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What is… Iterative
Proportional Fitting?
Nik Lomax
School of Geography, University of Leeds
British Society for Population Studies Annual Conference, Cardiff
9 September 2019
Iterative Proportional Fitting (IPF) is…
•A technique for reweighting a known multidimensional array (e.g.
cross-tabulated data) to target marginal totals
•Used by demographers, transport planners economists and computer
scientists
•Can be done in a wide range of software, from Excel to bespoke
packages
You might know it by another name
•RAS in economics (see Bacharach 1965)
•Cross–Fratar (Fratar 1954) or Furness (Furness 1965) in transport
engineering
•Raking in in computer science and statistics (Cohen 2008)
•IPF has also been referred to as rim-weighting or structure-
preserving estimation (Simpson and Tranmer 2005).
Simply a way of reweighting a distribution
Lomax, N., Norman, P., Rees, P. et al. (2013) Subnational migration in the United Kingdom: producing a
consistent time series using a combination of available data and estimates, J Pop Research, 30: 265.
https://doi.org/10.1007/s12546-013-9115-z
Background
•First (demographic) use of IPF widely attributed to Deming and
Stephan (1940), who applied the technique to data from the 1940
U.S. census
•Although there were complete counts of the population for certain
characteristics, when these characteristics were cross-tabulated the
output was limited to a sample of the population.
•They used this sample as the starting distribution (the seeds) and
applied IPF to derive an estimate of these cross-tabulated
characteristics for the whole population.
Some examples of IPF in action
•To estimate the characteristics of residents of small geographical
areas Birkin and Clarke (1988)
•To updated the age and sex structure of small area populations in the
UK (Rees 1994)
•To estimate small area population counts of car ownership and tenure
type using 1991 Census data (Simpson and Tranmer, 2005)
•To disaggregate migration data by age and sex by (Willekens, Por, and
Raquillet, 1981; Willekens, 1982).
•To estimate missing cross-border migration data for the United
Kingdom (Lomax et al. 2013)
Example: estimating UK migration
Example: estimating UK migration
Example: estimating UK migration
Extension to multiple dimensions: Age –
Ethnicity - Health
Software solutions
•Modules and user-produced syntax are available for Excel, SAS,
Matlab, Stata, and SPSS.
•I like the mipfp package in R
References
•Bacharach, M. 1965. Estimating nonnegative matrices from marginal data. International Economic Review, 6(3): 294–310
•Birkin, M., and M. Clarke. 1988. SYNTHESIS—A synthetic spatial information system for urban and regional analysis: Methods and
examples.Environment and Planning A20(12): 1645–71
•Cohen, M. 2008. Raking. Encyclopedia of survey researchmethods, ed.P.Lavrakas, 672–74. Thousand Oaks, CA: Sage.
•Fratar, T. J. 1954. Vehicular trip distribution by successive approximations. Traffic Quarterly, 8(1): 53–65.
•Furness, K. P. 1965. Time function iteration. Traffic Engineering and Control, 7(7): 458–60.
•Lomax, N., Norman, P., Rees, P. et al. 2013. Subnational migration in the United Kingdom: producing a consistent time series using
a combination of available data and estimates, J Pop Research, 30: 265. https://doi.org/10.1007/s12546-013-9115-z
•Lomax, N & Norman, P. 2016. Estimating Population Attribute Values in a Table: “Get Me Started in” Iterative Proportional Fitting,
The Professional Geographer, 68:3,451-461, DOI: 10.1080/00330124.2015.1099
•Rees, P. 1994. Estimating and projecting the populations of urban communities. Environment & Planning A, 26:1671–97.
•Simpson, L., and M. Tranmer. 2005. Combining sample and census data in small area estimates: Iterative proportional fitting with
standard software. The Professional Geographer57 (2): 222–34.
•Willekens, F. 1982. Multidimensional population analysis with incomplete data. In Multidimensional mathematical demography,
ed. K. Land and A. Rogers, 43–111. NewYork: Academic.
•Willekens, F., A. Por, and R. Raquillet. 1981. Entropy, multiproportional, and quadratic techniques for inferring patterns of
migration from aggregate data. In: Advances in multiregional demography, ed. A. Rogers, 84–106. Laxenburg, Austria: International
Institute for Applied Systems Analysis.