Package 'fmerPack'

Finite Mixture Effects Model with Heterogeneity Pursuit

Description

Produce solution for specified lambda of regularized finite mixture effects model with lasso or adaptive lasso; compute the degrees of freeom, likelihood and information criteria (AIC, BIC and GIC) of the estimators. Model fitting is conducted by EM algorithm and Bregman coordinate descent.

Usage

fmrHP(y, X, m, intercept = FALSE, lambda, equal.var = FALSE,
      ic.type = c("ALL", "BIC", "AIC", "GIC"),
      B = NULL, prob = NULL, rho = NULL, w = NULL,
      control = list(), report = FALSE)

Arguments

y	a vector of response ($n \times 1$)
X	a matrix of covariate ($n \times p$)
m	number of components
intercept	indicating whether intercept should be included
lambda	value of tuning parameter
equal.var	indicating whether variances of different components are equal
ic.type	the information criterion to be used; currently supporting "AIC", "BIC", and "GIC".
B	initial values for the rescaled coefficients with first column being the common effect, and the rest m columns being the heterogeneity for corresponding components
prob	initial values for prior probabilitis for different components
rho	initial values for rho vector ($1 / \sigma$), the reciprocal of standard deviation
w	weight matrix for penalty function. Default option is NULL
control	a list of parameters for controlling the fitting process
report	indicating whether printing the value of objective function during EM algorithm for validation checking of initial value.

Details

The available elements for argument control include

• epsilon: Convergence threshold for generalized EM algorithm. Defaults value is 1E-6.

• maxit: Maximum number of passes over the data for all lambda values. Default is 1000.

• inner.eps: Convergence threshold for Bregman coordinate descent algorithm. Defaults value is 1E-6.

• inner.maxit: Maximum number of iteration for Bregman coordinate descent algorithm. Defaults value is 200.

• n.ini: Number of initial values for EM algorithm. Default is 10. In EM algorithm, it is preferable to start from several different initial values.

Value

A list consisting of

y	vector of response
X	matrix of covariates
m	number of components
B.hat	estimated rescaled coefficient ($p \times m + 1 \times nlambda$)
pi.hat	estimated prior probabilities ($m \times nlambda$)
rho.hat	estimated rho values ($m \times nlambda$)
lambda	lambda used in model fitting
plik	value of penalized log-likelihood
loglik	value of log-likelihood
conv	indicator of convergence of EM algorithm
IC	values of information criteria
df	degree of freedom

Examples

library(fmerPack)
## problem settings
n <- 100; m <- 3; p <- 5;
sigma2 <- c(0.1, 0.1, 0.4); rho <- 1 / sqrt(sigma2)
phi <- rbind(c(1, 1, 1), c(1, 1, 1), c(0, -3, 3), c(-3, 3, 0), c(3, 0, -3))
beta <- t(t(phi) / rho)
## generate response and covariates
z <- rmultinom(n, 1, prob= rep(1 / m, m))
X <- matrix(rnorm(n * p), nrow = n, ncol = p)
y <- MASS::mvrnorm(1, mu = rowSums(t(z) * X[, 1:(nrow(beta))] %*% beta), 
                   Sigma = diag(colSums(z * sigma2)))
fmrHP(y, X, m = m, lambda = 0.01, control = list(n.ini = 10))

---

Finite Mixture Model with lasso and adaptive penalty

Description

Produce solution for specific lambda of regularized finite mixture model with lasso or adaptive lasso penalty; compute the degrees of freeom, likelihood and information criteria (AIC, BIC and GIC) of the estimators. Model fitting is conducted by EM algorithm and coordinate descent.

Usage

fmrReg(y, X, m, intercept = FALSE, lambda, equal.var = FALSE, common.var = NULL,
       ic.type = c("ALL", "BIC", "AIC", "GIC"), 
       B = NULL, prob = NULL, rho = NULL, w = NULL, 
       control = list(), report = FALSE)

Arguments

y	a vector of response ($n \times 1$)
X	a matrix of covariate ($n \times p$)
m	number of components
intercept	indicating whether intercept should be included
lambda	value of tuning parameter
equal.var	indicating whether variances of different components are equal
common.var	indicating whether the effects over different components are the same for specific covariates
ic.type	the information criterion to be used; currently supporting "AIC", "BIC", and "GIC".
B	initial values for the rescaled coefficients with columns being the coefficients for different components
prob	initial values for prior probabilitis for different components
rho	initial values for rho vector ($1 / \sigma$), the reciprocal of standard deviation
w	weight matrix for penalty function. Default option is NULL
control	a list of parameters for controlling the fitting process
report	indicating whether printing the value of objective function during EM algorithm for validation checking of initial value.

Details

The available elements for argument control include

• epsilon: Convergence threshold for generalized EM algorithm. Defaults value is 1E-6.

• maxit: Maximum number of passes over the data for all lambda values. Default is 1000.

