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Nonstationary dynamical systems arise in applications, but little has been done in terms of the characterization of such systems, as most standard notions in nonlinear dynamics such as the Lyapunov exponents and fractal dimensions are developed for stationary dynamical systems. We propose a framework to characterize nonstationary dynamical systems. A natural way is to generate and examine ensemble snapshots using a large number of trajectories, which are capable of revealing the underlying fractal properties of the system. By defining the Lyapunov exponents and the fractal dimension based on a proper probability measure from the ensemble snapshots, we show that the Kaplan-Yorke formula, which is fundamental in nonlinear dynamics, remains valid most of the time even for nonstationary dynamical systems.

Multilayer perceptron neural network (MLPNN) is considered as one of the most efficient forecasting techniques which can be implemented for the prediction of weather occurrence. As with any machine learning implementation, the challenge on the utilization of MLPNN in rainfall forecasting lies in the development and evaluation of MLPNN models which delivers optimal forecasting performance. This research conducted performance analysis of MLPNN models through data preparation, model designing, and model evaluation in order to determine which parameters are the best-fit configurations for MLPNN model implementation in rainfall forecasting. During rainfall data preparation, imputation process and spatial correlation evaluation of weather variables from various weather stations showed that the geographical location of the chosen weather stations did not have a direct correlation between stations with respect to rainfall behavior leading to the decision of utilizing the weather station having the most complete weather data to be fed in the MLPNN. By conducting performance analysis of MLPNN models with different combinations of training algorithms, activation functions, learning rate, and momentum, it was found out that MLPNN model having 100 hidden neurons with Scaled Conjugate Gradient training algorithm and Sigmoid activation function delivered the lowest RMSE of 0.031537 while another MLPNN model having the same number of hidden neurons, the same activation function but Resilient Propagation as training algorithm had the lowest MAE of 0.0209. The results of this research showed that performance analysis of MLPNN models is a crucial process in model implementation of MLPNN for week-ahead rainfall forecasting.

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