Package 'subsampling'

Optimal Subsampling Methods for Generalized Linear Models

Description

Draw subsample from full dataset and fit a generalized linear model (GLM) on the subsample. For a quick start, refer to the vignette.

Usage

ssp.glm(
  formula,
  data,
  subset = NULL,
  n.plt,
  n.ssp,
  family = "binomial",
  criterion = "optL",
  sampling.method = "poisson",
  likelihood = "weighted",
  control = list(...),
  contrasts = NULL,
  ...
)

Arguments

formula	A model formula object of class "formula" that describes the model to be fitted.
data	A data frame containing the variables in the model. Denote $N$ as the number of observations in data.
subset	An optional vector specifying a subset of observations from data to use for the analysis. This subset will be viewed as the full data.
n.plt	The pilot subsample size (first-step subsample size). This subsample is used to compute the pilot estimator and estimate the optimal subsampling probabilities.
n.ssp	The expected size of the optimal subsample (second-step subsample). For sampling.method = 'withReplacement', The exact subsample size is n.ssp. For sampling.method = 'poisson', n.ssp is the expected subsample size.
family	family can be a character string naming a family function, a family function or the result of a call to a family function.
criterion	The choices include optA, optL(default), LCC and uniform. optA Minimizes the trace of the asymptotic covariance matrix of the subsample estimator. optL Minimizes the trace of a transformation of the asymptotic covariance matrix. The computational complexity of optA is $O(N d^2)$ while that of optL is $O(N d)$. LCC Local Case-Control sampling probability, used as a baseline subsampling strategy. uniform Assigns equal subsampling probability $\frac{1}{N}$ to each observation, serving as a baseline subsampling strategy.
sampling.method	The sampling method to use. Options include withReplacement and poisson (default). withReplacement draws exactly n.ssp subsamples from size $N$ full dataset with replacement, using the specified subsampling probabilities. poisson draws observations independently by comparing each subsampling probability with a realization of uniform random variable $U(0,1)$. Differences between methods: Sample size: withReplacement draws exactly n.ssp subsamples while poisson draws subsamples with expected size n.ssp, meaning the actual size may vary. Memory usage: withReplacement requires the entire dataset to be loaded at once, while poisson allows for processing observations sequentially (will be implemented in future version). Estimator performance: Theoretical results show that the poisson tends to get a subsample estimator with lower asymptotic variance compared to the withReplacement
likelihood	The likelihood function to use. Options include weighted (default) and logOddsCorrection. A bias-correction likelihood function is required for subsample since unequal subsampling probabilities introduce bias. weighted Applies a weighted likelihood function where each observation is weighted by the inverse of its subsampling probability. logOddsCorrection This likelihood is available only for logistic regression model (i.e., when family is binomial or quasibinomial). It uses a conditional likelihood, where each element of the likelihood represents the probability of $Y=1$, given that this subsample was drawn.
control	The argument control contains two tuning parameters alpha and b. alpha $\in [0,1]$ is the mixture weight of the user-assigned subsampling probability and uniform subsampling probability. The actual subsample probability is $\pi = (1-\alpha)\pi^{opt} + \alpha \pi^{uni}$. This protects the estimator from extreme small subsampling probability. The default value is 0.01. b is a positive number which is used to constraint the poisson subsampling probability. b close to 0 results in subsampling probabilities closer to uniform probability $\frac{1}{N}$. b=2 is the default value. See relevant references for further details.
contrasts	An optional list. It specifies how categorical variables are represented in the design matrix. For example, contrasts = list(v1 = 'contr.treatment', v2 = 'contr.sum').
...	A list of parameters which will be passed to svyglm().

Details

A pilot estimator for the unknown parameter $\beta$ is required because both optA and optL subsampling probabilities depend on $\beta$. There is no "free lunch" when determining optimal subsampling probabilities. Fortunately the pilot estimator only needs to satisfy mild conditions. For logistic regression, this is achieved by drawing a size n.plt subsample with replacement from full dataset. The case-control subsample probability is applied, that is, $\pi_i = \frac{1}{2N_1}$ for $Y_i=1$ and $\pi_i = \frac{1}{2N_0}$ for $Y_i=0$, $i=1,...,N$, where$N_0$ and $N_1$ are the counts of observations with $Y = 0$ and $Y = 1$, respectively. For other families, uniform subsampling probabilities are applied. Typically, n.plt is relatively small compared to n.ssp.

When criterion = 'uniform', there is no need to compute the pilot estimator. In this case, a size n.plt + n.ssp subsample will be drawn with uniform sampling probability and coef is the corresponding estimator.

As suggested by survey::svyglm(), for binomial and poisson families, use family=quasibinomial() and family=quasipoisson() to avoid a warning "In eval(family$initialize) : non-integer #successes in a binomial glm!". The quasi versions of the family objects give the same point estimates and suppress the warning. Since subsampling methods only rely on point estimates from svyglm() for further computation, using the quasi families does not introduce any issues.

