Transcript The Landmark Model: An Instance Selection Method for Time Series Data

The Landmark Model: An Instance
Selection Method for Time Series Data
C.-S. Perng, S. R. Zhang, and D. S. Parker
Instance Selection and Construction for Data Mining,
Chapter 7, pp. 113-130
Cho, Dong-Yeon
Introduction

Complexity
 Patterns: continuous time series segments with
particular features
 The reflection of events in time series is better
represented by patterns.
 The complexity of processing patterns
 The
number of all possible segments for a time series of length
N is N(N+1)/2.
 A simple inspection of each of these segments takes O(N3).
 Good instance selection algorithms are especially
helpful here, since they can greatly reduce complexity
by reducing the volume of data.

Similarity Model
 Euclidian distance does not match human intuition.
 1,2,3,4,3
and 3,4,5,6,5
 Previous works
 None
of these proposed techniques supports a similarity model
that can both capture the similarity and support efficient
pattern querying of time series.

Pattern Representation
 Two formats for temporal association rules to verify the
cause-effect relation
association: C1,…,Cn  E1,…,Em
 Backward association: C1,…,Cn  E1,…,Em
 Forward
 Association rules can be either formulated as
hypotheses and verified with data, or be discovered by
data mining process.
 It is sill not clear what kind of segments can
represented event.
 What
is the basic vocabulary for spelling association rule?

Noise Removal and Data Smoothing
 Commonly-used smoothing techniques, such as moving
averages, often lag or miss the most significant peaks and
bottoms.
 These
peaks and bottoms can be very meaningful, and smoothing
or removing them can lose a great deal of information.
 Little previous work takes smoothing as an integral part of
the process of pattern definition, index construction, and
query processing.
The Landmark Data Model and
Similarity Model

The Landmark Concept
 Episodic memory: human and animals depend on
landmarks in organizing their spatial memory
 Landmarks: (times, events)
 Using
landmarks instead of the raw data for processing
 N-th order landmark of a curve if the N-th order derivative is 0.
 Local maxima, local minima, and inflection points
 Tradeoff
 The
more different types of landmarks in use, the more
accurately a time series will be represented.
 Using fewer landmarks will result in storage savings and
smaller index trees.
 Stock market data
 Almost
half of the record
 The normalized error is reasonably small when the curve is
reconstructed from the landmarks.
 The more volatile the time series, the less significant the
higher-order landmarks.

Smoothing
 Minimal Distance/Percentage Principle (MDPP)
 A minimal
distance D and a minimal percentage P
 Remove landmarks (xi, yi) and (xi+1, yi+1) if
xi 1  xi  D and
| yi 1  yi |
P
(| yi |  | yi 1 |) / 2
 The effect of the MDPP
 Normalized error generated by the MDPP and DFT

Transformations
 Six kinds of transformations
 Shifting:
SHk(f) such that SHk(f(t))=f(t)+k where k is a constant.
 Uniform Amplitude Scaling: UASk(f) such that UASk(f(t))=kf(t)
where k is a constant.
 Uniform Time Scaling: UTSk(f) such that UTSk(f(t))=f(kt)
where k is a positive constant.
 Uniform Bi-scaling: UBSk(f) such that UBSk(f(t))=kf(t/k) where
k is a positive constant.
 Time Warping: TWg(f) such that TWg(f(t))=f(g(t)) where g is a
positive and monotonically increasing.
 Non-uniform Amplitude Scaling: NASg(f) such that
NASg(f(t))=g(t) where for every t, g´(t)=0 if and only if f´(t)=0.
 The more transformation included in a similarity model,
the more powerful the similarity model.
 These transformations can be composed to form new
transformations.
composition order is flexible: Fu  Gv  Gu  Fv
 The composition is idempotent: Fw  Fu  Fv
 The
 Two time series are defined to be similar if they differ
only by a transform.

Landmark Similarity
 Dissimilarity measure
two sequences of landmarks L= L1,…,Ln and L´=
L´1,…,L´n where Li=(xi, yi) and L´i=(x´i, y´i), the distance
between the k-th landmark is defined by
 k ( L, L)  ( ktime ( L, L),  kamp ( L, L)) where
 Given
 | ( xk  xk 1 )  ( xk  xk 1 ) |
if 1  k  n

 ktime ( L, L)   (| xk  xk 1 |  | xk  xk 1 |) / 2
0
otherwise
if yk  yk
0

 kamp ( L, L)   | yk  yk |
otherwise
 (| yk |  | yk |) / 2
 The
distance between the two sequences is
( L, L)   time ( L, L) ,  amp ( L, L)  ( time ,  amp )
time
 We define (
,  amp )  ( time,  amp ) if  time   time and  amp   amp


 A land mark similarity measure is a binary relation on
time series segments defined by a 5-tuple
LSM=D,P,T,time,amp.
 Given
two time series sequences s1
and s2, let L1 and L2 be the landmark
sequences after MDPP(D, P)
smoothing.
 (s1, s2)LMS if and only if |L1|=|L2|
and there exist two parameterized
transformations T1 and T2 of T
whose dissimilarity satisfies
time(T1(L1), T2(L2)) < time and
amp(T1(L1), T2(L2)) < amp.
Data Representation

Family of Time Series Segments
 Equivalent under the six transformations
 Replacing
naïve landmark coordinates with various features of
landmarks that are invariant under these transformations
 F = {y, h, v, hr, vr, vhr, pv}
hi=xi-xi-1 vi=yi-yi-1 hri=hi+1/hi vri=vi+1/vi vhri=vi/ hi pvi=vi/yi
 Invariant features under transformations
Conclusion

Landmark Model
 An instance selection system for time series
 This integrates similarity measures, data representation and
smoothing techniques in a single framework.
 Minimal
Distance/Percentage Principle (MDPP): The smoothing
method for the Landmark Model
 This also supports a generalized similarity model which can
ignore differences corresponding to six transformations.
 Intuitive to human