Lecture index: Introduction. / Solution of a conundrum with multivalent fuzzy logic. / An example - identifying minerals. / Fuzzy set shapes and some rules. / Geochemical discriminant diagrams and clusters as fuzzy sets an example of potential use. / Fuzzy classifications in remote sensing. / Fuzzy histograms or summed probability distributions. / Fuzzy sets in sedimentology. / Exercise 14 - in class.
Reading:
Kosko, B. & Isaka, S., 1993, Fuzzy Logic,
Scientific American, July, p. 76-81. Read this one carefully.
Spend an hour or two looking through the below:
This is definitely a non-traditional topic, and one that is still in its relative infancy it terms of geoscience applications. It is better known and more widely used in the engineering and artificial intelligence worlds. I strongly believe it is worth investigating in a geoscience framework, even at the undergraduate level, and that it will be fertile field of investigation in the future. The field has developed quite a bit of depth since I first became aware of it. If nothing else, I believe is conceptual understanding of fuzzy logic will help you to think more clearly.
In class exercise: Fuzzy logic has to do with the basic tenants of set theory, and set theory has everything to do with classification. Create a list of all the different classifications we use in geology (list at least 10). For which of these classifications is there one underlying variable. For which of these classifications are there multiple underlying variables. This will come in use during the weekly exercise.
Basic idea behind fuzzy logic: In basic set theory an item can belong to a given set partially. Fundamentally, it is that simple. However, you may not realize to what degree you have been taught to think in terms of true vs. false, black vs. white, on vs. off, and other bivalent value models. In a way this goes back to dichotomies. As Stephen J. Gould argues we love to think in dichotomies, but we also need to explore the places in between the end points. of a dichotomy That is what fuzzy logic does in a much more deliberate way.
One major distinction is between fuzzy sets and fuzzy numbers. We will focus on fuzzy sets. There is a fair bit of debate as to what the difference between fuzzy set theory and probability is. I do not have the expertise to comment on this debate. There seem to be some striking differences. For me the bottom line is that this perspective has opened up interesting new possibilities, and thus is worth conveying. Exactly what you call this approach isn't so important in the long run. This week's exercise is focused on new possibilities, instead of classic and established procedures for analyzing and modeling data. Remember that novelty is often resisted, sometimes with good reasons.
The following is adapted from Stewart (1993). There are a number of permutations on the following paired statements that demonstrate the basic idea.
It presents a conundrum because if the speaker's first statement is true his second statement indicates it is not, and if the first statement is false, the second statement makes it true. One approach is to consider this as dynamic or time changing logic. The truth value keeps oscillating as you go from one statement to another. This is sort of like Schrodinger's cat in the box - it doesn't change until you consider it. These conundrums are associated with self referent systems, meaning that statements refer to each other. This sets up the potential of a feedback loop, and via this line of thought there is a connection to chaos theory. Typically these types of statements have always been a fun example of where classical logic breaks down, and not much more. However, the important thing is that there is a solution, and this conundrum is also a initial portal into fuzzy set theory.
Consider the first statement as P. If a statement is true it has a truth value, p , equal to one. It follows then that for P being true p=1 then for not-P p' = 1 - p = 1 - 1 = 0. But the second statement asserts that p of P is equal to 0, or p=0. So we end up with the equation that would satisfy them both as:
If p can only be bivalent, 1 or 0, then you have a mathematical expression of the conundrum. But if you were given this equation without this background context it would be easy to solve. Relax the bivalence (that p can only be 0 or 10 to permit multivalence (or fractional values in this case) and p is then equal to .5. Another words, it works if you consider the statements to be half true! If you can not accept ambiguity then fuzzy logic is not for you. Indeed, the real world is not for you. You might ask if knowing something is half true is useful knowledge?
