Data Mining using Fractals (fractals for fun and profit)

1
Data Mining using Fractals(fractals for fun and
profit)

• Christos Faloutsos
• Carnegie Mellon University

2
Overview

• Goals/ motivation find patterns in large
  datasets
• (A) Sensor data
• (B) network/graph data
• Solutions self-similarity and power laws
• Discussion

3
Applications of sensors/streams

• Smart house monitoring temperature, humidity
  etc
• Financial, sales, economic series

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Applications of sensors/streams

• Smart house monitoring temperature, humidity
  etc
• Financial, sales, economic series

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Motivation - Applications

• Medical ECGs blood pressure etc monitoring
• Scientific data seismological astronomical
  environment / anti-pollution meteorological

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Motivation - Applications (contd)

• civil/automobile infrastructure
• bridge vibrations Oppenheim02
• road conditions / traffic monitoring

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Motivation - Applications (contd)

• Computer systems
• web servers (buffering, prefetching)
• network traffic monitoring
• ...

http//repository.cs.vt.edu/lbl-conn-7.tar.Z
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Problem definition

• Given one or more sequences
• x1 , x2 , , xt , (y1, y2, , yt, )
• Find
• patterns clusters outliers forecasts

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Problem 1
bytes

• Find patterns, in large datasets

time
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Problem 1
bytes

• Find patterns, in large datasets

time
Poisson indep., ident. distr
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Problem 1
bytes

• Find patterns, in large datasets

time
Poisson indep., ident. distr
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Problem 1
bytes

• Find patterns, in large datasets

time
Poisson indep., ident. distr
Q Then, how to generate such bursty traffic?
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Overview

• Goals/ motivation find patterns in large
  datasets
• (A) Sensor data
• (B) network/graph data
• Solutions self-similarity and power laws
• Discussion

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Problem 2 - network and graph mining

• How does the Internet look like?
• How does the web look like?
• What constitutes a normal social network?
• What is the network value of a customer?
• which gene/species affects the others the most?

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Network and graph mining
Food Web Martinez 91
Protein Interactions genomebiology.com
Friendship Network Moody 01
Graphs are everywhere!
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Problem2

• Given a graph

• which node to market-to / defend / immunize
  first?
• Are there un-natural sub-graphs? (eg.,
  criminals rings)?

from Lumeta ISPs 6/1999
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Solutions

• New tools power laws, self-similarity and
  fractals work, where traditional assumptions
  fail
• Lets see the details

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Overview

• Goals/ motivation find patterns in large
  datasets
• (A) Sensor data
• (B) network/graph data
• Solutions self-similarity and power laws
• Discussion

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What is a fractal?

• self-similar point set, e.g., Sierpinski
  triangle

zero area (3/4)inf infinite length! (4/3)inf
...
Q What is its dimensionality??
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What is a fractal?

• self-similar point set, e.g., Sierpinski
  triangle

zero area (3/4)inf infinite length! (4/3)inf
...
Q What is its dimensionality?? A log3 / log2
1.58 (!?!)
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Intrinsic (fractal) dimension

• Q fractal dimension of a line?

• Q fd of a plane?

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Intrinsic (fractal) dimension

• Q fractal dimension of a line?
• A nn ( lt r ) r1
• (power law yxa)

• Q fd of a plane?
• A nn ( lt r ) r2
• fd slope of (log(nn) vs.. log(r) )

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Sierpinsky triangle
correlation integral CDF of pairwise
distances
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Observations Fractals lt-gt power laws

• Closely related
• fractals ltgt
• self-similarity ltgt
• scale-free ltgt
• power laws ( y xa
• FK r-2)
• (vs ye-ax or yxab)

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Outline

• Problems
• Self-similarity and power laws
• Solutions to posed problems
• Discussion

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Solution 1 traffic

• disk traces self-similar (also Leland94)
• How to generate such traffic?

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Solution 1 traffic

• disk traces (80-20 law) multifractals

bytes
time
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80-20 / multifractals
20
80
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80-20 / multifractals
20
80

• p (1-p) in general
• yes, there are dependencies

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Overview

• Goals/ motivation find patterns in large
  datasets
• (A) Sensor data
• (B) network/graph data
• Solutions self-similarity and power laws
• sensor/traffic data
• network/graph data
• Discussion

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Problem 2 - topology

• How does the Internet look like? Any rules?

