A literature review on time series representations, distance measures and indexing techniques used for pattern mining in very large time series data.

1. M.Tech Seminar Presentation
2014-15
Presented by
Jitesh Khandelwal
10211015
Guided by
Dr. Durga Toshniwal
IIT Roorkee

2. Finance
e.g. Stock prices Medical
e.g. Electro Cardio Grams
Marketing
e.g. Forecasting product/brand demands
Operations
e.g. Monitoring control infrastructure
at LHC
Social Networks
e.g. Like count on a profile picture
based on gender
Almost everything is a time series!

3. Value Prediction Pattern Identification
2.1, 9.3, 4.5, 3, 6.7, 4.0, 18.8, 9.2, 5.8, ?
2.1, 9.3, 4.5, 3, 6.7, 4.0, 18.8, 9.2, 5.8
Based on mathematical models Based on human perception
What’s next?

4. Classification
Anomaly DetectionMotif discovery
Clustering
[16]
[16] [16]
Source: www.google.com

5. Raw time series
data
Similarity model
selection
Dimensionality
reduction
Index
construction
Mathematical formulation
of human perception
of similarity
High dimensionality
makes distance
calculation slow
Enables efficient querying
of big and fast incoming
time series data
1 32

6. Raw time series
data
Similarity model
selection
Dimensionality
reduction
Index
construction
1 32
Symbolic
Representation
Text Processing
Algorithms
Double is 4 byte, Char is 1 byte. Hence, lower memory footprint.

7. Lp Norms
DTW distance
Longest common
subsequence
Landmark
Similarity
ℒ 𝑝 𝑥 − 𝑦 =
𝑖=1
𝐿
𝑥𝑖 − 𝑦𝑖
𝑝
1
𝑝 ℒ1
ℒ2
- Manhattan distance
- Euclidean distance
Invariant to amplitude scaling when used with z-score normalization.
Source: www.google.com

8. Lp Norms
DTW distance
Longest common
subsequence
Landmark
Similarity
𝐷 𝑖, 𝑗 = 𝑥𝑖 − 𝑦𝑗
+ 𝑚𝑖𝑛 𝐷 𝑖 − 1, 𝑗 , 𝐷 𝑖 − 1, 𝑗 − 1 , 𝐷 𝑖, 𝑗 − 1
DTW is Dynamic Time Warping.
Allows comparison of variable length time series.
Computationally Expensive. Can be optimized using warping
window techniques and early abandoning using lower bounds.
Source: www.google.com

9. Lp Norms
DTW distance
Longest common
subsequence
Landmark
Similarity
Applicable only to symbolic representations of time series.
Non-metric because it does not satisfy triangle inequality.
𝑆𝑖𝑚 𝑥, 𝑦 = 𝐿𝐶𝑆 𝑥, 𝑦
A distance measure D is a metric if it satisfies the following properties:
1. Symmetry: D(X, Y) = D(Y, X)
2. Triangle Inequality: D(X, Y) + D(Y, Z) <= D(X, Z)
Threshold parameter, matching criteria for 2 points from x and y.
Warping threshold, constraint on matching of points along the
time axis.

10. Works the same ways as human remember patterns.
Definition of landmarks vary with application domains.
E.g. local minima, local maxima, inflection point etc.
Uses MDPP (Minimum Distance/Percentage Principle)
technique to eliminate noisy landmarks.
𝑥𝑖+1 − 𝑥𝑖 < 𝐷
𝑦𝑖+1 − 𝑦𝑖
𝑦𝑖 − 𝑦𝑖+1 2
< 𝑃
𝑀𝐷𝑃 𝐷, 𝑃 removes landmarks at and if𝑥𝑖 𝑥𝑖+1
Lp Norms
DTW distance
Longest common
subsequence
Landmark
Similarity

11. 𝐷𝑟𝑒𝑑𝑢𝑐𝑒𝑑 𝑠𝑝𝑎𝑐𝑒 𝐴, 𝐵 ≤ 𝐷𝑡𝑟𝑢𝑒 𝐴, 𝐵
False Alarms False Dismissals
Objects that appear close in index space are actually
distant.
Objects appear distant in index space but are actually
closer.
Removed in post-processing step. Unacceptable.

