Review

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Review
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Time Series Data
A time series is a collection of observations
made sequentially in time.
25.1750 25.1750 25.2250 25.2500
25.2500 25.2750 25.3250 25.3500
25.3500 25.4000 25.4000 25.3250
25.2250 25.2000 25.1750 .. ..
24.6250 24.6750 24.6750 24.6250
24.6250 24.6250 24.6750 24.7500
value axis
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)

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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

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Euclidean model
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Dynamic Time WarpingBerndt, 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

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Dynamic Time WarpingBerndt, Clifford, 1994
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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

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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)

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Basic LCS Idea

• X 3, 2, 5, 7, 4, 8, 10, 7
• Y 2, 5, 4, 7, 3, 10, 8, 6
• LCS 2, 5, 7, 10

Sim(X,Y) LCS or Sim(X,Y) LCS /n
Longest Common Subsequence Edit Distance is
another possibility
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Indexing Time Series using GEMINI

• (GEneric Multimedia INdexIng)
• Extract a few numerical features, for a quick
  and dirty test

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GEMINI - Pictorially
eg,. std
eg, avg
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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

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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

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Feature Extraction

• How to extract the features? How to define the
  feature space?
• Fourier transform
• Wavelets transform
• Averages of segments (Histograms or APCA)

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Piecewise Aggregate Approximation (PAA)
Original time series (n-dimensional
vector) Ss1, s2, , sn
n-segment PAA representation (n-d vector) S
sv1 , sv2, , svn
PAA representation satisfies the lower bounding
lemma (Keogh, Chakrabarti, Mehrotra and Pazzani,
2000 Yi and Faloutsos 2000)
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Can we improve upon PAA?
n-segment PAA representation (n-d vector) S
sv1 , sv2, , svN
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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!

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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

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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

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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

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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

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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

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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

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Problem Decomposition

• 1. Find all sets of items that have minimum
  support (frequent itemsets)
• 2. Use the frequent itemsets to generate the
  desired rules

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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

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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)

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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

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What is Cluster Analysis?

• Cluster a collection of data objects
• Similar to one another within the same cluster
• Dissimilar to the objects in other clusters
• Cluster analysis
• Grouping a set of data objects into clusters
• Typical applications
• As a stand-alone tool to get insight into data
  distribution
• As a preprocessing step for other algorithms

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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

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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 (MacQueen67) Each cluster is
  represented by the center of the cluster
• k-medoids or PAM (Partition around medoids)
  (Kaufman Rousseeuw87) Each cluster is
  represented by one of the objects in the cluster

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Optimization problem

• The goal is to optimize a score function
• The most commonly used is the square error
  criterion

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CLARANS (Randomized CLARA)

• CLARANS (A Clustering Algorithm based on
  Randomized Search) (Ng and Han94)
• 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

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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

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HAC

• Different approaches to merge clusters
• Min distance
• Average distance
• Max distance
• Distance of the centers

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BIRCH

• Birch Balanced Iterative Reducing and Clustering
  using Hierarchies, by Zhang, Ramakrishnan, Livny
  (SIGMOD96)
• 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

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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

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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. (KDD96)
• OPTICS Ankerst, et al (SIGMOD99).
• DENCLUE Hinneburg D. Keim (KDD98)
• CLIQUE Agrawal, et al. (SIGMOD98)

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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

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Classification

• Given old data about customers and payments,
  predict new applicants loan eligibility.

Previous customers
Classifier
Decision rules
Age Salary Profession Location Customer type
Salary 5 L
Good/ bad
Prof. Exec
New applicants data
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Decision trees

• Tree where internal nodes are simple decision
  rules on one or more attributes and leaf nodes
  are predicted class labels.

Salary
Prof teaching
Age
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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)

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Split Criteria

• Select the attribute that is best for
  classification.
• Information Gain
• Gini Index
• Gini(D) 1 - ?? pj2

Ginisplit(D) n1 gini(D1) n2 gini(D2)
n n
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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

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Bayesian Classification

• Probabilistic approach based on Bayes theorem
• MAP (maximum posteriori) hypothesis

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Naïve Bayes Classifier (I)

• A simplified assumption attributes are
  conditionally independent
• Greatly reduces the computation cost, only count
  the class distribution.

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Bayesian Belief Networks (I)
Age
FamilyH
(FH, A)
(FH, A)
(FH, A)
(FH, A)
M
0.7
0.8
0.5
0.1
Diabetes
Mass
M
0.3
0.2
0.5
0.9
The conditional probability table for the
variable Mass
Insulin
Glucose
Bayesian Belief Networks