# Applying spectral biclustering to mortality data

^{*}

## Abstract

**:**

## 1. Introduction

_{x,}

_{t}} of log-mortality data for the age x at time t in more detail: x is an integer number in the range [0, 120], and t an integer representing the year of observation (to make an example: m

_{58,2001}is the log-mortality rate recorded in 2001 for individuals 58 years old), so that M has 121 rows and a number of columns depending on the overall number of years for which log-mortality data are available. The model seeks to summarize an age-period surface of log–mortality rates in terms of the vectors

**a**and

**b**along the age dimension, and

**k**along the time dimension: log m

_{x,}

_{t}= a

_{x}+ b

_{x}k

_{t}+ ε

_{x,}

_{t}, for every x ∈ [0, 120], and for every t, with restrictions such that the “b”s are normalized to sum to one, the “k”s sum to zero, and the “a”s are average log rates. The vector

**a**can be interpreted as an average age profile, tracking mortality changes over time; the vector

**b**determines how much each age group changes when k

_{t}changes. Finally, the error term ε

_{x,}

_{t}reflects age-period effects eventually not captured by the model. In the basic model, the fit to historical data is made through the Singular Value Decomposition (SVD) [2] of M, and then the time-varying parameter is modeled and forecasted as an ARIMA process using standard Box-Jenkins methodology.

## 2. Methodological Aspects of Spectral Biclustering

#### 2.1. Biclustering

#### 2.2. Spectral Biclustering

^{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:

_{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}.

**Box 1.**Spectral biclustering: the algorithm.

_{N×M}: gene expression matrix.

**1**

_{M}) and C = diag(

**1**

_{M}

^{T}D)

^{−1/2}D C

^{−1/2}

^{−1}DC

^{−1}D

^{T}and C

^{−1}D

^{T}R

^{−1}D with the same eigenvalue do:

## 3. Discussion Case

#### 3.1. Demographical Settings and Notational Conventions

_{0}, l

_{1}, ... , 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 l

_{0}, l

_{1}, ... , lω is the longitudinal observation of the actual numbers of individuals alive at ages 1, 2, . . . , ω out of a given initial cohort of l

_{0}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.

#### 3.2. Empirical Results

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A: The Euclidean Hierarchical Clustering Method

- Assigning each item to a cluster, so that if you have N items, you now have N clusters, each containing just one item. The distances between the clusters are assumed to be the same as the distances (similarities) between the items they contain.
- Find the closest (i.e., the most similar) pair of clusters and merge them into a single cluster, hence reducing by one the original number of clusters, as defined in 1.
- Compute the distances between the new cluster and each of the old clusters.
- Repeat steps 2 and 3 until the items are clustered into the desired number k of clusters.

