Transcript cs412slides

Data Mining:
Concepts and Techniques
— Chapter 8 —
8.2 Mining time-series data
Jiawei Han and Micheline Kamber
Department of Computer Science
University of Illinois at Urbana-Champaign
www.cs.uiuc.edu/~hanj
©2010 Jiawei Han and Micheline Kamber. All rights reserved.
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April 10, 2016
Data Mining: Concepts and Techniques
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Mining Time-Series Data
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A time series is a sequence of data points, measured typically at
successive times, spaced at (often uniform) time intervals
Time series analysis: A subfield of statistics, comprises methods
that attempt to understand such time series, often either to
understand the underlying context of the data points or to make
forecasts (or predictions)
Methods for time series analyses
 Frequency-domain methods: Model-free analyses, well-suited to
exploratory investigations
 spectral analysis vs. wavelet analysis
 Time-domain methods: Auto-correlation and cross-correlation
analysis
 Motif-based time-series analysis
Applications
 Scientific: experiment results
 Financial: stock price, inflation
 Meteorological: precipitation
 Industry: power consumption
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Mining Time-Series Data
Regression Analysis
Trend Analysis
Similarity Search in Time Series Data
Motif-Based Search and Mining in Time
Series Data
Summary
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Time-Series Data Analysis: Prediction &
Regression Analysis
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(Numerical) prediction is similar to classification
 construct a model
 use model to predict continuous or ordered value for a given input
Prediction is different from classification
 Classification refers to predict categorical class label
 Prediction models continuous-valued functions
Major method for prediction: regression
 model the relationship between one or more independent or
predictor variables and a dependent or response variable
Regression analysis
 Linear and multiple regression
 Non-linear regression
 Other regression methods: generalized linear model, Poisson
regression, log-linear models, regression trees
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What is Regression?
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Modeling the relationship between one response
variable and one or more predictor variables
Analyzing the confidence of the model
E.g, height v.s weight
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Regression Yields Analytical Model
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Discrete data points →Analytical model
 General relationship
 Easy calculation
 Further analysis
Application - Prediction
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Application - Detrending
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Obtain the trend for irregular data series
Subtract trend
Reveal oscillations
trend
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Linear Regression - Single Predictor
w1
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Model is linear
y: response
variable
y = w0 + w1 x
where w0 (y-intercept) and w1
(slope) are regression
coefficients
w0
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Method of least squares:
| D|
w1 
 ( x  x )( y
i 1
i
i
 y)
| D|
 (x  x )
i 1
2
x: predictor
variable
w  y w x
0
1
i
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Linear Regression – Multiple Predictor
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Training data is of the form (X1, y1), (X2, y2),…, (X|D|,
y|D|)
E.g., for 2-D data or
y = w0 + w1 x1+ w2 x2
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y
Solvable by
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Extension of least square method
x2
(XTX ) W=Y →W = (XTX ) -1Y
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Commercial software (SAS, S-Plus)
x1
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Nonlinear Regression with Linear Method
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Polynomial regression model
2
3
 E.g., y = w0 + w1 x + w2 x + w3 x
Let x2 = x2, x3= x3
y = w0 + w1 x + w2 x2 + w3 x3
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Log-linear regression model
 E. g., y = exp(w0 + w1 x + w2 x2 + w3 x3 )
Let y’=log(y)
y’= w0 + w1 x + w2 x2 + w3 x3
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Generalized Linear Regression
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Response y
 Distribution function in the exponential family
 Variance of y depends on E( y), not a constant
E( y) = g-1( w0 + w1 x + w2 x2 + w3 x3 )
Examples
 Logistic regression (binomial regression): probability of
some event occurring
 Poisson regression: number of customers
 …
References: Nelder and Wedderburn, 1972; McCullagh and
Nelder, 1989
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Regression Tree (Breiman et al., 1984)
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Partition the domain space
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Leaf: (1) a continuous-valued
prediction; (2) average value
Figure source: http://www.stat.cmu.edu/~cshalizi/350-2006/lecture-10.pdf
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Model Tree (Quinlan, 1992)
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Leaf – a linear equation
More general than regression tree
Figure source: http://datamining.ihe.nl/research/model-trees.htm
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Regression Trees and Model Trees
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Regression tree: proposed in CART system (Breiman et al. 1984)
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CART: Classification And Regression Trees
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Each leaf stores a continuous-valued prediction
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It is the average value of the predicted attribute for the training tuples
that reach the leaf
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Model tree: proposed by Quinlan (1992)
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Each leaf holds a regression model—a multivariate linear equation for the
predicted attribute
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A more general case than regression tree
Regression and model trees tend to be more accurate than linear
regression when the data cannot be represented well by a simple
linear model
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Predictive Modeling in
Multidimensional Databases
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Predictive modeling: Predict data values or construct
generalized linear models based on the database data
One can only predict value ranges or category
distributions
Method outline
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Minimal generalization
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Attribute relevance analysis
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Generalized linear model construction
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Prediction
Determine the major factors which influence the prediction
 Data relevance analysis: uncertainty measurement,
entropy analysis, expert judgment, etc.
