Modeling and forecasting age-specific mortality: Lee-Carter method vs. Functional time series

Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecasting

Modeling and forecasting age-speciﬁc mortality:
Lee-Carter method vs. Functional time series

Han Lin Shang

Econometrics & Business Statistics

http://monashforecasting.com/index.php?title=User:Han


Outline

1 Lee-Carter model

2 Nonparametric smoothing

3 Functional principal component analysis

4 Functional time series forecasting


Lee-Carter model

1 Lee and Carter (1992) proposed one-factor principal
component method to model and forecast demographic data,
such as age-speciﬁc mortality rates.


Lee-Carter model

2 The Lee-Carter model can be written as

ln mx,t = ax + bx × kt + ex,t , (1)

where


Lee-Carter model



where
ln mx,t is the observed log mortality rate at age x in year t,


Lee-Carter model



where
ax is the sample mean vector,


Lee-Carter model



where
bx is the ﬁrst set of sample principal component,


Lee-Carter model



where
kt is the ﬁrst set of sample principal component scores,


Lee-Carter model



where
kt is the ﬁrst set of sample principal component scores,
ex,t is the residual term.


Lee-Carter model forecasts

1 There are a number of ways to adjust kt , which led to
extensions and modiﬁcation of original Lee-Carter method.



2 Lee and Carter (1992) advocated to use a random walk with
drift model to forecast principal component scores, expressed
as
kt = kt−1 + d + et , (2)
where



as
kt = kt−1 + d + et , (2)
where
d is known as the drift parameter, measures the average
annual change in the series,



as
kt = kt−1 + d + et , (2)
where
et is an uncorrelated error.



as
kt = kt−1 + d + et , (2)
where
et is an uncorrelated error.
3 From the forecast of principal component scores, the forecast
age-specific log mortality rates are obtained using the
estimated age effects ax and estimated first set of principal
component bx .


Construction of functional data
1 Functional data are a collection of functions, represented in
the form of curves, images or shapes.

France: male log mortality rate (1899−2005)
0
−2
Log mortality rate

−4
−6
−8
−10

0 20 40 60 80 100

Age


2 Let’s consider annual French male log mortality rates from
1816 to 2006 for ages between 0 and 100.

0
−2
Log mortality rate

−4
−6
−8
−10

0 20 40 60 80 100

Age


2 Let’s consider annual French male log mortality rates from
1816 to 2006 for ages between 0 and 100.
3 By interpolating 101 data points in one year, functional curves
can be constructed below.
0
−2
Log mortality rate

−4
−6
−8
−10

0 20 40 60 80 100

Age


Smoothed functional data

1 Age-speciﬁc mortality rates are ﬁrst smoothed using penalized
regression spline with monotonic constraint.



2 Assuming there is an underlying continuous and smooth
function {ft (x); x ∈ [x1 , xp ]} that is observed with errors at
discrete ages in year t, we can express it as

mt (xi ) = ft (xi ) + σt (xi )εt,i , t = 1, 2, . . . , n, (3)

where





where
mt (xi ) is the log mortality rates,





where
ft (xi ) is the smoothed log mortality rates,





where
σt (xi ) allows the possible presence of heteroscedastic error,





where
σt (xi ) allows the possible presence of heteroscedastic error,
εt,i is iid standard normal random variable.



1 Smoothness (also known ﬁltering) allows us to analyse
derivative information of curves.

0
−2
Log mortality rate

−4
−6
−8
−10

0 20 40 60 80 100

Age



1 Smoothness (also known ﬁltering) allows us to analyse
derivative information of curves.
2 We transform n × p data matrix to n vector of functions.
0
−2
Log mortality rate

−4
−6
−8
−10

0 20 40 60 80 100

Age


Functional principal component analysis (FPCA)
1 FPCA can be viewed from both covariance kernel function
and linear operator perspectives.


2 It is a dimension-reduction technique, with nice properties:


FPCA minimizes the mean integrated squared error,
K 2
E f c (x) − βk φk (x) dx, K < ∞, (4)
I k=1

where f c (x) = f (x) − µ(x) represents the decentralized
functional curves, and x ∈ [x1 , xp ].


K 2
E f c (x) − βk φk (x) dx, K < ∞, (4)
I k=1

FPCA provides a way of extracting a large amount of variance,
∞ ∞ ∞
Var[f c (x)] = Var(βk )φ2 (x) =
k λk φ2 (x) =
k λk , (5)
k=1 k=1 k=1

where λ1 ≥ λ2 , . . . , ≥ 0 is a decreasing sequence of
eigenvalues and φk (x) is orthonormal.


K 2
E f c (x) − βk φk (x) dx, K < ∞, (4)
I k=1

FPCA provides a way of extracting a large amount of variance,
∞ ∞ ∞
Var[f c (x)] = Var(βk )φ2 (x) =
k λk φ2 (x) =
k λk , (5)
k=1 k=1 k=1

where λ1 ≥ λ2 , . . . , ≥ 0 is a decreasing sequence of
eigenvalues and φk (x) is orthonormal.
The principal component scores are uncorrelated, that is
cov(βi , βj ) = E(βi βj ) = 0, for i = j.


