CISD 794

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CISD 794 Knowledge Discovery in Databases
Prof. Junping Sun

• DENCLUE - Clustering DNA Sequences using FFT
  (Fast Fourier Transform )
• Gilberto R dos Santos

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Clustering

• Cluster analysis is an unsupervised learning
  method that constitutes a cornerstone of an
  intelligent data analysis process.
• Some DNA sequences may reach several millions of
  base pairs. Any search method tried to have a
  direct hit comparing two DNA sequences may have
  an algorithm that grows Onn, being n the
  number of bases in the DNA sequence. Several
  methods are being combined to optimize sequence
  search and matching. One initial approach is to
  build clusters of DNA sequences.

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Functions in Time or Frequency

• A physical process can be described either in the
  time domain, by the values of some quantity h as
  a function of time t, e.g., h(t), or else in the
  frequency domain, where the process is specified
  by giving its amplitude H (generally a complex
  number indicating phase also) as function of
  frequency f, that is H(f), with - 8 lt f lt 8
• h(t) and H(f) are two different representations
  of the same function.
• H(f) h(t) e ( 2 p i f t ) dt
• h(t) H(f) e ( -2 p i f t ) dt

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FFT Fast Fourier Transform

• Fourier Equation
• Where An and Bn are

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DNA Sequence PDB 1b05
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DNA Sequence PDB 1b25
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DNA and FFT (DFT)

• DFT /FFT (Discrete Fourier Transformation) can
  be used to define a wave that is the closest
  representation of a DNA sequence function h(t).
  DFT transformation technique can be used to
  reduce the search time of range queries. This
  method can be used in conjunction with any other
  existing method, or can be used as a filtration
  technique. The challenge of this research is to
  define how DFT (FFT) can represent a DNA
  sequence. The proximity search seeks the
  sequences close enough to a given query sequence
  either through direct alignment or using other
  heuristics. The alignment of biological sequences
  ( pairwise or multiple alignment) is the
  operation to place nucleotide or amino acid
  residues in columns inferring the closest common
  ancestral relationships. The best alignment
  usually refers to the one demonstrating the most
  likely evolutionary scenario.

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Wave Composition
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Data Structures Used

• Data matrix
• (two modes)
• Dissimilarity matrix
• (one mode)

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Euclidean Distance

• The Euclidean Distance for the matrix d(i,j) is
•  
• Dx (i,j) sqrt ( (Xi1 Xj1 ) 2 (Xi2 Xj2
  ) 2 .. ( Xip Xjp ) 2 )
•  
• Dy (i,j) sqrt ( (Yi1 Yj1 ) 2 (Yi2 Yj2
  ) 2 .. ( Yip Yjp ) 2 )
•  
• Dz (i,j) sqrt ( (Zi1 Zj1 ) 2 (Zi2 Zj2
  ) 2 .. ( Zip Zjp ) 2 )

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Euclidean Distance

• cursor c1 is select from DNA_SEQUENCE_FREQ_XY
  where frequency gt 0 for update order by 1
• cursor c2 is select from DNA_SEQUENCE_FREQ_YZ
  where frequency gt 0 for update order by 1
• cursor c3 is select from DNA_SEQUENCE_FREQ_ZX
  where frequency gt 0 for update order by 1
• d_v_dist_a number0
• d_v_dist_b number0
• v_dist_a number0
• v_dist_b number0
• r2 dna_variance_coef_xyrowtype
• rec3 dna_variance_coef_yzrowtype
• rec4 dna_variance_coef_zxrowtype
• begin
• for r1 in c1 loop
• select into r2
• from dna_variance_coef_xy where frequency
  r1.frequency
• begin
• select / INDEX ( DNA_SEQUENCE_FREQ_XY,
  DNA_SEQUENCE_FREQ_XY_IDX1 ) /
• sum( ( r1.ACOEF - ACOEF ) exp ( -1 (
  power ( (r1.ACOEF - ACOEF) ,2) / r2.var_acoef ))
  ),
• sum( ( r1.BCOEF - BCOEF ) exp ( -1 (
  power ( (r1.BCOEF - BCOEF) ,2) / r2.var_bcoef ))
  )
• into v_dist_a,v_dist_b

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Euclidean Distance - sine coefficient Function
X ? Y

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Euclidean Distance - cosine coefficient
Function (X) ? Y
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DENCLUE Clustering Based on Density
Distribution Functions

• The influence of each data point can be formally
  modeled using a mathematical function (influence
  function), that describes the impact of a data
  point within its neighborhood.
• The overall density of the data space can be
  modeled analytically as the sum of the influence
  functions of all data points
• Clusters can then be determined mathematically by
  identifying density atractors, where density
  atractors are local maxima of the overall density
  functions.
• d(X,Y) ? should be reflexive and symetric ?
  Euclidean distance Function.
• Density Attractor/Density-Attracted Points
• -local maximum of the density function
• -density-attracted points are determined by a
  gradient-based hill-climbing method
• Center-Defined Cluster
• A center-defined cluster with density-attractor
  x ( ) is the subset of the database which is
  density-attracted by x.

