's rather specialistic stuff, adapted by myself to the specific situation and might well be the subject of a technical paper (I don't know if something similar has been done) or conversely of sensitive criticism.

Please note that the issue of means would apply more practically and simply to lab or field measurements of specific parameters, but here we should include a few more aspects like spatial correlation.

By the way, I think in our group there is one of the leading world researchers in statistical study of random fields (Gordon Fenton, who has also published detailed papers in elastic settlements, available in the web).
If he'd like to chime in his opinions would be most valuable (and please, feel free to criticize whatever I said above, I'll take no offence, guaranteed!)

 

[VAD]

I would agree with you that the choice of what value one uses is a matter of personal judgement. However, if your work is subject to thorough review then the premise on which it is chosen must be stated so as to make the reviewer and others understand. I know that in a good report that this is done. Others sometimes choose not to say why they have picked values etc. I look at geotech as telling a story and that story comes from the information we obtain and our reasoning about those values etc and why we are choosing so and so. We unfold the mystery in such dialogue.

In respect to the use of statistical methods, the lieterature has numerous methods and all have limitations. I see no problem using such methods if sample sizes allow us to use a particular method etc. In the end we can also decide to rely on our judgement. In such circumstances I would prefer to use optimistic and pessimistic values and use those to discuss the end result of the problem, as I believe as well that there is no singulat value that we can hang our hat on in this business.

These are some of my initial thoughts, though maybe not exactly to the point of your thread.

 

[Mccoy]

I would agree with you that the choice of what value one uses is a matter of personal judgement. However, if your work is subject to thorough review then the premise on which it is chosen must be stated so as to make the reviewer and others understand.

Vad, I perfectly agree with you. Our choices should always be justified and an explanation given, to the best of our ability.
Nevertheless, sometimes it's just not possible to make the reviewer and others understand. I would add, sometimes we are at the top of the decisional ladder, and we can do wathever we wish. I remember an instance when I used a Bayesian method to assign responsabilities in a court case (it was a landslide). I have all the way been sure that judge and lawyers, and also my colleagues could not grasp a single angle of the model: I just briefly illustrated the theory behind it, the method and clearly outlined and defended the results (which assigned probabilities of individual responsabilities hence legal liabilities).
I was at the top of the group, and could have been challenged only in formally and written way, by other technical people who had a sufficient knowledge in that field: there were none.
In other instances I was at an intermediate decisional level, I could only use methods which people hierarchically higher up deemed suitable, could propose but not decide. In such instances simpler and well-known methods are usually desirable.
In reports, well, not to justify one's choices is bad style and could lead to misunderstanding and trouble.
Sometimes two or more methods may be equally right, and it is up to you to decide which one to choose (and give explanations).
To get back to the settlements example:
Would you choose to filter an anomalously high value (geometric mean) or not to filter it (arithmetic mean) or rather to ignore it altogether (outlier)? All decisions would be right, you have just to judge which one would represent best the real settlement in your case (and yes, adding a few notes won't hurt!).
Sample size is a strong limitation in geotech practice.
Analysis of the extreme values is part of interval mathematics or statistics, which has developed its own theory. Also, a new field is emerging, that of "imprecise probabilities", which appears to be well suited to the substantial uncertainties often found in this field and others. I confess that I have yet to start with that one!

 

[jheidt2543]

Mccoy:

I think I understand the point you are making with the statement "I would add, sometimes we are at the top of the decisional ladder, and we can do wathever we wish". However, regardless of our position on the ladder, one still must justify the position taken even if it is above the understanding of the reviewer. The task is not only to provide a workable engineering solution, but be able to make it understandable at the level of the reviewer, the owner, or the opposing attorney. In this day and age, it is unwise to be misunderstood.

 

[dmoler]

I am of the opinion that geotechnical engineering is more of an art than a science. In that regard, that we can 'do whatever we wish' is realistic to some extent, but whatever we design must perform, or our reputation and legitamacy will vanish, as well as our clients.

There is some 'smoke and mirrors' associated to geo tech. The more difficult areas that I have found involve layered clay/rock/sand (combinations). The predictions get far more difficult and the spread of bearing capacities, foundation movements, ect. get more extreme by simply altering some assumption values a little. Stats may help more in those areas.

But I always say that one test result is worth a thousand opinions, and one performing foundation is worth a thousand theoretical predictions. (please dont beat me up too much on those simplistic statements)

 

[Mccoy]

dmoler says:

The more difficult areas that I have found involve layered clay/rock/sand (combinations)...

That's an interesting discussion topic and would well be worth a new thread!