Description Usage Arguments Details Value Author(s) References See Also Examples
Fit a mixed model with lasso, (fitted) group lasso, or (fitted)
sparsegroup lasso via proximal gradient descent. As this is an iterative
algorithm, the step size for each iteration is determined via backtracking
line search. The only exception to this is the fitted sparsegroup lasso. Its
step size needs determination prior to the call. A grid search for the
regularization parameter λ is performed using warm starts.
Depending on the input parameters alpha
and l2_fitted_values
this function subsequently calls one of the five implemented
lasso_variants
. Input variables will be standardized
if demanded by the user.
1 2 3 
y 
numeric vector of observations. 
X 
(optional) numeric design matrix relating y to fixed effects b. 
Z 
numeric design matrix relating y to random effects u. 
weights_u 
(optional) numeric vector of weights for the vector of
random effects u. If no weights are passed by the user, all weights for u are
set to 
groups 
(not required for the lasso) integer vector specifying which effect belongs to which group. If the lasso is called, this vector is without use and may remain empty. For group and sparsegroup lasso, at least each random effect variable needs to be assigned to one group. If in this case also fixed effect variables are present in the model, but without assigned group values, then the same value will be assigned to all fixed effect variables automatically. Float or double input values will be truncated. 
alpha 
mixing parameter for the sparsegroup lasso. Has to satisfy: 0 ≤ α ≤ 1. The penalty term looks as follows: α * '"lasso penalty" + (1α) * "group lasso penalty". If 
standardize 
if 
l2_fitted_values 
if 
step_size 
is a parameter, which is only used if

delta 
is a ridgetype parameter, which is only used if

rel_acc 
(optional) relative accuracy of the solution to stop the algorithm for the current value of λ. The algorithm stops after iteration m, if: sol^{(m)}  sol^{(m1)}_∞ < rel_acc * sol1{(m1)}_2. Default value is 
max_lambda 
(optional) maximum value for the penalty parameter. Default
option is an integrated algorithm to calculate this value. This is the start
value for the grid search of the penalty parameter λ. (More
details about the integrated algorithms are available here:

