1. basic association analysis (idm ch. 6) 1. review 2. maximal and closed itemsets 3. rule...
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1. Basic Association Analysis (IDM ch. 6)1. Review2. Maximal and Closed Itemsets3. Rule Generation4. Kuis
2. Support Vector Machines / SVM (IDM ch. 5.5 and DMBAR 14.4)3. Practical:
1. Association Rules Mining (DMBAR ch. 16) 1. online radio 2. predicting income
2. SVM1. “Support vector machines in R,” by Karatzoglou et al. (2006)
Pertemuan 9: 27 Oktober 2015
Association Rule Mining• Given a set of transactions, find rules that will predict the
occurrence of an item based on the occurrences of other items in the transaction
Market-Basket transactions
TID Items
1 Bread, Milk
2 Bread, Diaper, Beer, Eggs
3 Milk, Diaper, Beer, Coke
4 Bread, Milk, Diaper, Beer
5 Bread, Milk, Diaper, Coke
Example of Association Rules
{Diaper} {Beer},{Milk, Bread} {Eggs,Coke},{Beer, Bread} {Milk},
Implication means co-occurrence, not causality!
Definition: Association RuleTID Items
1 Bread, Milk
2 Bread, Diaper, Beer, Eggs
3 Milk, Diaper, Beer, Coke
4 Bread, Milk, Diaper, Beer
5 Bread, Milk, Diaper, Coke
Example:Beer}Diaper,Milk{
4.052
|T|)BeerDiaper,,Milk(
s
67.032
)Diaper,Milk()BeerDiaper,Milk,(
c
Association Rule– An implication expression of the form X
Y, where X and Y are itemsets– Example:
{Milk, Diaper} {Beer}
Rule Evaluation Metrics– Support (s)
Fraction of transactions that contain both X and Y
– Confidence (c) Measures how often items in Y
appear in transactions thatcontain X
Frequent Itemset Generation Strategies
• Reduce the number of candidates (M)– Complete search: M=2d
– Use pruning techniques to reduce M• Reduce the number of transactions (N)
– Reduce size of N as the size of itemset increases– Used by DHP and vertical-based mining algorithms
• Reduce the number of comparisons (NM)– Use efficient data structures to store the candidates or
transactions– No need to match every candidate against every
transaction
Reducing Number of Candidates• Apriori principle:
– If an itemset is frequent, then all of its subsets must also be frequent
• Apriori principle holds due to the following property of the support measure:
– Support of an itemset never exceeds the support of its subsets
– This is known as the anti-monotone property of support
)()()(:, YsXsYXYX
Found to be Infrequent
null
AB AC AD AE BC BD BE CD CE DE
A B C D E
ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE
ABCD ABCE ABDE ACDE BCDE
ABCDE
Illustrating Apriori Principlenull
AB AC AD AE BC BD BE CD CE DE
A B C D E
ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE
ABCD ABCE ABDE ACDE BCDE
ABCDE
Pruned supersets
Illustrating Apriori PrincipleItem CountBread 4Coke 2Milk 4Beer 3Diaper 4Eggs 1
Itemset Count{Bread,Milk} 3{Bread,Beer} 2{Bread,Diaper} 3{Milk,Beer} 2{Milk,Diaper} 3{Beer,Diaper} 3
Itemset Count {Bread,Milk,Diaper} 3
Items (1-itemsets)
Pairs (2-itemsets)
(No need to generatecandidates involving Cokeor Eggs)
Triplets (3-itemsets)Minimum Support = 3
If every subset is considered, 6C1 + 6C2 + 6C3 = 41
With support-based pruning,6 + 6 + 1 = 13
Apriori Algorithm
• Method: – Let k=1– Generate frequent itemsets of length 1– Repeat until no new frequent itemsets are identified
• Generate length (k+1) candidate itemsets from length k frequent itemsets
• Prune candidate itemsets containing subsets of length k that are infrequent
• Count the support of each candidate by scanning the DB• Eliminate candidates that are infrequent, leaving only those
that are frequent
Reducing Number of Comparisons• Candidate counting:
– Scan the database of transactions to determine the support of each candidate itemset
– To reduce the number of comparisons, store the candidates in a hash structure
• Instead of matching each transaction against every candidate, match it against candidates contained in the hashed buckets
TID Items 1 Bread, Milk 2 Bread, Diaper, Beer, Eggs 3 Milk, Diaper, Beer, Coke 4 Bread, Milk, Diaper, Beer 5 Bread, Milk, Diaper, Coke
N
Transactions Hash Structure
k
Buckets
Generate Hash Tree
2 3 45 6 7
1 4 51 3 6
1 2 44 5 7 1 2 5
4 5 81 5 9
3 4 5 3 5 63 5 76 8 9
3 6 73 6 8
1,4,7
2,5,8
3,6,9Hash function
Suppose you have 15 candidate itemsets of length 3:
{1 4 5}, {1 2 4}, {4 5 7}, {1 2 5}, {4 5 8}, {1 5 9}, {1 3 6}, {2 3 4}, {5 6 7}, {3 4 5}, {3 5 6}, {3 5 7}, {6 8 9}, {3 6 7}, {3 6 8}
You need:
• Hash function
• Max leaf size: max number of itemsets stored in a leaf node (if number of candidate itemsets exceeds max leaf size, split the node)
Association Rule Mining Task
• Given a set of transactions T, the goal of association rule mining is to find all rules having
• support ≥ minsup threshold• confidence ≥ minconf threshold
• Brute-force approach:• List all possible association rules• Compute the support and confidence for each rule• Prune rules that fail the minsup and minconf
thresholds Computationally prohibitive!
