Iterative proportional fitting routine for the indirect estimation of origin-destination-migrant type migration flow tables with known origin and destination margins.
Source:R/ipf3.R
ipf3.RdThe ipf3 function finds the maximum likelihood estimates for fitted values in the log-linear model:
$$ \log y_{ijk} = \log \alpha_{i} + \log \beta_{j} + \log \lambda_{k} + \log \gamma_{ik} + \log \kappa_{jk} + \log m_{ijk} $$
where \(m_{ijk}\) is a set of prior estimates for \(y_{ijk}\) and is no more complex than the matrices being fitted.
Arguments
- row_tot
Vector of origin totals to constrain the sum of the imputed cell rows.
- col_tot
Vector of destination totals to constrain the sum of the imputed cell columns.
- m
Array of auxiliary data. By default set to 1 for all origin-destination-migrant typologies combinations.
- tol
Numeric value for the tolerance level used in the parameter estimation.
- maxit
Numeric value for the maximum number of iterations used in the parameter estimation.
- verbose
Logical value to indicate the print the parameter estimates at each iteration. By default
FALSE.
Value
Iterative Proportional Fitting routine set up in a similar manner to Agresti (2002, p.343). The arguments row_tot and col_tot take the row-table and column-table specific known margins.
The user must ensure that the row and column totals in each table sum to the same value. Care must also be taken to allow the dimension of the auxiliary matrix (m) to equal those provided in the row and column totals.
Returns a list object with
- mu
Array of indirect estimates of origin-destination matrices by migrant characteristic
- it
Iteration count
- tol
Tolerance level at final iteration
References
Abel and Cohen (2019) Bilateral international migration flow estimates for 200 countries Scientific Data 6 (1), 1-13
Azose & Raftery (2019) Estimation of emigration, return migration, and transit migration between all pairs of countries Proceedings of the National Academy of Sciences 116 (1) 116-122
Abel, G. J. (2013). Estimating Global Migration Flow Tables Using Place of Birth. Demographic Research 28, (18) 505-546
Agresti, A. (2002). Categorical Data Analysis 2nd edition. Wiley.
Examples
## create row-table and column-table specific known margins.
dn <- LETTERS[1:4]
P1 <- matrix(c(1000, 100, 10, 0,
55, 555, 50, 5,
80, 40, 800 , 40,
20, 25, 20, 200),
nrow = 4, ncol = 4, byrow = TRUE,
dimnames = list(pob = dn, por = dn))
P2 <- matrix(c(950, 100, 60, 0,
80, 505, 75, 5,
90, 30, 800, 40,
40, 45, 0, 180),
nrow = 4, ncol = 4, byrow = TRUE,
dimnames = list(pob = dn, por = dn))
# display with row and col totals
addmargins(P1)
#> por
#> pob A B C D Sum
#> A 1000 100 10 0 1110
#> B 55 555 50 5 665
#> C 80 40 800 40 960
#> D 20 25 20 200 265
#> Sum 1155 720 880 245 3000
addmargins(P2)
#> por
#> pob A B C D Sum
#> A 950 100 60 0 1110
#> B 80 505 75 5 665
#> C 90 30 800 40 960
#> D 40 45 0 180 265