Karhunen-Lo`ve (KL) expansion
e
By KL expansion, a stochastic process f (x), x ∈ [x1 , xp ] can be
expressed as
∞
f (x) = µ(x) + βk φk (x), (6)
k=1
K
= µ(x) + βk φk (x) + e(x), (7)
k=1

where
1 µ(x) is the population mean,


e
expressed as
∞
f (x) = µ(x) + βk φk (x), (6)
k=1
K
= µ(x) + βk φk (x) + e(x), (7)
k=1

where
2 βk is the k th principal component scores,


e
expressed as
∞
f (x) = µ(x) + βk φk (x), (6)
k=1
K
= µ(x) + βk φk (x) + e(x), (7)
k=1

where
3 φk (x) is the k th functional principal components,


e
expressed as
∞
f (x) = µ(x) + βk φk (x), (6)
k=1
K
= µ(x) + βk φk (x) + e(x), (7)
k=1

where
4 e(x) is the error function, and


e
expressed as
∞
f (x) = µ(x) + βk φk (x), (6)
k=1
K
= µ(x) + βk φk (x) + e(x), (7)
k=1

where
4 e(x) is the error function, and
5 K is the number of retained principal components.


Empirical FPCA

1 Because the stochastic process f is unknown in practice, the
population mean and eigenfunctions can only be approximated
through realizations of {f1 (x), f2 (x), . . . , fn (x)}.


Empirical FPCA

2 A function ft (x) can be approximated by
K
¯
ft (x) = f (x) + βt,k φk (x) + e(x), (8)
k=1

where


Empirical FPCA

K
¯
ft (x) = f (x) + βt,k φk (x) + e(x), (8)
k=1

where
¯ 1 n
f (x) = n t=1 ft (x) is the sample mean function,


Empirical FPCA

K
¯
ft (x) = f (x) + βt,k φk (x) + e(x), (8)
k=1

where
¯ 1 n
βk is the k th empirical principal component scores,


Empirical FPCA

K
¯
ft (x) = f (x) + βt,k φk (x) + e(x), (8)
k=1

where
¯ 1 n
φk (x) is the k th empirical functional principal components,


Empirical FPCA

K
¯
ft (x) = f (x) + βt,k φk (x) + e(x), (8)
k=1

where
¯ 1 n
φk (x) is the k th empirical functional principal components,
e(x) is the residual function.


Decomposition

0.2

0.2
−1

0.2
0.20

0.1
0.1
−2

Basis function 1

Basis function 2

Basis function 3

Basis function 4
Mean function

0.15

0.0
0.1
−3

0.0

−0.1
0.10
−4

0.0
−0.1

−0.2
−5

0.05

−0.3
−0.1
−6

−0.2
0.00

0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100
Age Age Age Age Age

8
10

0.5
6

1
5
Coefficient 1

Coefficient 2

Coefficient 3

Coefficient 4
0

4

0.0
0
−5

2

−0.5
−1
−10

0
−15

−1.0
−2
−2

1850 1900 1950 2000 1850 1900 1950 2000 1850 1900 1950 2000 1850 1900 1950 2000
Year Year Year Year

1 The principal components reveal underlying characteristics
across age direction.


Decomposition

0.2

0.2
−1

0.2
0.20

0.1
0.1
−2

Basis function 1

Basis function 2

Basis function 3

Basis function 4
Mean function

0.15

0.0
0.1
−3

0.0

−0.1
0.10
−4

0.0
−0.1

−0.2
−5

0.05

−0.3
−0.1
−6

−0.2
0.00

0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100
Age Age Age Age Age

8
10

0.5
6

1
5
Coefficient 1

Coefficient 2

Coefficient 3

Coefficient 4
0

4

0.0
0
−5

2

−0.5
−1
−10

0
−15

−1.0
−2
−2

1850 1900 1950 2000 1850 1900 1950 2000 1850 1900 1950 2000 1850 1900 1950 2000
Year Year Year Year

1 The principal components reveal underlying characteristics
across age direction.
2 The principal component scores reveal possible outlying years
across time direction.


Point forecast

Because orthogonality of the estimated functional principal
components and uncorrelated principal component scores, point
forecasts are obtained by
K
¯
fn+h|n (x) = E[fn+h (x)|I, Φ] = f (x) + βn+h|n,k φk (x), (9)
k=1

where
1 fn+h|n (x) is the h-step-ahead point forecast,


Point forecast

K
¯
k=1

where
2 I represents the past data,


Point forecast

K
¯
k=1

where
3 Φ = (φ1 (x), . . . , φK (x)) is a set of ﬁxed estimated functional
principal components,


Point forecast

K
¯
k=1

where
3 Φ = (φ1 (x), . . . , φK (x)) is a set of ﬁxed estimated functional
principal components,
4 βn+h|n,k is the forecast of principal component scores by a
univariate time series method, such as exponential smoothing.


Point forecast

Point forecasts (2007−2026)
0
−2
Log mortality rate

−4
−6
−8

Past data
−10

Forecasts

0 20 40 60 80 100

Age

Figure: 20-step-ahead point forecasts. Past data are shown in gray,
whereas the recent data are shown in color.


Conclusion

1 We revisit the Lee-Carter model and functional time series
model for modeling age-speciﬁc mortality rates,


Conclusion

1 We revisit the Lee-Carter model and functional time series
model for modeling age-speciﬁc mortality rates,
2 We show how to compute point forecasts for both models.

Modeling and forecasting age-specific mortality: Lee-Carter method vs. Functional time series

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Modeling and forecasting age-specific mortality: Lee-Carter method vs. Functional time series