Constraints and the R language

(Jan 16, 2020)

This is the fourth and final blog on the use of constraints in the modelling and forecasting of mortality. The previous three blogs (here, here and here) demonstrated that there is no need to worry about which linear constraints to use: the fitted values of mortality and crucially their forecast values always come out the same. In this blog I thought it would be both interesting and instructive to see how the R language deals with models with non-identifiable parameters.

We will illustrate the ideas with the age-period-cohort or APC model fitted to male data from the Office for National Statistics. We have the number of deaths, \(d_{x,y}\), age \(x\) last birthday in year \(y\) and corresponding mid-year population…

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Tags: identifiability constraints, R

From magical thinking to statistical thinking

(Dec 4, 2019)

The Institute and Faculty of Actuaries in the UK has recently added mortality projection to its syllabus, so this year I have been teaching the subject for the first time to students at Heriot-Watt University.

As an exercise, I asked the students to imagine they were an actuary back in 1980, and to first fit a Lee-Carter model to data from the Human Mortality Database for males in England and Wales, 1940-1980.  Then the students had to project the fitted mortality surface forward 25 years, to 2005.  This is easily done using some R programs written by Iain Currie for our recent book, Macdonald, Richards and Currie (2018). The programs \(\tt Lee\_Carter.r\) and \(\tt Forecast\_LC.r\) referred to there are freely…

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Tags: mortality projections, deterministic models

Immune Response

(Nov 25, 2019)

The resurgence of measles in Europe signals something of a confidence crisis in the area of vaccination, and not for the first time. Mass panics of this sort are not new, but the reach of modern hysteria is aided by technology. Such rapid global spread has propelled vaccine hesitancy into the World Health Organisation's most critical global health threats of 2019.

Of course, like any medical intervention, vaccination comes with the possibility of side effects (even if precisely quantified and well understood). And by their very nature, such interventions are acts taken to prevent a disease an individual does not yet have and may not catch. Human beings can rationalise harms from random misfortune more successfully…

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Tags: longevity, vaccine

Matrix repair

(Nov 25, 2019)

When fitting a statistical model we want two things as a minimum:

  1. The parameter estimates, e.g. the maximum-likelihood estimates (MLEs), and
  2. The estimated variance-covariance matrix, \(\hat V\), for those estimates.

We can get both from the log-likelihood: the MLEs maximise the value of the log-likelihood function, and an approximation for the covariance matrix comes from inverting the negative information matrix, \(\mathcal J\), i.e. the matrix of second partial derivatives evaluated at the MLEs. However, the limitations of computer arithmetic can sometimes get in the way, as shown in the information matrix in Figure 1:

Figure 1. Information matrix, \(\mathcal J\), for a five-parameter model. Only…

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Tags: information matrix, covariance matrix

Turning the tables

(Nov 8, 2019)

Traditional actuarial mortality analysis was done by expressing a portfolio's mortality experience relative to a reference mortality table (a so-called A/E analysis).  In modern actuarial work the A/E analysis is supplemented (or even replaced) with a multi-factor statistical model; besides age and gender, common risk factors include pension size, geodemographic profile and early-retirement status.  However, when it comes to communicating results, it is still often necessary to express the result in terms of a reference mortality table.  Ideally this would be the same mortality table as the A/E analysis for comparison.  Another reason for needing a percentage of a reference table is that many pricing…

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Tags: A/E, standard table

One (more) time passcodes

(Nov 8, 2019)

Passwords seldom stand alone in modern applications, and for good reason. Perhaps they might be guessed or leak or otherwise be broken, so it is a bad idea to make them the only line of defence. This is why multi-factor authentication (MFA) is modern best practice. MFA stipulates that users of a system must combine an account name with:

  • something they know (such as a password), and
  • something they have (such as a token, smartphone or network address assigned to them), or,
  • something they are meaning a biometric factor, such as fingerprint or facial scan.

The Longevitas applications have supported SMS passcodes and account-specific network addresses for a long time. But in our last release we updated our enhanced…

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Tags: technology

All About The Base(line)

(Oct 16, 2019)

When we first developed a technique for putting longevity trend risk into a 1-in-200 framework consistent with Solvency II, we sought to accommodate model risk by supporting a wide range of stochastic projection models. To this end, our VaR approach from 2012 supported both time-series and penalty projections across a range of model families, each with their own properties and approach. Since forecasters cannot know which model will produce results closest to the path of future mortality, a range of different models will best inform actuarial judgement.

The 2D P-spline family proposed by Currie, Durban and Eilers (2004) is a distinctive family of projection models. Both Age-Period and Age-Cohort variants…

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Tags: VaR, smoothing, projections

Mortality convergence

(Sep 5, 2019)

In his blog on socio-economic differentials in England and Wales, Torsten Kleinow showed how mortality rates between sub-groups converge with age. And in his blog on ill-health retirements, Kai Kaufhold demonstrated how excess mortality relative to normal retirements reduces, then vanishes. Both are examples of the phenomenon known to demographers as the compensation law of mortality - regardless of the initial differentials between two contemporaneous sub-populations, their mortality rates will converge at an advanced age, often around 90-95.

[We will caveat this for casual readers by stating that we are talking about human populations in peacetime, where external causes of death such as accident…

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Tags: mortality convergence, compensation law of mortality, mortality plasticity

Auditing Firewalls

(Aug 17, 2019)

In a recent blog I discussed the security improvements brought by changing our certification authority, but that isn't our only recent change. Our v2.8 release contained a number of other technology changes and improvements and we'll discuss a couple of them here.

The first was our implementation of a Web Application Firewall (WAF) on all of our services. Just as a network firewall scrutinises and blocks traffic at the network layer, a WAF functions as a gatekeeper higher up the stack, at the level of the web application. A WAF can fully scrutinise the content of http-level requests and block any that violate defined security rules.

We chose the modsecurity WAF as it was the best fit with our existing platform,…

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Tags: technology, defence in-depth

Fun and games with constraints

(Aug 16, 2019)

I'm a statistician so I worry about standard errors just as much as I worry about point estimates. My blog Up close and intimate with the APCI model looked at the effect of different constraints on parameter estimates in models of mortality. This blog looks at the effect of constraints on the standard errors of the parameter estimates. The results for standard errors are equally surprising as those for parameter estimates; we even have an example where a standard error is identically zero. Both sets of results serve to remind us exactly what is meant by not identifiable.

We will illustrate the ideas with the age-period-cohort or APC model fitted to male data from the Office for National Statistics (ONS). We have…

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Tags: constraints, identifiability

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