Clustering-based extensions of the common age effect multi-population mortality model

Schnürch, Simon; Kleinow, Torsten; Korn, Ralf Article Clustering-based extensions of the common age effect multi-population mortality model Risks Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Schnürch, Simon; Kleinow, Torsten; Korn, Ralf (2021) : Clustering-based extensions of the common age effect multi-population mortality model, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 9, Iss. 3, pp. 1-32, https://doi.org/10.3390/risks9030045 This Version is available at: https://hdl.handle.net/10419/258134 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ risks Article Clustering-Based Extensions of the Common Age Effect Multi-Population Mortality Model Simon Schnürch 1,2,* , Torsten Kleinow 3,4 and Ralf Korn 1,2   Citation: Schnürch, Simon, Torsten Kleinow, and Ralf Korn. 2021. Clustering-Based Extensions of the Common Age Effect Multi-Population Mortality Model. Risks 9: 45. https://doi.org/10.3390/risks9030045 Academic Editor: Jackie Li Received: 7 January 2021 Accepted: 14 February 2021 Published: 1 March 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1Department of Financial Mathematics, Fraunhofer Institute for Industrial Mathematics ITWM, Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany; [email protected] 2Department of Mathematics, University of Kaiserslautern, Gottlieb-Daimler-Straße 48, 67663 Kaiserslautern, Germany 3Department of Actuarial Mathematics and Statistics, School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK; [email protected] 4The Maxwell Institute for Mathematical Sciences, School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK *Correspondence: simon.schnuer[email protected].de Abstract: We introduce four variants of the common age effect model proposed by Kleinow, which describes the mortality rates of multiple populations. Our model extensions are based on the assumption of multiple common age effects, each of which is shared only by a subgroup of all considered populations. This makes the models more realistic while still keeping them as parsimonious as possible, improving the goodness of fit. We apply different clustering methods to identify suitable subgroups. Some of the algorithms are borrowed from the unsupervised learning literature, while others are more domain-specific. In particular, we propose and investigate a new model with fuzzy clustering, in which each population’s individual age effect is a linear combination of a small number of age effects. Due to their good interpretability, our clustering-based models also allow some insights in the historical mortality dynamics of the populations. Numerical results and graphical illustrations of the considered models and their performance in-sample as well as out-of-sample are provided. Keywords: mortality modeling and forecasting; stochastic mortality model; mortality of multiple populations; common age effect model; cluster analysis; maximum likelihood estimation 1. Introduction All over the world, progress in medical research, technological developments and a general improvement of living conditions have lead to a steady decrease of mortality during the past century. While this development has several obvious benefits for society and individuals, it also poses a serious financial risk for many stakeholders (see Oppers et al. 2012) . This so-called longevity risk consists of an underestimation of future longevity and a resulting lack of assets to meet the liabilities that come with aging populations. In fact, in recent decades, longevity forecasts have been consistently too low, and the potential total size of the global market for longevity risk has been estimated to be trillions of dollars (see Zugic et al. 2010). One way of addressing longevity risk is to model future mortality stochastically, which is conceptually motivated by empirical observations (see Cairns et al. 2006). Stochastic mortality models have the advantage that they not only provide point forecasts but also allow an estimation of forecast uncertainty and tail risk, and they offer the possibility to easily simulate mortality scenarios. While, initially, such models were used to describe only one population, there are various situations in which it is useful or even necessary to model the mortality of multiple populations simultaneously, see for example Kleinow and Cairns (2013), Cairns (2014) and Chen et al. (2015). In this paper, we consider an established multi-population mortality model, the common age effect (CAE) model of Kleinow (2015). Its eponymous assumption is that the Risks 2021,9, 45. https://doi.org/10.3390/risks9030045 https://www.mdpi.com/journal/risks Risks 2021,9, 45 2 of 32 age effect, the influence of general mortality improvements over time on different ages, is identical or at least similar for all considered populations. This is justified if all these populations exhibit similar age structure, socio-economic characteristics, geographical location and life style factors. However, for arbitrary sets of populations, the assumption is too strong and might only hold on multiple smaller groups or clusters of populations. The benefits of applying clustering methods to heterogeneous mortality data have already been recognized by Hatzopoulos and Haberman (2013), Debón et al. (2017) and Guibert et al. (2020) , whose approaches we describe below. Further applications of cluster analysis in mortality modeling are presented by Danesi et al. (2015), who cluster period effects which describe the general mortality development over time instead of age effects, by Léger and Mazzuco (2020) , who apply functional cluster analysis to identify groups of populations with similar age distributions of death, and by Wen et al. (2020), who consider groups of Canadian pensioners by pension level and find that clustering can reduce noise in small population data. In contrast to most of these papers, we focus on age effects instead of period effects or death rates. Additionally, we employ cluster analysis not only as a tool for analyzing the data but also as an essential