| computePL {TDAvec} | R Documentation |
A Vector Summary of the Persistence Landscape Function
Description
Vectorizes the persistence landscape (PL) function constructed from a given persistence diagram. The kth order landscape function of a persistence diagram D=\{(b_i,d_i)\}_{i=1}^N is defined as
\lambda_k(t) = k\hbox{max}_{1\leq i \leq N} \Lambda_i(t), \quad k\in N,
where k\hbox{max} returns the kth 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.
Usage
computePL(D, homDim, scaleSeq, k=1)
Arguments
D |
matrix with three columns containing the dimension, birth and death values respectively |
homDim |
homological dimension (0 for |
scaleSeq |
numeric vector of increasing scale values used for vectorization |
k |
order of landscape function. By default, |
Value
A numeric vector whose elements are the values of the kth order landscape function evaluated at each point of scaleSeq=\{t_1,t_2,\ldots,t_n\}:
(\lambda_k(t_1),\lambda_k(t_2),\ldots,\lambda_k(t_n))
Author(s)
Umar Islambekov
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).
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 persistence landscape (PL) for homological dimension H_0 with order of landscape k=1
computePL(D,homDim=0,scaleSeq,k=1)
# compute persistence landscape (PL) for homological dimension H_1 with order of landscape k=1
computePL(D,homDim=1,scaleSeq,k=1)