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Patterns?

• avg degree is, say 3.3
• pick a node at random guess its degree, exactly
  (-gt mode)

count
?
avg 3.3
degree
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Patterns?

• avg degree is, say 3.3
• pick a node at random guess its degree, exactly
  (-gt mode)
• A 1!!

count
avg 3.3
degree
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Patterns?

• avg degree is, say 3.3
• pick a node at random - what is the degree you
  expect it to have?
• A 1!!
• A very skewed distr.
• Corollary the mean is meaningless!
• (and std -gt infinity (!))

count
avg 3.3
degree
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Solution2 Rank exponent R

• A1 Power law in the degree distribution
  SIGCOMM99

internet domains
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Power laws - discussion

• do they hold, over time?
• do they hold on other graphs/domains?

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Power laws - discussion

• do they hold, over time?
• Yes! for multiple years Siganos
• do they hold on other graphs/domains?
• Yes!
• web sites and links Tomkins, Barabasi
• peer-to-peer graphs (gnutella-style)
• who-trusts-whom (epinions.com)

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Time Evolution rank R
Domain level

• The rank exponent has not changed! Siganos

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The Peer-to-Peer Topology
count
Jovanovic
degree

• Number of immediate peers ( degree), follows a
  power-law

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epinions.com

• who-trusts-whom Richardson Domingos, KDD 2001

count
(out) degree
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Why care about these patterns?

• better graph generators BRITE, INET
• for simulations
• extrapolations
• abnormal graph and subgraph detection

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Recent discoveries KDD05

• How do graphs evolve?
• degree-exponent seems constant - anything else?

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Evolution of diameter?

• Prior analysis, on power-law-like graphs, hints
  that
• diameter O(log(N)) or
• diameter O( log(log(N)))
• i.e.., slowly increasing with network size
• Q What is happening, in reality?

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Evolution of diameter?

• Prior analysis, on power-law-like graphs, hints
  that
• diameter O(log(N)) or
• diameter O( log(log(N)))
• i.e.., slowly increasing with network size
• Q What is happening, in reality?
• A It shrinks(!!), towards a constant value

x
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Shrinking diameter
diameter

• Leskovec05a
• Citations among physics papers
• 11yrs _at_ 2003
• 29,555 papers
• 352,807 citations
• For each month M, create a graph of all citations
  up to month M

time
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Shrinking diameter

• Authors publications
• 1992
• 318 nodes
• 272 edges
• 2002
• 60,000 nodes
• 20,000 authors
• 38,000 papers
• 133,000 edges

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Shrinking diameter

• Patents citations
• 1975
• 334,000 nodes
• 676,000 edges
• 1999
• 2.9 million nodes
• 16.5 million edges
• Each year is a datapoint

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Shrinking diameter

• Autonomous systems
• 1997
• 3,000 nodes
• 10,000 edges
• 2000
• 6,000 nodes
• 26,000 edges
• One graph per day

diameter
N
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Temporal evolution of graphs

• N(t) nodes E(t) edges at time t
• suppose that
• N(t1) 2 N(t)
• Q what is your guess for
• E(t1) ? 2 E(t)

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Temporal evolution of graphs

• N(t) nodes E(t) edges at time t
• suppose that
• N(t1) 2 N(t)
• Q what is your guess for
• E(t1) ? 2 E(t)
• A over-doubled!

x
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Temporal evolution of graphs

• A over-doubled - but obeying
• E(t) N(t)a for all t
• where 1ltalt2

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Densification Power Law

• ArXiv Physics papers
• and their citations

1.69
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Densification Power Law

• ArXiv Physics papers
• and their citations

1
1.69
tree
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Densification Power Law

• ArXiv Physics papers
• and their citations

clique
2
1.69
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Densification Power Law

• U.S. Patents, citing each other

1.66
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Densification Power Law

• Autonomous Systems

1.18
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Densification Power Law

• ArXiv authors papers

1.15
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Outline

• problems
• Fractals
• Solutions
• Discussion
• what else can they solve?
• how frequent are fractals?

