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

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

Outline

1 Lee-Carter model

2 Nonparametric smoothing

3 Functional principal component analysis

4 Functional time series forecasting

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

Lee-Carter model

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

2 The Lee-Carter model can be written as

lnmx ,t = ax + bx × kt + ex ,t , (1)

where

lnmx,t is the observed log mortality rate at age x in year t,ax is the sample mean vector,bx is the first set of sample principal component,kt is the first set of sample principal component scores,ex,t is the residual term.

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

Lee-Carter model

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

2 The Lee-Carter model can be written as

lnmx ,t = ax + bx × kt + ex ,t , (1)

where

lnmx,t is the observed log mortality rate at age x in year t,ax is the sample mean vector,bx is the first set of sample principal component,kt is the first set of sample principal component scores,ex,t is the residual term.

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

Lee-Carter model

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

2 The Lee-Carter model can be written as

lnmx ,t = ax + bx × kt + ex ,t , (1)

where

lnmx,t is the observed log mortality rate at age x in year t,

ax is the sample mean vector,bx is the first set of sample principal component,kt is the first set of sample principal component scores,ex,t is the residual term.

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

Lee-Carter model

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

2 The Lee-Carter model can be written as

lnmx ,t = ax + bx × kt + ex ,t , (1)

where

lnmx,t is the observed log mortality rate at age x in year t,ax is the sample mean vector,

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

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

Lee-Carter model

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

2 The Lee-Carter model can be written as

lnmx ,t = ax + bx × kt + ex ,t , (1)

where

lnmx,t is the observed log mortality rate at age x in year t,ax is the sample mean vector,bx is the first set of sample principal component,

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

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

Lee-Carter model

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

2 The Lee-Carter model can be written as

lnmx ,t = ax + bx × kt + ex ,t , (1)

where

lnmx,t is the observed log mortality rate at age x in year t,ax is the sample mean vector,bx is the first set of sample principal component,kt is the first set of sample principal component scores,

ex,t is the residual term.

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

Lee-Carter model

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

2 The Lee-Carter model can be written as

lnmx ,t = ax + bx × kt + ex ,t , (1)

where

lnmx,t is the observed log mortality rate at age x in year t,ax is the sample mean vector,bx is the first set of sample principal component,kt is the first set of sample principal component scores,ex,t is the residual term.

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

Lee-Carter model forecasts

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

2 Lee and Carter (1992) advocated to use a random walk withdrift model to forecast principal component scores, expressedas

kt = kt−1 + d + et , (2)

where

d is known as the drift parameter, measures the averageannual change in the series,et is an uncorrelated error.

3 From the forecast of principal component scores, the forecastage-specific log mortality rates are obtained using theestimated age effects ax and estimated first set of principalcomponent bx .

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

Lee-Carter model forecasts

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

2 Lee and Carter (1992) advocated to use a random walk withdrift model to forecast principal component scores, expressedas

kt = kt−1 + d + et , (2)

where

d is known as the drift parameter, measures the averageannual change in the series,et is an uncorrelated error.

3 From the forecast of principal component scores, the forecastage-specific log mortality rates are obtained using theestimated age effects ax and estimated first set of principalcomponent bx .

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

Lee-Carter model forecasts

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

2 Lee and Carter (1992) advocated to use a random walk withdrift model to forecast principal component scores, expressedas

kt = kt−1 + d + et , (2)

where

d is known as the drift parameter, measures the averageannual change in the series,

et is an uncorrelated error.

3 From the forecast of principal component scores, the forecastage-specific log mortality rates are obtained using theestimated age effects ax and estimated first set of principalcomponent bx .

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

Lee-Carter model forecasts

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

2 Lee and Carter (1992) advocated to use a random walk withdrift model to forecast principal component scores, expressedas

kt = kt−1 + d + et , (2)

where

d is known as the drift parameter, measures the averageannual change in the series,et is an uncorrelated error.

3 From the forecast of principal component scores, the forecastage-specific log mortality rates are obtained using theestimated age effects ax and estimated first set of principalcomponent bx .