ingredient for modeling, which takes the possible heterogeneity of populations into account. For this purpose, we present several clustering algorithms, embed them into the CAE model and check their performance in an empirical study. Some of the clustering techniques we use are well-known from the general unsupervised learning literature. In particular, we employ k -means clustering and hierarchical clustering (see Hastie et al. 2017, chapter 14) . The latter is at the core of a likelihood-ratio based clustering algorithm we propose, which additionally includes a simple statistical test for checking whether any two populations have the same age effect. Another clustering algorithm we investigate is more domain-specific: We review the fitting method of the augmented common factor model by Li and Lee (2005) and demonstrate that it yields a grouping of the populations as a by-product in our formulation. Guibert et al. (2020) propose a different clustering-based variant of the Li and Lee (2005) model. They mention two simple clustering methods—expert judgment and hierarchical clustering of period effects—but their focus is not on the investigation of clustering methods but rather on the implications of using such a clustering-based mortality model for risk management. Finally, we propose a new model related to the CAE model, which to the best of our knowledge has not been investigated before. For the calibration of this model, we combine ideas from fuzzy clustering with maximum likelihood estimation and refer to this method as fuzzy maximum likelihood clustering. While in the CAE model a number of populations share a common age effect, this is in general not the case with the fuzzy clustering approach. Instead, our fuzzy maximum likelihood clustering model suggests that each population’s individual age effect is a linear combination of a small number of age effects. In particular, this implies that, in general, a population does not entirely belong to a unique cluster but rather is assigned a membership weight for each cluster, which describes how much the population-specific age effect is influenced by the age effect of the cluster. This allows for a more adequate description of the heterogeneity within the clusters. As we will demonstrate, it can be especially useful when considering a population which is hard to classify as similar to any of the other populations. A more general fuzzy clustering technique called fuzzy c -means clustering (Bezdek 1981) has been applied by Hatzopoulos and Haberman (2013) to analyze similarities in the period effects of 35 countries. They observe a clear separation between former communist and other populations like France, England and Wales, Canada or the USA, an effect which has already been discovered and extensively described by Meslé and Vallin (2002) using hierarchical clustering methods. In a second step, they use their results to choose 19 of these countries and construct a common age-period model for their mortality. Debón et al. (2017) also apply the fuzzy c -means algorithm as a tool for discovering patterns in mortality data. They start by performing a principal component analysis of the logit death probability Risks 2021,9, 45 3 of 32 surfaces of the male and female populations of 24 EU countries in order to reduce the data to the 5 most relevant dimensions before clustering the populations. In their analysis, they find significant differences between Western and Eastern European countries as well, and less pronounced differences between Southern and Northern populations in Western Europe. Our fuzzy clustering approach is different from these applications of the fuzzy c -means technique because it fully integrates the clusters into the model equation, allowing each population to have a unique age effect with only a modest increase in the number of parameters compared to an ordinary CAE model. This is achieved by calibrating all the model parameters including the cluster membership weights in the same step of the maximum likelihood estimation. In our empirical findings, we confirm some of the observations in the literature. For instance, our algorithms detect the distinct pattern Danish mortality has followed until around 1990, which has also been found by Hatzopoulos and Haberman (2013), whose algorithm puts Denmark in the same cluster as other Western European countries but with a relatively low degree of cluster membership, which accounts for the differences to the general trend of this cluster. We also provide a detailed comparison of our clusters and the ones obtained by Guibert et al. (2020) based on their hierarchical clustering in Section 4.2 , where we find some similarities—for example, Germany, France, Portugal and Spain are assigned to the same cluster by both methods—but also some differences, for example regarding the number of clusters. However, when comparing our results to those of other papers, one has to keep in mind that we are focusing on age effects here while the literature so far has mostly clustered by period effects or death rates. It is not surprising that the clusters obtained from these different target features do not always coincide. In total, our main contribution consists of enhancing the CAE model using cluster analysis methods and, additionally, proposing a new related model inspired by the concept of fuzzy clustering. As a consequence, we are able to address the following research questions: • How can one handle differences in the mortality of multiple populations which make the assumption of a single common age effect implausible? • How can one identify clusters of populations with similar age effects? • What information can be obtained on similarities and dissimilarities between the age effects of the considered populations by applying different clustering algorithms? • How do CAE-type models based on a cluster analysis of populations perform on different data sets compared to several benchmark models from the literature? Clustering age effects allows for more realistic, better fitting models which are still as parsimonious as possible. Furthermore, due