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Scholars mainly propose and establish theoretical models of cumulative fatigue damage for their research fields. This review aims to select the applicable model from many fatigue damage models according to the actual situation. However, relatively few models can be generally accepted and widely used.

This review introduces the development of cumulative damage theory. Then, several typical models are selected from linear and nonlinear cumulative damage models to perform data analyses and obtain the fatigue life for the metal.

Considering the energy law and strength degradation, the nonlinear fatigue cumulative damage model can better reflect the fatigue damage under constant and multi-stage variable amplitude loading. In the following research, the complex uncertainty of the model in the fatigue damage process can be considered, as well as the combination of advanced machine learning techniques to reduce the prediction error.

This review compares the advantages and disadvantages of various mainstream cumulative damage research methods. It provides a reference for further research into the theories of cumulative fatigue damage.

A large number of data have proved that under the same von Mises equivalent strain condition, the fatigue life under multiaxial non-proportional loading is often much lower than the life under multiaxial proportional loading. This is mainly due to the influence of the non-proportional loading path and the additional hardening effect, which lead to a sharp decrease in life.

The modulus attenuation effect is used to modify the static hardening coefficient, and the predicted value obtained is closer to the additional hardening coefficient obtained from the experiment. A fatigue life model can consider non-proportional paths, and additional hardening effects are proposed. And the model uses multiaxial fatigue test data to verify the validity and adaptability of the new model. The life prediction accuracy and material application range are satisfactory.

Because loading path and additional hardening of the material affect fatigue life, a new multiaxis fatigue life model based on the critical plane approach is proposed. And introducing a non-proportional additional damage coefficient, the joint influence of the load path and the additional hardening can be considered. The model's life prediction accuracy and material applicability were verified with multiaxial fatigue test data of eight materials and nine loads compared with the prediction accuracy of the Kandil–Brown–Miller (KBM) model and Fatemi–Socie (FS) model.

The physical meaning of the new model is clear, convenient for practical engineering applications.

The purpose of this study is to develop a precise Islamic securities index forecasting model using artificial neural networks (ANNs).

The data of daily closing prices of KMI-30 index span from Aug-2009 to Oct-2019. The data of 2,520 observations are divided into training and test data sets by using the 80:20 ratio, which corresponds to 2016 and 504 observations, respectively. In total, 25 features are used; however, in model selection step, based on maximum accuracy, top ten indicators are selected from several iterations of predictive models.

The results of feature selection show that top five influencing indicators on Islamic index include Bollinger Bands, Williams Accumulation Distribution, Aroon Oscillator, Directional Movement and Forecast Oscillator while Mesa Sine Wave is the least important. The findings show that the model captures much of the trend and some of the undulations of the original series.

The findings of this study may have important implications for investment and risk management by using index-based products.

Numerous studies proved that traditional econometric techniques face significant challenges in out-of-sample predictability due to model uncertainty and parameter instability. Recent studies show an upsurge of interest in machine learning algorithms to improve the prediction accuracy.