bess.one: Best subset selection/Best subset ridge regression with a specified model size and a shrinkage parameter

Description

Best subset selection with a specified model size for generalized linear models and Cox's proportional hazard model.

Usage

bess.one(
  x,
  y,
  family = c("gaussian", "binomial", "poisson", "cox"),
  type = c("bss", "bsrr"),
  s,
  lambda = 0,
  always.include = NULL,
  screening.num = NULL,
  normalize = NULL,
  weight = NULL,
  max.iter = 20,
  group.index = NULL
)

Arguments

Input matrix, of dimension $n \times p$ ; each row is an observation vector and each column is a predictor/feature/variable.

The response variable, of n observations. For family = "binomial" should be a factor with two levels. For family="poisson", y should be a vector with positive integer. For family = "cox", y should be a two-column matrix with columns named time and status.

family

One of the following models: "gaussian", "binomial", "poisson", or "cox". Depending on the response.

type

One of the two types of problems. type = "bss" for the best subset selection, and type = "bsrr" for the best subset ridge regression.

A specified model size

lambda

A shrinkage parameter for "bsrr".

always.include

An integer vector containing the indexes of variables that should always be included in the model.

screening.num

Users can pre-exclude some irrelevant variables according to maximum marginal likelihood estimators before fitting a model by passing an integer to screening.num and the sure independence screening will choose a set of variables of this size. Then the active set updates are restricted on this subset.

normalize

Options for normalization. normalize = 0 for no normalization. Setting normalize = 1 will only subtract the mean of columns of x. normalize = 2 for scaling the columns of x to have $\sqrt{n}$ norm. normalize = 3 for subtracting the means of the columns of x and y, and also normalizing the columns of x to have $\sqrt{n}$ norm. If normalize = NULL, by default, normalize will be set 1 for "gaussian", 2 for "binomial" and "poisson", 3 for "cox".

weight

Observation weights. Default is 1 for each observation.

max.iter

The maximum number of iterations in the bess function. In most of the case, only a few steps can guarantee the convergence. Default is 20.

group.index

A vector of integers indicating the which group each variable is in. For variables in the same group, they should be located in adjacent columns of x and their corresponding index in group.index should be the same. Denote the first group as 1, the second 2, etc. If you do not fit a model with a group structure, please set group.index = NULL. Default is NULL.

Value

A list with class attribute 'bess' and named components:

beta

The best fitting coefficients.

coef0

The best fitting intercept.

bestmodel

The best fitted model for type = "bss", the class of which is "lm", "glm" or "coxph".

loss

The training loss of the fitting model.

The model size.

lambda

The shrinkage parameter.

family

Type of the model.

nsample

The sample size.

type

Either "bss" or "bsrr".

Details

Given a model size $s$ , we consider the following best subset selection problem: $min_{β} - 2 \log L (β); s . t . ‖ β ‖_{0} = s .$ And given a model size $s$ and a shrinkage parameter $λ$ , consider the following best subset ridge regression problem: $min_{β} - 2 \log L (β) + λ ‖ β ‖_{2}^{2}; s . t . ‖ β ‖_{0} = s .$

In the GLM case, $\log L (β)$ is the log likelihood function; In the Cox model, $\log L (β)$ is the log partial likelihood function.

The best subset selection problem is solved by the primal dual active set algorithm, see Wen et al. (2017) for details. This algorithm utilizes an active set updating strategy via primal and dual variables and fits the sub-model by exploiting the fact that their support set are non-overlap and complementary.

References

Wen, C., Zhang, A., Quan, S. and Wang, X. (2020). BeSS: An R Package for Best Subset Selection in Linear, Logistic and Cox Proportional Hazards Models, Journal of Statistical Software, Vol. 94(4). doi:10.18637/jss.v094.i04.

Examples

Run this code

# NOT RUN {
#-------------------linear model----------------------#
# Generate simulated data
n <- 200
p <- 20
k <- 5
rho <- 0.4
seed <- 10
Tbeta <- rep(0, p)
Tbeta[1:k*floor(p/k):floor(p/k)] <- rep(1, k)
Data <- gen.data(n, p, k, rho, family = "gaussian", beta = Tbeta, seed = seed)
x <- Data$x[1:140, ]
y <- Data$y[1:140]
x_new <- Data$x[141:200, ]
y_new <- Data$y[141:200]
lm.bss <- bess.one(x, y, s = 5)
lm.bsrr <- bess.one(x, y, type = "bsrr", s = 5, lambda = 0.01)
coef(lm.bss)
coef(lm.bsrr)
print(lm.bss)
print(lm.bsrr)
summary(lm.bss)
summary(lm.bsrr)
pred.bss <- predict(lm.bss, newx = x_new)
pred.bsrr <- predict(lm.bsrr, newx = x_new)

