Projected Polya tree
Polya trees are a non-parametric distribution and can be described as random histograms, with a fixed number of classes or partitions and with a random probability mass associated with each one.
Their advantage lies in the fact that they can be constructed in such a way as to give probability one to the set of continuous or absolutely continuous probability measures. For the mathematical definition see: Nieto-Barajas and Nunez-Antonio (2021) https://arxiv.org/abs/1902.06020. Polya trees can be centered around a parametric probability measure \(F_0\). In this model, we will match the partition with the dyadic quantiles.
For the case of a bivariate Polya tree, we define the partition as the cross product of univariate partitions. In the model, the bivariate Polya tree is centered around a parametric probability measure \(F_0\). For simplicity, let us assume that \(F_0(x_1; x_2) = F_{1_0}(x_1)F_{2_0}(x_2)\) defines a bivariate distribution on \(\mathbb{R}^2\). Therefore, we proceed as in the univariate case by matching the partition with the dyadic quantiles of the marginals \(F_{1_0}\) and \(F_{2_0}\).
As we explained, we can project a bivariate distribution on \(\mathbb{R}^2\) into a circular distirbution. We construct the projected Polya tree. Let us assume a bivariate random vector \(X_0 = (X_1,X_2)\) such that \(X \sim f\). We project the random vector \(X\) to the unit circle by defining \(U = X/||X||\), or by using polar coordinates. In this model all the prior projected Polya trees will be centered on the projected Normal distribution .
Prior projected Polya Tree in the package
The function dsimpriorpppt() simulates paths of prior projected Polya tree distributions centered around a projected Normal distribution.
The main parameters are:
- nsim: integer indicating the number of simulations.
- mm: integer indicating the number of finite levels of the Polya tree.
- mu: mean vector of the projected bivariate normal distribution.
- units: units of the support: “radians”, “degrees” or “hours”.
There are other functions that allows us to visualize and summarize the Polya tree distribution:
- priorppt.plot: plots a linear or circular of the projected Polya paths, or a boxplot of the circular mean and concentration of the simulated paths, depending of the value of the parameter plot.type.
- priorppt.summary: returns a table with the mean, quantiles 2.5% and 97.5% of the mean direction and concentration.
z1 <- dsimpriorppt(nsim = 10, mm = 4, mu = c(0,0), sig = 1, ll = 100,
aa = 1, delta = 1.1, units = "radians")
class(z1)
#> [1] "priorppt.circ"
priorppt.plot(z1, plot.type = "line")
priorppt.plot(z1, plot.type = "circle", tol = 8, shrink = .13)
priorppt.summary(z1)
#> mean.direction concentration
#> mean 2.3388108 0.4111898
#> quantile2.5 0.6942642 0.1909665
#> quantile97.5 4.6798731 0.7481354Depending of the value of the mean vector of the centered projected bivariate normal distribution, the paths will change, the location parameter of the centering measure not only will control the shape of the densities but also the variability of the paths. The larger the magnitude of mu, the greater the concentration:
mat_mu <- matrix(c(0,0,-1,0,0,1,2,2), nrow = 4, ncol = 2, byrow = TRUE)
for(i in 1:4){
z <- dsimpriorppt(nsim = 10, mm = 4, mu = mat_mu[i,], sig = 1, ll = 100,
aa = 1, delta = 1.1, units = "radians")
priorppt.plot(z, plot.type = "line")
}
