Properties of Kruithof's Projection Method

J. Kruithofs method1 for projecting from measured point-to-point teletraffic data qtJ to some future values ptJ is based upon estimates of total originating and terminating traffic only. While the original publication (in Flemish) did not receive as much attention as it may deserve, the idea was good enough that it has been independently reinvented numerous times in the intervening years. Related techniques have turned up in economics, statistics, biophysics, pattern recognition, and vehicular traffic studies, for instance. Such repetition largely seems due to a scientific "Babel" effect: workers in different technical disciplines can no longer read each other's work and recognize the same problem in a new context. While Kruithof showed that his method had certain properties which are clearly desirable in a projection scheme, he did not investigate the underlying mathematical problems. Subsequent workers, such as Bear,2 Kullback,3 Sinkhorn,4'5 Theil,6 and particularly Csiszar,7 have thrown much light on these matters. This paper seeks to give a unified picture of Kruithof's method and its many generalizations, with some practical details and recommendations for implementation. Much of the more intricate mathematics is relegated to an appendix, including proofs of existence, uniqueness, convergence, and continuity. These are cited as needed in the main text, which contains information on 517