EM algorithm for a univariate Gaussian mixture

Description

Fits a univariate Gaussian mixture model using the Expectation-Maximization (EM) algorithm. The function is intended as a lightweight fallback implementation (e.g., when mixtools is unavailable or fails).

Usage

EM(X, K = 2, max_iter = 100, tol = 1e-05)

Arguments

Details

The algorithm is initialized using the k-means clustering procedure and then alternates between:

1. E-step: computing the expectation of the complete log-likelihood function.

2. M-step: maximizing the expectation of the complete log-likelihood function.

Value

A list with the following components:

mu

Numeric vector of estimated component means (length K).

sigma

Numeric vector of estimated component standard deviations (length K).

pi

Numeric vector of estimated mixing proportions (length K).

num_iteraciones

Number of iterations performed.

posterior

Matrix of posterior probabilities (responsibilities) with dimension length(X) by K.

Examples

set.seed(1)
x <- c(rnorm(100, -2, 1), rnorm(100, 2, 1))
fit <- EM(x, K = 2)
fit$mu
fit$pi

Fisher information matrix for a two-component Gaussian mixture (working approximation).

Description

Computes a Fisher information matrix approximation based on the outer product of gradients for a two-component univariate Gaussian mixture model.

Usage

FIM_mixture_normals(data, params)

Arguments

Value

Fisher information approximation

OVL.BCAN

Description

Parametric approach using a bootstrap-based approach to estimate the variance.

Usage

OVL.BCAN(x, y, alpha = 0.05, B = 100, h_ini = -0.6)

Arguments

Value

confidence interval.

Examples

controls = rnorm(50,6,1)
cases = rnorm(100,6.5,0.5)
OVL.BCAN (controls,cases)

OVL.BCPB

Description

Parametric approach using a bootstrap percentil approach.

Usage

OVL.BCPB(x, y, alpha = 0.05, B = 100, h_ini = -0.6)

Arguments

Value

confidence interval.

Examples

controls = rnorm(50,6,1)
cases = rnorm(100,6.5,0.5)
OVL.BCPB (controls,cases)

OVL.BCbias

Description

Parametric approach using a bootstrap bias-corrected approach.

Usage

OVL.BCbias(x, y, alpha = 0.05, B = 100, h_ini = -0.6)

Arguments

Value

confidence interval.

Examples

controls = rnorm(50,6,1)
cases = rnorm(100,6.5,0.5)
OVL.BCAN (controls,cases)

OVL.D

Description

Parametric approach using the delta method.

Usage

OVL.D(x, y, alpha = 0.05)

Arguments

Value

confidence interval.

Examples

controls = rnorm(50,6,1)
cases = rnorm(100,6.5,0.5)
OVL.D (controls,cases)

OVL.DBC

Description

Parametric approach using the delta method after the Box-Cox transformation.

Usage

OVL.DBC(x, y, alpha = 0.05, h_ini = -0.6)

Arguments

Value

confidence interval.

Examples

controls = rnorm(50,6,1)
cases = rnorm(100,6.5,0.5)
OVL.DBC (controls,cases)

OVL.DBCL

Description

Parametric approach using the delta method after the Box-Cox transformation taking into account the variability of the estimated transformation parameter.

Usage

OVL.DBCL(x, y, alpha = 0.05, h_ini = -0.6)

Arguments

Value

confidence interval.

Examples

controls = rnorm(50,6,1)
cases = rnorm(100,6.5,0.5)
OVL.DBCL (controls,cases)

EM-Delta

Description

Computes a confidence interval for the OVL between two populations under Gaussian and two-component Gaussian mixture models, or both populations modeled as two-component Gaussian mixtures, using EM-based estimation and the delta method.

Usage

OVL.Delta.mix(
  x,
  y,
  alpha = 0.05,
  h = 10^(-5),
  interv = c(0, 20),
  all_mix = FALSE
)

Arguments

Value

A list containing a confidence interval. Additional elements (e.g., var_OVL, parameter estimates, OVL_hat) may also be returned.

Examples

set.seed(1)
x <- ifelse(runif(100) < 0.5, rnorm(100, mean = 0, sd = 1), rnorm(100, mean = 2, sd = 1))
y <- ifelse(runif(100) < 0.5, rnorm(100, mean = 2.5, sd = 1), rnorm(100, mean = 2, sd = 1))
res <- OVL.Delta.mix(x, y, all_mix = TRUE, interv = c(-10, 10))
res$IC1
res$IC2

OVL.GPQ

Description

Parametric approach based on generalized inference.