to their good interpretability, our clusteringbased common age effect models also yield some insights in the historical mortality dynamics of the populations under consideration. Finally, we demonstrate that, depending on the data, some of our model extensions achieve a superior out-of-sample performance compared to the original common age effect model by providing a detailed forecasting performance evaluation for our proposed and several benchmark models. We proceed as follows: In Section 2, we introduce some notations, give a brief overview of existing mortality models and compile some methodological prerequisites. Based on that, in Section 3, we present four clustering-based extensions of the CAE model. In Section 4 , we describe the obtained clusters and the results of an empirical comparison of the models. In Section 5, we conclude and list possible directions for future research. For computing the numerical results and for creating all the figures in this paper, we have used the statistical software environment R by R Core Team (2019) as well as the data visualization package ggplot2 by Wickham (2016). Risks 2021,9, 45 4 of 32 2. Methodological Preliminaries 2.1. Notation Let n , m∈N , where N:={ 1, 2, 3, . . . } is the set of natural numbers. We denote a matrix A∈Rn×m in terms of its elements by aijij . For matrices A , B∈Rn×m , A<B means aij ≥bij for all i∈ { 1, . . . , n} , j∈ { 1, . . . , m} . By 0n×m , we denote an n×m -matrix containing only zeros, and by 1 n×m an n×m -matrix containing only ones. The group of invertible matrices of dimension n×n is denoted by GL(n) . By In , we denote the n×n identity matrix. For vectors, where applicable, we use analogous notation. We are going to investigate models for human mortality data, which depend on the age x , whose possible values we denote by x1 , . . . , xA∈N∪ { 0 } , where xj−xj−1= 1 for all j∈ { 2, . . . , A} . Similarly, the data also depend on the calendar year, which we denote by t=t1 , . . . , tY∈N , and for which we also assume tl−tl−1= 1 for all l∈ { 2, . . . , Y} . Moreover, when considering the mortality experience of multiple populations, we denote them by i= 1, . . . , P . We mainly focus on the (central) death rate mi x,t . It is defined as the ratio mi x,t:=Di x,t Ei x,t (1) of the number Di x,t of lives belonging to population i who die in year t at age x to the corresponding total number of person-years lived during that year, the so-called exposure Ei x,t . More details on the mathematical description of mortality data can be found in Pitacco et al. (2008, chapters 2 and 3.3). 2.2. Overview of Existing Models Lee and Carter (1992) propose a very influential mortality model called the Lee–Carter (LC) model. They additively decompose logarithmic death rates into an age-specific base level αx , a time-varying component (period effect) κt —which is multiplicatively affected by the age-modulating parameter (age effect) βx —and random error terms εx,t , which they assume to be zero in expectation and homoskedastic: log mx,t=αx+βxκt+εx,t. (2) Here, some identifiability constraints have to be imposed so that the parameters are uniquely determinable. Lee and Carter (1992) propose xA ∑ x=x1 βx=1 and tY ∑ t=t1 κt=0, (3) but other constraints like κt1= 0 have been used as well. Nielsen and Nielsen (2010) provide a more general consideration of identifiability constraints in the Lee–Carter model and also prove that condition (3) indeed makes the model identifiable. To calibrate the model, Lee and Carter (1992) set ˆ αx:=1 Y tY ∑ t=t1 log mx,t(4) and determine estimates for the remaining parameters βx and κt using singular value decomposition (SVD). After the model has been fit, mortality forecasts are obtained by modeling κt as an ARIMA process—often simply a random walk with drift—and extrapolating this process. In retrospective, Lee (2000) notes that this basic approach works quite well for US mortality data between 1989 and 1997, especially in comparison to official forecasts by the government based on expert opinions. Risks 2021,9, 45 5 of 32 A simple mortality modeling approach for multiple populations, which is quite natural on first glance, consists in separately describing each population with its own LC model, log mi x,t=αi x+βi xκi t+εi x,t, (5) and assuming independence of the (κi t)t time series for i= 1, . . . , P . However, this does not account for dependence between the populations because there are no shared parameters and each population’s mortality is driven by a different stochastic factor. Zhou et al. (2013) demonstrate that using such an approach may lead to erroneous conclusions, for example when analyzing mortality bonds. Nonetheless, this individual Lee–Carter (ILC) model can serve as a basis and as a benchmark for more sophisticated multi-population mortality models. An influential model which imposes coherence of mortality between populations is proposed by Li and Lee (2005): log mi x,t=αi x+βxκt+βi xκi t+εi x,t, (6) see Section 3.2 for details on how to fit this model. Coherence means that long-term mortality developments of different populations are constrained to maintain a constant ratio to one another, i.e., lim t→∞ mi x,t mj x,t ∈R(7) in expectation for each age x and populations i6=j (see Cairns et al. 2011;Hyndman et al. 2013). This concept is motivated by the fact that the mortality of populations which are to a certain degree similar to each other cannot reasonably be expected to diverge in the long term. Li and Lee (2005) find their model, which they call augmented common factor (ACF) model, to fit well to the mortality data of a group of 15 low-mortality countries. Kleinow (2015) considers a special case of the ILC model where all populations share the same age-modulating parameters, i.e., log mi x,t=αi x+βxκi t+εi x,t, (8) which he calls the common age effect (CAE) model. He finds it to achieve better in-sample fit than the ACF model on a selection of 10 countries. Additionally, it allows for an easier comparison of the period effects because they are scaled with the same factors (age effects) for every population. Even though