Entropy multiproportional and quadratic techniques for inferring patterns of migration from aggregate data

Data on migration flows are not often disaggregated by regions of origin and destination for subgroups of the population. These detailed flows must be inferred from aggregate information. The 2 methods presented for deducing these estimates are conceptualized as mathematical optimization problems with nonlinear objective functions and linear constraints. The individual techniques differ in the objective functions used. Bi- and multiproportional adjustment methods integrate information divergence minimizing problems and entropy maximization (a special formulation of the information divergence method). In the entropy problem no initial values of the elements to be estimated are available and they are uniformly set to equal unity. A single solution algorithm is developed for both problems. The entropy method is suited for estimating detailed migration flows on the basis of aggregate data only; the information divergent method is useful for upgrading migration tables. The 2nd method is quadratic adjustment where weighted squared deviations between estimates and initial guesses are minimized. The guesses may be derived from outdated migration tables or may represent "a priori" information on the migration pattern. A modified Friedlander method is presented which yields estimates of the appropriate sign. Proofs of the 5 theorems and solutions for 3 special entropy cases are provided. Validity of the estimation methods is measured using chi-square and absolute percentage error techniques examining the accuracy of the estimates when compared to observed data. Illustrations of the procedures using migration data from Austria and Sweden uncovered an interesting observation. A certain amount of data on the age composition of migrants is necessary to yield good estimates of detailed flow--additional initial information on age structure does not add significantly to the quality of the estimates. Additional research will increase the utility of estimation methodology. For example determination of the amount of initial information needed strategies for improvement of initial guesses and the potential of contingency table analysis are being explored for their contribution toward improved estimation. Further research of the methodologys application to updating migration tables would be valuable. Improvement of validity measures that are less susceptible to smaller flows is also needed. Application of the estimation methodology is not limited to migration data by can be applied to all cross-classified data or multidimensional contingency tables.