Topological Data Analysis
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1 Mikael Vejdemo-Johansson School of Computer Science University of St Andrews Scotland 22 November 2011 M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
2 e Shape of Data Outline 1 e Shape of Data 2 Persistent Homology Signal Analysis Robotics Biomedicine Natural images analysis M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
3 e Shape of Data Shape of data Fundamentally, data analysis is the task of describing the shape of data: What is data? Data is a collection of observations gives three commonly used categories: Quantitative Measured by some (real) number Categorical Assigned to one of several possible categories Qualitative Measured by presence or absence of some characteristic A datum will be some collection of such observations There are interesting metrics for all, which allows us to de ne: A data set is a nite metric space M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
4 e Shape of Data Shape of data Fundamentally, data analysis is the task of describing the shape of data: Tasks of data analysis Summarize Provide a description that is, preferably, smaller than the dataset Model Provide a description that allows for predictions of the behaviour of the source of the data Highlight Provide emphasis on certain interesting properties of the data M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
5 e Shape of Data Shape of data Fundamentally, data analysis is the task of describing the shape of data: Fundamental data analysis techniques Mean (centroid) tells us where the data is located + M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
6 e Shape of Data Shape of data Fundamentally, data analysis is the task of describing the shape of data: Fundamental data analysis techniques Standard deviation tells us how spread out the data is + M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
7 e Shape of Data Shape of data Fundamentally, data analysis is the task of describing the shape of data: Fundamental data analysis techniques Regression analyses t the data to an easy to analyze model + M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
8 e Shape of Data Shape of data Fundamentally, data analysis is the task of describing the shape of data: Fundamental data analysis techniques Cluster analysis divides the data into its connected components M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
9 e Shape of Data Shape of data Fundamentally, data analysis is the task of describing the shape of data: Fundamental data analysis techniques Principal Component Analysis (and other dimension reduction techniques) give a new coordinate frame that more faithfully represent the data M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
10 Outline 1 e Shape of Data 2 Persistent Homology Signal Analysis Robotics Biomedicine Natural images analysis M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
11 Wherefore topology? Issues with classical data analysis Unreliable metric The metrics in use may well not be accurate measures of dissimilarity as distances grow Ill motivated metric The metrics in use may be arbitrarily chosen, not well anchored as distances grow Noisy data Data may be very noisy High-dimensional data Data may be very high-dimensional, and thus slow to process Topology only depends on a notion of nearness Produces dimension-agnostic qualitative features M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
12 Persistent Homology Discrete made continuous De nition The Vietoris-Rips complex is an abstract simplicial complex VR ϵ (X) for ϵ R + and X a nite metric space: Contains one vertex for each element in X Contains a simplex (x 0,, x k ) exactly when d(x i, d j ) < ϵ for all i, j [k] M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
13 Persistent Homology Discrete made continuous De nition The Vietoris-Rips complex is an abstract simplicial complex VR ϵ (X) for ϵ R + and X a nite metric space: Contains one vertex for each element in X Contains a simplex (x 0,, x k ) exactly when d(x i, d j ) < ϵ for all i, j [k] M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
14 Persistent Homology Discrete made continuous De nition The Vietoris-Rips complex is an abstract simplicial complex VR ϵ (X) for ϵ R + and X a nite metric space: Contains one vertex for each element in X Contains a simplex (x 0,, x k ) exactly when d(x i, d j ) < ϵ for all i, j [k] M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
15 Persistent Homology Discrete made continuous De nition The Vietoris-Rips complex is an abstract simplicial complex VR ϵ (X) for ϵ R + and X a nite metric space: Contains one vertex for each element in X Contains a simplex (x 0,, x k ) exactly when d(x i, d j ) < ϵ for all i, j [k] M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
16 Persistent Homology Discrete made continuous De nition The Vietoris-Rips complex is an abstract simplicial complex VR ϵ (X) for ϵ R + and X a nite metric space: Contains one vertex for each element in X Contains a simplex (x 0,, x k ) exactly when d(x i, d j ) < ϵ for all i, j [k] M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
17 Persistent Homology Functoriality and persistent homology M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
18 Persistent Homology Functoriality and persistent homology 1=3 M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
19 Persistent Homology Functoriality and persistent homology M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
20 Persistent Homology Functoriality and persistent homology 1=2 M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
21 Persistent Homology Functoriality and persistent homology Homology is a functor: if f : X Y is a function, there is an induced function H (f) : H X H Y If ε < ε, then VR ε (X) VR ε (X) De nition The persistent homology space H ε,ε n (VR (X)) is the subspace of the vector space H n (VR ε (X)) consisting of the image of the induced map H n (VR ε (X)) H n (VR ε (X)) M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
22 Persistent Homology Barcodes We can summarize, pictorially, the collection of persistent homology spaces as a barcode De nition The persistence barcode for a ltered complex VR (X) is a collection of pairs (s, t) (s, t) is in the barcode if there is a basis element of Hn s,t (VR (X)) not in Hn s ε,t (VR (X)) nor in H s,t+ε n (VR (X)) s and t can take values in positive reals, and The dendrogram of single linkage clustering is (almost) exactly the barcode for H 0 M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
23 Signal Analysis Delay embedding quality Delay Embedding {a x } {(a x, a x+ε,, a x+(d 1)ε } Converts a 1-dimensional signal into a d-dimensional signal Problem Choose appropriate parameters ε, d Topology helps for periodic signals: closed curves are embeddings of S 1, recognizable by Betti numbers M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
24 Signal Analysis Detecting good delay embeddings (de Silva Skraba VJ) Clarinet middle E tone in R 2 : M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
25 Signal Analysis Detecting good delay embeddings (de Silva Skraba VJ) Clarinet middle E tone in R 2 : M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
