Formal Concept Analysis II Closure Systems and Implications Robert - PowerPoint PPT Presentation

Formal Concept Analysis II Closure Systems and Implications Robert J¨ aschke Asmelash Teka Hadgu FG Wissensbasierte Systeme/L3S Research Center Leibniz Universit¨ at Hannover slides based on a lecture by Prof. Gerd Stumme Robert J¨ aschke (FG KBS) Formal Concept Analysis 1 / 36

Agenda Closure Systems 3 Concept Intents as Closed Sets Next Closure Algorithm Iceberg Concept Lattices Titanic Algorithm Robert J¨ aschke (FG KBS) Formal Concept Analysis 2 / 36

Closure Systems On the blackboard: closure system A closure operator ϕ closure systems and closure operators (Th. 1) closure systems and complete lattices (Prop. 3) examples (subtrees, subintervals, convex sets, equivalence relations) For every formal context ♣ G, M, I q holds: The extents form a closure system on G . The intents form a closure system on M . ✷ is a closure operator. Robert J¨ aschke (FG KBS) Formal Concept Analysis 3 / 36

Concept Intents as Closed Sets abce the line diagram of the powerset of t a, b, c, e ✉ classes of attributes that abc abe ace bce describe the same set of objects ac ab ae be bc ce unique representatives: concept intents (=closed sets) a c b e minimal generator c be a b c e 2 1 ✂ ✂ a 2 ✂ ✂ 1 3 3 ✂ ✂ ✂ Robert J¨ aschke (FG KBS) Formal Concept Analysis 4 / 36

Next Closure Algorithm Developed 1984 by Bernhard Ganter. Can be used to compute the concept lattice, or to compute the concept lattice together with the stem base, or for interactive knowledge exploration. The algorithm computes the concept intents in the lectic order . Robert J¨ aschke (FG KBS) Formal Concept Analysis 5 / 36

Next Closure Algorithm: Lectic Order Let M ✏ t 1 , . . . , n ✉ . We say that A ❸ M is lectically smaller than B ❸ M , if B ✘ A and the smallest element in which A and B differ belongs to B : A ➔ B : ô ❉ i P B ③ A : A ❳ t 1 , 2 , . . . , i ✁ 1 ✉ ✏ B ❳ t 1 , 2 , . . . , i ✁ 1 ✉ 12345 2345 1345 1234 1245 1235 345 234 124 235 134 123 145 135 124 125 34 23 45 35 13 24 25 14 12 15 3 4 2 5 1 ❍ Robert J¨ aschke (FG KBS) Formal Concept Analysis 6 / 36

Next Closure Algorithm: Theorem Some definitions before we start: A ➔ i B : ô i P B ③ A ❫ A ❳ t 1 , 2 , . . . , i ✁ 1 ✉ ✏ B ❳ t 1 , 2 , . . . , i ✁ 1 ✉ A � i : ✏ ♣ A ❳ t 1 , 2 , . . . , i ✁ 1 ✉q ❨ t i ✉ Theorem The smallest concept intent larger than a given set A ⑨ M with respect to the lectic order is A ❵ i : ✏ ♣ A � i q ✷ , with i being the largest element of M with A ➔ i A ❵ i . Robert J¨ aschke (FG KBS) Formal Concept Analysis 7 / 36

Next Closure Algorithm The Next Closure algorithm to compute all concept intents: 1 The lectically smallest concept intent is ❍ ✷ . 2 If A is a concept intent, we find the lectically next intent by checking all attributes i P M ③ A (starting with the largest), continuing in descending order until for the first time A ➔ i A ❵ i . Then A ❵ i is the lectically next intent. 3 If A ❵ i ✏ M , we stop. Otherwise we set A : ✏ A ❵ i and go to step 2. Robert J¨ aschke (FG KBS) Formal Concept Analysis 8 / 36

Next Closure Algorithm: Example mobile (1) phone (2) fax (3) paper fax (4) Sinus 44 ✂ Nokia 6110 ✂ ✂ T-Fax 301 ✂ ✂ T-Fax 360 PC ✂ A i A � i A ❵ i : ✏ ♣ A � i q ✷ A ➔ i A ❵ i ? new intent Robert J¨ aschke (FG KBS) Formal Concept Analysis 9 / 36

Next Closure Algorithm: Lectic Order 1234 234 123 134 124 23 12 34 24 13 14 2 3 1 4 ❍ Robert J¨ aschke (FG KBS) Formal Concept Analysis 10 / 36

Iceberg Concept Lattices veil type: partial gill attachment: free 100 % ring number: one veil color: white 92.30 % 97.62 % 97.43 % 90.02 % 97.34 % 89.92 % The seven most general concepts (for minsupp = 85%) of the 32086 concepts of the mushroom database ( http://kdd.ics.uci.edu/ ). Robert J¨ aschke (FG KBS) Formal Concept Analysis 11 / 36

Iceberg Concept Lattices veil type: partial 100 % gill attachment: free ring number: one veil color: white minsupp = 85% 92.30 % 97.62 % 97.43 % veil type: partial gill attachment: free 90.02 % 97.34 % ring number: one 100 % veil color: white 89.92 % gill spacing: close 92.30 % 97.43 % 97.62 % 81.08 % 90.02 % 97.34 % 76.81 % 78.80 % 89.92 % 78.52 % minsupp = 70% 74.52 % Robert J¨ aschke (FG KBS) Formal Concept Analysis 12 / 36

