1. Deterministic Error Guarantees for Queries on Compressed Time Series
Chunbin Lin
Joint with Etienne Boursier, Jacque Brito, Korhan Demirkaya, Joshua Lapacik, Yannis Papakonstantinou

2. Compute the correlation of the foreign-exchange CAD/JPY and AUD/JPY
Motivation
Fast analytic query processing over historical time series is necessary
Future prediction
Abnormally detection
Similarity matching
CAD/JPY
AUD/JPY
Compute the correlation of the foreign-exchange CAD/JPY and AUD/JPY
Motivation: data is big. Online query processing is required. Two directions: distribute computation and approximate computation. We follow the second one
public health analyst

3. Challenge Historical time series is big
1 billion data points for each forex*1
8 TB operational data per day for each oil drilling rig*2
Motivation: data is big. Online query processing is required. Two directions: distribute computation and approximate computation. We follow the second one *1 *2

4. Solutions Distributed query processing in many machines
Approximate query processing in a single machine
sampling methods
Probabilistic error guarantees
E.g., the actual answer is within with 95% confidence
Motivation: data is big. Online query processing is required. Two directions: distribute computation and approximate computation. We follow the second one
our goal
Deterministic error guarantees
E.g., the actual answer is within with 95% confidence

5. Data Time Series: a sequence of (timestamp, value) pairs
Assume queries involve time series with the same resolution
Omit timestamps
1, 10000, [
115.80,
115.90,
116.25,
116.30,
116.11,
116.15,
116.16,
116.06,
115.72,
...... ] [
( , ),
( , ),
( , ),
( , ),
( , ),
( , ),
( , ),
( , ),
( , ), ]
How to normalize data into same resolutions is not a topic of this work. There are several prior work
Apple stock price

6. Query Time subseries operators Arithmetic operators (+,−×,÷,√ )
Time series operators.
Arithmetic operators (+,−×,÷,√ )
E.g., , , 100*20, 100/20…

7. Query Statistic queries Covariance, Correlation, Cross-correlation, ……
Time series operators.

8. Query Statistic queries Covariance, Correlation, Cross-correlation, ……
base time series
time series produced by time series operators
Time series operators.

9. Offline precomputation phase – building indexes
FL and SW

10. Segment list index Index: a list of compressed time series segments
f(x) = a x + b
Forex CAD/JPY (the Canadian Dollar and the Japanese Yen)
For each segment, we store:
Estimation function (minimize Euclidean distance)
Error measures
Note that, we are not really store (a,b) but the (a,b) in its orthogonal basis
(a , b)
L2-norm of errors:
Reconstruction error:
L2-norm of estimated values:

11. no limitation on estimation functions
Segment list index
Estimation function families
no limitation on estimation functions
logarithmic function family
polynomial function family
exponential function family
logistic function family
gaussian function family
sin/cos function family
Functions. Minimize Eucliean distance. And show the function families.

12. depends on the data values
Segment list index
Error guarantees
L2-norm of errors:
Reconstruction error:
L2-norm of estimated values:
depends on the data values
3.0
4.8
5.4
f(x) = 1.2x + 2.0
L2-norm of estimated values is usually big

13. Segment list index Existing index building algorithms
Fix-length segmentation (FL) : control segment size
Sliding-window segmentation (SW): control reconstruction error ……
CAD/JPY
AUD/JPY
CAD/JPY
AUD/JPY
Functions. Minimize Eucliean distance. And show the function families.
E. Keogh, S. Chu, D. Hart, and M. Pazzani. An online algorithm for segmenting time series. In ICDM, pages 289–296, 2001.

14. Offline precomputation phase – building indexes
We build a segment list index for each time series
We store an estimation function and error measures for each segment
FL and SW

15. Online query processing – providing deterministic error guarantees
FL and SW

16. Error guarantees
Actual error: the absolute difference between the true answer R and the estimated answer 𝑅 , i.e.,
Error guarantee: the upper bound of the actual error, i.e.,
Error measures.

