Estimate Migration Flows to Match Net Totals via Iterative Proportional Fitting

The net_matrix_ipf function finds the maximum likelihood estimates for a flow matrix under the multiplicative log-linear model: $$\log y_{ij} = \log \alpha_i + \log \alpha_j^{-1} + \log m_{ij}$$ where \(y_{ij}\) is the estimated migration flow from origin \(i\) to destination \(j\), and \(m_{ij}\) is the prior flow. The function iteratively adjusts origin and destination scaling factors (\(\alpha\)) to match directional net migration totals.

Usage

net_matrix_ipf(
  net_tot,
  m,
  zero_mask = NULL,
  maxit = 500,
  tol = 1e-06,
  verbose = FALSE
)

Arguments

net_tot

A numeric vector of net migration totals for each region. Must sum to zero.

m

A square numeric matrix providing prior flow estimates. Must have dimensions length(net_tot) × length(net_tot).

zero_mask

A logical matrix of the same dimensions as m, where TRUE indicates forbidden (structurally zero) flows. Defaults to disallowing diagonal flows.

maxit

Maximum number of iterations to perform. Default is 500.

tol

Convergence tolerance based on maximum change in \(\alpha\) between iterations. Default is 1e-6.

verbose

Logical flag to print progress and \(\alpha\) updates during iterations. Default is FALSE.

Value

A named list with components:

n

Estimated matrix of flows satisfying the net constraints.

it

Number of iterations used.

tol

Convergence tolerance used.

value

Sum of squared residuals between actual and target net flows.

convergence

Logical indicator of convergence within tolerance.

message

Text description of convergence result.

Details

The function avoids matrix inversion by updating \(\alpha\) using a closed-form solution to a quadratic equation at each step. Only directional net flows (column sums minus row sums) are matched, not marginal totals. Flows are constrained to be non-negative. If multiple positive roots are available when solving the quadratic, the smaller root is selected for improved stability.

See also

net_matrix_entropy() for entropy-based estimation minimizing KL divergence, net_matrix_lp() for L1-loss linear programming, and net_matrix_optim() for least-squares (L2) optimization.

Author

Guy J. Abel, Peter W. F. Smith

Examples

m <- matrix(c(0, 100, 30, 70,
              50,   0, 45,  5,
              60,  35,  0, 40,
              20,  25, 20,  0),
            nrow = 4, byrow = TRUE,
            dimnames = list(orig = LETTERS[1:4], dest = LETTERS[1:4]))
addmargins(m)
#>      dest
#> orig    A   B  C   D Sum
#>   A     0 100 30  70 200
#>   B    50   0 45   5 100
#>   C    60  35  0  40 135
#>   D    20  25 20   0  65
#>   Sum 130 160 95 115 500
sum_region(m)
#> # A tibble: 4 × 5
#>   region out_mig in_mig  turn   net
#>   <chr>    <dbl>  <dbl> <dbl> <dbl>
#> 1 A          200    130   330   -70
#> 2 B          100    160   260    60
#> 3 C          135     95   230   -40
#> 4 D           65    115   180    50

net <- c(30, 40, -15, -55)
result <- net_matrix_ipf(net_tot = net, m = m)
result$n |>
  addmargins() |>
  round(2)
#>      dest
#> orig       A      B      C     D    Sum
#>   A     0.00  80.58  26.05 34.78 141.41
#>   B    62.05   0.00  48.48  3.08 113.62
#>   C    69.11  32.48   0.00 22.89 124.48
#>   D    40.25  40.55  34.95  0.00 115.75
#>   Sum 171.41 153.62 109.48 60.75 495.26
sum_region(result$n)
#> # A tibble: 4 × 5
#>   region out_mig in_mig  turn   net
#>   <chr>    <dbl>  <dbl> <dbl> <dbl>
#> 1 A         141.  171.   313.  30.0
#> 2 B         114.  154.   267.  40.0
#> 3 C         124.  109.   234. -15.0
#> 4 D         116.   60.8  177. -55.0