Package 'inca'

Integer Calibration

Description

Specific functions are provided for rounding real weights to integers and performing integer programming algorithms for calibration problems.

Details

Package:	inca
Type:	Package
Version:	0.0.4
Date:	2019-09-18
License:	GPL (>= 2)

Calibration forces the weighted estimates of calibration variables to match known totals. This improves the quality of the design-weighted estimates. It is used to adjust for non-response and/or under-coverage. The commonly used methods of calibration produce non-integer weights. In cases where weighted estimates must be integers, one must "integerize" the calibrated weights. However, this procedure often produces final weights that are very different for the "sample" weights. To counter this problem, the inca package provides specific functions for rounding real weights to integers, and performing an integer programming algorithm for calibration problems with integer weights.

For a complete list of exported functions, use library(help = "inca").

This research was supported in part by the U.S. Department of Agriculture, National Agriculture Statistics Service. The findings and conclusions in this publication are those of the authors and should not be construed to represent any official USDA or U.S. Government determination or policy.

Author(s)

Luca Sartore [email protected] and Kelly Toppin [email protected]

Maintainer: Luca Sartore [email protected]

References

Theberge, A. (1999). Extensions of calibration estimators in survey sampling. Journal of the American Statistical Association, 94(446), 635-644.

Little, R. J., & Vartivarian, S. (2003). On weighting the rates in non-response weights.

Kish, L. (1992). Weighting for unequal Pi. Journal of Official Statistics, 8(2), 183.

Rao, J. N. K., & Singh, A. C. (1997). A ridge-shrinkage method for range-restricted weight calibration in survey sampling. In Proceedings of the section on survey research methods (pp. 57-65). American Statistical Association Washington, DC.

Horvitz, D. G., & Thompson, D. J. (1952). A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47(260), 663-685.

Kalton, G., & Flores-Cervantes, I. (2003). Weighting methods. Journal of Official Statistics, 19(2), 81-98.

Sartore, L., Toppin, K., Young, L., Spiegelman, C. (2019). Developing integer calibration weights for the Census of Agriculture. Journal of Agricultural, Biological, and Environmental Statistics, 24(1), 26-48.

Examples

library(inca)

---

Function for Weights Adjustments

Description

This function provides a trimming procedure to force the weights to be within the provided boundaries

Usage

adjWeights(weights, lower = -Inf, upper = +Inf)

Arguments

weights	A numerical vector of weights
lower	A numerical vector of lower bounds
upper	A numerical vector of upper bounds

Details

The function produces trimmed weights, which will be the input for the rounding technique before integer calibration. When the weights are bounded, the function rounds-up the lower bounds and rounds-down the upper. If the condition upper > lower + 1, an error is returned.

Value

A vector of adjusted weights

Examples

library(inca)
w <- rnorm(150, 0, 2)
aw <- adjWeights(w, runif(150, -3, -1), runif(150, 1, 3))
hist(aw, main = "Adjusted weights")

---

Integer Calibration Function

Description

This function performs an integer programming algorithm developed for calibrating integer weights, in order to reduce a specific objective function

Usage

intcalibrate(weights, formula, targets, objective = c("L1", "aL1", "rL1",
  "LB1", "rB1", "rbLasso1", "L2", "aL2", "rL2", "LB2", "rB2", "rbLasso2"),
  tgtBnds = NULL, lower = -Inf, upper = Inf, scale = NULL,
  sparse = FALSE, data = environment(formula))

Arguments

weights	A numerical vector of real or integer weights to be calibrated. If real values are provided, they will be rounded before applying the calibration algorithm
formula	A formula to express a linear system for hitting the targets
targets	A numerical vector of point-targets to hit
objective	A character specifying the objective function used for calibration. By default "L1". See details for more information
tgtBnds	A two-column matrix containing the bounds for the point-targets
lower	A numerical vector or value defining the lower bounds of the weights
upper	A numerical vector or value defining the upper bounds of the weights
scale	A numerical vector of positive values
sparse	A logical value denoting if the linear system is sparse or not. By default it is FALSE
data	A data.frame or matrix object containing the data to be used for calibration

Details

The integer programming algorithm for calibration can be performed by considering one of the following objective functions:

