Help for package SSLfmm

Type:

Package

Title:

Semi-Supervised Learning under a Mixed-Missingness Mechanism in Finite Mixture Models

Version:

0.1.0

Maintainer:

Jinran Wu <jinran.wu@uq.edu.au>

Description:

Implements a semi-supervised learning framework for finite mixture models under a mixed-missingness mechanism. The approach models both missing completely at random (MCAR) and entropy-based missing at random (MAR) processes using a logistic–entropy formulation. Estimation is carried out via an Expectation–-Conditional Maximisation (ECM) algorithm with robust initialisation routines for stable convergence. The methodology relates to the statistical perspective and informative missingness behaviour discussed in Ahfock and McLachlan (2020) <doi:10.1007/s11222-020-09971-5> and Ahfock and McLachlan (2023) <doi:10.1016/j.ecosta.2022.03.007>. The package provides functions for data simulation, model estimation, prediction, and theoretical Bayes error evaluation for analysing partially labelled data under a mixed-missingness mechanism.

License:

GPL-3

Encoding:

UTF-8

RoxygenNote:

7.3.3

Depends:

R (≥ 4.2.0)

Imports:

stats, mvtnorm, matrixStats

NeedsCompilation:

Packaged:

2025-12-04 00:22:06 UTC; uqjwu15

Author:

Jinran Wu

[aut, cre], Geoffrey J. McLachlan

[aut]

Repository:

CRAN

Date/Publication:

2025-12-09 16:30:28 UTC

EM for Semi-Supervised FMM with a Mixed-Missingness Mechanism (MCAR + entropy-based MAR)

Description

Runs an EM-like procedure that models a mixed-missingness mechanism: unlabeled indicator m_j follows a mixed-missingness mechanism with MCAR (prob \alpha) and entropy-based MAR via a logistic link q_j = \text{logit}^{-1}(\xi_0 + \xi_1 \log e_j). Supports shared (ncov = 1) or class-specific (ncov = 2) covariance.

Usage

EM_FMM_SemiSupervised(
  data,
  g = 2,
  init_res,
  max_iter = 5,
  tol = 1e-06,
  ncov = 1,
  verbose = FALSE
)

Arguments

data

A data.frame or matrix with p+2 columns: first p are features, then missing (0 = labelled, 1 = unlabelled), and z (class label for labelled rows; ignored otherwise).

g

Integer, number of mixture components (classes).

init_res

A list with initial parameters:

pi: numeric length-g (mixture weights, sum to 1)
mu: list of length g, each length-p mean vector
Sigma: if ncov = 1, a p x p matrix; if ncov = 2, a list of g p x p matrices
alpha: scalar in (0,1)
xi: numeric length-2, logistic coefficients (xi0, xi1)

max_iter

Integer, max EM iterations.

tol

Convergence tolerance on log-likelihood increase.

ncov

Integer covariance structure: 1 = shared/equal, 2 = class-specific/unequal.

verbose

Logical; if TRUE, progress messages are printed using message(). Default is FALSE.

Details

This function expects the following helpers to be available:

pack_theta(pi_k, mu_k, Sigma_k, g, p, ncov)
unpack_theta(theta, g, p, ncov)
neg_loglik(theta, Y_all, m_j, Z_all, d2_yj, xi, alpha_k, unpacker)
get_entropy(dat, n, p, g, paralist) returning per-observation entropy-like values

Value

A list with elements: pi, mu, Sigma, xi, alpha, loglik, and ncov.

Examples


  ## Toy example using a simple partially-labelled dataset
  set.seed(1)

  ## 1) Construct an n x (p+2) partially-labelled dataset:
  ##    first p columns = features, then 'missing' and 'z'
  n <- 100; p <- 2; g <- 2

  X <- matrix(rnorm(n * p), nrow = n, ncol = p)

  ## missing: 0 = labelled, 1 = unlabelled
  missing <- rbinom(n, size = 1, prob = 0.3)

  ## z: observed class labels for labelled rows, NA for unlabelled
  z <- rep(NA_integer_, n)
  z[missing == 0] <- sample(1:g, sum(missing == 0), replace = TRUE)

  sim_dat <- data.frame(X, missing = missing, z = z)

  ## 2) Warm-up initialisation using the complete-data initializer
  init <- EM_FMM_SemiSupervised_Complete_Initial(
    data = sim_dat,
    g    = g,
    ncov = 1
  )

  ## 3) Run the main EM algorithm (small number of iterations)
  fit <- EM_FMM_SemiSupervised(
    data     = sim_dat,
    g        = g,
    init_res = init,
    ncov     = 1,
    max_iter = 5,
    verbose  = FALSE
  )

  str(fit)

Complete-Data Warm-Up Initialization for Semi-Supervised FMM with a Mixed-Missingness Mechanism

Description

Uses both labeled and unlabeled subsets of the data to obtain quick initial estimates for mixture parameters and missingness mechanism parameters (alpha, xi) via a warm-up EM procedure.