Kleinow (2015) finds a sum of two age-period interaction terms β1 xκi,1 t+β2 xκi,2 tto fit well for his data, in this paper we only consider one such interaction term to keep our proposed model extensions simpler. Kleinow (2015) uses a generalization of SVD, so-called common principal components analysis (cPCA), to fit the model and imposes slightly different identifiability constraints than Lee and Carter (1992) . However, in this paper, we will use analogous identifiability constraints as for the ILC model, i.e., xA ∑ x=x1 βx=1 and tY ∑ t=t1 κi t=0 for all i∈ {1, . . . , P}. (9) Enchev et al. (2017) further investigate the CAE model and reach the conclusion that it performs in a more satisfactory way than the ACF model with respect to several criteria for the mortality data of a group of six countries. A broader survey of multi-population stochastic mortality models is provided by Villegas et al. (2017), who evaluate and compare these models with respect to a variety of qualitative and quantitative criteria. They reach the conclusion that among the considered models, a combination of the M7 and the M5 model (Cairns et al. 2009) or, alternatively, a variant of the CAE model provide the most suitable balance between flexibility, goodness of fit and forecasting performance. In a further study on English mortality data divided into 10 socio-economic groups, Wen et al. (2020) find the CAE model to fit quite well with respect to the Bayesian Information Criterion Risks 2021,9, 45 6 of 32 ( see Section 2.4 ). More precisely, they obtain superior goodness of fit compared to around 10 other models for an extension of the CAE model with two age-period interaction terms which additionally assumes common additive age effects αx . It has to be noted that the CAE model generally does not ensure coherence (see (7)). Of course, this depends on the technique which is used for projecting the κi ttime series. 2.3. Poisson Maximum Likelihood Estimation Instead of normally distributed error terms εi x,t as in the original LC model, a broad stream of literature assumes Poisson distributed death counts for several reasons (see Booth et al. 2002;Brouhns et al. 2002;Delwarde et al. 2006). Other distributional assumptions for the death counts would be possible but we would not expect a different underlying distribution assumption (for example, negative binomial) to cause any significant changes of our results. As the optimal choice of death count distribution is not the main focus of this work, we will rely on the Poisson assumption, whose applicability for mortality modeling has been thoroughly investigated both theoretically and empirically by Brillinger (1986). In this case, the ILC model for multiple populations reads Di x,t|θ∼PoissonEi x,t·expαi x+βi xκi t, (10) and the parameters θ:=αi xi x,βi xi x,κi ti t are calibrated such that they maximize the log-likelihood function L(θ)= P ∑ i=1 xA ∑ x=x1 tY ∑ t=t1Di x,t·αi x+βi xκi t−Ei x,t·expαi x+βi xκi t+K, (11) where K∈R is some constant which only depends on the data. Maximization is carried out numerically, usually by applying gradient-based algorithms like the Newton–Raphson method. Here, we use the Limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGSB) algorithm, a Quasi-Newton optimization method, as implemented in R Core Team (2019) with the parameters obtained by SVD or cPCA as starting points for the optimization. As Equation (11) immediately carries over to the CAE model and its clustering-based extensions (see Section 3), we can fit these models by Poisson maximum likelihood estimation as well. Moreover, we will specifically make use of maximum likelihood methods for likelihood-ratio-based clustering (Section 3.3) and fuzzy maximum likelihood clustering (Section 3.4). 2.4. The Bayesian Information Criterion For optimizing hyperparameters in several variants of the clustering-based CAE model in Section 3, we employ the Bayesian Information Criterion ( BIC ), which is commonly used as a measure for model selection in the actuarial literature on mortality modeling (see for example Kleinow 2015;Li et al. 2015). It is defined as BIC :=−2 Lmax +log(nobs)·npar (12) for models estimated by maximum likelihood (see Section 2.3), where Lmax is the maximal value of the log-likelihood function, nobs :=Y·A·P(13) is the number of observations and npar is the number of free parameters. Note that this is usually not the same as the total number of model parameters because the identifiability constraints applied when fitting the model already determine some of its parameters (see Appendix Afor details). Risks 2021,9, 45 7 of 32 For models which are not (directly) fit by maximum likehood, we define BIC :=nobs ·log(MSE)+log(nobs)·npar (14) with the mean squared error MSE :=1 nobs P ∑ i=1 xA ∑ x=x1 tY ∑ t=t1log mi x,t−log ˆ mi x,t2. (15) For example, we have log ˆ mi x,t=ˆ αi x+ˆ βxˆ κi t in the CAE model. If we assume that the error terms εi x,t (see for example (8)) follow a normal distribution with mean 0 and variance σ2 ε , it is easily seen that definition (12) and definition (14) coincide up to an additive constant which only depends on the data (see Hastie et al. 2017, p. 233). If we compare two models fit on the same data in terms of their BIC, the definitions are therefore interchangeable. Even if the normality assumption is violated, definition (14) is still plausible because it aims to achieve a balance between goodness of fit (small MSE ) and parsimony (small npar ), both of which are desirable qualities of any statistical model. 