Applied Soft Computing 65 (2018) 47–57 Contents lists available at ScienceDirect Applied Soft Computing j ourna l h o mepage: www.elsevier.com/locate/asoc Bernstein polynomials for adaptive evolutionary prediction of short-term time series Kristina Lukoseviciute a , Rita Baubliene a , Daniel Howard b , Minvydas Ragulskis a,∗ a Research Group for Mathematical and Numerical Analysis of Dynamical Systems, Kaunas University of Technology, Studentu 50–147, Kaunas LT–51368, Lithuania b Howard Science Limited, Malvern, UK a r t i c l e i n f o Article history: Received 5 March 2017 Received in revised form 27 December 2017 Accepted 2 January 2018 Available online 7 January 2018 Keywords: Bernstein polynomial Time series prediction Evolutionary algorithms a b s t r a c t We introduce a short-term time series prediction model by means of evolutionary algorithms and Bern- stein polynomials. This adapts Bernstein-type algebraic skeletons to extrapolate and predict short time series. A mixed smoothing strategy is used to achieve the necessary balance between the roughness of the algebraic prediction and the smoothness of the moving average. Computational experiments with standardized real world time series illustrate the accuracy of this approach to short-term prediction. © 2018 Elsevier B.V. All rights reserved. 1. Introduction Forecasting is a modelling challenge that relies on a time series analysis. Its aim is to identify a model in time-stamped data pre- sumably generated by some process. Extrapolating by means of this model it makes reliable predictions for unseen data. Recent decades have delivered various models and techniques that are suited to long-term or short-term time series forecasting [1]. Unfortunately, the sheer amount of data needed for training, validating and test- ing mostly renders long-term time series analysis implausible. Yet, a one-step forward future horizon is adequate for short-term time series forecasting [1] delivering methods which are widely used in high frequency time series analysis with intra-daily data values [2]. Short-term time series predictors are used in finance [3–5]; electricity demand and the associated price forecasting problem [6–8]; wind power; passenger demand [9] and many other indus- trial applications. Time series forecasting techniques can be coarsely grouped into classical linear modelling, such as simple exponential smoothing [10], Holt-Winte’s methods [11] or Autoregressive Integrated Mov- ing Average (ARIMA) [12], and modern non-linear modelling that is based on soft computing. The latter includes regime-switching models comprising a wide variety of threshold autoregressive mod- ∗ Corresponding author. E-mail addresses: kristina.lukoseviciute@ktu.lt (K. Lukoseviciute), rita.palivonaite@ktu.lt (R. Baubliene), dr.daniel.howard@gmail.com (D. Howard), minvydas.ragulskis@ktu.lt (M. Ragulskis). els [13–15]: self exciting models [15–17], smooth transition models [18] and continuous-time models [19,20]. Hybrid forecasting meth- ods combine regression, data smoothing, and other techniques to produce forecasts that make up for the comparative deficiencies of individual methods. A large number of linear and non-linear methods of forecast- ing appear in the literature, with some methods claiming to do a better job than others under competing assumptions, for example: when given only a short series of input data, or if applied to long- term forecasting [1]. The literature [3–23] covers a wide-variety of techniques that include various flavours of signal processing, sup- port vector machines, ARIMA, Artificial Neural Network (ANN) and Evolutionary Algorithms (EA). The reader may wonder why there is a continued and strong interest in a plethora of algorithms. The no-free-lunch theorems [24,25] lead to the conclusion that a problem can always be found to defeat any algorithm. Indeed, practical interest in the develop- ment of new and hybrid algorithms is warranted because of this reality that no single method will outperform all others in every single situation. At the same time, as Stafford Beer once observed [26], problems of practical interest cannot take an algorithm com- pletely by surprise because the regularities that they comprise are of this world. Real-world problems have neither been designed nor contrived to defeat a popular algorithm. A taxonomy of prac- tical problems, therefore, exists, and it motivates the search for improved algorithms that suit different classes of problems. In our earlier work [27–29], special EA schemes for the identifi- cation of near-optimal algebraic skeleton sequences based on Prony interpolants (represented as linear recurrence sequences (LRS) in https://doi.org/10.1016/j.asoc.2018.01.002 1568-4946/© 2018 Elsevier B.V. All rights reserved.