人工智能英文版课件：Information granularity

1、Information granularity, information granules, and Granular ComputingInformation granularity and information granules Granular Computing- examples and key observationsFormal platforms of information granularityInformation granules of higher type and higher orderGranulation- degranulation principle O

2、utlineInformation granularityInformation granules as intuitively appealing constructs playing a pivotal role in human cognitive and decision-making activities. We perceive complex phenomena by organizing existing knowledge along with available experimental evidence.Knowledge is structured in a form

3、of some meaningful, semantically sound entities, which are central to all ensuing processes of describing the world, reasoning about the environment and support decision-making activities. Information granules and Granular ComputingIn general, by information granule one regards a collection of eleme

4、nts drawn together by their closeness (resemblance, proximity, functionality, etc.) articulated in terms of some useful spatial, temporal, or functional relationships. Granular Computing is about -representing, -constructing, and -processing information granules. Information granules and levels of a

5、bstractioninformation granules permeate almost all human endeavorsNo matter which problem is taken into consideration, we usually set it up in a certain conceptual framework composed of some generic and conceptually meaningful entities - information granules, which we regard to be of relevance to th

6、e problem formulation, further problem solving, and a way in which the findings are communicated to the community. Information granules realize a framework in which we formulate generic concepts by adopting a certain level of abstraction. Information granules examples (1)Image processing. In spite o

7、f the continuous progress in the area, a human being assumes a dominant and very much uncontested position when it comes to understanding and interpreting images. We do not focus our attention on individual pixels and process them as such but group them together into semantically meaningful construc

8、ts familiar objects we deal with in everyday life. Such objects involve regions that consist of pixels or categoriesof pixels drawn together because of their proximity in the image, similar texture, color, etc. This remarkable and unchallenged ability of humans dwells on our effortless ability to co

9、nstruct information granules, manipulate them and arrive at sound conclusions. Information granules examples (2)Processing and interpretation of time series. From our perspective we can describe time series in a semi-qualitative manner by pointing at specific regions of such signals. One distinguish

10、es some segments of temporal signals and interpret their combinations. E.g., in stock market, one analyzes numerous time series by looking at existing amplitudes, trends, patterns, and relationships among them. Information granules examples (3)Granulation of time Time is another important and omnipr

11、esent variable that is subjected to granulation. We use seconds, minutes, days, months, and years. Depending upon a specific problem we have in mind who the user is, the size of information granules (time intervals) could vary quite significantly. Information granules examples (4)Design of software

12、systems We develop software artifacts by admitting a modular structure of an overall architecture of the designed system.Each module is a result of identifying essential functional closeness of some components of the overall system. Modularity (granularity) is a holy grail of the systematic software

13、 design supporting a production of high quality software products. Information granules some general observationsinformation granules are the key components of knowledge representation and processing, the level of granularity of information granules (their size, to be more descriptive) becomes cruci

14、al to the problem description and an overall strategy of problem solving,hierarchy of information granules supports an important aspect of perception of phenomena and delivers a tangible way of dealing with complexity of the system by focusing on the most essential facets of the problem, there is no

15、 universal level of granularity of information; commonly the size of granules is problem-oriented and user dependent.Human-centricity and information granulesHuman-centricity comes as an inherent feature of intelligent systems. It is anticipated that a two-way effective human-machine communication i

16、s imperative. Human perceive the world, reason, and communicate at some level of abstraction. Abstraction comes hand in hand with non-numeric constructs, which embrace collections of entities characterized by some notions of closeness, proximity, resemblance, or similarity. Algorithmic realization G

17、ranular ComputingIn algorithmic realization of Granular Computing, implicit nature of information granules has to be translated into constructs that are explicit in their nature, viz. described formally in which information granules can be efficiently processed. Granular Computing main observations

18、(1)It identifies the essential commonalities between the surprisingly diversified problems and technologies used there, which could be cast into a unified framework known as a granular world. fully operational processing entity that interacts with the external world (that could be another granular o

19、r numeric world) by collecting necessary granular information and returning the outcomes of the granular computingGranular Computing main observations (2)existing plethora of formalisms of set theory (interval analysis) rough setsfuzzy setsshadowed setsprobabilityrandom sets placed under the same ro

20、of by clearly visualizing that in spite of their visibly distinct underpinnings (and ensuing processing), they exhibit some fundamental commonalities. In this sense, Granular Computing establishes a highly stimulating environment of synergy between the individual approaches. Granular Computing main

21、observations (3)By building upon the commonalities of the existing formal approaches, Granular Computing helps assemble heterogeneous and multifaceted models of processing of information granules by clearly recognizing the orthogonal nature of some of the existing and well established frameworks. Gr

22、anular Computing main observations (4)Granular Computing fully acknowledges a notion of variable granularity whose range could cover detailed numeric entities and very abstract and general information granules. It looks at the aspects of compatibility of such information granules and ensuing communi

