Transcript 1 - University of California, Irvine

ICS 278: Data Mining
Lecture 11: Pattern Discovery Algorithms
Padhraic Smyth
Department of Information and Computer Science
University of California, Irvine
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Project Progress Report
• Written Progress Report:
– Due Tuesday May 18th in class
– Expect at least 3 pages (should be typed not handwritten)
– Hand in written document in class on Tuesday May 18th
• 1 Powerpoint slide:
– 1 slide that describes your project
– Use lecture slides as template “master” (i.e., white background, etc)
– Should contain:
•
•
•
•
•
•
Your name (top right corner)
Clear description of the main task
Some visual graphic of data relevant to your task
1 bullet or 2 on what methods you plan to use
Preliminary results or results of exploratory data analysis
Make it graphical (use text sparingly)
– Submit by 12 noon Monday May 17th
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
List of Sections for your Progress Report
•
Clear description of task (reuse original proposal if needed)
•
Discussion of relevant literature
•
Preliminary data evaluation
•
Preliminary algorithm work
–
–
–
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Basic task + extended “bonus” tasks (if time allows)
Discuss prior published/related work (if it exists)
Exploratory data analysis relevant to your task
Include as many of plots/graphs as you think are useful/relevant
–
Summary of your progress on algorithm implementation so far
–
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Relevant information about other code/algorithms you have downloaded, some preliminary
testing on, etc.
Difficulties encountered so far
–
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Algorithm implementation
Experimental methods
Evaluation, validation
•
If you are not at this point yet, say so
•
Plans for the remainder of the quarter
•
Approximately ½ page to 1 page of text per section (graphs/plots don’t count –
include as many of these as you like).
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Pattern-Based Algorithms
• “Global” predictive and descriptive modeling
– “global” models in the sense that they “cover” all of the data space
• “Patterns”
– More local structure, only describe certain aspects of the data
– Examples:
• A single small very dense cluster in input space
– e.g., a new type of galaxy in astronomy data
• An unusual set of outliers
– e.g., indications of an anomalous event in time-series climate data
• Associations or “rules”
– If bread is purchased and wine is purchased then cheese is purchased
with probability p
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
General Ideas for Patterns
•
Many patterns can be described in the general form:
–
if condition 1 then condition 2 (with some certainty)
• Probabilistic rules:
If Age > 40 and education > college then income > $50k with probability p
• “Bumps”
If Age > 40 and education > college then mean income = $73k
–
–
if antecedent then consequent
if j then v
•
•
•
where j is generally some “box” in the input space
where v is a statement about a variable of interest, e.g., p(y | j ) or E [ y | j ]
Pattern support
– “Support” = p( j ) or p(j , w )
– Fraction of points in input space where the condition applies
– Often interested in patterns with larger support
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
How Interesting is a Pattern?
• Note: “interestingness” is inherently subjective
– Depends on what the data analyst already knows
• Difficult to quantify prior knowledge
– How interesting a pattern is, can be a function of
• How surprising it is relative to prior knowledge?
• How useful (actionable) it is?
– This is a somewhat open research problem
– In general pattern “interestingness” is difficult to quantify
• => Use simple “surrogate” measures in practice
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
How Interesting is a Pattern?
