The combination of forecasts is a well established procedure for improving forecast performance and decreasing the risk of selecting an inferior model out of an existing pool of models. Work in this area mainly focuses on combining several functionally different models, but some publications also deal with combining forecasts with the same functional approach. In the latter case, individual forecasts are generated by diversifying one or more model parameters or, if dealing with hierarchical data, by using forecasts from different levels. This work looks at multi-dimensional data from airline industry, with the aim of improving the forecast of cancellation rates for bookings. Three different methods are employed for the generation of individual forecasts. Forecast combinations are usually implemented in a more or less static structure, either including all available forecasts or trimming a fixed percentage of the worst performing models. For a big number of individual forecasts, this procedure can become inefficient. In this paper, a dynamic approach of pooling and trimming is applied to the generated forecasts for airline cancellation data.

In this paper, we provide a theoretical analysis of effects of applying different forecast diversification methods on the structure of the forecast error covariance matrices and decomposed forecast error components based on the bias-variance-Bayes error decomposition of James and Hastie. We express the "diversityrdquo of different forecasts in relation to different error components and propose a measure in order to quantify it. We illustrate and discuss typical inhomogeneities frequently occurring in the forecast error covariance matrices and show that previously proposed pooling based only on error variances cannot fully exploit the complementary information present in a set of diverse forecasts to be combined. If covariance values could be reliably calculated, they could be taken into account during the pooling process. We study the difficult case in which covariance information cannot be measured properly and propose a novel simplified representation of the covariance matrix, which is only based on knowledge about the forecast generation process. Finally, we propose a new pooling approach that avoids inhomogeneities in the forecast error covariance matrix by considering the information contained in the simplified covariance representation and compare it with the error-variance-based pooling approach introduced by Aiolfi and Timmermann. Applying our approach more than once leads to the generation of multistep and multilevel forecast combination structures, which have generated significantly improved forecasts in our previous extensive experimental work; the summary of which is also provided.

In this paper we provide experimental results and extensions to our previous theoretical findings concerning the combination of forecasts that have been diversified by three different methods: with parameters learned at different data aggregation levels, by thick modeling and by the use of different forecasting methods. An approach of error variance based pooling as proposed by Aiolfi and Timmermann has been compared with flat combinations as well as an alternative pooling approach in which we consider information about the used diversification. An advantage of our approach is that it leads to the generation of novel multi step multi level forecast generation structures that carry out the combination in different steps of pooling corresponding to the different types of diversification. We describe different evolutionary approaches in order to evolve the order of pooling of the diversification dimensions. Extensions of such evolutions allow the generation of more flexible multi level multi step combination structures containing better adaptive capabilities. We could prove a significant error reduction comparing results of our generated combination structures with results generated with the algorithm of Aiolfi and Timmermann as well as with flat combination for the application of Revenue Management seasonal forecasting.