"L1"

for the summation of absolute errors

"aL1"

for the asymmetric summation of absolute errors

"rL1"

for the summation of absolute relative errors

"LB1"

for the summation of absolute errors if outside the boundaries

"rB1"

for the summation of absolute relative errors if outside the boundaries

"rbLasso1"

for the summation of absolute relative errors if outside the boundaries plus a Lasso penalty based on the distance from the provided weights

"L2"

for the summation of square errors

"aL2"

for the asymmetric summation of square errors

"rL2"

for the summation of square relative errors

"LB2"

for the summation of square errors if outside the boundaries

"rB2"

for the summation of square relative errors if outside the boundaries

"rbLasso2"

for the summation of square relative errors if outside the boundaries plus a Lasso penalty based on the distance from the provided weights

A two-column matrix must be provided to tgtBnds when objective = "LB1", objective = "rB1", objective = "rbLasso1", objective = "LB2", objective = "rB2", and objective = "rbLasso2".

The argument scale must be specified with a vector of positive reals number when objective = "rL1" or objective = "rL2".

Value

A numerical vector of calibrated integer weights.

Examples

library(inca)
set.seed(0)
w <- rpois(150, 4)
data <- matrix(rbinom(150000, 1, .3) * rpois(150000, 4), 1000, 150)
y <- data %*% w
w <- runif(150, 0, 7.5)
print(sum(abs(y - data %*% w)))
cw <- intcalibrate(w, ~. + 0, y, lower = 1, upper = 7, sparse = TRUE, data = data)
print(sum(abs(y - data %*% cw)))
barplot(table(cw), main = "Calibrated integer weights")

---

Function for Rounding Weights

Description

This function performs an optimal rounding of the provided real weights, in order to reduce a specific objective function

Usage

roundWeights(weights, formula, targets, objective = c("L1", "aL1", "rL1",
  "LB1", "rB1", "rbLasso1", "L2", "aL2", "rL2", "LB2", "rB2", "rbLasso2"),
  tgtBnds = NULL, lower = -Inf, upper = Inf, scale = NULL,
  sparse = FALSE, data = environment(formula))

Arguments

weights	A numerical vector of real weights to be rounded
formula	A formula to express a linear system for hitting the targets
targets	A numerical vector of point-targets to hit
objective	A character specifying the objective function used for calibration. By default, it is "L1". See details for more information
tgtBnds	A two-column matrix containing the bounds for the point-targets
lower	A numerical vector or value defining the lower bounds of the weights
upper	A numerical vector or value defining the upper bounds of the weights
scale	A numerical vector of positive values
sparse	A logical value denoting if the linear system is sparse or not. By default, it is FALSE
data	A data.frame or matrix object containing the data to be used for calibration

Details

The optimal rounding can be performed by considering one of the following objective functions:

"L1"

for the summation of absolute errors

"aL1"

for the asymmetric summation of absolute errors

"rL1"

for the summation of absolute relative errors

"LB1"

for the summation of absolute errors if outside the boundaries

"rB1"

for the summation of absolute relative errors if outside the boundaries

"rbLasso1"

for the summation of absolute relative errors if outside the boundaries plus a Lasso penalty based on the distance from the provided weights

"L2"

for the summation of square errors

"aL2"

for the asymmetric summation of square errors

"rL2"

for the summation of square relative errors

"LB2"

for the summation of square errors if outside the boundaries

"rB2"

for the summation of square relative errors if outside the boundaries

"rbLasso2"

for the summation of square relative errors if outside the boundaries plus a Lasso penalty based on the distance from the provided weights

A two-column matrix must be provided to tgtBnds when objective = "LB1", objective = "rB1", objective = "rbLasso1", objective = "LB2", objective = "rB2", and objective = "rbLasso2".

The argument scale must be specified with a vector of positive reals number when objective = "rL1" or objective = "rL2".

Value

A vector of integer weights to be the input of the calibration algorithm

Examples

library(inca)
set.seed(0)
w <- rpois(150, 4)
data <- matrix(rbinom(150000, 1, .3) * rpois(150000, 4), 1000, 150)
y <- data %*% w
w <- runif(150, 0, 7.5)
rw <- roundWeights(w, ~. + 0, y, lower = 1, upper = 7, sparse = TRUE, data = data)
barplot(table(rw), main = "Rounded weigths")

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