Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-10T22:09:44.761Z Has data issue: false hasContentIssue false
  • English
  • Français

Forecasting mortality rates with functional signatures

Published online by Cambridge University Press:  09 January 2025

Zhong Jing Yap Open the ORCID record for Zhong Jing Yap [Opens in a new window] ,
Dharini Pathmanathan Open the ORCID record for Dharini Pathmanathan [Opens in a new window]  and
Sophie Dabo-Niang Open the ORCID record for Sophie Dabo-Niang [Opens in a new window]
Zhong Jing Yap
Affiliation:
Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, 50603, Kuala Lumpur, Malaysia
Dharini Pathmanathan*
Affiliation:
Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, 50603, Kuala Lumpur, Malaysia Universiti Malaya Centre for Data Analytics, Universiti Malaya, 50603, Kuala Lumpur, Malaysia Center of Research for Statistical Modelling and Methodology, Faculty of Science, Universiti Malaya, 50603, Kuala Lumpur, Malaysia
Sophie Dabo-Niang
Affiliation:
UMR8524–Laboratoire Paul Painlevé, Inria-MODAL, University of Lille, CNRS, Lille, 59000, France CNRS–Université de Montréal, CRM–CNRS, Montréal, Canada
*
Corresponding author: Dharini Pathmanathan; Email: dharini@um.edu.my
Get access Rights & Permissions [Opens in a new window]

Abstract

This study introduces an innovative methodology for mortality forecasting, which integrates signature-based methods within the functional data framework of the Hyndman–Ullah (HU) model. This new approach, termed the Hyndman–Ullah with truncated signatures (HUts) model, aims to enhance the accuracy and robustness of mortality predictions. By utilizing signature regression, the HUts model is able to capture complex, nonlinear dependencies in mortality data which enhances forecasting accuracy across various demographic conditions. The model is applied to mortality data from 12 countries, comparing its forecasting performance against variants of the HU models across multiple forecast horizons. Our findings indicate that overall the HUts model not only provides more precise point forecasts but also shows robustness against data irregularities, such as those observed in countries with historical outliers. The integration of signature-based methods enables the HUts model to capture complex patterns in mortality data, making it a powerful tool for actuaries and demographers. Prediction intervals are also constructed with bootstrapping methods.

Keywords

Hyndman-Ullah modelLee-Carter modelprincipal component analysisfunctional data analysis
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , First View , pp. 1 - 24
Copyright
© The Author(s), 2025. 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

