high performance discovery from time series streams dennis shasha joint work with yunyue zhu...

High Performance Discovery from Time Series Streams

Dennis Shasha

Joint work with Yunyue Zhu

[email protected] [email protected]

Courant Institute, New York University

Overall Outline• Data mining – both classical and

activist

• Algorithmic tools for time series

• Surprise.

Goal of this work• Time series are important in so many

applications – biology, medicine, finance, music, physics, …

• A few fundamental operations occur all the time: burst detection, correlation, pattern matching.

• Do them fast to make data exploration faster, real time, and more fun.

• Extend functionality for music and science.

StatStream (VLDB,2002): Example

• Stock prices streams– The New York Stock Exchange (NYSE) – 50,000 securities (streams); 100,000 ticks (trade and quote)

• Pairs Trading, a.k.a. Correlation Trading

• Query:“which pairs of stocks were correlated with a value of over 0.9 for the last three hours?”

XYZ and ABC have been correlated with a correlation of 0.95 for the last three hours.Now XYZ and ABC become less correlated as XYZ goes up and ABC goes down.They should converge back later.I will sell XYZ and buy ABC …

Online Detection of High Correlation• Given tens of thousands of high speed time series data streams, to

detect high-value correlation, including synchronized and time-lagged, over sliding windows in real time.

• Real time– high update frequency of the data stream– fixed response time, online

Correlated!




Correlated!

StatStream: Algorithm

• Naive algorithm– N : number of streams

– w : size of sliding window

– space O(N) and time O(N2w) VS space O(N2) and time O(N2) .

• Suppose that the streams are updated every second.– With a Pentium 4 PC, the exact computing method can only monitor 700

streams with a delay of 2 minutes.

• Our Approach – Use Discrete Fourier Transform to approximate correlation

– Use grid structure to filter out unlikely pairs

– Our approach can monitor 10,000 streams with a delay of 2 minutes.

StatStream: Stream synoptic data structure• Three level time interval hierarchy

– Time point, Basic window, Sliding window• Basic window (the key to our technique)

– The computation for basic window i must finish by the end of the basic window i+1

– The basic window time is the system response time.• Digests

Basic window digests:

sum

DFT coefs

Sliding window

Basic window

Time point


sum

DFT coefs

StatStream: Stream synoptic data structure


sum

DFT coefs

Sliding window

Basic window

Time point


sum

DFT coefs


sum

DFT coefs

• Three level time interval hierarchy– Time point, Basic window, Sliding window

• Basic window (the key to our technique)– The computation for basic window i must finish by the end of the

basic window i+1– The basic window time is the system response time.

• Digests


Sliding window digests:

sum

DFT coefs


sum

DFT coefs

Sliding window

Basic window

Time point


sum

DFT coefs


sum

DFT coefs




• Digests



sum

DFT coefs

Sliding window

Basic window

Time point


sum

DFT coefs


sum

DFT coefs




• Digests

Synchronized Correlation Uses Basic Windows

w

i i

w

i i

w

i iiw

rrss

rsrsrscorr

1

2

1

2

11

)()(),(

• Inner-product of aligned basic windows

Stream x

Stream y

Sliding window

Basic window

• Inner-product within a sliding window is the sum of the inner-products in all the basic windows in the sliding window.

• Approximate with an orthogonal function family (e.g. DFT)

Approximate Synchronized Correlation

x1 x2 x3 x4 x5 x6 x7 x8

f1(1) f1(2) f1(3) f1(4) f1(5) f1(6) f1(7) f1(8)

f2(1) f2(2) f2(3) f2(4) f2(5) f2(6) f2(7) f2(8)

f3(1) f3(2) f3(3) f3(4) f3(5) f3(6) f3(7) f3(8)

x

x

x

c

c

c

3

2

1



x1 x2 x3 x4 x5 x6 x7 x8xxx ccc 321 ,,



x1 x2 x3 x4 x5 x6 x7 x8xxx ccc 321 ,,

y1 y2 y3 y4 y5 y6 y7 y8yyy ccc 321 ,,


• Inner product of the time series Inner product of the digests

• The time and space complexity is reduced from O(b) to O(n).– b : size of basic window– n : size of the digests (n<<b)

• e.g. 120 time points reduce to 4 digests


x1 x2 x3 x4 x5 x6 x7 x8xxx ccc 321 ,,

y1 y2 y3 y4 y5 y6 y7 y8yyy ccc 321 ,,

i

pm pm

pmmVifif

,0

),()()(

Approximate lagged Correlation

• Inner-product with unaligned windows

• The time complexity is reduced from O(b) to O(n2) , as opposed to O(n) for synchronized correlation. Reason: terms for different frequencies are non-zero in the lagged case.

sliding window

sliding window

2

Grid Structure(to avoid checking all pairs)

• The DFT coefficients yields a vector.