Usage

OVL.GPQ(x, y, alpha = 0.05, K = 2500, h_ini = -1.6, BC = FALSE)

Arguments

Value

confidence interval.

Examples

controls = rnorm(50,6,1)
cases = rnorm(100,6.5,0.5)
OVL.GPQ (controls,cases)

GPQ-Mix

Description

Computes a confidence interval for the OVL between two populations under Gaussian and two-component Gaussian mixture models, or both populations modeled as two-component Gaussian mixtures, using generalized inference.

Usage

OVL.GPQ.mix(x, y, alpha = 0.05, interv = c(0, 20), k = 1000, all_mix = FALSE)

Arguments

Value

confidence interval.

Examples

set.seed(1)
x <- ifelse(runif(100) < 0.5,
            rnorm(100, mean = 0, sd = 1),
            rnorm(100, mean = 2, sd = 1))
y <- ifelse(runif(100) < 0.5,
            rnorm(100, mean = 2.5, sd = 1),
            rnorm(100, mean = 2, sd = 1))
res <- OVL.GPQ.mix(x, y, all_mix = TRUE, interv = c(-10, 10))
res$IC1
res$IC2

OVL.K

Description

Kernel approach estimating the variance via bootstrap.

Usage

OVL.K(x, y, alpha = 0.05, B = 100, k = 1, h = 1)

Arguments

Value

confidence interval.

Examples

controls = rnorm(50,6,1)
cases = rnorm(100,6.5,0.5)
OVL.K (controls,cases)

OVL.KPB

Description

Kernel approach using a bootstrap percentile approach.

Usage

OVL.KPB(x, y, alpha = 0.05, B = 100, k = 1, h = 1)

Arguments

Value

confidence interval.

Examples

controls = rnorm(50,6,1)
cases = rnorm(100,6.5,0.5)
OVL.KPB (controls,cases)

OVL.LogitBCAN

Description

BCAN procedure carried out in the logit scale and back-transformed.

Usage

OVL.LogitBCAN(x, y, alpha = 0.05, B = 100, h_ini = -0.6)

Arguments

Value

confidence interval.

Examples

controls = rnorm(50,6,1)
cases = rnorm(100,6.5,0.5)
OVL.LogitBCAN (controls,cases)

OVL.LogitD

Description

Parametric approach using the delta method after switching to a logit scale and then transforming back.

Usage

OVL.LogitD(x, y, alpha = 0.05)

Arguments

Value

confidence interval.

Examples

controls = rnorm(50,6,1)
cases = rnorm(100,6.5,0.5)
OVL.LogitD (controls,cases)

OVL.LogitDBC

Description

Parametric approach using the delta method after the Box-Cox transformation after switching to a logit scale and then transforming back.

Usage

OVL.LogitDBC(x, y, alpha = 0.05, h_ini = -0.6)

Arguments

Value

confidence interval.

Examples

controls = rnorm(50,6,1)
cases = rnorm(100,6.5,0.5)
OVL.LogitDBC (controls,cases)

OVL.LogitDBCL

Description

OVL.LogitDBCL

Usage

OVL.LogitDBCL(x, y, alpha = 0.05, h_ini = -0.6)

Arguments

Value

confidence interval.

Examples

controls = rnorm(50,6,1)
cases = rnorm(100,6.5,0.5)
OVL.LogitDBCL (controls,cases)

OVL.LogitK

Description

Kernel approach estimating the variance via bootstrap in the logit scale and back-transformed.

Usage

OVL.LogitK(x, y, alpha = 0.05, B = 100, k = 1, h = 1)

Arguments

Value

confidence interval.

Examples

controls = rnorm(50,6,1)
cases = rnorm(100,6.5,0.5)
OVL.LogitK (controls,cases)

Numerical derivatives of the overlap functional (normal vs. 2-component mixture).

Description

Computes finite-difference approximations of the partial derivatives of the overlap coefficient with respect to the parameters in the case where the first group is modeled as a normal distribution and the second group as a two-component Gaussian mixture.

Usage

OVL_derivates_1(
  mu1_hat,
  mu2_hat,
  sigma1_hat,
  sigma2_hat,
  pi2_hat,
  h,
  intersec,
  OVL_mix
)

Arguments

Details

This function is intended for internal use (delta-method variance estimation).

