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Time Series Data
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A time series is a collection of observations
made sequentially in time.
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value
axis
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0
50
100
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time axis
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Time Series Problems
(from a databases perspective)
• The Similarity Problem
X = x1, x2, …, xn and Y = y1, y2, …, yn
• Define and compute Sim(X, Y)
– E.g. do stocks X and Y have similar movements?
• Retrieve efficiently similar time series (Similarity
Queries)
Similarity Models
•
•
•
•
•
Euclidean and Lp based
Dynamic Time Warping
Edit Distance and LCS based
Probabilistic (using Markov Models)
Landmarks
• How appropriate a similarity model is depends on
the application
Euclidean model
Query Q
Database
Distance
Rank
0.98
4
0.07
1
0.21
2
0.43
3
n datapoints
Euclidean Distance between
two time series Q = {q1, q2, …, qn}
and S = {s1, s2, …, sn}
Q
S
DQ, S  
 qi  si 
n
i 1
2
n datapoints
Dynamic Time Warping
[Berndt, Clifford, 1994]
• Allows acceleration-deceleration of signals along the time
dimension
• Basic idea
– Consider X = x1, x2, …, xn , and Y = y1, y2, …, yn
– We are allowed to extend each sequence by repeating
elements
– Euclidean distance now calculated between the
extended sequences X’ and Y’
Dynamic Time Warping
[Berndt, Clifford, 1994]
j=i+w
warping path
j=i–w
Y
X
Restrictions on Warping Paths
• Monotonicity
– Path should not go down or to the left
• Continuity
– No elements may be skipped in a sequence
• Warping Window
| i – j | <= w
Formulation
• Let D(i, j) refer to the dynamic time warping
distance between the subsequences
x1, x2, …, xi
y1, y2, …, yj
D(i, j) = | xi – yj | + min {
D(i – 1, j),
D(i – 1, j – 1),
D(i, j – 1) }
Basic LCS Idea
X
Y =
LCS
=
3, 2, 5, 7, 4, 8, 10, 7
2, 5, 4, 7, 3, 10, 8, 6
=
2, 5, 7, 10
Sim(X,Y) = |LCS| or Sim(X,Y) = |LCS| /n
Longest Common Subsequence
Edit Distance is another possibility
Indexing Time Series using
‘GEMINI’
(GEneric Multimedia INdexIng)
Extract a few numerical features, for a ‘quick
and dirty’ test
‘GEMINI’ - Pictorially
eg,. std
S1
F(S1)
1
365
day
Sn
F(Sn)
eg, avg
1
365
day
GEMINI
Solution: Quick-and-dirty' filter:
• extract n features (numbers, eg., avg., etc.)
• map into a point in n-d feature space
• organize points with off-the-shelf spatial
access method (‘SAM’)
• discard false alarms
GEMINI
Important: Q: how to guarantee no false
dismissals?
A1: preserve distances (but: difficult/impossible)
A2: Lower-bounding lemma: if the mapping
‘makes things look closer’, then there are no
false dismissals
Feature Extraction
• How to extract the features? How to define
the feature space?
• Fourier transform
• Wavelets transform
• Averages of segments (Histograms or
APCA)
Piecewise Aggregate Approximation (PAA)
Original time series
(n-dimensional vector)
S={s1, s2, …, sn}
value
axis
time axis
n’-segment PAA representation
(n’-d vector)
S = {sv1 , sv2, …, svn’ }
sv6
sv1
sv2 sv sv4
3
sv5
sv7
sv8
PAA representation satisfies the lower bounding lemma
(Keogh, Chakrabarti, Mehrotra and Pazzani, 2000; Yi and Faloutsos 2000)
Can we improve upon PAA?
sv6
n’-segment PAA representation
(n’-d vector)
S = {sv1 , sv2, …, svN }
sv1
sv2 sv sv4
3
sv7
sv5
sv8
Adaptive Piecewise Constant
Approximation (APCA)
n’/2-segment APCA representation
(n’-d vector)
S= { sv1, sr1, sv2, sr2, …, svM , srM }
(M is the number of segments = n’/2)
sv3
sv1
sv2
sr1
sv4
sr2
sr3
sr4
Dimensionality Reduction
• Many problems (like time-series and image
similarity) can be expressed as proximity problems in
a high dimensional space
• Given a query point we try to find the points that are
close…
• But in high-dimensional spaces things are different!
