Hostname: page-component-848d4c4894-ttngx Total loading time: 0 Render date: 2024-06-08T02:00:26.237Z Has data issue: false hasContentIssue false
  • English
  • Français

Expressive mortality models through Gaussian process kernels

Published online by Cambridge University Press:  15 February 2024

Jimmy Risk Open the ORCID record for Jimmy Risk [Opens in a new window]  and
Mike Ludkovski Open the ORCID record for Mike Ludkovski [Opens in a new window]
Jimmy Risk*
Affiliation:
Mathematics & Statistics, Cal Poly Pomona, Pomona, CA 91676, USA
Mike Ludkovski
Affiliation:
Statistics & Applied Probability, University of California, Santa Barbara, CA 93106-3110, USA
*
Corresponding author: Jimmy Risk; Email: jrisk@cpp.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We develop a flexible Gaussian process (GP) framework for learning the covariance structure of Age- and Year-specific mortality surfaces. Utilizing the additive and multiplicative structure of GP kernels, we design a genetic programming algorithm to search for the most expressive kernel for a given population. Our compositional search builds off the Age–Period–Cohort (APC) paradigm to construct a covariance prior best matching the spatio-temporal dynamics of a mortality dataset. We apply the resulting genetic algorithm (GA) on synthetic case studies to validate the ability of the GA to recover APC structure and on real-life national-level datasets from the Human Mortality Database. Our machine learning-based analysis provides novel insight into the presence/absence of Cohort effects in different populations and into the relative smoothness of mortality surfaces along the Age and Year dimensions. Our modeling work is done with the PyTorch libraries in Python and provides an in-depth investigation of employing GA to aid in compositional kernel search for GP surrogates.

Keywords

Gaussian process kernel engineeringgenetic algorithmsmortality surfacesHuman Mortality Database
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 54 , Issue 2 , May 2024 , pp. 327 - 359
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The International Actuarial Association

