| computeECC {TDAvec} | R Documentation |
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 dimeansions 0,1,\ldots,d respectively, all computed from the same filtration
Usage
computeECC(D, maxhomDim, scaleSeq)
Arguments
D |
matrix with three columns containing the dimension, birth and death values respectively |
maxhomDim |
maximum homological dimension considered (0 for |
scaleSeq |
numeric vector of increasing scale values used for vectorization |
Value
A numeric vector whose elements are the average values of the Euler characteristic curve computed between each pair of
consecutive scale points of scaleSeq=\{t_1,t_2,\ldots,t_n\}:
\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),
where \Delta t_k=t_{k+1}-t_k
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
set.seed(123)
# sample N points uniformly from unit circle and add Gaussian noise
X <- TDA::circleUnif(N,r=1) + rnorm(2*N,mean = 0,sd = 0.2)
# compute a persistence diagram using the Rips filtration built on top of X
D <- TDA::ripsDiag(X,maxdimension = 1,maxscale = 2)$diagram
scaleSeq = seq(0,2,length.out=11) # sequence of scale values
# compute ECC
computeECC(D,maxhomDim=1,scaleSeq)