#-------------------logistic model----------------------#
#Generate simulated data
Data <- gen.data(n, p, k, rho, family = "binomial", beta = Tbeta, seed = seed)

x <- Data$x[1:140, ]
y <- Data$y[1:140]
x_new <- Data$x[141:200, ]
y_new <- Data$y[141:200]
logi.bss <- bess.one(x, y, family = "binomial", s = 5)
logi.bsrr <- bess.one(x, y, type = "bsrr", family = "binomial", s = 5, lambda = 0.01)
coef(logi.bss)
coef(logi.bsrr)
print(logi.bss)
print(logi.bsrr)
summary(logi.bss)
summary(logi.bsrr)
pred.bss <- predict(logi.bss, newx = x_new)
pred.bsrr <- predict(logi.bsrr, newx = x_new)

#-------------------poisson model----------------------#
Data <- gen.data(n, p, k, rho=0.3, family = "poisson", beta = Tbeta, seed = seed)

x <- Data$x[1:140, ]
y <- Data$y[1:140]
x_new <- Data$x[141:200, ]
y_new <- Data$y[141:200]
poi.bss <- bess.one(x, y, family = "poisson", s=5)
lambda.list <- exp(seq(log(5), log(0.1), length.out = 10))
poi.bsrr <- bess.one(x, y, type = "bsrr", family = "poisson", s = 5, lambda = 0.01)
coef(poi.bss)
coef(poi.bsrr)
print(poi.bss)
print(poi.bsrr)
summary(poi.bss)
summary(poi.bsrr)
pred.bss <- predict(poi.bss, newx = x_new)
pred.bsrr <- predict(poi.bsrr, newx = x_new)

#-------------------coxph model----------------------#
#Generate simulated data
Data <- gen.data(n, p, k, rho, family = "cox", beta = Tbeta, scal = 10)

x <- Data$x[1:140, ]
y <- Data$y[1:140, ]
x_new <- Data$x[141:200, ]
y_new <- Data$y[141:200, ]
cox.bss <- bess.one(x, y, family = "cox", s = 5)
cox.bsrr <- bess.one(x, y, type = "bsrr", family = "cox", s = 5, lambda = 0.01)
coef(cox.bss)
coef(cox.bsrr)
print(cox.bss)
print(cox.bsrr)
summary(cox.bss)
summary(cox.bsrr)
pred.bss <- predict(cox.bss, newx = x_new)
pred.bsrr <- predict(cox.bsrr, newx = x_new)
#----------------------High dimensional linear models--------------------#
# }
# NOT RUN {
data <- gen.data(n, p = 1000, k, family = "gaussian", seed = seed)

# Best subset selection with SIS screening
fit <- bess.one(data$x, data$y, screening.num = 100, s = 5)
# }
# NOT RUN {
#-------------------group selection----------------------#
beta <- rep(c(rep(1,2),rep(0,3)), 4)
Data <- gen.data(200, 20, 5, rho=0.4, beta = beta, seed =10)
x <- Data$x
y <- Data$y

group.index <- c(rep(1, 2), rep(2, 3), rep(3, 2), rep(4, 3),
                rep(5, 2), rep(6, 3), rep(7, 2), rep(8, 3))
lm.group <- bess.one(x, y, s = 5, type = "bss", group.index = group.index)
lm.groupbsrr <- bess.one(x, y, type = "bsrr", s = 5, lambda = 0.01, group.index = group.index)
coef(lm.group)
coef(lm.groupbsrr)
print(lm.group)
print(lm.groupbsrr)
summary(lm.group)
summary(lm.groupbsrr)
pred.group <- predict(lm.group, newx = x_new)
pred.groupl0l2 <- predict(lm.groupbsrr, newx = x_new)
#-------------------include specified variables----------------------#
Data <- gen.data(n, p, k, rho, family = "gaussian", beta = Tbeta, seed = seed)
lm.bss <- bess.one(Data$x, Data$y, s = 5, always.include = 2)

# }
# NOT RUN {
#-------------------code demonstration in doi: 10.18637/jss.v094.i04----------------------#
Tbeta <- rep(0, 20)
Tbeta[c(1, 2, 5, 9)] <- c(3, 1.5, -2, -1)

data <- gen.data(n = 200, p = 20, family = "gaussian", beta = Tbeta,
rho = 0.2, seed = 123)
fit.one <- bess.one(data$x, data$y, s = 4, family = "gaussian")
print(fit.one)
summary(fit.one)
coef(fit.one, sparse = FALSE)
pred.one <- predict(fit.one, newdata = data$x)
bm.one <- fit.one$bestmodel
summary(bm.one)
# }

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