Solving for the B-spline weights.

<div><p>The majority of migration moves globally are internal within national borders. This makes internal migration intensities an important component for understanding the dynamics of population change according to size, composition and across geographies. While incorporating migration into demography’s quantitative framework allows a description of population change across both time and space, and mathematical and conceptual frameworks for migration have been developed, researchers lack a public repository of historical age-origin-destination-specific migration probabilities that is in a common format and spans a range of countries. Addressing this requires a robust method for inferring migration probabilities from census and survey data when there are significant levels of uncertainty from small-sample noise and age aggregation. In this paper we extend the P-TOPALS and P-spline methods for smoothing migration probabilities to apply to grouped data by ages to develop a methods protocol for a harmonised, homogeneous format and multi-nation Human Internal Migration Database. We find our method out-performs a hybrid spline-parametric method in terms of both accuracy and plausibility. We illustrate the method by estimating complete age-origin-destination migration probabilities for more than 50 countries using microdata samples from IPUMS International. This work advances the stock of migration data from which demographers and others can draw from in the analysis and projection of population change.</p></div>