Maximal Frequent Itemset
null
AB AC AD AE BC BD BE CD CE DE
A B C D E
ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE
ABCD ABCE ABDE ACDE BCDE
ABCDE
BorderInfrequent Itemsets
Maximal Itemsets
An itemset is maximal frequent if none of its immediate supersets is frequent
Closed Itemset
TID Items1 {A,B}2 {B,C,D}3 {A,B,C,D}4 {A,B,D}5 {A,B,C,D}
• An itemset is closed if none of its immediate supersets has the same support as the itemset
Itemset Support{A} 4{B} 5{C} 3{D} 4
{A,B} 4{A,C} 2{A,D} 3{B,C} 3{B,D} 4{C,D} 3
Itemset Support{A,B,C} 2{A,B,D} 3{A,C,D} 2{B,C,D} 3
{A,B,C,D} 2
Maximal vs Closed Itemsets
TID Items
1 ABC
2 ABCD
3 BCE
4 ACDE
5 DE
null
AB AC AD AE BC BD BE CD CE DE
A B C D E
ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE
ABCD ABCE ABDE ACDE BCDE
ABCDE
124 123 1234 245 345
12 124 24 4 123 2 3 24 34 45
12 2 24 4 4 2 3 4
2 4
Transaction Ids
Not supported by any transactions
Maximal vs Closed Frequent Itemsets
null
AB AC AD AE BC BD BE CD CE DE
A B C D E
ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE
ABCD ABCE ABDE ACDE BCDE
ABCDE
124 123 1234 245 345
12 124 24 4 123 2 3 24 34 45
12 2 24 4 4 2 3 4
2 4
Minimum support = 2
# Closed = 9
# Maximal = 4
Closed and maximal
Closed but not maximal
Maximal vs Closed Itemsets
FrequentItemsets
ClosedFrequentItemsets
MaximalFrequentItemsets
Rule Generation
• Given a frequent itemset L, find all non-empty subsets f L such that f L – f satisfies the minimum confidence requirement• If {A,B,C,D} is a frequent itemset, candidate rules:
ABC D, ABD C, ACD B, BCD A, A BCD, B ACD, C ABD, D ABC,AB CD, AC BD, AD BC, BC AD, BD AC, CD AB,
• If |L| = k, then there are 2k – 2 candidate association rules (ignoring L and L)
Rule Generation• How to efficiently generate rules from
frequent itemsets?– In general, confidence does not have an anti-
monotone propertyc(ABC D) can be larger or smaller than c(AB D)
– But confidence of rules generated from the same itemset has an anti-monotone property
– e.g., L = {A,B,C,D}:c(ABC D) c(AB CD) c(A BCD)
• Confidence is anti-monotone w.r.t. number of items on the RHS of the rule
Rule Generation for Apriori Algorithm
ABCD=>{ }
BCD=>A ACD=>B ABD=>C ABC=>D
BC=>ADBD=>ACCD=>AB AD=>BC AC=>BD AB=>CD
D=>ABC C=>ABD B=>ACD A=>BCD
Lattice of rules
ABCD=>{ }
BCD=>A ACD=>B ABD=>C ABC=>D
BC=>ADBD=>ACCD=>AB AD=>BC AC=>BD AB=>CD
D=>ABC C=>ABD B=>ACD A=>BCD
Pruned Rules
Low Confidence Rule
Rule Generation for Apriori Algorithm
• Candidate rule is generated by merging two rules that share the same prefixin the rule consequent
• Join(CD=>AB,BD=>AC)would produce the candidaterule D => ABC
• Prune rule D=>ABC if itssubset AD=>BC does not havehigh confidence
BD=>ACCD=>AB
D=>ABC
Effect of Support Distribution
• Many real data sets have skewed support distribution
Support distribution of a retail data set
Effect of Support Distribution
• How to set the appropriate minsup threshold?– If minsup is set too high, we could miss
itemsets involving interesting rare items (e.g., expensive products)
– If minsup is set too low, it is computationally expensive and the number of itemsets is very large
• Using a single minimum support threshold may not be effective
Multiple Minimum Support• How to apply multiple minimum supports?
