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Fast Algorithms for Time Series with applications to Finance, Physics, Music and other Suspects

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Title: Fast Algorithms for Time Series with applications to Finance, Physics, Music and other Suspects


1
Fast Algorithms for Time Series with applications
to Finance, Physics, Music and other Suspects
Dennis Shasha Joint work with Yunyue Zhu,
Xiaojian Zhao, Zhihua Wang, and Alberto
Lerner shasha,yunyue, xiaojian, zhihua,
lerner_at_cs.nyu.edu Courant Institute, New
York University
2
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.

3
Sample Needs
  • Pairs Trading in Finance find two stocks that
    track one another closely. When they go out of
    correlation, buy one and sell the other.
  • Match a persons humming against a database of
    songs to help him/her buy a song.
  • Find bursts of activity even when you dont know
    the window size over which to measure.
  • Query and manipulate ordered data.

4
Why Speed Is Important
  • Person on the street As processors speed up,
    algorithmic efficiency no longer matters
  • True if problem sizes stay same.
  • They dont. As processors speed up, sensors
    improve e.g. satellites spewing out a terabyte
    a day, magnetic resonance imagers give higher
    resolution images, etc.
  • Desire for real time response to queries.

5
Surprise, surprise
  • More data, real-time response, increasing
    importance of correlation
  • IMPLIES
  • Efficient algorithms and data management more
    important than ever!

6
Corollary
  • Important area, lots of new problems.
  • Small advertisement High Performance Discovery
    in Time Series (Springer 2004). At this
    conference.

7
Outline
  • Correlation across thousands of time series
  • Query by humming correlation shifting
  • Burst detection when you dont know window size
  • Aquery a query language for time series.

8
Real-time Correlation Across Thousands (and
scaling) of Time Series
9
Scalable Methods for Correlation
  • Compress streaming data into moving synopses.
  • Update the synopses in constant time.
  • Compare synopses in near linear time with respect
    to number of time series.
  • Use transforms simple data structures.
  • (Avoid curse of dimensionality.)

10
GEMINI framework
Faloutsos, C., Ranganathan, M. Manolopoulos,
Y. (1994). Fast subsequence matching in
time-series databases. In proceedings of the ACM
SIGMOD Int'l Conference on Management of Data.
Minneapolis, MN, May 25-27. pp 419-429.
11
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
  • Querywhich 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
12
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

13
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

14
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

15
StatStream Naïve Approach
  • Goal find most highly correlated stream pairs
    over sliding windows
  • N number of streams
  • w size of sliding window
  • space O(N) and time O(N2w) .
  • Suppose that the streams are updated every
    second.
  • With a Pentium 4 PC, the exact computing method
    can monitor only 700 streams, where each result
    is produced with a separation of two minutes.
  • Note Punctuated result model not continuous,
    but online.

16
StatStream Our Approach
  • Use Discrete Fourier Transform to approximate
    correlation as in Gemini approach.
  • Every two minutes (basic window size), update
    the DFT for each time series over the last hour
    (window size)
  • Use grid structure to filter out unlikely pairs
  • Our approach can report highly correlated pairs
    among 10,000 streams for the last hour with a
    delay of 2 minutes. So, at 202, find highly
    correlated pairs between 1 PM and 2 PM. At 204,
    find highly correlated pairs between 102 and
    202 PM etc.

17
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 i1
  • The basic window time is the system response
    time.
  • Digests

18
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 i1
  • The basic window time is the system response
    time.
  • Digests

Basic window digests sum DFT coefs
19
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 i1
  • The basic window time is the system response
    time.
  • Digests

Basic window digests sum DFT coefs
Sliding window digests sum DFT coefs
20
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 i1
  • The basic window time is the system response
    time.
  • Digests

Basic window digests sum DFT coefs
Sliding window digests sum DFT coefs
21
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 i1
  • The basic window time is the system response
    time.
  • Digests

Basic window digests sum DFT coefs
Basic window digests sum DFT coefs
Basic window digests sum DFT coefs
Time point
Basic window
22
How general technique is applied
  • Compress streaming data into moving synopses
    Discrete Fourier Transform.
  • Update the synopses in time proportional to
    number of coefficients basic window idea.
  • Compare synopses in real time compare DFTs.
  • Use transforms simple data structures (grid
    structure).

