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Encyclopedia of Complexity and Systems Science pp 1789–1810Cite as

Data-Mining and Knowledge Discovery: Case-Based Reasoning, Nearest Neighbor and Rough Sets

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Definition of the Subject

Basic ideas of Rough Set Theory were proposed by Zdzisław Pawlak [60,62], as a formal set of notions aimed at carrying out tasks of reasoning, in particular about classification of objects, in conditions of uncertainty. Conditions of uncertainty are imposed by incompleteness, imprecision and ambiguity of knowledge. Originally, the basic notion proposed was that of a knowledge base, understood as a collection \( { \mathcal{R} } \) of equivalence relations on a universe of objects; each relation \( { r \in {\mathcal{R}} } \) induces on the set U a partition \( { \operatorname{ind}_{r} } \)into equivalence classes. Knowledge so encoded is meant to represent the classification ability. As objects for analysis and classification come most often in the form of data, a useful notion of an information system is commonly used in knowledge representation; knowledge base in that case is defined as the collection of indiscernibility relations. Exact concepts are defined...

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Abbreviations

Knowledge:

This is a many‐faceted and difficult notion to define and it is used very frequently without any attempt at definition as a notion that explains per se. One can follow J. M. Bocheński in claiming that the world is a system of states of things, related to themselves by means of the network of relations; things, their features, relations among them and states are reflected in knowledge: things in objects or notions, features and relations in notions (or, concepts), states of things in sentences. Sentences constitute knowledge. Knowledge allows its possessor to classify new objects, model processes, make predictions etc.

Reasoning:

Processes of reasoning include an effort by means of which sentences are created; various forms of reasoning depend on the chosen system of notions, symbolic representation of notions, forms of manipulating symbols etc.

Knowledge representation:

This is a chosen symbolic system (language) by means of which notions are encoded and reasoning is formalized.

Boolean functions:

An n‑ary Boolean function is a mapping \( { f \colon \{0,1\}^n\rightarrow \{0,1\} } \) from the space of binary sequences of length n into the doubleton \( { \{0,1\} } \). An equivalent representation of the function f is as a formula \( { \phi_f } \) of propositional calculus; a Boolean function can be thus represented either in DNF form or in CNF form. The former representation: \( { \vee_{i \in I} \wedge_{j \in J_i} l^i_j } \), where \( { l^i_j } \) is a literal, i. e., either a propositional variable or its negation, is instrumental in Boolean Reasoning: when the problem is formulated as a Boolean function, its solutions are searched for as prime implicants \( { \bigwedge_{j\in J_i} l^i_j } \). Applications are to be found in various reduct induction algorithms.

Information systems:

One of the languages for knowledge representation is the attribute‐value language in which notions representing things are described by means of attributes (features) and their values; information systems are pairs of the form (U, A) where U is a set of objects – representing things – and A is a set of attributes; each attribute a is modeled as a mapping \( { a \colon U\rightarrow V_a } \) from the set of objects into the value set V a . For an attribute a and its value v, the descriptor \( { (a=v) } \) is a formula interpreted in the set of objects U as \( { [(a=v)]=\{u\in U \colon a(u)=v\} } \). Descriptor formulas are the smallest set containing all descriptors and closed under sentential connectives \( { \vee, \wedge, \neg, \Rightarrow } \). Meanings of complex formulas are defined recursively: \( { [\alpha\vee\beta]=[\alpha]\cup [\beta] } \), \( { [\alpha\wedge\beta]=[\alpha]\cap [\beta] } \), \( { [\neg \alpha]=U\setminus [\alpha] } \), \( { [\alpha\Rightarrow\beta]=[\neg\alpha\vee\beta] } \). In descriptor language each object \( { u\in U } \) can be encoded over a set B of attributes as its information vector \( { \operatorname{Inf}_B(u)=\{(a=a(u)) \colon a\in B\} } \).

Indiscernibility:

The Leibnizian Principle of Identity of Indiscernibles affirms that two things are identical in case they are indiscernible, i. e., no available operator acting on both of them yields distinct values; in the context of information systems, indiscernibility relations are induced from sets of attributes: given a set \( { B\subseteq A } \), the indiscernibility relation relative to B is defined as \( \operatorname{Ind}(B)=\{(u,u^{\prime}) \colon a(u)=a(u^{\prime})\ \textrm{for each}\ a\in B\} \). Objects \( { u, u^{\prime} } \) in relation Ind(B) are said to be B‑indiscernible and are regarded as identical with respect to knowledge represented by the information system (U, B). The class \( { [u]_B=\{u^{\prime} \colon (u,u^{\prime})\in \operatorname{Ind}(B)\} } \) collects all objects identical to u with respect to B.

Exact, inexact notion:

An exact notion is a set of objects in the considered universe which can be represented as the union of a collection of indiscernibility classes; otherwise, the set is inexact. In this case, there exist a boundary about the notion consisting of objects which can be with certainty classified neither into the notion nor into its complement (Pawlak, Frege).

Decision systems:

A particular form of an information system, this is a triple \( { (U,A,d) } \) in which d is the decision, the attribute not in A, that does express the evaluation of objects by an external oracle, an expert. Attributes in A are called conditional in order to discern them from the decision d.