• inner.maxit: Maximum number of iteration for flexmix package to compute initial values. Defaults value is 200.

• n.ini: Number of initial values for EM algorithm. Default is 10. In EM algorithm, it is preferable to start from several different initial values.

Value

A list consisting of

y	vector of response
X	matrix of covariates
m	number of components
B.hat	estimated rescaled coefficient ($p \times m \times nlambda$)
pi.hat	estimated prior probabilities ($m \times nlambda$)
rho.hat	estimated rho values ($m \times nlambda$)
lambda	lambda used in model fitting
plik	value of penalized log-likelihood
loglik	value of log-likelihood
conv	indicator of convergence of EM algorithm
IC	values of information criteria
df	degree of freedom

Examples

library(fmerPack)
## problem settings
n <- 100; m <- 3; p <- 5;
sigma2 <- c(0.1, 0.1, 0.4); rho <- 1 / sqrt(sigma2)
phi <- rbind(c(1, 1, 1), c(1, 1, 1), c(0, -3, 3), c(-3, 3, 0), c(3, 0, -3))
beta <- t(t(phi) / rho)
## generate response and covariates
z <- rmultinom(n, 1, prob= rep(1 / m, m))
X <- matrix(rnorm(n * p), nrow = n, ncol = p)
y <- MASS::mvrnorm(1, mu = rowSums(t(z) * X[, 1:(nrow(beta))] %*% beta), 
                   Sigma = diag(colSums(z * sigma2)))
fmrReg(y, X, m = m, lambda = 0.01, control = list(n.ini = 10))

---

Finite Mixture Effects Model with Heterogeneity Pursuit

Description

Produce solution paths of regularized finite mixture effects model with lasso or adaptive lasso penalty; compute the degrees of freeom, likelihood and information criteria (AIC, BIC and GIC) of the estimators. Model fitting is conducted by EM algorithm and Bregman coordinate descent.

Usage

path.fmrHP(y, X, m, equal.var = FALSE, 
           ic.type = "ALL", B = NULL, prob = NULL, rho = NULL, 
           control = list(), modstr = list(), report = FALSE)

Arguments

y	a vector of response ($n \times 1$)
X	a matrix of covariate ($n \times p$)
m	number of components
equal.var	indicating whether variances of different components are equal
ic.type	the information criterion to be used; currently supporting "AIC", "BIC", and "GIC".
B	initial values for the rescaled coefficients with first column being the common effect, and the rest m columns being the heterogeneity for corresponding components
prob	initial values for prior probabilitis for different components
rho	initial values for rho vector ($1 / \sigma$), the reciprocal of standard deviation
control	a list of parameters for controlling the fitting process
modstr	a list of model parameters controlling the model fitting
report	indicating whether printing the value of objective function during EM algorithm for validation checking of initial value.

Details

Model parameters can be specified through argument modstr. The available include

• lambda: A vector of user-specified lambda values with default NULL.

• lambda.min.ratio: Smallest value for lambda, as a fraction of lambda.max, the (data derived) entry value.

• nlambda: The number of lambda values.

• w: Weight matrix for penalty function. Default option is NULL, which means lasso penailty is used for model fitting.

• intercept: Should intercept(s) be fitted (default=TRUE) or set to zero (FALSE).

• common.only: A vector of user-specified indicators of the variables only with common effects.

• common.no.penalty: A vector of user-specified indicators of the variables with no penalty on the common effect.

• cluster.no.penalty: A vector of user-specified indicators of the variables with no penalty on the cluster-specific effects.

• select.ratio: A user-specified ratio indicating the ratio of variables to be selected.

The available elements for argument control include

• epsilon: Convergence threshold for generalized EM algorithm. Defaults value is 1E-6.

• maxit: Maximum number of passes over the data for all lambda values. Default is 1000.

• inner.eps: Convergence threshold for Bregman coordinate descent algorithm. Defaults value is 1E-6.

• inner.maxit: Maximum number of iteration for Bregman coordinate descent algorithm. Defaults value is 200.

• n.ini: Number of initial values for EM algorithm. Default is 10. In EM algorithm, it is preferable to start from several different initial values.

Value

A list consisting of

lambda	vector of lambda used in model fitting
lambda.used	vector of lambda in model fitting after truncation by select.ratio
B.hat	estimated rescaled coefficient ($p \times m + 1 \times nlambda$)
pi.hat	estimated prior probabilities ($m \times nlambda$)
rho.hat	estimated rho values ($m \times nlambda$)
IC	values of information criteria

References

Li, Y., Yu, C., Zhao, Y., Yao, W., Aseltine, R. H., & Chen, K. (2021). Pursuing Sources of Heterogeneity in Modeling Clustered Population.