Usage

EM_FMM_SemiSupervised_Complete_Initial(
  data,
  g = 2,
  ncov = 1,
  alpha_init = 0.01,
  warm_up_iter = 200,
  tol = 1e-06
)

Arguments

data

A data frame containing:

The first p columns: numeric variables used in the FMM.
A column missing: indicator (0 = labeled, 1 = unlabeled/missing).
A column z: class labels for labeled rows (1:g); NA for unlabeled.

g

Integer, number of mixture components (default 2).

ncov

Integer, covariance structure: 1 = shared (equal), 2 = class-specific (unequal).

alpha_init

Numeric in (0,1), initial MCAR proportion (default 0.01).

warm_up_iter

Integer, number of warm-up EM iterations (default 200).

tol

Convergence tolerance on alpha (default 1e-6).

Details

This function first calls initialestimate to get initial \pi, \mu, \Sigma.
Then it calls EM_FMM_SemiSupervised_Initial with these values for a short warm-up run.
Covariance structure (equal vs. unequal) is determined by ncov.

Value

A list with initial values for EM_FMM_SemiSupervised:

pi - mixture weights.
mu - list of component mean vectors.
Sigma - covariance matrix/matrices.
alpha - MCAR proportion.
xi - logistic regression coefficients for MAR mechanism.

Quick Initializer for alpha, xi, and Mixture Parameters

Description

Provides rough initial estimates of the missingness parameters alpha and xi, together with mixture parameters pi, mu, and covariance matrices, using a lightweight EM-style routine. The covariance structure is chosen automatically based on Sigma_init:

If Sigma_init is a p \times p matrix, a shared (equal) covariance is used.
If Sigma_init is a list of length g of p \times p matrices or a p \times p \times g array, class-specific (unequal) covariances are used.
If Sigma_init is NULL, a shared covariance is estimated from the labeled data.

This function is intended as a fast, heuristic initializer rather than a final estimator for the mixed missingness model.

Usage

EM_FMM_SemiSupervised_Initial(
  Y_labelled,
  Z_labelled,
  Y_unlabelled,
  g = 2,
  pi_init = NULL,
  mu_init = NULL,
  Sigma_init = NULL,
  alpha_init = 0.01,
  warm_up_iter = 50,
  tol = 1e-06
)

Arguments

Y_labelled

Numeric matrix of labeled observations (n_L \times p).

Z_labelled

Integer vector of class labels in 1:g for Y_labelled.

Y_unlabelled

Numeric matrix of unlabeled observations (n_U \times p).

g

Integer, number of mixture components (default 2).

pi_init

Optional numeric length-g vector of initial mixing proportions.

mu_init

Optional list of length g of initial mean vectors (each of length p).

Sigma_init

Optional initial covariance: a p \times p matrix (shared), or a list of g p \times p matrices, or a p \times p \times g array (class-specific).

alpha_init

Numeric in (0,1), initial MCAR proportion (default 0.01).

warm_up_iter

Integer, number of warm-up EM iterations used to refine the quick initial estimates (default 50).

tol

Convergence tolerance on alpha (default 1e-6).

Value

A list with elements:

pi - length-g vector of mixing proportions.
mu - list of g mean vectors.
Sigma - shared p \times p matrix (equal-Sigma) or list of g matrices (unequal-Sigma).
xi - length-2 numeric vector c(xi0, xi1) from the logistic MAR model.
alpha - estimated MCAR proportion.
gamma - n \times g responsibility matrix.
d2_yj - numeric vector of entropy-based scores used in the missingness model.

Note

This is a heuristic warm-up routine. It requires helper functions normalise_logprob() and get_entropy() to be available in the package namespace.

Bayes' Rule Classifier

Description

Classifier specified by Bayes' rule. Assigns \arg\max_i \{ \log(\pi_i) + \log \mathcal{N} (y_j \mid \mu_i, \Sigma_i) \}.