Classification task:

The problem of assigning to each element in a set of objects (test sample) of a class (of a decision) to which the given element should belong; it is effected on the basis of knowledge induced from the given collection of examples (the training sample). To perform this task, objects are mapped usually onto vectors in a multi‐dimensional real vector space (feature space).

Decision rule:

A formula in descriptor language that does express a particular relation among conditional attributes in the attribute set A and the decision d, of the form: \( { \bigwedge_{a\in A}(a=v_a)\Rightarrow (d=v) } \) with the semantics defined in (Glossary: “Indiscernibility”). The formula is true in case \( [\bigwedge_{a\in A}(a=v_a)]=\bigcap_{a\in A}[(a=v_a)]\subseteq [(d=v)] \). Otherwise, the formula is partially true. An object o which matches the rule, i. e., \( { a(o)=v_a } \) for \( { a\in A } \) can be classified to the class \( { [(d=v)] } \); often a partial match based on a chosen distance measure has to be performed.

Distance functions (metrics):

A metric on a set X is a non‐negative valued function \( { \rho \colon X\times X \rightarrow R } \) where R is the set of reals, which satisfies conditions: 1. \( \rho (x,y)=0 \) if and only if \( { x=y } \). 2. \( \rho (x,y)=\rho (y,x) \). 3. \( \rho (x,y)\leq \rho (x,z) + \rho (z,y) \) for each z in X (the triangle inequality); when in 3. \( { \rho (x, y) } \) is bound by \( \max \{\rho (x,z), \rho (z,y)\} \) instead of by the sum of the two, one says of non‐archimedean metric.

Object closest to a set:

For a metric ρ on a set X, and a subset Y of X, the distance from an object x in X and the set Y is defined as \( \operatorname{dist}(x,Y)=\operatorname{inf} \{\rho (x,y) \colon y \in Y\} \); when Y is a finite set, then infimum inf is replaced with minimum min.

A nearest neighbor:

For an object \( { x_0 \in X } \), this is an object \( { n(x_0) } \) such that \( { \rho (x_0, n(x_0)) = dist (x_0, X\setminus \{x_0\}) } \); \( { n(x_0) } \) may not be unique. In plain words, \( { n(x_0) } \) is the object closest to x 0 and distinct from it.

K-nearest neighbors:

For an object \( { x_0 \in X } \), and a natural number \( { K\geq 1 } \), this is a set \( { n(x_0, K)\subseteq X\setminus \{x_0\} } \) of cardinality K such that for each object \( y\in X\setminus [n(x_0,K)\cup \{x_0\}] \) one has \( { \rho(y, x_0) \geq \rho (z,x_0) } \) for each \( { z\in n(x_0,K) } \). In plain words, objects in \( { n(x_0,K) } \) for a set of K objects that are closest to x 0 among objects distinct from it.

Rough inclusion:

A ternary relation μ on a set \( U\times U\times [0,1] \) which satisfies conditions: 1. \( { \mu(x,x,1) } \). 2. \( { \mu(x,y,1) } \) is a binary partial order relation on the set X. 3. \( { \mu(x,y,1) } \) implies that for each object z in U: if \( { \mu(z,x,r) } \) then \( { \mu(z,y,r) } \). 4. \( { \mu(x,y,r) } \) and \( { s<r } \) imply that \( { \mu(x,y,s) } \). The formula \( { \mu(x,y,r) } \) is read as “the object x is a part in object y to a degree at least r”. The partial containment idea does encompass the idea of an exact part, i. e., mereological theory of concepts (Leśniewski, Leonard–Goodman).

Similarity:

An extension and relaxing of an equivalence relation of indiscernibility. Among similarity relations, we single out the class of tolerance relations τ (Poincaré, Zeeman) which are reflexive, i. e., \( { \tau(x,x) } \), and symmetric, i. e., \( { \tau(x,y) } \) implies necessarily \( { \tau(y,x) } \). The basic distinction between equivalences and tolerances is that the former induce partitions of their universes into disjoint classes whereas the latter induce covering of their universes by their classes; for this reason they are more difficult in analysis. The symmetry condition may be over‐restrictive, for instance, rough inclusions are usually not symmetric.

Case:

A case is informally a part of knowledge that describes a certain state of the world (the context), along with a query (problem) and its solution, and a description of the outcome, i. e., the state of the world after the solution is applied.

Retrieve:

A process by means of which a case similar in a sense to the currently considered is recovered from the case base.

Reuse:

A process by means of which the retrieved case's solution is re-used in the current new problem.

Revise:

A process by means of which the retrieved solution is adapted in order to satisfactorily solve the current problem.

Retain:

A process by means of which the adapted solution is stored in the case base as the solution to the current problem.

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  1. Polish–Japanese Institute of Information Technology, Warsaw, Poland

    Lech Polkowski

  2. Department of Mathematics and Computer Science, University of Warmia and Mazury, Olsztyn, Poland

    Lech Polkowski

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Polkowski, L. (2009). Data-Mining and Knowledge Discovery: Case-Based Reasoning, Nearest Neighbor and Rough Sets. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_114

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