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

Lee-Carter model forecasts

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

2 Lee and Carter (1992) advocated to use a random walk withdrift model to forecast principal component scores, expressedas

kt = kt−1 + d + et , (2)

where

d is known as the drift parameter, measures the averageannual change in the series,et is an uncorrelated error.

3 From the forecast of principal component scores, the forecastage-specific log mortality rates are obtained using theestimated age effects ax and estimated first set of principalcomponent bx .

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

Construction of functional data

1 Functional data are a collection of functions, represented inthe form of curves, images or shapes.

2 Let’s consider annual French male log mortality rates from1816 to 2006 for ages between 0 and 100.

3 By interpolating 101 data points in one year, functional curvescan be constructed below.

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

Construction of functional data

1 Functional data are a collection of functions, represented inthe form of curves, images or shapes.

2 Let’s consider annual French male log mortality rates from1816 to 2006 for ages between 0 and 100.

3 By interpolating 101 data points in one year, functional curvescan be constructed below.

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

Construction of functional data

1 Functional data are a collection of functions, represented inthe form of curves, images or shapes.

2 Let’s consider annual French male log mortality rates from1816 to 2006 for ages between 0 and 100.

3 By interpolating 101 data points in one year, functional curvescan be constructed below.

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

Smoothed functional data

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

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

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

where

mt(xi ) is the log mortality rates,ft(xi ) is the smoothed log mortality rates,σt(xi ) allows the possible presence of heteroscedastic error,εt,i is iid standard normal random variable.

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

Smoothed functional data

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

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

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

where

mt(xi ) is the log mortality rates,ft(xi ) is the smoothed log mortality rates,σt(xi ) allows the possible presence of heteroscedastic error,εt,i is iid standard normal random variable.

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

Smoothed functional data

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

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

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

where

mt(xi ) is the log mortality rates,

ft(xi ) is the smoothed log mortality rates,σt(xi ) allows the possible presence of heteroscedastic error,εt,i is iid standard normal random variable.

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

Smoothed functional data

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

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

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

where

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

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

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

Smoothed functional data

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

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

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

where

mt(xi ) is the log mortality rates,ft(xi ) is the smoothed log mortality rates,σt(xi ) allows the possible presence of heteroscedastic error,

εt,i is iid standard normal random variable.

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

Smoothed functional data

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

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

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

where

mt(xi ) is the log mortality rates,ft(xi ) is the smoothed log mortality rates,σt(xi ) allows the possible presence of heteroscedastic error,εt,i is iid standard normal random variable.

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

Smoothed functional data

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

2 We transform n × p data matrix to n vector of functions.

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

Smoothed functional data

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

2 We transform n × p data matrix to n vector of functions.

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

Functional principal component analysis (FPCA)

1 FPCA can be viewed from both covariance kernel functionand linear operator perspectives.

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

FPCA minimizes the mean integrated squared error,

E

∫I

[f c(x)−

K∑k=1

βkφk(x)]2dx , K <∞, (4)

where f c(x) = f (x)− µ(x) represents the decentralizedfunctional curves, and x ∈ [x1, xp].FPCA provides a way of extracting a large amount of variance,

Var[f c(x)] =∞∑k=1

Var(βk)φ2k(x) =∞∑k=1

λkφ2k(x) =

∞∑k=1

λk , (5)

where λ1 ≥ λ2, . . . ,≥ 0 is a decreasing sequence ofeigenvalues and φk(x) is orthonormal.The principal component scores are uncorrelated, that iscov(βi , βj) = E(βiβj) = 0, for i 6= j .

