Trip-distribution calculations and sampling error: some theoretical aspects

Part 1 of this paper discusses the treatment of zero observations in growth-factor methods; part 2 discusses the calibration of gravity models when the data are of different sampling variability. The problem discussed in part 1 is that zero observations are preserved at zero value in the forecast year, no matter how much growth takes place. Since the probability of making a trip is not zero but very small for those matrix cells that are empty by chance, it is shown how to estimate nonzero values for these cells by use of a method of smoothing reported in the statistical literature. It is argued in part 2 that the usual practice of calibrating against `grossed-up' data is incorrect. If the survey method is the same for all trips, but the sampling fraction varies from one cell to another, it is shown, by use of maximum-likelihood methods, that it is the raw sample data against which the model should be calibrated. If the matrix is made up of data from different surveys, it is shown, by use of least-squares methods, that the raw sample data should be modified before estimating the parametes, leading to a minimum chi 2 fitting criterion.