Value

A list with components deriv1–deriv7.

Numerical derivatives of the overlap functional (2-component mixture vs. 2-component mixture).

Description

Computes finite-difference approximations of the partial derivatives of the overlap coefficient with respect to the parameters when both groups are modeled as two-component Gaussian mixtures.

Usage

OVL_derivates_2(
  mu1_hat,
  mu2_hat,
  sigma1_hat,
  sigma2_hat,
  pi1_hat,
  pi2_hat,
  h,
  intersec,
  OVL_mix
)

Arguments

Details

This function is intended for internal use (delta-method variance estimation).

Value

A list with components deriv1–deriv10.

Computes the overlap coefficient (OVL) between two cumulative distribution functions corresponding to finite mixtures of two normal distributions.

Description

Computes the overlap coefficient (OVL) between two cumulative distribution functions corresponding to finite mixtures of two normal distributions.

Usage

OVL_mix(mu1, mu2, sigma1, sigma2, pi_H, pi_D, x)

Arguments

Details

Mixing proportions equal to 0 or 1 are allowed, in which case the corresponding mixture reduces to a single normal distribution.

Value

A numeric value corresponding to the OVL between the two mixture distributions.

Evaluates an auxiliary function.

Description

Evaluates an auxiliary function.

Usage

U(mu1, mu2, sigma1, sigma2)

Arguments

Value

evaluation of an auxiliary function.

Computes the probability density function of a finite mixture of normal distributions at a given point or vector of points.

Description

Computes the probability density function of a finite mixture of normal distributions at a given point or vector of points.

Usage

dnorm_mixture(x, mu, sigma, pi)

Arguments

Value

A numeric vector containing the values of the mixture density evaluated at x.

Identifies the roots of a univariate function over a given interval by subdividing the interval into smaller subintervals and applying uniroot on those subintervals where a sign change is detected.

Description

Identifies the roots of a univariate function over a given interval by subdividing the interval into smaller subintervals and applying uniroot on those subintervals where a sign change is detected.

Usage

encontrar_raices(intersection_function, interval, n_subintervals = 10)

Arguments

Value

A numeric vector containing the distinct roots found within interval. If no roots are detected, an empty numeric vector is returned.

Evaluates the Epanechnikov kernel.

Description

Evaluates the Epanechnikov kernel.

Usage

kernel.e(u)

Arguments

Value

evaluation of the Epanechnikov kernel.

Estimates the density function using the Epanechnikov kernel.

Description

Estimates the density function using the Epanechnikov kernel.

Usage

kernel.e.density(data, points, h)

Arguments

Value

density estimation.

Evaluates the Gaussian kernel.

Description

Evaluates the Gaussian kernel.

Usage

kernel.g(u)

Arguments

Value

evaluation of the Gaussian kernel.

Estimates the density function using the Gaussian kernel.

Description

Estimates the density function using the Gaussian kernel.

Usage

kernel.g.density(data, points, h)

Arguments

Value

density estimation.

Computation of the likelihood function of the BoxCox transformation.

Description

Computation of the likelihood function of the BoxCox transformation.

Usage

likbox(h, data, n)

Arguments

Value

the likelihood function of the BoxCox transformation.

Simulated data with normal and mixture of normal distributions

Description

Contains control and case samples generated from a normal distribution and a two-component normal mixture distribution, respectively.

Usage

data(mixnorm_data)

Format

A data frame with 100 rows and 2 variables:

controls

Simulated data from a N(5,1) normal distribution.

cases

Simulated data from a two-component normal mixture distribution: 0.8N(2,1) + 0.2N(3,1).

References

This dataset was artificially generated for the OVL.CI package.

Examples

data(mixnorm_data)

Computes the sample variance of a vector of observations.

Description

Computes the sample variance of a vector of observations.

Usage

ssdd(x)

Arguments

Value

the sample variance.

Simulated data with normal distributions

Description

Contains controls and cases data from normal distributions.

Usage

data(test_data)

Format

A data frame with 100 rows and 2 variables:

controls

Simulated data from a N(10,1)distribution for the control group.

cases

Simulated data from a N(10.5,0.5)distribution for the case group.

References

This data set was artificially created for the OVL.CI package.

Examples

data(test_data)