12. DFT
DWT
PAA
eAPCA
APCA
Discrete Fourier Transform
𝑋 𝑓 =
1
𝑛
𝑡=0
𝑛−1
𝑥𝑡 𝑒
−𝑗2𝜋𝑡𝑓
𝑛
1. Choose coefficients corresponding to a few low values of
frequencies.
2. Choose coefficients corresponding to frequencies with higher
values of coefficients.
Based on Parseval’s Relation, Euclidean
distance is preserved in the Frequency
domain.

13. DFT
DWT
PAA
eAPCA
APCA
Discrete Fourier Transform
[6]

14. DFT
DWT
PAA
eAPCA
APCA
Discrete Fourier Transform
[6]

15. DFT
DWT
PAA
eAPCA
APCA
Discrete Wavelet Transform
𝜓𝑗,𝑘
= 2
𝑗
2 𝜓 2𝑗
𝑡 − 𝑘
Used with Haar wavelets as the basis function. Applicable only for time
series with lengths which are a power of 2.
Lower bound is tighter than DFT.

16. DFT
DWT
PAA
eAPCA
APCA
Discrete Wavelet Transform
Using Haar Wavelet as the basis function.
[6]

17. DFT
DWT
PAA
eAPCA
APCA
Piecewise Aggregate Approximation
𝑥𝑖 =
𝑁
𝑛
𝑗=
𝑛
𝑁 𝑖−1 +1
𝑛
𝑁 𝑖
𝑥𝑗
Reconstruction quality and estimated distance in index space is same as DWT
with the Haar Wavelet. With no restriction on length.
[6]

18. DFT
DWT
PAA
eAPCA
APCA
Piecewise Aggregate Approximation
𝐷 𝑋, 𝑌 =
𝑛
𝑁 𝑖=1
𝑁
𝑥𝑖 − 𝑦𝑖
2
A lower bound on the Euclidean distance in the PAA space.
N = actual number of points
n = number of PAA segments

19. DFT
DWT
PAA
eAPCA
APCA
Adaptive Piecewise Constant Approximation
Data adaptive. Shorter segments for areas of high activity.
An extension of PAA.
𝑋 = < 𝑥1, 𝑟1 >, < 𝑥2, 𝑟2 > ⋯ < 𝑥 𝑛, 𝑟 𝑛 >
𝑥𝑖 = 𝑚𝑒𝑎𝑛(𝑥 𝑟𝑖−1+1 … 𝑥 𝑟𝑖
)
[7]

20. DFT
DWT
PAA
eAPCA
APCA
Adaptive Piecewise Constant Approximation
𝐷 𝐶, 𝑄 =
𝑖=1
𝑀
𝑐𝑟𝑖 − 𝑐𝑟𝑖−1 𝑞𝑥𝑖 − 𝑐𝑥𝑖
2
M = number of APCA segments
A lower bound on the Euclidean distance in the APCA space.

21. DFT
DWT
PAA
eAPCA
APCA
Extended APCA
𝑆 = 𝜇1, 𝜎1, 𝑟1 , … , 𝜇 𝑚, 𝜎 𝑚, 𝑟 𝑚
Also stores variance along with mean for the segments.
As a result, it gives both a lower and upper bound on the Euclidean distance.
Formulas are very ugly!

22. SAX
iSAX
SFA
Based on PAA.
Symbolic Aggregate Approximation
Static alphabet size.
“Desirable to have a
discretization technique that
produce symbols with equal
probability.”
Can leverage run length encoding compression.
Breakpoints
[9]

23. SAX
iSAX
SFA
Symbolic Aggregate Approximation
Supports a lower bound distance measure to Euclidean distance.
𝐷𝑆𝐴𝑋 ≤ 𝐷 𝑃𝐴𝐴 ≤ 𝐷𝑡𝑟𝑢𝑒
Can be calculated in a streaming fashion.
[9]

24. SAX
iSAX
SFA
Indexable SAX
a, b, c, d (SAX)
00, 01, 10, 11
(iSAX)
0 00, 01, 10, 11
1 00, 01, 10, 11
1 00, 01, 0 10, 11
1 00, 01, 1 10, 11
Fixed number of segments. Dynamic alphabet size.
iSAX Notation: iSAX(T, segment count, alphabet size)
e.g. iSAX(T, 4, 8)

25. SAX
iSAX
SFA
Indexable SAX
Comparison of iSAX words with different alphabet size.
iSAX(A, 4, 8) = { 110, 110, 011, 000 }
iSAX(B, 4, 2) = { 0 , 0 , 1 , 1 }
Replace 0 with either of
{ 0 00, 0 01, 0 10, 0 11 }
whichever is closest to 110.
Similarly for all segments.
{ 011, 011, 100, 100} != iSAX(B, 4, 8)
Just a lower bound estimate. We
cannot undo lossy compression.