Andrès, H., Boumezoued, A. and Jourdain, B. (2024) Signature-based validation of real-world economic scenarios. ASTIN Bulletin, 54(2), 410–440. https://doi.org/10.1017/asb.2024.12 CrossRefGoogle Scholar
Basellini, U., Camarda, C.G. and Booth, H. (2023) Thirty years on: A review of the Lee–Carter method for forecasting mortality. International Journal of Forecasting, 39(3), 1033–1049. https://doi.org/10.1016/j.ijforecast.2022.11.002 CrossRefGoogle Scholar
Beyaztas, U. and Shang, H.L. (2022) Robust bootstrap prediction intervals for univariate and multivariate autoregressive time series models. Journal of Applied Statistics, 49(5), 1179–1202. https://doi.org/10.1080/02664763.2020.1856351 CrossRefGoogle ScholarPubMed
Bjerre, D.S. (2022) Tree-based machine learning methods for modeling and forecasting mortality. ASTIN Bulletin, 52(3), 765–787. https://doi.org/10.1017/asb.2022.11 CrossRefGoogle Scholar
Bonnier, P., Kidger, P., Arribas, I.P., Salvi, C. & Lyons, T. (2019) Deep signature transforms. https://doi.org/10.48550/arXiv.1905.08494 CrossRefGoogle Scholar
Booth, H., Maindonald, J. and Smith, L. (2002) Applying Lee–Carter under conditions of variable mortality decline. Population Studies, 56(3), 325–336. https://doi.org/10.1080/00324720215935 CrossRefGoogle ScholarPubMed
Booth, H. and Tickle, L. (2008) Mortality modelling and forecasting: A review of methods. Annals of Actuarial Science, 3(1–2), 3–43. https://doi.org/10.1017/S1748499500000440 CrossRefGoogle Scholar
Chang, L. and Shi, Y. (2023) Forecasting mortality rates with a coherent ensemble averaging approach. ASTIN Bulletin, 53(1), 2–28. https://doi.org/10.1017/asb.2022.23 CrossRefGoogle Scholar
Chatfield, C. (1993) Calculating interval forecasts. Journal of Business & Economic Statistics, 11(2), 121–135. https://doi.org/10.2307/1391361 CrossRefGoogle Scholar
Chen, K.-T. (1957) Integration of paths, geometric invariants and a generalized Baker–Hausdorff formula. Annals of Mathematics, 65(1), 163–178. https://doi.org/10.2307/1969671 CrossRefGoogle Scholar
Cohen, S. N., Lui, S., Malpass, W., Mantoan, G., Nesheim, L., de Paula, Á., Reeves, A., Scott, C., Small, E. and Yang, L. (2023) Nowcasting with signature methods. https://doi.org/10.48550/arXiv.2305.10256 CrossRefGoogle Scholar
Compagnoni, E.M., Scampicchio, A., Biggio, L., Orvieto, A., Hofmann, T. and Teichmann, J. (2023) On the effectiveness of randomized signatures as reservoir for learning rough dynamics. 2023 International Joint Conference on Neural Networks (IJCNN), pp. 1–8. https://doi.org/10.1109/IJCNN54540.2023.10191624 Google Scholar
Cuchiero, C., Gazzani, G., Möller, J. and Svaluto-Ferro, S. (2024) Joint calibration to SPX and VIX options with signature-based models. Mathematical Finance. https://doi.org/10.1111/mafi.12442 CrossRefGoogle Scholar
Cuchiero, C., Gazzani, G. and Svaluto-Ferro, S. (2023) Signature-based models: Theory and calibration. SIAM Journal on Financial Mathematics, 14(3), 910–957. https://doi.org/10.1137/22M1512338 CrossRefGoogle Scholar
Cuchiero, C., Gonon, L., Grigoryeva, L., Ortega, J.-P. and Teichmann, J. (2020) Discrete-time signatures and randomness in reservoir computing. IEEE Transactions on Neural Networks and Learning Systems, 33, 6321–6330. https://api.semanticscholar.org/CorpusID:225094517 CrossRefGoogle Scholar
Cuchiero, C. and Möller, J. (2024) Signature methods in stochastic portfolio theory. https://doi.org/10.48550/arXiv.2310.02322 CrossRefGoogle Scholar
Czado, C., Delwarde, A. and Denuit, M. (2005) Bayesian Poisson log-bilinear mortality projections. Insurance: Mathematics and Economics, 36(3), 260–284. https://doi.org/10.1016/j.insmatheco.2005.01.001 Google Scholar
Deprez, P., Shevchenko, P. and Wüthrich, M. (2017) Machine learning techniques for mortality modeling. European Actuarial Journal, 7, 337–352. https://doi.org/10.1007/s13385-017-0152-4 CrossRefGoogle Scholar
Dokumentov, A., Hyndman, R.J. and Tickle, L. (2018) Bivariate smoothing of mortality surfaces with cohort and period ridges. Stat, 7(1), e199. https://doi.org/10.1002/sta4.199 CrossRefGoogle Scholar
Fermanian, A. (2021) Embedding and learning with signatures. Computational Statistics & Data Analysis, 157, 107148. https://doi.org/10.1016/j.csda.2020.107148 CrossRefGoogle Scholar