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What else can they solve?

• separability KDD02
• forecasting CIKM02
• dimensionality reduction SBBD00
• non-linear axis scaling KDD02
• disk trace modeling PEVA02
• selectivity of spatial/multimedia queries
  PODS94, VLDB95, ICDE00
• ...

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Problem 3 - spatial d.m.

• Galaxies (Sloan Digital Sky Survey w/ B. Nichol)

• - spiral and elliptical galaxies
• - patterns? (not Gaussian not uniform)
• attraction/repulsion?
• separability??

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Solution3 spatial d.m.
CORRELATION INTEGRAL!
log(pairs within ltr )
- 1.8 slope - plateau! - repulsion!
ell-ell
spi-spi
spi-ell
log(r)
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Solution3 spatial d.m.
w/ Seeger, Traina, Traina, SIGMOD00
log(pairs within ltr )
- 1.8 slope - plateau! - repulsion!
ell-ell
spi-spi
spi-ell
log(r)
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Solution3 spatial d.m.
Heuristic on choosing of clusters
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Solution3 spatial d.m.
log(pairs within ltr )
- 1.8 slope - plateau! - repulsion!
ell-ell
spi-spi
spi-ell
log(r)
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Outline

• problems
• Fractals
• Solutions
• Discussion
• what else can they solve?
• how frequent are fractals?

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Fractals power laws

• appear in numerous settings
• medical
• geographical / geological
• social
• computer-system related
• ltand many-many more! see Mandelbrotgt

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Fractals Brain scans

• brain-scans

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More fractals

• periphery of malignant tumors 1.5
• benign 1.3
• Burdet

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More fractals

• cardiovascular system 3 (!) lungs 2.9

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Fractals power laws

• appear in numerous settings
• medical
• geographical / geological
• social
• computer-system related

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More fractals

• Coastlines 1.2-1.58

1.1
1
1.3
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More fractals

• the fractal dimension for the Amazon river is
  1.85 (Nile 1.4)
• ems.gphys.unc.edu/nonlinear/fractals/examples.htm
  l

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More fractals

• the fractal dimension for the Amazon river is
  1.85 (Nile 1.4)
• ems.gphys.unc.edu/nonlinear/fractals/examples.htm
  l

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GIS points

• Cross-roads of Montgomery county
• any rules?

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GIS

• A self-similarity
• intrinsic dim. 1.51

log(pairs(within lt r))
log( r )
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ExamplesLB county

• Long Beach county of CA (road end-points)

log(pairs)
log(r)
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More power laws

• Energy of earthquakes (Gutenberg-Richter law)
  simscience.org

Energy released
log(count)
Magnitude log(energy)
day
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Fractals power laws

• appear in numerous settings
• medical
• geographical / geological
• social
• computer-system related

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A famous power law Zipfs law
log(freq)
a

• Bible - rank vs. frequency (log-log)

the
Rank/frequency plot
log(rank)
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TELCO data
count of customers
best customer
of service units
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SALES data store96
count of products
aspirin
units sold
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Olympic medals (Sidney00, Athens04)
log(medals)
log( rank)
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Olympic medals (Sidney00, Athens04)
log(medals)
log( rank)
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Even more power laws

• Income distribution (Paretos law)
• size of firms
• publication counts (Lotkas law)

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Even more power laws

• library science (Lotkas law of publication
  count) and citation counts (citeseer.nj.nec.com
  6/2001)

log(count)
Ullman
log(citations)
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Even more power laws

• web hit counts w/ A. Montgomery

yahoo.com
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Fractals power laws

• appear in numerous settings
• medical
• geographical / geological
• social
• computer-system related

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Power laws, contd

• In- and out-degree distribution of web sites
  Barabasi, IBM-CLEVER

log indegree
from Ravi Kumar, Prabhakar Raghavan, Sridhar
Rajagopalan, Andrew Tomkins
- log(freq)
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Power laws, contd

• In- and out-degree distribution of web sites
  Barabasi, IBM-CLEVER

log(freq)
from Ravi Kumar, Prabhakar Raghavan, Sridhar
Rajagopalan, Andrew Tomkins
log indegree
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Power laws, contd

