Topological data analysis

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Topological data analysis

  1. 1. Sunghyon Kyeong
 Severance Biomedical Science Institute, 
 Yonsei University College of Medicine Topological Data Analysis -Methods and Examples-
  2. 2. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p Contents • Brief overview of topological data analysis • About Ayasdi 
 Startup company providing solutions for data analytics • Topological data analysis - Methods • Applications to medical science data • Applications to social science data 2
  3. 3. Brief Overview of 
 Topological Data Analysis
  4. 4. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p Machine Learning 4 •Supervised Machine Learning 
 → Classification of new input data
 (LDA, bayesian, support vector machine, neural network, and so on) •Unsupervised Machine Learning 
 → Clustering of given dataset / Community detection
 (k-means clustering, modularity optimisation, ICA, PCA, and so on) •Topological Data Analysis
 → partial clustering with allowing overlaps among clusters
  5. 5. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p World Interests for TDA 5 Heat map for viewers of my TDA slide at sliceshare (for 2500 viewers during 2015.2.14. - 2014.11.31.)
  6. 6. Data has Shape
  7. 7. Raw Data 
 (diabetes related data) An example
  8. 8. Shape has Meaning
  9. 9. Normal Type II 
 diabetes Type I 
 diabetes An example
  10. 10. Meaning drives “Values”
  11. 11. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p When to use TDA? 11 • To study complex high-dimensional data
 : feature selections are not required in TDA • Extracting shapes (patterns) of data • Insights qualitative information is needed. • Summaries are more valuable than individual parameter choices.
  12. 12. Geometric Topology Algebraic Betti0: 8 Betti1: 5 Betti0: 10 Betti1: 5 Topology Betti0: clusters Betti1: holes Betti2: voids mathematically 
 defined “holes” in data
  13. 13. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 13 Persistent homology is a spatial type of homology that is useful for data analysis. Betti numbers, which come from computing homology, reflect the topological properties of an object. B O ㅂ q b : the connected components : the number of holes Algebraic Topology Quantitive Information
  14. 14. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 14 Homology homeomorphic to homeomorphic to , Betti2 = 1 , Betti2 = 0 Ref) Xiaojin Zhu, IJCAI 2013 presentation slide
  15. 15. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 15 ? Ref) Figures are obtained from Y.P. Lum et al (2013) Scientific Reports | 3: 1236 points cloud data topology Geometric Topology Extracting Shapes of Data
  16. 16. Topological Data Analysis using Mapper Two input functions
 - filter is to collapse high-dimensional data set into a single point
 - distance as a measure of distance between data points Resolution Parameters - Intervals, overlap, magic fudge
  17. 17. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 17 Filter Filter Range Intervals Overlap Interval Length : 0.0~2.0 : 5 : 50% : 0.4 0 0.4 0.8 1.2 1.6 2.0 0.2 0.6 1.0 1.4 1.8 NumberofNodesinEachFilterBin 0 10 20 30 40 50 Filter Metric 0 1 2single cluster one or two cluster(s)? : Divide point clouds into each filter bin
  18. 18. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 18 Filter Function - Filter function is not necessarily linear projections on a data matrix. - People often uses functions that depend only on the distance function itself, such as a measure of centrality. - Some filter functions may not produce any interesting shapes. Ref) Figures are obtained from Y.P. Lum et al (2013) Scientific Reports | 3: 1236
  19. 19. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 19 Distance Function -distance between all pairs of data points. -both euclidean or geodesic distances could be used.
  20. 20. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 0 4 8 12 1 2 3 4 5 20 Distance & Clustering • Single linkage dengrogram is used for clustering point clouds based on distance between two nodes. 0 3 6 9 1 2 3 4 5 Magic Fudge is the number of bins in the distribution of the distance obtained from single linkage dendrogram. No. of clusters are estimated from the number of continuous bins having zero elements. 1 Cluster 2 Clusters
  21. 21. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 21 Nodes, Edges, Colors 1,2,3,
 4,5 0,1,2,7 5,6,9 10 Nodes are groups of similar objects Edges connect similar nodes Colors let you see values of interest 8,11 3,4,8,12 12 1412,14 1311,13 indices for points cloud: 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14
  22. 22. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 22 Topology extraction Filter Range Intervals Overlap Magic Fudge : 0 ~ 2 : 5 : 50% : 5 Filter Partial Clustering The size of node represents the number of point clouds in each node.