• MS(i): minimum support for item i • e.g.:
• MS(Milk)=5%, • MS(Coke) = 3%, • MS(Broccoli)=0.1%, • MS(Salmon)=0.5%
• MS({Milk, Broccoli}) = min (MS(Milk), MS(Broccoli))= 0.1%
• Challenge: Support is no longer anti-monotone• Suppose: Support(Milk, Coke) = 1.5% and
Support(Milk, Coke, Broccoli) = 0.5%• {Milk,Coke} is infrequent but {Milk,Coke,Broccoli} is frequent
Multiple Minimum Support
A
Item MS(I) Sup(I)
A 0.10% 0.25%
B 0.20% 0.26%
C 0.30% 0.29%
D 0.50% 0.05%
E 3% 4.20%
B
C
D
E
AB
AC
AD
AE
BC
BD
BE
CD
CE
DE
ABC
ABD
ABE
ACD
ACE
ADE
BCD
BCE
BDE
CDE
Multiple Minimum Support
A
B
C
D
E
AB
AC
AD
AE
BC
BD
BE
CD
CE
DE
ABC
ABD
ABE
ACD
ACE
ADE
BCD
BCE
BDE
CDE
Item MS(I) Sup(I)
A 0.10% 0.25%
B 0.20% 0.26%
C 0.30% 0.29%
D 0.50% 0.05%
E 3% 4.20%
Multiple Minimum Support (Liu 1999)• Order the items according to their minimum support
(in ascending order)– e.g.: MS(Milk)=5% MS(Coke) = 3%,
MS(Broccoli)=0.1% MS(Salmon)=0.5%– Ordering: Broccoli, Salmon, Coke, Milk
• Need to modify Apriori such that:– L1 : set of frequent items
– F1 : set of items whose support is MS(1)where MS(1) is mini( MS(i) )
– C2 : candidate itemsets of size 2 is generated from F1
instead of L1
Multiple Minimum Support (Liu 1999)• Modifications to Apriori:
• In traditional Apriori, • A candidate (k+1)-itemset is generated by merging two
frequent itemsets of size k• The candidate is pruned if it contains any infrequent
subsets of size k
• Pruning step has to be modified:• Prune only if subset contains the first item• e.g.: Candidate={Broccoli, Coke, Milk} (ordered
according to minimum support)• {Broccoli, Coke} and {Broccoli, Milk} are frequent but
{Coke, Milk} is infrequent• Candidate is not pruned because {Coke,Milk} does not contain
the first item, i.e., Broccoli.
Pattern Evaluation• Association rule algorithms tend to produce
too many rules – many of them are uninteresting or redundant– Redundant if {A,B,C} {D} and {A,B} {D}
have same support & confidence• Interestingness measures can be used to
prune/rank the derived patterns• In the original formulation of association
rules, support & confidence are the only measures used
Application of Interestingness Measure
Feature
Pro
duct
Pro
duct
Pro
duct
Pro
duct
Pro
duct
Pro
duct
Pro
duct
Pro
duct
Pro
duct
Pro
duct
FeatureFeatureFeatureFeatureFeatureFeatureFeatureFeatureFeature
Selection
Preprocessing
Mining
Postprocessing
Data
SelectedData
PreprocessedData
Patterns
KnowledgeInterestingness Measures
Computing Interestingness Measure
• Given a rule X Y, information needed to compute rule interestingness can be obtained from a contingency table
Y Y
X f11 f10 f1+
X f01 f00 fo+
f+1 f+0 |T|
Contingency table for X Yf11: support of X and Yf10: support of X and Yf01: support of X and Yf00: support of X and Y
Used to define various measures support, confidence, lift, Gini, J-measure, etc.
