Conditional Maximization Routine for the Indirect Estimation of Origin-Destination Migration Flow Table with Known Margins.

Description

The cm2 function finds the maximum likelihood estimates for parameters in the log-linear model:

\log y_{ij} = \log α_i + \log β_j + \log m_{ij}

as introduced by Willekens (1999). The α_i and β_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

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cm2(rtot = NULL, ctot = NULL, m = matrix(1, length(rtot), length(ctot)), 
    tol = 1e-05, maxit = 500, verbose = TRUE)

Arguments

rtot

Origin (row) totals to constrain indirect estimates to.

ctot

Destination (column) totals to constrain indirect estimates to.

m

Auxiliary matrix. By default set to 1 for all origin-destination combinations.

tol

Tolerance level for parameter estimation.

maxit

Maximum number of iterations for parameter estimation.

verbose

Print the parameter estimates at each iteration. By default TRUE.

Details

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 (rtot) and column (ctot) arguments.

Value

Returns a list object with

N

Origin-Destination matrix of indirect estimates

theta

Collection of parameter estimates

Author(s)

Guy J. Abel

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.

See Also

ipf2

Examples

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## with Willekens (1999) data
dn <- LETTERS[1:2]
y <- cm2(rtot = c(18, 20), ctot = c(16, 22), m = matrix(c(5, 1, 2, 7), ncol = 2, 
        dimnames = list(orig = dn, dest = dn)))
y

## with all elements of offset equal (independence fit)
y <- cm2(rtot = c(18, 20), ctot = c(16, 22))
y

## with bigger matrix
dn <- LETTERS[1:3]
y <- cm2(rtot = c(170, 120, 410), ctot = c(500, 140, 60), 
        m = matrix(c(50, 10, 220, 120, 120, 30, 545, 0, 10), ncol = 3, 
        dimnames = list(orig = dn, dest = dn)))
# display with row and col totals
round(addmargins(y$N))