  23. 23. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 23 A) C) F1 F2 Filter Extract position information of y-axis F3 F4 F5 F6 F7 F8 F9 F10 B) F1 F2 F3 F4 F5 F6 F7 F8 F9 F9 F10 F10 0 10 20 30 1 2 3 4 5 9th Filter bin 1 1 1st Filter bin 0 10 20 1 2 3 4 5 11 Topology of Y-shape points cloud
  24. 24. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 24 interval = 5
 overlap = 20% interval = 5
 overlap = 50% interval = 5
 overlap = 80% interval = 10 overlap = 20% interval = 10 overlap = 50% interval = 10 overlap = 80% interval = 15 overlap = 20% interval = 15 overlap = 50% interval = 15
 overlap = 80%
  25. 25. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p Mathematical details can be found at 25 • Gurjeet Singh et al. (2007), Topological methods for the analysis of high dimensional data sets and 3D object recognition. • Gunnar Carlsson (2009), Topology and data, Bull. Amer. Math. Soc. 46. 255-308. Gurjeet Singh and Gunnar Carlsson are 
 co-founders of Ayasdi
  26. 26. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 27 US Patent by Ayasdi
  27. 27. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 28
  28. 28. Applications to 
 Neurobiological/Clinical Data International Neuroimaging Data-sharing Initiative (INDI) - URL: http://fcon_1000.projects.nitrc.org/index.html - Healthy control, ADHD, ASD data sets are available. - resting state fMRI, diffusion tensor imaging, - phenotype information such as intelligence scale and ADHD symptom severity are available.
  29. 29. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p Dataset : ADHD Symptoms & IQ 30 1 2 3 4 5 1 0.0 61.5 77.3 51.6 77.0 2 61.5 0.0 62.2 55.9 69.0 3 77.3 62.2 0.0 47.2 11.9 4 51.6 55.9 47.2 0.0 52.4 5 77.0 69.0 11.9 52.4 0.0 Data set: L2-Distance (or Euclidean distance) for all pairwise subjects: Distance Matrix: (or Euclidean) Filter Function: f(x) = max d(x, y) y 2 x L-infinity eccentricity
  30. 30. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 31 Phenotypic subgroups of ADHD AIQ-TDC
 (Average IQ) Low High TDC
 (Low IQ) Distance function: L2-distance Filter function: L-infinity eccentricity N of subjects = 204 Low resolution AIQ-ADHD
 (Average IQ & 
 Borderline Sx ) HIQ-TDC
 (High IQ) LIQ-ADHD
 (Low IQ & Ordinary Sx)
  31. 31. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 32 30 60 90 ADHD 60 30 60 90 Inattentive 59 30 60 90 Hyper/Impulsivity 59 70 100 130 FSIQ 108 70 100 130 VIQ 109 70 100 130 PIQ 105 HIQ-TDC
 (High IQ) AIQ-TDC 
 (Average IQ) LIQ-ADHD
 (Ordinary Sx) AIQ-ADHD
 (Borderline Sx) All subjects 
 (Avg. Sx & Avg. IQ) 13 (13 / 0) 21 (17 / 4) 12 (1 / 11) 19 (2 / 17) 204 (90 / 114)
  32. 32. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 33 Functional Modular Architectures AIQ-TDC 
 (TDC with Average IQ) LIQ-ADHD 
 (ADHD with Ordinary Sx.) AIQ-ADHD 
 (ADHD with Borderline Sx.) DMN module Occipital module Central module Parietofrontal module BG/THL module Temporal/Limbic module HIQ-TDC 
 (TDC with High IQ) ※ Only positive weights of the functional connectivity are used for module analysis and visualised.