xi 
(optional) multiplicative parameter to determine the minimum value
of λ for the grid search, i.e. λ_{min} = ξ *
λ_{max}. Has to satisfy: 0 < ξ ≤ 1. If 
loops_lambda 
(optional) number of lambdas for the grid search between
λ_{max} and ξ * λ_{max}. Loops are performed on a
logarithmic grid. Float or double input values will be truncated. Default
value is 
max_iter 
(optional) maximum number of iterations for each value of the
penalty parameter λ. Determines the end of the calculation if the
algorithm didn't converge according to 
gamma_bls 
(optional) multiplicative parameter to decrease the step
size during backtracking line search. Has to satisfy: 0 < γ_{bls}
< 1. Default value is 
trace_progress 
(optional) if 
The underlying mixed model has the form:
y = X b + Z u + residual,
where b is the vector of fixed effects and u is the vector of random effects. The penalty of the sparsegroup lasso (without additional weights for features) is then:
α λ u_1 + (1  α) λ ∑_l ω^G_l u^{(l)}_2.
The variable ω^G_l is a weight for group l. If α = 1, this leads to the lasso. If α = 0, this leads to the group lasso.
The above penalty can be enhanced by introducing additional weights ω^F for features in the lasso penalty:
α λ ∑_j  ω^F_j u_j  + (1  α) λ ∑_l ω^G_l u^{(l)}_2.
The group weights ω^G_l are implemented as follows: ω^G_l = ∑_{j, j \in G} √{ω^F_{j}}. So, in the case that the weights for features in a particular group are all equal to one, this term collapses to the square root of the group size.
Furthermore, if instead of applying the l_2norm on u^{(l)} but on the fitted values Z^{(l)} u^{(l)} two more algorithms may be called: either the fitted group lasso or the fitted sparsegroup lasso.
In addition, the above penalty can be formally rewritten by including the fixed effects and assigning weights equal to zero to all these features. This is how the algorithms are implemented. Consequently, if a weight for any random effect is set to zero by the user, the function drops an error and the calculation will not start.
A list of estimates and parameters relevant for the computation:
estimate for the general intercept. Only present, if the
parameter standardize
was manually set to TRUE
.
estimates for the fixed effects b, if present in the model. Each row corresponds to a particular value of λ.
predictions for the random effects u. Each row corresponds to a particular value of λ.
all values for λ which were used during the grid search.
a sequence of actual iterations for each value of
λ. If an occurring number is equal to max_iter
, then the
algorithm most likely did not converge to rel_acc
during the
corresponding run of the grid search.
The following parameters are also returned. But primarily for the purpose of
comparison and repetition: alpha
(only for the sparsegroup lasso),
max_iter
, gamma_bls
, xi
, and loops_lambda
.
Jan Klosa
Simon, N., Friedman, J., Hastie T., and Tibshirani, R. (2011): A SparseGroup Lasso, http://faculty.washington.edu/nrsimon/SGLpaper.pdf, Journal of Computational and Graphical Statistics, Vol. 22(2), 231245, April 2013
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 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62  set.seed(62)
n < 50 ## observations
p < 8 ## variables
## Create a design matrix X for fixed effects to model the intercept:
X < matrix(1, nrow = n, ncol = 1)
## Create a design matrix Z for random effects:
Z < matrix(rnorm(p * n, mean = 0, sd = 1), nrow = n)
## Intercept b, random effect vector u, and response y:
b < 0.2
u < c(0, 1.5, 0, 0.5, 0, 0, 2, 1)
y < X%*%b + Z%*%u + rnorm(n, mean = 0, sd = 1)
## Create a vector of three groups corresponding to vector u:
group_indices < c(1L, 2L, 2L, 2L, 1L, 1L, 3L, 1L)
## Calculate the solution:
fit_l < seagull(y = y, X = X, Z = Z, alpha = 1.0)
fit_gl < seagull(y = y, X = X, Z = Z, alpha = 0.0, groups = group_indices)
fit_sgl < seagull(y = y, X = X, Z = Z, groups = group_indices)
## Combine the estimates for fixed and random effects:
estimates_l < cbind(fit_l$fixed_effects, fit_l$random_effects)
estimates_gl < cbind(fit_gl$fixed_effects, fit_gl$random_effects)
estimates_sgl < cbind(fit_sgl$fixed_effects, fit_sgl$random_effects)
true_effects < c(b, u)
## Calculate mean squared errors along the solution paths:
MSE_l < rep(as.numeric(NA), fit_l$loops_lambda)
MSE_gl < rep(as.numeric(NA), fit_l$loops_lambda)
MSE_sgl < rep(as.numeric(NA), fit_l$loops_lambda)
for (i in 1:fit_l$loops_lambda) {
MSE_l[i] < t(estimates_l[i,]  true_effects)%*%(estimates_l[i,]  true_effects)/(p+1)
MSE_gl[i] < t(estimates_gl[i,]  true_effects)%*%(estimates_gl[i,]  true_effects)/(p+1)
MSE_sgl[i] < t(estimates_sgl[i,]  true_effects)%*%(estimates_sgl[i,]  true_effects)/(p+1)
}
## Plot a fraction of the results of the MSEs:
plot(x = seq(1, fit_l$loops_lambda, 1)[25:50], MSE_l[25:50], type = "l", lwd = 2)
points(x = seq(1, fit_l$loops_lambda, 1)[25:50], MSE_gl[25:50], type = "l", lwd = 2, col = "blue")
points(x = seq(1, fit_l$loops_lambda, 1)[25:50], MSE_sgl[25:50], type = "l", lwd = 2, col = "red")
## A larger example with simulated genetic data:
data(seagull_data)
fit_l1 < seagull(y = phenotypes[,1], Z = genotypes, alpha = 1.0)
fit_l2 < seagull(y = phenotypes[,2], Z = genotypes, alpha = 1.0)
fit_l3 < seagull(y = phenotypes[,3], Z = genotypes, alpha = 1.0)
fit_gl1 < seagull(y = phenotypes[,1], Z = genotypes, alpha = 0.0,
groups = groups)
fit_gl2 < seagull(y = phenotypes[,2], Z = genotypes, alpha = 0.0,
groups = groups)
fit_gl3 < seagull(y = phenotypes[,3], Z = genotypes, alpha = 0.0,
groups = groups)
fit_sgl1 < seagull(y = phenotypes[,1], Z = genotypes, groups = groups)
fit_sgl2 < seagull(y = phenotypes[,2], Z = genotypes, groups = groups)
fit_sgl3 < seagull(y = phenotypes[,3], Z = genotypes, groups = groups)

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