MDS (multidimensional scaling)
• Input: a set of N items, the pair-wise (dis) similarities and the
dimensionality k
• Optimization criterion:
stress = (ij(D(Si,Sj) - D(Ski, Skj) )2 / ijD(Si,Sj) 2) 1/2
– where D(Si,Sj) be the distance between time series Si, Sj, and
D(Ski, Skj) be the Euclidean distance of the k-dim
representations
• Steepest descent algorithm:
– start with an assignment (time series to k-dim point)
– minimize stress by moving points
FastMap
[Faloutsos and Lin, 1995]
• Maps objects to k-dimensional points so that distances are
preserved well
• It is an approximation of Multidimensional Scaling
• Works even when only distances are known
• Is efficient, and allows efficient query transformation
Other DR methods
• PCA (Principle Component Analysis)
Move the center of the dataset to the center
of the origins. Define the covariance matrix
ATA. Use SVD and project the items on the
first k eigenvectors
• Random projections
What is Data Mining?
• Data Mining is:
(1) The efficient discovery of previously unknown,
valid, potentially useful, understandable
patterns in large datasets
(2) The analysis of (often large) observational
data sets to find unsuspected relationships and
to summarize the data in novel ways that are
both understandable and useful to the data
owner
What is Data Mining?
• Data Mining is:
(1) The efficient discovery of previously unknown,
valid, potentially useful, understandable
patterns in large datasets
(2) The analysis of (often large) observational
data sets to find unsuspected relationships and
to summarize the data in novel ways that are
both understandable and useful to the data
owner
Association Rules
• Given: (1) database of transactions, (2) each transaction is
a list of items (purchased by a customer in a visit)
• Find: all association rules that satisfy user-specified
minimum support and minimum confidence interval
• Example: 30% of transactions that contain beer also
contain diapers; 5% of transactions contain these items
– 30%: confidence of the rule
– 5%: support of the rule
• We are interested in finding all rules rather than verifying if
a rule holds
Problem Decomposition
1. Find all sets of items that have
minimum support (frequent itemsets)
2. Use the frequent itemsets to generate
the desired rules
Mining Frequent Itemsets
• Apriori
– Key idea: A subset of a frequent itemset must
also be a frequent itemset (anti-monotonicity)
• Max-miner:
– Idea: Instead of checking all subsets of a long
pattern try to detect long patterns early
FP-tree
• Compress a large database into a compact,
Frequent-Pattern tree (FP-tree) structure
– highly condensed, but complete for frequent
pattern mining
• Create the tree and then run recursively the
algorithm over the tree (conditional base for
each item)
Association Rules
• Multi-level association rules: each attribute
has a hierarchy. Find rules per level or at
different levels
• Quantitative association rules
– Numerical attributes
• Other methods to find correlation:
– Lift, correlation coefficient
corrA, B 
P( A B)
P( A) P( B)
Major Clustering Approaches
• Partitioning algorithms: Construct various partitions and then
evaluate them by some criterion
• Hierarchical algorithms: Create a hierarchical decomposition of
the set of data (or objects) using some criterion
• Density-based algorithms: based on connectivity and density
functions
• Model-based: A model is hypothesized for each of the clusters
and the idea is to find the best fit of that model to each other
Partitioning Algorithms: Basic
Concept
• Partitioning method: Construct a partition of a database D of n objects into a
set of k clusters
• Given a k, find a partition of k clusters that optimizes the chosen partitioning
criterion
– Global optimal: exhaustively enumerate all partitions
– Heuristic methods: k-means and k-medoids algorithms
– k-means (MacQueen’67): Each cluster is represented by the center of the
cluster
– k-medoids or PAM (Partition around medoids) (Kaufman &
Rousseeuw’87): Each cluster is represented by one of the objects in the
cluster
