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Citation: Forecasting Canadian Age-Specific Mortality Rates: Application of Functional Time Series Analysis. mathematics Forecasting Canadian Age-Specific Mortality Rates: Application of Functional Time Series Analysis

September 2023
Mathematics 11:1-14

Authors:

University of Manitoba

In the insurance and pension industries, as well as in designing social security systems, forecasted mortality rates are of major interest. The current research provides statistical methods based on functional time series analysis to improve mortality rate prediction for the Canadian population. The proposed functional time series-based model was applied to the three-mortality series: total, male and female age-specific Canadian mortality rate over the year 1991 to 2019. Descriptive measures were used to estimate the overall temporal patterns and the functional principal component regression model (fPCA) was used to predict the next ten years mortality rate for each series. Functional autoregressive model (fAR (1)) was used to measure the impact of one year age differences on mortality series. For total series, the mortality rates for children have dropped over the whole data period, while the difference between young adults and those over 40 has only been falling since about 2003 and has leveled off in the last decade of the data. A moderate to strong impact of age differences on Canadian age-specific mortality series was observed over the years. Wider application of fPCA to provide more accurate estimates in public health, demography, and age-related policy studies should be considered.

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Citation: Rahman, A.; Jiang, D.

Forecasting Canadian Age-Speciﬁc

Mortality Rates: Application of

Functional Time Series Analysis.

Mathematics 2023,11, 3808. https://

doi.org/10.3390/math11183808

Academic Editor: Mustapha Rachdi

Received: 25 July 2023

Revised: 2 September 2023

Accepted: 4 September 2023

Published: 5 September 2023

Licensee MDPI, Basel, Switzerland.

This article is an open access article

distributed under the terms and

conditions of the Creative Commons

Attribution (CC BY) license (https://

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4.0/).

mathematics

Article

Forecasting Canadian Age-Speciﬁc Mortality Rates: Application

of Functional Time Series Analysis

Azizur Rahman * and Depeng Jiang

Department of Community Health Sciences, University of Manitoba, Winnipeg, MB R3T 2N2, Canada;

depeng.jiang@umanitoba.ca

*Correspondence: rahmana1@myumanitoba.ca

Abstract:

AbstractIn the insurance and pension industries, as well as in designing social security

systems, forecasted mortality rates are of major interest. The current research provides statistical

methods based on functional time series analysis to improve mortality rate prediction for the Canadian

population. The proposed functional time series-based model was applied to the three-mortality

series: total, male and female age-speciﬁc Canadian mortality rate over the year 1991 to 2019.

Descriptive measures were used to estimate the overall temporal patterns and the functional principal

component regression model (fPCA) was used to predict the next ten years mortality rate for each

series. Functional autoregressive model (fAR (1)) was used to measure the impact of one year age

differences on mortality series. For total series, the mortality rates for children have dropped over the

whole data period, while the difference between young adults and those over 40 has only been falling

since about 2003 and has leveled off in the last decade of the data. A moderate to strong impact of age

differences on Canadian age-speciﬁc mortality series was observed over the years. Wider application

of fPCA to provide more accurate estimates in public health, demography, and age-related policy

studies should be considered.

Keywords:

functional time series analysis; age-speciﬁc mortality; functional principal component;

functional autoregressive model

MSC: G2R10; 91B84

1. Introduction

Over the previous century, most developed countries have seen signiﬁcant demo-

graphic shifts, including a signiﬁcant decrease in fertility rates and a signiﬁcant drop in

mortality, owing in part to the changing nature of primary causes of death. In Canada, the

life expectancy at birth for both sexes rose, from 57.0 year in 1921 to 81.7 in 2011 (Canadian

Human Mortality Database [

]). Moreover, Canada’s age-speciﬁc mortality rate for both

sexes combined rose, from 7.0 per 1000 population in 1991 to 7.6 in 2019 (Statistics Canada,

Table 13a [

]). Bourbeau and Quellette [

] gave an excellent historical summary of mortality

trends and patterns in Canada from 1921 to 2011, demonstrating the country’s remarkable

achievement in mortality management and efforts to promote health. In addition, there is

literature on the epidemiology of population change due to cause-of-death analysis (Bah

and Rajulton [

]) and changes in life-expectancy (Bourbeau [

]; Lussier et al. [

]) for an

overview see Mandich and Margolis [7].

