Transcript lecture19b_pattern_discovery

Project Presentations
• Thursday this week, each student will make a 4-minute presentation
on their project in class (with 1 or 2 minutes for questions)
• Email me your Powerpoint or PDF slides, with your name (e.g.,
joesmith.ppt), before 10am next Thursday
• Suggested content:
–
–
–
–
–
Data Mining Lectures
Definition of the task/goal
Description of data sets
Description of algorithms
Experimental results and conclusions
Be visual where possible! (i.e., use figures, graphs, etc)
Lecture 19: Pattern Discovery
Padhraic Smyth, UC Irvine
Final Project Reports
• Must be submitted as an email attachment (PDF, Word, etc) by
12 noon Tuesday next week
– Use “ICS 278 final project report” in the subject line of your email
• Report should be self-contained
– Like a short technical paper
– A reader should be able to repeat your results
• Include details in appendices if necessary
• Approximately 1 page of text per section (see next slide)
–
graphs/plots don’t count – include as many of these as you like.
• Can re-use material from proposal and from midterm progress
report if you wish
Data Mining Lectures
Lecture 19: Pattern Discovery
Padhraic Smyth, UC Irvine
Suggested Outline of Final Project Report
•
Introduction:
•
Discussion of relevant literature
•
Technical approach
–
–
–
Clear description of task/goals of the project
Motivation: why is this problem interesting and/or important?
Summarize relevant aspects of prior published/related work
–
Data used in your project
–
Algorithms used in your project
•
•
Exploratory data analysis relevant to your task
Include as many of plots/graphs as you think are useful/relevant
•
•
Clear description of all algorithms used
Credit appropriate sources if you used other implementations
•
Experimental Results
•
Discussion and Conclusions
•
Also: References and Appendices (if needed)
–
–
–
Clear description of your experimental methodology
Detailed description of your results (graphs, tables, etc)
What you learned from this project
Data Mining Lectures
Lecture 19: Pattern Discovery
Padhraic Smyth, UC Irvine
ICS 278: Data Mining
Lecture 19: Pattern Discovery Algorithms
Padhraic Smyth
Department of Information and Computer Science
University of California, Irvine
Data Mining Lectures
Lecture 19: 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
• Motif-finding in sequences, e.g.,
– motifs in DNA sequences ~ noisy words in random background
Data Mining Lectures
Lecture 19: 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 19: 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 19: 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 19: 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:
•
Statistical issues
–
–
–
–
Huge combinatorial search space
How many patterns to return to the user
How to avoid problems with redundant patterns
–
–
Even in random noise, if we search over a very large number of patterns, we are likely to
find something that looks significant
This is known as “multiple hypothesis testing” in statistics
One approach that can help is to conduct randomization tests
–
Ideally, also need a 2nd data set to validate patterns
Data Mining Lectures
•
•
e.g., for matrix data randomly permute the values in each column
Run pattern-discovery algorithm – resulting scores provide a “null distribution”
Lecture 19: 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 19: 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 19: 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 19: Pattern Discovery
Padhraic Smyth, UC Irvine
“Bump Hunting”: Extrema Regions for Target f(x)
• let Sj be set of all possible values 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) >> 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 19: Pattern Discovery
Padhraic Smyth, UC Irvine
Greedy Covering
• a generic greedy covering algorithm
– first box B1 induced 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 19: 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 optimize 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 19: Pattern Discovery
Padhraic Smyth, UC Irvine
Data Mining Lectures
Lecture 19: 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 19: 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 19: 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 19: 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 19: Pattern Discovery
Padhraic Smyth, UC Irvine
Market Basket Analysis
• 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 19: 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 19: Pattern Discovery
Padhraic Smyth, UC Irvine
Association Rules: Problem Definition
•
given: set I of items, set T transactions, t T, t  I
–
•
support count (Z) = num transactions containing Z
–
•
s(R) = s(XY) = (XY)/|T| = p(XY)
confidence:
–
•
R=“X  Y [s,c]”, X,Y  I, XY=
support:
–
•
given any itemset Z  I, (Z) = | { t | t T, Z  t } |
association rule:
–
•
Itemset Z = a set of items (any subset of I)
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 19: 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 19: 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 19: 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 19: 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 19: 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 19: 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 19: 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 19: 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 19: 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 19: 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 19: 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 19: 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 19: 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 19: 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 19: Pattern Discovery
Padhraic Smyth, UC Irvine