Iceberg Concept Lattices veil type: partial gill attachment: free ring number: one 100 % veil color: white gill size: broad gill spacing: close 69.87 % 92.30 % 97.43 % 97.62 % 81.08 % 62.17 % 67.59 % 90.02 % 97.34 % 76.81 % 78.80 % stalk color above ring: white 67.30 % 89.92 % 56.37 % 78.52 % stalk color below ring: white 59.89 % 55.13 % 74.52 % stalk shape: tapering stalk surface above ring: smooth stalk surface below ring: smooth 63.17 % 57.79 % 60.31 % 57.94 % 55.09 % no bruises 60.88 % 58.03 % With decreasing 59.89 % minimal support 55.66 % minsupp = 55% 55.70 % 57.32 % 57.51 % more information is revealed. 57.22 % Robert J¨ aschke (FG KBS) Formal Concept Analysis 13 / 36

Iceberg Concept Lattices veil type: partial gill attachment: free ring number: one veil color: white gill size: broad gill spacing: close 100% 92.30% 97.62% 81.08% 69.87% 97.43% 78.80% 67.59% 90.02% 97.34% 62.17% 67.30% 89.92% 78.52% 58.89% stalk color below ring: white stalk color above ring: white 55.13% 56,37% stalk surface below ring: smooth stalk surface above ring: smooth 63.17% 60.31% 57.94% 55.09% 58.03% 60.88% 55.66% no bruises stalk shape: tapering In a nested line 57.79% 58.89% diagram we can 55.70% minsupp = 55% read off 57.22% implications. Robert J¨ aschke (FG KBS) Formal Concept Analysis 14 / 36

Iceberg Concept Lattices: Support Def.: The support of a set X ❸ M of attributes is defined as supp ♣ X q : ✏ ⑤ X ✶ ⑤ ⑤ G ⑤ Def.: The iceberg concept lattice of a formal context ♣ G, M, I q for a given minimal support value minsupp is the set t♣ A, B q P B ♣ G, M, I q ⑤ supp ♣ B q ➙ minsupp ✉ The iceberg concept lattice can be computed using the Titanic algorithm. (Stumme et al., 2001) Robert J¨ aschke (FG KBS) Formal Concept Analysis 15 / 36

Titanic Algorithm Titanic computes the closure system of all ( frequent ) concept intents using the support function supp ♣ X q : ✏ ⑤ X ✶ ⑤ ⑤ G ⑤ (for a set X ❸ M of attributes). frequent : only concept intents above a threshold minsupp P r 0 , 1 s Robert J¨ aschke (FG KBS) Formal Concept Analysis 16 / 36

Titanic Algorithm Titanic employs some simple properties of the support function: Lemma 4. Let X, Y ❸ M . 1 X ❸ Y ù ñ supp ♣ X q ➙ supp ♣ Y q 2 X ✷ ✏ Y ✷ ù ñ supp ♣ X q ✏ supp ♣ Y q ñ X ✷ ✏ Y ✷ 3 X ❸ Y ❫ supp ♣ X q ✏ supp ♣ Y q ù Robert J¨ aschke (FG KBS) Formal Concept Analysis 17 / 36

Titanic Algorithm abce Lemma 4. Let X, Y ❸ M . abc abe ace bce X ❸ Y ù ñ supp ♣ X q ➙ supp ♣ Y q 1 X ✷ ✏ Y ✷ ù ñ supp ♣ X q ✏ supp ♣ Y q 2 ac ab ae be bc ce X ❸ Y ❫ supp ♣ X q ✏ supp ♣ Y q ù ñ 3 X ✷ ✏ Y ✷ a c b e c be 2 a 1 3 a b c e 1 ✂ ✂ 2 ✂ ✂ 3 ✂ ✂ ✂ Robert J¨ aschke (FG KBS) Formal Concept Analysis 18 / 36

Titanic Algorithm Titanic tries to optimize the following three questions: 1 How can we compute the closure of an attribute set using only the support values? 2 How can we compute the closure system such that we need to compute as few closures as possible? 3 How can we derive as many support values as possible from already known support values? Robert J¨ aschke (FG KBS) Formal Concept Analysis 19 / 36

Titanic Algorithm 1 How can we compute the closure of an attribute set using only the support values? X ✷ ✏ X ❨ t m P M ③ X ⑤ supp ♣ X q ✏ supp ♣ X ❨ t m ✉q✉ Example: t b, c ✉ ✷ ✏ t b, c, e ✉ , since abce supp ♣t b, c ✉q ✏ 1 3 abc abe ace bce a b c e and 1 ✂ ✂ supp ♣t a, b, c ✉q ✏ 0 ac ab ae be bc ce 3 2 ✂ ✂ supp ♣t b, c, e ✉q ✏ 1 3 3 ✂ ✂ ✂ a c b e Robert J¨ aschke (FG KBS) Formal Concept Analysis 20 / 36

Titanic Algorithm abce 2 How can we compute the closure system such that we need to compute as few closures abc abe ace bce as possible? We compute only the closures of the ac ab ae be bc ce minimal generators. a c b e c be 2 a 1 3 a b c e For this example 1 ✂ ✂ Titanic needs two 2 ✂ ✂ runs (Apriori four). 3 ✂ ✂ ✂ Robert J¨ aschke (FG KBS) Formal Concept Analysis 21 / 36

Titanic Algorithm abce 2 How can we compute the closure system such that we need to compute as few closures abc abe ace bce as possible? We compute only the closures of the ac ab ae be bc ce minimal generators. A set is a minimal generator , iff its a c b e support is unequal to the support of its lower covers. The minimal generators form an order ideal (i.e., if a set is not a a b c e minimal generator, then none of its For this example 1 ✂ ✂ Titanic needs two supersets is either) 2 ✂ ✂ runs (Apriori four). ➞ approach similar to Apriori 3 ✂ ✂ ✂ Robert J¨ aschke (FG KBS) Formal Concept Analysis 21 / 36