mappeR

2025-02-16

This is an implementation of the mapper algorithm by Singh, Mémoli, and Carlsson (2007).

Setup

To install and use the most recent CRAN upload of this package, run the following:

install.packages("mappeR")

library(mappeR)

To install the latest development version of this package from Github, run the following commands:

install.packages("devtools")

library(devtools)

devtools::install_github("https://github.com/Uiowa-Applied-Topology/mappeR/tree/dev", upgrade=FALSE)

library(mappeR)

If you’re installing from Github, you might need to do some more stuff:

• Windows: install Rtools (https://cran.r-project.org/bin/windows/Rtools/)
• OS X: install Xcode (from the Mac App Store)
• Linux: run apt-get install r-base-dev (or similar).

Mathy Overview

Mapper is a way to view a point cloud \(P\) through a “lens” of our choice.

Consider a function

f: P \to \mathbb{R}

Cover \(\mathbb{R}\) in a set of intervals \(`\{I_i\}_{i=1}^n`\), so that every point in \(P\) is contained in some level set \(L_i = f^{-1}(I_i)\). We may then construct a graph

G = (V,E)

based on this cover to reflect the original data, where

V = \{L_i \mid L_i \neq \varnothing\}

and

E = \{\{L_i, L_j\}\mid L_i\cap L_j \neq \varnothing,\ i\neq j\}

This is the basic idea of the mapper algorithm, with the addition that each level set is first refined into clusters based on the intrinsic pairwise distances of the data according to some clustering algorithm. That is, we partition each level set \(L_i\) into \(k_i\) disjoint clusters

L_i = \bigsqcup_{j=1}^{k_i} C_j

and build a new graph \(G' = (V', E')\) that is homomorphic to \(G\) defined by

V' = \bigsqcup_{i=1}^{n}\{C_j\}_{j=1}^{k_i}

and

E' = \{\{C_i, C_j\}\mid C_i\cap C_j \neq \varnothing\}

So in general, the ingredients to construct a mapper graph are

• A data set, along with their pairwise distances
• A lens function with the data as its domain (above this was real-valued, but it does not have to be)
• A cover of the codomain of the lens function
• A clustering algorithm

Example 1: 1D Mapper

num_points = 5000

P.data = data.frame(
  x = sapply(1:num_points, function(x)
    sin(x) * 10) + rnorm(num_points, 0, 0.1),
  y = sapply(1:num_points, function(x)
    cos(x) ^ 2 * sin(x) * 10) + rnorm(num_points, 0, 0.1),
  z = sapply(1:num_points, function(x)
    10 * sin(x) ^ 2 * cos(x)) + rnorm(num_points, 0, 0.1)
)

P.dist = dist(P.data)

Here is a point cloud \(P\) formed by adding a bit of uniform noise to 5000 points regularly sampled from the parametric curve

\gamma(t) = \begin{cases}x = 10\ \sin(t)\\ y=10\ \sin(t)\ \cos^2(t)\\ z=10\ \sin^2(t)\ \cos(t) \end{cases}

This seems to form a kind of figure-8 curve just based on this projection. But as we can see from the 2D projections, the “shape” of the data set we’re seeing really does depend on how we’re looking at it:

We will build graphs using the outline of the mapper algorithm described, with real-valued lens functions.

Parameters:

• Data: figure-8
• Lens: Projection to the \(x\)-coordinate
• Cover: A cover of \(\mathbb{R}\) (just up to the extremes of the function values) using 10 equally spaced intervals with 25% overlap between each consecutive interval
• Clustering method: Single-linkage hierarchical clustering

# lens function
projx = P.data$x

# cover parameters to generate a width-balanced cover
num_bins = 10
percent_overlap = 25

# generate the cover
xcover = create_width_balanced_cover(min(projx), max(projx), num_bins, percent_overlap)

# bin tester machine machine
check_in_interval <- function(endpoints) {
  return(function(x) (endpoints[1] - x <= 0) & (endpoints[2] - x >= 0))
}

# each of the "cover" elements will really be a function that checks if a data point lives in it
xcovercheck = apply(xcover, 1, check_in_interval)

# build the mapper objects
xmapper = create_mapper_object(
  data = P.data,
  dists = P.dist,
  filtered_data = projx,
  cover_element_tests = xcovercheck,
  clusterer = local_hierarchical_clusterer("single") # built-in mappeR method
)

The object returned by create_mapper_object is a list of two dataframes containing vertex and edge information.

