A neural network approach for the mortality analysis of multiple populations: a case study on data of the Italian population

Since the seminal work of Lee and Carter [17], several stochastic models for the estimation and projection of mortality rates were developed during the last decades, see, e.g., the contributions of Brouhns et al. [2], Renshaw and Haberman [31], Cairns et al. [3], and Plat [28]. While these pioneering approaches analyzed single populations, models for multiple populations gained considerable importance in subsequent years; after it was found that the mortality profiles of multiple populations tended to converge (cf. Wilson [40]). Indeed, the multi-population paradigm often has advantages over modelling mortality rates for each population separately. Most notably, multi-population mortality models can capture common features of the mortality profile of similar populations such as neighbouring countries, populations showing similar socio-economic-, environmental-, or biological characteristics, while simultaneously reflecting population-specific features. This motivates their use for producing coherent projections of mortality rates.

Recently, Neural Network (NN)-based approaches for mortality modelling were proposed as an appealing alternative to classical stochastic models. Among the first examples is the work of Richman and Wüthrich [33], who analyses the Swiss population, and Hainaut [11] who uses French, UK, and US mortality rates to compare a NN approach to the Lee–Carter model w/wo cohort effects. Another contribution is Nigri et al. [23], who use a deep learning algorithm based on a two-step Recurrent Neural Network (RNN) to enhance the forecasts obtainable under the Lee–Carter model. Indeed, there have been many recent developments in the use of NNs in the context of multi-population mortality modelling, such as Perla et al. [27], which considers the use of one-dimensional (period effect only) RNN with Long Short-Term Memory (LSTM) and of Convolutional Neural Networks (CNN) to provide direct forecasts of the mortality rates compared to the two-step approach of Nigri et al. [23]. Lindholm and Palmborg [19] consider similar models with a focus on the optimal use of data for projection. Schnürch and Korn [35] extend the RNN and CNN by proposing a two-dimensional approach involving age and period. In a slightly different fashion, Scognamiglio [36] proposes a NN architecture for the joint calibration of individual Lee–Carter models based on its classical log-normal representation as well as the Poisson Lee–Carter version of Brouhns et al. [2]. An approach that allows for coherent predictions within sub-groups of similar population is provided in Perla and Scognamiglio [26]. Finally, Wang et al. [38] develop a framework which ‘augments’ the mortality dataset to construct an image of neighborhood mortality data around the central death rate and use two CNN approaches for projecting mortality rates.

A key application of multi-population mortality modelling approaches is the analysis of the mortality levels based on socio-economic characteristics. Among others, understanding mortality is relevant to policymakers in order to propose and plan sustainable state pension reforms and budgets, or to address disparities between socio-economic groups. Understanding mortality from a statistical point-of-view is also relevant for private sector players such as insurance companies and pension funds, offering mortality-linked products like annuities and pensions and designing effective solutions for longevity risk management and transfer.

Considering specific populations that have already been investigated in the literature, let us mention Wen et al. [39], who compare several stochastic mortality models to fit mortality rates of small geographic areas in the UK (Lower Layer Super Output Areas), grouped in deciles of their Index of Multiple Deprivation. Cairns et al. [6] develop a multi-population mortality modelling approach for the analysis of the Danish population on the basis of the deciles of a newly created affluence index, which accounts for individual information on income and wealth.

This paper investigates the use of NNs to jointly model the mortality rates of multiple populations and compares the empirical results to classical stochastic mortality models. The model for the dynamics of mortality rates draws on the work of Richman and Wüthrich [32], where they propose the use of Recurrent Neural Networks (RNN) such as Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU). Although computationally expensive, we focus on RNNs to exploit their suitability to deal with the time-series structure of mortality rates. Indeed, due to their recurrent connections, RNNs allow to maintain memory of past observations, since these let the information pass from past time steps to the current one. In this way, it is possible to model the information throughout the entire time series. Furthermore, RNNs can handle time series of variable length, due to their sequential processing of the data, cf. Hsu [12]. The NN approaches are then compared to well-established stochastic mortality models for multiple populations such as the Lee–Carter extension of Li and Lee [18] and the Common Age Effect model of Kleinow [16], as well as to the single-population approach represented by the Plat [28] model.

