An internal function to create an A matrix for calibration of quantiles

joint_calib_create_matrix is function that creates an \(A = [a_{ij}]\) matrix for calibration of quantiles. Function allows to create matrix using logistic interpolation (using stats::plogis, default) or linear (as in Harms and Duchesne (2006), i.e. slightly modified Heavyside function).

In case of logistic interpolation elements of \(A\) are created as follows

\[a_{i j} = \frac{1}{(1 + \exp\left(-2l\left(x_{ij}-Q_{x_j, \alpha}\right)\right))N},\]

where \(x_{ij}\) is the \(i\)th row of the auxiliary variable \(X_j\), \(N\) is the population size, \(Q_{x_j, \alpha}\) is the known population \(\alpha\)th quantile, and \(l\) is set to -1000 (by default).

In case of linear interpolation elements of \(A\) are created as follows

\[a_{i j}= \begin{cases} N^{-1}, & x_{i j} \leqslant L_{x_{j}, r} \left(Q_{x_j, \alpha}\right), \cr N^{-1} \beta_{x_{j}, r}\left(Q_{x_j, \alpha}\right), & x_{i j}=U_{x_{j}, r}\left(Q_{x_j, \alpha}\right), \cr 0, & x_{i j}>U_{x_{j}, r} \left(Q_{x_j, \alpha}\right), \end{cases}\]

\(i=1,...,r\), \(j=1,...,k\), where \(r\) is the set of respondents, \(k\) is the auxiliary variable index and

\[L_{x_{j}, r}(t) = \max \left\lbrace\left\lbrace{x_{i j}}, i \in s \mid x_{i j} \leqslant t\right\rbrace \cup \lbrace-\infty\rbrace \right\rbrace,\] \[U_{x_{j}, r}(t) = \min \left\lbrace\left\lbrace{x_{i j}}, i \in s \mid x_{i j}>t\right\rbrace \cup \lbrace\infty\rbrace \right\rbrace,\] \[\beta_{x_{j}, r}(t) = \frac{t-L_{x_{j}, s}(t)}{U_{x_{j}, s}(t)-L_{x_{j}, s}(t)},\]

\(i=1,...,r\), \(j=1,...,k\), \(t \in \mathbb{R}\).

Usage

joint_calib_create_matrix(X_q, N, pop_quantiles, control = control_calib())

Arguments

X_q: matrix of variables for calibration of quantiles,
N: population size for calibration of quantiles,
pop_quantiles: a vector of population quantiles for X_q,
control: a control parameter for creation of X_q matrix.

Value

Return matrix A

References

Harms, T. and Duchesne, P. (2006). On calibration estimation for quantiles. Survey Methodology, 32(1), 37.

Author

Maciej Beręsewicz

Examples

# Create matrix for one variable and 3 quantiles
set.seed(123)
N <- 1000
x <- as.matrix(rnorm(N))
quants <- list(quantile(x, c(0.25,0.5,0.75)))
A <- joint_calib_create_matrix(x, N, quants)
head(A)
#>               [,1]         [,2]  [,3]
#> [1,]  3.417662e-33 1.000000e-03 0.001
#> [2,] 1.221974e-176 1.000000e-03 0.001
#> [3,]  0.000000e+00 0.000000e+00 0.000
#> [4,] 3.168423e-307 2.389406e-30 0.001
#> [5,]  0.000000e+00 7.091618e-56 0.001
#> [6,]  0.000000e+00 0.000000e+00 0.000
colSums(A)
#> [1] 0.2502323 0.4999654 0.7498815

# Create matrix with linear interpolation
A <- joint_calib_create_matrix(x, N, quants, control_calib(interpolation="linear"))
head(A)
#>      [,1]  [,2]  [,3]
#> [1,]    0 0.001 0.001
#> [2,]    0 0.001 0.001
#> [3,]    0 0.000 0.000
#> [4,]    0 0.000 0.001
#> [5,]    0 0.000 0.001
#> [6,]    0 0.000 0.000
colSums(A)
#> [1] 0.24975 0.49950 0.74925

# Create matrix for two variables and different number of quantiles

set.seed(123)
x1 <- rnorm(N)
x2 <- rchisq(N, 1)
x <- cbind(x1, x2)
quants <- list(quantile(x1, 0.5), quantile(x2, c(0.1, 0.75, 0.9)))
B <- joint_calib_create_matrix(x, N, quants)
head(B)
#>              [,1]          [,2]          [,3]  [,4]
#> [1,] 1.000000e-03  7.889542e-30  1.000000e-03 0.001
#> [2,] 1.000000e-03 1.614292e-242  1.000000e-03 0.001
#> [3,] 0.000000e+00  0.000000e+00  0.000000e+00 0.001
#> [4,] 2.389406e-30  0.000000e+00  0.000000e+00 0.000
#> [5,] 7.091618e-56  0.000000e+00 1.360616e-132 0.001
#> [6,] 0.000000e+00  0.000000e+00  1.000000e-03 0.001
colSums(B)
#> [1] 0.49996541 0.09961593 0.74980652 0.89987264