TDAvec (version 0.1.41)

computeAlgebraicFunctions: Compute Algebraic Functions from a Persistence Diagram

Description

For a given persistence diagram \(D=\{(b_i,d_i)\}_{i=1}^N\) (corresponding to a specified homological dimension), computeAlgebraicFunctions() computes the following four algebraic functions based on the birth and death values:

• \(f_1=\sum_i b_i (d_i - b_i).\)

• \(f_2=\sum_i (d_{\max} - d_i) (d_i - b_i)\), where \(d_{\max} = \max(d_i)\).

• \(f_3=\sum_i b_i^2 (d_i - b_i)^4\).

• \(f_4=\sum_i (d_{\max} - d_i)^2 (d_i - b_i)^4\).

Points in \(D\) with infinite death values are ignored.

Usage

computeAlgebraicFunctions(D, homDim)

Value

A (named) numeric vector \((f_1,f_2,f_3,f_4)\).

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.

Author

Umar Islambekov

Details

The function extracts rows from D where the first column equals homDim, and computes the four algebraic functions based on the filtered data. If D does not contain any points corresponding to homDim, a vector of zeros is returned.

References

1. Adcock, A., Carlsson, E. and Carlsson, G., 2013. The ring of algebraic functions on persistence bar codes. Homology, Homotopy Appl., 18:381–402, 2016.

2. Ali, D., Asaad, A., Jimenez, M.J., Nanda, V., Paluzo-Hidalgo, E. and Soriano-Trigueros, M., (2023). A survey of vectorization methods in topological data analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence.

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)

# Compute algebraic functions for homological dimension H_0
computeAlgebraicFunctions(D, homDim = 0)

# Compute algebraic functions for homological dimension H_1
computeAlgebraicFunctions(D, homDim = 1)