Drawback of Confidence
Coffee Coffee
Tea 15 5 20
Tea 75 5 80
90 10 100
Association Rule: Tea CoffeeConfidence= P(Coffee|Tea) = 0.75
but P(Coffee) = 0.9
Þ Although confidence is high, rule is misleading
Þ P(Coffee|Tea) = 0.9375
Statistical Independence• Population of 1000 students
• 600 students know how to swim (S)• 700 students know how to bike (B)• 420 students know how to swim and bike (S,B)
• P(SB) = 420/1000 = 0.42• P(S) P(B) = 0.6 0.7 = 0.42
• P(SB) = P(S) P(B) => Statistical independence• P(SB) > P(S) P(B) => Positively correlated• P(SB) < P(S) P(B) => Negatively correlated
Statistical-based Measures• Measures that take into account statistical
dependence
)](1)[()](1)[(
)()(),(
)()(),(
)()(
),(
)(
)(
)(
)|(
YPYPXPXP
YPXPYXPtcoefficien
YPXPYXPPS
YPXP
YXPInterest
Ys
YXc
YP
XYPLift
Example: Lift/Interest
Coffee Coffee
Tea 15 5 20
Tea 75 5 80
90 10 100
Association Rule: Tea CoffeeConfidence= P(Coffee|Tea) = 0.75
but P(Coffee) = 0.9
Þ Lift = 0.75/0.9= 0.8333 (< 1, therefore is negatively associated)
Drawback of Lift & Interest
Y Y
X 10 0 10
X 0 90 90
10 90 100
Y Y
X 90 0 90
X 0 10 10
90 10 100
10)1.0)(1.0(
1.0 Lift 11.1)9.0)(9.0(
9.0 Lift
Statistical independence:
If P(X,Y)=P(X)P(Y) => Lift = 1
There are lots of measures proposed in the literature
Some measures are good for certain applications, but not for others
What criteria should we use to determine whether a measure is good or bad?
What about Apriori-style support based pruning? How does it affect these measures?
Quiz: Mining Association Rules
1. Berapakah nilai support untuk itemset {B,D,E}?2. Berapakah nilai support dan confidence untuk aturan BD E?3. Gunakan algoritma Apriori untuk membentuk itemset dengan nilai minimal
frekuensi 3. Perhatikan bahwa dalam algoritma Apriori sebuah transaksi akan di-scan beberapa kali, dengan iterasi scan ke-i membangkitkan himpunan frekuensi i-itemset. Berikan semua himpunan itemset yang terbentuk dalam setiap iterasi.
4. Berikan aturan-aturan yang memiliki nilai support minimal 3 dan confidence ¾, berdasarkan itemset yang terbentuk pada nomor 3.
Support Vector Machines
• Find a linear hyperplane (decision boundary) that will separate the data
Support Vector Machines
• One Possible Solution
B1
Support Vector Machines
• Another possible solution
B2
Support Vector Machines
• Other possible solutions
B2
Support Vector Machines
• Which one is better? B1 or B2?• How do you define better?
B1
B2
Support Vector Machines
• Find hyperplane maximizes the margin => B1 is better than B2
B1
B2
b11
b12
b21
b22
margin
Support Vector MachinesB1
b11
b12
0 bxw
1 bxw 1 bxw
1bxw if1
1bxw if1)(
xf 2||||
2 Margin
w
Support Vector Machines
• We want to maximize:
– Which is equivalent to minimizing:
– But subjected to the following constraints:
• This is a constrained optimization problem– Numerical approaches to solve it (e.g., quadratic programming)
2||||
2 Margin
w
1bxw if1
1bxw if1)(
i
i
ixf
2
||||)(
2wwL
Support Vector Machines
• What if the problem is not linearly separable?
Support Vector Machines
• What if the problem is not linearly separable?– Introduce slack variables
• Need to minimize:
• Subject to:
ii
ii
1bxw if1
-1bxw if1)(
ixf
N
i
kiC
wwL
1
2
2
||||)(
Nonlinear Support Vector Machines
• What if decision boundary is not linear?
Nonlinear Support Vector Machines
• Transform data into higher dimensional space
Practical Work
• Association Rules Mining (DMBAR ch. 16) – online radio – predicting income
• SVM (package: kernlab)– “Support vector machines in R,” by Karatzoglou et
al. (2006), section 4 ksvm in kernlab– “Introduction to SVM in R” (Jean-Philippe Vert):
questions 1-10