computePES {TDAvec}

A Vector Summary of the Persistent Entropy Summary Function

Description

For a given persistence diagram D=\{(b_i,d_i)\}_{i=1}^N, computePES() vectorizes the persistent entropy summary (PES) function

S(t)=-\sum_{i=1}^N \frac{l_i}{L}\log_2{(\frac{l_i}{L}})\bold 1_{[b_i,d_i]}(t),

where l_i=d_i-b_i and L=\sum_{i=1}^Nl_i. Points of D with infinite death value are ignored

Usage

computePES(D, homDim, scaleSeq)

Arguments

D

matrix with three columns containing the dimension, birth and death values respectively

homDim

homological dimension (0 for H_0, 1 for H_1, etc.)

scaleSeq

numeric vector of increasing scale values used for vectorization

Value

A numeric vector whose elements are the average values of the persistent entropy summary function 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}S(t)dt,\frac{1}{\Delta t_2}\int_{t_2}^{t_3}S(t)dt,\ldots,\frac{1}{\Delta t_{n-1}}\int_{t_{n-1}}^{t_n}S(t)dt\Big),

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

Author(s)

Umar Islambekov

References

1. Atienza, N., Gonzalez-Díaz, R., & Soriano-Trigueros, M. (2020). On the stability of persistent entropy and new summary functions for topological data analysis. Pattern Recognition, 107, 107509.

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 PES for homological dimension H_0
computePES(D,homDim=0,scaleSeq)

# compute PES for homological dimension H_1
computePES(D,homDim=1,scaleSeq)