Examples

library(fmerPack)
## problem settings
n <- 100; m <- 3; p <- 5;
sigma2 <- c(0.1, 0.1, 0.4); rho <- 1 / sqrt(sigma2)
phi <- rbind(c(1, 1, 1), c(1, 1, 1), c(1, 1, 1), c(-3, 3, 0), c(3, 0, -3))
beta <- t(t(phi) / rho)
## generate response and covariates
z <- rmultinom(n, 1, prob= rep(1 / m, m))
X <- matrix(rnorm(n * p), nrow = n, ncol = p)
y <- MASS::mvrnorm(1, mu = rowSums(t(z) * X[, 1:(nrow(beta))] %*% beta), 
                   Sigma = diag(colSums(z * sigma2)))
## lasso
fit1 <- path.fmrHP(y, X, m = m, modstr = list(nlambda = 10), control = list(n.ini = 1))
## adaptive lasso
fit2 <- path.fmrHP(y, X, m = m, 
                   modstr = list(w = abs(select.tuning(fit1)$B + 1e-6)^2))

---

Finite Mixture Model with lasso and adaptive penalty

Description

Produce solution paths of regularized finite mixture model with lasso or adaptive lasso penalty; compute the degrees of freeom, likelihood and information criteria (AIC, BIC and GIC) of the estimators. Model fitting is conducted by EM algorithm and coordinate descent.

Usage

path.fmrReg(y, X, m, equal.var = FALSE,
            ic.type = "ALL", B = NULL, prob = NULL, rho = NULL, 
            control = list(), modstr = list(), report = FALSE)

Arguments

y	a vector of response ($n \times 1$)
X	a matrix of covariate ($n \times p$)
m	number of components
equal.var	indicating whether variances of different components are equal
ic.type	the information criterion to be used; currently supporting "ALL", "AIC", "BIC", and "GIC".
B	initial values for the rescaled coefficients with columns being the columns being the coefficient for different components
prob	initial values for prior probabilitis for different components
rho	initial values for rho vector ($1 / \sigma$), the reciprocal of standard deviation
control	a list of parameters for controlling the fitting process
modstr	a list of model parameters controlling the model fitting
report	indicating whether printing the value of objective function during EM algorithm for validation checking of initial value.

Details

Model parameters can be specified through argument modstr. The available include

• lambda: A vector of user-specified lambda values with default NULL.

• lambda.min.ratio: Smallest value for lambda, as a fraction of lambda.max, the (data derived) entry value.

• nlambda: The number of lambda values.

• w: Weight matrix for penalty function. Default option is NULL, which means lasso penailty is used for model fitting.

• intercept: Should intercept(s) be fitted (default=TRUE) or set to zero (FALSE).

• no.penalty: A vector of user-specified indicators of the variables with no penalty.

• common.var: A vector of user-specified indicators of the variables with common effect among different components.

• select.ratio: A user-specified ratio indicating the ratio of variables to be selected.

The available elements for argument control include

• epsilon: Convergence threshold for generalized EM algorithm. Defaults value is 1E-6.

• maxit: Maximum number of passes over the data for all lambda values. Default is 1000.

• inner.maxit: Maximum number of iteration for flexmix package to compute initial values. Defaults value is 200.

• n.ini: Number of initial values for EM algorithm. Default is 10. In EM algorithm, it is preferable to start from several different initial values.

Value

A list consisting of

lambda	vector of lambda used in model fitting
B.hat	estimated rescaled coefficient ($p \times m \times nlambda$)
pi.hat	estimated prior probabilities ($m \times nlambda$)
rho.hat	estimated rho values ($m \times nlambda$)
IC	values of information criteria

Examples

library(fmerPack)
## problem settings
n <- 100; m <- 3; p <- 5;
sigma2 <- c(0.1, 0.1, 0.4); rho <- 1 / sqrt(sigma2)
phi <- rbind(c(1, 1, 1), c(1, 1, 1), c(1, 1, 1), c(-3, 3, 0), c(3, 0, -3))
beta <- t(t(phi) / rho)
## generate response and covariates
z <- rmultinom(n, 1, prob= rep(1 / m, m))
X <- matrix(rnorm(n * p), nrow = n, ncol = p)
y <- MASS::mvrnorm(1, mu = rowSums(t(z) * X[, 1:(nrow(beta))] %*% beta), 
                   Sigma = diag(colSums(z * sigma2)))
## lasso
fit1 <- path.fmrReg(y, X, m = m, modstr = list(nlambda = 10), control = list(n.ini = 1))
## adaptive lasso
fit2 <- path.fmrReg(y, X, m = m, 
                   modstr = list(w = abs(select.tuning(fit1)$B + 1e-6)^2))

---

Tuning paramater selection

Description

Select tuning parameter via AIC, BIC or GIC from objects generated by path.fmrHP.

Usage

select.tuning(object, figure = FALSE, criteria = c("BIC", "GIC", "AIC"))

Arguments

object	Object generated from path.fmrHP.
figure	incidator for showing plot of information criteria.
criteria	information criteria for selection of tuning parameter.

Value

list of parameters of selected model.

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