1. Locally adaptive dimensionality reduction for indexing large time series databases
Keogh, E.,  Chakrabarti, K., Pazzani, M. & Mehrotra, S. (2001). Locally adaptive dimensionality reduction for indexing large time series databases. In proceedings of ACM SIGMOD Conference on Management of Data, May. pp

2. Motivating Applications
Time Series Similarity Retrieval
Typical query: Find time series objects similar to a given pattern
Example Applications
Medical Diagnosis: doctor searching for pattern (that implies heart irregularity) in ECG database
Financial Data: stock analyst searching for stock price pattern for prediction
Time-Series Mining: similarity search a component inside mining algorithm
Other Applications
Scientific, Spatial/Spatio-temporal
A Time Series
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.19
.28
.38
.30
.25
.32
High Dimensional Vector Representation of Time Series

3. Similarity Search in Time Series Data
Query Q
n datapoints
Database
n datapoints
Distance
0.98
0.07
0.21
0.43
Rank 4 1 2 3 S Q
Euclidean Distance between
two time series Q = {q1, q2, …, qn}
and S = {s1, s2, …, sn}

4. Mapping to Multidimensional Space
Database
n datapoints S1 S1
Query Q
n datapoints Q S2 S2 S4 S3 S3
n-dimensional space
Index the n-d space using a multidimensional index structure to avoid slow sequential scanning
n ~ ; need to reduce dimensionality from n to n’ that can be handled by index structure S4

5. Dimensionality Reduction Techniques for Time Series Data
eigenwave 0
eigenwave 1
eigenwave 2
eigenwave 3
eigenwave 4
eigenwave 5
eigenwave 6
eigenwave 7
Korn, Jagadish, Faloutsos 1997 X X'
PCA 20 40 60 80
100
120
140 20 40 60 80
100
120
140 X X'
DFT X X'
DWT 20 40 60 80
100
120
140
Haar 0
Haar 1
Haar 2
Haar 3
Haar 4
Haar 5
Haar 6
Haar 7 1 2 3 4 5 6 7
Agrawal, Faloutsos, Swami 1993
Chan & Fu 1999

6. Piecewise Aggregate Approximation (PAA)
value
axis
time axis
Original time series
(n-dimensional vector)
S={s1, s2, …, sn}
sv1
sv2
sv3
sv4
sv5
sv6
sv7
sv8
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)

7. They are all global
All the dimensionality reduction techniques (DFT, DWT, PCA and PAA) are global - they choose a common reduced-representation for all items in the database
Is it possible to devise a representation that adapts locally to each time-series item and chooses the best representation for that item?
Researchers considered such a representation to be infeasible as it “does not allow for indexing due to its irregularity” (Yi and Faloutsos, VLDB 2000)

8. Locally Adaptive Representation
sv1
sv2
sv3
sv4
sv5
sv6
sv7
sv8
n’-segment PAA representation
(n’-d vector)
S = {sv1 , sv2, …, svN }
Adaptive Piecewise Constant
Approximation (APCA)
sv1
sv2
sv3
sv4
sr1
sr2
sr3
sr4
n’/2-segment APCA representation
(n’-d vector)
S= { sv1, sr1, sv2, sr2, …, svM , srM }
(M is the number of segments = n’/2)

9. Is it any good?
Question 1: Does APCA approximate the original signal better than PAA when M = n’/2?
Question 2: Can we compute the APCA representation efficiently?
Question 3: Can we come up with a lower bounding distance measure for APCA?
Question 4: Can we index APCA?
Question 5: If all the above, how does it perform compared to other dimensionality reduction techniques?
PAA
APCA

10. Answer 1: APCA approximates original signal better than PAA
Reconstruction error PAA Reconstruction error APCA
Improvement factor =
1.69
3.77
1.21
1.03
3.02
1.75

11. Answer 2: APCA Representation can be computed efficiently
Near-optimal representation can be computed in O(nlog(n)) time
Optimal representation can be computed in O(n2M)

12. Answer 3: Lower Bounding Distance MeasureQ
D(Q,S)
Exact (Euclidean) distance D(Q,S)
Lower bounding distance DLB(Q,S) S Q’ Q
DLB(Q’,S)
DLB(Q’,S)

13. Is it any good? Question 4: Can we index APCA?
Question 1: Does APCA approximate the original signal better than PAA when M = n’/2?
Question 2: Can we compute the APCA representation efficiently?
Question 3: Can we come up with a lower bounding distance measure for APCA?
Question 4: Can we index APCA?
Question 5: If all the above, how does it perform compared to other dimensionality reduction techniques?
PAA
APCA

14. Can we index APCA? R1 R3 R2 R4 S6 S5 S1 S2 S3 S4 S8 S7 S9 R2 R3 R4 R1
What does indexability mean?
If we build the index on the 2M-dimensional APCA space, can we retrieve nearest neighbors to a query time series using the index?
Nearest neighbors based on the true distance, i.e., Euclidean distance in the original n-dimensional space R1 R3 R2 R4
2M-dimensional APCA space S6 S5 S1 S2 S3 S4 S8 S7 S9
Any feature-based index structure can used (e.g., R-tree, X-tree, Hybrid Tree) R2 R3 R4 R1 S3 S4 S5 S6 S7 S8 S9 S2 S1
The k-nearest neighbor search traverse the nodes of the multidimensional index structure in the order of the distance from the query.
We define the node distance as MINDIST which is the minimum distance from the query to any point in the node boundary.
In case of Query Point Movement, MinDist is computed as
the distance between the centroid of relevant points and
the point P which is defined as in this equation.

