Fuzzy set theory and possible applications to geoscience.

: 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:

Introduction

This is definitely a non-traditional topic in a geology course, and one that is still in its relative infancy it terms of geoscience applications, although there are now books on fuzzy logic applications in geology. Fuzzy sets and logic 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 a conceptual understanding of fuzzy logic will simply 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 some of 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 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. Fuzzy logic does so 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. For me the bottom line is that this perspective has opened up interesting new possibilities, and thus is worth conveying. This week's exercise is focused on exploring new possibilities, instead of classic and established procedures for analyzing and modeling data.

Solution of a conundrum with multivalent fuzzy logic.

The following is adapted from Stewart (1993). There are a number of permutations on the following paired statements that demonstrate the basic idea.

"The following statement is true - I just told a lie."

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 often 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 consideration for us is that there is another 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, which we will call p', p' = 1 - p = 1 - 1 = 0. But the second statement asserts that p of P is equal to 0, or p=0. If 1-p = 0 and p = 0, we end up with the equation that would satisfy them both as:

p = 1 - p

If p can only be bivalent, 1 or 0, then you have a mathematical expression of the conundrum - it can not work. 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 1 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 perhaps fuzzy logic is not for you. Some might argue that the real world may not be for you. You might ask if knowing something is half true, or three quarters true is still useful knowledge?

An example - identifying minerals.

In introductory physical geology courses students learn to identify minerals. That skill is honed with practice, and when you learn to identify minerals you are gaining expert knowledge. Exactly what is the mental process used in recognizing mica, or feldspar? A series of guidelines or expert rules that are fairly complicated are developed. For example, 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. 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. Recognition has become internalized, and comes before a deliberative process of testing hardness, looking for cleavages, and considering color. Importantly, one uses combinations of different if-then statements, rules, to come up with an identification. 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.

In the diagram below we look at one property used to optically identify minerals in thin section - berefrigence (the color under cross nichols due to the mineral's polarizing ability). Rarely will someone remember the numerical value for a mineral's birefrigence or will a measurement of birefrigence be made. For an unknown mineral you estimate the birefrigence by eye using terms like low or high, or first order or second order. Quartz has a low birefrigence and calcite is high. In addition for a variety of reasons (like solid substitution series, true thin section thickness) the actual birefrigence of a given mineral varies. How can you take human observations (from an expert, someone with experience, or trained) and represent them, or turn them into input - how can words be 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). As described by the fuzzy sets below, 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? How about for the proposition that it is kyanite versus epidote? A fuzzy rule would then take an input variable such as medium birefrigence to produce a desired output, such as a mineral ID. More likely a series of rules (often of the if-then type) together would aid in mineral identification.

An example of how one optical characteristic of minerals under thin section can be captured as three fuzzy sets, that mimic the qualitative evaluation made when identifying minerals.
Fuzzy set shapes and some rules.

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?

Diagram depicting different shapes for set membership. These have been separated from each other to aid consideration, but remember that when partial membership is possible they will ny necessity overlap. The actual variety of possible shapes is of course infinite. Curved functions could also be used, and these could even be probability distributions.

Some simple rules for defining standard fuzzy sets.You can use the below as check list during your exercise.

• The horizontal axis is some measurable parameter such as size, Temperature, composition percentage. This represents your universal set.
• The vertical axis is degree of membership and ranges from 0 (not a member) to 1 (100% a member).
• The shape of your fuzzy set membership function should reflect the nature of uncertainty.
• For overlapping sets and a complete description of all possible sets, the sum of the memberships for an element should equal one. Therefore, the shapes of two overlapping sets are not independent.

Example of fuzzy set definition for a type of algal bloom from NOAA site - http://www.eco-check.org/forecast/chesapeake/2009/methods#_Prorocentrum_Blooms_-_HAB

One way fuzzy sets become useful is if they are incorporated into a fuzzly logic system. Such a system connects an input series of fuzzy sets to an output set of fuzzy sets with if-then statements (rules). Then a specific numeric input can have membership in more than one input fuzzy set that causes more than one if-then rule to 'fire' resulting in an aggregate output set. This can then be defuzzified into an output value by taking the centroid of the resulting shape. As a simple example, for the fuzzy set above one could also develop fuzzy sets for high surface dissolved oxyten content and low surface dissolved oxygen content. Then if then statements such as the following can link the two: If "no bloom" then dissolved oxygen contact is "high", if "bloom" then dissolved oxygen content is "low". Depending on the shapes of the output fuzzy sets this will result in a predictable model relationship between the concentration of Procentrum and the fissolved oxygen content.