In our case study, we investigate Italian mortality data that is grouped according to socio-economic characteristics. More precisely, we first propose a new deprivation index on the basis of five variables. This index allows separating the 106 Italian counties into nine groups of different socio-economic level, which are then used as populations for the empirical analysis. To the best of our knowledge, this study is the first to explicitly address the use of NNs in the context of the analysis of multiple populations on the basis of socio-economic characteristics.

The structure of the remaining paper is as follows: Sect. 2 introduces and explains the Italian mortality data used for our analysis and the creation of the new Index of Multiple Deprivation. Section 3 describes how NNs are used in our study. Section 4 briefly introduces the models used for comparison. Then, Sect. 5 shows the results of the empirical analysis and, finally, Sect. 6 concludes.

For each county, we originally collected a set of twelve indicators representing different aspects of the quality of life. Out of these indicators, a subset of five was ultimately selected, reflecting a proper mix of different types of socio-economic factors. These have been aggregated to a so-called ‘Index of Multiple Deprivation’ (IMD). In this way, the 106 provinces⁵ from twenty regions could be classified to different groups based on their socio-economic indicators. The chosen variables are: A detailed description of these variables can be found on the ‘Statbase portal’ of Istat, which provides access to a large amount of data on the Italian population,⁸ which is also the source of the underlying data (downloaded on \(1^{st}\) of March 2021 for year 2018).

The variables have been chosen based on a correlation analysis between them. The selection was then validated by a ranking measure, Kendall’s tau, to check whether the rank of provinces changes significantly when omitting a variable from the selection. We found that Kendall’s tau does not significantly change when omitting or adding a further variable to the five we selected.

Our index was constructed using z-scores to scale the five different variables to a comparable level and unit; the same approach is used in Osservatorio della salute [24]. This allowed to aggregate the single z-scores per province to a total value and finally permitted a ranking of the provinces. For each province, the z-score was calculated aswhere i indicates the respective socio-economic classifier, \(x_i^j\) its value, j the county, \(m_i\) the mean of this classifier over all counties, and \(s_i\) the (unbiased) sample deviation over all counties.

The interpretation is intuitive: the higher the value \(z^j\) of a county, the worse is its socio-economic situation, or in other words, the deprivation in that area is higher. Implicitly, we assume that the standardized value of each variable has the same impact on the z-score (which could be easily generalized by introducing weights).

Then, the aim is to rank the counties on the basis of the values of their z-score. Hence, we created nine groups of counties, each with homogeneous populations, ranging between 6 and 7 million people. This split does not account for sex and age of the Italian population. This implicitly assumes that the counties have similar distributed population by age and sex. If, by using this criterion, we found that two counties had the same z-score value,⁹ we allocate these in the same group. In this way, we aim at creating groups of comparable size, where the counties therein have similar socio-economic characteristics. The use of socio-economic indicators for the calendar year 2018 only, implies that the ranking and the corresponding groups do not change over time. This assumption may not reflect the socio-economic developments in Italy across the last 36 years. Nevertheless, we find this assumption reasonable in consideration of the results obtained in the preliminary analysis of the mortality and socio-economic data. Indeed, more deprived socio-economic counties are located in the south of Italy, compared to the historically wealthier areas in the north, as also demonstrated by the time series of the unemployment rate, and this evidence is further confirmed when analysing the evolution of the raw mortality rates.

We purposely opted for a simple and interpretable approach to create the index, that might be refined in further studies. Our main focus is to elaborate on mortality models as described later, and this simple and intuitive approach provides very plausible results. In any case, we remark how the creation of the groups was carried out on the basis of their order, rather than their index value.

To obtain a geographical impression of the index-based subdivision, the following map is reported, see Fig. 1. The color-scheme is as follows: Brighter colors indicate a lower index value, reflecting better socio-economic condition based on the IMD defined above. The original map was taken from the De Agostini website¹⁰ and colored with standard graphic tools.

