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Fuzzy Interpretation of Discretized Intervals Dr. Xindong Wu

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Presentation Transcript

Plan For Presentation

- Introduction to Problem, HCV
- Discretization Techniques/Fuzzy Borders
- A Hybrid Solution for HCV
- Experiments and Results
- Conclusion

Introduction

- Real-world data contains both numerical and nominal data, must be able to deal with different types of data.
- Existing systems discretize numerical domains into intervals and treat intervals as nominal values during induction.
- Problems occur if test examples are not covered in training data (no-match, multiple match)
- The solution is a hybrid approach using fuzzy intervals for no-match problem.

HCV

- Attribute based rule induction algorithm, extension matrix approach
- Divide positive examples into intersecting groups
- Find a heuristic conjunctive rule in each group that covers all PE and no NE
- HCV can find a rule in the form of variable-valued logic
- More compact than the decision trees/rules of ID3 and C4.5

Variable Valued Logic and Selectors

- Represents decisions where variables can take a range
- Selector:

[ X # R ]

X = attribute

# = relational operator ( = , <, >, . . . )

R = Reference, list of 1 or more values

e.g [ Windy = true] [Temp > 90]

HCV Software

- C++ implementation
- Can work with noisy and real-valued domains as well as nominal and noise-free databases
- Provides a set of deduction facilities for the user to test the accuracy of the produced rules on test examples

C4.5:The T class

X2 = b

X1 = 0 & X3 = a

X1 = 0 & X3 = b

X1 = 0 & X2 = a

C4.5 Results vs. HCV- HCV:The T class
- X2 = b
- X1 = 0 & X2 = a
- X1 = 0 & X4 = 0

- C4.5:The F class

X1 = 1 & X2 = a

- X1 = 1 & X2 = c
- X2 = c & X3 = c

Deduction of Induction Results

- Induction generates knowledge from existing data
- Deduction applies induction results to interpret new data.
- With real-world data, induction can not be assumed to be perfect
- Three cases:

1) no-match (measure of fit)

2) single-match

3) multiple-match (estimate of probability)

Discretization

- Occurs during rule induction
- Discretize numerical domains into intervals and treat similar to nominal values.
- The challenge is to find the right borders for the intervals
- Possible Methods:

1) Simplest Class-Separating Method

2) Information Gain Heuristic (implemented in HCV)

Simplest Class- Separating Method:

- Interval Borders are places between each adjacent pair of examples which have different classes.
- If attribute is very informative - method is efficient and useful.
- If attribute is not informative - method produces too many intervals

Information Gain Heuristic

Use IGH to find more informative border.

- x = (xi + xi+1)/2 for (i = 1, …, n-1)
- x is a possible cut point if xi and xi+1 are of different classes.
- Use IGH to find best x
- Recursively split on left and right
- To stop recursive splitting:

1) stop if IGH is same on all possible cut points.

2) stop if # of examples to split is less than a predefined number

3) limit the number of intervals

Fuzzy Borders

- Discretization of continuous domains does not always fit accurate interpretation.
- Instead of using sharp borders, use a membership function, measures the degree of membership.
- A value can be classified into a few different intervals at the same time (e.g. single to multiple match)

Fuzzy Borders (2)

- Fuzzy matching - deduction with fuzzy borders of discretized intervals.
- Take the interval with the greatest degree as the value’s discrete value.
- 3 functions to fuzzify borders:

1) linear

2) polynomial

3) arctan

- Definitions

s = spread parameter l = length of original

xleft, xright = left/right sharp borders

l

xleft xright

l

sl

xleft xright

Linear Membership Functiona = -kxleft + 1/2 b = kxright + 1/2

linleft(x) = kx + a

lin right(x) = -kx + b

lin(x) = MAX(0, MIN(1,linleft(x),linright(x)))

k = 1/2sl

*Polynomial Membership Function

polyside(x) = asidex3 + bsidex2 + csidex + dside

aside = 1/(4(ls)3)

bside = -3asidexside side {left,right}

cside = 3aside(xside2 - (ls)2)