23
Synchronized Correlation Uses Basic Windows
  • Inner-product of aligned basic windows

Stream x
Stream y
Basic window
Sliding window
  • Inner-product within a sliding window is the sum
    of the inner-products in all the basic windows in
    the sliding window.

24
Approximate Synchronized Correlation
  • Approximate with an orthogonal function family
    (e.g. DFT)

x1 x2 x3 x4 x5
x6 x7 x8
25
Approximate Synchronized Correlation
  • Approximate with an orthogonal function family
    (e.g. DFT)

x1 x2 x3 x4 x5
x6 x7 x8
26
Approximate Synchronized Correlation
  • Approximate with an orthogonal function family
    (e.g. DFT)

x1 x2 x3 x4 x5
x6 x7 x8
27
Approximate Synchronized Correlation
  • Approximate with an orthogonal function family
    (e.g. DFT)
  • 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 (nltltb)
  • e.g. 120 time points reduce to 4 digests

x1 x2 x3 x4 x5
x6 x7 x8
28
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.

29
Grid Structure(to avoid checking all pairs)
  • The DFT coefficients yields a vector.
  • High correlation gt 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.

30
Empirical Study Speed
Our algorithm is parallelizable.
31
Empirical Study Accuracy
  • Approximation errors
  • Larger size of digests, larger size of sliding
    window and smaller size of basic window give
    better approximation
  • The approximation errors (mistake in correlation
    coef) are small.

32
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 cant capture approximate white noise
    series because the energy is distributed across
    many frequency components.
  • Solution Sketches (a form of random landmark)
  • Sketch pool list of random vectors drawn from
    stable distribution
  • Sketch The list of inner products from a data
    vector to the sketch pool.
  • The Euclidean distance (correlation) between time
    series is approximated by the distance between
    their sketches with a probabilistic guarantee.
  • W.B.Johnson and J.Lindenstrauss. Extensions of
    Lipshitz mapping into hilbert space. Contemp.
    Math.,26189-206,1984
  • D. Achlioptas. Database-friendly random
    projections. In Proceedings of the twentieth ACM
    SIGMOD-SIGACT-SIGART symposium on Principles of
    database systems, ACM Press,2001

33
Sketches Intuition
  • You are walking in a sparse forest and you are
    lost.
  • You have an old-time cell phone without GPS.
  • You want to know whether you are close to your
    friend.
  • You identify yourself as 100 meters from the
    pointy rock, 200 meters from the giant oak etc.
  • If your friend is at similar distances from
    several of these landmarks, you might be close to
    one another.
  • The sketch is just the set of distances.

34
Sketches Random Projection
inner product
sketches
random vector
raw time series
35
Sketches approximate distance well (Real
distance/sketch distance)
(Sliding window size256 and sketch size80)
36
Empirical Study Sketch on Price and Return Data
  • DFT and DWT work well for prices (todays price
    is a good predictor of tomorrows)
  • But badly for returns (todayprice
    yesterdayprice)/todayprice.
  • Data length256 and the first 14 DFT coefficients
    are used in the distance computation, db2 wavelet
    is used here with coefficient size16 and sketch
    size is 64

37
Empirical Comparison DFT, DWT and Sketch
38
Empirical Comparison DFT, DWT and Sketch
39
Sketch Guarantees
  • Note Sketches do not provide approximations of
    individual time series window but help make
    comparisons.
  • Johnson-Lindenstrauss Lemma
  • For any and any integer n, let k be
    a positive integer such that
  • Then for any set V of n points in , there
    is a map such that for all
  • Further this map can be found in randomized
    polynomial time

40
Overcoming curse of dimensionality
  • May need many random projections.
  • Can partition sketches into disjoint pairs or
    triplets and perform comparisons on those.
  • Each such small group is placed into an index.
  • Algorithm must adapt to give the best results.

Idea from P.Indyk,N.Koudas, and S.Muthukrishnan.
Identifying representative trends in massive
time series data sets using sketches. VLDB 2000.
41
Inner product with random vectors
r1,r2,r3,r4,r5,r6
42
(No Transcript)
43
Further Performance Improvements
  • -- Suppose we have R random projections of window
    size WS.
  • -- Might seem that we have to do RWS work for
    each timepoint for each time series.
  • -- In ongoing work with colleague Richard Cole,
    we show that we can cut this down by use of
    convolution and an oxymoronic notion of
    structured random vectors.

