In the field of mortality, the Lee–Carter based approach can be considered the milestone to forecast mortality rates among stochastic models. We could define a “Lee–Carter model family” that embraces all developments of this model, including its first formulation (1992) that remains the benchmark for comparing the performance of future models. In the Lee–Carter model, the ${\kappa }_t$ parameter, describing the mortality trend over time, plays an important role about the future mortality behavior. The traditional ARIMA process usually used to model ${\kappa }_t$ shows evident limitations to describe the future mortality shape. Concerning forecasting phase, academics should approach a more plausible way in order to think a nonlinear shape of the projected mortality rates. Therefore, we propose an alternative approach the ARIMA processes based on a deep learning technique. More precisely, in order to catch the pattern of ${\kappa }_t$ series over time more accurately, we apply a Recurrent Neural Network with a Long Short-Term Memory architecture and integrate the Lee–Carter model to improve its predictive capacity. The proposed approach provides significant performance in terms of predictive accuracy and also allow for avoiding the time-chunks’ a priori selection. Indeed, it is a common practice among academics to delete the time in which the noise is overflowing or the data quality is insufficient. The strength of the Long Short-Term Memory network lies in its ability to treat this noise and adequately reproduce it into the forecasted trend, due to its own architecture enabling to take into account significant long-term patterns.

The time-series nature of mortality rates lends itself to processing through neural networks that are specialized to deal with sequential data, such as recurrent and convolutional networks. The aim of this work is to show how the structure of the Lee–Carter model can be generalized using a relatively simple shallow convolutional network model, allowing for its components to be evaluated in familiar terms. Although deep networks have been applied successfully in many areas, we find that deep networks do not lead to an enhanced predictive performance in our approach for mortality forecasting, compared to the proposed shallow one. Our model produces highly accurate forecasts on the Human Mortality Database, and, without further modification, generalizes well to the United States Mortality Database.

This paper introduces a neural network (NN) approach for fitting the Lee-Carter (LC) and the Poisson Lee-Carter model on multiple populations. We develop some NNs that replicate the structure of the individual LC models and allow their joint fitting by simultaneously analysing the mortality data of all the considered populations. The NN architecture is specifically designed to calibrate each individual model using all available information instead of using a population-specific subset of data as in the traditional estimation schemes. A large set of numerical experiments performed on all the countries of the Human Mortality Database shows the effectiveness of our approach. In particular, the resulting parameter estimates appear smooth and less sensitive to the random fluctuations often present in the mortality rates’ data, especially for low-population countries. In addition, the forecasting performance results significantly improved as well.

In this work, we investigate the application of Deep Learning in Portfolio selection in a Markowitz mean-variance framework. We refer to a $l_1$ regularized multi-period model; the choice of the $l_1$ norm aims at producing sparse solutions. A crucial issue is the choice of the regularization parameter, which must realize a trade-off between fidelity to data and regularization. We propose an algorithm based on neural networks for the automatic selection of the regularization parameter. Once the neural network training is completed, an estimate of the regularization parameter can be computed via forward propagation. Numerical experiments and comparisons performed on real data validate the approach.

Multi-criteria decision methods (MCDMs) are used as an effective tool to support decision makers (DMs) in critical decision processes. These methods are used in several fields of application by analyzing static decision-making problems in which it is assumed that the decision is made at a precise moment. By increasing the complexity of decision-making problems and operating in increasingly competitive production sectors, very often analyzing a decision-making problem in a static way is not enough. This paper deals with considering the temporal variable in the construction of a dynamic MCDM, which takes into account historical and current data in order to learn from the past; and prospective also allowing to have a forecasting perspective of future data through the use of techniques that work in this sense. Our approach was tested in a multinational company in the manufacturing sector. The results show that the use of dynamic approaches allows DMs to obtain more precise alternative rankings given the information they exploit from the past; furthermore, the use of the prospective model, integrated with the dynamic one, makes it possible to provide greater detail on the possible future rankings of the alternatives that update their positions based on the feedback received. The approach allows for drawing advantages from a management point of view as it defines a complete decision support tool for the choices related to the planning and control of production processes. Our approach can be implemented in corporate information systems. Furthermore, the involvement of the DM in the construction of the model helps to define a learning process that feeds the decision-making process by generating greater awareness of the DM on the choices to be made.