For Gamma family, ssp.glm returns only the estimation of coefficients, as the dispersion parameter is not estimated.

Value

ssp.glm returns an object of class "ssp.glm" containing the following components (some are optional):

model.call

The original function call.

coef.plt

The pilot estimator. See Details for more information.

coef.ssp

The estimator obtained from the optimal subsample.

coef

The weighted linear combination of coef.plt and coef.ssp. The combination weights depend on the relative size of n.plt and n.ssp and the estimated covariance matrices of coef.plt and coef.ssp. We blend the pilot subsample information into optimal subsample estimator since the pilot subsample has already been drawn. The coefficients and standard errors reported by summary are coef and the square root of diag(cov).

cov.ssp

The covariance matrix of coef.ssp.

cov

The covariance matrix of coef.

index.plt

Row indices of pilot subsample in the full dataset.

index.ssp

Row indices of of optimal subsample in the full dataset.

N

The number of observations in the full dataset.

subsample.size.expect

The expected subsample size, equals to n.ssp for ssp.glm. Note that for other functions, such as ssp.relogit, this value may differ.

comp.time

The total time of computing subsampling probabilities, drawing subsample and fitting model on the subsample.

terms

The terms object for the fitted model.

References

Wang, H. (2019). More efficient estimation for logistic regression with optimal subsamples. Journal of machine learning research, 20(132), 1-59.

Ai, M., Yu, J., Zhang, H., & Wang, H. (2021). Optimal subsampling algorithms for big data regressions. Statistica Sinica, 31(2), 749-772.

Wang, H., & Kim, J. K. (2022). Maximum sampled conditional likelihood for informative subsampling. Journal of machine learning research, 23(332), 1-50.

Examples

# logistic regression
set.seed(2)
N <- 1e4
beta0 <- rep(-0.5, 7)
d <- length(beta0) - 1
corr <- 0.5
sigmax  <- matrix(corr, d, d) + diag(1-corr, d)
X <- MASS::mvrnorm(N, rep(0, d), sigmax)
Y <- rbinom(N, 1, 1 - 1 / (1 + exp(beta0[1] + X %*% beta0[-1])))
colnames(X) <- paste0("X", 1:d)
data <- data.frame(Y = Y, X)
formula <- Y ~ .
n.plt <- 500
n.ssp <- 1000
subsampling.results <- ssp.glm(formula = formula, 
data = data, 
n.plt = n.plt,
n.ssp = n.ssp,
family = 'quasibinomial',
criterion = "optL",
sampling.method = 'poisson',
likelihood = "logOddsCorrection")
summary(subsampling.results)
subsampling.results <- ssp.glm(formula = formula, 
data = data, 
n.plt = n.plt,
n.ssp = n.ssp,
family = 'quasibinomial', 
criterion = "optL",
sampling.method = 'withReplacement', 
likelihood = "weighted")
summary(subsampling.results)
Uni.subsampling.results <- ssp.glm(formula = formula, 
data = data, 
n.plt = n.plt,
n.ssp = n.ssp,
family = 'quasibinomial', 
criterion = 'uniform')
summary(Uni.subsampling.results)
####################
# poisson regression
set.seed(1)
N <-  1e4
beta0 <- rep(0.5, 7)
d <- length(beta0) - 1
X <- matrix(runif(N * d), N, d)
epsilon <- runif(N)
lambda <- exp(beta0[1] + X %*% beta0[-1])
Y <- rpois(N, lambda)
colnames(X) <- paste0("X", 1:d)
data <- data.frame(Y = Y, X)
formula <- Y ~ .
n.plt <- 200
n.ssp <- 600
subsampling.results <- ssp.glm(formula = formula, 
data = data, 
n.plt = n.plt,
n.ssp = n.ssp,
family = 'poisson',
criterion = "optL", 
sampling.method = 'poisson',
likelihood = "weighted")
summary(subsampling.results)
subsampling.results <- ssp.glm(formula = formula, 
data = data, 
n.plt = n.plt,
n.ssp = n.ssp,
family = 'poisson', 
criterion = "optL", 
sampling.method = 'withReplacement',
likelihood = "weighted")
summary(subsampling.results)
Uni.subsampling.results <- ssp.glm(formula = formula, 
data = data, 
n.plt = n.plt,
n.ssp = n.ssp,
family = 'poisson', 
criterion = 'uniform')
summary(Uni.subsampling.results)
##################
# normal linear regression
set.seed(4)
N <- 1e4
beta0 <- rep(-0.5, 7)
d <- length(beta0) - 1
corr <- 0.5
sigmax  <- matrix(corr, d, d) + diag(1-corr, d)
X <- MASS::mvrnorm(N, rep(0, d), sigmax)
epsilon <- rnorm(N, sd = 1)
Y <- beta0[1] + X %*% beta0[-1] + epsilon
colnames(X) <- paste0("X", 1:d)
data <- data.frame(Y = Y, X)
formula <- Y ~ .
n.plt <- 200
n.ssp <- 1000
subsampling.results <- ssp.glm(
formula = formula,
data = data,
n.plt = n.plt,
n.ssp = n.ssp,
family = "gaussian",
criterion = "optA", 
sampling.method = 'poisson',
likelihood = "weighted"
)
summary(subsampling.results)
##################
# gamma regression
set.seed(1)
N <- 1e4
p <- 3
beta0 <- rep(0.5, p + 1)
d <- length(beta0) - 1
shape <- 2
X <- matrix(runif(N * d), N, d)
link_function <- function(X, beta0) 1 / (beta0[1] + X %*% beta0[-1])
scale <- link_function(X, beta0) / shape
Y <- rgamma(N, shape = shape, scale = scale)
colnames(X) <- paste0("X", 1:d)
data <- data.frame(Y = Y, X)
formula <- Y ~ .
n.plt <- 200
n.ssp <- 1000
subsampling.results <- ssp.glm(formula = formula, 
data = data, 
n.plt = n.plt,
n.ssp = n.ssp,
family = 'Gamma',
criterion = "optL", 
sampling.method = 'poisson',
likelihood = "weighted")
summary(subsampling.results)