The process of creating mortality assumptions for the short- and long-term future has

been hampered by recent disruptions in international mortality trends. In 2015, numerous

countries had a signiﬁcant drop in their natal life expectancies compared to the previous

year (Jasilionis [

]). The evaluation of mortality of a country, both in terms of level and age-

pattern, is reﬂected by a set of mortality rates by age, sex, and calendar year. Furthermore,

many countries are undergoing signiﬁcant shifts in welfare policy as a result of projections

Mathematics 2023,11, 3808. https://doi.org/10.3390/math11183808 https://www.mdpi.com/journal/mathematics

Mathematics 2023,11, 3808 2 of 14

of an ageing population (Hyndman and Ullah [

]). As a result, any advancements in

mortality and fertility forecasts have an immediate impact on policy decisions about

present and future resource allocation. Nowadays, the societal importance of accurate

mortality forecasts is greater than ever before.

Several authors have proposed new approaches to mortality forecasting utilizing

smoothing and statistical modeling. Lee and Carter ’s [

] publication is a signiﬁcant contri-

bution to the ﬁeld on death rate forecasting. They presented a method for modelling and

projecting long-term mortality patterns, and they utilized it to forecast US death through

2065 using the ﬁrst principal component of the log-mortality matrix. The methodology has

since become very widely used and there have been various extensions and modiﬁcations

proposed (e.g., Lee and Miller [

]; Booth et al. [

]; Renshaw and Haberman [

]). De

Jong and Tickle [

] proposed a generalized version of the LC model via the Kalman ﬁlter

to combine spline smoothing and estimation of mortality forecasting. Another signiﬁcant

expansion of Lee and Carter method in terms of functional data paradigm (Ramsay and

Silverman [

]) has been proposed by Hyndman and Ullah [

]. Their method combined

the ideas from functional time series analysis, nonparametric smoothing and robust statis-

tics. They used functional principal component regression (FPCR) proposed by Aguilera

et al. [

] to decompose smoothed univariate functional time series into a set of functional

principal components and their associated principal component scores. Shang and Hyn-

dman [

] used functional principal component regression (FPCR) technique to a large

multivariate set of functional time series via aggregation constraints to address the problem

of forecast reconciliation. Forecasting functional time series observed in different ﬁelds

has received increasing attention in the functional data analysis (Hyndman and Ullah [

];

Wang et al. [

], Abilash et al. [

]; Piotr et al. [

], Hyndman et al. [

]). Most recently, Hyn-

dman et al. [

] has forecasted the old-age dependency ratio for Australia under various

pension age proposal through forecasting multivariate set of functional time series such as

age-and sex-speciﬁc mortality rates, fertility rates and net migration. Hence, there has been

a surge of concern in developing new approach to accurately forecast age-speciﬁc mortality

rates in the last few years, driven by the need for good forecasts to inform government

policy and planning. There has been no study to our knowledge that uses a multivariate

functional time series approach to forecast Canadian age-speciﬁc mortality rates. The

purpose of this paper is to give descriptive metrics to estimate overall temporal patterns

and to anticipate Canadian age-speciﬁc death rates over the next 10 years for three separate

series, as well as to look at the impact of age group disparities across time. In addition,

the inﬂuence of one-year age disparities on death series for the Canadian population is

measured using the functional autoregressive model (fAR (1)).

The rest of the article is structured as follows. In Section 2, we describe the motivating

dataset as a functional time series, which is Canadian age-speciﬁc mortality rates for total,

male and female population observed annually along with the smoothing technique. In

addition, we describe functional principal component regression (FPCR) for producing

point and interval forecast and functional autoregressive model (fAR (1)) to measure the

impact of one year age differences on male and female mortality series. We apply the ideas

of Canadian mortality data in Section 3and ﬁnally in Section 4we provide discussion and

concluding remarks as well as identiﬁes the scope for future research.

2. Materials and Methods

2.1. Canadian Age-Speciﬁc Mortality Rates

In many developed countries such as Canada, increases in longevity and an aging pop-

ulation have led to concerns regarding the sustainability of pensions, health, and universal

health care systems (OECD 2014 [

]). These concerns have resulted in attention among

policy makers and planners in accurately modeling and forecasting age-speciﬁc mortality

rates. Any improvement in the forecast accuracy of mortality rates will be beneﬁcial for

determining the allocation of current and future resources at the national levels.