• Interestingness of a pattern
– Measures how “interesting” the pattern j -> v is
• Typical measures of interest
– Conditional probability: p( v | j )
– Change in probability: | p( v | j ) - p( v ) |
– “Lift” =
p( v | j ) / p( v )
(also log of this)
– Change in mean target response, e.g., E [y| j ]/E[y]
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Pattern-Finding Algorithms
• Typically… search a data set for the set of patterns that maximize
some score function
– Usually a function of both support and “interestingness”
– E.g.,
• Association rules
• Bump-hunting
• Issues:
– Huge combinatorial search space
– How many patterns to return to the user
– How to avoid problems with redundant patterns
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Generic Pattern Finding
Task
Find patterns
Representation
pattern language
Score Function
f(support,
interestingness)
Search/Optimization
greedy, branch-and-bound
Data
Management
Models,
Parameters
Data Mining Lectures
varies
list of K highest
scoring
patterns
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Two Pattern Finding Algorithms
1. Bump-hunting: the PRIM algorithm
Bump Hunting in High Dimensional Data
J. H. Friedman & N. I. Fisher
Statistics and Computing, 2000
2. Market basket data: association rule algorithms
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
“Bump-Hunting” (PRIM) algorithm
• Patient Rule Induction Method (PRIM)
– Friedman and Fisher, 2000
• Addresses “bump-hunting” problem:
– Assume we have a target variable Y
• Y could be real-valued or a binary class variable
– And we have p “input” variables
– We want to find “boxes” j in input space where E[Y| j ] >> E[Y]
• or where E[Y| j ] << E[Y] , i.e., “data holes”
– A box j is a “conjunctive sentence”, e.g.,
if Age < 22 and occupation = student
Example of a “box pattern”
if Age > 30 and education > bachelor then E[income | j ] = $120k
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
“Bump Hunting”: Extrema Regions for Target f(x)
• let Sj be set of all possible vals for input variable xj
– entire input domain is S = S1  S2  …  Sd
• goal: find subregion R  S for which
– mR = avg xR f(x) = [ xR f(x)p(x)dx ] / [ xR p(x)dx ] >> m
– where m =  f(x) p(x) dx (target mean, over all inputs)
• subregion size as fraction of full space (“support”):
– R = xR p(x) dx
• tradeoff between mR and R (increase R => reduce mR) ...
• sample-based estimates used in practice:
– R = (1/n) XiR 1(XiR),
yavgR = 1/(nR) XiR yi
– note: mR is true quantity of interest, not yavgR
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Greedy Covering
• a generic greedy covering algorithm
– first box B1 induced (somehow) from entire data set
– second box B2 induced from data not covered by B1
– … BK induced from remaining data {yi,Xi | Xi  j=1…K-1 Bj}
• do until either:
– estimated target mean f(x) in Bk becomes too small
• yavgK = avg[yi | Xi Bk & Xi  j=1…K-1 Bj] <  = (1/n) ni=1 yi
– support of Bk becomes too small
• K = (1/n) i=1…n 1(Xi Bk & Xi  j=1…K-1 Bj)
• then select set of boxes R = j Bj for some threshold
– for which each yavgj > some yavgthreshold or
– yield largest yavgR for which R = i i  some threshold
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
PRIM algorithm
• PRIM uses “patient greedy search” on individual variables
• Start with all training data and maximal box
• Repeat until minimal box (e.g., minimal support  or n<10)
– Shrink box by compressing one face of the box
– For each variable in input space
• “Peel” off a proportion  of observations to maximize or minimize
E[y |new box],
• typical =0.05 or =0.1
• Now “expand” the box if E[y|box] can be increased (“pasting”)
• Yields a sequence of boxes
– Use cross-validation (on E[y|box]) to select the best box
• Remove box from training data, then repeat process
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Comments on PRIM
•
Works one variable at a time
– So time-complexity is similar to tree algorithms, i.e.,
• Linear in p, and n log n for sorting
•
Nominal variables
– Can peel/paste on single values, subsets, negations, etc
•
Similar in some sense to CART….but
– More “patient” in search (removes only small fraction of data at each step)
•
Useful for finding “pockets” in the input space with high-response
– e.g., marketing data: small groups of consumers who spend much more on a
given product than the average consumer
– Medical data: patients with specific demographics whose response to a drug is
much better than the average patient
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Marketing Data Example (n=9409, p=502)
• freq air travel: y=num flights/yr, global mean(y)=1.7
• B1: mean(y1)=4.2, 1=0.08 (8% market seg)
– education >= 16 yrs; income > $50K &  missing
– occupation in {professional/manager, sales, homemaker}
– number of children (<18) in home <= 1
• B2: mean(y2)=3.2, 2=0.07
(~2x global mean)
– education > 12 yrs &  missing
– income > $30K &  missing; 18 < age < 54
– married / dual income in {single, married-one-income}
• these boxes intuitive: nothing really surprising ...
