1. GAMPS COMPRESSING MULTI SENSOR DATA BY GROUPING & AMPLITUDE SCALING
Sorabh Gandhi, UC Santa Barbara
Suman Nath, Microsoft Research
Subhash Suri, UC Santa Barbara
Jie Liu, Microsoft Research
GAMPS bigger remaining smaller. Problem, Current solution, what we are trying.

2. Fine Grained Sensing & Data Glut
Advances in sensing technology fine grained ubiquitous sensing of environment
Many applications, but the issue is data glut
Automated Data Center Cooling: [MSFT DCGenome project]
physical parameters ex. humidity, temperature etc
1000s of sensors, 10 bytes/sensors/sec  10s of GBs/day
Server Performance Monitoring: [MSFT server farm monitoring]
performance counters ex. cpu utilization, memory usage etc
100s of counters, 1000s of servers, few bytes/counter/sec TBs/day
Recent advances in sensing technologies have made possible, both technologically and economically, the deployment of densely distributed sensor networks. These networks provide fine grained ubiquotous
sensing of the environment.
Large scale phenomenon with lots of dynamic elements- data glut. Challenges in data management.

3. Focus and Objectives
Data archival + (reliable and fast) query processing
Centralized setting
Point query: report value for sensor x, time t
Similarity query: report sensors ‘similar’ to sensor x in time range
Obvious solution: compression, data is set of time series
Initial idea: approximate every time series individually
Many approximation techniques known ex. DFT, DCT, piecewise linear
Focus: L1 error [guarantee on point queries]
ex techniques wavelets, piecewise constant/linear approximations
Compression not enough!!
Gives upto an order of magnitude improvement, we want more
Our focus is data archival, for historical and trend analysis
Also, we want to archive this data in a format which enables fast query processing
Where the query could be a point query, or a similarity query.

4. Signals are Correlated!
Shifted/Scaled groups
Dynamic groups
Server dataset: 40 signals, 1 day, sampling once every 30 seconds, counter: # of connected users
# Connected Users
Similar signals in a group
Example dataset, Server, Server performance monitoring application. The signals are certainly correlated across time but they also seem correlated in space with each other
Time

5. Contributions
We propose GAMPS, which exploits linear correlations among multiple signals while compressing them together, and gives L1 guarantees
Compression both along time and across signals
We propose an index structure for compressed data which can give fast responses to a lot of relevant queries
Through simulations on real data, we show that on large datasets, GAMPS can achieve upto an order of magnitude improvement over state of the art compression techniques

6. State of the art: Single Signal Optimal L1 approximations
Problem: Given a time series S and input parameter ² approximate S with piecewise constant segments such that the L1 error is <= ²
Greedy algorithm (PCGreedy(S, ²))

7. State of the art: Single Signal Optimal L1 approximations
Problem: Given a time series S and input parameter ², approximate S with piecewise constant segments such that the L1 error is <= ²
Greedy algorithm (PCGreedy(S, ²)) 2²
The algorithm divides the time series into contiguous disjoint parts  buckets
The algorithms approximates each bucket by a segment
Let the magnitude of 2\epsilon be as shown by the segment on the left hand part of the slide.
The algorithm starts with an empty bucket, and processes every data point one by one.
The algorithms maintains the maximum and minimum values seen for every part, and the moment the difference of these values excceds 2\epsilon, the
Is there anything more we can hope to do ?
Original Time Series
ICDE’03 Lazardis et al.
Approximation

8. GAMPS Overview
GAMPS take as input, the set of time series and approximation parameter ²
Compression
Partition phase: partitions the data into contiguous time intervals
Group phase: divides a given partition into groups of similar signals
Amplitude scaling phase: compression happens with sharing of representations
Data
Amplitude Scaling Phase
Partition Phase
Grouping Phase
Compressed
Index Structure
Data
INDEXING
COMPRESSION

9. Compression by Amplitude Scaling
Given a group of k ‘similar’ signals
Let the signals be denoted by set X = {X1, X2, …, Xk}
Key idea: express all signals Xi as scaled function of some signal Xj: Xi = AiXj
Ai is the ratio/amplitude signal and Xj is the base signal
If signal Xi is a perfectly scaled version of Xj then Ai = constant
To reconstruct Xi, we only need to store the constant and Xj
In reality, no perfect correlation
However, we found that if there are enough linearly correlated signals smartly approximating Ais and Xj can give very good compression factors!
If our premise is true, that is, if signal Xi is a perfectly scaled/shifted/overlapping version of Xj, then we can
Achieve big compression gains. For instance, if Xi is a perfectly scaled version of Xj then Ai is constant
We found experimentally that if there are enough linearly correlated signals, By smartly approximating Ai and Xj
with PCGreedy() such that reconstruction error in Xi is less than target epsilon, we can achive …. Let us
have a look at an example of this with the help of a small part of our datacenter dataset