Mathematics 2023,11, 3808 3 of 14

We consider Canadian age-speciﬁc mortality rates for both sexes, male and female

series, observed annually from 1991 to 2019, obtained from Statistics Canada (2020) (Data|

Table 13-10-0710-01 [

]) as functional time series, where the continuum is the age variable.

These are deﬁned as the mortality rates per 1000 population during the calendar year,

according to the age at time of death. Details of the data can be found on Statistics Canada-

mortality rates by age group website (https://doi.org/10.25318/1310071001-eng, accessed

on 1 August 2022).

Three years of observed data for total (both sexes) mortality rates, are shown in

Figure 1. We take into consideration age-speciﬁc mortality series as functional time series,

where the continuum is the age variable. Here we then convert observed data to functional

data by estimating a smooth curve through the observations, taking the center of each age

group as the point of interpolation.

Mathematics 2023, 11, x FOR PEER REVIEW 3 of 14

We consider Canadian age-speciﬁc mortality rates for both sexes, male and female

series, observed annually from 1991 to 2019, obtained from Statistics Canada (2020) (Data|

Table 13-10-0710-01 [7]) as functional time series, where the continuum is the age variable.

These are deﬁned as the mortality rates per 1000 population during the calendar year,

according to the age at time of death. Details of the data can be found on Statistics Canada-

mortality rates by age group website (hps://doi.org/10.25318/1310071001-eng, accessed

on 1 August 2022).

Three years of observed data for total (both sexes) mortality rates, are shown in Fig-

ure 1. We take into consideration age-speciﬁc mortality series as functional time series,

where the continuum is the age variable. Here we then convert observed data to functional

data by estimating a smooth curve through the observations, taking the center of each age

group as the point of interpolation.

Figure 1. Plot of the total mortality rates in Canada.

Depending on whether or not the continuum is also a time variable, a functional time

series can be grouped into two categories: when separating an almost continuous time

record into natural consecutive intervals such as days, months or years (Hormann and

Kokoszka [23]) and when observations in a time period can be considered together as ﬁ-

nite realizations of an underlying continuous function such as annual age-speciﬁc mortal-

ity rates in demography (Hyndman and Ullah [9]). Suppose that we have a real-valued

random process 𝑦(𝑥), which can be expressed as:

𝑦(𝑥)=

𝑓

(𝑥)+ 𝜎(𝑥) 𝜖, for 𝑖 = 1, 2, ⋯ ,𝑝 ; 𝑡 = 1, 2,⋯ ,𝑛 (1)

where 𝑦(𝑥) was the observed mortality rate for age 𝑥 in year 𝑡. Here, 𝑡 denotes the ar-

gument of function such as number of functions and typically 𝑥,⋯,𝑥 denote the do-

main of a function (observed ages) and are usually considered single years of age or de-

note 5-year of age groups. We assume there is an underlying smooth function 𝑓(𝑥) that

we are observing with error and at set of discreate time points, 𝑥. Our observations are

󰇝𝑥,𝑦(𝑥) 󰇞 deﬁned in Equation (1), where 𝜖, is an iid normal random variables and

𝜎(𝑥) allows the amount of noise to vary with 𝑥.

In this article, we consider spline basis function for transforming discrete data to

functional object and smoothed functional mortality series using penalized regression

smoothing technique with monotonic constraint described in following section, and GCV

criterion was used to determine smoothing parameter. Then we decompose the smoothed

curves using functional principal component analysis technique described in Hyndman

and Ullah [9] for functional time series, and used for ℎ step ahead forecast of 𝑦 (𝑥),

discussed in detail in Section 3.

2.2. Smoothing Techniques

In practice, the observed data are often contaminated by random noise, referred to as

measurement errors (Wood [24]). As deﬁned by Wang, et al. [18], measurement error can

Figure 1. Plot of the total mortality rates in Canada.