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Pattern Finding Algorithms
1. Bump-hunting: the PRIM algorithm
2. Market basket data: association rule algorithms
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Transaction Data and Market Baskets
Transactions
Items
•
x
x
x
x
x
x
x
x x
x
x
x
x
x
x
x x
x
x
x
x
Supermarket example: (Srikant and Agrawal, 1997)
– #items = 50,000, #transactions = 1.5 million
•
Data sets are typically very sparse
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Market Basket Analysis
• historically, assoc rules in terms of market baskets
• given: a (huge) “transactions” database
– each transaction representing basket for 1 customer visit
– each transaction containing set of items (“itemset”)
• finite set of (boolean) items (e.g. wine, cheese, diaper, beer, …)
• Association rules
– classically used on supermarket transaction databases
– associations: Trader Joe’s customers frequently buy wine & cheese
• rule:
“people who buy wine also buy cheese 60% of time”
– infamous “beer & diapers” example:
• “in evening hours, beer and diapers often purchased together”
– generalize to many other problems, e.g.:
• baskets = documents, items = words
• baskets = WWW pages, items = links
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Market Basket Analysis: Complexity
• usually transaction DB too huge to fit in RAM
– common sizes:
• number of transactions: 105 to 108 (hundreds of millions)
• number of items:
102 to 106 (hundreds to millions)
• entire DB needs to be examined
– usually very sparse
• e.g. ~ 0.1% chance of buying random item
– subsampling often a useful trick in DM, but
• here, subsampling could easily miss the (rare) interesting patterns
• thus, runtime dominated by disk read times
– motivates focus on minimizing number of disk scans
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Association Rules: Problem Definition
• given: set I of items, set T transactions, t T, t  I
– Itemset Z = a set of items (any subset of I)
• support count (Z) = num transactions containing Z
– given any itemset Z  I, (Z) = | { t | t T, Z  t } |
• association rule:
– R=“X  Y [s,c]”, X,Y  I, XY=
• support:
– s(R) = s(XY) = (XY)/|T| = p(XY)
• confidence:
– c(R) = s(XY) / s(X) = (XY) / (X) = = p(X | Y)
• goal: find all R such that
– s(R)  given minsup
– c(R)  given minconf
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Comments on Association Rules
• association rule: R=“X  Y [s,c]”
– Strictly speaking these are not “rules”
• i.e., we could have “wine => cheese” and “cheese => wine”
• correlation is not causation
• The space of all possible rules is enormous
– O( 2p ) where p = the number of different items
– Will need some form of combinatorial search algorithm
• How are thresholds minsup and minconf selected?