Optimization problem
• The goal is to optimize a score function
• The most commonly used is the square error
criterion:
k
E    p mi
i  1 p  Ci
2
CLARANS (“Randomized” CLARA)
• CLARANS (A Clustering Algorithm based on Randomized
Search) (Ng and Han’94)
• CLARANS draws sample of neighbors dynamically
• The clustering process can be presented as searching a graph
where every node is a potential solution, that is, a set of k
medoids
• If the local optimum is found, CLARANS starts with new
randomly selected node in search for a new local optimum
• It is more efficient and scalable than both PAM and CLARA
Hierarchical Clustering
• Use distance matrix as clustering criteria. This
method does not require the number of clusters k as an
input, but needs a termination condition
Step 0
a
Step 1
Step 2 Step 3 Step 4
ab
b
abcde
c
cde
d
de
e
Step 4
agglomerative
(AGNES)
Step 3
Step 2 Step 1 Step 0
divisive
(DIANA)
HAC
• Different approaches to merge clusters:
–
–
–
–
Min distance
Average distance
Max distance
Distance of the centers
BIRCH
• Birch: Balanced Iterative Reducing and Clustering using
Hierarchies, by Zhang, Ramakrishnan, Livny
(SIGMOD’96)
• Incrementally construct a CF (Clustering Feature) tree, a
hierarchical data structure for multiphase clustering
– Phase 1: scan DB to build an initial in-memory CF tree (a
multi-level compression of the data that tries to preserve the
inherent clustering structure of the data)
– Phase 2: use an arbitrary clustering algorithm to cluster the
leaf nodes of the CF-tree
CURE (Clustering Using
REpresentatives )
• CURE: proposed by Guha, Rastogi & Shim, 1998
– Stops the creation of a cluster hierarchy if a level consists of
k clusters
– Uses multiple representative points to evaluate the distance
between clusters, adjusts well to arbitrary shaped clusters
and avoids single-link effect
Density-Based Clustering
Methods
• Clustering based on density (local cluster criterion), such as
density-connected points
• Major features:
–
–
–
–
Discover clusters of arbitrary shape
Handle noise
One scan
Need density parameters as termination condition
• Several interesting studies:
–
–
–
–
DBSCAN: Ester, et al. (KDD’96)
OPTICS: Ankerst, et al (SIGMOD’99).
DENCLUE: Hinneburg & D. Keim (KDD’98)
CLIQUE: Agrawal, et al. (SIGMOD’98)
Model based clustering
• Assume data generated from K probability
distributions
• Typically Gaussian distribution Soft or
probabilistic version of K-means clustering
• Need to find distribution parameters.
• EM Algorithm
Classification
• Given old data about customers and payments,
predict new applicant’s loan eligibility.
Previous customers
Age
Salary
Profession
Location
Customer type
Classifier
Decision rules
Salary > 5 L
Prof. = Exec
New applicant’s data
Good/
bad
Decision trees
• Tree where internal nodes are simple decision rules
on one or more attributes and leaf nodes are
predicted class labels.
Salary < 1 M
Prof = teaching
Good
Bad
Age < 30
Bad
Good
Building tree
GrowTree(TrainingData D)
Partition(D);
Partition(Data D)
if (all points in D belong to the same class) then
return;
for each attribute A do
evaluate splits on attribute A;
use best split found to partition D into D1 and
D2;
Partition(D1);
Partition(D2);
Split Criteria
• Select the attribute that is best for classification.
• Information Gain:
k
Entropy( S )  pi log pi
i 1
r
Gain( S , S1..S r )  Entropy( S )  
j 1
Sj
S
Entropy( S j )
• Gini Index:
Gini(D) = 1 -  pj2
Ginisplit(D) = n1* gini(D1) + n2* gini(D2)
n
n
SLIQ (Supervised Learning In Quest)
• Decision-tree classifier for data mining
• Design goals:
–
–
Able to handle large disk-resident training sets
No restrictions on training-set size
Bayesian Classification
• Probabilistic approach based on Bayes
theorem:
P(D | h)P(h)
P(h | D) 
P(D)
• MAP (maximum posteriori) hypothesis
h
 arg max P(h | D)  arg max P(D | h)P(h).
MAP hH
hH
Bayesian Belief Networks (I)
Age
FamilyH
(FH, A) (FH, ~A)(~FH, A)(~FH, ~A)
Diabetes
Mass
M
0.8
0.5
0.7
0.1
~M
0.2
0.5
0.3
0.9
The conditional probability table
for the variable Mass
Insulin
Glucose
Bayesian Belief Networks