Depending on whether or not the continuum is also a time variable, a functional time

series can be grouped into two categories: when separating an almost continuous time

record into natural consecutive intervals such as days, months or years (Hormann and

Kokoszka [

]) and when observations in a time period can be considered together as ﬁnite

realizations of an underlying continuous function such as annual age-speciﬁc mortality

rates in demography (Hyndman and Ullah [

]). Suppose that we have a real-valued random

process yt(x), which can be expressed as:

yt(xi)=ft(xi)+σt(xi)et,ifor i=1, 2, · · · ,p;t=1, 2, · · · ,n(1)

where

yt(x)

was the observed mortality rate for age

in year

. Here,

denotes the

argument of function such as number of functions and typically

x1,· · · ,xp

denote the

domain of a function (observed ages) and are usually considered single years of age or

denote 5-year of age groups. We assume there is an underlying smooth function

ft(x)

that

we are observing with error and at set of discreate time points,

. Our observations are

{xi,yt(xi)}

deﬁned in Equation (1), where

et,i

is an iid normal random variables and

σt(xi)

allows the amount of noise to vary with x.

In this article, we consider spline basis function for transforming discrete data to

functional object and smoothed functional mortality series using penalized regression

smoothing technique with monotonic constraint described in following section, and GCV

criterion was used to determine smoothing parameter. Then we decompose the smoothed

curves using functional principal component analysis technique described in Hyndman

and Ullah [

] for functional time series, and used for

step ahead forecast of

yn+h(x)

discussed in detail in Section 3.

2.2. Smoothing Techniques

In practice, the observed data are often contaminated by random noise, referred to as

measurement errors (Wood [

]). As deﬁned by Wang, et al. [

], measurement error can

Mathematics 2023,11, 3808 4 of 14

be viewed as random ﬂuctuations around a continuous and smooth function, or as actual

errors in the measurement. We observe that measurement errors are realized only at those

time points

where measurements are being taken. As a result, these errors are treated as

discretized data

et,i

deﬁned in Equation (1). However, to estimate the variance

σ2

t(xi)

, we

assume that there is latent smooth function σ2

t(x)observed at discrete time points.

For modeling age-speciﬁc log mortality rates, Hyndman and Ullah [

] proposed the

application of weighted penalized regression splines with a monotonic constraint for ages

above 65, where the weights are equal to the inverse variances,

ωt(xi)=1/ ˆ

σ2

t(xi)

. For

each year t,

ft(xi)=argmin

θt(xi)

∑P

i=1ωt(xi)|yt(xi)−θt(xi)|+λ∑P−1

i=1

θ0t(xi+1)−θ0t(xi)

(2)

where,

represents different ages (grid points) in a total of Pgrid points,

represents a

smoothing parameter,

θ0

denotes the ﬁrst derivative of smooth function

, which can be

both approximated by a set of B-splines (Wang [

]). The

loss function and

penalty

function (i.e., monotonic constraint) are used to obtain robust estimates for high ages

(D’Amato, et al. [14]).

2.3. Functional Principal Component Regression (FPCR)

The theoretical, methodological, and practical aspects of functional principal com-

ponent analysis (FPCA) have been extensively studied in the functional data analysis

literature, since it allows ﬁnite dimensional analysis of a problem that is intrinsically

inﬁnite-dimensional (Shang and Hyndman [

]). Numerous examples of using FPCA as

an estimation tool in regression problem can be found in different ﬁelds of applications,

such as breast cancer mortality rate modeling and forecasting (Erbas et al. [

], call vol-

ume forecasting Shen and Huang [

], Hall and H-NM [

]), climate forecasting (Shang

and Hyndman [

]), demographical modeling and forecasting (Shang and Hyndman [

];

Hyndman et al. [21]).

At a population level, under FPCA settings, a smoothed stochastic process denoted by

f(x)={f1(x),· · · ,fn(x)}

in Equation (1) can be decomposed into the mean function and

the sum of the products of orthogonal functional principal components and uncorrelated

principal component scores. It can be expressed as

ft(x)=µ(x)+

∞

∑

k=1

βt,k∅k(x)

=µ(x)+∑K

k=1βt,k∅k(x)+et(x)

(3)

where, underlying smooth function

ft(x)

belongs to squared integral Hilbert space and

a set of discrete time points,

µ(x)

is the mean function;

{∅1(x),· · · ,∅K(x)}

is a set of the

ﬁrst

functional principal components;

β1=(β1,1,· · · ,β1,n)0

and

(β1,· · · ,βK)

denotes a

set of principal component scores and

βk∼N(0, ηk)

, where

ηk

is the

kth

eigen value of

the covariance function

cf(r,s)=E{[f(r)−µ(r)][ f(s)−µ(s)]}(4)

And

et(x)

is model truncation error function with a mean of zero and a ﬁnite variance.

