computePersistenceImage: A Vector Summary of the Persistence Surface
In TDAvec: Vector Summaries of Persistence Diagrams
computePersistenceImage R Documentation
A Vector Summary of the Persistence Surface
Description
For a given persistence diagram D=\{(b_i,p_i)\}_{i=1}^N (corresponding to a specified homological dimension), computePersistenceImage() computes the persistence image - a vector summary of the persistence surface:
\rho(x,y)=\sum_{i=1}^N f(b_i,p_i)\phi_{(b_i,p_i)}(x,y),
where \phi_{(b_i,p_i)}(x,y) is
the Gaussian distribution with mean (b_i,p_i) and
covariance matrix \sigma^2 I_{2\times 2} and
f(b,p) = w(p)=\left\{
\begin{array}{ll}
0 & \quad p\leq 0 \\
p/p_{max} & \quad 0<p<p_{max}\\
1& \quad p\geq p_{max}
\end{array}
\right.
is the weighting function with p_{max} being the maximum persistence value among all persistence diagrams considered in the experiment. Points in D with infinite persistence values are ignored.
Usage
computePersistenceImage(D, homDim, xSeq, ySeq, sigma)
Arguments
D
a persistence diagram: a matrix with three columns containing the homological dimension, birth and persistence values respectively.
homDim
the homological dimension (0 for H_0, 1 for H_1, etc.). Rows in D are filtered based on this value.
xSeq
a numeric vector of increasing x (birth) values used for vectorization.
ySeq
a numeric vector of increasing y (persistence) values used for vectorization.
sigma
a standard deviation (\sigma) of the Gaussian.
Details
The function extracts rows from D where the first column equals homDim, and computes values based on the filtered data, xSeq and ySeq. If D does not contain any points corresponding to homDim, a vector of zeros is returned.
Value
A numeric vector whose elements are the average values of the persistence surface computed over each cell of the two-dimensional grid constructred from xSeq=\{x_1,x_2,\ldots,x_n\} and ySeq=\{y_1,y_2,\ldots,y_m\}:
\Big(\frac{1}{\Delta x_1\Delta y_1}\int_{[x_1,x_2]\times [y_1,y_2]}\rho(x,y)dA,\ldots,\frac{1}{\Delta x_{n-1}\Delta y_{m-1}}\int_{[x_{n-1},x_n]\times [y_{m-1},y_m]}\rho(x,y)dA\Big)\in\mathbb{R}^{d},
where d=(n-1)(m-1), dA=dxdy, \Delta x_k=x_{k+1}-x_k and \Delta y_j=y_{j+1}-y_j.
If homDim=0 and all the birth values are equal (e.g., zero), univariate Gaussians are used instead for vectorization:
\Big(\frac{1}{\Delta y_1}\int_{y_1}^{y_2}\rho(y)dy,\ldots,\frac{1}{\Delta y_{m-1}}\int_{y_{m-1}}^{y_m}\rho(y)dy\Big)\in\mathbb{R}^{m-1},
where \rho(y)=\sum_{i=1}^N f(p_i)\phi_{p_i}(y) and \Delta y_j=y_{j+1}-y_j.
Author(s)
Umar Islambekov
References
1. Adams, H., Emerson, T., Kirby, M., Neville, R., Peterson, C., Shipman, P., ... & Ziegelmeier, L. (2017). Persistence images: A stable vector representation of persistent homology. Journal of Machine Learning Research, 18.
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)
# Switch from the birth-death to the birth-persistence coordinates
D[,3] <- D[,3] - D[,2]
colnames(D)[3] <- "Persistence"
resB <- 5 # Resolution (or grid size) along the birth axis
resP <- 5 # Resolution (or grid size) along the persistence axis
# Compute persistence image for homological dimension H_0
minPH0 <- min(D[D[,1]==0,3]); maxPH0 <- max(D[D[,1]==0,3])
ySeqH0 <- seq(minPH0, maxPH0, length.out = resP + 1)
sigma <- 0.5 * (maxPH0 - minPH0) / resP
computePersistenceImage(D, homDim = 0, xSeq = NA, ySeq = ySeqH0, sigma = sigma)
# Compute persistence image for homological dimension H_1
minBH1 <- min(D[D[,1]==1,2]); maxBH1 <- max(D[D[,1]==1,2])
minPH1 <- min(D[D[,1]==1,3]); maxPH1 <- max(D[D[,1]==1,3])
xSeqH1 <- seq(minBH1, maxBH1, length.out = resB + 1)
ySeqH1 <- seq(minPH1, maxPH1, length.out = resP + 1)
sigma <- 0.5 * (maxPH1 - minPH1) / resP
computePersistenceImage(D, homDim = 1, xSeq = xSeqH1, ySeq = ySeqH1, sigma = sigma)
TDAvec documentation built on April 4, 2025, 1:37 a.m.
