# Finite Mixture Effects Model with Heterogeneity Pursuit 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. ```r 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. ## Returns 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 ## 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. ## Examples ```r 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)) ``` ## References Li, Y., Yu, C., Zhao, Y., Yao, W., Aseltine, R. H., & Chen, K. (2021). Pursuing Sources of Heterogeneity in Modeling Clustered Population.

path.fmrHP function