Idea from Dimitris Achlioptas,
Database-friendly Random Projections,
Proceedings of the twentieth ACM
SIGMOD-SIGACT-SIGART symposium on Principles of
database systems
44
Empirical Study Speed
  • Sketch/DFTGrid structure
  • Sliding Window Size3616, basic window size32
  • Correlationgt0.9

45
Empirical Study Breakdown
46
Empirical Study Breakdown
47
Query by humming Correlation Shifting
48
Query By Humming
  • You have a song in your head.
  • You want to get it but dont know its title.
  • If youre not too shy, you hum it to your friends
    or to a salesperson and you find it.
  • They may grimace, but you get your CD?

49
With a Little Help From My Warped Correlation
  • Karens humming Match
  • Denniss humming Match
  • What would you do if I sang out of tune?"
  • Yunyues humming Match

50
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
  • We use both.

51
Time Series Representation of Query
Segment this!
  • An example hum query
  • Note segmentation is hard!

52
How to deal with poor hummers?
  • No absolute pitch
  • Solution the average pitch is subtracted
  • Inaccurate pitch intervals
  • Solution return the k-nearest neighbors
  • Incorrect overall tempo
  • Solution Uniform Time Warping
  • Local timing variations
  • Solution Dynamic Time Warping
  • Bottom line timing variations take us beyond
    Euclidean distance.

53
Dynamic Time Warping
  • Euclidean distance sum of point-by-point
    distance
  • DTW distance allowing stretching or squeezing
    the time axis locally

54
Envelope Transform using Piecewise Aggregate
Approximation(PAA) Keogh VLDB 02
55
Envelope Transform using Piecewise Aggregate
Approximation(PAA)
  • Advantage of tighter envelopes
  • Still no false negatives, and fewer false
    positives

56
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.

57
Framework to Scale Up for Large Database
58
Improvement by Introducing Humming with ta
  • Solve the problem of note segmentation
  • Compare humming with la and ta

Idea from N. Kosugi et al A pratical
query-by-humming system for a large music
database ACM Multimedia 2000
59
Improvement by Introducing Humming with ta(2)
  • Still use DTW distance to tolerate poor humming
  • Decrease the size of time series by orders of
    magnitude.
  • Thus reduce the computation of DTW distance

60
Statistics-Based Filters
  • Low dimensional statistic feature comparison
  • Low computation cost comparing to DTW distance
  • Quickly filter out true negatives
  • Example
  • Filter out candidates whose note length is much
    larger/smaller than the querys note length
  • More
  • Standard Derivation of note value
  • Zero crossing rate of note value
  • Number of local minima of note value
  • Number of local maxima of note value

Intuition from Erling Wold et al Content-based
classification, search and retrieval of audio
IEEE Multimedia 1996 http//www.musclefish.com
61
Boosting Statistics-Based Filters
  • Characteristics of statistics-based filters
  • Quick but weak classifier
  • May have false negatives/false positives
  • Ideal candidates for boosting
  • Boosting
  • An algorithm for constructing a strong
    classifier using only a training set and a set of
    weak classification algorithm
  • A particular linear combination of these weak
    classifiers is used as the final classifier which
    has a much smaller probability of
    misclassification

Cynthia Rudin et al On the Dynamics of
Boosting In Advances in Neural Information
Processing Systems 2004
62
Verify Rhythm Alignment in the Query Result
  • Nearest-N search only used melody information
  • Will A. Arentz et al suggests combining rhythm
    and melody
  • Results are generally better than using only
    melody information
  • Not appropriate when the sum of several notes
    duration in the query may be related to duration
    of one note in the candidate
  • Our method
  • First use melody information for DTW distance
    computing
  • Merge durations appropriately based on the note
    alignment
  • Reject candidates which have bad rhythm alignment

Will Archer Arentz Methods for retrieving
musical information based on rhythm and pitch
correlation CSGSC 2003
63
Experiments Setup
  • Data Set
  • 1049 songs Beatles, American Rock and Pop, one
    Chinese song
  • 73,051 song segments
  • Query Algorithms Comparison
  • TS matching pitch-contour time series using DTW
    distance and envelope filters
  • NS matching ta-based note-sequence using DTW
    distance and envelope filters plus length and
    standard-derivation based filters
  • NS2 NS plus boosted statistics-based filters and
    alignment verifiers
  • Training Set Human humming query set
  • 17 ta-style humming clips from 13 songs (half
    are Beatles songs)
  • The number of notes varies from 8 to 25, average
    15
  • The matching song segments are labeled by
    musicians
  • Test Set Simulated queries
  • Queries are generated based on 1000 randomly
    selected segments in database
  • The following error types are simulated
    different keys/tempos, inaccurate pitch
    intervals, varying tempos, missing/extra notes
    and background noise

