ObjFunL2: The objective function using L2 loss with three penalty...

In ipapercodes/FGSPCA: Feature Grouping and Sparse Principal Component Analysis (FGSPCA)

Description

The objective function using L2-loss with three penalty functions. An extension of the elastic net regression.

Usage

ObjFunL2(
  x,
  y,
  tau_S1,
  tau_S2,
  lambda1,
  lambda2,
  lambda3,
  beta,
  Bjj,
  SF,
  SFc,
  SE,
  SEc,
  SN
)

Arguments

x	the data matrix X_{n\times p}, where n is the number of observations, p is the number of features.
y	the response vector Y_{n\times 1} with length n
tau_S1	τ_1, the controlling parameter corresponding to p_1(\cdot), which determines when the small values of |β_j| will be penalized.
tau_S2	τ_2, the controlling parameter corresponding to p_2(\cdot), which determines when the small difference values of |β_j - β_{j'}| will be penalized.
lambda1	λ_1, the tuning parameter corresponding to p_1(\cdot)
lambda2	λ_2, the tuning parameter corresponding to p_2(\cdot)
lambda3	λ_3, the tuning parameter corresponding to p_3(\cdot)
beta	β, the estimation of β
Bjj	a matrix of p\times p with element β_{jj'} = β_{j}-β_{j'} .
SF	the \mathcal{F} set, p-length vector of indicator 0-1. Its value is 1 if |β_j| ≤q τ_1 . Otherwise 0.
SFc	the \mathcal{F}^{c} set, p-length vector of indicator 0-1. Its value is 1 if |β_j| > τ_1 . Otherwise 0.
SE	the \mathcal{E} set, a matrix of p\times p with indicator 0-1. Its value is 1 if |β_{j}-β_{j'}| ≤q τ_2. Otherwise 0. Note if τ_2=0, j=j', then 0<=0 is true, the diagonal of \mathcal{E} is 1. We should set SEc <- (1-SE), SE <- SE - diag(p) after the calculation of \mathcal{E}.
SEc	the \mathcal{E}^{c} set, a matrix of p\times p with indicator 0-1.
SN	the \mathcal{N} set, p-length vector of indicator 0-1. Its value is 1 when β_j < 0, corresponding to \min(β_j, 0).

Details

The objective function using L2 loss is defined as follows,

\frac{1}{2n} \|Y- Xβ\|^2 + λ_{1}p_1(β) + λ_{2}p_2(β) + λ_{3}p_3(β) .

The three penalties are as follows,

p_1(β) = ∑_{j=1}^p \min\{\frac{|β_j|}{τ_1}, 1\} ,

p_2(β) = ∑_{j < j', (j, j') \in E} \min\{\frac{|β_j - β_{j'}|}{τ_2}, 1\},

p_3(β) = ∑_{j=1}^p (\min\{β_j, 0\})^2 .

Value

the value of the objective function

ipapercodes/FGSPCA documentation built on Dec. 20, 2021, 7:58 p.m.