Data with an array structure are common in statistics (mortality tables and spatio-temporal data are two important examples). Such data often require smoothing to remove noise and estimate trend. One natural and attractive approach is to use penalized regression where (a) the basis for the regression is a Kronecker product of B-splines and (b) the penalty is a roughness penalty on regression coefficients; this is the P-spline approach of Eilers & Marx. However, such an approach is particularly susceptible to runaway problems with (a) storage and (b) computational time. Generalized linear array models (GLAM) were developed precisely to address both these issues. In a conventional GLM you store the model matrix and then fit the model. Unfortunately, with large amounts of data this model matrix can get rather large: computation and even storage can be a problem. In GLAM the model matrix is not stored; the GLAM algorithm works sequentially with the factors of the Kronecker product. Further, the GLAM algorithm is very fast and can be orders of magnitude quicker than the usual GLM approach in a large problem. In this paper we first describe the GLAM algorithms and then give an introduction to a range of applications. These applications include various models for smoothing and forecasting of mortality tables, density estimation and spatio-temporal smoothing.
Keywords: Generalized linear array models; Kronecker products; P-splines; Smoothing
Biography: Iain Currie is Reader in Statistics in the Department of Actuarial Mathematics & Statistics of Heriot-Watt University in Edinburgh, Scotland. His main research interest is in multi-dimensional smoothing where his joint work with Paul Eilers and Maria Durban has led to the development of Generalized Linear Array Models (GLAM). He has been particularly interested in the application of these methods to the problem of modelling and forecasting of mortality tables.