10. Illustration: Amplitude Scaling on Real Dataset
DataCenter dataset
6 signals shown for ~3 days each, parameter: relative humidity
Input: X = {X1, X2, …, X6}, ² = 1%
Need to choose base signal and divide ² among base signal (²b) and ratio signal approximations (²r)
Oracle: X4 is base signal, also provides values ²b and ²r
Run PCGreedy(X4, ²b) and PCGreedy(Ai, ²r) for signals other than the base signal
DataCenter Dataset
Let the target error be 1% of the data value. Again, we will concentrate for now on signal A_i, which we call the
ratio signal, as it gives much better compression results for all our datasets as compared to Bi. Call X_j as base signal. Say for now oracle tell us that base signal is X_4 and it should be approximated with 40% of the total error.
Using this, one can determine e2 such that reconstruction error in approximation of X1 X2 X3 X5 X6 is less than
target error e. So we construct the ratio signal approximations.

11. Illustration: Amplitude Scaling on Real Dataset
Leftmost figure, all signals use PCGreedy() with ² = 1.0%
Middle figure, higher fidelity base signal, ²b =0.4%
Rightmost figure: Ratio signals
Very sparse (small number of segments to represent)
Individual approx
Y-axis: Relative Humidity
Base signal approx
Y-axis: Relative Humidity
Ratio signal approx
Y-axis: Ratio
The results for technique mentioned on the previous slide are shown here.
Leftmost figure show individual approximations with optimal single signal approximation algorithm with error 1%.
Middle figure shows approximation of base signal with 0.4e.
And the right figure shows ratio signal approximations such that reconstruction error is less than e.
The most interesting things to note here is that Ratio signals are very sparse, i.e. lot less memory than corresponding individual approximation in Figure 1. So, even though Base signal approximation takes more segments, overall compression seems better than individual approximations.

12. Quantitative Comparison for Amplitude Scaling
Compression factor = M1/M2
M1 = number of segments in individual signal approximations
M2 = number of segments in (base signal + ratio signal) approximations
For this illustrative dataset, compression factor
(1% error) is 1.9
Comparison with optimal individual approximations

13. Grouping and Amplitude Scaling by Facility Location
Facility location problem
Problem is modeled as a graph G(V, E)
Opening a facility at node j costs c(j)
Serving a demand point j using facility i
costs w(i,j)
Objective is to choose F µ V
Minimize j 2 F c(i) + i 2 V w(i,j)
Grouping & amplitude scaling is modeled as facility location
Complete graph, every signal is represented by a node
Cost opening a facility: # segments needed to represent base signal
Cost of serving a demand point: # segments needed to represent the ratio signal
Graph
Suppose the paritioning part provides us with a batch of data. We solve our grouping and compression problem
by using algorithms known for facility location problem. Let us first understand the facility location problem.
The Facility Location problem consists of a set of potential facility sites, represented by nodes V where facility can be opened, and a set of demand points also represented by V that must be serviced. The goal is to pick a subset F of facilities to open, to minimize the sum of cost of serving every demand point by one facility, plus the sum of opening costs of the facilities.

14. Implementation Setup
We set ²b = 0.4² [error allocation for base signal]
Facility location : NP hard
We show results with exact solution (integer linear program)
Approximation solutions are with 90% of the results shown
Time taken to solve the linear program is <= few seconds
We use three different datasets
Server dataset: 240 signals, 1 day data [CPU utilization counter]
DataCenter dataset: 24 signals, 3 days of data [humidity sensors]
IBT dataset: 45 signals, 1 day of data [temperature sensors in a building in Berkeley]
I will only show results with compression. For all the experiments shown ….
Certainly a parameter which can be tuned, but we find that we get good result even with this fixed value.