Equation (2) provides dimension reduction as the ﬁrst Kterms often serve a good approxi-

mation to the inﬁnite terms and thus smoothed function

f(x)

is adequately summarized

by the Kdimensional vector

(β1,· · · ,βK)

. Here, using the formula deﬁned in Shang and

Hyndman [

], the optimal value of Kis chosen as the minimum that reaches a certain level

of the proportion of total variance explained by the Kleading components such that

K=argmin

K≥1n∑K

k=1ˆ

ηk/∑∞

k=1ˆ

ηk1{ˆ

ηk>0}≥δo.

Mathematics 2023,11, 3808 5 of 14

where

δ=

0, 9,

1{ˆ

ηk>0}

is used to exclude possible zero eigenvalues with binary indica-

tor function

1{·}

. In a dense and regularly spaced functional time series, the estimated

mean function and the covariance function are shown to be consistent under the weak

dependency (Haberman and Kokoxzka [

]). From the empirical covariance function, we

can extract empirical functional principal component functions B={ˆ

∅1(x),· · · ,ˆ

∅K(x)}

using singular value decomposition. By conditioning on the set of smoothed functions

f(x)={f1(x),· · · ,fn(x)}

and the estimated functional principal component

, the h-step

ahead forecasts of yn+h(x)can be obtained as

yn+h|n(x)=E[yn+h(x)|f(x),B]=ˆ

µ(x)+∑K

k=1ˆ

βn+h|n,kˆ

∅k(x)(5)

where

βn+h|n,k

denotes the h-step ahead forecasts of

βn+h|k

using a univariate time series

forecasting method, which can handle non-stationarity of the principal component scores.

2.4. A Univariate Time Series Forecasting Method

To obtain

βn+h|n,k

, a univariate time series forecasting method namely autoregres-

sive integrated moving average (ARIMA) method has been considered (Hyndman and

Shang [

]). This method is helpful to deal with nonstationary series which contains

stochastic trend component. Since the annual age-speciﬁc mortality rates for total, male

and female series do not contain any seasonality, the ARIMA has a general form of

(1−ψ1B− · · · ,ψmBm)(1−B)dβk=α+(1+θ1B+· · · +ψnBn)wk

where

represents the intercept,

(ψ1,· · · ,ψm)

are the coefﬁcients associated with autore-

gressive part,

(θ1,· · · ,θn)

are the coefﬁcients associated with moving average part,

denote the backshift operator, ddenotes the differencing operator and

represents a

white-noise terms. To determine the optimal value of orders for autoregressive, moving

average and differencing part, we used automatic algorithm of Hyndman and Khan-

dakar [

]). Once we determine the value of d, the orders of mand nare selected based on

the optimal Akaike information criterion (AIC) with a correction for small sample sizes.

Once we identiﬁed the best ARIMA model, maximum likelihood method can then be used

to estimate the parameters of that model.

2.5. Functional Autoregressive Process (FAR) of Order One

Bosq [

] introduced the functional autoregressive (fAR) process of order 1, and

derived one-step ahead forecasts that are based on a regularized form of the Yule-Walker

equations. It is a direct extension of function-on-function linear model under the functional

data framework (Ramsay and Silverman [

]). In this model, we would like to use

yt−1(x)

to predict

yt(x)

i.e., to measure the impact of one year age differences on mortality series

for Canadian population. yt(x)is denoted as fAR (1) and deﬁned as

yt(x)=β0(x)+Zβ(z,x)yt−1(x)dx+∈t(x)(6)

Here,

yt(x)

was the observed mortality rate for age

in year

; we assume that

yt(x)

depends on

yt−1(x)

other than the current time and

β(z,x)

is the coefﬁcient function

deﬁned as

β(z,x)=∑J

j=1∑K

k=1bjk ϕj(z)∅k(x)

. Here, we have used two basis functions for

two different time points (ages)

and

. The two basis functions may be different but for

simplicity we used the same basis function for estimating the coefﬁcient function. Details of

estimation and inference procedure can be found in Bosq [

]. Here,

∈t(x)

is a stochastic

process with mean zero and covariance function γ(z,x)=cov{∈t(z),∈t(x)},∀t.