Usage

bayesclassifier(dat, p, g, pi = NULL, mu = NULL, sigma = NULL, paralist = NULL)

Arguments

dat

An n \times p numeric matrix (or a length-p numeric vector, treated as 1 \times p).

p

Integer; dimension of the observation vector.

g

Integer; number of Gaussian components/classes.

pi

Numeric length-g vector of mixing proportions (must sum to 1).

mu

Either:

A p \times g numeric matrix (column i is \mu_i), or
A length-g list of length-p numeric vectors.

sigma

Either:

A p \times p covariance matrix (shared), or
A length-g list of p \times p covariance matrices, or
A p \times p \times g numeric array.

paralist

Optional list with elements pi, mu, sigma (overrides explicit args if provided and non-NULL).

Value

An integer vector of length n with predicted class labels in 1:g.

Examples

# Minimal example with list-style mu and sigma:
set.seed(1)
p <- 2; g <- 2
pi <- c(0.5, 0.5)
mu <- list(`1`=c(0.96, 0.02), `2`=c(-1.02, -0.03))
sigma <- list(
  matrix(c(0.9417379,0.5447264, 0.5447264,0.9811853), 2, 2, byrow=TRUE),
  matrix(c(0.9984812,0.3314474, 0.3314474,1.1316865), 2, 2, byrow=TRUE)
)
X <- mvtnorm::rmvnorm(5, mean = mu[[1]], sigma = sigma[[1]])
bayesclassifier(X, p=p, g=g, pi=pi, mu=mu, sigma=sigma)

Squared Discriminant Score for Two-Group LDA (Equal Covariance)

Description

Computes d^2 where d = \beta_0 + y^\top \beta_1 for two classes under the common-covariance (LDA) model:

\beta_1 = \Sigma^{-1}(\mu_1 - \mu_2), \quad \beta_0 = \log(\pi_1/\pi_2) - \tfrac{1}{2}(\mu_1 + \mu_2)^\top \Sigma^{-1}(\mu_1 - \mu_2).

Usage

compute_d2(y, mu1, mu2, Sigma_inv, pi1, pi2)

Arguments

y

Numeric vector (length p) or numeric matrix with p columns; rows are observations.

mu1, mu2

Numeric vectors of length p: class means.

Sigma_inv

p \times p numeric precision matrix (inverse covariance).

pi1, pi2

Positive scalars: class prior probabilities (need not sum to 1).

Value

If y is a vector, a single numeric d^2. If y is a matrix, a numeric vector of d^2 values for each row.

Examples

set.seed(1)
mu1 <- c(0, 0); mu2 <- c(1, 1)
S <- matrix(c(1, .2, .2, 1), 2, 2)
Sigma_inv <- solve(S)
x <- c(0.5, -0.2)
compute_d2(x, mu1, mu2, Sigma_inv, pi1 = 0.6, pi2 = 0.4)

X <- matrix(rnorm(10 * 2), ncol = 2)
compute_d2(X, mu1, mu2, Sigma_inv, 0.5, 0.5)

Compute Theoretical Bayes' Error for a Binary Gaussian Mixture

Description

Computes the Bayes'classification error rate for a two-component Gaussian mixture given mu_hat, Sigma_hat, and pi_hat. If mu_hat is supplied as a list of length 2, it is converted to a p \times 2 matrix internally.

Usage

error_beta_classification(mu_hat, Sigma_hat, pi_hat)

Arguments

mu_hat

Either a numeric matrix of size p \times 2 whose columns are component means, or a list of two numeric vectors.

Sigma_hat

Numeric p \times p covariance matrix shared across components.

pi_hat

Numeric vector of length 2 with mixing proportions (\pi_1, \pi_2) that are non-negative and sum to 1.

Details

The linear discriminant is

\beta_1 = \Sigma^{-1}(\mu_1 - \mu_2), \qquad \beta_0 = -\frac{1}{2}(\mu_1 + \mu_2)^\top \Sigma^{-1}(\mu_1 - \mu_2) + \log(\pi_1 / \pi_2)

and the Bayes' error is

\mathrm{Err} = \sum_{k = 1}^2 \pi_k \, \Phi\left( \frac{(-1)^k \{\beta_0 + \beta_1^\top \mu_k\}} {\|\beta_1\|} \right),

where \Phi is the standard normal cdf.

Value

A numeric scalar giving the theoretical Bayes' classification error rate.

Examples

mu_hat <- matrix(c(1, 0, -1, 0), nrow = 2)  # columns are mu1, mu2
Sigma_hat <- diag(2)
pi_hat <- c(0.5, 0.5)
error_beta_classification(mu_hat, Sigma_hat, pi_hat)

Posterior Cluster Probabilities for a Gaussian Mixture

Description

Compute posterior membership probabilities \tau_{ij} for each observation under a Gaussian mixture with either a shared covariance matrix or component-specific covariances.

