1. Fast Approximate Query Answering over Sensor Data with Deterministic Error Guarantees
Chunbin Lin
Joint with Etienne Boursier, Jacque Brito, Yannis Katsis, Yannis Papakonstantinou

2. Challenge & Motivation
Sensor data is big
GB per minute
DELPHI project collects many environmental and biomedical data (spatiotemporal) ……
Public health analyst
Online analytical queries are needed
E.g., Cross-Correlation between heart rate and temperature (for various participant subsets)
Performance is too slow
Approximate answers with bounded errors are acceptable
Errors should be bounded by user provided budgets. E.g., budget = 10%
Computation of approximate answer should be efficient

3. PlatoDB Overview
PlatoDB provides efficient approximate query processing and deterministic error guarantees
Offline
Construct an segment tree data structure. Each node is a segment that is represented by a linear scalable function minimizing the Euclidean distance
Offline
Store error statistics for each segment.
Online
Use error statistics to get tight estimated error for each operator
Calculate query errors by combining estimated operator errors
Online
Navigate tree structures and incrementally update the estimated error

4. Data model Time series data a sequence of (timestamp, value) pairs
Assume fixed intervals
Omit timestamps
( :00:00,15.72),( :00:01,15.74),...,( :59:59,2.32)
(15.72, ,..., 2.32)
*Dina Q. Goldin, Paris C. Kanellakis: On Similarity Queries for Time-Series Data: Constraint Specification and Implementation. CP 1995:

5. Query Basic time series operators Serialize
Serialize(1.2, 5) creates a time series with 5 points whose values are 1.2
Shift
Shift(T, 2) moves i-th point to (i+2)-th position
Plus
Plus(T1, T2) creates a new time series where di = di(1) + di(2)
Minus
Minus(T1, T2) creates a new time series where di = di(1) - di(2)
Times
Times(T1, T2) creates a new time series where di = di(1) * di(2)
Aggregation operator
Sum
Sum(T, a, b) returns the value of
Arithmetic manipulation operator (e.g., +, -, x, / )

6. Arithmetic manipulation
Example Queries
Time series operator
Aggregation operator
Arithmetic manipulation

7. PlatoDB Architecture Pre-Processing Query Processing Query Q
Online
Error budget
Query Processor
Query Processing
Segment Tree Generator
Time Series Segment Trees
Sensor Data
Pre-Processing
Offline
Noise Removal
clean
data
error statistics
compressed data
Time series T1
Segment Tree for T1 S1
S1.1
S1.2
S1.1.1
S1.1.2
Segment S1
Segment S1.1
Segment S1.2
S1.1,1
Segment Tree for T2
Segment Tree for T1

8. Segment tree One tree structure for each time series
How to build such a tree?
Existing time series segmentation algorithms
Top-down
Bottom-up
Sliding-window
Tree may not be balance
Each node represents one segment
Lower level nodes represent smaller segments
How to represent each segment?
Existing time series representation algorithms
Piecewise Aggregate Approximation (PAA)
Adaptive Piecewise Constant Approximation (APCA)
Piecewise Linear Representation (PLR)
Discrete Wavelet Transform (DWT)
*Fu, Tak-chung. "A review on time series data mining." Engineering Applications of Artificial Intelligence 24.1 (2011):

9. Segment tree One tree structure for each time series
Linear scalable family: A family is linear scalable if for every segment S and every function f on this segment, for any subsegment S1, the restriction f|S1 is still a function in this family.
Tree may not be balance
Each node represents one segment
Lower level nodes represent smaller segments
Polynomial function family is linear scalable family
PlatoDB chooses a linear scalable function f to represent a segment where f minimizes the Euclidean Distance
Contribution 1:
PlatoDB provides O(n) algorithm to build the tree structure.

10. Segment tree One tree structure for each time series
Each node stores two kinds of values
Coefficients of the estimation function in the orthonormal basis
Error statistics
One tree structure for each time series
Tree may not be balance
Each node represents one segment
Lower level nodes represent smaller segments
PlatoDB stores by setting p = q =2

11. Query Processing Contribution 2:
PlatoDB provides a top-down query processing algorithm
Update one node at a time and update the current error
Terminate when the current error < error budget
Consider query = (Sum(Times(T1, T2)), 10%
Time series T1
Time series T2

12. Query Processing Contribution 2:
PlatoDB provides a top-down query processing algorithm
Update one node at a time and update the current error
Terminate when the current error < error budget
Consider query = (Sum(Times(T1, T2)), 10%
Time series T1
Time series T2

13. Error Estimation Contribution 3:
PlatoDB provides minimal error estimation for each time series operator by using error statistics
Store for each segment
Aligned segments (same length and same start point)
Based on the orthogonal projection property
In PlatoDB

14. Error Estimation Contribution 3:
PlatoDB provides minimal error estimation for each time series operator by using error statistics
Store for each segment
Unaligned segments (different lengths or different start points)
Symbolic computation
Contribution 4:
PlatoDB provides an optimal algorithm in O(K1+K2) to find the segmentation with minimal error

15. Further Optimizations
Performance-wise optimization
Contribution 5:
PlatoDB provides an incremental update segmentation algorithm that gives ratio compared with the optimal one.
1+ 2
Space-wise optimization
Contribution 6:
PlatoDB can avoid storing the coefficients for the right nodes.
Only red nodes store coefficients
Coefficients can be deduced from the parent node and the left sibling node via an invert basis matrix

16. Preliminary Experiment
Data: 1 billion temperature and ozone data points.
Query: Correlation of T1 and T2

17. Fast Approximate Query Answering over Sensor Data with Deterministic Error Guarantees
Thank you for your time!
Q & A

18. Techniques orthogonal to PlatoDB
Synopses
E.g., Wavelets, histograms, sketches
Tightly tied to specific classes of queries
Queries are assume to be known in advance
Samples
E.g., STRAT (single stratified sample),
SciBORQ (biased samples),
AQUA (single stratified sample),
BlinkDB (multiple samples)

19. Sampling vs Compression
Sampling: a subset of the original data
Compression: use functions to model data