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Fuzzy-rough data mining Richard Jensen Advanced Reasoning Group University of Aberystwyth [email protected] http://users.aber.ac.uk/rkj Outline • Knowledge discovery process • Fuzzy-rough methods – Feature selection and extensions – Instance selection – Classification/prediction – Semi-supervised learning Knowledge discovery • The process • The problem of too much data – Requires storage – Intractable for data mining algorithms – Noisy or irrelevant data is misleading/confounding Feature Selection Feature selection • Why dimensionality reduction/feature selection? High dimensional data Dimensionality Reduction Intractable Low dimensional data Processing System • Growth of information - need to manage this effectively • Curse of dimensionality - a problem for machine learning and data mining • Data visualisation - graphing data Why do it? • Case 1: We’re interested in features – We want to know which are relevant – If we fit a model, it should be interpretable • Case 2: We’re interested in prediction – Features are not interesting in themselves – We just want to build a good classifier (or other kind of predictor) Feature selection process • Feature selection (FS) preserves data semantics by selecting rather than transforming Feature set Generation Subset Evaluation Subset suitability Continue Stopping Criterion Stop Validation • Subset generation: forwards, backwards, random… • Evaluation function: determines ‘goodness’ of subsets • Stopping criterion: decide when to stop subset search Fuzzy-rough feature selection Fuzzy-rough set theory • Problems: – Rough set methods (usually) require data discretization beforehand – Extensions, e.g. tolerance rough sets, require thresholds – Also no flexibility in approximations • E.g. objects either belong fully to the lower (or upper) approximation, or not at all Fuzzy-rough sets Rough set t-norm Fuzzy-rough set implicator Fuzzy-rough feature selection • Based on fuzzy similarity (e.g.) | a ( x) a ( y ) | Ra ( x, y ) 1 | a max a min | RP ( x, y ) T {Ra ( x, y )} aP • Lower/upper approximations FRFS: evaluation function • Fuzzy positive region #1 • Fuzzy positive region #2 (weak) • Dependency function FRFS: finding reducts • Fuzzy-rough QuickReduct – Evaluation: use the dependency function (or other fuzzy-rough measure) – Generation: greedy hill-climbing – Stopping criterion: when maximal evaluation function is reached (or to degree α) FRFS • Other search methods – GAs, PSO, EDAs, Harmony Search, etc – Backward elimination, plus-L minus-R, floating search, SAT, etc • Other subset evaluations – Fuzzy boundary region – Fuzzy entropy – Fuzzy discernibility function Ant-based FS Boundary region Upper Approximation Set X Lower Approximation Equivalence class [x]B FRFS: boundary region • Fuzzy lower and upper approximation define fuzzy boundary region • For each concept, minimise the boundary region – (also applicable to crisp RSFS) • Results seem to show this is a more informed heuristic (but more computationally complex) Finding smallest reducts • Usually too expensive to search exhaustively for reducts with minimal cardinality • Reducts found via discernibility matrices through, e.g.: – Converting from CNF to DNF (expensive) – Hill-climbing search using clauses (non-optimal) – Other search methods - GAs etc (non-optimal) • SAT approach – Solve directly in SAT formulation – DPLL approach ensures optimal reducts Fuzzy discernibility matrices • Extension of crisp approach – Previously, attributes had {0,1} membership to clauses – Now have membership in [0,1] • Fuzzy DMs can be used to find fuzzy-rough reducts Formulation • Fuzzy satisfiability • In crisp SAT, a clause is fully satisfied if at least one variable in the clause has been set to true • For the fuzzy case, clauses may be satisfied to a certain degree depending on which variables have been assigned the value true Example DPLL algorithm Experimentation: results FRFS: issues • Problem – noise tolerance! Vaguely quantified rough sets Pawlak rough set VQRS y belongs to the lower approximation of A iff all elements of Ry belong to A y belongs to the upper approximation of A iff at least one element of Ry belongs to A y belongs to the lower approximation of A iff most elements of Ry belong to A y belongs to the upper approximation of A iff at least some elements of Ry belong to A VQRS-based feature selection • Use the quantified lower approximation, positive region and dependency degree – Evaluation: the quantified dependency (can be crisp or fuzzy) – Generation: greedy hill-climbing – Stopping criterion: when the quantified positive region is maximal (or to degree α) • Should be more noise-tolerant, but is nonmonotonic Progress Qualitative data Rough set theory Quantitative data Fuzzy rough set theory ... Noisy data VQRS Fuzzy VPRS Monotonic OWA-FRFS More issues... • Problem #1: how to choose fuzzy similarity? • Problem #2: how to handle missing values? Interval-valued FRFS • Answer #1: Model uncertainty in fuzzy similarity by interval-valued similarity IV fuzzy rough set IV fuzzy similarity Interval-valued FRFS • When comparing two object values for a given attribute – what to do if at least one is missing? • Answer #2: Model missing values via the unit interval Other measures • Boundary region • Discernibility function Initial experimentation Original Dataset Type-1 FRFS Cross-validation folds Data corruption IV-FRFS methods Reduced folds Reduced folds JRip JRip Initial experimentation Initial results: lower approx Instance Selection Instance selection: basic ideas Not needed Remove objects to keep the underlying approximations unchanged Instance selection: basic ideas Noisy objects Remove objects whose positive region membership is < 1 FRIS-I FRIS-II FRIS-III Fuzzy rough instance selection • Time complexity is a problem for FRIS-II and FRIS-III • Less complex: Fuzzy rough prototype selection – More on this later... Fuzzy-rough classification and prediction FRNN/VQNN FRNN/VQNN Further developments • FRNN and VQNN have limitations (for classification problems) – FRNN only uses one neighbour – VQNN equivalent to FNN if the same similarity relation is used • POSNN uses the positive region to also consider the quality of neighbours – E.g. instances in overlapping class regions are less interesting – More on this later... Discovering rules via RST • Equivalence classes – Form the antecedent part of a rule – The lower approximation tells us if this is predictive of a given concept (certain rules) • Typically done in one of two ways: – Overlaying reducts – Building rules by considering individual equivalence classes (e.g. LEM2) QuickRules framework • The fuzzy tolerance classes used during this process can be used to create fuzzy rules Feature set Generation Subset Evaluation and Rule Induction Subset suitability Continue Stopping Criterion Stop Validation • When a reduct is found the resulting rules cover all instances Harmony search approach • R. Diao and Q. Shen. A harmony search based approach to hybrid fuzzy-rough rule induction, Proceedings of the 21st International Conference on Fuzzy Systems, 2012. Harmony search approach Musicians a b c score Harmony 2 3 1 3 2 3 2 4 4 4 2 9 3 4 5 21 Notes Minimise ( a – 2 ) 2 + ( b – 3 ) 4 + ( c – 1 ) 2 + 3 Fitness Harmony Memory Key notion mapping Harmony Search Numerical Optimisation Hybrid Rule Induction Musician Variable Fuzzy rule rx Note Value Feature subset Harmony Solution Rule set Fitness Evaluation Combined evaluation Comparison vs QuickRules HarmonyRules 56.33±10.00 QuickRules 63.1±11.89 Rule cardinality distribution for dataset web of 2556 features Fuzzy-rough semi-supervised learning Semi-supervised learning (SSL) • Lies somewhere between supervised and unsupervised learning • Why use it? – Data is expensive to label/classify – Labels can also be difficult to obtain – Large amounts of unlabelled data available • When is SSL useful? – Small number of labelled objects but large number of unlabelled objects Semi-supervised learning • A number of methods for SSL – self-learning, generative models etc. – – – – Labelled data objects – usually small in number Unlabelled data objects – usually large in number A set of features describe the objects Class label tells us only which labelled objects belong to • SSL therefore attempts to learn labels (or structure) for data which has no labels – Labelled data provides ‘clues’ for the unlabelled data Co-training Labelled Dataset subset 1 subset 2 Unlabelled Data Learner 1 Learner 2 Predictions Predictions Self-learning Labelled Dataset Labelled data objects Learner Predictions Unlabelled Data Fuzzy-rough self learning (FRSL) • Basic idea is to propagate labels using the upper and lower approximations – Label only those objects which belong to the lower approximation of a class to a high degree – Can use upper approximation to decide on ties • Attempts to minimise mis-labelling and subsequent reinforcement • Paper: N. Mac Parthalain and R. Jensen. Fuzzy-Rough Set based Semi-Supervised Learning. Proceedings of the 20th International Conference on Fuzzy Systems (FUZZIEEE’11), pp. 2465-2471, 2011. FRSL Labelled data objects Labelled dataset Yes Lower approximation membership = 1? Fuzzy-rough learner Predictions Unlabelled Data No Experimentation (Problem 1) SS-FCM FNN FRSL Experimentation (Problem 2) cluster 1 cluster 2 labelled 1 labelled 2 SS-FCM FNN FRSL Conclusion • Looked at fuzzy-rough methods for data mining – – – – – Feature selection, finding optimal reducts Handling missing values and other problems Classification/prediction Instance selection Semi-supervised learning • Future work – Imputation, better rule induction and instance selection methods, more semi-supervised methods, optimizations, instance/feature weighting FR methods in Weka • Weka implementations of all fuzzy-rough methods can be downloaded from: http://users.aber.ac.uk/rkj/book/wekafull.jar • KEEL version available soon (hopefully!)