– Not that easy to know ahead of time how to select these
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Example
• simple example transaction database (|T|=4):
–
–
–
–
Transaction1
Transaction2
Transaction3
Transaction4
=
=
=
=
{A,B,C}
{A,C}
{A,D}
{B,E,F}
s({A}) = 3/4 = 75%
s({B}) = 2/4 = 50%
s({C}) = 2/4 = 50%
s({A,C}) = 2/4 = 50%
• with minsup=50%, minconf=50%:
– R1: A --> C [s=50%, c=66.6%]
• s(R1) = s({A,C}) , c(R1) = s({A,C})/s({A}) = 2/3
– R2: C --> A [s=50%, c=100%]
• s(R2) = s({A,C}), c(R2) = s({A,C})/s({C}) = 2/2
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Finding Association Rules
• two steps:
– step 1: find all “frequent” itemsets (F)
• F = {Z | s(Z)  minsup}
(e.g. Z={a,b,c,d,e})
– step 2: find all rules R: X --> Y such that:
• X  Y  F and X  Y=
• s(R)  minsup and c(R)  minconf
(e.g. R: {a,b,c} --> {d,e})
• step 1’s time-complexity typically >> step 2’s
• step 2 need not scan the data (s(X),s(Y) all cached in step 1)
• search space is exponential in |I|, filters choices for step 2
• so, most work focuses on fast frequent itemset generation
• step 1 never filters viable candidates for step 2
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Finding Frequent Itemsets
• frequent itemsets: {Z | s(Z)>minsup}
• “Apriori (monotonicity) Principle”: s(X)  s(XY)
– any subset of a frequent itemset must be frequent
• finding frequent itemsets:
– bottom-up approach:
s({A}) = 3/4 = 75%
s({B}) = 2/4 = 50%
s({C}) = 2/4 = 50%
s({A,C}) = 2/4 = 50%
• do level-wise, for k=1 … |I|
– k=1: find frequent singletons
– k=2: find frequent pairs (often most costly)
– step k.1: find size-k itemset candidates from the freq size-(k-1)’s of
prev level
– step k.2 prune candidates Z for which s(Z)<minsup
• each level requires a single scan over all the transaction data
– computes support counts (Z) = | { t | t T, Z  t } for all size-k Z
candidates
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
bottleneck:
Apriori Example (minsup=2)
transactions T
{1,3,4}
{2,3,5}
{1,2,3,5}
{2,5}
itemset
sup
{2,3,5}
2
filter
itemset
sup
{2,3,5}
2
count
(scan T)
Data Mining Lectures
C1 itemset sup
{1} 2
{2} 3
count
{3} 3
(scan T)
{4} 1
{5} 3
F3
filter
count
C3 knows can avoid gen
{1,2,3} (and {1,3,5}) apriori,
without counting, because
{1,2} ({1,5}) not freq
C3
notice how
|C3| << |C2|
C3
itemset
{2,3,5}
gen
C2
F1
itemset sup
{1} 2
{2} 3
{3} 3 gen
{5} 3
itemset
{1,2}
{1,3}
{1,5}
{2,3}
{2,5}
{3,5}
F2
itemset sup
{1,3} 2
{2,3} 2
{2,5} 3
{3,5} 2
Lecture 11: Pattern Discovery
C2
filter
(scan T)
itemset sup
{1,2} 1
{1,3} 2
{1,5} 1
{2,3} 2
{2,5} 3
{3,5} 2
Padhraic Smyth, UC Irvine
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Problems with Association Rules
• Consider 4 highly correlated items A, B, C, D
– Say p(subset i|subset j) > minconf for all possible pairs of disjoint
subsets
– And p(subset i  subset j) > minsup
– How many possible rules?
•
•
•
•
E.g., A->B, [A,B]=>C, [A,C]=>B, [B,C]=>A
All possible combinations: 4 x 23
In general for K such items, K x 2K-1 rules
For highly correlated items there is a combinatorial explosion of redundant
rules
• In practice this makes interpretation of association rule results difficult
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
References on Association Rules
• Chapter 13 in text (Sections 13.1 to 13.5)
• Early papers:
– R. Agrawal and R. Srikant, Fast algorithms for mining association rules, in
Proceedings of VLDB 1994, pp.487-499, 1994.
– R. Agrawal et al. Fast discovery of association rules, in Advances in Knowledge
Discovery and Data Mining, AAAI/MIT Press, 1996.
• More recent:
– Good review in Chapter 6 of Data Mining: Concepts and Techniques, J. Han and
M. Kamber, Morgan Kaufmann, 2001.
– J. Han, J. Pei, and Y. Yin, Mining frequent patterns without candidate generation,
Proceedings of SIGMOD 2000, pages 1-12.