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

Functional principal component analysis (FPCA)

1 FPCA can be viewed from both covariance kernel functionand linear operator perspectives.

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

FPCA minimizes the mean integrated squared error,

E

∫I

[f c(x)−

K∑k=1

βkφk(x)]2dx , K <∞, (4)

where f c(x) = f (x)− µ(x) represents the decentralizedfunctional curves, and x ∈ [x1, xp].FPCA provides a way of extracting a large amount of variance,

Var[f c(x)] =∞∑k=1

Var(βk)φ2k(x) =∞∑k=1

λkφ2k(x) =

∞∑k=1

λk , (5)

where λ1 ≥ λ2, . . . ,≥ 0 is a decreasing sequence ofeigenvalues and φk(x) is orthonormal.The principal component scores are uncorrelated, that iscov(βi , βj) = E(βiβj) = 0, for i 6= j .

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

Functional principal component analysis (FPCA)

1 FPCA can be viewed from both covariance kernel functionand linear operator perspectives.

2 It is a dimension-reduction technique, with nice properties:FPCA minimizes the mean integrated squared error,

E

∫I

[f c(x)−

K∑k=1

βkφk(x)]2dx , K <∞, (4)

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

FPCA provides a way of extracting a large amount of variance,

Var[f c(x)] =∞∑k=1

Var(βk)φ2k(x) =∞∑k=1

λkφ2k(x) =

∞∑k=1

λk , (5)

where λ1 ≥ λ2, . . . ,≥ 0 is a decreasing sequence ofeigenvalues and φk(x) is orthonormal.The principal component scores are uncorrelated, that iscov(βi , βj) = E(βiβj) = 0, for i 6= j .

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

Functional principal component analysis (FPCA)

1 FPCA can be viewed from both covariance kernel functionand linear operator perspectives.

2 It is a dimension-reduction technique, with nice properties:FPCA minimizes the mean integrated squared error,

E

∫I

[f c(x)−

K∑k=1

βkφk(x)]2dx , K <∞, (4)

where f c(x) = f (x)− µ(x) represents the decentralizedfunctional curves, and x ∈ [x1, xp].FPCA provides a way of extracting a large amount of variance,

Var[f c(x)] =∞∑k=1

Var(βk)φ2k(x) =∞∑k=1

λkφ2k(x) =

∞∑k=1

λk , (5)

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

The principal component scores are uncorrelated, that iscov(βi , βj) = E(βiβj) = 0, for i 6= j .

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

Functional principal component analysis (FPCA)

1 FPCA can be viewed from both covariance kernel functionand linear operator perspectives.

2 It is a dimension-reduction technique, with nice properties:FPCA minimizes the mean integrated squared error,

E

∫I

[f c(x)−

K∑k=1

βkφk(x)]2dx , K <∞, (4)

where f c(x) = f (x)− µ(x) represents the decentralizedfunctional curves, and x ∈ [x1, xp].FPCA provides a way of extracting a large amount of variance,

Var[f c(x)] =∞∑k=1

Var(βk)φ2k(x) =∞∑k=1

λkφ2k(x) =

∞∑k=1

λk , (5)

where λ1 ≥ λ2, . . . ,≥ 0 is a decreasing sequence ofeigenvalues and φk(x) is orthonormal.The principal component scores are uncorrelated, that iscov(βi , βj) = E(βiβj) = 0, for i 6= j .

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

Karhunen-Loeve (KL) expansion

By KL expansion, a stochastic process f (x), x ∈ [x1, xp] can beexpressed as

f (x) = µ(x) +∞∑k=1

βkφk(x), (6)

= µ(x) +K∑

k=1

βkφk(x) + e(x), (7)

where

1 µ(x) is the population mean,

2 βk is the kth principal component scores,

3 φk(x) is the kth functional principal components,

4 e(x) is the error function, and

5 K is the number of retained principal components.

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

Karhunen-Loeve (KL) expansion

By KL expansion, a stochastic process f (x), x ∈ [x1, xp] can beexpressed as

f (x) = µ(x) +∞∑k=1

βkφk(x), (6)

= µ(x) +K∑

k=1

βkφk(x) + e(x), (7)

where

1 µ(x) is the population mean,

2 βk is the kth principal component scores,

3 φk(x) is the kth functional principal components,

4 e(x) is the error function, and

5 K is the number of retained principal components.

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

Karhunen-Loeve (KL) expansion

By KL expansion, a stochastic process f (x), x ∈ [x1, xp] can beexpressed as

f (x) = µ(x) +∞∑k=1

βkφk(x), (6)