3. Clustering-Based Common Age Effect Models In the following, we describe four extensions of the common age effect model based on different clustering algorithms. The first three model extensions are two-step procedures, whose first goal is to find a number k∈ {1, . . . , P}and a surjective function C:{1, . . . , P} → {1, . . . , k} which assigns to every population i∈ { 1, . . . , P} a cluster number C(i)∈ { 1, . . . , k} . From this, the clusters are obtained via Cl:=C−1({l}) , l= 1, . . . , k . In the following, we will sometimes simply refer to a cluster Cl by its number l . In a second step, we formulate the resulting clustering-based CAE model, or—to emphasize the dependence on the clustering—the CAE(k,C)model log mi x,t=αi x+βC(i) xκi t. (16) We will introduce three different methods to identify a suitable clustering function C in Sections 3.1–3.3. Obviously, (16) interpolates between the two extreme cases of the ordinary CAE model—alias the CAE ( 1, C) model with C≡ 1—and the ILC model—alias the CAE (P , C) model with C=id . Given the clustering information k and C , the CAE (k , C) model is calibrated by fitting a CAE model on each cluster. In Section 3.4, we introduce a fourth, one-step model based on the fuzzy clustering paradigm. 3.1. k-Means Clustering The first step of this approach is to fit an ILC model to the populations, yielding one age effect vector βi xx∈RA per population i . These vectors can then be clustered by a standard cluster analysis method. Here, we choose the well-known k -means algorithm (see Hastie et al. 2017, chapter 13). For a given k∈ { 1, . . . , P} , this technique aims at minimizing the squared Euclidean distances between the vectors and the cluster centroids µl∈RA , i.e., min C k ∑ l=1 ∑ i:C(i)=l kβi xx−µlk2, (17) where minimization takes place over all surjective functions C:{ 1, . . . , P}→{ 1, . . . , k} . Given such a clustering function, the centroids are calculated as µl:=1 Cl ∑ i∈Clβi xx. (18) Risks 2021,9, 45 8 of 32 As the minimization problem (17) generally is not solvable exactly, the Hartigan– Wong algorithm as implemented in R Core Team (2019) is used to find an approximate solution. Note that for this approach to be applicable, the number of clusters k has to be specified. We do this by initially running the k -means algorithm for all k∈ { 1, . . . , P} and choosing that value of k for which the BIC of the resulting CAE (k , C) model is minimized (see Section 2.4). 3.2. Augmented Common Factor Clustering The ACF model has been introduced by Li and Lee (2005) with the goal to obtain coherent mortality forecasts (see Section 2.2). In the following, we slightly adapt their methodology and terminology so that it encompasses both the ACF-based clustering algorithm and the algorithm we use for fitting the ACF model. The goal is to find a cluster number k , a clustering function C and model parameters αi x , βi x , κi t , ˜ βC(i) x and ˜ κC(i) t , where the latter two only depend on the cluster C(i) to which a population i∈ { 1, . . . , P} is assigned. These parameters should be calibrated in such a way that death rates are explained well by the following variation of (6) from Section 2.2: log mi x,t=αi x+˜ βC(i) x˜ κC(i) t+βi xκi t+εi x,t. (19) The model parameters and the clusters are determined by a divisive algorithm, of which we give a summary here and further details in Appendix B. As a numerical measure for quantifying the goodness of fit, Li and Lee (2005) propose the explanation ratio Ri,l C:=1− xA ∑ x=x1 tY ∑ t=t1log mi x,t−αi x−˜ βl x˜ κl t2 xA ∑ x=x1 tY ∑ t=t1log mi x,t−αi x2, (20) which describes how well the death rates of a population i belonging to cluster l∈ { 1, . . . , k} are described by the common factor ˜ βl x˜ κl t of this cluster. The model fit can optionally be enhanced by including a population-specific factor βi xκi t . The benefit of including such a factor is measured by the population-specific explanation ratio Ri,l AC :=1− xA ∑ x=x1 tY ∑ t=t1log mi x,t−αi x−˜ βl x˜ κl t−βi xκi t2 xA ∑ x=x1 tY ∑ t=t1log mi x,t−αi x−˜ βl x˜ κl t2. (21) As our aim is to make forecasts with the model, we also have to specify how the period effects are projected into the future. For the estimated cluster-specific period effects ˆ ˜ κl t , we fit a random walk with drift analogously as in the LC model. For the estimated population-specific period effects ˆ κi t, we fit either a random walk without drift ˆ κi t+1=ˆ κi t+ei t+1with ei t+1∼ N 0, σRW i2i.i.d., (22) or an AR(1) process ˆ κi t+1=ci+ϕiˆ κi t+ei t+1with ei t+1∼ N 0, σAR i2i.i.d. (23) Risks 2021,9, 45 15 of 32 Figure 2. Death rates for males aged 53, 60, 67, 74, 81 and 87 in Denmark, France and the USA between 1948 and 2007 (note the differing scales). We split the considered data set into training and test sets so that we can analyze both the goodness of fit (Section 4.3) and the forecasting performance (Section 4.4). More precisely, the data set comprising the years 1948 to 2007 is split into training data from 1948 to 1987, on which the models are fit and goodness of fit is evaluated, and test data from 1988 to 2007, on which forecasting performance is evaluated. 4.2. Clustering Results Figure 3displays the population-specific age effects obtained by the ILC model (colored lines) and the cluster-specific age effects obtained by the k -means, ACF-based and likelihood-ratio-based clustering CAE models in comparison to the ordinary CAE model (black lines). Note that for the k -means and likelihood-ratio-based clustering algorithms the ILC age effects are, in fact, inputs. The ACF does not make direct use of the ILC age effects but these are displayed nonetheless for illustration purposes and for an easier graphical overview of the clustering results. The results of the fuzzy maximum likelihood clustering algorithm are displayed in Figure 4, which shows the cluster membership weights ωi,l in the top row. In the bottom row, we plot the cluster-specific age effects βl xx as well as the fitted population-specific age effects ∑k l=1ωi,lβl xx , where we assign each population i∈ { 1, . . . , 10 } to the cluster l∈ { 1, . . . , k} for which the corresponding weight ωi,l is largest. This clearly illustrates that the fitted age effects are different for all the populations, although for most of them it is also visible that they are a convex combination of a few basic shapes βl xx , l= 1, . . . , k . For k= 2, we observe that β1 xx looks similar to the age effect vector in the ordinary CAE model for both age groups, while β2 xx allows the model to express deviations from the general pattern for some populations. Risks 2021,9, 45 16 of 32 (a) CAE (MLE), y axis restricted (0 to 0.05) (b)k-means (c) ACF-based (d) Likelihood-ratio-based (average linkage) Figure 3. Age effects by cluster for males aged 53 to 87 in 10 countries between 1948 and 1987 obtained by the common age effect (CAE) model and three of its clustering-based extensions. The age effects of the clusters are displayed in black and the individual Lee–Carter (ILC) age effects of the populations are in different colors. We provide an overview of the clustering results in Table 1. Most of the clustering algorithms detect that the Danish age effects follow a very distinct pattern, which is due to the stagnation of Danish death rates in the considered time period, and put Denmark into its own cluster. This is in line with the results of Hatzopoulos and Haberman (2013) even though they consider a different data set (ages 0–90, years 1960–2006 and 35 countries). With fuzzy c -means clustering they obtain one West cluster and two East clusters, where all of the populations considered here are in the West cluster but the fuzzy membership weight of Denmark in this cluster is by far the lowest. Two other groups which could be inferred from our clustering results consist of (i) the English-speaking countries Australia, Canada, the UK, the USA and New Zealand and (ii) the remaining European countries France, Sweden, Switzerland and Austria. In fact, k -means identifies these three groups (Denmark, anglophone countries, remaining European countries) but it assigns Austria to its own cluster. The reason for this gets clearer when we look at the fuzzy maximum likelihood clustering with k= 2 in Figure 4, where we have imposed NNVM constraints to make the following interpretations sensible (see Section 3.4). Here, the Austrian age effects turn out to be almost a 50:50-mixture of the two fuzzy cluster centers, which can be interpreted as a demonstration of the higher degree Risks 2021,9, 45 17 of 32 of flexibility of the fuzzy maximum likelihood clustering when dealing with observations which are harder to classify. In general, the results are similar to k -means but there are some additional nuances we can recognize, for example, France and Sweden exhibit more similar age effects than France and Switzerland under this model. It is remarkable that Denmark does not get a separate cluster despite its quite special behavior, which might be due to the fact that maximum likelihood estimates are driven by high exposures and the Danish population is rather small. The number of fuzzy clusters which minimizes the BIC is k= 4, which is the same as for the k -means algorithm. The clustering results are similar as well: Denmark drives its own cluster with only very small shares of other populations (cluster 4), another cluster is driven by Canada with high shares of Australia, New Zealand and the UK (cluster 1). The USA, however, have a higher degree of membership in cluster 3, along with Switzerland, which is separated from France and Sweden, for which the weights are highest in cluster 2 (as well as for Austria). (a) Weights for k=2 (b) Weights for k=4 (chosen by BIC) (c) Age effects for k=2 (d) Age effects for k=4 (chosen by BIC) Figure 4. Weights ωi,l (top) and cluster-specific age effects βl xx as well as fitted population-specific age effects ∑k l=1ωi,lβl xx (bottom) of the fuzzy maximum likelihood clustering for males aged 53 to 87 in 10 countries between 1948 and 1987. Risks 2021,9, 45 18 of 32 Table 1. Comparison of clustering results obtained by different algorithms for males aged 53 to 87 in 10 countries between 1948 and 1987. Cluster k-Means ACF-Based LikelihoodRatio-Based (AL) Fuzzy ML Clustering (k=2) Fuzzy ML Clustering (k=4) 1 AUT AUS, AUT, CAN, CHE, FRA, SWE, UK, USA AUS AUS, CAN, DNK, NZL, UK, USA AUS, CAN, NZL, UK 2 CHE, FRA, SWE DNK AUT, DNK AUT, CHE, FRA, SWE AUT, FRA, SWE 3 AUS, CAN, NZL, UK, USA NZL CAN - CHE, USA 4 DNK - FRA, UK - DNK 5 - - NZL - - 6 - - SWE, USA - - 7 - - CHE - - Before moving on to the results of the ACF and the likelihood ratio clustering, we discuss some observations in the results of k -means and fuzzy clustering which might be counterintuitive at first glance. Looking at Figure 3b, we find that the cluster-specific age effect β2 xx of the second cluster is very similar to the LC age effect of France. Sweden and Switzerland also belong to this second cluster but their age effects seem to behave quite differently. However, one has to mind the scale of the plots: In fact, the variability of the age effects in cluster 2 is rather small compared to the other clusters. Therefore, while the Swedish and Swiss age effects seem to be very different to the French one on a relative scale, they are nevertheless most similar to it in terms of Euclidean distance compared to the other populations. The observation that the cluster-specific age effect is driven by France results from this age effect being estimated by Poisson maximum likelihood, which is heavily influenced by exposure sizes, and exposures in the French population are substantially larger than in the Swedish and Swiss populations. Furthermore, in Figure 4we make the striking observation that the cluster-specific age effect β2 xx of the second cluster is increasing over ages, whereas the population-specific age effects of all the cluster members (France, Sweden and Austria) are almost constant or even decreasing over ages. To understand this, one has to recall that cluster membership is fuzzy in this case and that all three population-specific age effects are to a significant part influenced by the other cluster-specific age effects as well. Note that β1 xx and β3 xx are mostly decreasing with age. Combining these with β2 xx therefore offsets its increase and yields the almost constant population-specific age effects for France and Sweden. The population-specific age effect of Austria also contains a non-negligible share of β4 xx , the age effect with by far the highest amplitude, which is sufficient to visibly influence the shape of the Austrian age effect in the direction of the age effect of the fourth cluster. In summary, these observations show that population-specific age effects in the fuzzy clustering