This paper proposes a new linear combination model to predict the closing prices on multivariate financial data sets. The new approach integrates two delays of deep learning methods called the two-delay combination model. The forecasts are derived from three different deep learning models: the multilayer perceptron (MLP), the convolutional neural network (CNN) and the long short-term memory (LSTM) network. Moreover, the weight combination of our proposed model is estimated using the differential evolution (DE) algorithm. The proposed model is built and tested for three high-frequency stock data in financial markets—Microsoft Corporation (MSFT), Johnson & Johnson (JNJ) and Pfizer Inc. (PFE). The individual and combination forecast methods are compared using the root mean square error (RMSE) and the mean absolute percentage error (MAPE). The state-of-the-art combination models used in this paper are the equal weight (EW), the inverse of RMSE (INV-RMSE) and the variance-no-covariance (VAR-NO-CORR) methods. These comparisons demonstrate that our proposed approach using DE weight’s optimization has significantly lower forecast errors than the individual model and the state-of-the-art weight combination procedures for all experiments. Consequently, combining two delay deep learning models using differential evolution weights can effectively improve the stock price prediction.

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The classical autoregressive (AR) model has been widely applied to predict future data using m past observations over five decades. As the classical AR model required m unknown parameters, this paper implements the AR model by reducing m parameters to two parameters to obtain a new model with an optimal delay called as the m-delay AR model. We derive the m-delay AR formula for approximating two unknown parameters based on the least squares method and develop an algorithm to determine optimal delay based on a brute-force technique. The performance of the m-delay AR model was tested by comparing with the classical AR model. The results, obtained from Monte Carlo simulation using the monthly mean minimum temperature in Perth Western Australia from the Bureau of Meteorology, are no significant difference compared to those obtained from the classical AR model. This confirms that the m-delay AR model is an effective model for time series analysis.

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