---

Balanced Subsampling Methods for Generalized Linear Models with Rare Features

Description

This function inherits the weighted objective function (likelihood = "weighted") and the poisson sampling method (sampling.method = "poisson") from ssp.glm, while additionally handling rare features in the model. Rare features refer to binary covariates with low prevalence of being one (expressed) and mostly zero. Such features create challenges for subsampling, because it is likely to miss these rare but informative observations. The balanced subsampling method upweights observations that contain expressed rare features, thereby preserving estimation efficiency for the coefficients of rare features.

A quick start guide is provided in the vignette: https://dqksnow.github.io/subsampling/articles/ssp-logit-rF.html.

Usage

ssp.glm.rF(
  formula,
  data,
  subset = NULL,
  n.plt,
  n.ssp,
  family = "binomial",
  criterion = "BL-Uni",
  sampling.method = "poisson",
  likelihood = "weighted",
  balance.plt = TRUE,
  balance.Y = FALSE,
  rareFeature.index = NULL,
  control = list(...),
  contrasts = NULL,
  ...
)

Arguments

formula	A model formula object.
data	A data frame containing the variables in the model.
subset	An optional vector specifying a subset of observations to be used as the full dataset.
n.plt	The pilot sample size for computing the pilot estimator and estimating optimal subsampling probabilities.
n.ssp	The expected size of the optimal subsample. For sampling.method = "poisson", the actual sample size may vary, but the expected size equals n.ssp.
family	A character string naming a family. It can be a character string naming a family function, a family function or the result of a call to a family function.
criterion	The subsampling criterion. Choices include: "BL-Uni" (default): probabilities proportional to the balance score. "R-Lopt": rareness-aware L-optimality. "Aopt": classical A-optimality, minimizing the trace of the asymptotic covariance matrix. "Lopt": classical L-optimality, minimizing a transformed trace of the asymptotic covariance. "BL-Lopt": balance score combined with L-optimality. "Uni": uniform sampling.
sampling.method	The sampling method. Currently "poisson" is supported, which avoids drawing repeated observations.
likelihood	The objective function used for estimation. Currently "weighted" is supported. Each sampled observation is weighted by the inverse of its subsampling probability.
balance.plt	Logical. Whether to use "BL-Uni" to draw the pilot sample for two-step subsampling methods. Default is TRUE. A good pilot estimator significantly improves the performance of the second-step subsampling; a poor pilot estimator may cause failure.
balance.Y	Logical. Whether to balance the binary response variable in logistic regression. If TRUE, the pilot sampling probability combines the balance score with the case-control method, and the negative subsampling will be performed for the second-step subsampling method. That is, automatically include all Y=1 observations to the subsample, while performing subsampling for Y=0 observations.
rareFeature.index	Column indices of binary rare features in the design matrix, coded as 1 for the rare case. For example, c(4, 9) indicates that columns 4 and 9 of the design matrix contain rare features.
control	A list passed to glm.control(). It includes: alpha: mixture weight between optimal and uniform probabilities, giving $\pi = (1 - \alpha)\pi^{opt} + \alpha \pi^{uni}$.
contrasts	Optional list specifying how categorical variables are encoded in the design matrix.
...	Additional arguments passed to svyglm().

Details

See the package vignette for more details and examples.