Usage

get_clusterprobs(
  dat,
  n,
  p,
  g,
  pi = NULL,
  mu = NULL,
  sigma = NULL,
  paralist = NULL
)

Arguments

dat

Numeric matrix n \times p. Data matrix (rows = observations).

n

Integer. Number of observations (checked against nrow(dat)).

p

Integer. Number of variables (checked against ncol(dat)).

g

Integer. Number of components.

pi

Optional numeric vector length g. Mixing proportions (sum to 1).

mu

Optional numeric matrix p \times g. Column i is mean of component i.

sigma

Optional numeric matrix p \times p (shared covariance) or array p \times p \times g (component-specific covariances).

paralist

Optional list with elements pi, mu, sigma. If provided, these take precedence over the corresponding explicit args.

Value

Numeric matrix n \times g of posterior probabilities \tau_{ij}.

Examples

  n <- 100; p <- 2; g <- 2
  X <- matrix(rnorm(n*p), n, p)
  pi <- c(0.6, 0.4)
  mu <- cbind(c(0,0), c(1,1))
  Sig <- array(0, dim = c(p,p,g)); Sig[,,1] <- diag(p); Sig[,,2] <- diag(p)
  tau <- get_clusterprobs(X, n, p, g, pi = pi, mu = mu, sigma = Sig)
  head(tau)

Per-Row Entropy of Posterior Cluster Probabilities

Description

Compute the Shannon entropy (in nats) of posterior membership probabilities for each observation, given a Gaussian mixture model. Posterior probabilities \tau_{ij} are obtained via get_clusterprobs().

Usage

get_entropy(dat, n, p, g, pi = NULL, mu = NULL, sigma = NULL, paralist = NULL)

Arguments

dat

Numeric matrix n \times p. The data matrix.

n

Integer. Number of rows in dat.

p

Integer. Number of columns (features) in dat.

g

Integer. Number of mixture components.

pi

Optional numeric vector length g. Mixing proportions (sum to 1).

mu

Optional numeric matrix p \times g. Column j is the mean of component j.

sigma

Optional numeric matrix p \times p (shared covariance) or array p \times p \times g (component-specific covariances).

paralist

Optional list with elements pi, mu, sigma. If supplied, these take precedence over the corresponding explicit arguments.

Value

Numeric vector of length n, the entropy per observation (in nats).

Examples

  # Suppose you have get_clusterprobs(), and a fitted/pi, mu, sigma:
  n <- 100; p <- 2; g <- 2
  X <- matrix(rnorm(n * p), n, p)
  pi <- c(0.6, 0.4)
  mu <- cbind(c(0,0), c(1,1))
  Sigma <- array(0, dim = c(p, p, g))
  Sigma[,,1] <- diag(p); Sigma[,,2] <- diag(p)
  ent <- get_entropy(dat = X, n = n, p = p, g = g, pi = pi, mu = mu, sigma = Sigma)
  summary(ent)

Initialize Parameters for a FMM from Labeled Subset

Description

Builds initial estimates (\pi, \mu, \Sigma) for a g-component Gaussian mixture using only rows with observed labels in zm. Supports either a shared covariance (ncov = 1) or class-specific covariances (ncov = 2).

Usage

initialestimate(dat, zm, g, ncov = 2, ridge = 1e-06)

Arguments

dat

A numeric matrix or data frame of features (n x p).

zm

Integer vector of length n with class labels in 1:g; use NA for unlabeled rows. Only labeled rows contribute to the initialization.

g

Integer, number of mixture components.

ncov

Integer, 1 for a shared covariance matrix, 2 for class-specific covariance matrices. Default 2.

ridge

Numeric, small diagonal ridge added to covariance(s) for numerical stability. Default 1e-6.

Details

If a class has zero or one labeled sample, its covariance is set to the global empirical covariance (from labeled data) with a small ridge. Class means for empty classes default to the global mean with a small jitter.

Value

A list with

pi: length-g vector of mixing proportions (summing to 1).
mu: p x g matrix of class means (column i is \mu_i).
sigma: if ncov = 1, a p x p shared covariance matrix; if ncov = 2, a p x p x g array of class-specific covariances.

