Decision Making Based on Satellite Images: Optimal Fuzzy Clustering Approach

In many real life decision making situations in particu lar in processing satellite images we have an enormous amount of information to process To speed up the in formation processing it is reasonable to rst classify the situations into a few meaningful classes clusters nd the best decision for each class and then for each new situation to apply the decision which is the best for the corresponding class One of the most e cient clustering methodologies is fuzzy clustering which is based on the use of fuzzy logic Usually heuristic clus terings are used i e methods which are selected based on their empirical e ciency rather than on their proven optimality Because of the importance of the corre sponding decision making situations it is therefore de sirable to theoretically analyze these empirical choices In this paper we formulate the problem of choosing the optimal fuzzy clustering as a precise mathematical problem and we show that in the simplest cases the empirically best fuzzy clustering methods are indeed optimal Fuzzy Clustering Existing Approaches and Formulation of the Problem For satellite imaging fuzzy clustering is im portant Decision making is especially important in geophysics because in many geophysical situations a wrong decision can be very costly be it digging a well where there is no oil or not preparing the building for the potential earthquakes or spending lost of ef fort on securing building against earthquakes which are not typical for this area To decrease the possi bility of a costly erroneous decision we must use as much information as possible One of the important sources of such information is satellite imaging How ever with satellite images we face a di erent problem each satellite image contains a huge amount of data A good photo contains up to a Gigabyte of information and with modern multi spectral satellite images we get several Gigabytes We do not know how to process all this information One of the known methods of ghting this information explosion is clustering Instead of analyzing each photo individually we do the following First we classify the photos into a few meaningful clusters Then for each cluster we nd the best decision Finally when we encounter a new situation we nd the cluster to which this situation belongs and make a decision which is the best for this cluster The idea of clustering is very natural in science The analysis of every new phenomenon starts with classi cation when instead of numerous di erent examples we have a few classes Classi cation helped to analyze chemical elements elementary particles living organ isms astronomical objects etc In some situations where assumptions about structure of data can be formulated in statistical terms statisti cal techniques see e g are appropriate if we have su ciently many data In other situations we must use heuristic classi cation methods in particular methods that use fuzzy logic The main idea of fuzzy clustering is described in The goal of fuzzy clustering typical repre sentatives and how to use them We start with objects which we want to classify i e to cluster To classify we use several numerical characteristics of these object Let us denote the total number of these characteristics by s The s real numbers that charac terize each object can be naturally viewed as a point in s dimensional space R Thus having n objects means that we have n points x xn in this space These n points are the input for clustering As a result of clustering we want to describe several clusters Each cluster can be characterized by its typ ical element tj R s After these typical elements t tq are found we can then classify each object x R according to which typical element it is closest to This classi cation is a fuzzy notion if an element x is very close to say t and not close to any other typical representative then it is reasonable to conclude that x belongs to class however if an object x R is almost equally close to two di erent representatives t and t then it is reasonable to conclude that this object belongs to some extent to both clusters and To express this idea in precise terms we select a func tion f x called potential function such that for every two point x and y fromR the value f x y describes to what extent x and y are close This function is usu ally non negative and the closer x and y the larger the value of the potential function Potentially as a poten tial function we can use a membership function which describes the relation x and y are close however from the mathematical viewpoint the choice of mem bership function would mean that we only allow f x to take values from the interval and sometimes more general values are needed in our main text we will explain why we need such values When the potential function is selected then we can say that an object x belongs to st cluster with a degree f x t to the nd cluster with the degree f x t and to q th cluster with the degree f x tq Since we do not require any normalization of the function f x it is convenient to normalize these values so that they will add up to in other words to describe the degree to which x belongs to j th cluster as dj x f x tj f x t f x tq How to nd typical representatives The most widely used approach We have described how to classify an object when the clusters or to be more precise their typical representatives have al ready been found How can we nd these representa tives The most widely used fuzzy clustering method is the method of Fuzzy C Means Fuzzy ISODATA This method is based on the natural idea that each characteristic of a typical representative should be equal to an average over all elements of the corresponding cluster If we have crisp clustering then we would simply take the arithmetic average How ever since we have fuzzy clustering it is natural to count in this average each element xi with the weight dj xi that is proportional to this element s degree of belonging to the cluster In other words it is natural to require that for each j tj dj x x dj xn xn dj x dj xn This method leads to good quality clustering Its main disadvantage is that since the values dj xi in their turn depend on tj the equation is actually a non linear system of equations for determining the cluster centers t tq and solving this system of equa tions often requires lots of computation time How to nd typical representatives Recent approaches To simplify computations a new method has been recently proposed see also This method is based on the following idea when we say that an element tj is a typical representative of the cluster that consists of elements xi xik we mean that for each element x R the degree f x tj with which x is close to tj is equal to the average of the degrees f x xi f x xik with which x is close to all elements of this cluster f x xi f x xik k f x tj If we have a crisp classi cation then each of the origi nal data points x xn belongs to one and only one cluster and therefore by adding equalities for all q clusters we would conclude that