26. SAX
iSAX
SFA
Symbolic Fourier Approximation
Uses MCB (multiple coefficient binning) discretization.
Based on DFT.
SAX - assumes a common distribution
for all the coefficients of the reduced
representation
MCB – histograms are built for all the
coefficients and then equi-width
binning is used.
Tighter lower
bound than iSAX
[12]

27. SAX
iSAX
SFA
Symbolic Fourier Approximation
Every SFA symbol has some global information since it is based on
DFT. Cannot be calculated in a streaming fashion.
Unlike iSAX, Fixed alphabet size. Dynamic segment count. Quality of
representation improves with segment count.

28. Pruning
Power
Tightness of
Lower bound
number of data points examined to answer the query
total number of data points in the database
Intuitively captures the measure of the quality of representation. Free from
implementation bias.
On a random walk dataset with query lengths 256 to 1024 and dimension of
representation 16 to 64 [7]
Lower is better.

29. Pruning
Power
Tightness of
Lower bound
lower bounding distance
true distance
Tightness of lower bounds for various time series representations on the Koski ECG
dataset [10]
Higher is better.

30. R/R* Trees
SFA Trie
DS Tree
ADS index
iSAX Tree
R trees are multi-dimensional index structures.
Encloses close objects in a MBR (Minimum Bounding Rectangle).
Individual objects are at the leaves and intermediate nodes are
MBRs enclosing other MBRs or the objects.
Used for indexing time series after dimensionality reduction using
DFT, PAA, APCA and other numeric representations.
Unlike R trees, In R* trees, there is no overlap between the
different MBRs due to which it also works for range queries rather
than only point queries.

31. R/R* Trees
SFA Trie
DS Tree
ADS index
iSAX Tree
Based on the dynamic alphabet size of iSAX representation.
Given the segment count, say d. The root node has 2^d children.
A Leaf node, when overflows, is converted to an intermediate node.
A segment is selected and its cardinality is increased to produce 2 child
leaf nodes that contain the iSAX representations of the time series.
iSAX 2.0 is an improvement over iSAX where the segment on which
split occurs is selected based on the distribution of time series so that
the splitting is balanced.

32. R/R* Trees
SFA Trie
DS Tree
ADS index
iSAX Tree
Based on the SFA representation.
Time series with common SFA prefix lie in common sub-tree.
SFA is computed for more number of Fourier coefficients. But not all
are used in the index. Hence, small index size.
Example:
SFA( T1 ) = abaacde | SFA( T2 ) = abbadef
SFA( T1 ) = abaacde | SFA( T2 ) = abbadef | SFA( T3 ) = abaagef

33. R/R* Trees
SFA Trie
DS Tree
ADS index
iSAX Tree
Based on the Extended APCA reduction method.
Intermediate nodes store
𝜇𝑖
𝑚𝑖𝑛
, 𝜇𝑖
𝑚𝑎𝑥
, 𝜎𝑖
𝑚𝑖𝑛
, 𝜎𝑖
𝑚𝑎𝑥
for all segments i = 1 to m.
It also stores the splitting strategy chose during splitting.
Dynamic Segmentation Tree

34. R/R* Trees
SFA Trie
DS Tree
ADS index
iSAX Tree
Dynamic Segmentation Tree
Splitting strategies are of 2 types: Horizontal and Vertical.
Horizontal: using mean and variance.
Vertical: using segment splitting.
Splitting strategy is chosen based on the value of a Quality Measure.
The one with maximum value is selected.
𝑄 =
𝑖=1
𝑚
𝑟𝑖 − 𝑟𝑖−1 𝜇𝑖
𝑚𝑎𝑥
− 𝜇𝑖
𝑚𝑖𝑛 2
+ 𝜎𝑖
𝑚𝑎𝑥2
𝑆𝑝𝑙𝑖𝑡𝑡𝑖𝑛𝑔 𝐵𝑒𝑛𝑒𝑓𝑖𝑡 = 𝑄 𝑝𝑎𝑟𝑒𝑛𝑡 − 𝑄𝑙 + 𝑄 𝑟 2

35. R/R* Trees
SFA Trie
DS Tree
ADS index
iSAX Tree
Dynamic Segmentation Tree
Apart from similarity search as supported by other indices, it also allows
distance histogram computation for a given query.
For e.g.
Given a query Q, a list L = [ ([10, 20], 10), ([15, 30], 15), ([40, 50], 2) ]
means that there are 3 leaf nodes: N1, N2 and N3. N1 includes 10 time
series, and their distance from Q is between [10, 20]. Similarly for N2
and N3.
This is due to the lower and upper bounds provided by eAPCA.

