Robust Long Term Forecasting of Seasonal Time Series J¨ org Wichard 1 and Christian Merkwirth 2 1 Institute of Molecular Pharmacology, Robert-R¨ ossle-Str. 10, D-13125 Berlin, Germany JoergWichard@web.de, 2 Applied Computer Science Department, Jagiellonian University Reymonta 4, PL 30-059 Krak ´ ow, Poland ChristianMerkwirth@web.de Abstract. We propose the usage of a simple difference equation for predicting seasonal, trended time series with clear periodicity. By computing several fore- casts for different settings of the method’s single free parameter we obtain an en- semble of forecasts. These ensemble is combined to the final forecast by taking the samplewise median of those forecasts that were generated by models showing low prediction errors on left-out parts of the time-series. We show the application of this approach to the Mauna Loa atmospheric carbon dioxide concentration (ACDC) time series. 1 Introduction Time series forecasting is a growing field of interest with applications in nearly any field of science. Since ancient times the prediction of the positions of sun, moon and the be- ginning of the seasons were of great importance for the early civilizations. The attempts to build proper forecasting models in Astronomy gave birth to the modern science when Newton developed his celestial mechanics and the Calculus. Nowadays, there is a huge interest to build long term market models in economics or climatic models that can give indication of global changes. In many cases we have only the measurement of one system variable over a longer pe- riod, but no other quantitative information of variables that may influence the system of interest. In these cases we can only use the time series itself to build a predictive model. This approach was introduced by Yule, who described a linear (autoregressive) model based on a measured time series in order to predict the sunspot cycle [1]. To- day the stochastic autoregressive process modelling performed by Yule belongs to the classics of linear time series analysis that is embedded in the theory of linear stochas- tic processes. In general the conventional linear methods for modeling and prediction fail if they are applied to time series originating from nonlinear systems. This seems to be evident because a nonlinear (deterministic) system should not be treated as a linear stochastic process. Since the discovery of deterministic chaos, many methods for nonlinear time series modeling and prediction have been suggested and refined. A common characteristic of these models is the reconstruction of the systems state space based on the embed- ding theorems given by Takens [2], Sauer et al. [3] and Stark [4]. From a equally sam- pled time series {y t } t=1,...,N we can construct the d-dimensional state space vector