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Joint Expected Hits

Source: R/generics.R
joint_expected_hits.Rd

Computes the matrix of pairwise expectations \(E(n_i n_j)\) for a with-replacement sampling design, where \(n_k\) is the number of times unit \(k\) is selected.

Usage

joint_expected_hits(x, ...)

# S3 method for class 'wr'
joint_expected_hits(x, sampled_only = FALSE, nsim = 10000L, ...)

# S3 method for class 'wor'
joint_expected_hits(x, ...)

# Default S3 method
joint_expected_hits(x, ...)

Arguments

x

A with-replacement design object (class "wr").

...

Additional arguments passed to methods (e.g., nsim for simulation-based methods).

sampled_only

If TRUE, return only the submatrix for units selected at least once (requires nrep = 1). Useful when N is large but the number of distinct selected units is manageable. Default FALSE.

nsim

Number of simulations for Chromy's pairwise expectations (default 10000).

Value

A symmetric N x N matrix (or n_s x n_s if sampled_only = TRUE, where n_s is the number of distinct selected units). Diagonal entries are \(E(n_i^2)\) and off-diagonal entries are \(E(n_i n_j)\).

Details

The computation depends on the sampling method:

Exact analytic

multinomial (\(E(n_i n_j) = n(n-1) p_i p_j\)) and srs (\(E(n_i n_j) = n(n-1)/N^2\)).

Simulation-based

chromy: pairwise expectations are estimated by Monte Carlo simulation (controlled by the nsim parameter, default 10 000). Increase nsim for more precise estimates at the cost of computation time.

When sampled_only = TRUE, the submatrix is indexed by population units that were selected at least once (i.e., units with hits > 0). All methods compute the n_s x n_s submatrix directly, avoiding the N x N allocation. For chromy, the simulation draws still iterate over the full population (O(N) per draw), but the accumulator is n_s x n_s.

See also

joint_inclusion_prob() for the without-replacement analogue, sampling_cov() for the covariance matrix.

Examples

x <- c(40, 80, 50, 60, 70)
hits <- expected_hits(x, n = 3)
s <- unequal_prob_wr(hits, method = "chromy")
joint_expected_hits(s)
#>        [,1]   [,2]   [,3]   [,4]   [,5]
#> [1,] 0.3934 0.2617 0.1633 0.2023 0.1595
#> [2,] 0.2617 0.8069 0.3425 0.4526 0.5570
#> [3,] 0.1633 0.3425 0.4987 0.1592 0.3324
#> [4,] 0.2023 0.4526 0.1592 0.5918 0.3695
#> [5,] 0.1595 0.5570 0.3324 0.3695 0.7092

# Only the submatrix for selected units
joint_expected_hits(s, sampled_only = TRUE)
#>        [,1]   [,2]   [,3]
#> [1,] 0.3963 0.1636 0.2111
#> [2,] 0.1636 0.4929 0.1659
#> [3,] 0.2111 0.1659 0.6058