17. Error guarantees
Providing the error guarantee for each Sum(T) is the key
base time series
time series produced by time series operators
Error measures.
If we can provide an error guarantee for each Sum(Ti), then we are able to give the error guarantee for general queries

18. Query over single segment
Error guarantees for time series operators T1 T2
Error measures.

19. Query over single segment
Error guarantee of Sum(T1 x T2)
Show how to get error guarantees for VS
𝐻 𝑜 𝑙𝑑𝑒𝑟 𝑖𝑛𝑒𝑞𝑢𝑎𝑙𝑖𝑡𝑦

20. Query over single segment
Error guarantee of Sum(T1 x T2)
= 0 if the estimation function family forms a vector space (VS)
Error measures.
Vector space: A set that is closed under finite vector addition and scalar multiplication
Polynomial function family is a vector space

21. Orthogonal projection property in VS
Show how to get error guarantees for ANY

22. Orthogonal projection property in VS
Show how to get error guarantees for ANY

23. =0 Query over single segment Orthogonal projection property in VS
Show how to get error guarantees for VS
𝐻 𝑜 𝑙𝑑𝑒𝑟 𝑖𝑛𝑒𝑞𝑢𝑎𝑙𝑖𝑡𝑦

24. Query over single segment
Error guarantee of Sum(T1 x T2)
Estimation function family is not VS
Estimation function family is VS
Show how to get error guarantees for ANY

25. Query over aligned segments
All the segments are perfectly aligned
CAD/JPY
AUD/JPY
Error guarantees
Sum of the error guarantees of each segment pair
Show how to get error guarantees for LSF

26. Query over aligned segments
CAD/JPY
AUD/JPY
Show how to get error guarantees for LSF

27. Query over misaligned segments
One segment overlaps with more than one segment
CAD/JPY
AUD/JPY
Segment combination selection algorithms.

28. Query over misaligned segments
Sum(T1 x T2)
Segment combination selection becomes an optimization problem
Minimize
CAD/JPY
AUD/JPY
Segment combination selection algorithms.

29. Query over misaligned segments
Segment combination selection
CAD/JPY
AUD/JPY
Intersection Strategy (IS)
Maximal number of segments
Optimal Strategy (OS)
Minimal error combination
Segment combination selection algorithms.

30. Query over misaligned segments
Orthogonal projection property
CAD/JPY
AUD/JPY
Cannot be applied, not aligned
Estimation function for a subsegment may not be in the family
Segment combination selection algorithms.
Linear scalable family (LSF): the restriction of any function in LSF to a smaller domain is still a function in LSF
ANY VS
LSF PF
LSF is a superset of the polynomial function family (PF)

31. Query over misaligned segments
Sum(T1 x T2)
If estimation functions are in LSF
CAD/JPY
AUD/JPY
Segment combination selection algorithms.

32. Error guarantee properties
Tightness
With the same error measures, no other error guarantee is smaller than it for queries on all the data
Amplitude-independence (AI)
Not using the amplitudes in the error guarantees
E.g., Changing from Celsius to Kelvin will not change the error guarantees
AI and tight.

33. Error guarantee properties
Function family
ANY\VS
VS\LSF
LSF
ANY
Queries on aligned segments
Queries on misaligned segments AI
Tight
Sum(T1 x T2)
Sum(T1 + T2)
Sum(T1 - T2)
Show dichotomies.

34. Error guarantee properties
Dichotomies of function families VS
LSF
ANY\VS
ANY\LSF
Show dichotomies. AI AI
non-AI
non-AI
Queries on aligned segments
Queries on misaligned segments

35. Avg # of data points in each time series
Experiments
Dataset
Avg # of data points in each time series
# of time series
Resolution
Historical Forex Data (HF)
126,059,817 15
millisecond
Historical IoT Data (HI)
2,676,311 14
second
Historical Bitcoin Exchanges Data (HB)
1,669,835 16
minute
Historical Air Quality Data (HA)
1,587,258 11

36. Experiments Estimation functions [1] [2] [3]
E. Keogh. Fast similarity search in the presence of longitudinal scaling in time series databases. In ICTAI, pages 578–584, 1997.
M. Tobita. Combined logarithmic and exponential function model for fitting postseismic gnss time series after 2011 tohoku-oki earthquake. Earth, Planets and Space, 68(1):41, 2016.
Z. Pan, Y. Hu, and B. Cao. Construction of smooth daily remote sensing time series data: a higher spatiotemporal resolution perspective. Open Geospatial Data, Software and Standards, 2(1):25, 2017.

