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Discretization in RSES

Discretization in RSES. Zbigniew Ras. Quantization Process (based on discernibility formulas). Table T. p1  [0.8, 1), p2  [1, 1.3), p3  [1.3, 1.4), p4  [1.4, 1.6], q1  [0.5, 1), q2  [1, 2), q3  [2, 3]. V a = [0, 2) V b = [0, 4)

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Discretization in RSES

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  1. Discretization in RSES ZbigniewRas

  2. Quantization Process (based on discernibility formulas) Table T p1[0.8, 1), p2[1, 1.3), p3[1.3, 1.4), p4[1.4, 1.6], q1[0.5, 1), q2[1, 2), q3[2, 3] Va = [0, 2) Vb = [0, 4) Dom(a) = {0.8, 1, 1.3, 1.4, 1.6} Dom(b) = {0.5, 1, 2, 3} Boolean Variables: p1, p2, p3, p4 Boolean Variables: q1,q2,q3

  3. Discernibility formulas (i, j) for d(xi)  d(xj) (2, 1) = p1 + q1 + q2 /objects differ on p1,q1,q2 / (2, 4) = p2 + p3 + q1 (2, 6) = p2 + p3 + p4 +q1+q2+q3 (2, 7) = p2+ q1 ……………….. • Discernibility formula for the Table T: (T) = (p1+q1+q2)*(p2+p3+q1)*(p2+p3+p4+q1+q2+q3)*(p2+q1)*... =(p2*p4*q2)+(p2*p3*q2*q3+(p3*q1*q2*q3)+(p1*p4*q1*q2). For the last prime implicant we have the following cuts: {(a, 0.9), (a, 1.5), (b, 0.75), (b, 1.5)}. Cut for (a, 0.9): 0.9 = (0.8 + 1)/2 since p1 [0.8, 1).

  4. Algorithm which finds semi-optimal set of cuts • Step 1. Construct table T* from T and set B:=T*.

  5. Step 2. Choose a column from B with the maximal number of occurrences of 1’s. • Step 3. Delete from B the column chosen in Step 2 and all rows marked in this column by 1. • Step 4. If B is non-empty go to step 2 else Stop. In our example S = {p2, p4, q2}. So, The set of cuts P(S) = {(a, 1.15), (a, 1.5), (b, 1.5)}.

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