– Z. Zheng, R. Kohavi, and L. Mason, Real World Performance of Association Rule
Algorithms, Proceedings of KDD 2001
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Study on Association Rule Algorithms
•
Z. Zheng, R. Kohavi, and L. Mason, Real World Performance of Association
Rule Algorithms, Proceedings of KDD 2001
•
Evaluated a variety of association rule algorithms
– Used both real and simulated transaction data sets
•
Typical real data set from Web commerce:
–
–
–
–
Data Mining Lectures
Number of transactions = 500k
Number of items = 3k
Maximum transaction size = 200
Average transaction size = 5.0
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Study on Association Rule Algorithms
•
Conclusions:
– Very narrow range of minsup yields interesting rules
• Minsup too small => too many rules
• Minsup too large => misses potentially interesting patterns
– Superexponential growth of rules on real-world data
– Real-world data is different to simulated transaction data used in research
papers, e.g.,
• Simulated transaction sizes have a mode away from 1
• Real transaction sizes have a mode at 1 and are highly skewed
– Speed-up improvements demonstrated on artificial data did not generalize to real
transaction data
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Beyond Binary Market Baskets
• counts (vs yes/no)
– e.g. “3 wines” vs “wine”
• quantitative (non-binary) item variables
– popular: discretize real variable into k binary variables
– e.g. {age=[30:39],incomeK=[42:48]}  buys_PC
• Item hierachies
– Common in practice, e.g., clothing -> shirts -> men’s shirts, etc
– Can learn rules that generalize across the hierarchy
• mining sequential associations/patterns and rules
– e.g. {1@0,2@5}  4@15
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Association Rule Finding
Task
Representation
Score Function
Search/Optimization
Data
Management
Models,
Parameters
Data Mining Lectures
Find association
rules
[A and B] => C
P(A,B,C) > minsup,
P(C|A, B) > minconf
Breadth-first candidate
generation
Linear scans
list of all rules
satisfying
thresholds
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Bump Hunting (PRIM)
Task
Find high score
bumps
Representation
[A,B] => E[y|A,B] > E[y]
Score Function
E[y|A,B] and p(A,B)
Search/Optimization
Greedy search
Data
Management
Models,
Parameters
Data Mining Lectures
None
Set of “boxes”
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Summary
• Pattern finding
– An interesting and challenging problem
– How to search for interesting/unusual “regions” of a high-dimensional
space
– Two main problems
• Combinatorial search
• How to define “interesting” (this is the harder problem)
– Two examples of algorithms
• PRIM for bump-hunting
• Apriori for association rule mining
– Many open problems in this research area (room for new ideas!)
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Backup slides from Dennis DeCoste’s
2003 lectures in 278
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Apriori Algorithm: Finds All Frequent Itemsets
function F=Apriori(T,I,minsup)
F1={{Ii}|s({Ii}) minsup};%% freq singletons
for k=2 to |I| do
if (Fk-1 = ) break; end
Ck = AprioriGen(Fk-1); %% gen candidates
forall t  T do
%% scan T
forall c  subset(Ck,t) do
c.count++;
end
end
Fk = {c  Ck | c.count  minsup}
end
F = k Fk
classic generate-and-test approach
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Itemset Candidate Generation
function Ck=AprioriGen(Fk-1)
%% ===== SQL-ish self-join (items assumed ordered):
insert into Ck
%% {p1,p2,p3,…pk-1,qk-1}
select p.item1,p.item2,…p.itemk-1,q.itemk-1
from Fk-1 p, Fk-1 q
where p.item1=q.item1,…,p.itemk-2=q.itemk-2,
p.itemk-1<q.itemk-1
%% grow by 1 item {qk-1}
%% ===== prune if violate Apriori monotonicity principle:
forall itemsets c  Ck do
forall (k-1) subsets s of c do
if (s  Fk-1)then delete c from Ck
end
F3={{1,2,3},{1,2,4},{1,2,5},{1,4,5},{2,4,5}}
C4={{1,2,3,4},{1,2,3,5},{1,2,4,5}}
Data Mining Lectures
Lecture 11: Pattern Discovery
{1,2,3,4} pruned: {2,3,4}  F3
{1,2,3,5} pruned: {2,3,5}  F3
Padhraic Smyth, UC Irvine
Counting Support of Candidates Ck
• key computational challenge for Apriori
–
–
–
–
for each c  Ck, need to count supports s(c) over all T
|Ck| can be huge -- so O(|Ck| |T|) often prohibitive
single transaction tT may contain many candidates in Ck
Apriori runtime tends to be dominated by support counting
• methods
t = {i1,i2,…im} and c={c1,c2,…,ck}
– simple string match each c  Ck against each tT
• markvectors: mark all itemst once, O(k) time check each cCk
– O(k |Ck| |T| + |T|), where k=|c| --- vs O(m k |Ck| |T|) for naïve set
intersection
– hash tree
forall t  T do
for c  subset(Ck,t) do
c.count++;
end
end
• implements subset(Ck,t)
• quickly finds all candidates in t
– near-optimal: O(| {ct|cCk} |) time
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Example Hash Tree (all 3-item candidates)
Will traverse HT multiple
times, from root to leafs,
hashing on every item in t.
