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S = ( X, A  {d[1],d[2],..,d[k]}, V ), where: - X is a finite set of objects,

Multi- H ierarchical D ecision S ystem. S = ( X, A  {d[1],d[2],..,d[k]}, V ), where: - X is a finite set of objects, - A is a finite set of classification attributes , - {d[1],d[2],..,d[k]} is a set of hierarchical decision attributes, and

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S = ( X, A  {d[1],d[2],..,d[k]}, V ), where: - X is a finite set of objects,

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  1. Multi-Hierarchical Decision System S= (X, A{d[1],d[2],..,d[k]}, V), where: - X is afinite set of objects, - A is a finite set of classification attributes, - {d[1],d[2],..,d[k]} is a set of hierarchical decision attributes, and V = {Va : a A{d[1],d[2],..,d[k]}}is a set of their values. We assume that: Va, Vbare disjoint for any a, b A{d[1],d[2],..,d[k]},such that a ≠ b, a : X →Vais a partial function for every a A{d[1],d[2],..,d[k]}. Decision queries (d-queries) for S- a least set TDsuch that: -0, 1 TD, - if w{Va: a {d[1],d[2],..,d[k]}} , then w, ~w TD, - if t1, t2TD, then (t1 + t2), (t1t2) TD. Incomplete Database Atomic Level

  2. Example 1 2 1 2 3 LevelI C[1] C[2] d[1] d[2] d[3] LevelII 1 2 1 2 C[2,1] C[2,2] d[3,1] d[3,2] Classification Attributes Decision Attributes

  3. Simple Term to Simple Query • Classification terms (c-terms) for S are defined as the least set TC: • 0, 1 TC, • if w {Va : a  A}, then w, ~w TC, • if t1, t2TC, then (t1 + 2), (t1t2)TC. • c-term t is called simpleif t = t1t2…tnand • (j  {1,2,…,n})[(tj{Va : a  A})] (tj = ~w  w{Va : a  A})]. • d-query t is called simple if t = t1t2…tn and • (j  {1,2,…,n})[(tj{Va : a {d[1],d[2],..,d[k]}})  • (tj = ~w  w{Va : a {d[1], d[2],.., d[k]})]. • By a classification rulewe mean an expression [t1 t2], • - both t1 and t2 are simple. Atomic Level 3

  4. Classifier-based Semantics • Semantics MSof c-terms in S is defined in a standard way as follows: • - MS(0) = 0, MS(1) = X, • - MS(w) = {x X : w = a(x)}for any w Va, a A, • - MS(~w) = {x X : (v Va)[v = a(x) v≠w]}for any w Va, a A, • - if t1, t2are terms, then • MS(t1 + t2) = MS(t1) MS(t2), MS(t1 t2) = MS(t1) MS(t2). • Classifier-based semantics MSof d-queries in • S = (X, A{d[1],d[2],..,d[k]}, V ), • if tis a simple d-query in S and {rj = [tj t]: j Jt} is a set of all • rules defining t which are extracted from S by classifier, then • MS(t) = {(x,px): (j Jt)(x  MS(tj)[px = • {conf(j)sup(j): x  MS(tj) & j Jt}/{sup(j): x  MS(tj) & jJt}]}, • where conf(j), sup(j) denote the confidence and the support of • [tj t], correspondingly. 4

  5. Standard Semantics of D-queries • Attribute value d[j1, j2,…jn] in S is dependent on an attribute • value which is either its ancestor or descendant in d[j1]. • d[j1, j2,…jn] is independent from any other attribute value in S. • Let S = (X, A{d[1],d[2],..,d[k]}, V),w Vd[i] , and IVd[i] be the set of • all attribute values in Vd[i]which areindependent from w. • Standard semantics NSof d-queries in S is defined as follows: • NS(0) = 0, NS(1) = X, • if w Vd[i] , then NS(w) = {x X : d[i](x)=w}, for any 1i k • if w Vd[i] , then NS(~w) = {x X : (v IVd[i])[ d[i](x)=v]}, • for any 1i k • if t1, t2 are terms, then • NS(t1 + t2) = NS(t1)NS(t2), NS(t1t2) = NS(t1) NS(t2). 5

  6. The Overlap of Semantics Let S =(X, A{d[1],d[2],..,d[k]}, V), tis a d-query in S, NS(t) is its meaning under standard semantics, and MS(t) is its meaning under classifier-based semantics. Assume that NS(t) = X  Y, where X = {xi, i  I1}, Y = {yi , i  I2}. Assume also that MS(t) = {(xi, pi): i  I1}  {(zi, qi): i  I3} and {yi , i  I2}  {zi , i I3}= . I2 I1 I3 NS MS 6

  7. Precision & Recall I2 I1 I3 NS MS • Precision Prec(MS, t) of a classifier-based semantics MS on a d-query t: Prec(MS, t) = [{pi : i  I1} + {(1 – qi ) : i  I3}] [card(I1) + card(I3)]. • Recall Rec(MS, t) of a classifier-based semantics MS on a d-query t: Rec(MS, t) = [{pi : i  I1}]____ [card(I1) + card(I2)].

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