#> Sum 1160 680 935 225 3000
# run ipf
y <- ipf3(row_tot = t(P1), col_tot = P2)
#> 1 996
#> 2 1.136868e-13
# display with row, col and table totals
round(addmargins(y$mu), 1)
#> , , pob = A
#>
#> dest
#> orig A B C D Sum
#> A 855.9 90.1 54.1 0 1000
#> B 85.6 9.0 5.4 0 100
#> C 8.6 0.9 0.5 0 10
#> D 0.0 0.0 0.0 0 0
#> Sum 950.0 100.0 60.0 0 1110
#>
#> , , pob = B
#>
#> dest
#> orig A B C D Sum
#> A 6.6 41.8 6.2 0.4 55
#> B 66.8 421.5 62.6 4.2 555
#> C 6.0 38.0 5.6 0.4 50
#> D 0.6 3.8 0.6 0.0 5
#> Sum 80.0 505.0 75.0 5.0 665
#>
#> , , pob = C
#>
#> dest
#> orig A B C D Sum
#> A 7.5 2.5 66.7 3.3 80
#> B 3.8 1.3 33.3 1.7 40
#> C 75.0 25.0 666.7 33.3 800
#> D 3.8 1.3 33.3 1.7 40
#> Sum 90.0 30.0 800.0 40.0 960
#>
#> , , pob = D
#>
#> dest
#> orig A B C D Sum
#> A 3.0 3.4 0 13.6 20
#> B 3.8 4.2 0 17.0 25
#> C 3.0 3.4 0 13.6 20
#> D 30.2 34.0 0 135.8 200
#> Sum 40.0 45.0 0 180.0 265
#>
#> , , pob = Sum
#>
#> dest
#> orig A B C D Sum
#> A 873.0 137.8 126.9 17.3 1155
#> B 159.9 436.0 101.3 22.8 720
#> C 92.6 67.3 672.8 47.3 880
#> D 34.5 39.0 33.9 137.6 245
#> Sum 1160.0 680.0 935.0 225.0 3000
#>
# origin-destination flow table
round(sum_od(y$mu), 1)
#> dest
#> orig A B C D Sum
#> A 0.0 137.8 126.9 17.3 282.0
#> B 159.9 0.0 101.3 22.8 284.0
#> C 92.6 67.3 0.0 47.3 207.2
#> D 34.5 39.0 33.9 0.0 107.4
#> Sum 287.0 244.0 262.2 87.4 880.6
## with alternative offset term
dis <- array(c(1, 2, 3, 4, 2, 1, 5, 6, 3, 4, 1, 7, 4, 6, 7, 1), c(4, 4, 4))
y <- ipf3(row_tot = t(P1), col_tot = P2, m = dis)
#> 1 990
#> 2 267.9774
#> 3 13.70557
#> 4 1.609235
#> 5 0.176649
#> 6 0.01924174
#> 7 0.002094159
#> 8 0.000227895
#> 9 2.480023e-05
#> 10 2.698833e-06
# display with row, col and table totals
round(addmargins(y$mu), 1)
#> , , pob = A
#>
#> dest
#> orig A B C D Sum
#> A 847.7 96.5 55.8 0 1000
#> B 93.3 2.7 4.1 0 100
#> C 9.1 0.9 0.1 0 10
#> D 0.0 0.0 0.0 0 0
#> Sum 950.0 100.0 60.0 0 1110
#>
#> , , pob = B
#>
#> dest
#> orig A B C D Sum
#> A 2.3 49.3 3.2 0.2 55
#> B 74.8 404.4 71.1 4.7 555
#> C 2.6 46.9 0.4 0.1 50
#> D 0.3 4.5 0.2 0.0 5
#> Sum 80.0 505.0 75.0 5.0 665
#>
#> , , pob = C
#>
#> dest
#> orig A B C D Sum
#> A 1.2 0.5 77.5 0.9 80
#> B 0.9 0.1 38.6 0.5 40
#> C 87.0 29.1 645.3 38.6 800
#> D 1.0 0.3 38.7 0.0 40
#> Sum 90.0 30.0 800.0 40.0 960
#>
#> , , pob = D
#>
#> dest
#> orig A B C D Sum
#> A 0.4 0.6 0 19.0 20
#> B 0.7 0.2 0 24.1 25
#> C 0.6 0.8 0 18.5 20
#> D 38.3 43.4 0 118.3 200
#> Sum 40.0 45.0 0 180.0 265
#>
#> , , pob = Sum
#>
#> dest
#> orig A B C D Sum
#> A 851.5 146.8 136.6 20.1 1155
#> B 169.6 407.4 113.8 29.3 720
#> C 99.3 77.7 645.8 57.3 880
#> D 39.6 48.2 38.9 118.4 245
#> Sum 1160.0 680.0 935.0 225.0 3000
#>
# origin-destination flow table
round(sum_od(y$mu), 1)
#> dest
#> orig A B C D Sum
#> A 0.0 146.8 136.6 20.1 303.5
#> B 169.6 0.0 113.8 29.3 312.6
#> C 99.3 77.7 0.0 57.3 234.2
#> D 39.6 48.2 38.9 0.0 126.6
#> Sum 308.5 272.6 289.2 106.6 977.0