Fuzzy-rough data mining

Richard JensenAdvanced Reasoning GroupUniversity of Aberystwyth

rkj@aber.ac.ukhttp://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?

• Growth of information - need to manage this effectively• Curse of dimensionality - a problem for machine learning and data

mining• Data visualisation - graphing data

High dimensionaldata

DimensionalityReduction

Low dimensionaldata

Processing System

Intractable

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

• Subset generation: forwards, backwards, random…• Evaluation function: determines ‘goodness’ of subsets• Stopping criterion: decide when to stop subset search

Generation Evaluation

StoppingCriterion

Validation

Feature set Subset

Subsetsuitability

Continue Stop

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

implicator

t-norm

Fuzzy-rough set

Rough set

Fuzzy-rough feature selection

• Based on fuzzy similarity

• Lower/upper approximations

|minmax|

|)()(|1),(

aa

yaxayxRa

)},({),( yxRTyxRaP

Pa

(e.g.)

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 regionUpperApproximation

Set X

LowerApproximation

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

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

Pawlakrough set

VQRS

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 non-monotonic

ProgressRough set

theory

Fuzzy rough set theory

VQRS

OWA-FRFS

Fuzzy VPRS

...

Qualitative data

Quantitative data

Noisy data

Monotonic

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 experimentationOriginal Dataset

Cross-validation folds

Type-1 FRFS

Reduced folds

JRip

Data corruption

IV-FRFS methods

Reduced folds

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

Remove objects whose positive region membership is < 1

Noisy objects

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

• When a reduct is found the resulting rules cover all instances

GenerationEvaluation and

StoppingCriterion

Validation

Feature set Subset

Subsetsuitability

Continue Stop

Rule Induction

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.

a b c score

2 3 1 3

2 3 2 4

4 4 2 9

3 4 5 21

Minimise ( a – 2 ) 2 + ( b – 3 ) 4 + ( c – 1 ) 2 + 3

Musicians

Notes

Harmony

HarmonyMemory

Harmony search approach

Fitness

Key notion mapping

Harmony Rule set

Fitness Combined evaluation

Musician Fuzzy rule rx

Note Feature subset

Solution

Evaluation

Variable

Value

HarmonySearch

Hybrid RuleInduction

NumericalOptimisation

Comparison vs QuickRulesHarmonyRules56.33±10.00

QuickRules63.1±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

Learner 1 Learner 2

subset 1 subset 2

PredictionsPredictions

Unlabelled Data

Self-learning

Labelled Dataset

Learner

Predictions

Unlabelled Data

Labelled data objects

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 (FUZZ-IEEE’11), pp. 2465-2471, 2011.

FRSL

Labelled dataset

Fuzzy-rough learner

Predictions

Unlabelled Data

Labelled data objects

Lower approximation

membership = 1?

Yes

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:

• KEEL version available soon (hopefully!)

http://users.aber.ac.uk/rkj/book/wekafull.jar