1. University of Macau, Macau
Quick-Motif: An Efficient and Scalable Framework for Exact Motif Discovery
Yuhong Li
Department of Computer and Information Science
University of Macau, Macau

2. Quick-Motif: What is Motif ?
Most similar subsequence pair in a Time Series
Applications
A core subroutine for activity discovery, e.g., elder care, surveillance and sports training.
Clustering enumerated motifs is more meaningful than clustering all the subsequences in a long time series.

3. Quick-Motif: Formal Definition
time series subsequence s 𝑖
time series 𝑠 𝑖
𝑖+ℓ−1
𝑚−1
Timeline
Exact Motif Discovery
Input: time series 𝑠 and target motif length ℓ
Output: most similar subsequence pair in terms of normalized Euclidean distance.
Avoid trivial match  Non-overlapping
Adjacent subsequence pairs are expected to similar to each other naturally.

4. Quick-Motif: Naïve Solution
Sliding window size = ℓ, Step size = 1
Subsequences of length ℓ
Subsequences of length ℓ
Test all subsequence pairs
normalize …
Motif 
most similar subsequence pair … …
Time complexity is O( 𝑚 2 ℓ).

5. Quick-Motif: Existing Solutions
Reference-based Index (MK) [Mueen & Keogh, SDM 2009]
Good: Prune unpromising pairs by batches.
Bad: 𝑂(ℓ) time distance computations.
Smart Brute Force (SBF) [Mueen, ICDM 2013]
Good: 𝑂(1) time distance computations.
Bad: examine all subsequence pairs. … … ?
𝑂(ℓ)
𝑂(1)

6. Quick-Motif: Fast Distance Computation
Incremental distance computation.
𝑠 0
𝑠 20 ……
𝑠 1
𝑠 21
𝑠 2
𝑠 22
𝑠 23
𝑠 3
𝑠 4 …
𝑠 24
𝑠 0
𝑠 1
𝑠 2
𝑠 3
𝑠 4
𝑠 20
𝑠 21
𝑠 22
9 subsequence pairs  𝑂 ℓ
16 subsequence pairs  𝑂(1)
𝑠 23
𝑠 24

7. Quick-Motif: Pruning of Subsequence Pairs
Group every w consecutive subsequences as a PAA MBR.
𝑤 = 5
𝑓 2
𝑀 3 5
𝑀 1 5
minDist
𝑀 2 5
PAA feature space
𝑓 1
Minimum distance between two PAA MBRs  Distance LBs.
If distance LB is smaller than 𝑏𝑠𝑓  Further refinement.

8. Quick-Motif: Filter-and-Refinement
Naïve Solution.
Check the distance LBs for all 𝑤-MBR pairs.
The time complexity is 𝑂( (𝑚/𝑤) 2 𝜙) , 𝜙 is the PAA dimensionality.
How to Efficiently Find Surviving 𝑤-MBR Pairs?
Enable batch pruning.
Discover the true motif as soon as possible to improve the pruning ability.

9. Quick-Motif: Filter-and-Refinement
Enable Batch Pruning  Hierarchical Structure
Offer reasonable grouping quality, thus good pruning ability.
Can be constructed very efficiently.
𝑓 2
𝑀 8 𝑤
𝑀 1 𝑤
Level 2
𝑀 3 𝑤
𝑀 𝑟𝑜𝑜𝑡
𝑀 6 𝑤
Level 1
𝑀 5 𝑤
𝑀 0 𝑤
𝑀 𝑎
𝑀 𝑏
𝑀 𝑐
𝑀 7 𝑤
minDist
𝑀 4 𝑤
𝑀 2 𝑤
𝑀 4 𝑤
𝑀 6 𝑤
𝑀 0 𝑤
𝑀 2 𝑤
𝑀 7 𝑤
𝑀 5 𝑤
𝑀 3 𝑤
𝑀 1 𝑤
𝑀 8 𝑤
PAA feature space
𝑓 1
Hilbert curve sort list

10. Quick-Motif: Filter-and-Refinement
Discover true motif as soon as possible  Locality-based Search Strategy
Level 2
𝑀 𝑟𝑜𝑜𝑡
Bad locality
Level 1
𝑀 𝑎
𝑀 𝑏
𝑀 𝑐
Hilbert curve sort list
Leaf nodes
Good locality
𝑀 4 𝑤
𝑀 6 𝑤
𝑀 0 𝑤
𝑀 2 𝑤
𝑀 7 𝑤
𝑀 5 𝑤
𝑀 3 𝑤
𝑀 1 𝑤
𝑀 8 𝑤
Locality-based search vs Best-first search
Locality-based
Best-first
Surviving pairs
0.1256M
0.1249M
Heap size
N/A
2.78M
# pushes
11.73 M (queue)
6.75 M (heap)
Resp. time
1.56 s
6.32 s

11. Quick-Motif: Experimental Evaluation
Programming
Language: C++
Machine: Ubuntu 12.04, 4GB RAM
Datasets
RW: Random generate.
EEG: Reflect the activity of neurons, length
ECG: The Koski ECG. Length
EPG: Sequence that traces insect behaviour, length
TAO: Sea surface temperatures, length

12. Quick-Motif: Performance Evaluation
(a), Effect of ℓ on ECG
(b), Effect of ℓ on EEG
(c), Effect of ℓ on EPG
(d), Effect of ℓ on TAO

13. Thanks Q A
input
hidden
output