To obtain a first impression of mortality rates, the crude period death rates \({\hat{m}}(x,t,i) = \frac{d(x,t,i)}{E^c(x,t,i)}\) between 1982 and 2018 are plotted in Fig. 2 for the male and female population aged 68 and 83 years old, respectively. We observe:Throughout the analysis, we use the mortality rates of the first 33 calendar years 1982–2014 when training the models (training set), while those of 2015–2018 are used for predictions and forecasts (test set). We opted for this split in about 90%–10% due to the short length of the time-series of available data. We believe this split is a reasonable compromise between the need of sufficient data to fit the models (especially on a restricted age-range of a 50–95 years old population), whilst using the latest years to assess their predictive ability.

For RNNs, one has given input variables \((\tilde{x}_1, \dots , \tilde{x}_T)\) in time series structure with components \(\tilde{{{\textbf {x}}}_t}\in {\mathbb {R}}^{q_0}\) for \(t = 1, \dots , T\). For the k-th layer, we define the mappings as follows:whereThe individual neurons \(1 \le j \le q_k\) are modeled aswhere \(\phi\) is the (non-linear) activation function, which will be the same for all neurons. The weights are \(W^{(k)} = ({{\textbf {w}}}_1^{(k)}, \dots , {{\textbf {w}}}_{q_k}^{(k)})^T \in {\mathbb {R}}^{q_k \times (1+q_{k-1})}\) (including an intercept, see above) and \(U^{(k)} = ({{\textbf {u}}}_1^{(k)}, \dots , {{\textbf {u}}}_{q_k}^{(k)})^T \in {\mathbb {R}}^{q_k \times q_k}\) (excluding an intercept) which are identical for all time points t, as these weight matrices are homogeneous over time.

Note, \(\tilde{{{\textbf {x}}}}_t \in {\mathbb {R}}^{q_0}\) is an input in layer \(k=1\).

RNNs were implemented in R (R Core Team [30]). Training and forecasting was performed using the package keras Chollet et al. [8]. The choice of the parameters is motivated by the work of Richman and Wüthrich [33], as number of layers, number of neurons, type of activation functions, and so on. When experimenting with different hyper-parameters we did not notice any substantial differences or improvements in the results. The R code for data pre-processing, similar in spirit to Richman and Wüthrich [33], can be found in the Github repository https://​github.​com/​maxeuthum/​Multipopulation-Mortality-Models¹². This repository also includes the code for fitting the models, with a detailed description of the performed operations line-by-line.

Input values were smoothed over 5 neighbouring ages to predict mortality at central age x. Therefore, for group i we obtain:where, for general variables \(x, y \in {\mathbb {R}}\), \(x \vee y = \text {max}\{x,y\}\), and \(x \wedge y = \text {min}\{x,y\}\).

Furthermore, a look-back period of \(T=10\) years was determined. This means that the previous 10 years were taken as an input to predict mortality rates for year t. This is the same look-back period as in Richman and Wüthrich [33].

The training data set \({\mathcal {T}}\) was defined aswhere \({{\textbf {y}}}_{t,x}\) denote the log-mortality rates. Hence, we have log-mortality rates for 46 1-year age groups (50 to 95), denoted by variable x, over 23 calendar years (\(2014-(1982+10)+1=23\)), and nine deprivation groups. This gives \(46 \times 23 \times 9 = 9,522\) training samples of dimension \(10 \times 5\) as input x (10 for the look-back period, and 5 for the neighbouring ages). In R, this was stored in 5 arrays of dimension \(9,522 \times 10\). Last, \({{\textbf {y}}}_{t,x}\) was stored in an array of dimension 9, 522 (for this array of course no look-back period or smoothing takes place).

As a final step for training data pre-processing, scaling was applied on the prediction data using the MinMax-scaler. In the end, the group indicator has been mapped to the data to obtain the input data consisting of these two parts returning the total input of the NN. We remark that several choices for the input dimension are reasonable, allowing for further features to be included in the model.