The error model is experimental and the error
parameters are based on the analysis of training
set
64
Experiments Result
  • NS performs better than TS when the scale is
    large
  • NS2 is roughly 610 times faster than NS with a
    small loss of hit rate at about 1
  • NS2 can achieve
  • A high hit rate 97.0 (Top10)
  • Quick responses 0.42 seconds on average , 2
    seconds in a worst case scenario

65
Query by Humming Demo
  • 1039 songs (73051 note/duration sequences)

66
Burst detection when window size is unknown
67
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).

68
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.

69
Burst Detection Challenge
  • Single stream problem.
  • What makes it hard is we are looking at multiple
    window sizes at the same time.
  • Naïve approach is to do this one window size at a
    time.

70
Elastic Burst Detection Problem Statement
  • Problem Given a time series of positive numbers
    x1, x2,..., xn, and a threshold function f(w),
    w1,2,...,n, find the subsequences of any size
    such that their sums are above the thresholds
  • all 0ltwltn, 0ltmltn-w, such that xm xm1 xmw-1
    f(w)
  • Brute force search O(n2) time
  • Our shifted binary tree (SBT) O(nk) time.
  • k is the size of the output, i.e. the number of
    windows with bursts

71
Burst Detection Data Structure and Algorithm
  • Define threshold for node for size 2k to be
    threshold for window of size 1 2k-1

72
Burst Detection Example
73
Burst Detection Example
True Alarm
False Alarm
74
Burst Detection Algorithm
  • In linear time, determine whether any node in SBT
    indicates an alarm.
  • If so, do a detailed search to confirm (true
    alarm) or deny (false alarm) a real burst.
  • In on-line version of the algorithm, need keep
    only most recent node at each level.

75
False Alarms (requires work, but no errors)
76
Empirical Study Gamma Ray Burst
77
Case Study Burst Detection(1)
Background Motivation 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. An unusual
event burst may signal an event interesting to
physicists.
Technical Overview 1.The sky is partitioned into
1800900 buckets. 2.14 Sliding window lengths are
monitored from 0.1s to 39.81s 3.The original
code implements the naive algorithm.
78
Case Study Burst Detection(2)
  • The challenges
  • 1.Vast amount of data
  • 1800900 time series, so any trivial overhead
    may be accumulated to become a nontrivial
    expense.
  • 2. Unavoidable overheads of data transformations
  • Data pre-processing such as fetching and storage
    requires much work.
  • SBT trees have to be built no matter how many
    sliding windows to be investigated.
  • Thresholds are maintained over time due to the
    different background noises.
  • Hit on one bucket will affect its neighbours as
    shown in the previous figure

79
Case Study Burst Detection(3)
  • Our solutions
  • 1. Combine near buckets into one to save space
    and processing time. If any alarms reported for
    this large bucket, go down to see each small
    components (two level detailed search).
  • 2. Special implementation of SBT tree
  • Build the SBT tree only including those levels
    covering the sliding windows
  • Maintain a threshold tree for the sliding
    windows and update it over time.
  • Fringe benefits
  • 1. Adding window sizes is easy.
  • 2. More sliding windows monitored also benefit
    physicists.

80
Case Study Burst Detection(4)
Experimental results 1. Benefits improve with
more sliding windows. 2. Results consistent
across different data files. 3. SBT algorithm
runs 7 times faster than current algorithm. 4.
More improvement possible if memory limitations
are removed.
81
Extension to other aggregates
  • SBT can be used for any aggregate that is
    monotonic
  • SUM, COUNT and MAX are monotonically increasing
  • the alarm threshold is aggregateltthreshold
  • MIN is monotonically decreasing
  • the alarm threshold is aggregateltthreshold
  • Spread MAX-MIN
  • Application in Finance
  • Stock with burst of trading or quote(bid/ask)
    volume (Hammer!)
  • Stock prices with high spread

82
Empirical Study Stock Price Spread Burst
83
Extension to high dimensions

84
Elastic Burst in two dimensions
  • Population Distribution in the US

85
Can discover numeric thresholds from probability
threshold.
  • 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 PrSo(w) f(w) p.

