Revisiting fitting monotone polynomials to data

We revisit Hawkins' (Comput Stat 9(3):233-247, 1994) algorithm for fitting monotonic polynomials and discuss some practical issues that we encountered using this algorithm, for example when fitting high degree polynomials or situations with a sparse design matrix but multiple observations per x-value. As an alternative, we describe a new approach to fitting monotone polynomials to data, based on different characterisations of monotone polynomials and using a Levenberg-Marquardt type algorithm. We consider different parameterisations, examine effective starting values for the non-linear algorithms, and discuss some limitations. We illustrate our methodology with examples of simulated and real world data. All algorithms discussed in this paper are available in the R Development Core Team (A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, 2011) package MonoPoly.