Data Analysis with Intersection Graphs Valter Martins Vairinhos 1 Victor Lobo 1,2 Purificación Galindo Villardón 3

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1 Data Analysis with Intersection Graphs Valter Martins Vairinhos 1 Victor Lobo 1,2 Purificación Galindo Villardón 3 1 Escola Naval, 2 ISEGI-UNL, 3 Universidad de Salamanca, Departamento de Estadística 1- Introduction Formalization of DAIG Some useful results and techniques Constructing DAGT from tables using intersection graphs Data objects and cliques Multivalued characteristics Representing contingency tables Sparse datasets Conclusions and comparisons with other approaches Bibliography... 6 Abstract This paper presents a formalization for multivariate data analysis, based on graph theory, that we have named DAIG Data Analysis with Intersection Graphs. The basic principles of this approach are presented, followed by a brief example and comparison with the standard tablebased approach. A few advantages of this approach are presented, and the paper concludes with a brief comparison with other approaches. 1- Introduction This paper presents a formalization for multivariate data analysis, based on graph theory, that we have named DAIG Data Analysis with Intersection Graphs, first proposed in (Vairinhos 2003). This approach to data analysis breaks away from the traditional table based representation of data, and uses intersection graphs as the basic representation. In this approach, the vertices are not observations or variables, but sets of observations represented by pairs of variables and values (atoms). We will try to show that this approach can in some cases be more flexible and powerful than tables, and can make good use of the extensive recent work in graph theory. To illustrate what DAIG is, we shall use example taken from the well known zoo dataset from the UC-Irvine Machine Learning Repository (UCI 2008), that is composed of 17 characteristics of 101 different animals. Most of these characteristics are binary: hair, feathers, eggs, milk, airborne, aquatic, predator, toothed, backbone, breathes, venomous, fins, tail,, catsize. The two remaining characteristics, while numerical, can be treated as categorical since they have only 6 or 7 values: number of legs, and type (mammal, bird, fish, etc). 2- Formalization of DAIG The main building blocks, or atoms, for our representation of data are sets of observations represented by pairs of variables/values. An observation o i (i = 1n) is formed by p values of 1

2 characteristics or variables X j (j = 1..p), each with a particular value x ij. We can thus represent the observation o i by p pairs of the form (X j = x ij ),(i=1..n; j=1..p), called atoms. We then take these atoms as vertices of a graph, and since these atoms are connected in the sense that they form an observation, we connect them with edges, as can be seen in Figure 1. Data pattern x i (tabular form) Data graph X 1 =x i1 X 2 =x i2 X 1 X 2 x i1 x i2 X j x ij X m xim X m =x im X j =x ij Figure 1- Tabular and data graph representation of an observation On this data graph, an observation or data pattern is a clique, i.e., a set of atoms that are connected amongst themselves. Let us see a practical example. Suppose we observe 3 characteristics of a chicken, and present them in the traditional tabular form. It s representation on a data graph will be a 3-clique, as seen in Figure 2. Data pattern x i Data graph (tabular form) Has fins Is Chicken = fins legs No fins No legs 2 legs 4 legs Figure 2 -Tabular and data graph representation of chicken As more observations are made, their representation can be added to the graph. There are various ways to do this, and depending on what sort of data analysis is desired, we may prefer one or the other. When we have various observations, these will probably share some atoms, i.e., some observations will have the same values for the same variable. This information may be explicitly 2

3 recorded by each atom (vertex) by associating with it a list of the observations that contain them, as seen in Figure 3. This graph is an intersection graph (McKee and McMorris 1999), because the edges correspond to non-empty intersections of the sets of observations that form the vertices. We may or may not associate weights to the edges of this intersection graph, depending on the objective. When necessary those weights can be defined as a function of sets corresponding to adjacent vertices. Those weights may be, for example, the cardinality of the intersection of those sets. Another way of defining weighs may be using the function f(a,b) = a b 2 / a * b where a and b are the sets of adjacent vertices. This function corresponds to the proximity measure P(a b) * P(b a), which can useful in many analysis (Vairinhos 2003). Thus, depending of the objective, after adding various observations, we will have: a) An intersection graph, where each vertex has an associated set of observations. This is the preferred representation, because it retains all the information about individual observations, and thus allows all sorts of analysis to be performed. b) A weighted intersection graph, where ach edge has an associated weight. For example, the weight may be the number of co-occurrences of the atoms in the observations a b or any other function as convenient for the objective of the study. Data patterns (tabular form) Data graph No fins Antelope Bear Chicken Clam Crow Duck Goat Worm Fins Carp Catfish Sealion No legs Carp Catfish Clam Worm Figure 3 - Data graph of a set of observations 2 legs Chicken Crow Duck Sealion Is Carp Chicken Goat 4 legs Antelope Bear Goat Antelope Bear Catfish Clam Crow Duck Sealion Worm Vertices that correspond to a given variable (which we will call variable set) will normally (but not necessarily) form an independent set, since the observations will usually have a single value for each variable. 3

