In this example, we generate observations from the Poisson kernel-based distribution on the sphere, \(S^{d-1}\). We consider mean direction \(\mu=(0,0,1)\), \(d=3\) and the concentration parameter is \(\rho = 0.8\).

We sampled \(n=1000\) observations for each method available.

```
n <- 1000
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 function returns a list with the following components

```
## Length Class Mode
## x 3000 -none- numeric
## numAccepted 1 -none- numeric
## numTries 1 -none- numeric
```

We extract the generated data sets and create a unique data set.

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()
```

```
## null
## 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()
```