Mathematics 2023,11, 3808 6 of 14

2.6. Prediction Interval and Forecast Accuracy

To construct prediction interval, we calculate the forecast variance that follows from

Equations (1) and (3). Because of orthogonality, the forecast variance can be approximated

by the sum of component variances.

ξn+h(x)=Var[yn+h(x)|f(x),B]=σ2

µ(x)+∑K

k=1un+h,kˆ

∅2

k(x)+υ(x)+σ2

n+h(x)

where

un+h,k=Var(βn+h,k

β1,k,· · · ,βn,k)

can be obtained from the time series, and the

model error variance

υ(x)

is estimated by averaging error terms for each

, and

σ2

µ(x)

and

σ2

n+h(x)

can be obtained from the nonparametric smoothing method used. Based on the

normality assumption, the 100

(1−α)

% prediction interval for

yn+h(x)

is constructed as

yn+h|n(x)±zαpξn+h(x).

Again, let

en,h(x)=yn+h(x)−ˆ

yn+h|n(x)

denote the forecast error for Equation (5). We

could then obtain the minimum value of the integrated squared forecast error which is

deﬁned as

ISFEn(h)=Ze2

n,h(x)dx

We ﬁt the model to data up to time t and predict the next s period to obtain

ISFEn(h)

· · ·

. We have used a readily available R addon packages “demography” (Hynd-

man et al. [

]), “forecast” (Hyndman et al. [

]), “rainbow” (Shang and Hyndman [

]),

“fda”(Ramsay et al. [

]) and “ftsa”(Hyndman and Shang [

]) to produce ﬁgures and

analyzing the data respectively.

2.7. Evaluation of Interval Forecast Accuracy

To assess interval forecast accuracy, we use the interval score of Gneiting and Raftery [

For each year in the forecasting period, one-year-ahead to 10 years ahead prediction intervals

were calculated at the

(1−α)×

100% prediction interval, with lower and upper bounds that

are predictive quantiles at

α/

2 and 1

−α/2

. A scoring rule defined in Geniting and Rafery [

]

for the interval forecast at age xiis as follows:

Mα(xl,xu,xi)=(xu−xl)+2

α(xl−xi)I(xi<xl)+2

α(xi−xu)I(xi>xu)

where,

I{·}

represents the binary indicator function,

and

denote the lower and upper

predictive quantiles at

α/

2 and 1

−α/2

, and

denotes the level of signiﬁcance. A forecaster

is looking for narrow prediction intervals. For different ages and years in the forecasting

period, we consider the mean interval scores as deﬁned in Gneiting and Katzfuss [38].

3. Results

In this study we demonstrate the methodology using demographic data-annual age-

speciﬁc mortality rates of three different series: total, male and female population in Canada.

For this we have

yt(xi)=log(mt(xi))

where

mt(xi)

denotes the mortality rate for age

year t.

Figure 1shows three years of total death rates in Canada. The dynamic behavior of

the underlying curve is relatively complicated and is clear from Figure 1. For example, the

‘bump’ around 18–19 years of age higher relative to nearby ages in 1991 than in the other

years plotted. Furthermore, the general drop in mortality over time is not uniform over

ages or time as observed in nearby ages before 20, for three time periods over ten years

apart. First, we transformed our three different mortality series: total, male and female into

functional objects. We represented our discrete data with B-spline basis functions since our

data shows monotonicity. Next, we obtained the best estimated smoothed curves

f(t)

eliminating the contribution of the errors and noise presented in the functional objects. To

do that we used a penalized regression spline, constrained to be monotonic. We weight

the penalized regression splines using the inverse of variance

σ2

t(xi)

. The order-selection

Mathematics 2023,11, 3808 7 of 14

procedure described in functional principal component regression section led to a model

with K=2 basis functions. The robustness parameter was set via cross validation.

Figure 2shows rainbow plots of the total, male and female age-speciﬁc smoothed

log mortality rates in Canada from 1991 to 2019. The time ordering of the curves follows

the color order of a rainbow, where curves from the distant past are shown in red and

the more recent curves are shown in purple. The changing shape of the curves over time

and relatively complex dynamic nature around early ages are clear from the ﬁgure as well.