23、cation mechanisms of the granular worlds.Information granules arise as an evident realization of the fundamental paradigm of abstraction.Formal platforms of Granular Computing sets and intervalsSets (intervals) realize a concept of abstraction by introducing a notion of dichotomy: we admit element t

24、o belong to a given information granule or to be excluded from it. Along with set theory comes a well-developed discipline of interval analysisSets are described by characteristic functions taking on values in 0,1. Formal platforms of Granular Computing sets and intervalsCalculus of characteristic f

25、unctionsIntersection and unionFormal platforms of Granular Computing fuzzy setsimportant conceptual and algorithmic generalization of sets. By admitting partial membership of an element to a given information granule we bring an important feature which makes the concept to be in rapport with reality

26、. It helps working with the notions where the principle of dichotomy is neither justified nor advantageous. Formal platforms of Granular Computing shadowed setsdescription of information granules by distinguishing among elements which fully belong to the concept, are excluded from it, and whose belo

27、ngingness is completely unknown. Formally, these information granules are described as a mapping X: X 1, 0, 0,1 where the elements with the membership quantified as the entire 0,1 interval are used to describe a shadow of the construct. Formal platforms of Granular Computing probability oriented inf

28、ormation granulesexpressed in the form of some probability density functions or probability functions. They capture a collection of elements resulting from some experiment. In virtue of the concept of probability, the granularity of information becomes a manifestation of occurrence of some elements.

29、For instance, each element of a set comes with a probability density function truncated to 0,1, which quantifies a degree of membership to the information granule. Formal platforms of Granular Computing rough setsemphasize a roughness of description of a given concept X when being realized in terms

30、of the indiscernibility relation provided in advance. The roughness of the description of X is manifested in terms of itslower and upper approximations associated with a certain rough set. Information granules symbolic and numeric viewsymbolic perspective A concept - information granule is viewed as

31、 a single symbol (entity). This view very much present in the AI community, where computing revolves around symbolic processing numeric perspective information granules are associated with a detailed numeric characterization. Fuzzy sets are profound examples with this regard. We start with numeric m

32、embership functions. All ensuing processing involves numeric membership grades so in essence it focuses on number crunching. Information granularity and its quantificationlevel of abstraction supported by information granules is associated with the number of elements embraced by the granule. A certa

33、in measure of cardinality, which counts the number of elements in the information granule forms a sound descriptor of information granularity A principle of the least commitmentInstead committing a hustle decision made on a basis of perhaps initial insufficient information, the low level of specific

34、ity indicates postponement of any action until more evidence has been collected and the obtained results become specific enough. Information granules of higher typeThe quantification of levels of belongingness to a given information granule is granular itself rather than numeric as encountered in se

35、ts or fuzzy sets. This type of quantification is of interest in situations it is not quite justifiable or technically sound to quantify the grade of membership in terms of a single numeric value. These situations give rise to ideas of type-2 fuzzy sets or interval-valued fuzzy sets. Information gran

36、ules of higher orderThe notion of higher order of information granules points at a space in which an information granule is defined. Here the universe of discourse is composed of a family of information granules. fuzzy set of order 2 is constructed in the space of a family of so-called reference fuz

37、zy sets Hybrid models of information granules (1)fuzzy probabilities Probability and fuzzy sets are orthogonal concepts and as such they could be considered together as a single entity. The concepts of a fuzzy event and fuzzy probabilities (viz. probabilities whose values are quantified in terms of

38、fuzzy sets, say high probability, very low probability, negligible probability, Hybrid models of information granules (2)fuzzy rough and rough fuzzy information granules indiscernibility relation can be formed on a basis of fuzzy sets. Fuzzy sets, rather than sets are the entities that are described

39、 in terms of the elements of the indiscernibility relation Design of information granulesConstruction of a single information granule construction of a family of information granules Granulation-degranulation principle construction of information granule(granulation)reconstruction of information gra

40、nule(degranulation)original datareconstructed dataInformation granules in data representationOptimal allocation of information granularity Numeric mappingGranular parametersGranular mappingKey formalisms of information granules and processing mechanismsSets and their basic conceptsInterval analysis

41、and interval calculusFuzzy setsRough setsShadowed setsOutlineSet and interval analysisSets are fundamental notions of mathematics, science, and engineeringDichotomy as the underlying fundamental notion - belongingness- exclusion Sets and characteristic functionsCharacteristic functionsCharacteristic

42、 functionFormally, characteristic function of A expressed as mappingA: X 0,1Characteristic functions- main propertiesEmpty set Universe of discourse Operations on sets = operations on characteristic functionsIntersection min(A(x), B(x)Union max (A(x), B(x)Complement 1- A(x)RelationsTwo spaces X and

43、YCartesian product of the spaces X x YRelation R-a collection of pairs (x,y) Characteristic function of RExampleX=Y = 1, 4, 8, 9Relation “equal” n-ary relation“n” spaces X1 X2 XnInterval analysisNumeric intervals as generic information granulesA = a, b B =c, dProcessing concerns the use of set theor