Fermanian, A. (2022) Functional linear regression with truncated signatures. Journal of Multivariate Analysis, 192, 105031. https://doi.org/10.1016/j.jmva.2022.105031 CrossRefGoogle Scholar
Frévent, C. (2023) A functional spatial autoregressive model using signatures. https://doi.org/10.48550/arXiv.2303.12378 CrossRefGoogle Scholar
Haberman, S. and Renshaw, A. (2012) Parametric mortality improvement rate modelling and projecting. Insurance: Mathematics and Economics, 50(3), 309–333. https://doi.org/10.1016/j.insmatheco.2011.11.005 Google Scholar
Hainaut, D. (2018) A neural-network analyzer for mortality forecast. ASTIN Bulletin, 48(2), 481–508. https://doi.org/10.1017/asb.2017.45 CrossRefGoogle Scholar
Hambly, B. and Lyons, T. (2010) Uniqueness for the signature of a path of bounded variation and the reduced path group. Annals of Mathematics, 109–167. https://www.jstor.org/stable/27799199 CrossRefGoogle Scholar
Hubert, M., Rousseeuw, P.J. and Verboven, S. (2002) A fast method for robust principal components with applications to chemometrics. Chemometrics and Intelligent Laboratory Systems, 60(1), 101–111. https://doi.org/10.1016/S0169-7439(01)00188-5 CrossRefGoogle Scholar
Human Mortality Database. (2024) www.mortality.org Google Scholar
Hyndman, R.J. and Booth, H. (2008) Stochastic population forecasts using functional data models for mortality, fertility and migration. International Journal of Forecasting, 24(3), 323–342. https://doi.org/10.1016/j.ijforecast.2008.02.009 CrossRefGoogle Scholar
Hyndman, R.J., Booth, H. and Yasmeen, F. (2012) Coherent mortality forecasting: The product-ratio method with functional time series models. Demography, 50(1), 261–283. https://doi.org/10.1007/s13524-012-0145-5 CrossRefGoogle Scholar
Hyndman, R.J. and Shang, H.L. (2009) Forecasting functional time series. Journal of the Korean Statistical Society, 38(3), 199–211. https://doi.org/10.1016/j.jkss.2009.06.002 CrossRefGoogle Scholar
Hyndman, R.J. and Ullah, M.S. (2007) Robust forecasting of mortality and fertility rates: A functional data approach. Computational Statistics & Data Analysis, 51(10), 4942–4956. https://doi.org/10.1016/j.csda.2006.07.028 CrossRefGoogle Scholar
Jaber, E.A. and Gérard, L.-A. (2024) Signature volatility models: Pricing and hedging with fourier. https://doi.org/10.48550/arXiv.2402.01820 CrossRefGoogle Scholar
Jiménez-Varón, C.F., Sun, Y. and Shang, H.L. (2024) Forecasting high-dimensional functional time series: Application to sub-national age-specific mortality. Journal of Computational and Graphical Statistics, 1–15. https://doi.org/10.1080/10618600.2024.2319166 CrossRefGoogle Scholar
Kalsi, J., Lyons, T. and Arribas, I.P. (2020) Optimal execution with rough path signatures. SIAM Journal on Financial Mathematics, 11(2), 470–493. https://doi.org/10.1137/19M1259778 CrossRefGoogle Scholar
Kidger, P. and Lyons, T. (2020) Universal approximation with deep narrow networks. Proceedings of Thirty Third Conference on Learning Theory (eds. Abernethy, J. and Agarwal, S.), vol. 125, pp. 2306–2327. PMLR. https://proceedings.mlr.press/v125/kidger20a.html Google Scholar
Kiraly, F.J. and Oberhauser, H. (2019) Kernels for sequentially ordered data. Journal of Machine Learning Research, 20(31), 1–45. http://jmlr.org/papers/v20/16-314.html Google Scholar
Lee, R.D. and Carter, L.R. (1992) Modeling and forecasting U.S. mortality. Journal of the American Statistical Association, 87(419), 659–671. https://doi.org/10.2307/2290201 Google Scholar
Lee, R.D. and Miller, T. (2001) Evaluating the performance of the Lee–Carter method for forecasting mortality. Demography, 38(4), 537–549. https://doi.org/10.2307/3088317 CrossRefGoogle ScholarPubMed
Levin, D., Lyons, T. and Ni, H. (2016) Learning from the past, predicting the statistics for the future, learning an evolving system. https://doi.org/10.48550/arXiv.1309.0260 CrossRefGoogle Scholar
Lyons, T., Caruana, M. and Lévy, T. (2007) Differential equations driven by moderately irregular signals. In Differential Equations Driven by Rough Paths: École d’été de probabilités de saint-flour xxxiv - 2004, pp. 1–24. Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-540-71285-5_1.CrossRefGoogle Scholar