Multi-level prediction: drill-down and roll-up analysis
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Predictive Modeling in Multidimensional Databases
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Predictive modeling: Predict data values or construct
generalized linear models based on the database data
One can only predict value ranges or category
distributions
Method outline:
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Minimal generalization
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Attribute relevance analysis
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Generalized linear model construction
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Prediction
Determine the major factors which influence the
prediction
 Data relevance analysis: uncertainty measurement,
entropy analysis, expert judgment, etc.
Multi-level prediction: drill-down and roll-up analysis
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Prediction: Numerical Data
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References
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Nelder, J.A. and Wedderburn, R.W.M. (1972). Generalized linear models.
Journal of the Royal Statistical Society A, 135, 370-384.
C. Chatfield. The Analysis of Time Series: An Introduction, 3rd ed. Chapman &
Hall, 1984.
McCullagh, P. and Nelder, J.A. (1989). Generalized linear models, 2nd ed.
Chapman and Hall, London.
Breiman L, Friedman JH, Olshen RA, Stone CJ. (1984). Classification and
Regression Trees. Chapman &Hall (Wadsworth, Inc.): New York.
Quinlan, J. R. (1992). Learning with continuous classes. In: Adams, , Sterling,
(Eds.), Proceedings of artificial intelligence'92, World Scientific, Singapore. pp.
343-348.
Acknowledgment

This presentation integrates Xiaopeng Li’s slides in his CS 512 class
presentation
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Mining Time-Series Data
Regression Analysis
Trend Analysis
Similarity Search in Time Series Data
Motif-Based Search and Mining in Time
Series Data
Summary
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A time series can be illustrated as a time-series graph
which describes a point moving with the passage of time
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Categories of Time-Series Movements
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Categories of Time-Series Movements
 Long-term or trend movements (trend curve): general direction in
which a time series is moving over a long interval of time
 Cyclic movements or cycle variations: long term oscillations about
a trend line or curve
 e.g., business cycles, may or may not be periodic
 Seasonal movements or seasonal variations
 i.e, almost identical patterns that a time series appears to
follow during corresponding months of successive years.
 Irregular or random movements
Time series analysis: decomposition of a time series into these four
basic movements
 Additive Modal: TS = T + C + S + I
 Multiplicative Modal: TS = T  C  S  I
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Estimation of Trend Curve
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The freehand method
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Fit the curve by looking at the graph
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Costly and barely reliable for large-scaled data mining
The least-square method
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Find the curve minimizing the sum of the squares of
the deviation of points on the curve from the
corresponding data points
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The moving-average method
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Moving Average
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Moving average of order n
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Smoothes the data
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Eliminates cyclic, seasonal and irregular movements
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Loses the data at the beginning or end of a series
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Sensitive to outliers (can be reduced by weighted
moving average)
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Trend Discovery in Time-Series (1):
Estimation of Seasonal Variations
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Seasonal index
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Set of numbers showing the relative values of a variable during
the months of the year
E.g., if the sales during October, November, and December are
80%, 120%, and 140% of the average monthly sales for the
whole year, respectively, then 80, 120, and 140 are seasonal
index numbers for these months
Deseasonalized data
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Data adjusted for seasonal variations for better trend and cyclic
analysis
Divide the original monthly data by the seasonal index numbers
for the corresponding months
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Seasonal Index
Seasonal Index
160
140
120
100
80
60
40
20
0
1
2
3
4
5
6
7
Month
8
9
10
11
12
Raw data from
http://www.bbk.ac.uk/man
op/man/docs/QII_2_2003
%20Time%20series.pdf
April 10, 2016
Data Mining: Concepts and Techniques
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Trend Discovery in Time-Series (2)
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Estimation of cyclic variations
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Estimation of irregular variations
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If (approximate) periodicity of cycles occurs, cyclic
index can be constructed in much the same manner as
seasonal indexes
By adjusting the data for trend, seasonal and cyclic
variations
With the systematic analysis of the trend, cyclic, seasonal,
and irregular components, it is possible to make long- or
short-term predictions with reasonable quality
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Mining Time-Series Data
Regression Analysis
Trend Analysis
Similarity Search in Time Series Data
Motif-Based Search and Mining in Time
Series Data
Summary
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Similarity Search in Time-Series Analysis
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Normal database query finds exact match
Similarity search finds data sequences that differ only
slightly from the given query sequence
Two categories of similarity queries
 Whole matching: find a sequence that is similar to the
query sequence
 Subsequence matching: find all pairs of similar
sequences
Typical Applications
 Financial market
 Market basket data analysis
 Scientific databases
 Medical diagnosis
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Data Transformation
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Many techniques for signal analysis require the data to
be in the frequency domain
Usually data-independent transformations are used
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The transformation matrix is determined a priori
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discrete Fourier transform (DFT)
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discrete wavelet transform (DWT)
The distance between two signals in the time domain is
the same as their Euclidean distance in the frequency
domain
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Discrete Fourier Transform
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DFT does a good job of concentrating energy in the first
few coefficients
If we keep only first a few coefficients in DFT, we can
compute the lower bounds of the actual distance
Feature extraction: keep the first few coefficients (F-index)
as representative of the sequence
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DFT (continued)
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Parseval’s Theorem
n 1
n 1
t 0
f 0
2
2
|
x
|

|
X
|
 t  f
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The Euclidean distance between two signals in the time
domain is the same as their distance in the frequency
domain
Keep the first few (say, 3) coefficients underestimates the
distance and there will be no false dismissals!