In this work, different types of RNN models have been modelled, namely LSTM and GRU networks for all groups simultaneously, for all three sex groups (males, females, and combined) separately. Respectively one, two, and three layers have been specified (Figs. 3, 4).

We opted for running 500 epochs¹³ of the data with a batch size of 100 (again based on the work of Richman and Wüthrich [33]). To prevent overfitting, the training data set has been split into a training and validation set with a proportion of 80% and 20%, respectively. Furthermore, we implemented callbacks in order to select the calibration with the minimum loss on validation data among the 500 epochs.

As in other cases of NN-modelling, gradient descent algorithms were used to solve the optimization problem of the NN. Usually, these algorithms run until some stopping criterion is met such as the case where the error which we seek to minimize lies within a certain range. As stated in Richman and Wüthrich [33], an issue using early stopped solutions of gradient descent methods is that the resulting calibrations depend on the chosen seed of the algorithm. As a consequence, the results from different runs of the NN may be substantially deviating. For example, a certain run may lead to really good predictions, the next run to quite bad predictions. Hence, in our case study each model has been run 40 times, each time with different starting values. Then, we average the outcomes to assure not to have an outlying model by sheer coincidence. Hence, for each model the output (the log-mortality rates) has been transformed by the exponential function to obtain the mortality rates, which are then finally averaged.

This model extends the single-population Lee and Carter [17] model to the analysis of multiple populations. The Li and Lee [18] model introduces a set of parameters which are common to the set of analysed groups, as well as group-specific parameters capturing the unexplained variance.

Here, m(x, t, i) can be approximated as the ratio between the death counts, denoted as D(x, t, i), and the central exposure at risk \(E^c(x,t,i)\).

\(\alpha (x,i)\), \(\beta (x,i)\), and \(\kappa (t,i)\) are group-specific parameters. \(\alpha (x,i)\) indicates the average over time of the log mortality rate. The common function K(t) explains the evolution of the mortality over time for all groups, and B(x) is a global age modulating parameter, indicating how rates change by age for changes in the time factor K(t). \(\beta (x,i)\) and \(\kappa (t,i)\) have the same role as B(x) and K(t), but act on a group-specific level.

The CAE model of Kleinow [16] assumes that age effects are common to all populations, following the assumption that age effects may be very similar in countries sharing a similar socio-economic structure.

The order p follows from the allowance of further age–time interaction parameters. In our analysis we set \(p=2\).

The third model we use for comparison was initially proposed by Plat [28]. However, in this work we use a simplified version, which includes two period-specific factors, without accounting for the cohort effect.

where \(\bar{x}\) denotes the average age of the observed age range. The first stochastic component \(\kappa ^{(1)}\) represents changes in the level of mortality for all ages, while \(\kappa ^{(2)}\) allows for changes in mortality to vary between ages.

We first compare the three competing stochastic mortality models, that we briefly introduced in Sect. 4, in terms of their explanation ratio. Then we compare these to the rates obtained by using the NN models of this paper.

Table 1 shows the explanation ratios for the models of Li & Lee, Kleinow, and Plat, defined for group i and model M aswhere \(\alpha ^c(x,i):= \frac{1}{T} \sum \limits _{t} \log \frac{d(x,t,i)}{E^c(x,t,i)}\) is the average log crude death rate over time. The explanation ratio is useful for analysing how much information about the crude death rates \(\frac{d(x,t,i)}{E^c(x,t,i)}\) is explained by the respective model.

We observe how the Li & Lee model performs best in terms of explanation ratios for eight out of nine females subgroups, and the same is noted for the Kleinow model when looking at the male population. The Plat model, analysed at the level of a single population, still yield comparable explanation ratios, despite being lower compared to the Li & Lee and the CAE model of Kleinow. Furthermore, the Plat model has a lower number of parameters, which may in part explain these results. Our results are further confirmed by the analysis of their Akaike Information Criterion (AIC) (Akaike [1]), shown in Table 2, which indicates the relative quality of a statistical model with a penalty term for the number of parameters. The AIC is calculated aswhere \(\ell ({\hat{\theta }})\) indicates the log-likelihood value at (optimal) parameter \({\hat{\theta }}\) and p denotes the number of parameters of the respective model (see Table 9).