15. Quantitative Evaluation: GAMPS
Figure on the left shows compression factor over raw data
For 1.5% error, 300 for server data, 50 for the other two
Figure on the right: compression factor over individual approximations
For 1.5% error, between factor 2-10
Compression factor high for Server dataset
Average group size is highest (60 as compared to 4.5 & 6)

16. Scaling versus Group size
We extracted 60 signals in the same group for the Server dataset
Compression factor (versus individual approximations) increases as group size increases

17. Advantage of Grouping
Demonstrate the advantage of having multiple groups
Datasets IBT and Server
Hybrid: algorithm which allows only 1 group
Every signal is either in the group or approximated individually
For both datasets, for all errors, grouping gives great advantage
Compression Factor: 1.5 (IBT) - 9 (Server) [Error 1.5%]

18. Grouping: Geographical Locality
IBT dataset, 1 day, error = 1.5%
GAMPS runs the grouping on entire days data
Picture on left shows sensor layout in the Intel Berkeley lab
Hexagons are sensor positions, crosses are sensors without data for the one day, rectangles are outliers (individual approximations)
Simple region boundaries conform our intuition
Grouping algorithm has no information about geographical locations
Sensor Layout
Group Layout

19. Indexing Compressed Data
Skip-list of groups 1 2 3
Ptr. to base signal 4
Skip-list of approx. lines for ratio signal 5
Propose Skip list based index structure
Point query: log(n)
Range query : log(n) + range
Similarity query : log(n) + #groups in range

20. Future Work How to distribute error among base and ratio signals ?
How about generic linear transformations ?
We use only ratio signal (scaling) : Xi = AiXj
Maybe we can get much better compression by using Xi = AiXj + Bi
How about piecewise linear signals ?
Underlying algorithm is not so trivial (convex hulls)
Can we apply this technique to 2D signals ?
Consider a video, every pixel value in time  time series
Every pixel-time-series, correlated with neighboring pixel-time-series

21. Thanks for your attention

22. Example Query: Similarity Query
Based on grouping we can define similarity coefficient for a given time range (t1, t2)
= 1, if signals Si and Sj are in the same group at time t
Part of IBT dataset
Similarity Query

23. Compression by Interval Sharing
Key Idea: If two sensors have near overlapping time series they can share a part of the approximation
Let number of signals be k and desired error be ²
(®, ¯) approximation algorithm
For given error ² say optimal algorithm taken OPT
(®, ¯) algorithm has error no more than ®² and uses no more than ¯OPT segments
We propose polynomial time (5, log k + log OPT) approximation algorithm for approximation with PC segments using interval sharing
Signal 1
Signal 2
Representation can be shared

24. Multiple Correlated Signals: Example 1
Instant messaging service – Server dataset
240 servers, 2 weeks, >= 100 performance counters
40 signals shown (normalized) for one day, counter: #connected users, sampling rate once in 30 seconds
Signals are correlated (almost overlapping) with each other, can we exploit this in compression ?
Server Dataset
Hope is that many signals are related and if so, we want a technique which can exploit it.

25. Multiple Correlated Signals: Example 2
Data center monitoring
24 sensors, 2 years, 2 parameters: humidity, temperature
6 signals shown for ~3 days each, parameter: relative humidity, sampling rate once every 30 seconds
Signals not overlapping, but still correlated
Shifting or scaling may help
Question: Can we exploit this correlation ?
We propose a technique to compress multiple signals along both time and across signals
DataCenter Dataset
GAMPS overview: Grouping and compression (linear transform) in practice sclaing is pretty effectve

26. Partition Determination
Use double-half-same size heuristic
Start with some initial batch size (say 100 data points)
For next batch run group and compress with 200, 100 & 50 data points
For 200, compare with two batches of size 100, whichever one takes less memory is chosen
Similarly for 50, compare two batch sizes of 50 with one batch size 100
Memory taken = # segments + Cluster delta
Cluster delta: Every time clusters change, we need to update the base signals and base-ratio signal relationships

27. (Similar signals together) Select Base and Ratio Signals
GAMPS Illustration 1
Partition 1 2 2 3 3 4 4 5 5
Grouping
(Similar signals together)
Base signals
Select Base and Ratio Signals 2 4 1 3 5 1 2
Ratio signals 3 4 5

28. GAMPS Compression Illustration1
Partition 1 2 2 3
(To overcome varying correlations) 3 4 4 5 5
Grouping
(Similar signals together)
Compress by Amplitude Scaling 1 2 3 4 5