n
3
| S[t ]  Q[t ] |    | F (S )[ f ]  F (Q)[ f ] |  
2
t 0
2
f 0
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Multidimensional Indexing in Time-Series
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Multidimensional index construction
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Constructed for efficient accessing using the first few
Fourier coefficients
Similarity search
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Use the index to retrieve the sequences that are at
most a certain small distance away from the query
sequence
Perform post-processing by computing the actual
distance between sequences in the time domain and
discard any false matches
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Subsequence Matching
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Break each sequence into a set of
pieces of window with length w
Extract the features of the
subsequence inside the window
Map each sequence to a “trail” in
the feature space
Divide the trail of each sequence
into “subtrails” and represent each
of them with minimum bounding
rectangle
Use a multi-piece assembly
algorithm to search for longer
sequence matches
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Analysis of Similar Time Series
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Enhanced Similarity Search Methods
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Allow for gaps within a sequence or differences in offsets
or amplitudes
Normalize sequences with amplitude scaling and offset
translation
Two subsequences are considered similar if one lies within
an envelope of  width around the other, ignoring outliers
Two sequences are said to be similar if they have enough
non-overlapping time-ordered pairs of similar
subsequences
Parameters specified by a user or expert: sliding window
size, width of an envelope for similarity, maximum gap,
and matching fraction
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Steps for Performing a Similarity Search
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Atomic matching
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Window stitching
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Find all pairs of gap-free windows of a small length that
are similar
Stitch similar windows to form pairs of large similar
subsequences allowing gaps between atomic matches
Subsequence Ordering
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Linearly order the subsequence matches to determine
whether enough similar pieces exist
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Similar Time Series Analysis
VanEck International Fund
Fidelity Selective Precious Metal and Mineral Fund
Two similar mutual funds in the different fund group
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Query Languages for Time Sequences
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Time-sequence query language
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Should be able to specify sophisticated queries like
Find all of the sequences that are similar to some sequence in class
A, but not similar to any sequence in class B
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Should be able to support various kinds of queries: range queries,
all-pair queries, and nearest neighbor queries
Shape definition language
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Allows users to define and query the overall shape of time
sequences
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Uses human readable series of sequence transitions or macros
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Ignores the specific details
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E.g., the pattern up, Up, UP can be used to describe
increasing degrees of rising slopes
Macros: spike, valley, etc.
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Mining Time-Series Data
Regression Analysis
Trend Analysis
Similarity Search in Time Series Data
Motif-Based Search and Mining in Time
Series Data
Summary
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Sequence Distance
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A function that measures the differentness of two
sequences (of possibly unequal length)
Example: Euclidean Distance between TS Q,C
D(Q, C ) 

n
2
(
q

c
)
i
i
i 1
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Motif: Basic Concepts
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What is a motif? A previously unknown, frequently
occurring sequential pattern
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Match: Given subsequences Q,C ⊆ T,
C is a match for Q iff D(Q, C )  R for some R
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Non-Trivial Match: C = T[p..*], Q = T[q..*] and C match
Q. If p = q or ∄ non-match N = T[s..*] such that s between
p,q then match is non-trivial.
(i.e. C,Q must be separated by a non-match)
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1-Motif: the subsequence with most non-trivial matches
(least variance decides ties)
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k-Motif: Ck such that D(Ck,Ci) > 2R ∀i ∈ [1,k)
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SAX: Symbolic Aggregate approXimation
Dim. Reduction/Compression
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“Symbolic Aggregate approXimation”
SAX : ℝ → ∑
SAX :
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↦ ccbaabbbabcbcb
Essentially an alphabet over the Piecewise Aggregate
Approximation (PAA) rank
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Faster, simpler, more compression, yet on par with DFT,
DWT and other dim. reductions
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SAX Illustration
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SAX Algorithm
Parameters: alphabet size, word (segment) length (or output
rate)
1.