  33. 33. Applications to Medical science data
  34. 34. Diabetes Subtypes based on six quantities: - age - relative weight - fasting plasma glucose - area under the plasma glucose curve for the three hour glucose tolerance test (OGTT) - area under the plasma insulin curve for the OGTT - steady state plasma glucose (SSPG) response SSPG Glucose area Insulin area Ref) Eurographics Symposium on Point-Based Graphics, Singh G et al. (2007)
  35. 35. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 36 Subtypes of diabetes Type I 
 diabetes Type II 
 diabetes Type I 
 diabetes Type II 
 diabetes Low-resolution High-resolution Interval: 3
 Overlay: 50% Interval: 4
 Overlay: 80% Type I: adult onset TypeII: juvenile onset Distance function: L2-distance Filter function: density kernel with e=130,000
  36. 36. Subtypes of Breast Cancer using breast cancer microarray gene expression data set - disease specific genomic analysis (DSGA) transformed data Application to Biological Data, PNAS, Monica Nicolau et al. (2010)
  37. 37. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 38 Breast Cancer Subtype ER: Estrogen Receptor PNAS, Monica Nicolau et al. (2010)
  38. 38. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p Diabetes Mellitus (DM) • Commonly referred to as diabetes, a group of metabolic disease. • If left untreated, diabetes can cause many complications: 
 cardiovascular disease, stroke, chronic kidney failure, foot ulcers, and damage to eyes. • Diabetes is due to either the pancreas not producing enough insulin or the cells of the body not responding properly to the insulin produced. • Type 1 DM: results from the pancreas’s failure to produce enough insulin. • Type 2 DM: begins with insulin resistance, a condition in which cells fail to respond to
 insulin properly. heavy weight and not enough exercise are causes of T2D. • Gestational diabetes, is the third main form and occurs when pregnant women without a previous history of diabetes develop a high blood-sugar level. 40
  39. 39. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p Why subgroups of type 2 DM? • Risk factors of Type 2 DM are:
 obesity, family history of diabetes, physical inactivity, ethnicity, and advanced age. • Type 2 DM is heterogenous complex disease affecting 
 more than 29 million in American (9.3%). 2 million in Korea. • Increasing needs for early prevention and clinical management of Type 2 DM. 41
  40. 40. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p Methods • High dimensional EMRs and genotype data from 11,210 individuals from Mount Sinai Medical Center (MSMC)’s outpatient population (46% Hispanic, 32% African american, 20% European white, and 2% others). • Type 2 DM and non-Type 2 DM were defined by an electronic phenotyping algorithm (eMERGE network) based on ICM-9-CM diagnosis codes, laboratory test, prescribed medications (RxNorm), physician notes (natural language processing), and family history. • Form of preprocessed data matrix is n patients by P medical variables. 
 Medical variables included 505 clinical variable, 7097 unique ICM-9-CM codes (1 to 218 per patients). On average, 64 clinical variables per patients. To avoid overfitting, select the variable with at least 50% of patients who had the variables, resulting in 73 (of 505) variables to perform the analysis. 42
  41. 41. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p TDA pipeline • Distance metric: cosine distance metric was used to assess the similarity of the data points. • Filter metric: L-infinity centrality and principal metric singular value decomposition (SVD1). 
 L-infinity centrality is defined for each data point y to be the maximum distance from y to any other data point in the data set. Large values of this function correspond to points that are far from the centre of data set. 43
  42. 42. 3889 patients 7321 patients 44
  43. 43. TDA with only 
 2551 Type II DM 762
 Subtype 1 617
 Subtype 2 1096
 Subtype 3 Gender is not an organising factor in the topology Reproducibility test (2/3 training, 1/3 test data set, 10 times) revealed that the overall accuracy was 96% for a subtype classification. 45
  44. 44. Applications to 
 Social Science Data - Classification of (basket ball) player types - Partial clustering of personality using temperamental traits - Relationship among welfare, civil construction, and suicide rate
  45. 45. Classification of player types based on their in-game performance such as: - rates (per minute played) of rebounds, assists, turnovers, steals, blocked shots, personal fouls, and points scored (7 performance measures)
  46. 46. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 48 Map of Players low resolution map at 20 intervals high resolution map at 30 intervals Points Per Game Low High Filter function: 
 Principal and secondary SVD values Distance function: 
 Variance normalised L2-distance Traditionally, basket ball players are categorised into guards, forwards, and center.
