Smooth Models of Mortality with Period Shocks Iain Currie & - PowerPoint PPT Presentation

Smooth Models of Mortality with Period Shocks Iain Currie & James Kirkby Heriot Watt University Barcelona, July 2007

Swedish Mortality Data Year 1900 2000 10 Deaths : D Age Exposures : E D , E : 81 × 101 90

−2 −4 log(mortality) −6 −8 −10 2000 80 1980 60 1960 Year Age 1940 40 1920 20 1900

B−spline basis 0.7 0.6 0.5 0.4 B−spline 0.3 0.2 0.1 0.0 1900 1920 1940 1960 1980 2000 Year

B−spline regression −3.0 −3.1 −3.2 log(mortality) −3.3 −3.4 −3.5 Age = 70 Npar = 23 −3.6 DF = 23 1900 1920 1940 1960 1980 2000 Year

B−spline regression −3.0 −3.1 −3.2 log(mortality) −3.3 −3.4 −3.5 Age = 70 Npar = 23 −3.6 DF = 23 1900 1920 1940 1960 1980 2000 Year

B−spline regression with penalties −3.0 −3.1 −3.2 log(mortality) −3.3 −3.4 −3.5 Age = 70 Npar = 23 −3.6 DF = 13.6 1900 1920 1940 1960 1980 2000 Year

GLMs & penalized GLMs Estimation in GLMs uses the scoring algorithm B ′ W δ B ˆ θ = B ′ W δ z where B is the regression matrix, W δ is the diagonal matrix of weights and z is the working vector. Estimation in penalized GLMs uses the penalized scoring algorithm ( B ′ W δ B + P )ˆ θ = B ′ W δ z where P is the penalty matrix.

GLMs & penalized GLMs Estimation in GLMs uses the scoring algorithm B ′ W δ B ˆ θ = B ′ W δ z where B is the regression matrix, W δ is the diagonal matrix of weights and z is the working vector. Estimation in penalized GLMs uses the penalized scoring algorithm ( B ′ W δ B + P )ˆ θ = B ′ W δ z where P is the penalty matrix.

2d B−spline basis 0.5 0.4 0.3 b ( ) 0.2 0.1 0.0 2000 1980 1960 Year 1940 80 60 1920 Age 40 20 1900

GLMs & GLAMs Estimation in 2-d penalized GLMs uses the same penalized scoring algorithm ( B ′ W δ B + P )ˆ θ = B ′ W δ z where B = B y ⊗ B a is the regression matrix. A generalized linear array model or GLAM is • structure • computational procedure when data lies on an array and the model matrix is a Kronecker product. Structure D = E exp( B a Θ B ′ y ) + Ψ Computational procedure B a Θ B ′ Bθ ≡ y B ′ W δ B G ( B a ) W G ( B y ) ′ ≡

GLMs & GLAMs Estimation in 2-d penalized GLMs uses the same penalized scoring algorithm ( B ′ W δ B + P )ˆ θ = B ′ W δ z where B = B y ⊗ B a is the regression matrix. A generalized linear array model or GLAM is • structure • computational procedure when data lies on an array and the model matrix is a Kronecker product. Structure D = E exp( B a Θ B ′ y ) + Ψ Computational procedure B a Θ B ′ Bθ ≡ y B ′ W δ B G ( B a ) W G ( B y ) ′ ≡

GLMs & GLAMs Estimation in 2-d penalized GLMs uses the same penalized scoring algorithm ( B ′ W δ B + P )ˆ θ = B ′ W δ z where B = B y ⊗ B a is the regression matrix. A generalized linear array model or GLAM is • structure • computational procedure when data lies on an array and the model matrix is a Kronecker product. Structure D = E exp( B a Θ B ′ y ) + Ψ Computational procedure B a Θ B ′ Bθ ≡ y B ′ W δ B G ( B a ) W G ( B y ) ′ ≡

GLMs & GLAMs Estimation in 2-d penalized GLMs uses the same penalized scoring algorithm ( B ′ W δ B + P )ˆ θ = B ′ W δ z where B = B y ⊗ B a is the regression matrix. A generalized linear array model or GLAM is • structure • computational procedure when data lies on an array and the model matrix is a Kronecker product. Structure D = E exp( B a Θ B ′ y ) + Ψ Computational procedure B a Θ B ′ Bθ ≡ y B ′ W δ B G ( B a ) W G ( B y ) ′ ≡

GLAM • conceptually attractive • low footprint • very fast

Modelling shocks Want separate B -spline basis for each year I n y ⊗ ˘ B a Additive model: smooth surface + smooth period shocks � � B y ⊗ B a : I n y ⊗ ˘ 8181 × 1346 B a , y + ˘ B a ˘ B a Θ B ′ Additive GLAM: Θ    P 0 Penalty matrix:  ˘ 0 P • P penalizes roughness in rows and columns • ˘ P is a ridge penalty