Value

An object of class "ssp.glm.rF" containing the following fields.

model.call

The original function call.

coef.plt

Pilot estimator obtained from the pilot subsample.

coef.ssp

Estimator obtained from the optimal subsample.

coef.cmb

Combined estimator using the union of pilot and optimal subsamples. If criterion = "BL-Uni" or "Uni", this equals the pilot and subsample estimators because only one sampling step is used.

cov.plt

Estimated covariance matrix of coef.plt.

cov.ssp

Estimated covariance matrix of coef.ssp.

cov.cmb

Estimated covariance matrix of coef.cmb.

N

Number of observations in the full dataset.

subsample.size.expect

Expected subsample size (n.ssp).

subsample.size.actual

Actual number of subsampled observations.

full.rare.count

For each rare feature, return the counts of ones in the full dataset.

rare.count.plt

Vector of length K giving, for each rare feature, the number of ones in the pilot subsample.

rare.count.ssp

Same as above, computed for the optimal subsample.

rare.count.cmb

Same as above, computed for the combined subsample.

index.plt

Row indices of the pilot subsample within the full dataset.

index.ssp

Row indices of the optimal subsample within the full dataset.

index.cmb

Union of pilot and optimal subsample row indices.

rareFeature.index

Column indices of rare features in the design matrix (same as the user input).

comp.time

Total computation time for computing sampling probabilities, drawing subsamples, and fitting the subsample estimator.

terms

The terms object for the fitted model.

Examples

# logistic regression
set.seed(2)
N <- 1e4
d_rare <- 3
d_cont <- 2
p_rare <- c(0.01, 0.02, 0.05)
beta0 <- c(0.5, rep(0.5, d_rare), rep(0.5, d_cont)) 
corr <- 0.5
sigmax  <- matrix(corr, d_cont, d_cont) + diag(1-corr, d_cont)
X <- MASS::mvrnorm(N, rep(0, d_cont), sigmax)
Z <- do.call(cbind, lapply(seq_along(p_rare), function(i) {
rbinom(N, 1, p_rare[i])
}))
X <- cbind(Z, X)
P <- 1 / (1 + exp(-(beta0[1] + X %*% beta0[-1])))
Y <- as.integer(rbinom(N, 1, P))
colnames(X) <- paste0("X", 1:(d_rare + d_cont))
rareFeature.index <- c(1:d_rare)
data <- data.frame(Y = Y, X)
formula <- Y ~ .
n.plt <- 300
n.ssp <- 2000
BL.Uni.results <- ssp.glm.rF(formula = formula, 
data = data, 
n.plt = n.plt,
n.ssp = n.ssp,
family = 'quasibinomial',
criterion = 'BL-Uni',
sampling.method = 'poisson',
likelihood = 'weighted',
balance.plt = TRUE,
balance.Y = FALSE,
rareFeature.index = rareFeature.index
)
summary(BL.Uni.results)
R.Lopt.results <- ssp.glm.rF(formula = formula, 
data = data, 
n.plt = n.plt,
n.ssp = n.ssp,
family = 'quasibinomial',
criterion = 'R-Lopt',
sampling.method = 'poisson',
likelihood = 'weighted',
balance.plt = TRUE,
balance.Y = FALSE,
rareFeature.index = rareFeature.index
)
summary(R.Lopt.results)

---

Optimal Subsampling Methods for Quantile Regression Model

Description

Draw subsample from full dataset and fit quantile regression model. For a quick start, refer to the vignette.

Usage

ssp.quantreg(
  formula,
  data,
  subset = NULL,
  tau = 0.5,
  n.plt,
  n.ssp,
  B = 5,
  boot = TRUE,
  criterion = "optL",
  sampling.method = "withReplacement",
  likelihood = c("weighted"),
  control = list(...),
  contrasts = NULL,
  ...
)