= µ(x) +K∑

k=1

βkφk(x) + e(x), (7)

where

1 µ(x) is the population mean,

2 βk is the kth principal component scores,

3 φk(x) is the kth functional principal components,

4 e(x) is the error function, and

5 K is the number of retained principal components.

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

Karhunen-Loeve (KL) expansion

By KL expansion, a stochastic process f (x), x ∈ [x1, xp] can beexpressed as

f (x) = µ(x) +∞∑k=1

βkφk(x), (6)

= µ(x) +K∑

k=1

βkφk(x) + e(x), (7)

where

1 µ(x) is the population mean,

2 βk is the kth principal component scores,

3 φk(x) is the kth functional principal components,

4 e(x) is the error function, and

5 K is the number of retained principal components.

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

Karhunen-Loeve (KL) expansion

By KL expansion, a stochastic process f (x), x ∈ [x1, xp] can beexpressed as

f (x) = µ(x) +∞∑k=1

βkφk(x), (6)

= µ(x) +K∑

k=1

βkφk(x) + e(x), (7)

where

1 µ(x) is the population mean,

2 βk is the kth principal component scores,

3 φk(x) is the kth functional principal components,

4 e(x) is the error function, and

5 K is the number of retained principal components.

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

Empirical FPCA

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

2 A function ft(x) can be approximated by

ft(x) = f (x) +K∑

k=1

βt,k φk(x) + e(x), (8)

where

f (x) = 1n

∑nt=1 ft(x) is the sample mean function,

βk is the k th empirical principal component scores,φk(x) is the k th empirical functional principal components,e(x) is the residual function.

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

Empirical FPCA

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

2 A function ft(x) can be approximated by

ft(x) = f (x) +K∑

k=1

βt,k φk(x) + e(x), (8)

where

f (x) = 1n

∑nt=1 ft(x) is the sample mean function,

βk is the k th empirical principal component scores,φk(x) is the k th empirical functional principal components,e(x) is the residual function.

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

Empirical FPCA

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

2 A function ft(x) can be approximated by

ft(x) = f (x) +K∑

k=1

βt,k φk(x) + e(x), (8)

where

f (x) = 1n

∑nt=1 ft(x) is the sample mean function,

βk is the k th empirical principal component scores,φk(x) is the k th empirical functional principal components,e(x) is the residual function.

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

Empirical FPCA

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

2 A function ft(x) can be approximated by

ft(x) = f (x) +K∑

k=1

βt,k φk(x) + e(x), (8)

where

f (x) = 1n

∑nt=1 ft(x) is the sample mean function,

βk is the k th empirical principal component scores,

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

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

Empirical FPCA

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

2 A function ft(x) can be approximated by

ft(x) = f (x) +K∑

k=1

βt,k φk(x) + e(x), (8)

where

f (x) = 1n

∑nt=1 ft(x) is the sample mean function,

βk is the k th empirical principal component scores,φk(x) is the k th empirical functional principal components,

e(x) is the residual function.

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

Empirical FPCA

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

2 A function ft(x) can be approximated by

ft(x) = f (x) +K∑

k=1

βt,k φk(x) + e(x), (8)

where

f (x) = 1n

∑nt=1 ft(x) is the sample mean function,

βk is the k th empirical principal component scores,φk(x) is the k th empirical functional principal components,e(x) is the residual function.

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

Decomposition

0 20 40 60 80 100

−6

−5

−4

−3

−2

−1

Age

Mea

n fu

nctio

n

0 20 40 60 80 100

0.00

0.05

0.10

0.15

0.20

Age

Bas

is fu

nctio

n 1

0 20 40 60 80 100

−0.

2−

0.1

0.0

0.1

0.2

Age

Bas

is fu

nctio

n 2

0 20 40 60 80 100

−0.