Tables 3 shows the mean squared error for model M and subpopulation i, calculated as follows:where \({\hat{m}}(x,t,i)^M\) denotes the fitted mortality rate derived from model M. Table 4 shows the MSE for the RNN models analysed in this paper.

From the results in Table 3, we observe that it is not possible to draw a decisive conclusion about the best performing model, since for different groups and sex there is a mixed evidence about the best performing model. When the data for both males and females are combined, then we observe that the Plat model has a better performance in the two extremes of the deprivation group, or in other words, shows a better performance for the least and most deprived county groups.

The two RNN models seem to sensibly improve the in-sample fit compared to the three competing stochastic models. In more detail, the two-layer LSTM outperforms the other two LSTM models over all socio-economic groups for males, females, and combined. On the other hand, the GRU models with three layers turn out to perform even better. For female and both sexes combined mortality rates, the most deprived groups show higher in-sample errors compared to less deprived ones. A similar evidence can be observed for the LL, CAE, and Plat models. This may be indicative of the issues of mortality models in general when fitting more deprived subpopulations via a multi-population approach. This can also be the effect of the created index in Sect. 2 and of the underlying data. Nevertheless, we note that the difference between females and males is smaller for the RNN approach compared to the competing models.

In conclusion, the multi-population RNN models outperform the well established stochastic mortality models when analysing county-based socio-economic subgroups of the Italian population, based on the mean squared error. The mean squared errors for males are higher compared to females and both sexes combined.

For a deeper inspection of these results we restrict the MSE analysis to a shorter age range for males and females, in a way such that they both have the same life expectancy. These ranges were chosen based on remaining life expectancy of Italian males at age 50 in 2017 (32.08 years) and Italian females at age 54 in year 2017 (approx. 32 years). Therefore, based on the Italian life tables from the Human Mortality Database [13], we restrict the analysis to the male population aged 50 to 82 and to the females aged 54 to 86.

Again, we recalculated the deprivation group-specific MSE for males and females for the restricted age-ranges, still based on the model fitted to the original dataset (males and females aged 50 to 95). The results are shown in Table 5 (LL, CAE, and Plat models) and Table 6 (for RNN).

For the selected age ranges, we observe that for the three competing stochastic mortality models, the mean squared error is less than 0.1 of the mean squared error for all ages, even if these reduced age ranges cover 72% of the modelled years.

Again, we observe that for the female population the MSE is larger for the most deprived socio-economic groups for both the competing models as well as for the RNN-based ones. A similar evidence is obtained for the males at a lesser extent. For the models fitted using female subpopulation data, we still face more difficulties when considering the most deprived groups compared to less deprived ones, especially for SMM models. It is also evident, that all models within one model class (SMM or RNN) have similar mean squared errors. This means, the models perform very well for these ages through all selected models. Let us also mention the fact that mean squared errors in the RNN case are by far smaller than in the SMM case.

Furthermore, it is interesting to observe an alignment of female and male rate fitting (as it was aimed through the analysis of different age ranges): the SMM mean squared error for the male population were on average approximately 3.02 times compared to females. When restricting the age range observation, this factor reduces to about 1.23, which means that for our data both male and female rates are almost fitted equally well on average for an age range with similar remaining life expectation. For the three selected RNN models from above, the factors are 1.52 and 1.01 respectively, that is the alignment of female and male rates fitting ability is almost perfect in the RNN case.

Concluding, it seems that the oldest people (of age 87–95) drive most of the mean squared error. In addition, one should be aware of looking at different age ranges for females and males based on the evidence that life expectancy significantly differs for these two groups. The point in time for older ages, where mortality is harder to fit due to more volatility in death and exposure data, starts earlier for males which could explain a higher mean squared error when looking at all ages.

Another possible reason for larger errors in the male population through all models may come from the difference in population sizes of the underlying deprivation groups. These are higher in the female case, since in the observed age ranges females outlive males. A larger sample size could give the statistical model more stability when estimating parameters. However, the difference is not very large¹⁵.