86
Find threshold for Elastic Bursts
  • F(x) is the normal cumulative dens funct, so
    little prob at ends
  • Red and blue line touch at negative x value.
    Symmetric above.

F(x)
x
p
F-1(p)
87
Summary of Burst Detection
  • Able to detect bursts on many different window
    sizes 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, its embarrassingly simple.

88
AQuery A Database System for Order
89
Time Series and DBMSs
  • Usual approach is to store time series as a User
    Defined Datatype (UDT) and provide methods for
    manipulating it
  • Advantages
  • series can be stored and manipulated by DBMS
  • Disadvantages
  • UDTs are somewhat opaque to the optimizer (it
    misses opportunities)
  • Operations that mix series and regular data may
    be awkward to write

90
The Case for Order1
  • Series are sequences! Support them.
  • Add order to multiset-based DBMS2,3,4
  • AQuery is a query language (conservative SQL
    extension) and a data model that supports order
    from the ground up

1 A Call to Order, David Maier and Bennet
Vance, PODS93 2 SRQL Sorted Relational Query
Language, Ramakrishnan et al., SSDBM98 3 SEQ
A Model for Sequence Databases, Seshadri et al.,
ICDE95 4 OLAP Amendment to SQL1999
91
An Arrable-based Data Model
  • Arrable a collection of named arrays with same
    cardinality
  • Arrays vector of basic types or vectors thereof
    (no further nesting!)
  • Arrables order may be declared
  • Two ways to accommodate series horizontally
    and vertically

Series
ts 1 2 5 9 13
ID IBM MSFT IBM IBM MSFT
Ticks
price 12.02 43.23 12.04 12.05 43.22
ts 1 5 9 2 13
ID IBM MSFT
price 12.02 12.04.12.05 43.23 43.22
ID,ts time series order
ts time series order
92
AQuerys New Language Elements
  • ASSUMING ORDER clause
  • Order is explicitly declared on a per-query basis
  • All other clauses can count on order (and
    preserve it)
  • Column-Oriented Semantics
  • AQuery R.col in a query binds to an array/vector
    (an entire column)
  • SQL R.col binds successively to the scalar field
    values in a column
  • Built-in functions for order-dependent aggs over
    columns
  • Can group by without aggregation

93
Moving Average over Arrables
sold 140 120 140 100 130
month 4 2 3 1 5
SELECT month, sold, avgs(3,sold) FROM Sales ASSUM
ING ORDER month
  • Each query defines the data ordering it wants to
    work with
  • Order is enforced in the beginning of the
    query, right after the FROM clause
  • All expressions and clauses are order preserving

sold 100 120 140 140 130
month 1 2 3 4 5
94
Moving Average over Arrables
sold 100 120 140 140 130
month 1 2 3 4 5
SELECT month, sold, avgs(3,sold) FROM Sales ASSUM
ING ORDER month
  • Variables are bound to an entire column at once,
    as opposed to a succession of its values
  • Expressions are mappings from a list of columns
    (arrays) into a column

95
Moving Average over Arrables
sold 100 120 140 140 130
3-avg 100 110 120 133 136
month 1 2 3 4 5
SELECT month, sold, avgs(3,sold) FROM Sales ASSUM
ING ORDER month
  • Several built in vector-to-vector functions
  • Avgs, for instance, takes a window size and a
    column and returns the resulting moving average
    column
  • Other v2v functions prev, next, first, last,
    sums, mins, ..., and UDFs

96
Best-Profit Query
Find the best profit one could make by buying a
stock and selling it later in the same day using
Ticks(ID, price, tradeDate, ts)
  • price 15 19 16 17 15 13 5 8 7 13 11 14 10 5
    2 5

97
Formulating the Best-Profit Query
Find the best profit one could make by buying a
stock and selling it later in the same day using
Ticks(ID, price, tradeDate, ts)
Sell at 14
Buy at 5
  • price 15 19 16 17 15 13 5 8 7 13 11 14 10 5
    2 5
  • mins(price)15 15 15 15 15 13 5 5 5 5 5 5 5 5
    2 2
  • 0 4 1 2 0 0 0 3 2 8 6 9 0 0
    0 3

