mix_mode: Mode estimation

In BayesMultiMode: Bayesian Mode Inference

Mode estimation

Description

Mode estimation in univariate mixture distributions. The fixed-point algorithm of \insertCitecarreira-perpinan_mode-finding_2000;textualBayesMultiMode is used for Gaussian mixtures. The Modal EM algorithm of \insertCiteli_nonparametric_2007;textualBayesMultiMode is used for other continuous mixtures. A basic algorithm is used for discrete mixtures, see \insertCiteCross2024;textualBayesMultiMode.

Usage

mix_mode(
  mixture,
  tol_mixp = 0,
  tol_x = 1e-06,
  tol_conv = 1e-08,
  type = "all",
  inside_range = TRUE
)

Arguments

mixture	An object of class mixture generated with mixture().
tol_mixp	Components with a mixture proportion below tol_mixp are discarded when estimating modes; note that this does not apply to the biggest component so that it is not possible to discard all components; should be between 0 and 1; default is 0.
tol_x	(for continuous mixtures) Tolerance parameter for distance in-between modes; default is 1e-6; if two modes are closer than tol_x the first estimated mode is kept.
tol_conv	(for continuous mixtures) Tolerance parameter for convergence of the algorithm; default is 1e-8.
type	(for discrete mixtures) Type of modes, either "unique" or "all" (the latter includes flat modes); default is "all".
inside_range	Should modes outside of mixture$range be discarded? Default is TRUE. This sometimes occurs with very small components when K is large.

Details

This function finds modes in a univariate mixture defined as:

p(.) = \sum_{k=1}^{K}\pi_k p_k(.),

where p_k is a density or probability mass/density function.

Fixed-point algorithm Following \insertCitecarreira-perpinan_mode-finding_2000;textualBayesMultiMode, a mode x is found by iterating the two steps:

(i) \quad p(k|x^{(n)}) = \frac{\pi_k p_k(x^{(n)})}{p(x^{(n)})},

(ii) \quad x^{(n+1)} = f(x^{(n)}),

with

f(x) = (\sum_k p(k|x) \sigma_k)^{-1}\sum_k p(k|x) \sigma_k \mu_k,

until convergence, that is, until abs(x^{(n+1)}-x^{(n)})< \text{tol}_\text{conv}, where \text{tol}_\text{conv} is an argument with default value 1e-8. Following Carreira-perpinan (2000), the algorithm is started at each component location. Separately, it is necessary to identify identical modes which diverge only up to a small value; this tolerance value can be controlled with the argument tol_x.

MEM algorithm Following \insertCiteli_nonparametric_2007;textualBayesMultiMode, a mode x is found by iterating the two steps:

(i) \quad p(k|x^{(n)}) = \frac{\pi_k p_k(x^{(n)})}{p(x^{(n)})},

(ii) \quad x^{(n+1)} = \text{argmax}_x \sum_k p(k|x) \text{log} p_k(x^{(n)}),

until convergence, that is, until abs(x^{(n+1)}-x^{(n)})< \text{tol}_\text{conv}, where \text{tol}_\text{conv} is an argument with default value 1e-8. The algorithm is started at each component location. Separately, it is necessary to identify identical modes which diverge only up to a small value. Modes which are closer then tol_x are merged.

Discrete method By definition, modes must satisfy either:

p(y_{m}-1) < p(y_{m}) > p(y_{m}+1);

p(y_{m}-1) < p(y_{m}) = p(y_{m}+1) = \ldots = p(y_{m}+l-1) > p(y_{m}+l).

The algorithm evaluate each location point with these two conditions.

Value

A list of class mix_mode containing:

mode_estimates	estimates of the mixture modes.
algo	algorithm used for mode estimation.
dist	from mixture.
dist_type	type of mixture distribution, i.e. continuous or discrete.
pars	from mixture.
pdf_func	from mixture.
K	from mixture.
nb_var	from mixture.

References

\insertRef

Cross2024BayesMultiMode

\insertAllCited

Examples

# Example with a normal distribution ====================================
mu = c(0,5)
sigma = c(1,2)
p = c(0.5,0.5)

params = c(eta = p, mu = mu, sigma = sigma)
mix = mixture(params, dist = "normal", range = c(-5,15))
modes = mix_mode(mix)

# summary(modes)
# plot(modes)

# Example with a skew normal =============================================
xi = c(0,6)
omega = c(1,2)
alpha = c(0,0)
p = c(0.8,0.2)
params = c(eta = p, xi = xi, omega = omega, alpha = alpha)
dist = "skew_normal"

mix = mixture(params, dist = dist, range = c(-5,15))
modes = mix_mode(mix)
# summary(modes)
# plot(modes)

# Example with an arbitrary continuous distribution ======================
xi = c(0,6)
omega = c(1,2)
alpha = c(0,0)
nu = c(3,100)
p = c(0.8,0.2)
params = c(eta = p, mu = xi, sigma = omega, xi = alpha, nu = nu)

pdf_func <- function(x, pars) {
  sn::dst(x, pars["mu"], pars["sigma"], pars["xi"], pars["nu"])
}

mix = mixture(params, pdf_func = pdf_func,
dist_type = "continuous", loc = "mu", range = c(-5,15))
modes = mix_mode(mix)

# summary(modes)
# plot(modes, from = -4, to = 4)

# Example with a poisson distribution ====================================
lambda = c(0.1,10)
p = c(0.5,0.5)
params = c(eta = p, lambda = lambda)
dist = "poisson"


mix = mixture(params, range = c(0,50), dist = dist)

modes = mix_mode(mix)

# summary(modes)
# plot(modes)

# Example with an arbitrary discrete distribution =======================
mu = c(20,5)
size = c(20,0.5)
p = c(0.5,0.5)
params = c(eta = p, mu = mu, size = size)


pmf_func <- function(x, pars) {
  dnbinom(x, mu = pars["mu"], size = pars["size"])
}

mix = mixture(params, range = c(0, 50),
pdf_func = pmf_func, dist_type = "discrete")
modes = mix_mode(mix)

# summary(modes)
# plot(modes)

BayesMultiMode documentation built on May 29, 2024, 11:01 a.m.