model are not necessarily similar to any of the cluster-specific age effects. This illustrates the high flexibility of this approach, which allows for individual age effects obtained as linear combinations of a few basic shapes. The ACF clustering algorithm (with explanation ratio threshold η= 0.5 and improvement threshold ρ= 1 as chosen by the validation described in Section 3.2) assigns Denmark and New Zealand to separate clusters. All other populations are put into one group together. Likelihood-ratio-based clustering (with average linkage distances and Risks 2021,9, 45 19 of 32 significance level σ= 10 −6 ) yields the highest number of groups with separate clusters for Australia, New Zealand, Canada and Switzerland, respectively, and three clusters with two populations each (Austria and Denmark, France and the UK, Sweden and the USA). We have also performed some experiments on the robustness of the clustering to small changes in the training data. Switching the considered ages from 53–87 to 55–85, we find that the groupings obtained via k -means, ACF clustering and fuzzy clustering for k= 2 do not change at all while the likelihood-ratio-based clustering and fuzzy maximum likelihood clustering with k= 4 exhibit slight changes. Adding one more year to the training data (so that the clustering is calibrated on the years 1948 to 1988), we observe that the groupings obtained via k -means and fuzzy clustering do not change at all and ACF clustering merges the cluster consisting of New Zealand into the larger cluster consisting of all other populations except Denmark but remains unchanged apart from that. However, there are again some changes in the results of the likelihood-ratio-based clustering. In conclusion, all of the algorithms with the exception of likelihood-ratio-based clustering exhibit some robustness to small variations of the training data. We have also run our algorithms for the 16 male populations of the European countries considered by Guibert et al. (2020) (ages 45–90, years 1960–2014). Their hierarchical clustering method, which they apply simultaneously to male and female populations, assigns the male populations to five different clusters. This is exactly the number which is identified as optimal for our fuzzy clustering algorithm albeit with some differences in the composition of the obtained clusters. Populations which are clustered together by both their and our method are (i) West Germany, France, Portugal and Spain, (ii) Finland and Switzerland, (iii) Italy and Luxembourg and (iv) Norway and the Netherlands. If we consider fuzzy clustering with k= 2, we obtain a cluster consisting of West Germany, Belgium, France, Portugal and Spain—which are the male populations in the first cluster reported by Guibert et al. (2020)—and, additionally, Austria and Switzerland. There are also some similarities between the results of Guibert et al. (2020) and the k -means or likelihood-ratio-based clustering, for example mortality of Danish males being identified as an outlier and assigned its own cluster. However, we also observe several differences in the clustering results. First, this might be due to the fact that our methodology for choosing the number of clusters differs from theirs so that we obtain different cluster sizes at least for the non-fuzzy clustering algorithms ( k -means: 8, ACF clustering: 1, likelihood ratio clustering: 12). More importantly, one should recall that, apart from using a different clustering method, Guibert et al. (2020) cluster period effects while we cluster age effects, which may exhibit differing behaviors between populations and, thus, indicate different optimal clustering results. 4.3. Goodness of Fit For comparing the goodness of fit numerically, we consider the BIC as the main criterion and display it in Table 2. For completeness, we also provide its components, i.e., the maximal value of the log-likelihood function Lmax and the free number of parameters npar (see Section 2.4). Note that the BIC values for the models fitted by Poisson MLE are not directly comparable to those for the models fitted by SVD/cPCA. We observe that, using maximum likelihood estimation under the assumption of Poisson distributed death counts, the ILC model fits better than the CAE model with respect to the BIC . Interestingly, this is reversed if we calibrate the models via SVD/cPCA, which can also be interpreted as maximum likelihood estimation but under the assumption of a normal distribution of the residuals, see Section 2.4. Among the clustering-based CAE models, k -means and fuzzy maximum likelihood clustering achieve the lowest BIC values, which are in particular considerably lower than those for the ILC model or the CAE model (fitted by Poisson MLE). The BIC penalizes the ILC model for its high number of parameters, while the CAE model has fewer parameters but also a significantly lower log-likelihood value. These findings suggest that the clustering-based CAE models strike a Risks 2021,9, 45 20 of 32 better balance between goodness of fit and parsimony compared to the two extreme cases ILC and ordinary CAE. Table 2. The BIC and its components (maximal log-likelihood Lmax and free number of parameters npar ) for males aged 53 to 87 in 10 countries between 1948 and 1987. The BIC values for the models fitted by Poisson MLE are not directly comparable to those for the models fitted by singular value decomposition (SVD)/common principal components analysis (cPCA). Model Lmax npar BIC ACF (SVD) — 1299 −75,684 CAE (cPCA) — 774 −78,483 ILC (SVD) — 1080 −77,550 ILC (MLE) −86,791 1080 183,893 CAE (MLE) −91,299 774 189,988 CAE(k,C), k-means −87,584 876 183,532 CAE(k,C), ACF-based −90,986 842 190,009 CAE(k,C), LR (av. linkage) −88,233 978 185,803 CAE Fuzzy, k=2−87,983 816 183,756 CAE Fuzzy, k=4 (chosen by BIC) −87,054 894 182,643 We illustrate the goodness of fit graphically by plotting for Denmark and France the actual (red, dashed) and the fitted (black, solid) central death rates at age 67 in Figure 5(the models were calibrated on all 10 countries, but we depict only two of them for a clearer illustration). We observe that all models show a reasonable fit on visual inspection. 