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| computePersistenceImage | R Documentation |
A Vector Summary of the Persistence Surface
Description
For a given persistence diagram D=\{(b_i,p_i)\}_{i=1}^N (corresponding to a specified homological dimension), computePersistenceImage() computes the persistence image - a vector summary of the persistence surface:
\rho(x,y)=\sum_{i=1}^N f(b_i,p_i)\phi_{(b_i,p_i)}(x,y),
where \phi_{(b_i,p_i)}(x,y) is
the Gaussian distribution with mean (b_i,p_i) and
covariance matrix \sigma^2 I_{2\times 2} and
f(b,p) = w(p)=\left\{
\begin{array}{ll}
0 & \quad p\leq 0 \\
p/p_{max} & \quad 0<p<p_{max}\\
1& \quad p\geq p_{max}
\end{array}
\right.
is the weighting function with p_{max} being the maximum persistence value among all persistence diagrams considered in the experiment. Points in D with infinite persistence values are ignored.
Usage
computePersistenceImage(D, homDim, xSeq, ySeq, sigma)
Arguments
D |
a persistence diagram: a matrix with three columns containing the homological dimension, birth and persistence values respectively. |
homDim |
the homological dimension (0 for |
xSeq |
a numeric vector of increasing x (birth) values used for vectorization. |
ySeq |
a numeric vector of increasing y (persistence) values used for vectorization. |
sigma |
a standard deviation ( |
Details
The function extracts rows from D where the first column equals homDim, and computes values based on the filtered data, xSeq and ySeq. If D does not contain any points corresponding to homDim, a vector of zeros is returned.
Value
A numeric vector whose elements are the average values of the persistence surface computed over each cell of the two-dimensional grid constructred from xSeq=\{x_1,x_2,\ldots,x_n\} and ySeq=\{y_1,y_2,\ldots,y_m\}:
\Big(\frac{1}{\Delta x_1\Delta y_1}\int_{[x_1,x_2]\times [y_1,y_2]}\rho(x,y)dA,\ldots,\frac{1}{\Delta x_{n-1}\Delta y_{m-1}}\int_{[x_{n-1},x_n]\times [y_{m-1},y_m]}\rho(x,y)dA\Big)\in\mathbb{R}^{d},
where d=(n-1)(m-1), dA=dxdy, \Delta x_k=x_{k+1}-x_k and \Delta y_j=y_{j+1}-y_j.
If homDim=0 and all the birth values are equal (e.g., zero), univariate Gaussians are used instead for vectorization:
\Big(\frac{1}{\Delta y_1}\int_{y_1}^{y_2}\rho(y)dy,\ldots,\frac{1}{\Delta y_{m-1}}\int_{y_{m-1}}^{y_m}\rho(y)dy\Big)\in\mathbb{R}^{m-1},
where \rho(y)=\sum_{i=1}^N f(p_i)\phi_{p_i}(y) and \Delta y_j=y_{j+1}-y_j.
Author(s)
Umar Islambekov
References
1. Adams, H., Emerson, T., Kirby, M., Neville, R., Peterson, C., Shipman, P., ... & Ziegelmeier, L. (2017). Persistence images: A stable vector representation of persistent homology. Journal of Machine Learning Research, 18.
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)
# Switch from the birth-death to the birth-persistence coordinates
D[,3] <- D[,3] - D[,2]
colnames(D)[3] <- "Persistence"
resB <- 5 # Resolution (or grid size) along the birth axis
resP <- 5 # Resolution (or grid size) along the persistence axis
# Compute persistence image for homological dimension H_0
minPH0 <- min(D[D[,1]==0,3]); maxPH0 <- max(D[D[,1]==0,3])
ySeqH0 <- seq(minPH0, maxPH0, length.out = resP + 1)
sigma <- 0.5 * (maxPH0 - minPH0) / resP
computePersistenceImage(D, homDim = 0, xSeq = NA, ySeq = ySeqH0, sigma = sigma)
# Compute persistence image for homological dimension H_1
minBH1 <- min(D[D[,1]==1,2]); maxBH1 <- max(D[D[,1]==1,2])
minPH1 <- min(D[D[,1]==1,3]); maxPH1 <- max(D[D[,1]==1,3])
xSeqH1 <- seq(minBH1, maxBH1, length.out = resB + 1)
ySeqH1 <- seq(minPH1, maxPH1, length.out = resP + 1)
sigma <- 0.5 * (maxPH1 - minPH1) / resP
computePersistenceImage(D, homDim = 1, xSeq = xSeqH1, ySeq = ySeqH1, sigma = sigma)
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