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Recurrent neural networks are dynamical systems that provide for memory capabilities to recall past behaviour, which is necessary in the prediction of time series. In this paper, a novel neural network architecture inspired by the immune algorithm is presented and used in the forecasting of naturally occurring signals, including weather big data signals. Big Data Analysis is a major research frontier, which attracts extensive attention from academia, industry and government, particularly in the context of handling issues related to complex dynamics due to changing weather conditions. Recently, extensive deployment of IoT, sensors, and ambient intelligence systems led to an exponential growth of data in the climate domain. In this study, we concentrate on the analysis of big weather data by using the Dynamic Self Organized Neural Network Inspired by the Immune Algorithm. The learning strategy of the network focuses on the local properties of the signal using a self-organised hidden layer inspired by the immune algorithm, while the recurrent links of the network aim at recalling previously observed signal patterns. The proposed network exhibits improved performance when compared to the feedforward multilayer neural network and state-of-the-art recurrent networks, e.g., the Elman and the Jordan networks. Three non-linear and non-stationary weather signals are used in our experiments. Firstly, the signals are transformed into stationary, followed by 5-steps ahead prediction. Improvements in the prediction results are observed with respect to the mean value of the error (RMS) and the signal to noise ratio (SNR), however to the expense of additional computational complexity, due to presence of recurrent links.