The goal of clustering is to partition the elements into sets (clusters), while trying to both optimize groups homogeneity (i.e., elements of a cluster should be highly similar to each other), and group separation, which is to say: elements from different clusters should have low similarity to each other. Clustering may be a very powerful tool to automatically detect relevant sub–groups when one does not have prior knowledge about the hidden structure of these data. However, an important issue related to clustering techniques was firstly investigated in the past decade while identifying local patterns in gene expression data: it was highlighted that cluster analysis makes several a priori assumptions [16] that may not be perfectly adequate in all circumstances. As a matter of fact, clustering can be applied to either genes or samples, implicitly directing the analysis to a peculiar aspect of the system under study; in addition, clustering algorithms usually seek a disjoint cover of the set of elements, requiring that no gene or sample belongs to more than one cluster.

The notion of bicluster then gave rise to a more flexible computational framework. Given a gene expression matrix, we can characterize the biological phenomena that it embodies by a collection of biclusters, each representing a different type of joint behavior of a set of genes in a corresponding set of samples. Moreover, as clustering can be separately applied to either the rows or the columns of the data matrix; biclustering, on the other hand, performs clustering in these two dimensions simultaneously. This, in turn, means that while clustering derives a global model, biclustering produces a local model [19]: likewise, in genomics, this aspect is of importance within the mortality context because of the interest of actuaries for age-specific mortality patterns.

As already mentioned in Section 1, in fact, scholars are intended to capture three relevant aspects: the period, the age and the cohort effects, as their knowledge is a key factor in understanding actuarial liabilities of private life insurance companies, pension funds and national pension systems.

Following the work of [16], biclustering has quickly become popular in analyzing gene expression data and various biclustering algorithms have been proposed [17,18,19,20,21,22,23,24,25,26]. Since each algorithm focuses on identification of different bicluster patterns, it is a very challenging task to thoroughly evaluate these algorithms.

The rationale inside the spectral biclustering method [27] is that checkerboard structures may emerge in matrices of expression data once they have been properly arranged using a linear algebra approach, by means of the Singular Value Decomposition (SVD). Using SVD, the data matrix D of dimensions N × M can be decomposed as D = UΛV^T, where Λ is a diagonal matrix with decreasing non-negative entries, and U and V are orthonormal column matrices with dimensions N × min(N,M) and M × min(N,M), respectively. If the data matrix has a block diagonal structure (with all elements outside the blocks equal to zero), then each block can be associated with a bicluster. Specifically, if the data matrix is of the form: (1) D = [ D 1 0   … 0 0 D 2   … 0 ⋮ ⋮   ⋱ ⋮ 0 0   … D k ] , where D_i (i = 1, . . . , k) are arbitrary matrices, then, for each D_i, there will be a singular vector pair (u_i, v_i) such that a nonzero component of u_i corresponds to rows occupied by D_i, and a nonzero component of v_i corresponds to columns occupied by D_i.

In a less idealized case, when the elements outside the diagonal blocks are not necessarily zeros but the diagonal blocks still contain dominating values, the SVD can reveal the biclusters as dominating components in the singular vector pairs.

Box 1 illustrates the algorithmic details of the procedure, as reported in [27].

In our case, the choice of the biclustering method was suggested by the analogies existing between spectral biclustering and the Singular Value Decomposition on which the Lee–Carter method [1]—LCm—is based.

Furthermore, the starting point of the introduction of biclustering method in mortality context is to design a clustering approach suitable for the kind of data we are working on. Most biclustering algorithms, in fact, are apt to binary data only, and, therefore, they are not appropriate for mortality rates. Instead, spectral biclustering appears suitable for death rates matrix because it finds the checkerboard structure in data through the extraction of significant eigenvectors using the SVD associated to some normalization steps; this approach is also in line with the most used approach for fitting mortality data, the LCm, based on the SVD of the mortality matrix. Moreover, the normalization step reduces the effects of random differences in experimental conditions and levels of genes, and it is useful to highlight biclusters, if a structure exists. Finally, as a common practice in biclustering analysis is transforming the data by taking logarithms, we believe the log-normalization is appropriate in a mortality context because two or more ages are experiencing the same evolution if their mortality profiles are constantly proportional, and the log-transformation is not affected by an additive constant.

In the context of mortality data, demographers and actuaries work on life tables. A life table is a finite decreasing sequence l₀, l₁, ... , lω where l_x refers to an integer age x and represents the estimated number of people alive at x in a given population composed by l ₀ individuals aged 0 at inception; note that ω commonly indicates the so-called extreme age, representing the age at which it occurs: lω = 0: for sake of convenience, we can assume ω = 120. A cohort table is obtained if the sequence l₀, l₁, ... , lω is the longitudinal observation of the actual numbers of individuals alive at ages 1, 2, . . . , ω out of a given initial cohort of l₀ newborns. If we consider an existing population and observe the frequency of death at different ages in a given period, for example, one year, then we obtain the period table. Finally, M = {log m_ij} is usually employed to denote the matrix of death rates of a particular population, where the data are organized in rows by age, and in columns by years, so that m_i,j is the mortality rate at the age i (i = 0, . . . ,ω) in the year j.