37. Experiments Segment list building algorithms
Fix-length segmentation (FL)
Sliding-window segmentation (SW)
Queries
Correlation query
cross-correlation query
tree

38. Experiments Error guarantees for queries on aligned time series
20 correlation queries
FL segment lists building
Power of orthogonal property
tree
VS uses less space than ANY
VS uses 0.035% while ANY uses 0.06%

39. Experiments Error guarantees for queries on misaligned time series
20 correlation queries
SW segment lists building 1 2
LSF uses 0.02% while ANY uses 0.023%. ANY has less segments, but more parameters. 1
Effect of LSF (~100x)
Effect of optimal segment combination selection (~10x) 2

40. Experiments Aligned vs. misaligned
Fix space, compare error guarantees for aligned and misaligned
K segments in misaligned case, N data points involved in the query, then set segment size in FL to be N/K 1
Misaligned produces smaller true errors 2
Misaligned produces smaller error guarantees
~ 3x for ANY 2
tree 1

41. Experiments Aligned vs. misaligned
Fix space, compare error guarantees for aligned and misaligned
K segments in misaligned case, N data points involved in the query, then set segment size in FL to be N/K 1
Misaligned produces smaller error guarantees
~ 8.2 x for LSF 1
tree

42. Experiments
Index building time
Query processing time
tree

43. Experiments Compare with sampling method
uniform random sampling scheme with a global seed
Sampling size to provide same error guarantees with those of VS
Sampling size to provide same error guarantees with those of ANY
tree
confidence

44. Study the properties – AI and tight – of the proposed error guarantees
Conclusion
Provide deterministic error guarantees for statistic queries over aligned segments and misaligned segments.
Provide optimizations to reduce the error guarantees in both scenarios.
Study the properties – AI and tight – of the proposed error guarantees
Conduct experiments to evaluate the error guarantees
Time series operators.

45. Future work
Deterministic error guarantees for interactive analytic queries over compressed time series
Time series operators.

46. Architecture Build segment tree index for each time series (offline)
A node refers to a compressed segment
Each segment, we store estimation function and error measures
Tree may not be a balanced tree
Navigate trees to access minimal number of nodes to get answers with error guarantees less than given threshold value (online)
In the offline processing, we partition each time series into segments. And for each time series segment, we precompute some error parameters. The segments are organized as a hierarchy structure.
In the online processing, a user gives a query and an error budget. Instead of accessing the original data, ApproPlato access the hierarchy structure and returns an approximate answer. The approximate answer has an error guarantee, which means the error is no greater than the error budget.
In the following slides, I will introduce the error parameters, as shown in 1. Then describe the hierarchy structure as shown in 2. And finally introduce the hierarchy structure as shown in 3.
I will also keep this architecture graph in the following slides in order to indicate where we are now. But make it smaller.

47. Segment tree index One tree structure for each time series
A node refers to a compressed segment
Each segment, we store estimation function and error measures
Tree may not be a balanced tree
Segment tree building algorithms:
Top-down algorithm
Bottom-up method
Sliding-window approach
*Fu, Tak-chung. "A review on time series data mining." Engineering Applications of Artificial Intelligence 24.1 (2011):

48. Query processing algorithm
Given query and error budget, access minimal number of nodes to get approximate answers with error guarantees less than the error budgets
Consider query = (Agg(Times(T1, T2)), 10%
Time series T1
Time series T2

49. Query processing algorithm
Performance-wise optimization
An incremental update segmentation algorithm that gives ratio compared with the optimal one.
1+ 2
Space-wise optimization
Avoid storing the estimation functions for the right nodes.
Only red nodes store estimation functions
Estimation function can be deduced from the parent node and the left sibling node via an invert basis matrix

50. Thank you
Q&A

51. Backup slides

52. Application New compression consideration
Not only consider the reconstruction error but also the query error guarantees
Reconstruction error
Query error
Show dichotomies.