Only the subset of
leafs reachable from
root via hashing will
be compared to t.
If reach internal
node after hashing
on item i, will hash
on every item after
i in t, and recurse
over all nodes kids.
Build tree for all cCk, hashing
on c(i) for kid choice at level i.
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Subset Operation at Root of Hash Tree
1,4
Data Mining Lectures
2,5
3,6
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Subset Operation at Left-Most Subtree of Root
*
Assumes: items ordered, so internals
only need hash on items after i
Data Mining Lectures
Lecture 11: Pattern Discovery
Why works:
If ct then first
item in c is in t.
By hashing at
root on every
item in t, only
kids ignored will
be those leading
to itemsets
starting with
item not in t.
Similarly for
internals.
Padhraic Smyth, UC Irvine
Example Rule Generation
• assume only 1-conseq rules with minconf are:
– ACDE  B and ABCE  D
freq itemset = {A,B,C,D,E}
• potential candiate 2-conseq rules might be:
– ACD  BE
• cannot hold, E  BE and ABCD  E did not have minconf
– ADE  BC
• cannot hold, C  BC and ABDE  C did not have minconf
– CDE  AB
• cannot hold, A  AB and BCDE  A did not have minconf
– ACE  BD
• only rule that needs to be gen/test for minconf by ap_genrules
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Rule Generation
• 1st find all freq itemsets: F=Apriori(T,I,minsup);
• consider each f  F & cand rule R=“{f - c} c”, |c| 1
– all s(R) for R based on same f are same
– all c(R) computable from known s(g), g  F
• e.g. R= {1,2}  {3}, so c(R) = s({1,2,3}) / s({1,2})
• thus, no (costly!) DB scans required here if saved s(g) for g  F
• if some c(a  {f - a}) < minconf, then
– any ã  a also has c(ã  {f - ã}) < minconf
• e.g. f={1,2,3}: c({1,2}  {3}) < m  c({1}  {2,3}) < m
• why: c(XY)  s(XY) / s(X), num same, denom s(ã)  s(a)
– similarly: c({f-a}a)m requires ã  a c({f- ã}ã)m
• mononocity principle for conf; idea: try {f- ã}ã before try {f-a}a
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
forall fk  F, k Rule
 2 doGeneration%Algorithm
try each freq itemset in turn
H1={h | h  fk & |h|=1 & c({fk - h}h) minconf};
output rules H1; call ap_genrules(fk,H1,minconf,minsup);
end
function ap_genrules(fk , Hm , minconf, minsup)
if (k  m+1) return;
% stop before considering {}  fk
Hm+1=AprioriGen(Hm,minsup);
% expand/filter conseqs
forall hm+1  Hm+1 do
% recall: s(R) = s(XY)
c = s(fk) / s({fk-hm+1});
if (c  minconf) then output rule “{fk - hm+1}  hm+1”
else delete hm+1 from Hm+1;
% use conf monotonicity
call ap_genrules(fk , Hm+1, minconf, minsup);
end
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Tricks to Improve Apriori’s Efficiency
• speed scans>k: remove t once no f in t with f in Fk
• key: faster 2-itemset generation via hashing
– C2 is major bottleneck, since many frequent 1-items
• could generate nearly |I|2 C2 candidates, each slowing the T scan
– yields most of speedup given by Apriori’s complex “tids”
– main idea:
• when counting support of 1-items of each t T during full T scan
– forall 2-item subsets x of t do HT2[hash2(x)]++; end
• then C2=AprioriGen(F1) can just gather all x with HT2(x)>minsup
– good for common case where most transactions are small
• i.e. |t| << |T| and scales O(|T| mean(|t|)2) vs O(|T|2)
– see “An Effective Hash-Based Algorithm for Mining Association Rules”,
by Park et al, 1995. Many others ...
