A Generalized Version Space Learning Algorithm for Noisy and Uncertain Data T.-P. Hong, S.-S. Tseng IEEE Transactions on Knowledge and Data Engineering, Vol. 9, No. 2, 임희웅

Introduction Generalized learning strategy of VS Noisy & uncertain training data Searching & pruning Trade-off between including positive training instances and excluding negative ones Trade-off between computational time consumed and the accuracy by pruning factors

New Definition of S/G Addition Information : Count Sum of positive/negative information implicit in the training instances presented so far. S/G boundary S A set of the first i maximally consistent hypotheses. No other hypothesis in S exists which is both more specific than another and has equal or larger count. G A set of the first j maximally consistent hypotheses. No other hypothesis in G exists which is both more general than another and has equal or larger count.

FIPI Factor of Including Positive Instances Trade-off between including positive training instances vs. excluding negative ones 0~1, real number 0: only to include positive training example 1: only to exclude negative training example 0.5: same importance

Certainty Factor (CF) A measure for positiveness -1~1, real number -1: negative example 1: positive example In case of new training example of CF S  (1+CF)/2 positive example G  (1-CF)/2 negative example

Learning Process Searching & Pruning Searching Generate and collects possible candidates into a large set Pruning Prune above set according to the degree of consistency of the hypotheses

Learning Process

Input & Output Input A set of n training instances each with CF FIPI i: the max # of hypotheses in S J: the max # of hypotheses in G Output The hypotheses in sets S and G that are maximally consistent with the training instances.

Step 1 & 2 Step 1 Initialize S= , & G= with count 0 Step 2 For each training instance with uncertainty CF, do Step 3 to Step 7.

Step 3 – Search 1 Generalize/Specialize each hypothesis in S/G c k : count of hypothesis in S/G Attach new count c k +(1+CF)/2 / c k +(1-CF)/2  S ’ /G ’

Step 4 – Search 2 Find the set S ” /G ” Which Include/exclude only the new training instance itself Set the count of each hypothesis in S ” /G ” to be (1+CF)/2 / (1-CF)/2

Step 5 – Pruning 1 Combine S/G, S ’ /G ’, and S ” /G ” Identical hypotheses only with maximum count is retained If a particular hypothesis is both more general/specific than another and has an equal or smaller count, discard that.

Step 6 – Confidence Calc. Confidence of each new hypothesis For each hypothesis s with count c s in the new S Find the hypothesis g in the new G that is more general than s and has the maximum count c g Confidence = FIPI  c s + (1-FIPI)  c g For each hypothesis g with count c g in the new G Do the same.

s (count=c s ), … specific general S g (count=c g ), … G g is more general than s Confidence of s = FIPI  c s + (1-FIPI)  max(c g ) Confidence of g = FIPI  c s + (1-FIPI)  max(c g )

Step 7 – Pruning 2 Select only i/j hypotheses with highest confidence in the new S/G

Another Papers GA L. De Raedt, et al., “ A Unifying Framework for Concept- Learning Algorithms ”, Knowledge Engineering Rev., vol. 7, no. 3, 1989 R. G. Reynolds, et al., “ The Use of Version Space Controlled Genetic Algorithms to Solve the Boole Problem ”. Int ’ l J. Artificial Intelligence Tools, vol. 2, no. 2, 1993 Fuzzy C. C. Lee, “ Fuzzy Logic in Control Systems: Fuzzy Logic Controller Part1&2 ”, IEEE Trans. Systems, Man, and Cybernetics, vol. 20, no. 2, 1990 L. X. Wang, et al., “ Generating Fuzzy Rules by Learning from Examples ”, Proc. IEEE Conf. Fuzzy Systems, 1992