• High correlation => closeness in the vector space

– We can use a grid structure and look in the neighborhood, this will return a super set of highly correlated pairs.

x

Empirical Study : Speed

Comparison of processing time

0

100

200

300400

500

600

700

800

200 400 600 800 1000 1200 1400 1600

Number of Streams

Wa

ll C

loc

k T

ime

(s

ec

on

ds

)

Exact

DFT

Our algorithm is parallelizable.

Empirical Study: Precision• Approximation errors

– Larger size of digests, larger size of sliding window and smaller size of basic window give better approximation

– The approximation errors are small for the stock data.

0.51

2

0.5

1

23

0

0.001

0.002

0.003

0.004

0.005

Ave

rag

e A

pp

roxi

mat

ion

E

rro

r

Sliding windows (Hours)

Bas

ic

win

do

ws

(Min

ute

s)

Sketches : Random Projection

• Correlation between time series of the returns of stock – Since most stock price time series are close to random walks, their return

time series are close to white noise

– DFT/DWT can’t capture approximate white noise series because there is no clear trend (too many frequency components).

• Solution : Sketches (a form of random landmark)– Sketches pool: matrix of random variables drawn from stable distribution

– Sketches : The random projection of all time series to lower dimensions by multiplication with the same matrix

– The Euclidean distance (correlation) between time series is approximated by the distance between their sketches with a probabilistic guarantee.

Burst Detection

Burst Detection: Applications

• Discovering intervals with unusually large numbers of events.

– In astrophysics, the sky is constantly observed for high-energy particles. When a particular astrophysical event happens, a shower of high-energy particles arrives in addition to the background noise. Might last milliseconds or days…

– In telecommunications, if the number of packages lost within a certain time period exceeds some threshold, it might indicate some network anomaly. Exact duration is unknown.

– In finance, stocks with unusual high trading volumes should attract the notice of traders (or perhaps regulators).

Bursts across different window sizes in Gamma Rays

• Challenge : to discover not only the time of the burst, but also the duration of the burst.

Elastic Burst Detection: Problem Statement

• Problem: Given a time series of positive numbers x1, x2,..., xn, and a threshold function f(w), w=1,2,...,n, find the subsequences of any size such that their sums are above the thresholds:

– all 0<w<n, 0<m<n-w, such that xm+ xm+1+…+ xm+w-1 ≥ f(w)

• Brute force search : O(n^2) time

• Our shifted wavelet tree (SWT): O(n+k) time.

– k is the size of the output, i.e. the number of windows with bursts

Burst Detection: Data Structure and Algorithm

– Define threshold for node for size 2k to be threshold for window of size 1+ 2k-1

Burst Detection: Example

4 5 1 2 20 3 6 4 1 0 9 1 2 1 3 5

9 3 236 22 9

10 15 9

10 3 89 3 4

1226

3311

11 1012

4544

21

Window Size 2 3 4 5Threshold 24 26 47 50

Burst Detection: Example

4 5 1 2 20 3 6 4 1 0 9 1 2 1 3 5

9 3 236 22 9

10 15 9

10 3 89 3 4

1226

3311

11 1012

4544

21

True AlarmFalse Alarm

Window Size 2 3 4 5Threshold 24 26 47 50

False Alarms (requires work, but no errors)

p=0.000001

0

0.01

0.02

0.03

0.04

0.05

0.06

1 1.2 1.4 1.6 1.8 2

T=W/w

Fal

se A

larm

Rat

es

Empirical Study : Gamma Ray Burst

Processing time vs. Number of Windows

01000020000300004000050000600007000080000

0 10 20 30 40 50

Number of Windows

Pro

cess

ing

time

(ms)

SWT Algorithm

Direct Algorithm

Extension to other aggregates

• SWT can be used for any aggregate that is monotonic

– SUM, COUNT and MAX are monotonically increasing

• the alarm threshold is aggregate<threshold

– MIN is monotonically decreasing

• the alarm threshold is aggregate<threshold

– Spread =MAX-MIN

• Application in Finance

– Stock with burst of trading or quote(bid/ask) volume (Hammer!)

