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Fuzzy System - Coggle Diagram

- - - - a mathematical framework for dealing with uncertainty, imprecision, and approximation, providing a foundation for a logical system that allows for degrees of truth rather than absolute truth or falsehood
    - - Fuzzy Sets: a fundamental concept in fuzzy logic, providing a means of representing uncertainty and vagueness in the categorization of data.
      - Fuzzy Rules: Fuzzy rules are fundamental in constructing a fuzzy logic system. They are formulated as if-then statements that guide the decision-making process, especially in a fuzzy inference system. Fuzzy rules map input fuzzy sets to output fuzzy sets by establishing relationships between different linguistic variables.
      - Fuzzy Inference: Fuzzy inference is a process that interprets the values in the input vector and, based on user-defined rules, assigns values to the output vector. It's a method that maps from a given input to an output using fuzzy logic.
      - Defuzzification: Defuzzification is the process of converting a fuzzy output set into a crisp value. After the fuzzy inference process, the aggregated fuzzy output set represents degrees of membership across a range of output values. Defuzzification translates this fuzzy set into a single, specific output value that can be used to make decisions or control a system.
    - - Tolerance for Imprecision: It is robust and can manage imprecise and noisy data.
      - Suitability for Non-linear Functions: It can model non-linear functions of arbitrary complexity.
      - Ease of Design: Fuzzy logic systems can be easily designed without the need for a precise quantitative analytical model.
    - - Lack of Precision: It may not provide precise solutions for problems that have accurate mathematical models.
      - Computational Complexity: The computational cost can be high due to the processes of fuzzification, rule evaluation, and defuzzification.
      - Subjectivity: Designing membership functions and rule bases may involve a degree of subjectivity and can be application-specific.
  - - - Fuzzy controller integrated with neural network framework
        
        Benefit
        
        Addressed the knowledge acquisition and representation problem of classic fuzzy control systems by utilizing neural network methods.
        
        Reduce the computing burden by combining neural network structure and fuzzy logic framework.
        
        Model structure
        
        Fuzzification
        
        Linearly combine trigonometric functions with data driven method tuning weights to simulate the membership function
        
        Utilize a fully connected layer to realize the fuzzification
        
        Neural network based rule mapping
        
        Map the fuzzification linguistic output to parameters of triples which are inputs of the defuzzifier unit.
        
        Defuzzification
        
        A modified center of area method is used to calculate the numerical control signal output
        
        Model training scheme
    - - Benefits
        
        Handling Ambiguity: Capable of handling ambiguous and imprecise information effectively.
        
        Intuitive: Allows experts to encode their knowledge and expertise in a human-understandable manner
        
        Robust: Provides meaningful decisions even in scenarios with uncertainties and partial information.
      - Challenges
        
        Subjectivity: The choice of linguistic terms, membership functions, and rules may involve subjectivity.
        
        Computational Cost: Depending on the number of rules and variables, the fuzzy inference process can be computationally intensive.
        
        Complexity: Developing a comprehensive fuzzy decision-making model can be complex and require expertise
      - Real-world Applications
        
        Investment Analysis: Evaluating investment opportunities considering factors like risk, return, and market conditions, which might be described in fuzzy terms.
        
        Supplier Selection: Choosing suppliers based on criteria like reliability, cost, and quality, where assessments may be subjective and imprecise.
        
        Medical Diagnosis: Making diagnostic decisions considering symptoms and test results, which may be vague or ambiguous.
        
        Customer Relationship Management: Making decisions related to customer interaction and service levels, considering fuzzy customer feedback.
    - - Benefits
        
        Human-like Decision Making: Enables AI systems to make decisions in a way that is similar to human reasoning.
        
        Handling Uncertainty: Effectively manages uncertainties and imprecisions in various applications.
        
        Enhanced User Interaction: Improves interaction with users by understanding and working with vague and imprecise inputs.
      - Challenges
        
        Design Complexity: Developing fuzzy systems can be complex and may require expert knowledge.
        
        Computational Cost: Fuzzy inference, especially in systems with numerous rules and variables, can be computationally intensive.
        
        Optimization: Tuning membership functions and rule bases for optimal performance can be challenging.
      - Real-world Applications
        
        Natural Language Processing (NLP): Handling the imprecision and ambiguity in human language.
        
        Robotics: Implementing control systems for robots that allow them to navigate and interact with environments that have uncertain and variable characteristics.
        
        Game Development: Implementing AI in games to create characters that can make decisions and respond to player actions in a realistic and human-like manner.
  - - - Benefit
        
        Modeling human thinking based on premises with no complete truth value.
        