36. R/R* Trees
SFA Trie
DS Tree
ADS index
iSAX Tree
Adaptive data series index
Based in iSAX representation.
Delays the construction of leaf nodes to query time.
Also, leaf nodes contain only the iSAX representations and the actual
data series remain in the disk. Even during splits, only the iSAX
representations are shuffled.
Trade off:
Small leaf size require splits that costs disk IO time, whereas
Big leaf size leads to increased query time for linear scan.
So, ADS+ uses adaptive leaf size. A bigger build time leaf size and a
much smaller query time leaf size.

37. [1] Agrawal, R., Faloutsos, C., & Swami, A. (1993). “Efficient similarity search in sequence databases”.
Proceedings of the 4th Conference on Foundations of Data Organization and Algorithms.
[2] Antonin Guttman, (1984). “R-trees: a dynamic index structure for spatial searching”. Proceedings of the
1984 ACM SIGMOD international conference on Management of data.
[3] Yi, B.K., Faloutsos, C. (2000) “Fast Time Sequence Indexing for Arbitrary Lp-Norms”. Proceedings of the
26th International Conference on Very Large Data Bases.
[4] Keogh, E. (2002) “Exact Indexing of Dynamic Time Warping”. Proceedings of the 28th international
conference on Very Large Data Bases.
[5] Perng, C., Wang H., Zhang S. R., Parker, D.S. (2000). “Landmarks: A New Model for Similarity-Based Pattern
Querying in Time Series Databases”. Proceedings of the 16th International Conference on Data Engineering
[6] Keogh, E., Chakrabarti, K., Pazzani, M. & Mehrotra, S. (2000). “Dimensionality Reduction for Fast
Similarity Search in Large Time Series Databases”. Published in Journal Knowledge and Information Systems.

38. [7] Keogh, E., Chakrabarti, K., Mehrotra, S., Pazzani, M. (2002) “Locally Adaptive Dimensionality Reduction for
Indexing Large Time Series Databases”. Published in Journal ACM Transactions on Database Systems.
[8] Wang, Y., Wang, P., Pei, J., Wang, W., Huang, S. (2013) “A Data-adaptive and Dynamic Segmentation Index
for Whole Matching on Time Series”. Proceedings of the VLDB Endowment
[9] Lin, J., Keogh, E., Lonardi, S., Chiu, B. (2003). “A symbolic representation of time series, with implications
for streaming algorithms”. Proceedings of the 8th ACM SIGMOD workshop on Research issues in data mining
and knowledge discovery.
[10] Shieh, J., Keogh, E., (2008) “iSAX: Indexing and Mining Terabyte Sized Time Series”. Proceedings of the
14th ACM SIGKDD international conference on Knowledge discovery and data mining.
[11] Camerra, A., Palpanas, T., Shieh, J., Keogh, E. (2010) “iSAX 2.0: Indexing and Mining One Billion Time
Series”. Proceedings of the IEEE International Conference on Data Mining.

39. [12] Schäfer, P., Högqvist, M. (2012) “SFA: A Symbolic Fourier Approximation and Index for Similarity Search in
High Dimensional Datasets”. Proceedings of the 15th International Conference on Extending Database
Technology.
[13] Beckmann, N., Kriegel, H., Schneider, R., Seeger, B. (1990) “The R*-tree: an efficient and robust access
method for points and rectangles”, Proceedings of the ACM SIGMOD international conference on
Management of data.
[14] Zoumpatianos, K., Idreos, S., Palpanas, T. (2014) “Indexing for Interactive Exploration of Big Data Series”.
Proceedings of the ACM SIGMOD International Conference on Data Management.
[15] Wu, Y.L., Agrawal, D., Abbadi, A.E., (2000) “A comparison of DFT and DWT based Similarity Search in
Time-Series Databases”. Proceedings of the ninth international conference on Information and knowledge
management.
[16] Esling, P., Agon, C. (2012) “Time-Series Data Mining”. Published in Journal ACM Computing Surveys
(CSUR).