Package {PTT}

PTT: Pólya Tree Type Models

Description

Bayesian nonparametric density and conditional-density estimation with adaptive, Markov adaptive, and optional Pólya trees.

Author(s)

Maintainer: Li Ma mastatlab@gmail.com

Authors:

• Li Ma mastatlab@gmail.com

See Also

Useful links:

• https://github.com/MaStatLab/PTT

• Report bugs at https://github.com/MaStatLab/PTT/issues

Fit an Adaptive Pólya Tree

Description

Fits an adaptive Pólya tree (APT) or Markov APT to univariate or multivariate observations. The partition recursively bisects the sample space, while latent states control local shrinkage toward the uniform centering distribution.

Usage

apt(
  X,
  Xpred = NULL,
  Omega.type = "unit",
  max.resol = 10,
  rho0 = 0.2,
  rho0.mode = 0,
  tran.mode = 1,
  lognu.lb = -1,
  lognu.ub = 4,
  n.grid = 5,
  n.s = 5,
  beta = 0.1,
  n.post.samples = 0
)

Arguments

Value

A list with 'logrho' (the log posterior complete-shrinkage probability at the root), 'logphi' (the log marginal likelihood), 'part_points_hmap' (the hierarchical MAP partition), 'predictive_densities', and 'Omega'. If posterior samples are requested, 'part_points_post_samples' is also returned. Partition bounds are integer cell indices on the depth-'max.resol' grid; 'Omega' records the physical sample-space bounds.

References

Ma, L. (2017). Adaptive shrinkage in Pólya tree type models. *Bayesian Analysis*, 12(3), 779–805. doi:10.1214/16-BA1021.

See Also

[opt()], [cond.apt()]

Examples

set.seed(12345)
x <- c(rbeta(40, 3, 8), rbeta(40, 9, 3))
grid <- seq(0.05, 0.95, length.out = 25)
fit <- apt(x, Xpred = grid, max.resol = 4, n.grid = 3, n.s = 3)
head(fit$predictive_densities)

Fit a Conditional Adaptive Pólya Tree

Description

Fits the two-stage conditional-density model of Ma (2017): the predictor space is recursively partitioned and an adaptive Pólya tree models the response distribution within each predictor block.

Usage

cond.apt(
  X,
  Y,
  Xpred = NULL,
  Ypred = NULL,
  OmegaX.type = "unit",
  OmegaY.type = "unit",
  max.resX = 5,
  max.resY = 8,
  rho0.X = 0.2,
  rho0.mode.X = 0,
  rho0.Y = 0.2,
  rho0.mode.Y = 0,
  tran.mode = 1,
  lognu.lb = -1,
  lognu.ub = 4,
  n.grid = 5,
  n.s = 5,
  beta = 0.1,
  n.post.samples = 0
)

Arguments

Value

A list containing 'logrho', 'logphi', the predictor-space 'part_points_hmap', 'predictive_densities', 'OmegaX', and 'OmegaY'. If requested, 'part_points_post_samples' contains sampled predictor partitions. Partition bounds use integer cell indices at depth 'max.resX'.

References

Ma, L. (2017). Recursive partitioning and multi-scale modeling on conditional densities. *Electronic Journal of Statistics*, 11(1), 1297–1325. doi:10.1214/17-EJS1254.

See Also

[cond.opt()], [apt()]

Examples

set.seed(12345)
x <- runif(80)
y <- rbeta(80, 2 + 6 * x, 8 - 5 * x)
fit <- cond.apt(
  x, y, Xpred = c(0.25, 0.75), Ypred = c(0.3, 0.7),
  max.resX = 3, max.resY = 4, n.grid = 3, n.s = 3
)
fit$predictive_densities

Fit a Conditional Optional Pólya Tree

Description

A convenience wrapper around [cond.apt()] that uses an optional Pólya tree for the response distribution in each predictor block.

Usage

cond.opt(
  X,
  Y,
  Xpred = NULL,
  Ypred = NULL,
  OmegaX.type = "unit",
  OmegaY.type = "unit",
  max.resX = 7,
  max.resY = 7,
  rho0.X = 0.5,
  rho0.mode.X = 0,
  rho0.Y = 0.5,
  rho0.mode.Y = 0,
  n.post.samples = 0
)

Arguments

Value

A list with the same structure as [cond.apt()].

See Also

[cond.apt()], [opt()]

Examples

set.seed(12345)
x <- runif(60)
y <- ifelse(x < 0.5, rbeta(60, 3, 8), rbeta(60, 8, 3))
fit <- cond.opt(
  x, y, Xpred = c(0.25, 0.75), Ypred = c(0.3, 0.7),
  max.resX = 3, max.resY = 4
)
fit$predictive_densities

Fit an Optional Pólya Tree

Description

A convenience wrapper around [apt()] that uses one non-stopping state with zero beta precision, reproducing the optional Pólya tree (OPT).

Usage

opt(
  X,
  Xpred = NULL,
  Omega.type = "unit",
  max.resol = 10,
  rho0 = 0.5,
  rho0.mode = 0,
  n.post.samples = 0
)

Arguments

Value

A list with the same structure as [apt()].

References

Wong, W. H. and Ma, L. (2010). Optional Pólya tree and Bayesian inference. *The Annals of Statistics*, 38(3), 1433–1459. doi:10.1214/09-AOS755.

See Also

[apt()], [cond.opt()]

Examples

set.seed(12345)
x <- rbeta(60, 4, 7)
fit <- opt(x, Xpred = c(0.25, 0.5, 0.75), max.resol = 4)
fit$predictive_densities

Plot a Two-Dimensional Partition

Description

Draws rectangular partition cells with optional density-based fill colors. This helper uses grid graphics and is intended for two-dimensional HMAP or posterior partitions after their integer bounds have been converted to the desired plotting coordinates.

Usage

plot_partition(
  xy.part,
  border = FALSE,
  xlim = c(0, 1),
  ylim = c(0, 1),
  zlim = grDevices::extendrange(xy.part[, "den"]),
  main = "",
  nlevels = 100,
  plot.den = TRUE,
  color.fun = grDevices::terrain.colors,
  plot.scale = TRUE,
  color = NA,
  newpage = TRUE
)

## S3 method for class 'part'
plot(x, ...)

Arguments

Value

Invisibly, the clipped partition data with an added 'fill' column.

Examples

cells <- data.frame(
  xmin = c(0, 0.5), xmax = c(0.5, 1),
  ymin = c(0, 0), ymax = c(1, 1), den = c(0.6, 1.4)
)
plot_partition(cells, plot.scale = FALSE)

Plot an Empirical ROC Curve

Description

Uses empirical null-score quantiles as thresholds. The curve treats smaller scores as stronger evidence for the positive class.

Usage

plot_roc_curve(
  p,
  p0,
  fpr = seq(0, 1, by = 0.01),
  col = "black",
  xlim = c(0, 1),
  ylim = c(0, 1),
  main = "",
  lty = 1,
  lwd = 2
)

## S3 method for class 'roc'
plot(x, ...)

Arguments

Value

Invisibly, a data frame containing 'fpr', 'threshold', and 'tpr'.

Examples

set.seed(12345)
null <- rnorm(100)
alternative <- rnorm(100, mean = -1)
roc <- plot_roc_curve(alternative, null)
head(roc)