In our case study, we considered the death rates of the whole (male and female) Italian population; the data were downloaded from the Human Mortality Database 1 (HMD). The genes are represented by ages collected between 40 and 60, and the conditions by the years between 1950 and 2006. We focus on the ages in the range [40, 60] because those ages are the most important in pensions and actuarial product design.

We applied to data both the hierarchical Euclidean clustering (whose algorithm is provided in Appendix A, at the end of the paper) and the spectral biclustering, already explained in Box 1 of Section 2, and visualized the results through heatmaps.

The first step of our analysis consists in grouping the data through a Euclidean hierarchical cluster method; we applied the algorithm to the columns and rows of death rates matrix separately, and Figure 1 and Figure 2 show the corresponding heat maps.

Clustering performs data in one dimension at a time, so each gene in a given gene cluster is defined using all the conditions, and, similarly, each condition in a condition cluster is characterized by the activity of all the genes that belong to it. From a graphical point of view, this produces vertical regions that appear in Figure 1 and Figure 2, where the period effect and the age effect become visible: moving from yellow regions to red regions, the mortality decreases during years and increases across ages. We also highlight the presence of the cohort effect, as some generations have experienced specific mortality patterns differently from others. If the cohort effect is present, a kind of “ladder” appears in the heatmap: genes aged x in t, x + 1 in t + 1, and so on are grouped in the same cluster and are represented with the same color shade in the heatmap.

In more detail, in Figure 1, we may observe a diagonal line, corresponding to the individuals aged 40 in 1958, i.e., the generation born in 1918; such graphical evidence suggests that this group has experienced a particular reduction in mortality during the years.

However, it is not reasonable that groups of genes behave similarly under all conditions; it could occur, for example, that group of ages show similar features just during some years, and they cluster differently during other years. To capture this detailed structure, we introduce the biclustering as a useful tool to find meaningful checkerboard patterns.

In the second step of our application, we have therefore randomly applied the spectral biclustering algorithm to the matrix of mortality rates, normalized through the log-transformation, thus obtaining 20 clusters: main results are visually summarized in Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7. Above all, the heat map of Figure 3 shows not only vertical regions, but also rectangular areas produced by the interaction between age and period effects.

Moreover, Figure 4 and Figure 5 highlight the composition of each bicluster, summarized in Table 1: four groups of ages (40–47, 48–53, 54–57, 58–60) are defined, each of them characterized by significant patterns across particular years. For example, the first bicluster contains ages 58–60 during the years 1987–1993; the same ages act differently during the years 1970–1986 and are grouped in the second bicluster, and so on.

In order to provide the reader with additional interpretative keys, in Figure 6 and Figure 7, we show parallel coordinate graphs for mortality levels.

Let us take a look, for instance, at the first two clusters including the levels of mortality between the ages 58 and 60 during the years 1987–1993 and 1987–1993. In the parallel coordinate graphs, the light gray lines represent the mortality for each year of the whole dataset, while the bold lines represent the level of mortality for each year included in the bicluster. By a mutual comparison of the lines in the first two boxes of Figure 7, we can note that the mortality between the ages 58–60 moving from the period 1970–1986 to the period 1987–1993 has sensitively decreased. In particular, we can note the effect of the reduction of mortality trend during the period 1950–2006, looking in sequence at box 5-2-1-4 of Figure 7. The years included in the third bicluster represent an exception to the general trend, probably due to accidental deviations in the mortality trend. In the actuarial practice, it is well known that the demographic risk can be split into two components: the insurance risk and the longevity risk. The former arises from accidental deviations of the number of deaths from its expected trend, while the latter derives from systematic improvements in the mortality trend. It is possible to improve mortality forecasts removing the accidental deviations from the dataset on which the fitting and forecasts are implemented and the bicluster method can be useful to this scope.

Similar considerations can be easily extended to the other biclusters; for instance, as it is clear from biclusters 4-9-14-19, in Figure 6 and Figure 7, all ages have experimented a reduction from the mortality trend during the last two decays.

We also checked the statistical significance and the stability of the applied biclustering technique. To this aim, F-statistics are calculated from the two-way ANOVA mode with the row and column effect; low p-values denote non-random selection of columns for a given bicluster. We have calculated the F-statistics for each cluster and have obtained p-values of almost zero for both row and column effects. Table 2 shows the p-values of the F-statistics corresponding to the row and column effects.

Moreover, for each cluster, Constant and Additive Variances are calculated, as one can see in Table 3. They represent measures of how a bicluster is coherent, following [19], and return the corresponding variance of genes or conditions as the average of the sum of Euclidean distances between all rows and columns of the bicluster. The lower the values returned, the more constant or coherent the bicluster is. If the value returned is near 0, the bicluster is ideally constant or coherent. Usually, a value above 1–1.5 is enough to determine if the bicluster is not constant or coherent.