# Random sampling from the Poisson kernel-based density

    library(QuadratiK)

In this example, we generate observations from the Poisson kernel-based distribution on the sphere, \(S^{d-1}\). Given a vector \(\mathbf{\mu} \in \mathcal{S}^{d-1}\), where \(\mathcal{S}^{d-1}= \{x \in \mathbb{R}^d : ||x|| = 1\}\), and a parameter \(\rho\) such that \(0 < \rho < 1\), the probability density function of a \(d\)-variate Poisson kernel-based density is defined by:

\[ f(\mathbf{x}|\rho, \mathbf{\mu}) = \frac{1-\rho^2}{\omega_d ||\mathbf{x} - \rho \mathbf{\mu}||^d}, \]

where \(\mu\) is a vector orienting the center of the distribution, \(\rho\) is a parameter to control the concentration of the distribution around the vector \(\mu\), and it is related to the variance of the distribution. Furthermore, \(\omega_d =2\pi^{d/2} [\Gamma(d/2)]^{-1}\) is the surface area of the unit sphere in \(\mathbb{R}^d\) (see Golzy and Markatou, 2020). When \(\rho \to 0\), the Poisson kernel-based density tends to the uniform density on the sphere. Connection of the PKBDs to other distributions are discussed in detail in Golzy and Markatou (2020).

We consider mean direction \(\mu=(0,0,1)\), \(d=3\) and the concentration parameter is \(\rho = 0.8\).

    mu <- c(0, 0, 1)
    d <- 3
    n <- 1000
    rho <- 0.8

We sampled \(n=1000\) observations for each method available. Golzy and Markatou (2020) proposed an acceptance-rejection method for simulating data from a PKBD using von Mises-Fisher envelopes (`rejvmf` method). Furthermore, Sablica, Hornik, and Leydold (2023) proposed new ways for simulating from the PKBD, using angular central Gaussian envelopes (`rejacg`) or using the projected Saw distributions (`rejpsaw`).

    set.seed(2468)
    # Generate observations using the rejection algorithm with von-Mises
    # distribution envelopes
    dat1 <- rpkb(n = n, rho = rho, mu = mu, method = "rejvmf")
    # Generate observations using the rejection algorithm with angular central
    # Gaussian distribution envelopes
    dat2 <- rpkb(n = n, rho = rho, mu = mu, method = "rejacg")
    # Generate observations using the projected Saw distribution
    dat3 <- rpkb(n = n, rho = rho, mu = mu, method = "rejpsaw")

The number of observations generated is determined by `n` for `rpkb()`. This function returns the matrix of generated observations `x`. We create a unique data set.

    x <- rbind(dat1, dat2, dat3)

Finally, the observations generated through the different methods are compared graphically, by displaying the data points on the sphere colored with respect to the used method.

    library(rgl)
    # Legend information
    classes <- c("rejvmf", "rejacg", "rejpsaw")
    # Fix a color for each method
    colors <- c("red", "blue", "green")
    labels <- factor(rep(colors, each = 1000))
    # Element needed for the Legend position in the following plot
    offset <- 0.25
    # Coordinates for legend placement
    legend_x <- max(x[,1]) + offset
    legend_y <- max(x[,2]) + offset
    legend_z <- seq(min(x[,3]), length.out = length(classes), by = offset)

    open3d()

    ## wgl
    ##   1

    # Create the legend
    for (i in seq_along(classes)) {
       text3d(legend_x, legend_y, legend_z[i], texts = classes[i], adj = c(0, 0.5))
       points3d(legend_x-0.1, legend_y, legend_z[i], col = colors[i], size = 5)
    }
    title3d("", line = 3, cex = 1.5, font = 2, add = TRUE)
    # Plot the sampled observations colored with respect to the used method
    plot3d(x[,1], x[,2], x[,3], col = labels, size = 5, add = TRUE)
    title3d("", line = 3, cex = 1.5, font = 2, add = TRUE)
    # Add a Sphere as background
    rgl.spheres(0 , col = "transparent", alpha = 0.2)
    # Rotate and zoom the generated 3 dimensional plot to facilitate visualization
    view3d(theta = 10, phi = -25, zoom = 0.5)
    # rglwidget is added in order to display the generated figure into the html
    # replication file.
    rglwidget()

    close3d()

### Note

A limitation of the `rejvmf` method is that it does not ensure the computational feasibility of the sampler for \(\rho\) approaching 1.

## References

Golzy, M. and Markatou, M. (2020). Poisson Kernel-Based Clustering on the Sphere: Convergence Properties, Identifiability, and a Method of Sampling, Journal of Computational and Graphical Statistics, 29(4), 758-770. DOI: 10.1080/10618600.2020.1740713.

Sablica, L., Hornik, K. and Leydold, J. (2023). “Efficient sampling from the PKBD distribution”, Electronic Journal of Statistics, 17(2), 2180-2209. DOI: 10.1214/23-EJS2149