Arguments

formula	A model formula object of class "formula" that describes the model to be fitted.
data	A data frame containing the variables in the model. Denote $N$ as the number of observations in data.
subset	An optional vector specifying a subset of observations from data to use for the analysis. This subset will be viewed as the full data.
tau	The interested quantile.
n.plt	The pilot subsample size (first-step subsample size). This subsample is used to compute the pilot estimator and estimate the optimal subsampling probabilities.
n.ssp	The expected size of the optimal subsample (second-step subsample). For sampling.method = 'withReplacement', The exact subsample size is n.ssp. For sampling.method = 'poisson', n.ssp is the expected subsample size.
B	The number of subsamples for the iterative sampling algorithm. Each subsample contains n.ssp observations. This allows us to estimate the covariance matrix.
boot	If TRUE then perform iterative sampling algorithm and estimate the covariance matrix. If FALSE then only one subsample with size B*n.ssp is returned.
criterion	It determines how subsampling probabilities are computed. Choices include optL(default) and uniform. optL Minimizes the trace of a transformation of the asymptotic covariance matrix of the subsample estimator. uniform Assigns equal subsampling probability $\frac{1}{N}$ to each observation, serving as a baseline subsampling strategy.
sampling.method	The sampling method for drawing the optimal subsample. Choices include withReplacement and poisson(default). withReplacement draws exactly n.ssp subsamples from size $N$ full dataset with replacement, using the specified subsampling probabilities. poisson draws observations independently by comparing each subsampling probability with a realization of uniform random variable $U(0,1)$.
likelihood	The type of the maximum likelihood function used to calculate the optimal subsampling estimator. Currently weighted is implemented which applies a weighted likelihood function where each observation is weighted by the inverse of its subsampling probability.
control	The argument control contains two tuning parameters alpha and b. alpha $\in [0,1]$ is the mixture weight of the user-assigned subsampling probability and uniform subsampling probability. The actual subsample probability is $\pi = (1-\alpha)\pi^{opt} + \alpha \pi^{uni}$. This protects the estimator from extreme small subsampling probability. The default value is 0. b is a positive number which is used to constraint the poisson subsampling probability. b close to 0 results in subsampling probabilities closer to uniform probability $\frac{1}{N}$. b=2 is the default value. See relevant references for further details.
contrasts	An optional list. It specifies how categorical variables are represented in the design matrix. For example, contrasts = list(v1 = 'contr.treatment', v2 = 'contr.sum').
...	A list of parameters which will be passed to quantreg::rq().

Details

Most of the arguments and returned variables have the same meaning with ssp.glm. Refer to vignette

A pilot estimator for the unknown parameter $\beta$ is required because optL subsampling probabilities depend on $\beta$. There is no "free lunch" when determining optimal subsampling probabilities. For quantile regression, this is achieved by drawing a size n.plt subsample with replacement from full dataset, using uniform sampling probability.

If boot=TRUE, the returned value subsample.size.expect equals to B*n.ssp, and the covariance matrix for coef would be calculated. If boot=FALSE, the returned value subsample.size.expect equals to B*n.ssp, but the covariance matrix won't be estimated.

Value

ssp.quantreg returns an object of class "ssp.quantreg" containing the following components (some are optional):

model.call

The original function call.

coef.plt

The pilot estimator. See Details for more information.

coef

The estimator obtained from the optimal subsample.

cov

The covariance matrix of coef

index.plt

Row indices of pilot subsample in the full dataset.

index.ssp

Row indices of of optimal subsample in the full dataset.

N

The number of observations in the full dataset.

subsample.size.expect

The expected subsample size

terms

The terms object for the fitted model.

References

Wang, H., & Ma, Y. (2021). Optimal subsampling for quantile regression in big data. Biometrika, 108(1), 99-112.

Examples

#quantile regression
set.seed(1)
N <- 1e4
B <- 5
tau <- 0.75
beta.true <- rep(1, 7)
d <- length(beta.true) - 1
corr  <- 0.5
sigmax  <- matrix(0, d, d)
for (i in 1:d) for (j in 1:d) sigmax[i, j] <- corr^(abs(i-j))
X <- MASS::mvrnorm(N, rep(0, d), sigmax)
err <- rnorm(N, 0, 1) - qnorm(tau)
Y <- beta.true[1] + X %*% beta.true[-1] + 
err * rowMeans(abs(X))
data <- as.data.frame(cbind(Y, X))
colnames(data) <- c("Y", paste("V", 1:ncol(X), sep=""))
formula <- Y ~ .
n.plt <- 200
n.ssp <- 100
optL.results <- ssp.quantreg(formula,data,tau = tau,n.plt = n.plt,
n.ssp = n.ssp,B = B,boot = TRUE,criterion = 'optL',
sampling.method = 'withReplacement',likelihood = 'weighted')
summary(optL.results)
uni.results <- ssp.quantreg(formula,data,tau = tau,n.plt = n.plt,
n.ssp = n.ssp,B = B,boot = TRUE,criterion = 'uniform',
sampling.method = 'withReplacement', likelihood = 'weighted')
summary(uni.results)

---

Optimal Subsampling for Logistic Regression Model with Rare Events Data

Description

Draw subsample from full dataset and fit logistic regression model on subsample. For a quick start, refer to the vignette.