In introductory physical geology courses students learn to identify minerals. That skill is honed with practice. Exactly what is the mental process used in recognizing mica, or feldspar. Using fuzzy sets to identify minerals is an idea that is developed in the literature. The key terms expert system and fuzzy set theory and fuzzy logic are connected. When you learn to identify minerals you are gaining expert knowledge. This consists of a series of guidelines or expert rules. Color is useful in identifying micas, but not for many other minerals. Luster can be metallic or non-metallic, and if non-metallic vitreous or non-vitreous. Cleavage is a very useful diagnostic trait, but can be difficult to see in some specimens. Importantly, one uses combinations of different fuzzy rules to come up with an identification. Interestingly, after awhile it seems that when you look at a mineral the name surfaces in your mind first, and then if asked, you can convey the traits that support that identification. Mineral identification becomes internalized in some way with continuous practice.
In the diagram below we look at one property used to identify minerals in thin section - berefrigence. Rarely will someone remember the numerical value for a mineral's birefrigence. Instead they will use terms like low or high, or first order or second order. For a variety of reasons the actual birefrigence of a given mineral varies. Quartz has a low birefrigence and calcite is high. How can you take human observations (from an expert, someone with experience, or trained) and represent them, or turn them into input. Below is a depiction of three fuzzy sets for the terms low, medium and high birefrigence. In addition, the actual (crisp) values of some minerals has been mapped on to this set definition. Note that the y axis of a fuzzy set is always the degree of membership, from 0 (it doesn't belong) to 1 (it does belong). Kyanite would belong 35% to the set low birefrigence, and 65% to the set medium birefrigence. Talc, all the way to the right, clearly has high berefrigence. If a trained observer indicates that the birefrigence of a given mineral underneath the microscope is medium, what are the different implications for the proposition that the mineral is talc versus the proposition that it is kyanite?
Fuzzy sets cans have all sorts of shapes in membership space. Those shapes have distinct different meanings. Look at the diagram below and think about what the different shapes mean about the nature of the set. Which one is crisp and narrowly defined? Which one is very fuzzy and broadly defined? Which one is very fuzzy but narrowly defined? Which one has a crisper character for part of the variable range and a fuzzier character for another part? Most important why would you use one shape or the other?
Some simple rules for defining standard fuzzy sets.You can use the below as check list during your exercise.
Geochemical discriminant diagrams are plots where one elemental or oxide abundance or ratio of a rock is plotted against another. Plotting a population of samples produces a cloud of data points. They come in a wide variety or types and are often used when analyzing basaltic rocks, which to the eye and to some degree are otherwise amazingly homogeneous.
These diagrams are typically used in one of four ways:
We need to consider the science of what is going on some, and we can do that by addressing the question -what goes into determining the chemical mix of a volcanic rock?
When it comes to tectonic signature, present practice mainly sees a crisp world lines defining polygons divide one "field" (i.e. set) from the other. This can be seen in numerous examples. But consider Iceland everything would suggest that these basalts are something like half ridge half hot spot. Taking a fuzzy approach could allow us to make much more precise models and comparisons. In practice an informal fuzzy approach is taken, and similarities of populations are discussed. Fuzzy logic would allow one to formalize that practice and be more rigorous. You might then imagine fuzzy clusters defined by contour lines of membership in a space defined by two variables. Experience in modern environments could be used to define the tectonic signature. One approach would be to map data density distribution. Then in some manner this could be taken to represent the degree to which an unknown was similar to or belonged to that set. Data density distribution of geochemical diagrams has already been investigated by one senior thesis here at UNO (by Jeff Salazar).
Sui (1994) develops the potential here a little bit. We discussed the effort to develop classifications in the analysis of remote sensing imagery. In a classification the computer must decide whether a pixel is in one class or another. Consider the scale of some satellite imagery. Why can't it belong to one class to some degree, and to another class another degree? Can you think of examples that might produce such a mixed pixel?? Such an approach could be much more accurate by reflecting the accuracy of class assignment. It is simply more sophisticated assignment rules that permit the program to realize that a pixel may reflect a mixture of classes.
Another use is in assigning corridors to GIS elements in site selection. The present practice is to have the program delineate a corridor around a road of a given width. The simple rule uses is that inside that corridor the site may be acceptable, and outside not acceptable. But in many cases it seems obvious that if your corridor is 1 mile, a site that is one quarter mile is better than one that is three quarters away. Yet the in or out of corridor approach won't permit this nuance. Considering a fuzzy set where corridor membership is a linear or square function of distance from the road allows this nuance to be incorporated into site selection.
