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Stochastic population forecasts using functional data models for mortality, fertility and migration
Abstract
Age–sex-specific population forecasts are derived through stochastic population renewal using forecasts of mortality, fertility and net migration. Functional data models with time series coefficients are used to model age-specific mortality and fertility rates. As detailed migration data are lacking, net migration by age and sex is estimated as the difference between historic annual population data and successive populations one year ahead derived from a projection using fertility and mortality data. This estimate, which includes error, is also modeled using a functional data model. The three models involve different strengths of the general Box–Cox transformation chosen to minimise out-of-sample forecast error. Uncertainty is estimated from the model, with an adjustment to ensure that the one-step-forecast variances are equal to those obtained with historical data. The three models are then used in a Monte Carlo simulation of future fertility, mortality and net migration, which are combined using the cohort-component method to obtain age-specific forecasts of the population by sex. The distribution of the forecasts provides probabilistic prediction intervals. The method is demonstrated by making 20-year forecasts using Australian data for the period 1921–2004. The advantages of our method are: (1) it is a coherent stochastic model of the three demographic components; (2) it is estimated entirely from historical data with no subjective inputs required; and (3) it provides probabilistic prediction intervals for any demographic variable that is derived from population numbers and vital events, including life expectancies, total fertility rates and dependency ratios.
... Overall, life tables and death rates are key instruments spanning many disciplines and with multiple applications. It is not surprising, therefore, that over recent years much attention has been given to evolving life expectancy and mortality rate forecasting models, including stochastic, functional data and Bayesian models [3][4][5], and to improving the estimators of death probabilities [6][7][8]. ...
... For example, to measure the number of people dying with integer age x in the (r, s)-quarter of year a, r s D a x , all that is necessary is to count the number of crosses recorded in the Lexis diagram in that particular surface (example: see quarter (1,4) in the lower square of Fig 1-middle). In a similar fashion, we can define the number of people immigrating with integer age x in the (r, s)-quarter of year a, r s I a x , by counting the number of circles (example: see quarter (2,1) in the lower square of Fig 1-middle) and the number of people emigrating with integer age x in the (r, s)-quarter of year a, r s E a x , by counting the number of blades (example: see quarter (3,3) in the lower square of Fig 1-middle). ...
... For example, to measure the number of people dying with integer age x in the (r, s)-quarter of year a, r s D a x , all that is necessary is to count the number of crosses recorded in the Lexis diagram in that particular surface (example: see quarter (1,4) in the lower square of Fig 1-middle). In a similar fashion, we can define the number of people immigrating with integer age x in the (r, s)-quarter of year a, r s I a x , by counting the number of circles (example: see quarter (2,1) in the lower square of Fig 1-middle) and the number of people emigrating with integer age x in the (r, s)-quarter of year a, r s E a x , by counting the number of blades (example: see quarter (3,3) in the lower square of Fig 1-middle). ...
The big data revolution has greatly expanded the availability of microdata on vital statistics, providing researchers with unprecedented access to large and complex datasets on birth, death, migration, and population, sometimes even including exact dates of demographic events. This has led to the development of a novel methodology for estimating sub-annual life tables that offers new opportunities for the insurance industry, also potentially impacting on the management of pension funds and social security systems. This paper introduces the qlifetable package, an R implementation of this methodology. It begins by detailing how basic summary statistics are computed by the package from detailed individual records, including the length of age years, which should be observed as relative (subjective) to ensure congruency between age and calendar time when measuring exposure times and exact ages of individuals at events. This is a new result that compels the observation of time as relative in the disciplines of actuarial science, risk management and demography. Afterwards, the paper demonstrates the use of the package, which integrates a set of functions for estimating crude quarterly (and annual) death rates, calculating seasonal-ageing indexes (SAIs) and building quarterly life tables for a (general or insured) population by exploiting either microdata of dates of births and events or summary statistics.
... Until 2022, Statistics Iceland employed a mixture of several classes of methods for producing population projections. These were based on: (i) functional models [1,2], for predicting fertility and mortality rates (ii) Bayesian models [3] and econometric/ARDL models [4] for short term migration numbers (iii) assumptions/scenarios (as used by numerous statistical institutes) for long term migration predictions. In addition, modeling the time correlation between emigration and lagged immigration has been used in order to further improve the predictions. ...
... In the statistical demography literature, the following models for demographic components such as migration, mortality or fertility were recently tested [1][2][3][5][6][7][8]: generalized linear models (GLMs), generalized additive models (GAMs), state space models (SSM) or even Gaussian Process (GP) models. The proposed models for these components are simpler than the ones we employ for our projections, since they include either: (i) time and age smooth dependencies, when using GP or SSM, but no other attributes, or (ii) more characteristics in addition to time and age, such as gender and region in the case of hierarchical GLMs, although in that case the rates were linear functions of these attributes/interactions and the time priors were damped linear trends (e.g. in [5]). ...
... In addition, not all studies update the exposed population at each out-of-sample time-step when calculating the predicted demographic rates based on GAMs/GLMs. Examples of studies which do this operation are based on functional models or on expert assumptions (see [1][2][3]). ...