Vertex information:

• id: vertex ID
• cluster_size: number of datapoints in cluster
• mean_dist_to_medoid: mean distance to medoid of cluster
• data: names of datapoints in cluster
• patch: level set ID

Edge information:

• source: vertex ID of edge source
• target: vertex ID of edge target
• weight: Jaccard index of edge; intersection divided by union
• overlap_data: names of datapoints in overlap
• overlap_size: number of datapoints overlap

Example 2: ball mapper

By toying with the general mapper parameters, we can obtain different flavors of the algorithm. In the ball mapper flavor, we simply use the inclusion into the ambient space of the data as our lens function, and let the cover do the work. Specifically, we cover the ambient space with \(\varepsilon\)-balls by creating a \(\varepsilon\)-net, which can be done with a greedy algorithm.

Parameters:

• Data: figure-8
• Cover: set of \(\varepsilon\)-balls in \(\mathbb{R^3}\)
• Lens function: inclusion from \(P\hookrightarrow\mathbb{R}^3\)
• Clustering method: none (or, “any data set is one big cluster”-type clustering)

# creates a cover using a greedy algorithm
balls = create_balls(data = P.data, dists = P.dist, eps = .25)

# ball tester machine machine
is_in_ball <- function(ball) {
  return(function(x) x %in% ball)
}

# filtering is just giving back the data (row names because my balls are lists of data point names, so the filter should match)
ballmapper = create_mapper_object(P.data, P.dist, rownames(P.data), lapply(balls, is_in_ball))

Built-ins

Mapper Flavors

mappeR has built-in methods for:

1D mapper

create_1D_mapper_object(data, dists, filtered_data, cover, clusterer)

• Lens: \(P \to \mathbb{R}\)
• Cover: intervals
• Clustering: yes

Ball mapper

create_ball_mapper_object(data, dists, eps)

• Lens: \(P \to P\) by identity
• Cover: \(\varepsilon\)-balls in ambient \(P\)-space
• Clustering: no

Clusterball mapper

create_clusterball_mapper_object(data, ball_dists, clustering_dists, eps, clusterer)

• Lens: \(P \to P\) by identity
• Cover: \(\varepsilon\)-balls in ambient \(P\)-space
• Clustering: yes

Clustering

mappeR has two built-in clusterers, each of which implement agglomerative hierarchical clustering using fastcluster. The first is local_hierarchical_clusterer(method), which will cut each dendrogram according to its longest unbroken branches — it cuts in a “local” context. The second is global_hierarchical_clusterer(method, dists), which will run hierarchical clustering on dists to obtain a uniform cutting height for each dendrogram — it cuts in a “global” context.

Any of the linkage methods below will work:

• "single": single linkage
• "complete": complete linkage
• "average": average linkage
• "mcquitty": McQuitty linkage
• "centroid": centroid linkage
• "median": median linkage
• "ward.D": Ward linkage (assumes squared distances)
• "ward.D2": Ward linkage (assumes nonsquared distances)

You may also build your own function to use for clustering; this function must be able to accept a list of distance matrices and return a list of cluster information vectors for each matrix. A “cluster information vector” means a named vector (a vector whose elements have names) with datapoint IDs for names and cluster IDs for values. The cluster IDs should be positive integers but do not need to be unique across all of the information vectors; mappeR will handle this for you.