# A Deep Learning Integrated Lee–Carter Model

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## Abstract

**:**

## 1. Introduction

## 2. Lee–Carter Model

## 3. Neural Network Model

**Back-propagation**

- in the forward step, the prediction $\widehat{y}$ are computed fixing the synaptic weights,
- in the backward step, the weights are adjusted in order to reduce the error E of the network.

#### Recurrent Neural Network with Long Short-Term Memory Architecture

## 4. Numerical Application

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | More precisely, E also depends on the biases vectors ${\mathbf{b}}_{1},{\mathbf{b}}_{2},\dots ,{\mathbf{b}}_{g}$ of all hidden layers. For simplicity of notation, from now, for a generic layer $k\in [1,g],$ we indicate with ${W}_{k}$ the complete matrix ${\left({W}_{k}^{T}\right|{\mathbf{b}}_{k})}^{T}$. |

**Figure 1.**Schematical view of neural network (NN): the circles represent neurons and lines represent synapses. Synapses take the input and multiply it by a “weight” (the “strength” of the input in determining the output). Neurons add the outputs from all synapses and apply an activation function.

**Figure 3.**A representation of a vanilla Long Short-Term Memory (LSTM) block structure and is internal information forward flow.

**Figure 4.**A simplified representation on sequential operation of a one-hidden layered Recurrent Neural Network (RNN) with LSTM architecture for each time step.

**Figure 7.**Life expectancy at birth (on the left) and at age 70 (on the right): ARIMA(1,1,0) vs. LSTM. Australian male population.

**Figure 8.**Distribution of deaths: ARIMA(1,1,0) (on the left) vs. LSTM (on the right). Australian male population.

**Figure 9.**Logarithm of central death rates: ARIMA(1,1,0) (on the left) vs. LSTM (on the right). Australian male population.

Output | Input | |||
---|---|---|---|---|

${\kappa}_{t}$ | ${\kappa}_{t-1}$ | ${\kappa}_{t-2}$ | … | ${\kappa}_{t-J}$ |

${\kappa}_{t+1}$ | ${\kappa}_{t}$ | ${\kappa}_{t-1}$ | … | ${\kappa}_{t-J+1}$ |

${\kappa}_{t+2}$ | ${\kappa}_{t+1}$ | ${\kappa}_{t}$ | … | ${\kappa}_{t-J+2}$ |

… | … | … | … | … |

${\kappa}_{t+n}$ | ${\kappa}_{t+n-1}$ | ${\kappa}_{t+n-2}$ | … | ${\kappa}_{t-J+n}$ |

Country | Total Years | Testing Set Years |
---|---|---|

Australia | 1921–2014 | 1996–2014 |

Denmark | 1835–2016 | 1981–2016 |

Italy | 1872–2014 | 1987–2014 |

Spain | 1908–2016 | 1995–2016 |

USA | 1933–2016 | 2001–2016 |

Japan | 1947–2016 | 2003–2016 |

Country | Best ARIMA (p,d,q) |
---|---|

Australia | |

Male | ARIMA (1,1,0) |

Female | ARIMA (1,1,0) |

Denmark | |

Male | ARIMA (0,1,3) |

Female | ARIMA (0,1,3) |

Italy | |

Male | ARIMA (0,1,3) |

Female | ARIMA (0,1,1) |

Spain | |

Male | ARIMA (1,1,0) |

Female | ARIMA (1,1,0) |

USA | |

Male | ARIMA (0,1,0) |

Female | ARIMA (0,1,0) |

Japan | |

Male | ARIMA (0,1,3) |

Female | ARIMA (0,1,3) |

**Table 4.**Performance of Long Short-Term Memory (LSTM) and ARIMA in the testing set for each country.

Country | Male | Female | ||
---|---|---|---|---|

Australia | MAE | RMSE | MAE | RMSE |

${\kappa}_{t}$ ARIMA | 24.75 | 28.04 | 13.95 | 15.55 |

${\kappa}_{t}$ LSTM | 2.57 | 3.24 | 3.12 | 3.83 |

Denmark | MAE | RMSE | MAE | RMSE |

${\kappa}_{t}$ ARIMA | 11.10 | 16.10 | 7.99 | 10.70 |

${\kappa}_{t}$ LSTM | 2.97 | 3.84 | 6.62 | 8.18 |

Italy | MAE | RMSE | MAE | RMSE |

${\kappa}_{t}$ ARIMA | 55.41 | 63.46 | 41.12 | 45.74 |

${\kappa}_{t}$ LSTM | 4.63 | 5.48 | 8.04 | 10.69 |

Spain | MAE | RMSE | MAE | RMSE |

${\kappa}_{t}$ ARIMA | 20.24 | 26.33 | 26.10 | 33.95 |

${\kappa}_{t}$ LSTM | 7.69 | 8.96 | 15.61 | 17.67 |

the USA | MAE | RMSE | MAE | RMSE |

${\kappa}_{t}$ ARIMA | 8.39 | 9.48 | 10.81 | 12.54 |

${\kappa}_{t}$ LSTM | 2.31 | 2.86 | 3.32 | 4.18 |

Japan | MAE | RMSE | MAE | RMSE |

${\kappa}_{t}$ ARIMA | 9.61 | 10.50 | 15.12 | 17.52 |

${\kappa}_{t}$ LSTM | 4.71 | 5.24 | 7.89 | 10.23 |

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**MDPI and ACS Style**

Nigri, A.; Levantesi, S.; Marino, M.; Scognamiglio, S.; Perla, F.
A Deep Learning Integrated Lee–Carter Model. *Risks* **2019**, *7*, 33.
https://doi.org/10.3390/risks7010033

**AMA Style**

Nigri A, Levantesi S, Marino M, Scognamiglio S, Perla F.
A Deep Learning Integrated Lee–Carter Model. *Risks*. 2019; 7(1):33.
https://doi.org/10.3390/risks7010033

**Chicago/Turabian Style**

Nigri, Andrea, Susanna Levantesi, Mario Marino, Salvatore Scognamiglio, and Francesca Perla.
2019. "A Deep Learning Integrated Lee–Carter Model" *Risks* 7, no. 1: 33.
https://doi.org/10.3390/risks7010033