Examples

set.seed(1)
n <- 50; p <- 3; g <- 2
X <- matrix(rnorm(n*p), n, p)
z <- sample(c(1:g, NA), n, replace = TRUE, prob = c(0.4, 0.4, 0.2))
init <- initialestimate(X, z, g, ncov = 2)
str(init)

Numerically Stable Log-Sum-Exp

Description

Computes \log(\sum_i \exp(x_i)) in a numerically stable way by subtracting the maximum value before exponentiation.

Usage

logsumexp(x)

Arguments

x

A numeric vector.

Value

A single numeric value: the log-sum-exp of x.

Examples

logsumexp(c(1000, 1001, 1002))

Negative Log-Likelihood for Semi-Supervised FMM with a Mixed-Missingness Mechanism

Description

Computes the negative log-likelihood for a semi-supervised Gaussian mixture model under a mixed missingness mechanism (MCAR + entropy-based MAR). Assumes a covariance matrix \Sigma across all mixture components.

Usage

neg_loglik(theta, Y, m_j, Z, d2_yj, xi, alpha_k, unpack_fn)

Arguments

theta

Numeric vector of packed model parameters to be unpacked by unpack_fn.

Y

Numeric matrix of observations (n x p).

m_j

Integer or logical vector of length n indicating missingness: 0 for observed (labeled block), 1 for unlabeled/missingness block.

Z

Integer vector of length n with class labels for labeled samples (1..g); use NA for unlabeled rows.

d2_yj

Numeric vector of length n with the entropy-like score used in the MAR mechanism (e.g., posterior entropy or any scalar proxy).

xi

Numeric length-2 vector c(xi0, xi1) for the logistic MAR model q_j = \text{logit}^{-1}(\xi_0 + \xi_1 d2_{yj}).

alpha_k

Numeric scalar in (0,1), the MCAR mixing proportion in the missingness mechanism.

unpack_fn

Function that takes theta and returns a list with elements:

pi: Numeric vector of length g with mixture weights.
mu: List of length g; each element is a numeric mean vector (length p).
sigma: Shared covariance matrix (p x p).

Details

The total log-likelihood is composed of three parts:

Labeled samples (m_j=0) with observed class labels Z_j.
Unlabeled samples attributed to MCAR with probability mass m_{1j}.
Unlabeled samples attributed to MAR with probability mass m_{2j}.

The MAR probability for each sample is q_j = \text{logit}^{-1}(\xi_0 + \xi_1 d2_{yj}). Internally, the function uses a numerically stable logSumExp.

Value

A single numeric value: the negative log-likelihood.

Note

This implementation is for the equal covariance case (shared \Sigma).

Examples

# Minimal example (illustrative only):
library(mvtnorm)
set.seed(1)
n <- 20; p <- 2; g <- 2
Y <- matrix(rnorm(n*p), n, p)
Z <- sample(c(1:g, rep(NA, n - g)), n, replace = TRUE)
m_j <- ifelse(is.na(Z), 1L, 0L)
d2_yj <- runif(n)
xi <- c(-1, 2)
alpha_k <- 0.4
unpack_fn <- function(theta) {
  list(pi = c(0.6, 0.4),
       mu = list(c(0,0), c(1,1)),
       sigma = diag(p))
}
theta <- numeric(1) # not used in this toy unpack_fn
neg_loglik(theta, Y, m_j, Z, d2_yj, xi, alpha_k, unpack_fn)

Normalise Log-Probabilities

Description

Converts log-probabilities into a probability distribution by exponentiating in a numerically stable way.

Usage

normalise_logprob(log_probs)

Arguments

log_probs

A numeric vector of log-probabilities.

Value

A numeric vector summing to 1 (the normalised probabilities).

Examples

lp <- c(-1000, -999, -998)
normalise_logprob(lp)

Pack FMM Parameters into a Vector

Description

Packs mixture weights, means, and covariance(s) into a single numeric vector. Uses the last component as the baseline for mixture weights (g-1 logits stored).

Usage

pack_theta(pi_k, mu_k, Sigma, g, p, ncov = 1)

Arguments

pi_k

Numeric vector of length g with mixture weights (positive, sum to 1).

mu_k

List of length g; each element a numeric vector of length p (component means).

Sigma

Covariance: if ncov = 1, a single p x p matrix; if ncov = 2, a list of g p x p matrices.

g

Integer: number of components.

p

Integer: dimension.

ncov

Integer: covariance structure; 1 for shared covariance, 2 for class-specific.

Value

Numeric vector with parameters packed.