Select probability distribution for TS
2.
z-Normalize TS
3.
PAA: Within each time interval, calculate aggregated value
(mean) of the segment
4.
Partition TS range by equal-area partitioning the PDF into
n partitions (eq. freq. binning)
5.
Label each segment with arank ∈∑ for aggregate’s
corresponding partition rank
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Finding Motifs in a Time Series
EMMA Algorithm: Finds 1-(k-)motif of fixed length n
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SAX Compression (Dim. Reduction)
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Possible to store D(i,j) ∀(i,j) ∈ ∑∑
Allows use of various distance measures (Minkowski, Dynamic Time
Warping)
Multiple Tiers
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Tier 1: Uses sliding window to hash length-w SAX subsequences
(aw addresses, total size O(m)).
Bucket B with most collisions & buckets with
MINDIST(B) < R form neighborhood of B.
Tier 2: Neighborhood is pruned using more precise ADM
algorithm. Ni with max. matches is 1-motif. Early stop if |ADM
matches| > maxk>i(|neighborhoodk|)
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Hashing
…
2 4 2 0 1 1 2 1 0 2
…
5
2 2 2 2 1 1 2 2 3 2
…c e c a b b c b a c … c c c c b b c c d c
…
2 4 2 0 1 1 2 1 0 2
5
5
…
c e c a b b c b a c …
…
…
w
…
…
n
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Classification in Time Series
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Application: Finance, Medicine
0
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200
400
600
800
1000
1200
1-Nearest Neighbor
 Pros: accurate, robust, simple
 Cons: time and space complexity (lazy learning);
results are not interpretable
stinging nettles
false nettles
Shapelet Dictionary
5.
1
Shapelet
I
0 yes
false nettles
I
Leaf Decision Tree
no
1
stinging
nettles
false nettles
stinging nettles
Testing the utility of a candidate shapelet
Information gain
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Arrange the time series objects
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based on the distance from candidate
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Find the optimal split point (maximal information gain)
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Pick the candidate achieving best utility as the shapelet
candidate
0
Split Point
false nettles
stinging nettles
false nettles
Shapelet Dictionary
5.
1
Shapelet
I
I
0
yes
false nettles
Classification
Leaf Decision Tree
no
1
stinging nettles
false nettles
stinging nettles
Mining Time-Series Data
Regression Analysis
Trend Analysis
Similarity Search in Time Series Data
Motif-Based Search and Mining in Time
Series Data
Summary
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Summary
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Time series analysis is an important research field
in data mining
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Regression Analysis
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Trend Analysis
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Similarity Search in Time Series Data
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Motif-Based Search and Mining in Time Series
Data
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References on Time-Series Similarity Search
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R. Agrawal, C. Faloutsos, and A. Swami. Efficient similarity search in sequence databases. FODO’93
(Foundations of Data Organization and Algorithms).
R. Agrawal, K.-I. Lin, H.S. Sawhney, and K. Shim. Fast similarity search in the presence of noise,
scaling, and translation in time-series databases. VLDB'95.
R. Agrawal, G. Psaila, E. L. Wimmers, and M. Zait. Querying shapes of histories. VLDB'95.
C. Faloutsos, M. Ranganathan, and Y. Manolopoulos. Fast subsequence matching in time-series
databases. SIGMOD'94.
J. Lin, E. Keogh, S. Lonardi, and B. Chiu, “A Symbolic Representation of Time Series, with Implications
for Streaming Algorithms”, Data Mining and Knowledge discovery, 2003
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P. Patel, E. Keogh, J. Lin, and S. Lonardi, “Mining Motifs in Massive Time Series Databases”, ICDM’02
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D. Rafiei and A. Mendelzon. Similarity-based queries for time series data. SIGMOD'97.
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Y. Moon, K. Whang, W. Loh. Duality Based Subsequence Matching in Time-Series Databases, ICDE’02
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B.-K. Yi, H. V. Jagadish, and C. Faloutsos. Efficient retrieval of similar time sequences under time
warping. ICDE'98.
B.-K. Yi, N. Sidiropoulos, T. Johnson, H. V. Jagadish, C. Faloutsos, and A. Biliris. Online data mining for
co-evolving time sequences. ICDE'00.
D. Shasha and Y. Zhu. High Performance Discovery in Time Series: Techniques and Case Studies,
SPRINGER, 2004
L. Ye and E. Keogh, “Time Series Shapelets: A New Primitive for Data Mining”, KDD’09
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Data Mining: Concepts and Techniques
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