Conditional maximization routine for the indirect estimation of origin-destination migration flow table with known margins

The cm2 function finds the maximum likelihood estimates for parameters in the log-linear model: $$ \log y_{ij} = \log \alpha_i + \log \beta_j + \log m_{ij} $$ as introduced by Willekens (1999). The $\alpha_i$ and $\beta_j$ represent background information related to the characteristics of the origin and destinations respectively. The $m_{ij}$ factor represents auxiliary information on migration flows, which imposes its interaction structure onto the estimated flow matrix.

Usage

cm2(
  row_tot = NULL,
  col_tot = NULL,
  m = matrix(data = 1, nrow = length(row_tot), ncol = length(col_tot)),
  tol = 1e-06,
  maxit = 500,
  verbose = TRUE,
  rtot = row_tot,
  ctot = col_tot
)

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: Matrix of auxiliary data. By default set to 1 for all origin-destination 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.
rtot: Depreciated. Use row_tot
ctot: Depreciated. Use col_tot

Value

Parameter estimates are obtained using the EM algorithm outlined in Willekens (1999). This is equivalent to a conditional maximization of the likelihood, as discussed by Raymer et. al. (2007). It also provides identical indirect estimates to those obtained from the ipf2 routine.

The user must ensure that the row and column totals are equal in sum. Care must also be taken to allow the dimension of the auxiliary matrix (m) to equal those provided in the row (row_tot) and column (col_tot) arguments.

Returns a list object with

N: Origin-Destination matrix of indirect estimates
theta: Collection of parameter estimates

References

Raymer, J., G. J. Abel, and P. W. F. Smith (2007). Combining census and registration data to estimate detailed elderly migration flows in England and Wales. Journal of the Royal Statistical Society: Series A (Statistics in Society) 170 (4), 891--908.

Willekens, F. (1999). Modelling Approaches to the Indirect Estimation of Migration Flows: From Entropy to EM. Mathematical Population Studies 7 (3), 239--78.

Author

Guy J. Abel

Examples

## with Willekens (1999) data
r <- LETTERS[1:2]
y <- cm2(row_tot = c(18, 20), col_tot = c(16, 22), 
         m = matrix(c(5, 1, 2, 7), ncol = 2, dimnames = list(orig = r, dest = r)))
#> iteration: 0 
#> alpha parameters: 
#> beta parameters: 1 1 
#> 
#> iteration: 1 
#> alpha parameters: 2.571429 2.5 
#> beta parameters: 1.04186 0.9716088 
#> max difference: 1.571429 
#> 
#> iteration: 2 
#> alpha parameters: 2.516596 2.550005 
#> beta parameters: 1.057293 0.9614029 
#> max difference: 0.05483302 
#> 
#> iteration: 3 
#> alpha parameters: 2.496785 2.568346 
#> beta parameters: 1.062963 0.9576881 
#> max difference: 0.01981086 
#> 
#> iteration: 4 
#> alpha parameters: 2.489561 2.57507 
#> beta parameters: 1.065042 0.95633 
#> max difference: 0.00722337 
#> 
#> iteration: 5 
#> alpha parameters: 2.486919 2.577535 
#> beta parameters: 1.065805 0.9558326 
#> max difference: 0.00264242 
#> 
#> iteration: 6 
#> alpha parameters: 2.485951 2.578438 
#> beta parameters: 1.066084 0.9556504 
#> max difference: 0.000967792 
#> 
#> iteration: 7 
#> alpha parameters: 2.485597 2.578769 
#> beta parameters: 1.066187 0.9555837 
#> max difference: 0.0003546105 
#> 
#> iteration: 8 
#> alpha parameters: 2.485467 2.578891 
#> beta parameters: 1.066224 0.9555592 
#> max difference: 0.0001299542 
#> 
#> iteration: 9 
#> alpha parameters: 2.485419 2.578935 
#> beta parameters: 1.066238 0.9555502 
#> max difference: 4.762717e-05 
#> 
#> iteration: 10 
#> alpha parameters: 2.485401 2.578951 
#> beta parameters: 1.066243 0.9555469 
#> max difference: 1.745534e-05 
#> 
#> iteration: 11 
#> alpha parameters: 2.485395 2.578957 
#> beta parameters: 1.066245 0.9555457 
#> max difference: 6.397427e-06 
#> 
#> iteration: 12 
#> alpha parameters: 2.485393 2.57896 
#> beta parameters: 1.066246 0.9555453 
#> max difference: 2.34468e-06 
#> 
#> iteration: 13 
#> alpha parameters: 2.485392 2.57896 
#> beta parameters: 1.066246 0.9555451 
#> max difference: 8.593346e-07 
#> 
y
#> $n
#>     dest
#> orig         A         B
#>    A 13.250194  4.749808
#>    B  2.749806 17.250192
#> 
#> $theta
#>    alpha1    alpha2     beta1     beta2 
#> 2.4853919 2.5789604 1.0662459 0.9555451 
#> 

