Iterative Proportional Fitting Routine for the Indirect Estimation of Origin-Destination Migration Flow Table with Known Margins.

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Description

The ipf2 function finds the maximum likelihood estimates for fitted values in the log-linear model:

\log y_{ij} = \log α_{i} + \log β_{j} + \log m_{ij}

where m_{ij} is a set of prior estimates for y_{ij} and itself is no more complex than the one being fitted.

Usage

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ipf2(rtot = NULL, ctot = NULL, m = matrix(1, length(rtot), length(ctot)), tol = 1e-05, 
        maxit = 500, verbose = FALSE)

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

Iterative Proportional Fitting routine set up in a similar manner to Agresti (2002, p.343). This is equivalent to a conditional maximization of the likelihood, as discussed by Willekens (1999), and hence provides identical indirect estimates to those obtained from the cm2 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 and column totals.

If only one of the margins is known, the function can still be run. The indirect estimates will correspond to the log-linear model without the α_{i} term if (rtot = NULL) or without the β_{j} term if (ctot = NULL)

Value

Returns a list object with

mu

Origin-Destination matrix of indirect estimates

it

Iteration count

tol

Tolerance level at final iteration

Author(s)

Guy J. Abel

References

Agresti, A. (2002). Categorical Data Analysis 2nd edition. Wiley.

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

cm2, ipf3

Examples

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## with Willekens (1999) data
dn <- LETTERS[1:2]
y <- ipf2(rtot = c(18, 20), ctot = c(16, 22), m = matrix(c(5, 1, 2, 7), ncol = 2, 
        dimnames = list(orig = dn, dest = dn)))
round(addmargins(y$mu),2)

## with all elements of offset equal
y <- ipf2(rtot = c(18, 20), ctot = c(16, 22))
round(addmargins(y$mu),2)

## with bigger matrix
dn <- LETTERS[1:3]
y <- ipf2(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$mu))

## only one margin known
dn <- LETTERS[1:2]
y <- ipf2(rtot = c(18, 20), ctot = NULL, m = matrix(c(5, 1, 2, 7), ncol = 2, 
        dimnames = list(orig = dn, dest = dn)))
round(addmargins(y$mu))