Lyons, T., Nejad, S. and Arribas, I.P. (2020) Non-parametric pricing and hedging of exotic derivatives. Applied Mathematical Finance, 27(6), 457–494. https://doi.org/10.1080/1350486X.2021.1891555 CrossRefGoogle Scholar
Marino, M., Levantesi, S. and Nigri, A. (2023) A neural approach to improve the Lee–Carter mortality density forecasts. North American Actuarial Journal, 27(1), 148–165. https://doi.org/10.1080/10920277.2022.2050260 CrossRefGoogle Scholar
Miyata, A. and Matsuyama, N. (2022) Extending the Lee–Carter model with variational autoencoder: A fusion of neural network and Bayesian approach. ASTIN Bulletin, 52(3), 789–812. https://doi.org/10.1017/asb.2022.15 CrossRefGoogle Scholar
Morrill, J., Fermanian, A., Kidger, P. and Lyons, T. (2021) A generalised signature method for multivariate time series feature extraction. https://doi.org/10.48550/arXiv.2006.00873 CrossRefGoogle Scholar
Morrill, J., Kormilitzin, A., Nevado-Holgado, A. J., Swaminathan, S., Howison, S.D. and Lyons, T.J. (2020) Utilization of the signature method to identify the early onset of sepsis from multivariate physiological time series in critical care monitoring. Critical Care Medicine, 48(10). https://doi.org/10.1097/CCM.0000000000004510 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. https://doi.org/10.3390/risks7010033 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. https://doi.org/10.1080/03461238.2020.1867232 CrossRefGoogle Scholar
Ramsay, J.O. and Silverman, B.W. (2005) Functional Data Analysis. Springer. http://www.worldcat.org/isbn/9780387400808 CrossRefGoogle Scholar
Renshaw, A. and Haberman, S. (2003) Lee–Carter mortality forecasting: A parallel generalized linear modelling approach for England and Wales mortality projections. Journal of the Royal Statistical Society. Series C (Applied Statistics), 52(1), 119–137. http://www.jstor.org/stable/3592636 CrossRefGoogle 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. https://doi.org/10.1017/S1748499519000071 CrossRefGoogle Scholar
Schnürch, S. and Korn, R. (2022) Point and interval forecasts of death rates using neural networks. ASTIN Bulletin, 52(1), 333–360. https://doi.org/10.1017/asb.2021.34 CrossRefGoogle Scholar
Scognamiglio, S. (2022) Calibrating the Lee–Carter and the Poisson Lee–Carter models via neural networks. ASTIN Bulletin, 52(2), 519–561. https://doi.org/10.1017/asb.2022.5 CrossRefGoogle Scholar
Shang, H.L. (2012) Point and interval forecasts of age-specific life expectancies: A model averaging approach. Demographic Research, 27(21), 593–644. https://doi.org/10.4054/DemRes.2012.27.21 CrossRefGoogle Scholar
Shang, H.L. and Booth, H. (2020) Synergy in fertility forecasting: improving forecast accuracy through model averaging. Genus, 76(1), 27. https://doi.org/10.1186/s41118-020-00099-y CrossRefGoogle Scholar
Shang, H.L., Haberman, S. and Xu, R. (2022) Multi-population modelling and forecasting life-table death counts. Insurance: Mathematics and Economics, 106, 239–253. https://doi.org/10.1016/j.insmatheco.2022.07.002 Google Scholar
Shang, H.L. and Hyndman, R.J. (2017) Grouped functional time series forecasting: An application to age-specific mortality rates. Journal of Computational and Graphical Statistics, 26(2), 330–343. https://doi.org/10.1080/10618600.2016.1237877 CrossRefGoogle Scholar
Wang, B., Wu, Y., Taylor, N., Lyons, T., Liakata, M., Nevado-Holgado, A. J. and Saunders, K.E. (2020) Learning to detect bipolar disorder and borderline personality disorder with language and speech in non-clinical interviews. Proc. Interspeech 2020, pp. 437–441. https://doi.org/10.21437/Interspeech.2020-3040 CrossRefGoogle Scholar
Xie, Z., Sun, Z., Jin, L., Ni, H. and Lyons, T. (2018) Learning spatial-semantic context with fully convolutional recurrent network for online handwritten Chinese text recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 40(8), 1903–1917. https://doi.org/10.1109/TPAMI.2017.2732978 CrossRefGoogle Scholar
Supplementary material: File

Yap et al. supplementary material

Yap et al. supplementary material
Download Yap et al. supplementary material(File)
File 532.9 KB