## with all elements of offset equal (independence fit)
y <- cm2(row_tot = c(18, 20), col_tot = c(16, 22))
#> iteration: 0 
#> alpha parameters: 
#> beta parameters: 1 1 
#> 
#> iteration: 1 
#> alpha parameters: 9 10 
#> beta parameters: 0.8421053 1.157895 
#> max difference: 9 
#> 
#> iteration: 2 
#> alpha parameters: 9 10 
#> beta parameters: 0.8421053 1.157895 
#> max difference: 0 
#> 
y
#> $n
#>          [,1]     [,2]
#> [1,] 7.578947 10.42105
#> [2,] 8.421053 11.57895
#> 
#> $theta
#>     alpha1     alpha2      beta1      beta2 
#>  9.0000000 10.0000000  0.8421053  1.1578947 
#> 

## with bigger matrix
r <- LETTERS[1:4]
y <- cm2(row_tot = c(250, 100, 140, 110), col_tot = c(150, 150, 180, 120),
         m = matrix(data = c(0, 100, 30, 70, 50, 0, 45, 5, 60, 35, 0, 40, 20, 25, 20, 0),
                    nrow = 4, ncol = 4, dimnames = list(orig = r, dest = r), byrow = TRUE))
#> iteration: 0 
#> alpha parameters: 
#> beta parameters: 1 1 1 1 
#> 
#> iteration: 1 
#> alpha parameters: 1.25 1 1.037037 1.692308 
#> beta parameters: 1.026916 0.7367243 1.547107 0.8956462 
#> max difference: 0.6923077 
#> 
#> iteration: 2 
#> alpha parameters: 1.367758 0.7971692 1.136122 1.573709 
#> beta parameters: 1.075269 0.6948215 1.66083 0.8265955 
#> max difference: 0.2028308 
#> 
#> iteration: 3 
#> alpha parameters: 1.411084 0.7539557 1.148494 1.525817 
#> beta parameters: 1.093902 0.6835234 1.685758 0.80816 
#> max difference: 0.04789161 
#> 
#> iteration: 4 
#> alpha parameters: 1.424531 0.7429695 1.148635 1.513457 
#> beta parameters: 1.100226 0.6802975 1.691114 0.8033342 
#> max difference: 0.01344736 
#> 
#> iteration: 5 
#> alpha parameters: 1.428599 0.7400386 1.147943 1.510279 
#> beta parameters: 1.10226 0.6793635 1.692281 0.8020325 
#> max difference: 0.004067991 
#> 
#> iteration: 6 
#> alpha parameters: 1.429821 0.7392305 1.147592 1.509436 
#> beta parameters: 1.102895 0.6790904 1.692545 0.8016713 
#> max difference: 0.001221511 
#> 
#> iteration: 7 
#> alpha parameters: 1.430186 0.7390021 1.147459 1.509205 
#> beta parameters: 1.103089 0.67901 1.692608 0.8015688 
#> max difference: 0.0003654829 
#> 
#> iteration: 8 
#> alpha parameters: 1.430296 0.7389364 1.147414 1.50914 
#> beta parameters: 1.103148 0.6789863 1.692623 0.8015392 
#> max difference: 0.0001091228 
#> 
#> iteration: 9 
#> alpha parameters: 1.430328 0.7389173 1.1474 1.509122 
#> beta parameters: 1.103166 0.6789792 1.692627 0.8015306 
#> max difference: 3.253749e-05 
#> 
#> iteration: 10 
#> alpha parameters: 1.430338 0.7389117 1.147396 1.509116 
#> beta parameters: 1.103171 0.6789772 1.692628 0.8015281 
#> max difference: 9.693573e-06 
#> 
#> iteration: 11 
#> alpha parameters: 1.430341 0.73891 1.147394 1.509115 
#> beta parameters: 1.103173 0.6789765 1.692629 0.8015273 
#> max difference: 2.886352e-06 
#> 
#> iteration: 12 
#> alpha parameters: 1.430342 0.7389095 1.147394 1.509114 
#> beta parameters: 1.103173 0.6789764 1.692629 0.8015271 
#> max difference: 8.591439e-07 
#> 
                    
# display with row and col totals
round(addmargins(y$n)) 
#>      dest
#> orig    A   B   C   D Sum
#>   A     0  97  73  80 250
#>   B    41   0  56   3 100
#>   C    76  27   0  37 140
#>   D    33  26  51   0 110
#>   Sum 150 150 180 120 600