– Stock prices with high spread

Empirical Study : Stock Price Spread Burst

Processing time vs. Number of Windows

1

10

100

1000

10000

100000

1000000

0 10 20 30 40 50

Number of Windows

Pro

cess

ing

time

(ms)

SWT Algorithm

Direct Algorithm

Extension to high dimensions

Elastic Burst in two dimensions

• Population Distribution in the US

How to find the threshold for Elastic Burst?

• Suppose that the moving sum of a time series is a random variable from a normal distribution.

• Let the number of bursts in the time series within sliding window size w be So(w) and its expectation be Se(w).

– Se(w) can be computed from the historical data.

• Given a threshold probability p, we set the threshold of burst f(w) for window size w such that Pr[So(w) ≥ f(w)] ≤p.

Find threshold for Elastic Bursts

• Φ(x) is the normal cdf, so symmetric around 0:

• Therefore

)(1)()( 1 pwSwf e

Φ(x)

x p

Φ-1(p)

ppX

ppX

)](Pr[

)](Pr[1

1

ppwS

wSwS

e

eo

)()(

)()(Pr 1

Summary• Able to detect bursts of many different durations in

essentially linear time.• Can be used both for time series and for spatial

searching.• Can specify thresholds either with absolute numbers or

with probability of hit.• Algorithm is simple to implement and has low

constants (code is available).• Ok, it’s embarrassingly simple.

With a Little Help From My Warped Correlation

• Karen’s humming Match:

• Dennis’s humming Match:

• “What would you do if I sang out of tune?"• Yunyue’s humming Match:

Related Work in Query by Humming

• Traditional method: String Matching [Ghias et. al. 95, McNab et.al. 97,Uitdenbgerd and Zobel 99]– Music represented by string of pitch directions: U, D, S (degenerated

interval)– Hum query is segmented to discrete notes, then string of pitch directions – Edit Distance between hum query and music score

• Problem– Very hard to segment the hum query– Partial solution: users are asked to hum articulately

• New Method : matching directly from audio [Mazzoni and Dannenberg 00]

• Problem– slowed down by DTW

Time Series Representation of Query

• An example hum query

• Note segmentation is hard!

0

10

20

30

40

50

60

70

0 1 2 3 4 5 6 7 8 9 10 11

time (seconds)

pit

ch v

alu

esSegment this!

How to deal with poor hum queries?

• No absolute pitch– Solution: the average pitch is subtracted

• Incorrect tempo– Solution: Uniform Time Warping

• Inaccurate pitch intervals– Solution: return the k-nearest neighbors

• Local timing variations– Solution: Dynamic Time Warping

Dynamic Time Warping

• Euclidean distance: sum of point-by-point distance

• DTW distance: allowing stretching or squeezing the time axis locally

Time Series 1

Time Series 2

Envelope Transform using Piecewise Aggregate Approximation(PAA) [Keogh VLDB 02]

Original time series

Upper envelope

Lower envelope

U_Keogh

L_Keogh

Envelope Transform using Piecewise Aggregate Approximation(PAA)


Upper envelope

Lower envelope

U_Keogh

L_Keogh


Upper envelope

Lower envelope

U_new

L_new

• Advantage of tighter envelopes – Still no false negatives, and fewer false positives

Container Invariant Envelope Transform

• Container-invariant A transformation T for envelope such that

• Theorem: if a transformation is Container-invariant and Lower-bounding, then the distance between transformed times series x and transformed envelope of y lower bound their DTW distance.

Feature Space

The VisionAbility to match time series quickly may

open up entire new application areas, e.g. fast reaction to external events, music by humming and so on.

Main problems: accuracy, excessive specification.

Reference (advert): High Performance Discovery in Time Series (Springer 2004)

high performance discovery from time series streams dennis shasha joint work with yunyue zhu...

Documents

window basic window

digests basic window

size of sliding window

system response time

work time series

time series surprise

sum dft coefs slide

basic windows innerproduct