15. k-NN Algorithm Q R1 S7 R3 R2 R4 S1 S2 S3 S5 S4 S6 S8 S9
MINDIST(Q,R2)
MINDIST(Q,R3) R1 S7 R3 R2 R4 S1 S2 S3 S5 S4 S6 S8 S9 Q
MINDIST(Q,R4)
The k-nearest neighbor search traverse the nodes of the multidimensional index structure in the order of the distance from the query.
We define the node distance as MINDIST which is the minimum distance from the query to any point in the node boundary.
In case of Query Point Movement, MinDist is computed as
the distance between the centroid of relevant points and
the point P which is defined as in this equation.
Correctness criteria:
Distance DLB(Q, S) Q and APCA point:
DLB(Q, S) £ Euclidean (true) distance D(Q, original(S) )
Distance MINDIST(Q,R) between Q and MBR R of a node U of the index:
MINDIST(Q,R) £ Euclidean distance D(Q, original(S)) for any data item S under U

16. Index Modification for MINDIST Computation
APCA point S= { sv1, sr1, sv2, sr2, …, svM, srM }
smax3
smin3 R1
sv3 S2 S5 R3 S3
smax1
smin1
smax2
smin2 S1 S6 S4
sv1 R2
smax4
smin4 R4 S8
sv2 S9
sv4 S7
sr1
sr2
sr3
sr4
APCA rectangle S= (L,H) where
L= { smin1, sr1, smin2, sr2, …, sminM, srM } and
H = { smax1, sr1, smax2, sr2, …, smaxM, srM }

17. MBR Representation in time-value space
We can view the MBR R=(L,H) of any node U as two APCA representations
L= { l1, l2, …, l(N-1), lN } and H= { h1, h2, …, h(N-1), hN }
REGION 2
H= { h1, h2, h3, h4 , h5, h6 } h1 h2 h3 h4 h5 h6
time axis
value
axis l3 l4 l6 l5
REGION 1 l1 l2
REGION 3
L= { l1, l2, l3, l4 , l5, l6 }

18. Regions l(2i-1) h(2i-1) h2i l(2i-2)+1 h3 h5 h2 h4 h6 l3 l1 l2 l4 l6 l5
REGION i
l(2i-1)
h(2i-1)
h2i
l(2i-2)+1
M regions associated with each MBR; boundaries of ith region: h3 h1 h5 h2 h4 h6
value
axis
time axis l3 l1 l2 l4 l6 l5
REGION 1
REGION 3
REGION 2

19. Regions t1 t2 h3 l3 h5 l1 l5 l2 l4 h2 h4 h6 l6
ith region is active at time instant t if it spans across t
The value st of any time series S under node U at time instant t must lie in one of the regions active at t (Lemma 2)
REGION 2 t1 t2 h3
value
axis h1 l3
REGION 3 h5 l1 l5
REGION 1 l2 l4 h2 h4 h6 l6
time axis

20. MINDIST Computation t1 MINDIST(Q,R,t1) =min(MINDIST(Q, Region1, t1),
For time instant t, MINDIST(Q, R, t) = minregion G active at t MINDIST(Q,G,t)
MINDIST(Q,R,t1)
=min(MINDIST(Q, Region1, t1),
MINDIST(Q, Region2, t1))
=min((qt1 - h1)2 , (qt1 - h3)2 )
=(qt1 - h1)2 t1
REGION 2 h3 l3 h1
REGION 3 h5 l1 l5
REGION 1 l2 l4 h2 h4 h6
MINDIST(Q,R) = l6
Lemma3: MINDIST(Q,R) £ D(Q,C) for any time series C under node U

21. Other Queries
Range Search
Approximate Search

22. Answer 5: Comparison of APCA with other techniques (Pruning Power)
Dataset: Electrocardiogram data, 100,000 objects, n (query length) varying from 256 to 1024
Pruning Power = Number of objects examined for 1-NN query
Fourier
Wavelet/ PAA
APCA #
objects
examined
Original
dimensionality (n)
( )
Reduced
dimensionality (n’)
(16-64)

23. Comparison of APCA with other techniques (Index Performance)
Electrocardiogram data, Hybrid Tree to index reduced space
Linear Scan
Fourier
Wavelet/ PAA
APCA #
random
disk
access
CPU
time
(sec)
Original dimensionality (n)
( )
Reduced dimensionality (n’) (16-64) 22

24. Summary of APCA
APCA is a new dimensionality reduction technique for time series data that
Approximates the original data better by adapting to each data item “locally”
Can be indexed using a multidimensional index structure
Outperforms existing techniques by one to two orders of magnitude in terms of search performance
As of 2005, has been referenced well over 100 times, and implemented at least 20 times.