  47. 47. TDA versus K-means clustering Grouping subjects into personality types
  48. 48. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 50 Novelty seeking Harm avoidance Reward dependance Persistence Temperament and character inventory (TCI) scores from 40 normal subjects
  49. 49. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 51 the mean of points in Si V = 2X i=1 X xj 2Si ||xj µi||2 S = {S1, S2} 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Novelty Seeking HarmAvoidance Introverts Extraverts Centroids 20 30 40 50 60 70 Novelty Seeking HarmAvoidance 70 60 50 40 30 20 10 0 Subject Grouping high HA & low NS low HA & high NS Centroids× B. k-means ClusteringA. Input Dataset The goal of k-means clustering is to minimise the within-cluster 
 sum of squares.
  50. 50. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 52 TDA to extract personality groups 20 30 40 50 60 NS HA RD P Group1 Group2 Distance function: L2-distance Filter function: L-infinity eccentricity Low High Low resolution (5 intervals, 50% overlap) Group 1
 (n=7) Group 2
 (n=4) HA,NS,RD,P=(32,65,58,49) 1 1 1 High resolution (5 intervals, 70% overlap)
  51. 51. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 53 Welfare / civil engineering / suicide ratio Data Download: http://newstapa.com/news/201411935
  52. 52. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 54 Nation’s public data analysis Distance function: L2-distance Filter function: L-infinity eccentricity Input data: ratio of welfare to civil engineering (2009), ratio of welfare to civil engineering (2012), Suicide rate (2012) 부산(남구)
 인천(연수구, 남동구, 서구)
 광주(동구)
 강원(본청) 광주(북구) 전북(본청) 대구(북구) 서울(노원구) 대구(달서구) 대전(서구) 강원(홍천, 양양) 충북(단양) 전북(장수) 전남(함평) 경남(함양) Group 1 Group 3 Group 2 Group 4 Low High
  53. 53. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 55 0 1 2 3 4 5 6 7 Group1 Group3Group2 Group4 Ratio of Welfare to Civil Eng. (2009) Ratio of Welfare to Civil Eng. (2012) Group1 Group2 Group3 Group4 Suicide ratio (2012) 70 60 50 40 30 20 10 0 광주(북구) 전북(본청) 대구(북구) 서울(노원구) 대구(달서구) 대전(서구) 강원(홍천, 양양) 충북(단양) 전북(장수) 전남(함평) 경남(함양) 부산(남구)
 인천(연수구, 남동구, 서구)
 광주(동구)
 강원(본청) Blog posting: http://skyeong.tistory.com/136/ Welfare | Civil Eng. | Suicide ratio
  54. 54. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p 56 Topological Data Analysis,
 new weapon for 
 discovering new insight from data.
  55. 55. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p Conclusion • TDA can be applied to various dataset and has a coordinate free characteristic. • Useful for analysing non-linear dataset. • Selection and optimisation of distance and filter metric are important issue. • TDA will be a powerful weapon for those who want to find a new insight from data • It’s possible to make a supervised machine learning system using TDA. 57
  56. 56. Sunghyon Kyeong (Yonsei University) | sunghyon.kyeong@gmail.com | Topological Data Analysis: Methods and Examples | p References 1. Gurgeek Singh et al., Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition, Eurographics Symposium on Point-Based Graphics, 2007. 2. Gunnar Carlsson, Topology and Data, Bull. Amer. Math. Soc. 46(2):255-308, 2009. 3. Monica Nicolau et al., Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival, PNAS 108(17):7265-7270, 2011. 4. P. Y. Lum et al., Extracting insights from the shapes of complex data using topology, Nature Scientific Reports 3:1236, 2013. 5. Li Li et al., Identification of type 2 diabetes subgroups through topological analysis of patients similarity, Science Translational Medicine 7(311):311ra174, 2015. 6. Jessica L. Nielson et al., Topological data analysis for discovery in preclinical spinal cord injury and traumatic brain injury, Nature communications 6:8581, 2015. 7. AYASDI, a commercial software for TDA, http://www.ayasdi.com/ 58

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