The data sets used in the present study have been published as auxiliary materials to R11 and are available at ftp:/ftp.agu.org/apend/ja/2010ja015735. A comprehensive description of the data set preparation methodology is also given in the same paper and will not be repeated here. As mentioned above, this data set covers the time period from 22 September 1989 to 31 December 2009 i.e. 7405 days. However only 7187 daily averaged values of the high energy electron fluxes are available due data gaps. In the present study energy fluxes in the range 1.8–3.5 MeV are used as the system output and is the same range as used in R11. The solar wind parameters were obtained from the OMNI data sets. The initial sets of solar wind parameters included 1 day averages of the magnetic field components Bx, By, and Bz, the solar wind velocity v, the density n, and their combination in the form of the dynamic pressure p. The complete set of inputs included values for current day of all the above parameters together with their values for the previous four days.
 ERR constitutes the initial part of NARMAX (Nonlinear Autoregressive Moving Average modelling) algorithm, presently the most powerful methodology used for data based approaches to complex nonlinear systems [Billings et al., 1989; Boaghe et al., 2001; Balikhin et al., 2001]. The most important feature of NARMAX in comparison to purely forecasting techniques such as Neural Networks is that NARMAX provides physically interpretable results that can be directly compared with and which can be used to validate and enhance analytical models. The NARMAX approach enables the identification, from input-output data sets, of mathematical relations that describe the evolution of complex, nonlinear, dynamical systems for which analytical models derived from the first principles would be incomplete due to assumptions and omissions made in their derivation. The NARMAX approach has 3 stages: Structure detection, Estimation of parameters, and Model validation [Billings et al., 1989]. In the initial analysis stage the mathematical structure of the system model is identified from input output data. The ERR concept [Billings et al., 1989] plays a key role in structure identification. The ERR quantifies the contribution of model terms (inputs) to the evolution of the nonlinear dynamical system. Detailed descriptions and the rigorous mathematical foundations of NARMAX and ERR are given in original NARMAX papers [see, e.g., Billings et al., 1989] and are beyond the scope the present letter. Here we present a brief, very simplified explanation (following Balikhin et al. ) of the method for the readers convenience. The cornerstone of NARMAX is a very general assumption that the output of the system O(ti) can be expressed as some function of all previous values of inputs I(ti), outputs and some error e(ti): O(ti) = Ft[I(ti), I(ti−1), I(ti−2), …, O(ti−1), O(ti−2), ..e(ti), e(ti−1)…]. The error function e(ti) accounts not only for measurements errors, but for the effects of unknown inputs as well. Instead of the direct identification of explicit function F[…], the NARMAX approach is based on the search of the decomposition of this function F[…] in any orthogonal basis. In previous studies polynomials, RBF wavelets, and many other functions that constitute an orthogonal basis have been used. The rigorous mathematical derivation of NARMAX takes into account both measurement noise and error resulting from incomplete knowledge of all inputs [Billings et al., 1989]. The current illustrative review of NARMAX given here is limited to the polynomial basis and does not account for these errors. In this oversimplified case the NARMAX approach can be formulated as a technique to find coefficients sk in the decomposition of an unknown function F expressed using a polynomial basis qk: O(ti) = Σk sk qk, where the sum Σk sk qk is a representation of the unknown function F, and qk are the monomials with respect to values of inputs I and previous values of the output O [Balikhin et al., 2010]. As the first step of NARMAX, orthogonalisation of the basis qk is performed resulting in the new basis functions wk such that 〈wlwj〉 = Σti wl(ti)wj(ti) = 0 if l ≠ j. In the new orthogonal basis F = Σk gkwk, and the coefficients gk can be estimated separately one by one, as gk = .
 In order to implement the ERR approach, a set of 8 subintervals that possess at least 250 continuous data points have been extracted from the initial data set that contains data gaps. By using data gaps to break up the original data set into 8 subintervals there is no need to interpolate across data gaps [Qin et al., 2007] which may affect the results of the analysis. For each subinterval the ERR have been calculated for all possible second order monomials of the initial set of inputs (Bx, By Bz, v, n, p). Only monomials that are composed of simultaneous data have been considered. This means that for example the product of the density measured on the current day i and velocity measured on a previous day i − 1 n(i)v(i − 1) have been excluded. For each subinterval the sum of all possible monomials ERRsum have been found and the ERR for a particular monomial mp have been expressed as a percentage of ERRsum for this interval. The values of ERR were then averaged over the 8 different intervals.