Further analyses such as descriptive and forecasting are based on these functional data.

Mathematics 2023, 11, x FOR PEER REVIEW 7 of 14

Figure 2 shows rainbow plots of the total, male and female age-speciﬁc smoothed log

mortality rates in Canada from 1991 to 2019. The time ordering of the curves follows the

color order of a rainbow, where curves from the distant past are shown in red and the

more recent curves are shown in purple. The changing shape of the curves over time and

relatively complex dynamic nature around early ages are clear from the ﬁgure as well.

Further analyses such as descriptive and forecasting are based on these functional data.

(a) (b)

(c)

Figure 2. Plot of smoothed functional mortality series: (a) Female mortality series; (b) male series

and (c) total series (both sexes). Curves from the distant past are shown in red and the more recent

curves are shown in purple.

3.1. Temporal Paerns of Mortality Rates

Figure 3 displays the mean functions of the functional data for three diﬀerent series.

From Figure 3, it is apparent that the general increase of mean function in mortality over

Figure 2.

Plot of smoothed functional mortality series: (

) Female mortality series; (

) male series

and (

) total series (both sexes). Curves from the distant past are shown in red and the more recent

curves are shown in purple.

3.1. Temporal Patterns of Mortality Rates

Figure 3displays the mean functions of the functional data for three different series.

From Figure 3, it is apparent that the general increase of mean function in mortality over

time is observed for male group of population compared to total and female population

just after age of 20. Furthermore, the ‘bumps’ was around the age of 20 for male group,

however it was around the age of 15–17 years for female group and around 18–19 years for

total population respectively. The high variation was also found nearby 15–20 years of ages

compared to other discreate points for each series.

Mathematics 2023,11, 3808 8 of 14

Mathematics 2023, 11, x FOR PEER REVIEW 8 of 14

time is observed for male group of population compared to total and female population

just after age of 20. Furthermore, the ‘bumps’ was around the age of 20 for male group,

however it was around the age of 15–17 years for female group and around 18–19 years

for total population respectively. The high variation was also found nearby 15–20 years of

ages compared to other discreate points for each series.

Figure 3. Mean functions for smoothed functional mortality series: total mortality series (black);

male mortality series(red) and female mortality series (green).

3.2. Age-Speciﬁc Mortality Forecasting for Canada

Figure 4 shows the forecasts of Canadian mortality rate data for: total, male and fe-

male groups from 2020 to 2029 (ten years) highlighted in rainbow color. The time ordering

of the curves follows the color order of a rainbow, where curves from the distant past are

shown in red and the more recent curves are shown in purple. Compared to total and

female group, highest bumps occur around age 20 years for male group. However, a com-

plex dynamic behavior for next ten years was found around early ages for female groups

with relatively a similar paern at nearby 20 years of ages. Besides these, after middle

years of age (around 40 years), forecasted mortality rates for next ten years for all series

show general increase over the time and ages.

Figure 4. Ten year forecasted mortality rate of Canada for both sexes (left), male (middle) and fe-

male (right). Red color represents year 2020 and blue color represents year 2029.

Forecasts of mortality rates in 2020 for each series are shown in Figure 5 along with

95% prediction interval computed using the variance given in prediction interval and fore-

cast accuracy section. Clearly the greatest forecast variation observed around early and

middle years of ages for all the series. In addition, we obtained forecasts of mortality rates

in 2020 for each series based on the traditional univariate time series forecasting method

Figure 3.

Mean functions for smoothed functional mortality series: total mortality series (black); male

mortality series(red) and female mortality series (green).

3.2. Age-Speciﬁc Mortality Forecasting for Canada

Figure 4shows the forecasts of Canadian mortality rate data for: total, male and female

groups from 2020 to 2029 (ten years) highlighted in rainbow color. The time ordering of the

curves follows the color order of a rainbow, where curves from the distant past are shown

in red and the more recent curves are shown in purple. Compared to total and female

group, highest bumps occur around age 20 years for male group. However, a complex

dynamic behavior for next ten years was found around early ages for female groups with

relatively a similar pattern at nearby 20 years of ages. Besides these, after middle years

of age (around 40 years), forecasted mortality rates for next ten years for all series show

general increase over the time and ages.