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Alternatives to Apriori
• Frequent-Pattern (FP) trees [Han et al]
– compresses large T -- only requires couple DB scans
– no candidate generation
• accumulate prefix paths through FP tree
– uses divide-and-conquer, vs Apriori’s generate-and-test
– some random data studies show 10x faster than Apriori
• especially on often/overly-studied IBM random benchmark data
• many others: CLOSET [Pei and Han], etc.
• recent studies on real data show no clear winner
– “Real World Performance of Association Rule Algorithms” [Kohavi et al,
KDD 2001]
• Apriori is seminal and key starting point for others ...
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Measures of Rule Interestingness
• apply as preprocessing
– remove “too common” items (e.g. stopwords, S&H,...)
• e.g. use maxsup threshold as well, over singleton itemsets
• apply during mining
– minsup (and minconf) are the common way to do this
– not all interesting
• apply as post-processing
– filter or rank the high support, high confidence rules found
– this is where most interestingness measures apply
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Interestingness Measures
• subjective -- pattern interesting if either:
– unexpected
• surprise user (beliefs), diff in conf vs “user-defined similar” rules,...
– actionable (can do something useful with it)
• objective
– 3 principles (“3p”) of objective measures of rule XY
• M=0 if X,Y are statistically independent (e.g. 0 if conf = prior(Y))
• M monotonically increases with (XY), all else (e.g. |T|) steady
• M monotonically decreases with (X), all else steady
– recall c(XY) = (XY) / (X)
– often diff measures obeying these give same rankings
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Problems with Support and Confidence
• example (among 5000 students):
– 3000 play basketball (60%), 3750 eat cereal (75%)
– 2000 (40%) both play basketball and eat cereal
– “play basketball”  “eat cereal” [s=40%,c=66.7%]
• misleading: (prior) % students eating cereal is 75% (>66.7%)
– “play basketball”  “not eat cereal” [s=20%,c=33.3%]
• lower s & c, yet more accurate description (w/ 33.3% > prior 25%)
• challenge: many uninteresting neg assocs (sparse T); find c»prior
Contingency Table:
basketball not basketball
cereal
2000
1750 =3750
not cereal
1000
250 =1250
=3000
=2000 =5000
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine
Statistical Dependence vs Confidence
X: 1 1 1 1 0 0 0
Y: 1 1 0 0 0 0 0
Z: 0 1 1 1 1 1 1
• example:
– X and Y positively correlated
– X and Z negatively correlated
– support and confidence of XZ dominates
• need measure of dependence
– corr(A,B) = P(AB) / [ P(A) P(B) ]
– corr(A,B)~1 if stat independent
– corr(A,B)>1 = positive dependence
s
c
XY: 25.0% 50%
XZ : 37.5% 75%
• P(B|A)/P(B) = “lift” of AB = “interest” = I(A,B)
– where P(B|A) = P(AB)/P(A) = c(AB);
{I-1 obeys 3p}
• lift(XY)=0.5/(2/8)=2 > lift(XZ)=0.75/(7/8)=0.86~1
Data Mining Lectures
Lecture 11: Pattern Discovery
Padhraic Smyth, UC Irvine