Usage

ssp.relogit(
  formula,
  data,
  subset = NULL,
  n.plt,
  n.ssp,
  criterion = "optL",
  likelihood = "logOddsCorrection",
  control = list(...),
  contrasts = NULL,
  ...
)

Arguments

formula	A model formula object of class "formula" that describes the model to be fitted.
data	A data frame containing the variables in the model. Denote $N$ as the number of observations in data.
subset	An optional vector specifying a subset of observations from data to use for the analysis. This subset will be viewed as the full data.
n.plt	The pilot subsample size (first-step subsample size). This subsample is used to compute the pilot estimator and estimate the optimal subsampling probabilities.
n.ssp	The expected subsample size (the second-step subsample size) drawn from those samples with Y=0. All rare events (Y=1) are included in the optimal subsample automatically.
criterion	The choices include optA, optL(default), LCC and uniform. optA Minimizes the trace of the asymptotic covariance matrix of the subsample estimator. optL Minimizes the trace of a transformation of the asymptotic covariance matrix. The computational complexity of optA is $O(N d^2)$ while that of optL is $O(N d)$. LCC Local Case-Control sampling probability, used as a baseline subsampling strategy. uniform Assigns equal subsampling probability $\frac{1}{N}$ to each observation, serving as a baseline subsampling strategy.
likelihood	The likelihood function to use. Options include weighted and logOddsCorrection (default). A bias-correction likelihood function is required for subsample since unequal subsampling probabilities introduce bias. weighted Applies a weighted likelihood function where each observation is weighted by the inverse of its subsampling probability. logOddsCorrection This likelihood is available only for logistic regression model (i.e., when family is binomial or quasibinomial). It uses a conditional likelihood, where each element of the likelihood represents the probability of $Y=1$, given that this subsample was drawn.
control	The argument control contains two tuning parameters alpha and b. alpha $\in [0,1]$ is the mixture weight of the user-assigned subsampling probability and uniform subsampling probability. The actual subsample probability is $\pi = (1-\alpha)\pi^{opt} + \alpha \pi^{uni}$. This protects the estimator from extreme small subsampling probability. The default value is 0. b is a positive number which is used to constraint the poisson subsampling probability. b close to 0 results in subsampling probabilities closer to uniform probability $\frac{1}{N}$. b=2 is the default value. See relevant references for further details.
contrasts	An optional list. It specifies how categorical variables are represented in the design matrix. For example, contrasts = list(v1 = 'contr.treatment', v2 = 'contr.sum').
...	A list of parameters which will be passed to svyglm().

Details

'Rare event' stands for the number of observations where $Y=1$ is rare compare to the number of $Y=0$ in the full data. In the face of logistic regression with rare events, @wang2021nonuniform shows that the available information ties to the number of positive instances instead of the full data size. Based on this insight, one can keep all the rare instances and perform subsampling on the non-rare instances to reduce the computational cost. When criterion = optA, optL or LCC, all observations with $Y=1$ are preserved and it draw n.ssp subsamples from observations with Y=0. When criterion = uniform, it draws (n.plt+n.ssp) subsmples from the full sample with equal sampling probability.

A pilot estimator for the unknown parameter $\beta$ is required because both optA and optL subsampling probabilities depend on $\beta$. This is achieved by drawing half size subsample from rare observations and half from non-rare observations.

Most of the arguments and returned variables have similar meaning with ssp.glm. Refer to vignette

Value

ssp.relogit returns an object of class "ssp.relogit" containing the following components (some are optional):

model.call

The original function call.

coef.plt

The pilot estimator. See Details for more information.

coef.ssp

The estimator obtained from the optimal subsample.

coef

The weighted linear combination of coef.plt and coef.ssp. The combination weights depend on the relative size of n.plt and n.ssp and the estimated covariance matrices of coef.plt and coef.ssp. We blend the pilot subsample information into optimal subsample estimator since the pilot subsample has already been drawn. The coefficients and standard errors reported by summary are coef and the square root of diag(cov).

cov.ssp

The covariance matrix of coef.ssp.

cov

The covariance matrix of beta.cmb.

index.plt

Row indices of pilot subsample in the full dataset.

index.ssp

Row indices of of optimal subsample in the full dataset.

N

The number of observations in the full dataset.

subsample.size.expect

The expected subsample size.

terms

The terms object for the fitted model.

References

Wang, H., Zhang, A., & Wang, C. (2021). Nonuniform negative sampling and log odds correction with rare events data. Advances in Neural Information Processing Systems, 34, 19847-19859.

Examples

set.seed(1)
N <- 2 * 1e4
beta0 <- c(-5, -rep(0.7, 6))
d <- length(beta0) - 1
X <- matrix(0, N, d)
corr <- 0.5
sigmax <- corr ^ abs(outer(1:d, 1:d, "-"))
sigmax <- sigmax / 4
X <- MASS::mvrnorm(n = N, mu = rep(0, d), Sigma = sigmax)
Y <- rbinom(N, 1, 1 - 1 / (1 + exp(beta0[1] + X %*% beta0[-1])))
print(paste('N: ', N))
print(paste('sum(Y): ', sum(Y)))
n.plt <- 200
n.ssp <- 1000
data <- as.data.frame(cbind(Y, X))
colnames(data) <- c("Y", paste("V", 1:ncol(X), sep=""))
formula <- Y ~ .
subsampling.results <- ssp.relogit(formula = formula,
                                     data = data,
                                     n.plt = n.plt,
                                     n.ssp = n.ssp,
                                     criterion = 'optA',
                                     likelihood = 'logOddsCorrection')
summary(subsampling.results)

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Optimal Subsampling Method for Softmax (multinomial logistic) Regression Model

Description

Draw subsample from full dataset and fit softmax(multinomial logistic) regression model on the subsample. Refer to vignette for a quick start.