When doing a normal type of histogram each data point is treated the same, and the question that is asked over and over again is - does this data point fall in this histogram interval. Clearly, the histogram interval is being treated as a set. However, not all data points are of equal quality. Note that the error information has been lost - is not used in the analysis. A simple solution exists - which elementally consists of partial data point assignment to a histogram interval set. Any series of rules can be used to assign fractional membership. If you have an error bar and can assume an error distribution, that distribution can be used for fractional assignment. The portion of the distribution curve for a given histogram interval would be the fraction to which the data point falls in that particular histogram.
The results is a summed probability distribution. Basically,
a simple rule is that if your 2 sigma or other error is = to or
> than the histogram interval for a significant percentage
of your data then you should construct a fuzzy histogram. A fuzzy
histogram includes the error information and is more truthful.
Other rules for fractional assignment could be used in cases where
traditional error information is not available. I would suggest
that when you use this approach don't call it a fuzzy histogram
even though that is what it is. Just refer to it as a summed
probability distribution. However, if you consider a histogram
interval as a set and consider that you are asking the question
of whether a data point belongs in this interval or another, then
this is basically partial assignment to a set. Ask me for the
story behind this statement.
Diagram showing how the error curve (area underneath = 1) associated with a single measurement can be imposed on a histogram, and then partial membership of that one sample assigned to the different histogram intervals.
I have used this approach fruitfully when looking at the K-Ar ages of Cretaceous diabase intrusions in Svalbard. The problem is that these dates were done when the technology wasn't as good, and the error bars associated with mafic rocks tends to be large simply because of the small amounts of potassium in the rock. You can see the results below.
The potential may be best shown with an example. Consider water depth - a prime factor of interest in reconstructing depositional environments. You might imagine water depth separated into the following sets: emergent, tidal, nearshore, offshore, deep water. A little bit of thought suggests these can be envisioned as fuzzy sets. Tides vary in their extent (especially considering storm tides), as do wave phenomena. Think of an emergent area that is flooded by storm tides with a recurrence interval of 10 years or the like. It could be conceived of as 95% emergent and 5% tidal. It is fairly easy to fuzz up the boundaries between the rest of these.
More importantly consider the process by which we decide whether a given sedimentary body was deep water or offshore. We look for various traits that are indicative of a given environment and the more of different traits we find the more we are sure of our interpretation. This is known as convergent validity, which is a very important type of argument in science. It is related by an inverse relationship to Occum's razor of simplicity, in that we then have one interpretation that 'explains' several observations. For example, if we find glauconite and Cruzianas feeding traces these both indicate deep water. Add to that organic rich shales and the picture gets clearer. In one way we are keeping score of traits consistent with our hypothesis. Again, this is done on an informal basis. Now the trick would be develop some fuzzy sets and rules of association ( the fuzzy patches in your reading), so that by inputting one you can output the other.
Convergent validity and fuzzy scores: We could just keep track of traits, but some traverse more than one of the fuzzy water depth states. So we would need to have partial assignment. Basically, some traits are more diagnostic than others, and we would be formally incorporating that into the analysis. Then for a given rock body we can compute cumulative fuzzy scores from our fuzzy sets and theories. The higher the score for a given water depth the more convergent validity it has, the greater the aggregate truth value. Weighting of traits could also be used to develop a more nuanced score. Arguments could be about set and rule definitions, and would be much more focused than on informal expert interpretation. You can approach this as mapping out more complex truth functions similar to the conundrum discussed above.
For next time read the assignment and then give thought to what fuzzy sets might exist in geosciences and how you might define them. You can use some of the examples above as a guide. We will split up into groups and each group will try to define fuzzy sets in a particular geoscience endeavor. Perhaps use some of the examples above to seed your creative mind. You may want to find some supporting material associated with the classification you are working with. After spending 45 or so minutes or so exploring possibilities, you will share your ideas with the others in the class.
Each group should hand in a document with:
References:
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