        A lattice system that provide an adequate representation for the complexities of fuzzy logic
        
        Special arithmetical structure with interactive binary operations
        
        Alignment with intuitive expectations
      - Axiom system
        
        Heuristic thinking and logical operations
        
        Logical conjunctions, disjunctions and logic distributivity
        
        Logical quantifiers and qualifiers
    - - Scheme details
        
        Generalization of modus ponens
        
        Rule modeling
        
        Analogical scheme integration
      - Benefit
        
        Less constrained
        
        Preserve main properties of deductive reasoning
        
        Adequation with the analogical scheme
    - - Benefit
        
        Provide a systematic approach for linguistic hedge measurement
      - Detials of the algebra
        
        Intuitive properties of linguistic hedges
        
        Effect on fuzzy concept and hedge itself
        
        Structuring of sets of linguistic truth values
        
        Odering operation for hedge effect measurement
  - - - Definition
        
        Deductive reasoning, also known as deduction or top-down reasoning, starts with a general statement or hypothesis and examines the possibilities to reach a specific, logical conclusion. It's the process of drawing specific conclusions from general principles or premises.
      - Benefit
        
        Certainty
        
        Structured approach
        
        Widely applicable
        
        Consistency
      - Limitation
        
        Dependence on premises
        
        Lack of new information
        
        Rigidity
    - - Definition
        
        Analogical reasoning is a method of reasoning in which similarities between two or more things are used to draw a conclusion about another thing. It's based on the principle that if two things (A and B) are alike in one way, they are alike in other ways as well.
      - Benefit
        
        Intuitive
        
        Flexible
        
        Generative
        
        Useful in novel situations
      - Limitation
        
        Potential for misleading analogies
        
        Strength varies
        
        Not definitive
- - - - Optimal Structure search (Sugeno et al. FSS 1987)
        
        Alternatively optimize structure and premise parameters
        
        Proposed a criterion or metric to quantify the goodness of a fuzzy structure (UC)
        
        Incrementally insert rules to divide input spaces until best UC is achieved
        
        Solve premise parameters using Least-square method
      - Improved scalability on large input space (Manfred et al. SMC 2001)
        
        Reduce structure search space via shortcut one-step search
        
        Use gradient descent methods instead of Least-square to solve parameter.
    - - Interpolate the semantics when fuzzy variable is gradual or metric (allow middle states)
        
        Assign linguistic modifiers to fuzzy input
        
        Consequence (fuzzy output) still cannot be interpolated
        
        (Laszlo IS 1993)
        
        Constructs lattice and partial ordering on the values of multi-dimensional fuzzy variables forming a power set
        
        Defines a distance metrics between two possible combinations of fuzzy set within the lattice.
        
        Interpolate/extrapolate the rules that originally defined over the sparse fuzzy inputs
- - - - In 1980, Baldwin and Guild jointly propose feasible algorithms for approximate reasoning where the process is handled entirely in terms of truth values.
        
        Conjunction
        
        Disjunction
        
        Modus Ponens
        
        Modus Tollens
        
        In 1990, Turksen and Zhong propose the Approximate Analogical Reasoning Schema (AARS)
        
        Similarity Measure of Fuzzy Sets
        
        Disconsistency Measure
        
        Hausdorff Measure
        
        Kaufmann & Gupta Measure
        
        Threshold to determine whether a rule should be fired
        
        Modification Function to deduce consequent
        
        In 1991, Nafarieh and Keller propose to model implications using exponential operations.
      - Benefits
        
        Computationally efficient and simple (even for multiple compound implication statements).
        
        Easily extended to more complex arguments.
      - Applications
        
        Pattern recognition
        
        Hypothesis testing
        
        Medical diagnosis
        
        Fuzzy control synthesis
        
        Approximate theorem proving
        
        Decision theory
        
        AI
- - - - General Concept: Let us discuss a crisp relation R from set A to set B, or (A,B). The membership function of a crisp relation R maps A x B onto the set {0, 1} ![GeneralConcept of FuzzyRelation] μR(x, y) is either 1 or 0 where x ∈ A and y ∈ B.
      - Fuzzy Causality: Causality can have negative sign. In a FCM, the negative fuzzy relation (or causality) between two concept nodes is the degree of a relation with "negation" of a concept node.
    - - Definition
        
        FCM is a fuzzy signed digraph with feedback, which can model the real world as a collection of concept variables and causal
        relationships
      - Application
        
        Has been used in various fields for representing causal
        knowledge
        
        Has been used as a new knowledge acquisition scheme in expert systems domain, especially
        in abstract and fuzzy domains where the relationships between concept variables are causal
      - Proposed fuzzy implications
        
        Fuzzy causal relationship
        
        Fuzzy Partially Causal Relationship(FPCR)
        
        Uni-quantitative concept
        
        Bi-quantitative concept
    - - Properties
      - Forward Implication
      - Backward Implication
      - Definition of Expectation, Variance, Distribution
    - - Interrelationships between the properties of fuzzy implications
        
        Give examples of fuzzy implications I such that (i) I satisfies only CP(N), (ii) I satisfies only L-CP(N), (iii) I satisfies both CP(N) and L-CP(N) but not R-CP(N), (iv) I satisfies both CP(N) and R-CP(N) but not L-CP(N), with some fuzzy negation N. BACZY ´NSKI and ŁUKASIK [6] analyzed this problem and they found examples for the first two points.
        
