Group Models of Artificial Polynomial and Trigonometric Higher Order Neural Networks

HONN Models for Simulation

Artificial Higher Order Neural Network (HONN) is powerful technique to simulate data. Ghosh and Shin (1992) develop efficient higher-order neural networks for function approximation and classification. Zhang, Murugesan, and Sadeghi (1995) design polynomial higher order neural network for economic data simulation. Zhang, Fulcher, and Scofield (1996) study neural network group models for estimating rainfall from satellite images. Zhang, Zhang, and Fulcher (1997) explore financial simulation system using a higher order trigonometric polynomial neural network group model. Lu, Qi, Zhang, and Scofield (2000) provide PT-HONN models for multi-polynomial function simulation. Zhang and Lu (2001) research financial data simulation by using Multi-Polynomial Higher Order Neural Network (M-PHONN) model. Zhang (2001) deliver a new model, called Adaptive Multi-Polynomial Higher Order Neural Network (A-PHONN), for financial data simulation. Qi and Zhang (2001) test the rainfall estimation using M-PHONN model and find that the M-PHONN model for estimating heavy convective rainfall from satellite data has 5% to 15% more accuracy than the polynomial higher order neural network. Zhang and Scofield (2001) expand an adaptive multi-polynomial high order neural network (A-PHONN) model for heavy convective rainfall estimation and find that the A-PHONN model has 6% to 16% more accuracy than the polynomial higher order neural network. Zhang (2003) intends PL-HONN model for financial data simulation. Rovithakis, Chalkiadakis, and Zervakis (2004) learn the high-order neural network structure selection for function approximation applications using genetic algorithms. Crane and Zhang (2005) discover the data simulation using SINCHONN Model. Zhang (2005) gives a data simulation system using SINX/X and SINX polynomial higher order neural networks. Zhang (2006) seeks a data simulation system using CSINC polynomial higher order neural networks. Zhang (2009a) delivers general format of Higher Order Neural Networks (HONNs) for nonlinear data analysis and six different HONN models. This paper mathematically proves that HONN models could converge and have mean squared errors close to zero. This paper illustrates the learning algorithm with update formulas. HONN models are compared with SAS Nonlinear (NLIN) models and results show that HONN models are 3 to 12% better than SAS Nonlinear models. Zhang (2009b) develops a new nonlinear model, Ultra high frequency Trigonometric Higher Order Neural Networks (UTHONN), for time series data analysis. Results show that UTHONN models are 3 to 12% better than Equilibrium Real Exchange Rates (ERER) model, and 4 – 9% better than other Polynomial Higher Order Neural Network (PHONN) and Trigonometric Higher Order Neural Network (THONN) models. This study also uses UTHONN models to simulate foreign exchange rates and consumer price index with error approaching 0.0000%. Murata (2010) finds that a Pi-Sigma higher order neural network (Pi-Sigma HONN) is a type of higher order neural network, where, as its name implies, weighted sums of inputs are calculated first and then the sums are multiplied by each other to produce higher order terms that constitute the network outputs. This type of higher order neural networks has accurate function approximation capabilities.