computeEulerCharacteristic: A Vector Summary of the Euler Characteristic Curve

In TDAvec: Vector Summaries of Persistence Diagrams

A Vector Summary of the Euler Characteristic Curve

Description

Vectorizes the Euler characteristic curve

\chi(t)=\sum_{k=0}^d (-1)^k\beta_k(t),

where \beta_0,\beta_1,\ldots,\beta_d are the Betti curves corresponding to persistence diagrams D_0,D_1,\ldots,D_d of dimensions 0,1,\ldots,d respectively, all computed from the same filtration. The evaluation method depends on the argument evaluate.

Usage

computeEulerCharacteristic(D, scaleSeq, maxhomDim = -1, evaluate = "intervals")

Arguments

D	a persistence diagram: a matrix with three columns containing the homological dimension, birth and death values respectively.
scaleSeq	a numeric vector of increasing scale values used for vectorization.
maxhomDim	the maximum homological dimension considered (0 for H_0, 1 for H_1, etc.). If set to -1 (default), it uses the maximum dimension found in D.
evaluate	a character string indicating the evaluation method. Must be either "intervals" (default) or "points".

Value

A numeric vector containing elements computed using scaleSeq=\{t_1,t_2,\ldots,t_n\} according to the method specified by evaluate. If D does not contain any points corresponding to homDim, a vector of zeros is returned.

• "intervals": Computes average values of the Euler characteristic curve over intervals defined by consecutive elements in scaleSeq:

  \Big(\frac{1}{\Delta t_1}\int_{t_1}^{t_2}\chi(t)dt,\frac{1}{\Delta t_2}\int_{t_2}^{t_3}\chi(t)dt,\ldots,\frac{1}{\Delta t_{n-1}}\int_{t_{n-1}}^{t_n}\chi(t)dt\Big)\in\mathbb{R}^{n-1},

  where \Delta t_k=t_{k+1}-t_k.

• "points": Computes values of the Euler characteristic curve at each point in scaleSeq:

  (\chi(t_1),\chi(t_2),\ldots,\chi(t_n))\in\mathbb{R}^n.

Author(s)

Umar Islambekov

References

1. Richardson, E., & Werman, M. (2014). Efficient classification using the Euler characteristic. Pattern Recognition Letters, 49, 99-106.

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 a vector summary of the Euler characteristic curve
computeEulerCharacteristic(D, scaleSeq)

TDAvec documentation built on April 4, 2025, 1:37 a.m.