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Time Series DataA time series is a collection of observations made sequentially in time.

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value

axis

time axis

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

n datapoints

Database

Distance

Rank

Query Q

4

0.98

0.07

1

Euclidean Distance between

two time series Q = {q1, q2, …, qn}

and S = {s1, s2, …, sn}

Q

2

0.21

S

3

0.43

Euclidean modelDynamic 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’

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 = 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

Indexing Time Series using ‘GEMINI’

(GEneric Multimedia INdexIng)

Extract a few numerical features, for a ‘quick and dirty’ test

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)

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axis

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time axis

Piecewise Aggregate Approximation (PAA)Original time series

(n-dimensional vector)

S={s1, 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)

sv6

sv1

sv7

sv5

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Adaptive Piecewise Constant

Approximation (APCA)

sv3

n’/2-segment APCA representation

(n’-d vector)

S= { sv1, sr1, sv2, sr2, …, svM , srM }

(M is the number of segments = n’/2)

sv1

sv2

sv4

sr1

sr2

sr3

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Can we improve upon PAA?n’-segment PAA representation

(n’-d vector)

S = {sv1 ,sv2, …, svN}

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

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:

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

Step 1

Step 2

Step 3

Step 4

agglomerative

(AGNES)

a

a b

b

a b c d e

c

c d e

d

d e

e

divisive

(DIANA)

Step 3

Step 2

Step 1

Step 0

Step 4

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

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

Classifier

Decision rules

Age

Salary

Profession

Location

Customer type

Salary > 5 L

Good/

bad

Prof. = Exec

New applicant’s data

Good

Bad

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

Age < 30

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:
- 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:
- MAP (maximum posteriori) hypothesis

Bayesian Belief Networks (I)

Age

FamilyH

(FH, A)

(FH, ~A)

(~FH, A)

(~FH, ~A)

M

0.7

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0.1

Diabetes

Mass

~M

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The conditional probability table for the variable Mass

Insulin

Glucose

Bayesian Belief Networks

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