Usage

ssp.softmax(
  formula,
  data,
  subset,
  n.plt,
  n.ssp,
  criterion = "MSPE",
  sampling.method = "poisson",
  likelihood = "MSCLE",
  constraint = "summation",
  control = list(...),
  contrasts = NULL,
  ...
)

Arguments

formula	A model formula object of class "formula" that describes the model to be fitted.
data	A data frame containing the variables in the model. Denote $N$ as the number of observations in data.
subset	An optional vector specifying a subset of observations from data to use for the analysis. This subset will be viewed as the full data.
n.plt	The pilot subsample size (first-step subsample size). This subsample is used to compute the pilot estimator and estimate the optimal subsampling probabilities.
n.ssp	The expected size of the optimal subsample (second-step subsample). For sampling.method = 'withReplacement', The exact subsample size is n.ssp. For sampling.method = 'poisson', n.ssp is the expected subsample size.
criterion	The criterion of optimal subsampling probabilities. Choices include optA, optL, MSPE(default), LUC and uniform. MSPE Minimizes the mean squared prediction error between subsample estimator and full data estimator. optA Minimizes the trace of the asymptotic covariance matrix of the subsample estimator. optL Minimizes the trace of a transformation of the asymptotic covariance matrix, which reduces computational costs than optA. LUC Local uncertainty sampling method, serving as a baseline subsampling strategy. See Wang and Kim (2022). uniform Assigns equal subsampling probability $\frac{1}{N}$ to each observation, serving as a baseline subsampling strategy.
sampling.method	The sampling method to use. Choices include withReplacement and poisson(default). withReplacement draws exactly n.ssp subsamples from size $N$ full dataset with replacement, using the specified subsampling probabilities. poisson draws observations independently by comparing each subsampling probability with a realization of uniform random variable $U(0,1)$. Differences between methods: Sample size: withReplacement draws exactly n.ssp subsamples while poisson draws subsamples with expected size n.ssp, meaning the actual size may vary. Memory usage: withReplacement requires the entire dataset to be loaded at once, while poisson allows for processing observations sequentially (will be implemented in future version). Estimator performance: Theoretical results show that the poisson tends to get a subsample estimator with lower asymptotic variance compared to the withReplacement
likelihood	A bias-correction likelihood function is required for subsample since unequal subsampling probabilities introduce bias. Choices include weighted and MSCLE(default). weighted Applies a weighted likelihood function where each observation is weighted by the inverse of its subsampling probability. MSCLE It uses a conditional likelihood, where each element of the likelihood represents the density of $Y_i$ given that this observation was drawn.
constraint	The constraint for identifiability of softmax model. Choices include baseline and summation(default). The baseline constraint assumes the coefficient for the baseline category are $0$. Without loss of generality, we set the category $Y=0$ as the baseline category so that $\boldsymbol{\beta}_0=0$. The summation constraint $\sum_{k=0}^{K} \boldsymbol{\beta}_k$ is also used in the subsampling method for the purpose of calculating subsampling probability. These two constraints lead to different interpretation of coefficients but are equal for computing $P(Y_{i,k} = 1 \mid \mathbf{x}_i)$. The estimation of coefficients returned by ssp.softmax() is under baseline constraint.
control	A list of parameters for controlling the sampling process. There are two tuning parameters alpha and b. Default is list(alpha=0, b=2). alpha $\in [0,1]$ is the mixture weight of the user-assigned subsampling probability and uniform subsampling probability. The actual subsample probability is $\pi = (1-\alpha)\pi^{opt} + \alpha \pi^{uni}$. This protects the estimator from extreme small subsampling probability. The default value is 0. b is a positive number which is used to constraint the poisson subsampling probability. b close to 0 results in subsampling probabilities closer to uniform probability $\frac{1}{N}$. b=2 is the default value. See relevant references for further details.
contrasts	An optional list. It specifies how categorical variables are represented in the design matrix. For example, contrasts = list(v1 = 'contr.treatment', v2 = 'contr.sum').
...	A list of parameters which will be passed to nnet::multinom().

Details

A pilot estimator for the unknown parameter $\beta$ is required because MSPE, optA and optL subsampling probabilities depend on $\beta$. There is no "free lunch" when determining optimal subsampling probabilities. For softmax regression, this is achieved by drawing a size n.plt subsample with replacement from full dataset with uniform sampling probability.