## References

- D. Lee, and L. Carter. “Modeling and Forecasting U.S. Mortality.” J. Am. Stat. Assoc. 87 (1992): 659–671. [Google Scholar] [CrossRef]
- G. Golub, and C. Van Loan. Matrix Computations, 3rd ed. Baltimore MD, USA: Johns Hopkins University Press, 1996. [Google Scholar]
- H. Booth, R. Hyndman, and L. Tickle. “Lee-Carter mortality forecasting: A multi-country comparison of variants and extensions.” Demogr. Res. 15 (2006): 289–310. [Google Scholar] [CrossRef]
- H. Booth, J. Maindonald, and L. Smith. “Applying Lee-Carter under conditions of variable mortality decline.” Popul. Stud. 56 (2002): 325–336. [Google Scholar] [CrossRef] [PubMed]
- Z. Butt, S. Haberman, R. Verrall, and V. Wass. “Calculating compensation for loss of earnings: Estimating and using work life expectancy.” J. R. Stat. Soc. Ser. A 171 (2008): 763–805. [Google Scholar] [CrossRef]
- A. Delwarde, M. Denuit, and P. Eilers. “Smoothing the Lee-Carter and Poisson log-bilinear models for mortality forecasting. A penalized loglikelihood approach.” Stat. Model. 7 (2007): 29–48. [Google Scholar] [CrossRef]
- I. Currie, M. Durban, and P. Eilers. “Smoothing and forecasting mortality rates.” Stat. Model. 4 (2004): 279–298. [Google Scholar] [CrossRef]
- R. Hyndman, and S. Ullah. “Robust forecasting of mortality and fertility rates: A functional data approach.” Comput. Stat. Data Anal. 51 (2007): 4942–4956. [Google Scholar] [CrossRef]
- S. Richards. “Understanding Pensioner Longevity.” The Actuary Magazine, 26–27 May 2007. [Google Scholar]
- A. Renshaw, and S. Haberman. “A cohort-based extension to the Lee-Carter model for mortality reduction factors.” Insur. Math. Econ. 38 (2006): 556–570. [Google Scholar] [CrossRef]
- R. Willets. “The cohort effect: Insights and explanations.” Br. Actuar. J. 10 (2004): 833–877. [Google Scholar] [CrossRef]
- A. Renshaw, and S. Haberman. Lee-Carter Mortality Forecasting Incorporating Bivariate Time Series. Actuarial Research Paper 153; London, UK: Faculty of Actuarial Science Insurance, City University London, 2003. [Google Scholar]
- Y. Leong, and J. Yu. “A Spatial Cluster Modification of the Lee-Carter Model.” In Proceedings of the Longevity Risks 8, Cass Business School, London, UK, 7–8 September 2012. [Google Scholar]
- C. Skiadas, and C. Skiadas. “A Modeling Approach to Life Table Data Sets.” In Recent Advances in Stochastic Modeling and Data Analysis. Singapore, Singapore: World Scientific, 2007, pp. 350–359. [Google Scholar]
- P. Hatzopoulos, and S. Haberman. “Common mortality modelling and coherent forecasts. An empirical analysis of worldwide mortality data.” Insur. Math. Econ. 52 (2013): 320–337. [Google Scholar] [CrossRef]
- Y. Cheng, and G. Church. “Biclustering of expression data.” In Proceedings of the International Conference on Intelligent Systems for Molecular Biology, San Diego, CA, USA, 19–23 August 2000; Volume 8, pp. 93–103. [Google Scholar]
- J. Ihmels, G. Friedlander, S. Bergmann, O. Sarig, and Y. Ziv. “Revealing Modular Organization in the Yeast Transcriptional Network.” Nat. Genet. 31 (2002): 370–377. [Google Scholar] [CrossRef] [PubMed]
- T. Murali, and S. Kasif. “Extracting conserved gene expression motifs from gene expression data.” In Proceedings of the Pacific Symposium on Biocomputing, Kauai, HI, USA, 3–7 January 2003; Volume 8, pp. 77–88. [Google Scholar]
- S. Madeira, and A. Oliveira. “Biclustering algorithms for biological data analysis: A survey.” IEEE/ACM Trans. Comput. Biol. Bioinform. 1 (2004): 24–45. [Google Scholar] [CrossRef] [PubMed]
- E. Segal, B. Taskar, A. Gasch, N. Friedman, and D. Koller. “Rich probabilistic models for gene expression.” Bioinformatics 17 (2001): S243–S252. [Google Scholar] [CrossRef] [PubMed]
- A. Tanay, R. Sharan, and R. Shamir. “Discovering statistically significant biclusters in gene expression data.” Bioinformatics 18 (2002): 136–144. [Google Scholar] [CrossRef]
- H. Wang, W. Wang, J. Yang, and P. Yu. “Clustering by Pattern Similarity in Large Data Sets.” In Proceedings of the 2002 ACM SIGMOD International Conference on Management of Data, Madison, WI, USA, 3–6 June 2002; pp. 394–405. [Google Scholar]
- D. Jiang, J. Pei, and A. Zhang. “DHC: A density-based hierarchical clustering method for time series gene expression data.” In Proceedings of the IEEE International Symposium on Bioinformatics and Bioengineering, Bethesda, MD, USA, 12 March 2003; pp. 393–400. [Google Scholar]
- J. Liu, and W. Wang. “Op-cluster: Clustering by tendency in high dimensional space.” In Proceedings of the IEEE International Conference on Data Mining, Melbourne, FL, USA, 22 November 2003; p. 187. [Google Scholar]
- J. Gu, and J. Liu. “Bayesian biclustering of gene expression data.” BMC Genom. 9 (2008): S4. [Google Scholar] [CrossRef] [PubMed]
- G. Li, Q. Ma, H. Tang, A. Paterson, and Y. Xu. “QUBIC: A qualitative biclustering algorithm for analyses of gene expression data.” Nucleic Acids Res. 37 (2009): e1015. [Google Scholar] [CrossRef] [PubMed]
- Y. Kluger, R. Basri, J. Chang, and M. Gerstein. “Spectral biclustering of microarray data: Coclustering genes and conditions.” Genome Res. 13 (2003): 703–716. [Google Scholar] [CrossRef] [PubMed]
- S.C. Johnson. “Hierarchical clustering schemes.” Psychometrika 3 (1967): 241–254. [Google Scholar] [CrossRef]