Draw from a Gaussian Mixture Model

Description

Generate i.i.d. samples from a finite Gaussian mixture with either a shared covariance matrix or component-specific covariance matrices.

Usage

rmix(n, pi, mu, sigma, seed_number)

Arguments

n

Integer. Number of observations to generate.

pi

Numeric vector of length g. Mixing proportions (must sum to 1).

mu

Numeric matrix p \times g. Column j is the mean for component j.

sigma

Either a numeric matrix p \times p (shared covariance), or a numeric array p \times p \times g (component-specific covariances).

seed_number

Integer. Seed for reproducibility.

Value

A list with:

Y

Numeric matrix n \times p of generated features.

Z

Numeric matrix n \times g of one-hot component indicators.

clust

Integer vector n, the component labels in 1:g.

Examples

  set.seed(1)
  g  <- 2; p <- 2
  pi <- c(0.5, 0.5)
  mu <- cbind(c(1,0), c(-1,0))
  Sigma <- diag(p)
  out <- rmix(500, pi, mu, Sigma, seed_number = 123)
  str(out)

Simulate a Gaussian Mixture Dataset with a Mixed-Missingness Mechanism (MAR + MCAR)

Description

Simulate a Gaussian Mixture Dataset with a Mixed-Missingness Mechanism (MAR + MCAR)

Usage

simulate_mixed_missingness(
  n = 500,
  pi,
  mu,
  sigma,
  xi0 = 2,
  xi1 = 3,
  alpha = 0.1,
  seed_id = 123
)

Arguments

n

Integer; sample size.

pi

Numeric vector; mixing proportions (sum to 1).

mu

Matrix (p x K); component means, columns = components.

sigma

Array (p x p x K); component covariance matrices.

xi0

Numeric; MAR logit intercept.

xi1

Numeric; MAR logit slope on entropy.

alpha

Numeric in [0,1]; MCAR rate applied within both MAR and observed groups.

seed_id

Integer; seed passed to rmix() (your generator).

Details

Requires user-provided functions:

rmix(n, pi, mu, sigma, seed_number)
get_entropy(dat, n, p, g, paralist)

Missingness mechanism codes:

0 = fully observed
1 = MCAR
2 = MAR (entropy-based)

Value

A list with:

data: data.frame with columns x1..xp, en, missing, label, truth
true_setup: list(pi, mu, sigma)
groups: list(mar_group, obs_group, mcar_in_mar, mcar_in_obs)
probs: vector prob_mar
raw: original rmix output dat augmented with en and labels

Unpack FMM Parameter Vector

Description

Unpacks mixture weights, means, and covariance(s) from a parameter vector.

Usage

unpack_theta(theta, g, p, ncov = 1)

Arguments

theta

Numeric vector as returned by pack_theta().

g

Integer: number of components.

p

Integer: dimension.

ncov

Integer: covariance structure; 1 for shared covariance, 2 for class-specific.

Value

A list with:

pi: Mixture weights (length g).
mu: List of g mean vectors.
sigma: Shared covariance matrix (ncov=1) or list of g covariance matrices (ncov=2).

EM for Semi-Supervised FMM with a Mixed-Missingness Mechanism (MCAR + entropy-based MAR)

Description

Usage

Arguments

Details

Value

Examples

Complete-Data Warm-Up Initialization for Semi-Supervised FMM with a Mixed-Missingness Mechanism

Description

Usage

Arguments

Details

Value

Quick Initializer for alpha, xi, and Mixture Parameters

Description

Usage

Arguments

Value

Note

Bayes' Rule Classifier

Description

Usage

Arguments

Value

Examples

Squared Discriminant Score for Two-Group LDA (Equal Covariance)

Description

Usage

Arguments

Value

Examples

Compute Theoretical Bayes' Error for a Binary Gaussian Mixture

Description

Usage

Arguments

Details

Value

Examples

Posterior Cluster Probabilities for a Gaussian Mixture

Description

Usage

Arguments

Value

Examples

Per-Row Entropy of Posterior Cluster Probabilities

Description

Usage

Arguments

Value

See Also

Examples

Initialize Parameters for a FMM from Labeled Subset

Description

Usage

Arguments

Details

Value

Examples

Numerically Stable Log-Sum-Exp

Description

Usage

Arguments

Value

Examples

Negative Log-Likelihood for Semi-Supervised FMM with a Mixed-Missingness Mechanism

Description

Usage

Arguments

Details

Value

Note

Examples

Normalise Log-Probabilities

Description

Usage

Arguments

Value

Examples

Pack FMM Parameters into a Vector

Description