Fit a Dirichlet Kernel Process (DKP) Model

Fits a DKP model for multinomial response data by locally smoothing observed counts to estimate latent Dirichlet parameters.

Usage

fit.DKP(
  X,
  Y,
  Xbounds = NULL,
  prior = c("noninformative", "fixed", "adaptive"),
  r0 = 2,
  p0 = NULL,
  kernel = c("gaussian", "matern52", "matern32"),
  loss = c("brier", "log_loss"),
  n_multi_start = NULL
)

Arguments

X: A numeric input matrix of size \(n \times d\), where each row represents a covariate vector.
Y: Matrix of observed multinomial counts, with dimension \(n \times q\).
Xbounds: Optional \(d \times 2\) matrix specifying the lower and upper bounds of each input dimension. Used to normalize inputs to \([0,1]^d\). If Xbounds is NULL, the input is assumed to have already been normalized, and the default bounds are set to \([0,1]^d\).
prior: Type of prior to use. One of "noninformative", "fixed", or "adaptive".
r0: Global prior precision (only used when prior = "fixed" or "adaptive").
p0: Global prior mean vector (only used when prior = "fixed"). Must be of length \(q\).
kernel: Kernel function for local weighting. Choose from "gaussian", "matern52", or "matern32".
loss: Loss function for kernel hyperparameter tuning. One of "brier" (default) or "log_loss".
n_multi_start: Number of random initializations for multi-start optimization. Default is 10 × d.

Value

A list of class "DKP" representing the fitted DKP model, with the following components:

theta_opt: Optimized kernel hyperparameters (lengthscales).
kernel: Kernel function used, as a string.
loss: Loss function used for hyperparameter tuning.
loss_min: Minimum loss value achieved during optimization.
X: Original (unnormalized) input matrix of size n × d.
Xnorm: Normalized input matrix scaled to \([0,1]^d\).
Xbounds: Matrix specifying normalization bounds for each input dimension.
Y: Observed multinomial counts of size n × q.
prior: Type of prior used.
r0: Prior precision parameter.
p0: Prior mean (for fixed priors).
alpha0: Prior Dirichlet parameters at each input location (scalar or matrix).
alpha_n: Posterior Dirichlet parameters after kernel smoothing.

Examples

#-------------------------- 1D Example ---------------------------
set.seed(123)

# Define true class probability function (3-class)
true_pi_fun <- function(X) {
  p <- (1 + exp(-X^2) * cos(10 * (1 - exp(-X)) / (1 + exp(-X)))) / 2
  return(matrix(c(p/2, p/2, 1 - p), nrow = length(p)))
}

n <- 30
Xbounds <- matrix(c(-2, 2), nrow = 1)
X <- tgp::lhs(n = n, rect = Xbounds)
true_pi <- true_pi_fun(X)
m <- sample(100, n, replace = TRUE)

# Generate multinomial responses
Y <- t(sapply(1:n, function(i) rmultinom(1, size = m[i], prob = true_pi[i, ])))

# Fit DKP model
model1 <- fit.DKP(X, Y, Xbounds = Xbounds)
print(model1)
#> --------------------------------------------------
#>            Dirichlet Kernel Process (DKP) Model        
#> --------------------------------------------------
#> Number of observations (n):  30
#> Input dimensionality (d):    1
#> Output dimensionality (q):   3
#> Kernel type:                 gaussian
#> Loss function used:          brier
#> Optimized kernel parameters: 0.0407
#> Minimum achieved loss:       0.00058
#> 
#> Prior specification:
#>   Noninformative prior: Dirichlet(1,...,1).
#> --------------------------------------------------


#-------------------------- 2D Example ---------------------------
set.seed(123)

# Define latent function and transform to 3-class probabilities
true_pi_fun <- function(X) {
  if (is.null(nrow(X))) X <- matrix(X, nrow = 1)
  m <- 8.6928; s <- 2.4269
  x1 <- 4 * X[,1] - 2
  x2 <- 4 * X[,2] - 2
  a <- 1 + (x1 + x2 + 1)^2 *
    (19 - 14*x1 + 3*x1^2 - 14*x2 + 6*x1*x2 + 3*x2^2)
  b <- 30 + (2*x1 - 3*x2)^2 *
    (18 - 32*x1 + 12*x1^2 + 48*x2 - 36*x1*x2 + 27*x2^2)
  f <- (log(a * b) - m) / s
  p <- pnorm(f)
  return(matrix(c(p/2, p/2, 1 - p), nrow = length(p)))
}

n <- 100
Xbounds <- matrix(c(0, 0, 1, 1), nrow = 2)
X <- tgp::lhs(n = n, rect = Xbounds)
true_pi <- true_pi_fun(X)
m <- sample(100, n, replace = TRUE)

# Generate multinomial responses
Y <- t(sapply(1:n, function(i) rmultinom(1, size = m[i], prob = true_pi[i, ])))

# Fit DKP model
model2 <- fit.DKP(X, Y, Xbounds = Xbounds)
print(model2)
#> --------------------------------------------------
#>            Dirichlet Kernel Process (DKP) Model        
#> --------------------------------------------------
#> Number of observations (n):  100
#> Input dimensionality (d):    2
#> Output dimensionality (q):   3
#> Kernel type:                 gaussian
#> Loss function used:          brier
#> Optimized kernel parameters: 0.1245, 0.0706
#> Minimum achieved loss:       0.00061
#> 
#> Prior specification:
#>   Noninformative prior: Dirichlet(1,...,1).
#> --------------------------------------------------

Fit a Dirichlet Kernel Process (DKP) Model

Usage

Arguments

Value

See also

Examples