Geochemical discriminant diagrams and clusters as fuzzy sets ­ an example of potential use.

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:

• To identify a compositional trend in the geochemical space (e.g. calc-alkaline trend).
• For fingerprinting, to discriminate between different magma bodies. Amazingly, each separate magma chamber has its own fingerprint.
• Tectonic signature analysis -to propose tectonic affiliation for an unknown population by comparison with established holotypic patterns for known provinces. These tectonic affiliations, such as continental rift, can be thought of as fuzzy sets. You can have a rift with a hot-spot influence, such as Iceland.
• Characterization of source ­ to look for patterns that theory indicates is source composition and conditions sensitive. One example is that recently some have argued you can detect a tiny bit of core material in plume related volcanism, which would indicate whole mantle convection instead of upper-lower mantle stratified convection.

Trace element discriminant diagram with data from Mt. Pinatubo plotted. Source: Petrology and Geochemistry of the 1991 Eruption Products of Mount Pinatubo By Alain Bernard, Ulrich Knittel, Bernd Weber, Dominique Weis, Achim Albrecht, Keiko Hattori, Jeffrey Klein, and Dietmar Oles - http://pubs.usgs.gov/pinatubo/bernard/ .

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?

• composition of parent material (e.g. depleted and undepleted mantle).
• percentage of partial melt and related pressure and temperature conditions.
• volatile gain or loss.
• magmatic differentiation.
• crustal contamination.

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. Again, consider Iceland ­ everything would suggest that these basalts are something like half ridge ­ half hot spot. Taking a fuzzy approach could allow for accommodation of these hybrid possibilities. 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).

Fuzzy classifications in remote sensing.

Sui (1994) develops the potential here. 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.

Examples of use of fuzzy sets and logic in GIS projects now abound. Just type in those keywords in Google to get an idea.

Fuzzy histograms or summed probability distributions.

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, the bin, 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 accurate. Other rules for fractional assignment could be used in cases where traditional error information is not available. Some might object if you 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. This doesn't even speak to the accuracy issue (excess argon). You can see the results below.

Summed probability chart approximation (fuzzy histogram with 1 Ma year intervals) of K-Ar data for HALIP basalts and diabases with a n of 97 (Maher, 2001). The blue line represents a value of 1 data point. The conclusion that could be drawn is that the age is only broadly known, and that distinct pulses of magmatism could not be based on this data, contrary to earlier assertions in the literature.

Fuzzy sets in sedimentology.

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 mostly emergent and a bit 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. This is an example of 'capturing' expert knowledge in a clear and consistent way. However, it also gives a chance for the expert to test their understanding and through cycles of testing and feedback, to improve their knowledge.

Exercise 14 - in class.

For next time read the assignment and then give thought to what sets in geosciences (think of those associated with classifications) might be usefully fuzzified. You can use some of the examples above as an initial guide. You will work in groups of 2 to 3, and each group will try to define fuzzy versions of some suite of geoscience sets. You will want to find some supporting material associated with the geoscience classification you are working with. If you find articles where fuzzy classification systems that have been built already, then report on and build on what has been developed. After spending some time exploring possibilities, you will share your ideas with the others in the class.

Each group should hand in a document with:

• names of participants in the group.
• a paragraph or bulleted description of the geoscience sets being worked with, including diagrams and references were appropriate.
• a paragraph description of why these 'crisp' sets may be fuzzy in geologic reality.
• clearly drawn diagrams of membership rules (fuzzy sets shapes). These should not be schematic but have labeled axes, and the boundaries and positions of the sets along the horizontal axis should be geologically reasonable.
• a verbal description of why the particular set 'shape' was used.
• discussion/listing of possible applications of use of the explored fuzzy sets, and the advantages and disadvantages of using a fuzzy set approach versus crisp sets.

References:

• Cagnoli, B, 1998, Fuzzy logic in volcanology, Episodes, 19, 94-96.
• Demicco, R. V. & Klir, G. J., 2004, Fuzzy Logic in Geology, Elsevier, 347 p.
• Maher, H. D. Jr, 2001, Manifestations of the Cretaceous High Arctic Large Igneous Province in Svalbard; Journal of Geology, v. 109, p. 91-104.
• McNeill, D & Freiberger, P, 1994, Fuzzy Logic - The revolutionary Computer Technology that is Changing our World. Touchstone - Simon and Shuster, 319 p.
• Stewart, I, 1993 - Feb., Mathematical Recreations - A partly True Story; Sci. American, p. 110-113.
• Sui, D. Z., 1994, Fuzzy Logic can Help GIS Cope with Reality, GIS World, Fort Collins, CO, Sept, 50-53.