10.

00.

10.

2

Age

Bas

is fu

nctio

n 3

0 20 40 60 80 100

−0.

3−

0.2

−0.

10.

00.

10.

2

Age

Bas

is fu

nctio

n 4

1850 1900 1950 2000

−15

−10

−5

05

10

Year

Coe

ffici

ent 1

1850 1900 1950 2000

−2

02

46

8

Year

Coe

ffici

ent 2

1850 1900 1950 2000

−2

−1

01

Year

Coe

ffici

ent 3

1850 1900 1950 2000

−1.

0−

0.5

0.0

0.5

Year

Coe

ffici

ent 4

1 The principal components reveal underlying characteristicsacross age direction.

2 The principal component scores reveal possible outlying yearsacross time direction.

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

Decomposition

0 20 40 60 80 100

−6

−5

−4

−3

−2

−1

Age

Mea

n fu

nctio

n

0 20 40 60 80 100

0.00

0.05

0.10

0.15

0.20

Age

Bas

is fu

nctio

n 1

0 20 40 60 80 100

−0.

2−

0.1

0.0

0.1

0.2

Age

Bas

is fu

nctio

n 2

0 20 40 60 80 100

−0.

10.

00.

10.

2

Age

Bas

is fu

nctio

n 3

0 20 40 60 80 100

−0.

3−

0.2

−0.

10.

00.

10.

2

Age

Bas

is fu

nctio

n 4

1850 1900 1950 2000

−15

−10

−5

05

10

Year

Coe

ffici

ent 1

1850 1900 1950 2000

−2

02

46

8

Year

Coe

ffici

ent 2

1850 1900 1950 2000

−2

−1

01

Year

Coe

ffici

ent 3

1850 1900 1950 2000

−1.

0−

0.5

0.0

0.5

Year

Coe

ffici

ent 4

1 The principal components reveal underlying characteristicsacross age direction.

2 The principal component scores reveal possible outlying yearsacross time direction.

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

Point forecast

Because orthogonality of the estimated functional principalcomponents and uncorrelated principal component scores, pointforecasts are obtained by

fn+h|n(x) = E[fn+h(x)|I,Φ] = f (x) +K∑

k=1

βn+h|n,k φk(x), (9)

where

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

2 I represents the past data,

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

4 βn+h|n,k is the forecast of principal component scores by aunivariate time series method, such as exponential smoothing.

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

Point forecast

Because orthogonality of the estimated functional principalcomponents and uncorrelated principal component scores, pointforecasts are obtained by

fn+h|n(x) = E[fn+h(x)|I,Φ] = f (x) +K∑

k=1

βn+h|n,k φk(x), (9)

where

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

2 I represents the past data,

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

4 βn+h|n,k is the forecast of principal component scores by aunivariate time series method, such as exponential smoothing.

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

Point forecast

Because orthogonality of the estimated functional principalcomponents and uncorrelated principal component scores, pointforecasts are obtained by

fn+h|n(x) = E[fn+h(x)|I,Φ] = f (x) +K∑

k=1

βn+h|n,k φk(x), (9)

where

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

2 I represents the past data,

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

4 βn+h|n,k is the forecast of principal component scores by aunivariate time series method, such as exponential smoothing.

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

Point forecast

Because orthogonality of the estimated functional principalcomponents and uncorrelated principal component scores, pointforecasts are obtained by

fn+h|n(x) = E[fn+h(x)|I,Φ] = f (x) +K∑

k=1

βn+h|n,k φk(x), (9)

where

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

2 I represents the past data,

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

4 βn+h|n,k is the forecast of principal component scores by aunivariate time series method, such as exponential smoothing.

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

Point forecast

0 20 40 60 80 100

−10

−8

−6

−4

−2

0

Point forecasts (2007−2026)

Age

Log

mor

talit

y ra

te

Past dataForecasts

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

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

Conclusion

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

2 We show how to compute point forecasts for both models.

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

Conclusion

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

2 We show how to compute point forecasts for both models.