TDAvec (version 0.1.41)

computePersistenceLandscape: Vector Summaries of the Persistence Landscape Functions

Description

For a given persistence diagram \(D=\{(b_i,d_i)\}_{i=1}^N\) (corresponding to a specified homological dimension), computePersistenceLandscape() vectorizes the first \(k\) persistence landscape functions $$\lambda_j(t) = j\hbox{max}_{1\leq i \leq N} \Lambda_i(t), \quad j=1,\dots,k,$$ where \(j\hbox{max}\) returns the \(j\)th largest value and $$\Lambda_i(t) = \left\{ \begin{array}{ll} t-b_i & \quad t\in [b_i,\frac{b_i+d_i}{2}] \\ d_i-t & \quad t\in (\frac{b_i+d_i}{2},d_i]\\ 0 & \quad \hbox{otherwise} \end{array} \right.$$ based on a scale sequence scaleSeq. For generalized persistence landscape functions, we instead take $$\Lambda_i(t) = \left\{ \begin{array}{ll} \frac{d_i-b_i}{2K_h(0)}K_h(t-\frac{b_i+d_i}{2}) & \hbox{for } |\frac{t-\frac{b_i+d_i}{2}}{h}| \leq 1 \\ 0 & \hbox{otherwise} \end{array} \right.$$ where \(K_h(\cdot)\) is a kernel function with the bandwidth parameter \(h.\)

Usage

computePersistenceLandscape(D, homDim, scaleSeq, k = 1, generalized = FALSE, 
  kernel = "triangle", h = NULL)

Value

A matrix where the \(j\)th column contains the values of the \(j\)th order persistence landscape function evaluated at each point of scaleSeq=\(\{t_1,t_2,\ldots,t_n\}\):

$$ \begin{bmatrix} \lambda_1(t_1) & \lambda_2(t_1) & \cdots & \lambda_k(t_1) \\ \lambda_1(t_2) & \lambda_2(t_2) & \cdots & \lambda_k(t_2)\\ \vdots & \vdots& \ddots & \vdots \\ \lambda_1(t_n) & \lambda_2(t_n) & \cdots & \lambda_k(t_n) \end{bmatrix} $$

Arguments

D

a persistence diagram: a matrix with three columns containing the homological dimension, birth and death values respectively.

homDim

the homological dimension (0 for \(H_0\), 1 for \(H_1\), etc.). Rows in D are filtered based on this value.

scaleSeq

a numeric vector of increasing scale values used for vectorization.

k

an integer specifying the number of persistence landscape functions to consider (default is 1).

generalized

a logical value indicating whether to use a generalized persistence landscape or not. Default is FALSE.

kernel

a string specifying the kernel type to use if generalized = TRUE. Options are "triangle", "epanechnikov", or "tricubic". Default is "triangle".

h

a positive numeric value specifying the bandwidth for the kernel. Must be provided if generalized = TRUE.

Author

Umar Islambekov

Details

The function extracts rows from D where the first column equals homDim, and computes values based on the filtered data and scaleSeq. If D does not contain any points corresponding to homDim, a vector of zeros is returned. If generalized = TRUE, three different kernel functions are currently supported:

• Triangle kernel: \(K_h(t) = \frac{1}{h} \max(0, 1 - \frac{|t|}{h})\)

• Epanechnikov kernel: \(K_h(t) = \frac{3}{4h} \max(0, 1 - \frac{t^2}{h^2})\)

• Tricubic kernel: \(K_h(t) = \frac{70}{81h} \max(0, (1 - \frac{|t|^3}{h^3})^3)\)

References

1. Bubenik, P. (2015). Statistical topological data analysis using persistence landscapes. Journal of Machine Learning Research, 16(1), 77-102.

2. Chazal, F., Fasy, B. T., Lecci, F., Rinaldo, A., & Wasserman, L. (2014, June). Stochastic convergence of persistence landscapes and silhouettes. In Proceedings of the thirtieth annual symposium on Computational geometry (pp. 474-483).

3. Berry, E., Chen, Y. C., Cisewski-Kehe, J., & Fasy, B. T. (2020). Functional summaries of persistence diagrams. Journal of Applied and Computational Topology, 4(2):211–262.

Examples

N <- 100 # The number of points to sample

set.seed(123) # Set a random seed for reproducibility

# Sample N points uniformly from the unit circle and add Gaussian noise
theta <- runif(N, min = 0, max = 2 * pi)
X <- cbind(cos(theta), sin(theta)) + rnorm(2 * N, mean = 0, sd = 0.2)

# Compute the persistence diagram using the Rips filtration built on top of X
# The 'threshold' parameter specifies the maximum distance for building simplices
D <- TDAstats::calculate_homology(X, threshold = 2)

scaleSeq = seq(0, 2, length.out = 11) # A sequence of scale values

# Compute vector summaries of the first three persistence landscape functions 
# for homological dimension H_1

computePersistenceLandscape(D, homDim = 1, scaleSeq, k = 3)

# Compute vector summaries of the first three generalized persistence 
# landscape functions (with triangle kernel and bandwidth h=0.2) 
# for homological dimension H_1

computePersistenceLandscape(D, homDim = 1, scaleSeq, generalized = TRUE, k = 3, h = 0.2)