Check footprint conservation against direct extensions. — check_footprint

Verify the master input-output accounting identity for a footprint: the environmental pressure embodied across all final demand and traced back to an origin sector should equal the direct extension (the source pressure) of that sector.

The footprint engine zeroes negative coefficients, caps column sums, and drops near-zero-output sectors (the FABIO conventions), so the identity holds only approximately. This check quantifies the discrepancy per origin sector instead of asserting exact equality. Crucially it detects under-tracing (pressure that silently disappears and never reaches final demand), which compute_footprint()'s conserve_extensions bounding never reports because it only rescales results downward.

Only the positive side of the extensions is traced, matching the engine, which traces pmax(extensions, 0) and ignores sectors with output <= output_tol.

Usage

check_footprint_conservation(
  footprint,
  extensions,
  labels,
  x_vec,
  output_tol = 1e-08,
  tol = 0.01
)

Arguments

footprint: Footprint tibble from compute_footprint(), with origin_area, origin_item and value columns.
extensions: Numeric vector of environmental extensions per sector, as passed to compute_footprint().
labels: Tibble with area_code and item_cbs_code mapping each sector to its meaning, as passed to compute_footprint().
x_vec: Numeric vector of total output per sector.
output_tol: Minimum output for a sector to be traceable. Sectors with x_vec <= output_tol contribute zero direct pressure, matching compute_footprint().
tol: Relative tolerance for the conservation status. Discrepancies within tol of the direct pressure are "ok".

Value

A tibble with one row per origin sector:

origin_area: Country where the pressure occurs.
origin_item: Item causing the pressure.
direct: Direct (source) extension for the sector.
embodied: Footprint traced back to the sector.
discrepancy: embodied - direct.
rel_discrepancy: discrepancy / direct (NA when direct is zero).
status: One of "ok", "under_traced", "dropped" (embodied is zero while direct is positive) or "over_traced". Rows are ordered by descending absolute relative discrepancy.

Examples

z_mat <- matrix(c(0, 5, 10, 0), nrow = 2)
x_vec <- c(100, 200)
y_mat <- matrix(c(85, 195), ncol = 1)
extensions <- c(50, 30)
labels <- tibble::tibble(
  area_code = c(1L, 1L),
  item_cbs_code = c(1L, 2L)
)
fp <- compute_footprint(
  x_vec = x_vec, y_mat = y_mat, extensions = extensions,
  labels = labels, z_mat = z_mat
)
#> ℹ Computing footprint for 2 sectors.
#>   2 sectors have non-zero extensions.
#>   Final demand: 1 column.
#> Sparse solve path (no dense Leontief inverse).
#> Computing footprints...
#> ✔ Footprint complete: 2 non-zero flows.
check_footprint_conservation(fp, extensions, labels, x_vec)
#> # A tibble: 2 × 7
#>   origin_area origin_item direct embodied discrepancy rel_discrepancy status    
#>         <int>       <int>  <dbl>    <dbl>       <dbl>           <dbl> <chr>     
#> 1           1           1     50     47.5     -2.51          -0.0501  under_tra…
#> 2           1           2     30     30.0     -0.0376        -0.00125 ok