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adler, R.J. (2010) The Geometry of Random Fields. SIAM.CrossRefGoogle Scholar
Ahmadi, S.S. and Gaillardetz, P. (2014) Two factor stochastic mortality modeling with generalized hyperbolic distribution. Journal of Data Science, 12, 1–18.CrossRefGoogle Scholar
Azman, S. and Pathmanathan, D. (2022) The GLM framework of the Lee–Carter model: A multi-country study. Journal of Applied Statistics, 49(3), 752–763.CrossRefGoogle Scholar
Berlinet, A. and Thomas-Agnan, C. (2011) Reproducing Kernel Hilbert Spaces in Probability and Statistics. New York: Springer Science & Business Media.Google Scholar
Brouhns, N., Denuit, M. and Vermunt, J.K. (2002) A Poisson log-bilinear regression approach to the construction of projected lifetables. Insurance: Mathematics and Economics, 31(3), 373–393.Google Scholar
Cairns, A.J., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D., Ong, A. and Balevich, I. (2009) A quantitative comparison of stochastic mortality models using data from England and Wales and the United States. North American Actuarial Journal, 13(1), 1–35.CrossRefGoogle Scholar
Cairns, A.J., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D. and Khalaf-Allah, M. (2011) Mortality density forecasts: An analysis of six stochastic mortality models. Insurance: Mathematics and Economics, 48(3), 355–367.Google Scholar
Cole, D.A., Gramacy, R.B. and Ludkovski, M. (2022) Large-scale local surrogate modeling of stochastic simulation experiments. Computational Statistics & Data Analysis, 107537.CrossRefGoogle Scholar
Dittrich, D., Leenders, R.T.A. and Mulder, J. (2019) Network autocorrelation modeling: A Bayes factor approach for testing (multiple) precise and interval hypotheses. Sociological Methods & Research, 48(3), 642–676.CrossRefGoogle Scholar
Dowd, K., Cairns, A.J. and Blake, D. (2020) CBDX: A workhorse mortality model from the Cairns–Blake–Dowd family. Annals of Actuarial Science, 14(2), 445–460.CrossRefGoogle Scholar
Duvenaud, D. (2014) Automatic model construction with Gaussian processes. Ph.D. Thesis, University of Cambridge.Google Scholar
Duvenaud, D., Lloyd, J., Grosse, R., Tenenbaum, J. and Zoubin, G. (2013) Structure discovery in nonparametric regression through compositional kernel search. International Conference on Machine Learning, pp. 1166–1174. PMLR.Google Scholar
Gardner, J., Pleiss, G., Weinberger, K.Q., Bindel, D. and Wilson, A.G. (2018) GpyTorch: Blackbox matrix-matrix Gaussian Process inference with GPU acceleration. Advances in Neural Information Processing Systems, 31.Google Scholar
Gelman, A., Carlin, J.B., Stern, H.S. and Rubin, D.B. (1995) Bayesian Data Analysis. Boca Raton, FL: Chapman and Hall/CRC.CrossRefGoogle Scholar
Genton, M.G. (2001) Classes of kernels for machine learning: A statistics perspective. Journal of Machine Learning Research, 2, 299–312.Google Scholar
Hastie, T. and Tibshirani, R. (1990) Generalized Additive Models. Monographs on Statistics & Applied Probability. Boca Raton, FL: Chapman and Hall/CRC.Google Scholar
Hunt, A. and Blake, D. (2014) A general procedure for constructing mortality models. North American Actuarial Journal, 18(1), 116–138.CrossRefGoogle Scholar
Huynh, N. and Ludkovski, M. (2021a) Joint models for cause-of-death mortality in multiple populations. arXiv preprint arXiv:2111.06631.Google Scholar
Huynh, N. and Ludkovski, M. (2021b) Multi-output Gaussian processes for multi-population longevity modelling. Annals of Actuarial Science, 15(2), 318–345.CrossRefGoogle Scholar
Jähnichen, P., Wenzel, F., Kloft, M. and Mandt, S. (2018) Scalable generalized dynamic topic models. International Conference on Artificial Intelligence and Statistics, pp. 1427–1435. PMLR.Google Scholar
Jeffreys, H. (1961) The Theory of Probability. Oxford: Oxford University Press.Google Scholar
Jin, S.-S. (2020) Compositional kernel learning using tree-based genetic programming for Gaussian process regression. Structural and Multidisciplinary Optimization, 62(3), 1313–1351.CrossRefGoogle Scholar
Kanagawa, M., Hennig, P., Sejdinovic, D. and Sriperumbudur, B.K. (2018) Gaussian processes and kernel methods: A review on connections and equivalences. arXiv preprint arXiv:1807.02582.Google Scholar
Kingma, D.P. and Ba, J. (2014) Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980.Google Scholar
Lee, M.D. and Wagenmakers, E.-J. (2014) Bayesian Cognitive Modeling: A Practical Course. Cambridge University Press.CrossRefGoogle Scholar
Lee, R. (2000) The Lee-Carter method for forecasting mortality, with various extensions and applications. North American Actuarial Journal, 4(1), 80–91.CrossRefGoogle Scholar
Ludkovski, M., Risk, J. and Zail, H. (2018) Gaussian process models for mortality rates and improvement factors. ASTIN Bulletin: The Journal of the IAA, 48(3), 1307–1347.CrossRefGoogle Scholar
Luke, S. and Panait, L. (2006) A comparison of bloat control methods for genetic programming. Evolutionary Computation, 14(3), 309–344.CrossRefGoogle ScholarPubMed
Mehler, F.G. (1866) Ueber die entwicklung einer function von beliebig vielen variablen nach laplaceschen functionen höherer ordnung.CrossRefGoogle Scholar
Murphy, M. (2010) Reexamining the dominance of birth cohort effects on mortality. Population and Development Review, 36(2), 365–390.CrossRefGoogle ScholarPubMed
Nigri, A., Levantesi, S., Marino, M., Scognamiglio, S. and Perla, F. (2019) A deep learning integrated Lee–Carter model. Risks, 7(1), 33.CrossRefGoogle Scholar
Noack, M.M. and Sethian, J.A. (2021) Advanced stationary and non-stationary kernel designs for domain-aware Gaussian processes. arXiv preprint arXiv:2102.03432.Google Scholar
Parzen, E. (1961) An approach to time series analysis. The Annals of Mathematical Statistics, 32(4), 951–989.CrossRefGoogle Scholar
Perla, F., Richman, R., Scognamiglio, S. and Wüthrich, M.V. (2021) Time-series forecasting of mortality rates using deep learning. Scandinavian Actuarial Journal, 2021(7), 572–598.CrossRefGoogle Scholar
Poli, R., Langdon, W.B., McPhee, N.F. and Koza, J.R. (2008) A Field Guide to Genetic Programming. Springer.Google Scholar
Renshaw, A.E. and Haberman, S. (2006) A cohort-based extension to the Lee–Carter model for mortality reduction factors. Insurance: Mathematics and Economics, 38(3), 556–570.Google Scholar
Richman, R. and Wüthrich, M.V. (2021) A neural network extension of the Lee–Carter model to multiple populations. Annals of Actuarial Science, 15(2), 346–366.CrossRefGoogle Scholar
Roman, I., Santana, R., Mendiburu, A. and Lozano, J.A. (2021) Evolving Gaussian process kernels from elementary mathematical expressions for time series extrapolation. Neurocomputing, 462, 426–439.CrossRefGoogle Scholar
Roustant, O., Ginsbourger, D. and Deville, Y. (2012) Dicekriging, diceoptim:n Two r packages for the analysis of computer experiments by kriging-based metamodeling and optimization. Journal of Statistical Software, 51, 1–55.CrossRefGoogle Scholar
Schölkopf, B., Smola, A.J. and Bach, F. (2002) Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. Cambridge, MA: MIT Press.Google Scholar
Shawe-Taylor, J. and Cristianini, N. (2004) Kernel Methods for Pattern Analysis. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Sipper, M., Fu, W., Ahuja, K. and Moore, J.H. (2018) Investigating the parameter space of evolutionary algorithms. BioData Mining, 11(1), 1–14.CrossRefGoogle ScholarPubMed
Villegas, A., Kaishev, V.K. and Millossovich, P. (2015) StMoMo: An R package for stochastic mortality modelling. 7th Australasian Actuarial Education and Research Symposium.CrossRefGoogle Scholar
Wang, C.-W., Huang, H.-C. and Liu, I.-C. (2011) A quantitative comparison of the Lee-Carter model under different types of non-Gaussian innovations. The Geneva Papers on Risk and Insurance-Issues and Practice, 36, 675–696.CrossRefGoogle Scholar
Willets, R.C. (2004) The cohort effect: Insights and explanations. British Actuarial Journal, 10(4), 833–877.CrossRefGoogle Scholar
Williams, C.K. and Rasmussen, C.E. (2006) Gaussian Processes for Machine Learning, Vol. 2. Cambridge, MA: MIT Press.Google Scholar
Yaglom, A.M. (1957) Some classes of random fields in n-dimensional space, related to stationary random processes. Theory of Probability & Its Applications, 2(3), 273–320.CrossRefGoogle Scholar
Supplementary material: File

Riskand Ludkovski supplementary material 1

Riskand Ludkovski supplementary material
Download Riskand Ludkovski supplementary material 1(File)
File 524.3 KB
Supplementary material: File

Riskand Ludkovski supplementary material 2

Riskand Ludkovski supplementary material
Download Riskand Ludkovski supplementary material 2(File)
File 100.2 KB