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

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

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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

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SLIDE 1

Smooth Models of Mortality with Period Shocks Iain Currie & James Kirkby

Heriot Watt University Barcelona, July 2007

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SLIDE 2

Swedish Mortality Data

Age 10 90 Year 1900 2000

Deaths : D Exposures : E D, E : 81 × 101

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SLIDE 3

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

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SLIDE 4

1900 1920 1940 1960 1980 2000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

B−spline basis

Year B−spline

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SLIDE 5

1900 1920 1940 1960 1980 2000 −3.6 −3.5 −3.4 −3.3 −3.2 −3.1 −3.0

B−spline regression

Year log(mortality)

Age = 70 Npar = 23 DF = 23

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SLIDE 6

1900 1920 1940 1960 1980 2000 −3.6 −3.5 −3.4 −3.3 −3.2 −3.1 −3.0

B−spline regression

Year log(mortality)

Age = 70 Npar = 23 DF = 23

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SLIDE 7

1900 1920 1940 1960 1980 2000 −3.6 −3.5 −3.4 −3.3 −3.2 −3.1 −3.0

B−spline regression with penalties

Year log(mortality)

Age = 70 Npar = 23 DF = 13.6

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SLIDE 8

GLMs & penalized GLMs

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

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SLIDE 9

GLMs & penalized GLMs

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

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SLIDE 10

Age 20 40 60 80 Year 1900 1920 1940 1960 1980 2000 b ( ) 0.0 0.1 0.2 0.3 0.4 0.5

2d B−spline basis

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SLIDE 11

GLMs & GLAMs

Estimation in 2-d penalized GLMs uses the same penalized scoring algorithm (B′WδB + P )ˆ θ = B′Wδz where B = By ⊗ Ba 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(BaΘB′

y) + Ψ

Computational procedure Bθ ≡ BaΘB′

y

B′WδB ≡ G(Ba)W G(By)′

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SLIDE 12

GLMs & GLAMs

Estimation in 2-d penalized GLMs uses the same penalized scoring algorithm (B′WδB + P )ˆ θ = B′Wδz where B = By ⊗ Ba 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(BaΘB′

y) + Ψ

Computational procedure Bθ ≡ BaΘB′

y

B′WδB ≡ G(Ba)W G(By)′

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SLIDE 13

GLMs & GLAMs

Estimation in 2-d penalized GLMs uses the same penalized scoring algorithm (B′WδB + P )ˆ θ = B′Wδz where B = By ⊗ Ba 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(BaΘB′

y) + Ψ

Computational procedure Bθ ≡ BaΘB′

y

B′WδB ≡ G(Ba)W G(By)′

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SLIDE 14

GLMs & GLAMs

Estimation in 2-d penalized GLMs uses the same penalized scoring algorithm (B′WδB + P )ˆ θ = B′Wδz where B = By ⊗ Ba 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(BaΘB′

y) + Ψ

Computational procedure Bθ ≡ BaΘB′

y

B′WδB ≡ G(Ba)W G(By)′

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SLIDE 15

GLAM

  • conceptually attractive
  • low footprint
  • very fast
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SLIDE 16

Modelling shocks

Want separate B-spline basis for each year Iny ⊗ ˘ Ba Additive model: smooth surface + smooth period shocks

  • By ⊗ Ba : Iny ⊗ ˘

Ba

  • ,

8181 × 1346 Additive GLAM: BaΘB′

y + ˘

Ba ˘ Θ Penalty matrix:   P ˘ P  

  • P penalizes roughness in rows and columns
  • ˘

P is a ridge penalty

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SLIDE 17

Modelling shocks

Want separate B-spline basis for each year Iny ⊗ ˘ Ba Additive model: smooth surface + smooth period shocks

  • By ⊗ Ba : Iny ⊗ ˘

Ba

  • ,

8181 × 1346 Additive GLAM: BaΘB′

y + ˘

Ba ˘ Θ Penalty matrix:   P ˘ P  

  • P penalizes roughness in rows and columns
  • ˘

P is a ridge penalty

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SLIDE 18

Modelling shocks

Want separate B-spline basis for each year Iny ⊗ ˘ Ba Additive model: smooth surface + smooth period shocks

  • By ⊗ Ba : Iny ⊗ ˘

Ba

  • ,

8181 × 1346 Additive GLAM: BaΘB′

y + ˘

Ba ˘ Θ Penalty matrix:   P ˘ P  

  • P penalizes roughness in rows and columns
  • ˘

P is a ridge penalty

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SLIDE 19

Modelling shocks

Want separate B-spline basis for each year Iny ⊗ ˘ Ba Additive model: smooth surface + smooth period shocks

  • By ⊗ Ba : Iny ⊗ ˘

Ba

  • ,

8181 × 1346 Additive GLAM: BaΘB′

y + ˘

Ba ˘ Θ Penalty matrix:   P ˘ P  

  • P penalizes roughness in rows and columns
  • ˘

P is a ridge penalty

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SLIDE 20

Summary of results

(λa, λy, λs) DEV TR BIC 2-d (10, 6.5, ∞) 20918 285 23485 2-d + shocks (0.01, 2150, 800) 9354 455 13451

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SLIDE 21

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

Smooth + Shocks

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SLIDE 22

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

Smooth

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SLIDE 23

20 40 60 80 −0.10 0.00 0.10 0.20

Mortality shock − 1900

Age Mortality shock 20 40 60 80 −0.3 −0.2 −0.1 0.0

Mortality shock − 1909

Age Mortality shock 20 40 60 80 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Mortality shock − 1918

Age Mortality shock 20 40 60 80 −0.1 0.0 0.1 0.2 0.3 0.4

Mortality shock − 1919

Age Mortality shock

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SLIDE 24

20 40 60 80 −0.3 −0.2 −0.1 0.0

Mortality shock − 1923

Age Mortality shock 20 40 60 80 −0.25 −0.15 −0.05 0.05

Mortality shock − 1924

Age Mortality shock 20 40 60 80 0.0 0.2 0.4 0.6

Mortality shock − 1944

Age Mortality shock 20 40 60 80 −0.1 0.0 0.1 0.2 0.3 0.4

Mortality shock − 1945

Age Mortality shock

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SLIDE 25

Two level shock model

  • Single level shocks ⇒ λs
  • Two level shocks ⇒ λs1 and λs2

(λa, λy, λs) DEV TR BIC 2-d (10, 6.5, ∞) 20918 285 23485 2-d + 1-level shock (0.01, 2150, 800) 9354 455 13451 2-d + 2-level shock (0.01, 1500, 300, 5000) 9786 318 12652

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SLIDE 26

20 40 60 80 −0.10 0.00 0.10 0.20

Mortality shock − 1900

Age Mortality shock 20 40 60 80 −0.2 −0.1 0.0

Mortality shock − 1909

Age Mortality shock 20 40 60 80 0.0 0.4 0.8 1.2

Mortality shock − 1918

Age Mortality shock 20 40 60 80 −0.1 0.0 0.1 0.2 0.3 0.4

Mortality shock − 1919

Age Mortality shock

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SLIDE 27

20 40 60 80 −0.25 −0.15 −0.05 0.05

Mortality shock − 1923

Age Mortality shock 20 40 60 80 −0.20 −0.10 0.00

Mortality shock − 1924

Age Mortality shock 20 40 60 80 0.0 0.2 0.4 0.6

Mortality shock − 1944

Age Mortality shock 20 40 60 80 −0.1 0.0 0.1 0.2 0.3 0.4

Mortality shock − 1945

Age Mortality shock

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SLIDE 28

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

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SLIDE 29

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.