        Let I be a fuzzy implication. (i) For a given (continuous) t-norm T, characterize all fuzzy implications which satisfy the law of importation with T, i.e., the pair (I,T) satisfies (LI). (ii) Since T is commutative, we know that the law of importation implies the exchange principle (EP). a. Is the converse also true, i.e., does the exchange principle imply that there exists a t-norm such that the law of importation holds? b. If yes, can the t-norm be uniquely determined? c. If not, give an example and characterize all fuzzy implications for which such implication is true. A first partial answer to the above question, (Problem (ii) (a)), appeared in themonograph.
        
        Let I : [0,1]2 → [0,1] be any function that satisfies both (EP) and (OP). Then the following statements are equivalent: (i) I satisfies (I2). (ii) I is right-continuous in the second variable. Baczy´nski, M., Jayaram, B.: Fuzzy Implications. STUDFUZZ, vol. 231. Springer, Berlin(2008)
      - Properties or characterizations of specific families of fuzzy implications
        
        Characterize the families of f - and ggenerated implications. MASSANET and TORRENS [36] solved the above problem by using law of importation (LI).
        
        (i) Is the intersection non-empty? (ii) If yes, then characterize the intersection Massanet, S., Torrens, J.: Intersection of Yager’s implications with QL and Dimplications. Internat. J. Approx. Reason. 53, 467–479 (2012)
        
        Does there exist a continuous R-implication generated from non-(left)-continuous t-norm ? JAYARAM [18] answered the above poser in the negative, by showing that the continuity of an R-implication forces the left-continuity of the underlying t-norm and hence show that an R-implication IT is continuous if and only if T is a nilpotent t-norm.
        
        Characterize all left-continuous t-norms T which satisfy I(x,T(x,y)) = max(N(x),y), x,y ∈ [0,1]. where I is the residual operator linked to T given by (3) and N(x) = NT (x) = I(x,0), x ∈ [0,1]. And prove In the class of continuous t-norms, only nilpotent t-norms fulfill the above property.”
        JAYARAM further generalized the above statement by considering a t-subnorm instead of a t-norm and also dropping the condition of left-continuity.
        
        Give a necessary condition on a t-norm T for the corresponding IT to satisfy (EP)
        JAYARAM ET AL. [21] have partially solved the above problem for border-continuous t-norms. A complete characterization is not yet available.
- - - - Rule interpolation
        
        Fuzzy relation interpolation
        
        Fixed Point Law Fuzzy Set Interpolation (FPLI)
        
        Solid Cutting (SC) Fuzzy Set Interpolation
        
        Fixed Value Law Fuzzy Set Interpolation FVLI
        
        Fuzzy Relation Based Rule Interpolation
        
        Semantic relation interpolation
        
        Interpolation of Semantic Relation, IS-I
        
        Interpolation of Semantic Relation, IS-II
    - - Transformation of Fuzzy Relation
        
        Inference by Fuzzy Relation
      - Transformation of Semantic Relation
        
        Inference by Semantic Relation
    - - Extend to prediction and classification. Enable both interpolation and extrapolation which involve multiple fuzzy rules, with each rule consisting of multiple antecedents.
        
        Interpolation With Multiple Multiantecedent Rules
        
        Extrapolation
      - Backward Fuzzy Interpolation and Extrapolation
        with Multiple Multi-antecedent Rules (2012)
        
        The approach supports both interpolation and extrapolation which involve multiple intertwined fuzzy rules, with each having multiple antecedents. This approach is used to handle the case when a crucial antecedent of observation is absent.
        
        Backward fuzzy rule interpolation and extrapolation
        
        1.Determination of closest rules
        
        Construct the intermediate fuzzy terms
        
        Scale and Move Transformation
- - - - Nonlinear Dynamic System Approximation
        
        Goal: Approximate nonlinear dynamic systems
        
        Method: Interpolating between local affine dynamic models via fuzzy inference
      - Accurate Global and Local Predictions
        
        Global Aim: Precise global nonlinear predictions
        
        Local Aim: Ensure local models mirror local linearizations
    - - Approximation to Dynamic Linearization
        
        Capability: Approximate dynamic linearization about arbitrary trajectories
        
        Implication: Accurate global modeling and interpretable local models
    - - Virtual Equilibria and Stability Analysis
      - Transient Local Models
      - Affine Local Models
      - Membership Function Impact
    - - Least Squares Identification
        
        Perception: Views local models as approximators, not linearizations
        
        Result: Local models diverge from actual linearization.
    - - Experiment Design and Identification
        
        Strategies: Meticulous experiment design, constrained/regularized identification, and locally weighted identification
    - - Contributions: Theoretical insights and practical guidelines
      - Relevance: Applicable to other local model-based representations