44、etic operationsalgebraic operationsmappingsSet-theoretic operationsAlgebraic operationsUnary operation (f)Minimum (maximum) taken over “x” belonging to ADistance and metricreal numbers Hamming distanceHausdorff distancef(x,y) = |x-y| and A and B are compact, nonempty sets of real numbers Mapping int

45、ervalsMapping through functionMapping through relationFuzzy sets a departure from dichotomyConceptually and algorithmically, fuzzy sets arise as one of the most fundamental and influential notions in science and engineering. The notion of a fuzzy set is highly intuitive and transparent Fuzzy set cap

46、tures a way in which a real world is being perceived and described in our everyday activities.Description of objects whose belongingness to a given category (concept) is a matter of degree Fuzzy sets examplesHigh temperatureSafe speedLow humidityMedium inflationLow approximation errorFuzzy sets defi

47、nitionFuzzy set A defined in universe of discourse X in terms of itsmembership function A: X 0,1A(x) degree of membershipA(x) =1 full membership A(x) =0 exclusionA(x) in-between 0 and 1 partial membership Classes of membership functionsTriangular membership function Trapezoidal membership function C

48、lasses of membership functionsTrapezoidal membership function Classes of membership functionsS membership function Gaussian membership function Selected descriptors of fuzzy setsNormality fuzzy set is normal if the following condition is satisfiedHeight of fuzzy set Normalization of fuzzy setSelecte

49、d descriptors of fuzzy setsSupportCorea-cuts of fuzzy sets-cut of a fuzzy set A, denoted by A - a set consisting of the elements of the universe whose membership values are equal to or exceed acertain threshold level where in 0,1 strong a-cuta-cuts of fuzzy setsCardinality of fuzzy sets and specific

50、itySpecificity highest lowestRelationships between fuzzy setsEquality InclusionRepresentation theoremFuzzy set represented as a family of a-cutsTriangular norms as models of logic operatorst: 0,1 x 0,1 0,1Triangular norms as models of logic operators- examplesTriangular co-norms as models of logic o

51、peratorss: 0,1 x 0,1 0,1Triangular co-norms as models of logic operators-examplesTriangular norms and co-norms: dualityDe Morgan lawsRough setsVocabulary consisting of A1, A2, A20Describe X Rough sets- lower and upper boundLower boundUpper boundExample - detailsLower bound X- =A11 Upper bound X+Roug

52、h set definition and interpretationFor given Ai Rough set - definitionShadowed setsA three valued representation of fuzzy sets 0, 1, 0,1= SSets, fuzzy sets and shadowed setsOperations on shadowed setsDesign of shadowed setsConstruction of shadowed set for a given fuzzy setLocalization of membership

53、Membership condensation or “localization”. in fuzzy sets we encounter intermediate membership grades located in-between 0 and 1 and distributed across the entire space, in shadowed sets we “localize” the membership effect by building constrained and fairly compact shadows of the shadowed set. Locali

54、zation of membershipReduction of membership + Elevation of membership = shadowLocalization of membership- detailsLocalization of membership- optimizationTriangular membership function optimal aLocalization of membership optimal aTriangular membership functionGaussian membership functionInformation g

55、ranules of higher type, higher order, and hybrid information granules Fuzzy sets of higher orderRough fuzzy sets and fuzzy rough sets Fuzzy sets of higher type (type-2)Interval-valued fuzzy setsProbabilistic setsHybrid information granules- probability and fuzzy sets and their orthogonalitySystem mo

56、dels with information granules of higher order and type OutlineFuzzy sets of higher orderFuzzy sets defined in a universe of discourse whose elements are fuzzy sets rather than individual elementscomfortable temperature - fuzzy set of second order fuzzy sets as elements of the universeFuzzy sets of

57、higher order and fuzzy setsA nature of the universe of discourse fuzzy set of second order fuzzy setFuzzy sets of higher order- a hierarchical constructA hierarchical definition of fuzzy set a number of descriptorsexpressed as fuzzy sets fuzzy set of order 2Fuzzy rough setUpper boundLower boundFuzzy

58、 rough set - exampleIntervals as generic descriptorsExpress fuzzy set A in terms of generic descriptors resulting bounds: ARough fuzzy setsA family of descriptors realized in terms of fuzzy setsDescribe set XType-2 fuzzy setDetermination of numeric membership grades? Generalize: consider membership

59、grades as fuzzy sets defined in 0, 1Fuzzy sets of type-2Membership grades of different form (triangular, Gaussian membership)Intervals instead of fuzzy sets interval-valued fuzzy setsas a special case of type-2 fuzzy setsType-2 fuzzy sets origin and motivating exampleLocally developed fuzzy setaggregation of fuzzy setsleading to type-2 fuzzy setInterval-valued fuzzy sets and operationsProbabilistic setsA is