Cluster | Age | Years |
---|---|---|

1 | 58–60 | 1987–1993 |

2 | 58–60 | 1970–1986 |

3 | 58–60 | 1952, 1956, 1957, 1962 |

4 | 58–60 | 1994–2006 |

5 | 58–60 | 1950–1969 except for 1952, 1956, 1957, 1962 |

6 | 48–53 | 1987–1993 |

7 | 48–53 | 1970–1986 |

8 | 48–53 | 1952, 1956, 1957, 1962 |

9 | 48–53 | 1994–2006 |

10 | 48–53 | 1950–1969 except for 1952, 1956, 1957, 1962 |

11 | 40–47 | 1987–1993 |

12 | 40–47 | 1970–1986 |

13 | 40–47 | 1952, 1956, 1957, 1962 |

14 | 40–47 | 1994–2006 |

15 | 40–47 | 1950–1969 except for 1952, 1956, 1957, 1962 |

16 | 54–57 | 1987–1993 |

17 | 54–57 | 1970–1986 |

18 | 54–57 | 1952, 1956, 1957, 1962 |

19 | 54–57 | 1994–2006 |

20 | 54–57 | 1950–1969 except for 1952, 1956, 1957, 1962 |

Cluster | Row Effect | Column Effect |
---|---|---|

1 | 9.457631e-19 | 3.080849e-12 |

2 | 2.154947e-22 | 1.920493e-09 |

3 | 1.539265e-20 | 2.823244e-16 |

4 | 7.044707e-17 | 3.358952e-11 |

5 | 7.679044e-32 | 3.538135e-17 |

6 | 3.098577e-31 | 1.597127e-19 |

7 | 1.830569e-77 | 1.243124e-42 |

8 | 1.646694e-63 | 1.872372e-38 |

9 | 1.909664e-37 | 1.459496e-07 |

10 | 4.317890e-57 | 3.326384e-28 |

11 | 8.420389e-19 | 1.149809e-07 |

12 | 1.218959e-22 | 4.788635e-08 |

13 | 3.097022e-23 | 6.420247e-18 |

14 | 5.204899e-20 | 1.365834e-14 |

15 | 1.877757e-30 | 5.348786e-11 |

16 | 1.244019e-15 | 7.830883e-10 |

17 | 7.935750e-44 | 6.552596e-30 |

18 | 2.060731e-33 | 5.490419e-24 |

19 | 3.489601e-17 | 4.602981e-06 |

20 | 3.741736e-37 | 9.304545e-23 |

Cluster | Constant Variance | Additive Variance |
---|---|---|

1 | 0.0008312501 | 0.0001943193 |

2 | 0.0010608750 | 0.0007238702 |

3 | 0.0008112594 | 0.0002344328 |

4 | 0.0008110202 | 0.0002252854 |

5 | 0.0009152584 | 0.0003184399 |

6 | 0.0006558884 | 0.0002591269 |

7 | 0.0011777088 | 0.0004089964 |

8 | 0.0010524882 | 0.0001846966 |

9 | 0.0006908077 | 0.0001800295 |

10 | 0.0012443249 | 0.0005012390 |

11 | 0.0004307733 | 0.0001486275 |

12 | 0.0008466223 | 0.0002278785 |

13 | 0.0007208565 | 0.0002152609 |

14 | 0.0004183225 | 0.0001126892 |

15 | 0.0009253514 | 0.0004143284 |

16 | 0.0007953189 | 0.0002157055 |

17 | 0.0010980629 | 0.0006934491 |

18 | 0.0009243758 | 0.0002063028 |

19 | 0.0007612718 | 0.0002334575 |

20 | 0.0010023427 | 0.0003085179 |

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Piscopo, G.; Resta, M.
Applying spectral biclustering to mortality data. *Risks* **2017**, *5*, 24.
https://doi.org/10.3390/risks5020024

**AMA Style**

Piscopo G, Resta M.
Applying spectral biclustering to mortality data. *Risks*. 2017; 5(2):24.
https://doi.org/10.3390/risks5020024

**Chicago/Turabian Style**

Piscopo, Gabriella, and Marina Resta.
2017. "Applying spectral biclustering to mortality data" *Risks* 5, no. 2: 24.
https://doi.org/10.3390/risks5020024