dside = -a(xside3 -3xside(ls)2 + 2(ls)3)

polyleft(x), if xleft -ls x xleft + ls

poly(x) = polyright(x), if xright -ls x xright +ls

1, if xleft +ls x xright -ls

0, otherwise

Match Degree

- Selector method - take the max membership degree of the value in all the intervals involved. If 2 adjacent intervals have the same class, values close to the border will have low membership.
- Conjunction method - adds with fuzzy plus

ab=a + b - ab

No-Match Resolution

Largest Class

- Assign all no match examples to the largest class, the default class.
- Works well, if the number of classes in a training set is small and one class is clearly larger.
- Deteriorates if there is a larger number of classes and the examples are evenly distributed

No-Match Resolution

Measure of Fit

Calculate the measure of fit for each class:

1) calculate MF for each selector (sel)

MF(sel, e) = 1, if sel is satisfied by e

n/|x|, otherwise

2) calculate MF for each conjunctive rule(conj)

MF(conj, e) = MF(sel, e) * n(conj)/N

No-Match Resolution

Measure of Fit (2)

3) calculate MF for each class c

MF(c, e) = MF(conj1, e) + MF(conj2, e) - MF(conj1,e)MF(conj2,e)

* For more than two rules, apply formula recursively.

* Find maximum MF - determines which class is closest to the example

Multiple-Match

- Caused by over-generalization of the training examples at induction time
- Example
- (X1 = a, X2 = 1)
- All PE cover X1 = a
- All NE cover X2 = 1
- Multiple Match

Multiple-Match Resolution

First Hit

- Use first rule which classifies the example
- Produces reasonable results if the rules from induction have been ordered according to a measure of reliability
- Advantages - straightforward, efficient
- Disadvantages - have to sort rules at induction time

Multiple-Match Resolution

Largest Rule

- Similar to largest class method from no-match resolution
- Choose conjunctive rule that covers the most examples in the training set.

Multiple-Match Resolution

Estimation of Probability

- Assign EP value to each class based on the size of the satisfied conjunctive rules.

1) Find EP for each conjunctive rule (conj):

EP(conj, e)= { n(conj)/N, if conj is satisfied by e

0, otherwise

n(conj) = number of examples covered by conj

N = number of total examples

Multiple-Match Resolution

Estimation of Probability (2)

2) Find EP value for each class:

EP(c, e) = EP(conj1, e) + EP(conj2, e) - EP(conj1,e)EP(conj2,e).

* For more rules, apply formula recursively

* Choose class with highest EP value

Hybrid Interpretation

- Used because fuzzy borders only add conflicts because they don’t reduce the number rules that are applicable
- HCV - use sharp borders during induction and use fuzzy borders only during deduction
- Algorithm:

* Single match - use class indicated by rules

* Multiple match - use estimation probability (EP) with sharp borders

* No match - use fuzzy borders with polynomial membership function to find closest rule

The Data

- Used 17 databases from the Machine Learning Database Repository, U. of California, Irvine.
- Databases selected because:

1) All include numerical data

2) All lead to situations where no rules clearly apply.

Results (cont.)

- The results shown for C4.5 and NewID are the pruned ones
- These were usually better than the unpruned ones in this experiment
- HCV did not fine tune different parameters because this would be loss of generality and applicability of the conclusions

Accuracy Results

- HCV(hybrid) - 9 databases
- C4.5 (R 8) - 7 databases
- C4.5 (R 5) - 6 databases
- HVC (large) - 3 databases
- HCV (fuzzy) - 2 databases

HCV Comparison

- HCV (fuzzy) generally performs better than the simple largest class method
- HCV’s performance improves significantly when the fuzzy borders (for no match) are combined with probability estimation (for multiple match) in HCV (hybrid)

Conclusions

- Fuzzy borders are constructed and used at deduction time only when a no match case occurs.
- This hybrid method performs more accurately than several other current deduction programs.
- Fuzziness is strongly domain dependent, HCV allows the user to specify their own intervals and fuzzy functions.

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