• In- and out-degree distribution of web sites
  Barabasi, IBM-CLEVER

log(freq)
Q how can we use these power laws?
log indegree
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Power laws, contd

• In- and out-degree distribution of web sites
  Barabasi, IBM-CLEVER
• length of file transfers CrovellaBestavros
  96
• duration of UNIX jobs

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Conclusions

• Fascinating problems in Data Mining find
  patterns in
• sensors/streams
• graphs/networks

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Conclusions - contd

• New tools for Data Mining self-similarity
  power laws appear in many cases

Bad news lead to skewed distributions (no
Gaussian, Poisson, uniformity, independence, mean,
variance)
X
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Resources

• Manfred Schroeder Chaos, Fractals and Power
  Laws, 1991

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References

• vldb95 Alberto Belussi and Christos Faloutsos,
  Estimating the Selectivity of Spatial Queries
  Using the Correlation' Fractal Dimension Proc.
  of VLDB, p. 299-310, 1995
• Broder00 Andrei Broder, Ravi Kumar , Farzin
  Maghoul1, Prabhakar Raghavan , Sridhar
  Rajagopalan , Raymie Stata, Andrew Tomkins ,
  Janet Wiener, Graph structure in the web , WWW00
• M. Crovella and A. Bestavros, Self similarity in
  World wide web traffic Evidence and possible
  causes , SIGMETRICS 96.

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References

• J. Considine, F. Li, G. Kollios and J. Byers,
  Approximate Aggregation Techniques for Sensor
  Databases (ICDE04, best paper award).
• pods94 Christos Faloutsos and Ibrahim Kamel,
  Beyond Uniformity and Independence Analysis of
  R-trees Using the Concept of Fractal Dimension,
  PODS, Minneapolis, MN, May 24-26, 1994, pp. 4-13

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References

• vldb96 Christos Faloutsos, Yossi Matias and Avi
  Silberschatz, Modeling Skewed Distributions Using
  Multifractals and the 80-20 Law Conf. on Very
  Large Data Bases (VLDB), Bombay, India, Sept.
  1996.
• sigmod2000 Christos Faloutsos, Bernhard Seeger,
  Agma J. M. Traina and Caetano Traina Jr., Spatial
  Join Selectivity Using Power Laws, SIGMOD 2000

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References

• vldb96 Christos Faloutsos and Volker Gaede
  Analysis of the Z-Ordering Method Using the
  Hausdorff Fractal Dimension VLD, Bombay, India,
  Sept. 1996
• sigcomm99 Michalis Faloutsos, Petros Faloutsos
  and Christos Faloutsos, What does the Internet
  look like? Empirical Laws of the Internet
  Topology, SIGCOMM 1999

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References

• Leskovec 05 Jure Leskovec, Jon M. Kleinberg,
  Christos Faloutsos Graphs over time
  densification laws, shrinking diameters and
  possible explanations. KDD 2005 177-187

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References

• ieeeTN94 W. E. Leland, M.S. Taqqu, W.
  Willinger, D.V. Wilson, On the Self-Similar
  Nature of Ethernet Traffic, IEEE Transactions on
  Networking, 2, 1, pp 1-15, Feb. 1994.
• brite Alberto Medina, Anukool Lakhina, Ibrahim
  Matta, and John Byers. BRITE An Approach to
  Universal Topology Generation. MASCOTS '01

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References

• icde99 Guido Proietti and Christos Faloutsos,
  I/O complexity for range queries on region data
  stored using an R-tree (ICDE99)
• Stan Sclaroff, Leonid Taycher and Marco La
  Cascia , "ImageRover A content-based image
  browser for the world wide web" Proc. IEEE
  Workshop on Content-based Access of Image and
  Video Libraries, pp 2-9, 1997.

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References

• kdd2001 Agma J. M. Traina, Caetano Traina Jr.,
  Spiros Papadimitriou and Christos Faloutsos
  Tri-plots Scalable Tools for Multidimensional
  Data Mining, KDD 2001, San Francisco, CA.

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Thank you!

• Contact info
• christos ltatgt cs.cmu.edu
• www. cs.cmu.edu /christos
• (w/ papers, datasets, code for fractal dimension
  estimation, etc)