Modelling shocks Want separate B -spline basis for each year I n y ⊗ ˘ B a Additive model: smooth surface + smooth period shocks � � B y ⊗ B a : I n y ⊗ ˘ 8181 × 1346 B a , y + ˘ B a ˘ B a Θ B ′ Additive GLAM: Θ    P 0 Penalty matrix:  ˘ 0 P • P penalizes roughness in rows and columns • ˘ P is a ridge penalty

Modelling shocks Want separate B -spline basis for each year I n y ⊗ ˘ B a Additive model: smooth surface + smooth period shocks � � B y ⊗ B a : I n y ⊗ ˘ 8181 × 1346 B a , y + ˘ B a ˘ B a Θ B ′ Additive GLAM: Θ    P 0 Penalty matrix:  ˘ 0 P • P penalizes roughness in rows and columns • ˘ P is a ridge penalty

Modelling shocks Want separate B -spline basis for each year I n y ⊗ ˘ B a Additive model: smooth surface + smooth period shocks � � B y ⊗ B a : I n y ⊗ ˘ 8181 × 1346 B a , y + ˘ B a ˘ B a Θ B ′ Θ Additive GLAM:    P 0 Penalty matrix:  ˘ P 0 • P penalizes roughness in rows and columns • ˘ P is a ridge penalty

Summary of results ( λ a , λ y , λ s ) DEV TR BIC (10 , 6 . 5 , ∞ ) 2-d 20918 285 23485 (0 . 01 , 2150 , 800) 2-d + shocks 9354 455 13451

Smooth + Shocks −2 log(mortality) −4 −6 −8 2000 80 1980 60 1960 Year Age 1940 40 1920 20 1900

Smooth −2 log(mortality) −4 −6 −8 2000 80 1980 60 1960 Year Age 1940 40 1920 20 1900

Mortality shock − 1900 Mortality shock − 1909 0.20 0.0 Mortality shock Mortality shock 0.10 −0.1 0.00 −0.2 −0.10 −0.3 20 40 60 80 20 40 60 80 Age Age Mortality shock − 1918 Mortality shock − 1919 0.4 1.2 1.0 0.3 Mortality shock Mortality shock 0.8 0.2 0.6 0.4 0.1 0.2 0.0 0.0 −0.1 20 40 60 80 20 40 60 80 Age Age

Mortality shock − 1923 Mortality shock − 1924 0.05 0.0 Mortality shock Mortality shock −0.05 −0.1 −0.2 −0.15 −0.3 −0.25 20 40 60 80 20 40 60 80 Age Age Mortality shock − 1944 Mortality shock − 1945 0.6 0.4 0.3 0.4 Mortality shock Mortality shock 0.2 0.2 0.1 0.0 0.0 −0.1 20 40 60 80 20 40 60 80 Age Age

Two level shock model • Single level shocks ⇒ λ s • Two level shocks ⇒ λ s 1 and λ s 2 ( λ a , λ y , λ s ) DEV TR BIC (10 , 6 . 5 , ∞ ) 2-d 20918 285 23485 (0 . 01 , 2150 , 800) 2-d + 1-level shock 9354 455 13451 (0 . 01 , 1500 , 300 , 5000) 2-d + 2-level shock 9786 318 12652

Mortality shock − 1900 Mortality shock − 1909 0.20 0.0 Mortality shock Mortality shock 0.10 −0.1 0.00 −0.2 −0.10 20 40 60 80 20 40 60 80 Age Age Mortality shock − 1918 Mortality shock − 1919 0.4 1.2 0.3 Mortality shock Mortality shock 0.8 0.2 0.1 0.4 0.0 0.0 −0.1 20 40 60 80 20 40 60 80 Age Age

Mortality shock − 1923 Mortality shock − 1924 0.05 −0.05 0.00 Mortality shock Mortality shock −0.15 −0.10 −0.25 −0.20 20 40 60 80 20 40 60 80 Age Age Mortality shock − 1944 Mortality shock − 1945 0.6 0.4 0.3 0.4 Mortality shock Mortality shock 0.2 0.2 0.1 0.0 0.0 −0.1 20 40 60 80 20 40 60 80 Age Age

Uses for shock model • improves estimation of the underlying smooth surface • shocks are of interest in their own right • shocks as random effects can be used to simulate future mortality rates: smooth + shock + residual

References Eilers & Marx (1996) Statistical Science, 11, 758-783. Currie, Durban & Eilers (2004) Statistical Modelling, 4, 279-298. Currie, Durban & Eilers (2006) Journal of the Royal Statistical Society, Series B, 68, 259-280. Human Mortality Database University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available at www.mortality.org.