Topological Data Analysis Visual presentation of multidimensional data sets

Talk by Edward Kibardin, Head of Data and Analytics @Base79 at Data Science London meetup

Transcript

1. Topological   Data  Analysis Visual presentation of multidimensional data sets
2. Current  vs  New SQL Topological  Data  Analysis
3. Topology The  Seven  Bridges  of  Königsberg,  a  problem  solved  by  Leonard  Euler  (1736). The  study  of  qualitative  properties  of  certain   objects  (topological  spaces)  that  are  invariant   under  a  certain  kind  of  transformation   (continuous  map),  especially  those  properties   that  are  invariant  under  a  certain  kind  of   equivalence  (homeomorphism).
4. Topology  Data  Analysis  Pipeline a b a.  First  approximate  the  unknown  space  X  in   a  combinatorial  structure  K b.  Then  compute  topological  invariants  of  K
5. Combinatorial  Representations The  Čech  Complex
6. Combinatorial  Representations Alpha  Complex Vietoris-­‐‑Rips  Complex Cubical  Complex Witness  Complex
7. Topological  Invariants A  topological  invariant  is  a  map  f    that  assigns  the  same  object  to homeomorphic  spaces,  that  is: Homology:  is  a  machine  that   converts  local  data  about  a  space   into  global  algebraic  structure Reference:  Wikipedia,  2010.
8. Morse  Theory  and  Reeb  Graph Theorem:   Suppose  h  :  X  g        is  a  discrete   Morse  function. Then  X  is  homotopy  equivalent  to  a   CW-­‐‑complex  with  exactly  one  cell  of   dimension  p  for  each  critical  simplex   of  dimension  p. Reference:  Teng  Ma  ;  Zhuangzhi  Wu  ;  Pei  Luo  ;  Lu  Feng.  Reeb  graph  computation  through  spectral  clustering,  2011.
9. Case  study:  Demographics Data  shape: [220:45]
10. Case  study:  YT  channel  stats Data  shape: [1500:12]
11. Case  study:  Netflix  dataset Data  shape: [17770:480189] 8.5  billions  of  elements
12. Case  study:  Netflix  dataset Music Indian Anime French Honk   Kong US   Cartoons Kids Movie Ger man US Retro Horror
13. Case  study:  Netflix  comparison PCA Isomap LLE Spectral  Embedding LTSA Hessian  LLE
14. Case  study:  Netflix  (music)
15. Case  study:  Netflix  (kids  movie)
16. Case  study:  Netflix  (horror)
17. info@datarefiner.com www.datarefiner.com Please  sign  up  for  free  beta  access:
18. Questions?