26 Signal Analysis Detecting good delay embeddings (de Silva Skraba VJ) Clarinet middle E tone in R 2 : M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
27 Signal Analysis Detecting good delay embeddings (de Silva Skraba VJ) Clarinet middle E tone in R 2 : M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
28 Signal Analysis Detecting good delay embeddings (de Silva Skraba VJ) Clarinet middle E tone in R 2 : M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
29 Robotics Cyclo-octane con gurations Working to replicate an analysis in Topology of cyclo-octane energy landscape (Martin, Thompson, Coutsias, Watson; J Chem Phys 132, (2010)), we consider cyclo-octane as a linkage Martin-Thompson-Coutsias-Watson established the topology of this to be a sphere and a Klein bottle, fused along two circles They also computed the Betti numbers to be β 0 = 1, β 1 = 1, and β 2 = 2 Requiring rest-state distances between atoms, and rest-state planar angles for carbon-carbon bonds, the resulting linkage has only rotational joints at each carbon atom We sampled 34k points from the resulting variety, and are using this as a test-case for a systematic method for the computation M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
30 Robotics Worked example: Cyclo-octane con gurations Sphere Klein bottle Intersect in disjoint, unlinked pair of circles M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
31 Biomedicine A topological analysis method In a recent PhD thesis at Stanford [Singh, 08], a topological method for data analysis was introduced Fundamental topological result: Nerve lemma Suppose a space X is subdivided X = i X i into contractible (read simple) components Then X is equivalent to the nerve of the covering M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
32 Biomedicine e nerve of a covering M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
33 Biomedicine Topological application M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
34 Biomedicine Topological application M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
35 Biomedicine Topological application M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
36 Biomedicine Topological application M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
37 Biomedicine Topological application M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
38 Biomedicine Translate topology to statistics Continuous function Covering of target space Preimages Connected components Nerve complex Measurement function on datapoints Covering of datapoints Preimages Clusters Mapper diagram M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
39 Biomedicine Mapper algorithm M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
40 Biomedicine Mapper algorithm M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
41 Biomedicine Mapper algorithm M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
42 Biomedicine Mapper algorithm M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
43 Biomedicine Mapper algorithm M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
44 Biomedicine Mapper algorithm M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
45 Biomedicine Mapper algorithm Implementation This method is provided in a software package currently marketed by Ayasdi Startup company founded by Gurjeet Singh (original thesis on Mapper) and Gunnar Carlsson (thesis advisor) M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
46 Biomedicine Cancer data Carlsson Nicolau and the team at Ayasdi studied physiological data from around 170 breast cancer patients Mapper plot structured as a core with ares extending One are consisted exclusively of survivors (0% mortality) Cluster analyses and PCA techniques dispersed this group among high mortality patients M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
47 Biomedicine Political data Carlsson Lum Sandberg V-J and the team at Ayasdi have studied vote data from the US congress M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
48 Biomedicine Political data Carlsson Lum Sandberg V-J and the team at Ayasdi have studied vote data from the US congress M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
49 Natural images analysis Example: Spaces of natural images Lee-Mumford-Pedersen investigated whether a statistically signi cant difference exists between natural and random images Natural images form a subspace of all images Dimension of ambient space eg = This space of natural images should have: high dimension: there are many different images high codimension: random images look nothing like natural ones M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
50 Natural images analysis Natural 3x3 patches Instead of studying entire images, we consider the distribution of 3 3 pixel patches Most of these will be approximately constant in natural images Allowing these drowns out structure Lee-Mumford-Pedersen chose patches with high contrast from a collection of black-and-white images used in cognition research Each 3 3-patch is considered a vector in R 9 Normalised brightness: R 9 R 8 Normalised contrast: R 8 S 7 Subsequent topological analysis by Carlsson de Silva Ishkanov Zomorodian M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
51 Pixel patches in S 7 Natural images analysis The resulting patches are dense in S 7 so we consider high-density regions Pick out 25% densest points We can pick a parametrised method to measure density: De nition k-codensity δ k (x) of a point x is the distance to its kth nearest neighbour k-density d k (x) is 1/δ k (x) High k yields a smoothly changing density measure capturing global properties Low k yields a wilder density measure capturing local properties k acts as a kind of focus control M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
52 Natural images analysis 300-density M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
53 Natural images analysis 300-density M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
54 Natural images analysis 15-density M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
55 Natural images analysis 15-density M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
56 Natural images analysis ree circles M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
57 Natural images analysis Identifying the subspace of natural pixel patches Raising the cut-off bar yields, with coefficients in F 2 β 0 = 1 β 1 = 2 β 2 = 1 Assuming the shape is a surface, this corresponds to one of M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
58 Natural images analysis Identifying the subspace of natural pixel patches Raising the cut-off bar yields, with coefficients in F 3 Thus, the relevant shape is: β 0 = 1 β 1 = 1 M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
59 Natural images analysis Klein bottle of pixel patches M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
60 Natural images analysis Applications of this analysis Image compression A 3 3-cluster may be described using 4 values: Position of its projection onto the Klein bottle Original brightness Original contrast Texture analysis Textures yield distributions of occuring patches on the Klein bottle Rotating the texture corresponds to translating the distribution [J Perea] M Vejdemo-Johansson (St Andrews) TDA 22 November / 32
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