4.4. Forecasting Performance For evaluating the forecasting performance, we calculate several error measures which are defined in the following and should be minimized in absolute value by the most accurate forecasts: • Bias =1 N N ∑ j=1ˆ yj−yj, • mean absolute error, MAE =1 N N ∑ j=1ˆ yj−yj, • mean absolute percentage error, MAPE =1 N N ∑ j=1 |ˆ yj−yj| yj·100%, • root-mean-square error, RMSE =s1 N N ∑ j=1ˆ yj−yj2, where we denote the number of forecasts by N , the ground truth by yj and the forecast by ˆ yj and write j=J(x , t , i) with age x , year t , population i and a bijective function J from the set of all tuples (x,t,i)for which a forecast is made to {1, . . . , N}. Risks 2021,9, 45 21 of 32 (a) ACF (SVD) (b) ILC (MLE) (c) CAE (MLE) (d) CAE (k,C),k-means (e) CAE (k,C), ACF-based ( f ) CAE (k , C) , Likelihood-ratio-based (average linkage) Figure 5. Cont. Risks 2021,9, 45 22 of 32 (g) CAE Fuzzy (k=2) (h) CAE Fuzzy (k=4, chosen by BIC) Figure 5. Actual (red, dashed) and fitted (black, solid) central death rates for males aged 67 in Denmark and France (models calibrated on 10 countries) between 1948 and 1987. These out-of-sample error measures are displayed in Table 3. We generally observe that all models are biased upwards, indicating that mortality has decreased more than they predict based on the given training data. Generally, the ILC and ACF models perform worse than the other models. The CAE fuzzy clustering model with k= 2 performs best with regard to all error measures. This indicates that the decision to select the value for k by minimizing the BIC , which is a standard approach in the literature, is questionable for this application, as it results in k= 4 and a significantly worse out-of-sample performance (for lower ages, the optimal number of clusters indicated by the BIC is k= 3, and this also leads to inferior out-of-sample results compared to k= 2). Removing the population of Denmark, which as mentioned exhibits a very different fitted age effect pattern compared to the other populations, further enhances the performance of the CAE fuzzy clustering model with k=2 (see the corresponding table in the Supplementary Materials). Table 3. Out-of-sample error measures for males aged 53 to 87 in 10 countries between 1988 and 2007 (trained on 1948 to 1987). Best values in each column are marked in bold. Model Bias MAE MAPE RMSE ACF (SVD) 5.92h6.77h20.91% 9.65h CAE (cPCA) 5.97h6.72h19.95% 9.80h ILC (SVD) 5.94h6.76h20.40% 9.99h ILC (MLE) 6.01h6.75h20.38% 10.05h CAE (MLE) 6.14h6.75h19.64% 9.82h CAE(k,C), k-means 6.04h6.68h20.29% 9.94h CAE(k,C), ACF-based 6.06h6.68h19.98% 9.65h CAE(k,C), LR (av. linkage) 6.37h7.01h20.04% 10.58h CAE Fuzzy, k=25.90h6.47h19.63% 9.43h CAE Fuzzy, k=4 (chosen by BIC) 6.03h6.76h20.27% 10.16h Risks 2021,9, 45 23 of 32 As a graphical illustration of the projection results, we plot the forecast (black, solid) against the realized (red, dashed) central death rates for age 67 in Figure 6(the models were calibrated on all 10 countries, but we depict only Denmark and France for a clearer illustration). Additionally, we include 95% prediction intervals (black, dotted). The figures confirm that all models are biased upwards in their out-of-sample projections. Moreover, we find that the uncertainty about these projections differs depending on the model and on the population. The observed death rates for Canada, Denmark and New Zealand decline faster than most models anticipate, even when taking into account forecast uncertainty. In particular, we observe a change in the trend of Danish mortality starting around 1995 which, unsurprisingly, none of the models can foresee. To check the robustness of our results, we have evaluated the algorithms on three other data sets. Tables with the corresponding out-of-sample error measures can be found in the Supplementary Materials. There, we first present the model-specific error measures for the 21 countries which are considered by Li and Lee (2005) for their illustration of the ACF model. Unsurprisingly, the ACF model shows the best out-of-sample performance for these countries. Among the one-factor models, the fuzzy clustering model with k= 2 ranks first, in particular clearly outperforming the ordinary CAE model. Moreover, we show a further evaluation for the same 10 populations we have considered throughout this section but for different years, training on 1960 to 1999 and testing on 2000 to 2013. Here, the fuzzy clustering model works better with k= 3 than with k= 2 but it dominates the ordinary CAE model with respect to every error measure for both values of k . In addition, it has to be noted that the ILC model fitted via SVD performs surprisingly well for these data. Finally, we have evaluated our clustering-based CAE models and the benchmarks on the 10 populations we have considered in this section with the exception of New Zealand (data only available up to 2013) on even more recent data. We trained the models on the years 1958 to 1997 and evaluated them on the years 1998 to 2017. The results are qualitatively similar to the previous study. Bias and MAPE are minimized by the ILC model fitted via SVD. The fuzzy clustering model with k= 3 works slightly better than with k= 2 and clearly better than the ordinary CAE model. The CAE model based on k -means clustering also performs very well, yielding the lowest MAE and RMSE . In summary, we can conclude that no single model stands out as best for all data sets or all error measures. Depending on the data, the clustering-based extensions of the CAE model, in particular the fuzzy clustering model, can be a better alternative than the ordinary CAE model. As an additional robustness test, a comparison to the out-of-sample performance results of other papers employing clustering methods for mortality forecasting would be interesting. Unfortunately, neither Hatzopoulos and Haberman (2013) nor Guibert et al. (2020) provide such forecasting performance measures, and the results presented by Danesi et al. 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