98
Best-Profit Query Comparison
  • AQuery
  • SELECT max(pricemins(price))
  • FROM ticks ASSUMING timestamp
  • WHERE IDS
  • AND tradeDate1/10/03'
  • SQL1999
  • SELECT max(rdif)
  • FROM (SELECT ID,tradeDate,
  • price - min(price)
  • OVER
  • (PARTITION BY ID,
    tradeDate
  • ORDER BY timestamp
  • ROWS UNBOUNDED
  • PRECEDING) AS rdif
  • FROM Ticks ) AS t1
  • WHERE IDS
  • AND tradeDate1/10/03'

99
Best-Profit Query Comparison
  • AQuery
  • SELECT max(pricemins(price))
  • FROM ticks ASSUMING timestamp
  • WHERE IDS
  • AND tradeDate1/10/03'
  • SQL1999
  • SELECT max(rdif)
  • FROM (SELECT ID,tradeDate,
  • price - min(price)
  • OVER
  • (PARTITION BY ID,
    tradeDate
  • ORDER BY timestamp
  • ROWS UNBOUNDED
  • PRECEDING) AS rdif
  • FROM Ticks ) AS t1
  • WHERE IDS
  • AND tradeDate1/10/03'

AQuery optimizer can push down the selection
simply by testing whether selection is order
dependent or not SQL1999 optimizer has to
figure out if the selection would alter semantics
of the windows (partition), a much harder
question.
100
Best-Profit Query Performance
101
MicroArrays in a Nutshell
  • Device to analyze thousands of gene expressions
    at once
  • Base mechanism to acquire data in a number of
    genomic experiments
  • Generates large matrices and/or series that then
    have to be analyzed, quite often with statistical
    methods

each spot corresponds to a cDNA
Entire chip is exposed to stimuli (light,
nutrients, etc) and each spot reacts by producing
more mRNA (expressing) or less (repressing) than
base quantity
102
Generating Series in a Query
geneId g1 g2 g3 g4 ...
value 1.89 -0.5 1.3 0.8 ...
exprId 1 1 1 1 ...
SELECT geneID, value FROM RawExpr ASSUMING ORDER
geneID, exprID GROUP BY geneID
  • AQuery allows grouping without aggregation
  • Non-grouped columns become vector-fields
  • Queries can manipulate vector- fields in the
    same way it does other attributes

geneId g1 g2 g3 g4 ...
value 1.89 2.32 0.32 ... -0.5
1.22 1.03 ... 1.3 -0.38 -1.43 ... 0.8
0.79 3.23 ... ...
103
Taking Advantage of Ordering
SELECT geneID, value FROM RawExpr ASSUMING ORDER
geneID, exprID GROUP BY geneID
? geneID, value
each
Gby geneID
  • Group by and sort share prefix
  • Sort may be eliminated if RawExpr is in a
    convenient order1,2
  • Sort and Group by are commutative
  • either sort all and then slice the
  • groups, or group by and perform
  • one smaller sort per group

sort geneID, exprID
RawExpr
1 Bringing Order to Query Optimization,
Slivinskas et al., SIGMOD Record 31(2) 2
Fundamental Techniques for Order Optimization,
Simmen et al., SIGMOD96
104
Sort Cost Can Be Reduced
105
Manipulating Series
geneId g1 g2 g3 g4 ...
value 1.89 2.32 0.32 ... -0.5
1.22 1.03 ... 1.3 -0.38 -1.43 ... 0.8
0.79 3.23 ... ...
SELECT T1.geneID, T2.geneID FROM SeriesExpr T1,
SeriesExpr T2 WHERE corr(T1.value, T2.value)gt 0.9
  • corr() implements Pearsons correlation
  • It takes two columns of vector- fields as
    arguments
  • It returns a vector of correlation factors
  • AQuery can be easily extended with other UDFs

106
AQuery Summary
  • AQuery is a natural evolution from the Relational
    Model
  • AQuery is a concise language for querying order
  • Optimization possibilities are vast
  • Applications to Finance, Physics, Biology,
    Network Management, ...

107
Overall Summary
  • Improved technology implies better sensors and
    more data.
  • Real-time response is very useful and profitable
    in applications ranging from astrophysics to
    finance.
  • These require better algorithms and better data
    management.
  • This is a great field for research and teaching.
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