4 The degree of each vertex will usually be the number of variables minus one. However, it can be less if there are missing values, or greater if, for a given observation, there is more than one value for each variable. 3- Some useful results and techniques The proposed formalization leads to many interesting results and allows for efficient implementations of data analysis techniques. One of those implementations can be seen in (Vairinhos 2003) where this formalism has been used to suggest interpretations of concepts discovered with biplots. Each vertex and path along the graph corresponds to a concept described by intention. If the vertices that form this path have a non-empty intersection, that intersection is the extension of the corresponding concept. In this case, that path identifies also a non-empty clique. Let us now see some of the useful techniques and results developed for this formalism Constructing DAGT from tables using intersection graphs Conceptually, the construction of the data structure for DAGT is just like a chemical analysis of a compound: the observation is broken down into it s constituting atoms. For each variable that characterizes the observation, the existing graph is updated by adding the label of new observation to the vertex that corresponds to it s value. Edges will be added to the other vertices that compose the data pattern, if necessary. The edges can always be added later by finding nonempty intersections of the vertices. The weights of those edges can also be computed, since they are a function of the cardinality of those intersections. Adding the edges as the observations are introduced has the advantage of reducing the computing time later, since it avoids computing empty intersections. 3.1 Data objects and cliques As stated previously, observations will correspond to cliques in the graph. If the observations are characterized by p variables, they will be p-cliques. It must be noted that in many real problems there are missing values. In this case, if only m < p variables where measured for an observation, then the corresponding clique will also be of a lower order. A clique in the graph may correspond to an observation, a set of observations, or no observation at all. A p-clique corresponds to a complete specification of a data observation, and will thus correspond to single datum, a set of undistinguishable ones, or a non-existent object. Whether that observation exists or not can only be determined by computing the intersection of the vertices involved. If an empty set is obtained, then no observation with those characteristics was made. A m-clique (with m < p), will be a generalization, or concept, since some variables that characterize the observations are left out. It is important to note that concepts do not have to be cliques. The set of observations that correspond to a given concept is one again found by computing the intersection of the sets of the vertices. 3.2 Multivalued characteristics 4

5 One of the advantages of DAIG is the ease with which it deals with multi-valued characteristics. These types of variables are not very common, but they do occur, and are dealt with in a rather awkward manner by the traditional tabular approach. An example a possibly multi-valued variable is the nationality of a person. While most people have a single nationality, it is possible to have 2 or more. While this constitutes a problem for some representations, a DAIG will simple add the identifier of the observation to the various vertices that correspond to the different nationalities. The variable set will no longer be an independent set, but for most graph algorithms that is not important. This is also a case where the degree of the vertices can greater than the number of variables Representing contingency tables Another big advantage of DAIG is the ease with which multiway contingency tables can be represented. A cell in a multiway contingency table can be seen as the intersection of p atoms corresponding to the p variables of al-way table, having as absolute frequency the cardinal of that intersection. It is then easy to see that a cell in a p-way table corresponds to a non-empty p-clique and can be represented in a parallel coordinates graph as a path connecting all the variables and, in a radar graph as a polygon of p-edges. An example of this type of analysis it is show in Figure 4 where the 17-way contingency table of the 101 animals of the zoo dataset was filtered to show only the edges with a weight of more than 70. Instead of showing the numerical weighs, these are coded as thickness of the edges. The interpretation of this sketch is quite intuitive. Airbourne Has tail No feathers Venomous Has backbone Breaths Domestic Figure 4 - Sketch of the data graph of the 17 characteristics of 101 animals, filtered to inclide only edges with more than 70 ocurrences. In effect this corresponds to the most important information of the 17-way contiungency table. Fins 3.4 Sparse datasets The proposed DAIG is far more efficient than the traditional table-based representation when there are many missing values in the data. This a situation that is common in real data, for a number of reasons. A table based representation has to waste space with non-existing values, while the graph-based representation does not. 5

6 A data-analysis program has been developed using the DAIG framework, that has proved to be useful and efficient. The program was originally developed as part of a PhD thesis (Vairinhos 2003), and improved versions (Vairinhos 2004) have been used to analyse maintenance data of ships (Parreira, Vairinhos et al. 2007). 4. Conclusions and comparisons with other approaches An alternate way of looking at data analysis was explained. This approach is based on intersection graph theory providing an environment for reasoning about data and concepts based on points (vertices), lines (paths) that represent relationships amongst values of different variables. Graphs have been used for data analysis, but in a very different way. In (Agresi 1996) the concept of Formal Concept Analysis is developed, based in the algebraic concept of lattice, and the associated graphs. Association Graphs, for example (Agresi 1996) are used to represent correlations amongst variables. In (Whittaker 1990; Edwards 1995; Lauritzen 1996) graphical models are developed to represent the relations of conditional dependencies between variables using graphs. In those approaches, however, vertices correspond to variables (and not individual values of those variables), and edges are added when there is a conditional dependence between two variables. 5. Bibliography Agresi, A. (1996). An introduction to categorical data analysis. New York, John Wiley & Sons. Edwards, D. (1995). Introduction to Graphical Modeling, Springer. Lauritzen, S. L. (1996). Graphical Models, Oxford Science Publications. McKee, T. A. and F. R. McMorris (1999). Topics in intersection graph theory. Philadelphia, PA, Society for Industrial and Applied Mathematics (SIAM). Parreira, R. R., V. Vairinhos, et al. (2007). Análise de parâmetros de operação de máquinas marítimas. XIV Jornadas de Classificaçaão e Análise de Dados JOCALD Porto. UCI (2008). Machine Learning Repository, University of California at Irvine Vairinhos, V. M. (2003). Desarrollo de un Sistema para Minería de Datos Basado en los Metodos Biplot. Salamanca, Spain, Universidad de Salamanca. Vairinhos, V. M. (2004). BiplotsPMD- data Mining Cebtrada em Biplots. Apresentação de Um Protótipo. JOCLAD' XI Jornadas de Classificação e análise de Dados, Lisbon. Whittaker, J. (1990). Graphical Models in Applied Statistics, John Wiley. 6