Constrained additive ordination

For several decades now, ecologists have sought to determine the shape of species' response curves and how they are distributed along unknown underlying gradients, environmental latent variables, or ordination axes. Its determination has important implications for both continuum theory and community analysis because many theories and models in community ecology assume that responses are symmetric and unimodal. This article proposes a major new technique called constrained additive ordination (CAO) that solves this problem by computing the optimal gradients and flexible response curves. It allows ecologists to see the response curves as they really are, against the dominant gradients. With one gradient, CAO is a generalization of constrained quadratic ordination (CQO; formerly called canonical Gaussian ordination or CGO). It supplants symmetric bell-shaped response curves in CQO with completely flexible smooth curves. The curves are estimated using smoothers such as the smoothing spline. Loosely speaking, CAO models are generalized additive models (GAMs) fitted to a very small number of latent variables. Being data driven rather than model driven, CAO allows the data to "speak for itself" and does not make any of the assumptions made by canonical correspondence analysis. The new methodology is illustrated with a hunting spider data set and a New Zealand tree species data set.