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DNA Sequence 1b5s ALA Residue
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Denclue Technical Essence

• Uses grid cells but only keeps information about
  grid cells that do actually contain data points
  and manages these cells in a tree-based access
  structure.
• Influence function describes the impact of a
  data point within its neighborhood.
• Overall density of the data space can be
  calculated as the sum of the influence function
  of all data points.
• Clusters can be determined mathematically by
  identifying density attractors.
• Density attractors are local maximal of the
  overall density function.

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Gradient The steepness of a slope

• Example

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Density Attractor Function

• A point x Î is called a density-attractor
  for a given influence function, iff x a local
  maximum of the density-function
• A point x Î
• is density-attracted to a density attractor x,
• iff k Î N d ( , x) lt x with

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Center Defined Clusters

• A center-defined cluster (wrt to s , x ) for a
  density attractor x is a subset C Í D, with x Î
  C being density-attracted by x
• and (x) gt x. Points x Î D are
  called outliers if they are density-attracted by
  a local maximum
• with
• ( ) lt x

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DENCLUE AlgorithmLocal Density Function

• Two cubes c1, c2 Î Cp are connected if
• d(mean(c1) mean(c2)) lt 4s
• near(x) x1 Î c1 d(mean(c1),x) lt ks
• (note k4)

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Density Attractors
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DENCLUE Algorithm (cont.)

• After determining the density-attractor x for a
  point x and
• the point x is classified and attached to the
  cluster belonging to x.

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Gaussian Distance

• cursor c1 is select from DNA_SEQUENCE_FREQ_XY
  where frequency gt 0 for update order by 1
• cursor c2 is select from DNA_SEQUENCE_FREQ_YZ
  where frequency gt 0 for update order by 1
• cursor c3 is select from DNA_SEQUENCE_FREQ_ZX
  where frequency gt 0 for update order by 1
• d_v_dist_a number0
• d_v_dist_b number0
• v_dist_a number0
• v_dist_b number0
• r2 dna_variance_coef_xyrowtype
• rec3 dna_variance_coef_yzrowtype
• rec4 dna_variance_coef_zxrowtype
• begin
• for r1 in c1 loop
• select into r2
• from dna_variance_coef_xy where frequency
  r1.frequency
• begin
• select / INDEX ( DNA_SEQUENCE_FREQ_XY,
  DNA_SEQUENCE_FREQ_XY_IDX1 ) /
• sum(exp ( -1 ( power ( (r1.ACOEF -
  ACOEF) ,2) / r2.var_acoef )) ),
• sum( exp ( -1 ( power ( (r1.BCOEF
  - BCOEF) ,2) / r2.var_bcoef )) )

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Gaussian Distance - sine coefficient ? xy ? ALA
Residue
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Gaussian Distance - cosine coefficient ? xy ?
ALA Residue
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Gradient Distance

• cursor c1 is select from DNA_SEQUENCE_FREQ_XY
  where frequency gt 0 for update order by 1
• cursor c2 is select from DNA_SEQUENCE_FREQ_YZ
  where frequency gt 0 for update order by 1
• cursor c3 is select from DNA_SEQUENCE_FREQ_ZX
  where frequency gt 0 for update order by 1
• d_v_dist_a number0
• d_v_dist_b number0
• v_dist_a number0
• v_dist_b number0
• r2 dna_variance_coef_xyrowtype
• rec3 dna_variance_coef_yzrowtype
• rec4 dna_variance_coef_zxrowtype
• begin
• for r1 in c1 loop
• select into r2
• from dna_variance_coef_xy where frequency
  r1.frequency
• begin
• select / INDEX ( DNA_SEQUENCE_FREQ_XY,
  DNA_SEQUENCE_FREQ_XY_IDX1 ) /
• sum( ( r1.ACOEF - ACOEF ) exp ( -1 (
  power ( (r1.ACOEF - ACOEF) ,2) / r2.var_acoef ))
  ),
• sum( ( r1.BCOEF - BCOEF ) exp ( -1 (
  power ( (r1.BCOEF - BCOEF) ,2)/ r2.var_bcoef )) )

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Gradient Gaussian Distance Cos ? xy ? ALA Residue
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References

• Flexible Pattern Matching in Strings Practical
  On-line
• Search Algorithms for Texts and
  Biological Sequences
• Gonzalo Navarro and Mathieu Raffinot 
• Numerical Recipes in C
• William H. Press / Saul A. Teukolsky /
  William T. Vetterling /
• Brian P. Flannery - http//library.lanl
  .gov/numerical
• Introduction to Algorithms
• Thomas H. Cormen / Charles E.
  Leiserson / Ronald L. Rivest
• Introduction to Bioinformatics - Arthur M. Lesk
• Handbook of Algorithms and Data Structures,
• G.H. Gonnet R. Baeza-Yates.
• 6.  Data Mining Opportunities and
  Challenges - John Wang
• Who is Fourier? A Mathematical Adventure Alan
  Gleason
• 8. Mathematical Handbook for Scientists and
  Engineers
• Granino A. Korn and Theresa M. Korn
• 9. An Efficient Approach to Clustering in Large
  Multimedia Databases with Noise
• Alexander Hinneburg, Daniel A. Keim
• Data Mining Concepts and Techniques
• Jiawei Han and Micheline Kamber