Description Usage Arguments Value References Examples
This function implements group-regularized gamma regression with a known shape parameter ν and the log link. In gamma regression, we assume that y_i \sim Gamma(μ_i, ν), where
f(y_i | μ_i, ν ) = \frac{1}{Γ(ν)} (\frac{ν}{μ_i})^{ν} \exp(-\frac{ν}{μ_i}y_i) y_i^{ν-1}, y > 0.
Then E(y_i) = μ_i, and we relate μ_i to a set of p covariates x_i through the log link,
\log(μ_i) = β_0 + x_i^T β, i=1,..., n
If the covariates in each x_i are grouped according to known groups g=1, ..., G, then this function may estimate some of the G groups of coefficients as all zero, depending on the amount of regularization.
Our implementation for regularized gamma regression is based on the least squares approximation approach of Wang and Leng (2007), and hence, the function does not allow the total number of covariates p to be greater than sample size.
1 2 3 | grpreg.gamma(y, X, X.test, groups, gamma.shape=1,
penalty=c("gLASSO","gSCAD","gMCP"),
weights, taper, nlambda=100, lambda, max.iter=10000, tol=1e-4)
|
y |
n \times 1 vector of responses for training data. |
X |
n \times p design matrix for training data, where the jth column of |
X.test |
n_{test} \times p design matrix for test data to calculate predictions. |
groups |
p-dimensional vector of group labels. The jth entry in |
gamma.shape |
known shape parameter ν in Gamma(μ_i,ν) distribution for the responses. Default is |
penalty |
group regularization method to use on the groups of coefficients. The options are |
weights |
group-specific, nonnegative weights for the penalty. Default is to use the square roots of the group sizes. |
taper |
tapering term γ in group SCAD and group MCP controlling how rapidly the penalty tapers off. Default is |
nlambda |
number of regularization parameters L. Default is |
lambda |
grid of L regularization parameters. The user may specify either a scalar or a vector. If the user does not provide this, the program chooses the grid automatically. |
max.iter |
maximum number of iterations in the algorithm. Default is |
tol |
convergence threshold for algorithm. Default is |
The function returns a list containing the following components:
lambda |
L \times 1 vector of regularization parameters |
beta0 |
L \times 1 vector of estimated intercepts. The kth entry in |
beta |
p \times L matrix of estimated regression coefficients. The kth column in |
mu.pred |
n_{test} \times L matrix of predicted mean response values μ_{test} = E(Y_{test}) based on the test data in |
classifications |
G \times L matrix of classifications, where G is the number of groups. An entry of "1" indicates that the group was classified as nonzero, and an entry of "0" indicates that the group was classified as zero. The kth column of |
loss |
L \times 1 vector of negative log-likelihood of the fitted models. The kth entry in |
Breheny, P. and Huang, J. (2015). "Group descent algorithms for nonconvex penalized linear and logistic regression models with grouped predictors." Statistics and Computing, 25:173-187.
Wang, H. and Leng, C. (2007). "Unified LASSO estimation by least squares approximation." Journal of the American Statistical Association, 102:1039-1048.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 | ## Generate data
set.seed(12345)
X = matrix(runif(100*11), nrow=100)
n = dim(X)[1]
groups = c("a","a","a","b","b","b","c","c","d","e","e")
groups = as.factor(groups)
true.beta = c(-1,1,1,0,0,0,0,0,0,1.5,-1.5)
## Generate responses from gamma regression with known shape parameter 1
eta = crossprod(t(X), true.beta)
shape = 1
y = rgamma(n, rate=shape/exp(eta), shape=shape)
## Generate test data
n.test = 50
X.test = matrix(runif(n.test*11), nrow=n.test)
## Fit gamma regression models with the group LASSO penalty
gamma.mod = grpreg.gamma(y, X, X.test, groups, penalty="gLASSO")
## Tuning parameters used to fit models
gamma.mod$lambda
# Predicted n.test-dimensional vectors mu=E(Y.test) based on test data, X.test.
# The kth column of 'mu.pred' corresponds to the kth entry in 'lambda.'
gamma.mod$mu.pred
# Classifications of the 5 groups. The kth column of 'classifications'
# corresponds to the kth entry in 'lambda.'
gamma.mod$classifications
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