Value

ssp.softmax returns an object of class "ssp.softmax" containing the following components (some are optional):

model.call

The original function call.

coef.plt

The pilot estimator. See Details for more information.

coef.ssp

The estimator obtained from the optimal subsample.

coef

The weighted linear combination of coef.plt and coef.ssp, under baseline constraint. The combination weights depend on the relative size of n.plt and n.ssp and the estimated covariance matrices of coef.plt and coef.ssp. We blend the pilot subsample information into optimal subsample estimator since the pilot subsample has already been drawn. The coefficients and standard errors reported by summary are coef and the square root of diag(cov).

coef.plt.sum

The pilot estimator under summation constraint. coef.plt.sum = G %*% as.vector(coef.plt).

coef.ssp.sum

The estimator obtained from the optimal subsample under summation constraint. coef.ssp.sum = G %*% as.vector(coef.ssp).

coef.sum

The weighted linear combination of coef.plt and coef.ssp, under summation constraint. coef.sum = G %*% as.vector(coef).

cov.plt

The covariance matrix of coef.plt.

cov.ssp

The covariance matrix of coef.ssp.

cov

The covariance matrix of coef.cmb.

cov.plt.sum

The covariance matrix of coef.plt.sum.

cov.ssp.sum

The covariance matrix of coef.ssp.sum.

cov.sum

The covariance matrix of coef.sum.

index.plt

Row indices of pilot subsample in the full dataset.

index.ssp

Row indices of of optimal subsample in the full dataset.

N

The number of observations in the full dataset.

subsample.size.expect

The expected subsample size.

terms

The terms object for the fitted model.

References

Yao, Y., & Wang, H. (2019). Optimal subsampling for softmax regression. Statistical Papers, 60, 585-599.

Han, L., Tan, K. M., Yang, T., & Zhang, T. (2020). Local uncertainty sampling for large-scale multiclass logistic regression. Annals of Statistics, 48(3), 1770-1788.

Wang, H., & Kim, J. K. (2022). Maximum sampled conditional likelihood for informative subsampling. Journal of machine learning research, 23(332), 1-50.

Yao, Y., Zou, J., & Wang, H. (2023). Optimal poisson subsampling for softmax regression. Journal of Systems Science and Complexity, 36(4), 1609-1625.

Yao, Y., Zou, J., & Wang, H. (2023). Model constraints independent optimal subsampling probabilities for softmax regression. Journal of Statistical Planning and Inference, 225, 188-201.

Examples

# softmax regression
d <- 3 # dim of covariates
K <- 2 # K + 1 classes
G <- rbind(rep(-1/(K+1), K), diag(K) - 1/(K+1)) %x% diag(d)
N <- 1e4
beta.true.baseline <- cbind(rep(0, d), matrix(-1.5, d, K))
beta.true.summation <- cbind(rep(1, d), 0.5 * matrix(-1, d, K))
set.seed(1)
mu <- rep(0, d)
sigma <- matrix(0.5, nrow = d, ncol = d)
diag(sigma) <- rep(1, d)
X <- MASS::mvrnorm(N, mu, sigma)
prob <- exp(X %*% beta.true.summation)
prob <- prob / rowSums(prob)
Y <- apply(prob, 1, function(row) sample(0:K, size = 1, prob = row))
n.plt <- 500
n.ssp <- 1000
data <- as.data.frame(cbind(Y, X))
colnames(data) <- c("Y", paste("V", 1:ncol(X), sep=""))
head(data)
formula <- Y ~ . -1
WithRep.MSPE <- ssp.softmax(formula = formula,
 data = data, 
 n.plt = n.plt,
 n.ssp = n.ssp,
 criterion = 'MSPE', 
 sampling.method = 'withReplacement',
 likelihood = 'weighted',
 constraint = 'baseline')
summary(WithRep.MSPE)

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Optimal Subsampling Methods for Statistical Models

Description

Subsampling methods are utilized in statistical modeling for massive datasets. These methods aim to draw representative subsamples from the full dataset based on specific sampling probabilities, with the goal of maintaining inference efficiency. The sampling probabilities are tailored to particular objectives, such as minimizing the variance of the estimated coefficients or reducing prediction error. By using subsampling techniques, the package balances the trade-off between computational efficiency and statistical efficiency, making it a practical tool for massive data analysis.

Models Supported

• Generalized Linear Models (GLMs)

• Softmax (Multinomial) Regression

• Rare Event Logistic Regression

• Quantile Regression

Author(s)

Maintainer: Qingkai Dong [email protected] [copyright holder]

Authors:

• Yaqiong Yao

• Haiying Wang

Other contributors:

• Qiang Zhang [contributor]

• Jun Yan [contributor]

See Also

Useful links:

• https://github.com/dqksnow/subsampling

• https://dqksnow.github.io/subsampling/

• Report bugs at https://github.com/dqksnow/Subsampling/issues

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