Machine learning

Statistics done by computers

Machine learning is a set of methods for data analysis that automate statistical model building. Its algorithms “learn” from data enabling the computer discover its complex structure and associations. Learning is an iterative process of encoding these observations in a mathematical function through training and testing its fit to the data. The function can be a simple vector (like in k-means clustering) or a high-dimensional non-linear form (like in deep learning, below). The popularity of machine learning in various applications grows, despite its liability to overfitting and misinterpretations if used without sufficient caution and insight.

Forecasting the UK mortality rates using deep learning

Rates of survival were consistently improving throughout the UK population for more than 100 years, owing to medical progress and bettering standards of living. A reasonable projection should continue this trend. How much further improvement in the lifespans of future generations can be expected is, however, far from certain, with some forecasts predicting over 30% chance for a person born today of reaching the ranks of centenarians [1]. Inevitably, the further away in the future we project mortality rates, the higher is their uncertainty. Simple analytic models [2, 3] provide more stability, but may not be flexible enough to reflect all trends in the data. We propose a robust forecasting model based on the deep learning approach [4]. It uses a recurrent neural network, which is dedicated to the analysis of dynamic temporal behaviours, to exploit the information about mortality trends available from historical data [5]. Given the last N=40 mortality rates for a fixed age group, our model uses a recurrent neural network to predict the (N+1)-th one and then feed it back as part of the input used to predict the (N+2)-th rate, etc.

The presented model and its results have been employed in our microsimulation studies of the impact of state pension reforms on the British people and economy, presented at 2018 Asia-Pacific Regional Conference of the Intermational Microsimulation Association and (soon) PenCon2018. It is described, including the procedure of optimising N and other neural network parameters and more results, in my recent article “Forecasting the impact of state pension reforms in post-Brexit England and Wales using microsimulation and deep learning”, and available as a Python implementation.

The obtained forecast indicate a progressing decline of mortality for men and women at all age. The men's mortality rates fall faster than women's. This trend is particularly strong for male teenagers, young adults, as well as ages 60-85, where it significantly diverges from the ONS results. Conversely, mortality rates for senior men and women do not fall as dramatically and remain at higher levels than the ONS projections.

Move the coursor above the legend to display 95% CIs and click the items to remove/display data. Click in the axes to change the scale between logarithmic and linear or switch between complete and forecast period of the simulation.

Move the coursor above the legend to display 95% CIs and click the items to remove/display data. Click in the axes to change the scale between logarithmic and linear or switch between complete and forecast period of the simulation.

Click the items to remove/display data and the x axis to switch between complete and forecast period of the simulation.

The postponement of death to senectitude due to population ageing occurs with the rectangularisation of survival curves. Furthermore, the age distribution in deaths is more compressed for women than men, giving the latter a noticeably higher chance of becoming centenarians, but also a higher risk of dying younger. An akin effect of men outliving women has been already observed in the British population. Men have been catching women up in average lifespan over the last decades, the first benefiting from the industry move from physical labour to services and adapting healthier lifestyle, while the second often taking the toll of combining full-time jobs with housework [6]. At the same time, the positive correlation between life expectancy and socioeconomic status [7] combined with gender imbalance in the latter [8] may cause an uplift in the men's age of death and the observed widening of its distribution.

Click the items to remove/display data and the x axis to switch between complete and forecast period of the simulation.

The heatmap presents detailed historical and forecast patterns of relative mortality rates of men to women of the same age (in logarithmic scale). The bright regions indicate higher rates for women, while the dark ones higher rates for men. According to historical data, regions 1 and 2 are associated with four times higher prevalence of deaths from suicides, transport accidents and misuse of alcohol and drugs in young men, and twice higher prevalence of cardiovascular diseases in older men, as compared to their female peers, respectively [9]. Region 3 corresponds to an increased number of women's deaths at old age, mainly due to the Alzheimer disease and dementia. The forecast reveals mortality trends transforming those patterns. In region 4, women in their thirties have higher mortality rates than men (although very low for both sexes), possibly due to the commonly reported problem of young women adapting unhealthy behaviours (smoking and drinking). The weaker compression of mortality in men than women accounts for regions 5 and 6, which represent the lower and upper tails of men's distribution of age in deaths observed in the survival curves.

Fairness between generations requires that everybody spends a similar proportion of adult life contributing to and receiving a state pension. In the table below we compare the residual lifespan for people retiring under different legislations: at the age of 60 for women and 65 for men, at the equal age of 65 and next 68.

Year 60 women
/ 65 men
Equal age (65) Current (68)
194818.7 / 11.614.5 / 11.612.3 / 9.8
201828.5 / 22.223.4 / 22.220.7 / 19.3
204831.8 / 27.926.8 / 27.923.8 / 24.8
Life expectancies forecast by our model grow faster than the state pension age rises. In particular, women retiring in 2018 can expect to live 23 more years, which is almost 5 years longer than women retiring in 1948, despite their state pension age rising from 60 to 65. The difference is particularly striking for men (as it is not attenuated by the state pension age change), whose expected lifespan in retirement doubles. The following state pension age rise to 68 does not reduce the residual lifespan of those retiring in 2048, when it comes into effect (men gain another 1.5 year of life). While under the old legislation women and men could expect to spend 24% and 15% of their lives in retirement, respectively, the subsequent reforms guarantee a generous and fair ~26% percent to next generations. Furthermore, they can expect to enjoy more years in good health [10], as the general health improvements that lead to increasing life expectancy can also delay the onset and progress of diseases, perhaps replacing them with natural death upon reaching a hypothesised biological limit of human lifespan [11] thanks to the advances of medicine. This compression of morbidity scenario forms a more optimistic projection of the impact of population ageing on healthcare costs.


[1] Office for National Statistics. What are your chances of living to 100? 2016.
[2] R.D. Lee and L. Carter. Modelling and forecasting the time series of US mortality. J. Am. Stat. Assoc., 87:659-671, 1992.
[3] A.J.G. Cairns, D. Blake, K. Dowd, G.D. Coughlan, D. Epstein, A. Ong, and I. Balevich. A quantitative comparison of stochastic mortality models using data from England and Wales and the United States. N. Am. Actuar. J., 13(1):1-35, 2009.
[4] I. Goodfellow, Y. Bengio, and A. Courville. Deep Learning. MIT Press, 2016.
[5] Office for National Statistics. Historic and projected mortality rates (qx) from the 2014-base for England and Wales - males and females (user requested data). 2017.
[6] Donatien Hainaut. A neural network analyzer for mortality forecast. ASTIN Bulletin, pages 1-28, 2018.
[7] Office for National Statistics. Methodology used to produce the national population projections. 2016.
[8] Office for National Statistics. Health state life expectancies, UK: 2014 to 2016. 2017.
[8] Office for National Statistics. Most affluent man outlives the average woman for the first time. 2015.
[8] Office for National Statistics. 2011 Census: Approximated social grade by sex by age. 2014.
[9] Office for National Statistics. Deaths registered in England and Wales (Series DR): 2015. 2015 and earlier releases.
[10] Public Health England. Health profile for England, Chapter 1: life expectancy and healthy life expectancy. 2017.
[11] J.F. Fries. Aging, natural death, and the compression of morbidity. New England J. of Medicine, 303(3):130-135, 1980.