The population projections published by Statistics Iceland in 2022 are based on new statistical models of fertility, mortality and migration and their combined predictions. The projections are built by applying standard statistical and demographical methods. The new models and methods described in this paper allow us, in addition, to produce local projections and to incorporate the probability of overestimating the resident population. The overestimation effect is due to a lack of deregistration, e.g. the estimated resident population is about 2.5% smaller than the registered population. The results of the new population projections consist of predicted values as well as their associated uncertainty measures. The projection method does not include any effects due to possible crises caused by natural, social or economic factors, nor the number of refugees. For instance, the predicted population growth for the year 2022 is 2.02% while the observed, register based growth during 2022 was 3.06%. The difference is entirely explained by the unusually high refugee flow which accounted for a 0.96% additional change in population. The national and local projections will be updated as soon as auxiliary, prior information becomes available and users, planning and administration factors and policy makers are invited to provide feedback of qualitative and/or quantitative type to Statistics Iceland for that purpose. Statistical Series-Working papers 2
... Fundamental changes of welfare policies largely depend on the accurate forecast of longevity in any country. Stochastic modeling of mortality forecasting is gaining popularity in this context; United Nations and several industrialized countries already adapted stochastic forecasting techniques [4,5]. Several probabilistic approaches exist for mortality forecasting, both from Frequentist and Bayesian point of view. ...
... As mentioned before, the FDA method is essentially an extension of the LC method except for number of principal components and smoothing prior to model-fitting. LC model can explain almost 95% variation for most of the low mortality countries For post world war period, additional PCs increase the goodness-of-fit of the model [4,9]. For Matlab HDSS, the LC model can explain lower than that; only 85.4% variations for men and 88.2% for women for smoothed mortality rates (considering only the first PC in Table 1). ...
... This component shows a decline in mortality over time which is usually fastest at childhood and childbearing ages. The second PC models an increase and then decrease explaining 3% of mortality for low mortality countries [4]; for Matlab HDSS it is much higher ( Table 1). A large decrease took place at ages around 10 for women whereas for men it happens around age 20. ...
Mortality forecasts are essential part for policymaking in any aging society. In recent years, methods to model and forecast mortality have improved considerably. Among them, Lee-Carter method is one of the most influential method. In this paper, Lee-Carter method is applied to forecast mortality and life expectancy of Bangladesh. A functional data analysis approach is used to decompose the smoothed log-mortality rates in Lee-Carter framework for higher goodness-of-fit of the models and for longer forecast horizons. Bangladesh has been experiencing a mortality transition and has gained life expectancy in last few decades. The fitted model here showed higher pace of mortality decline for women in Bangladesh than that of men. The forecasts showed continuation of mortality improvement in long run and by 2060 life expectancy at birth is expected to reach over 80 years for both sexes in Bangladesh. The study also predicts the effect of reduction in infant mortality on the life expectancy in Bangladesh.
... Hyndman and Ullah (2007) addressed the problem of heteroscedasticity by smoothing the death rates (see 'Smoothing'). These authors assumed an approximate binomial distribution for deaths, but in later work Hyndman and Booth (2008) adopted the Poisson distribution. The random error in observed deaths is removed before model estimation and added back after estimation as one of the components of total error. ...
... For French mortality (single years of age from 1899 to 2001), four terms were required. Hyndman and Booth (2008) note that six terms is the maximum likely to be needed. ...
... Hyndman and Ullah (2007) accounted for all sources of error in the total analytical forecast variance, including random error (which was removed prior to estimation). On finding that this variance does not accord with the empirical forecast variance obtained by simulation, Hyndman and Booth (2008) derived a method for adjusting the one-step analytical variance to match the one-step empirical variance. ...
The introduction of the Lee-Carter (LC) method marked a breakthrough in mortality forecasting, providing a simple yet powerful data-driven stochastic approach. The method has the merit of capturing the dynamics of mortality change by a single time index that is invariably linear. This 30th anniversary review of its 1992 publication examines the LC method and the large body of research that it has since spawned. We first describe the method and present a 30-year ex-post evaluation of the original LC forecast for U.S.~mortality. We then review the most prominent extensions of the LC method in relation to the limitations that they sought to address. With a focus on the efficacy of the various extensions, we review existing evaluations and comparisons. To conclude, we juxtapose the two main statistical approaches used, discuss further issues, and identify several potential avenues for future research.
... The present work proposed a new method for coherent mortality forecasting that incorporates forecasting interpretable product and ratio functions of rates using the functional data paradigm introduced in Hyndman and Ullah (2007). The product-ratio functional forecasting method can be applied to two or more sub-populations, incorporates convenient calculation of prediction intervals as well as point forecasts and is suitable for use within a larger stochastic population modeling framework such as Hyndman and Booth (2008). The new method is simple to apply, flexible in its dynamics, and produces forecasts that are at least as accurate in overall terms as the comparable independent method. ...
... In the present research work, using a new method for coherent mortality forecasting which involves forecasting interpretable product and ratio functions of rates using the functional data paradigm introduced in Hyndman and Ullah (2007) [4] . The product-ratio functional forecasting method can be applied to two or more sub-populations, incorporates convenient calculation of prediction intervals as well as point forecasts and is suitable for use within a larger stochastic population modeling framework such as Hyndman and Booth (2008) [5] . ...
... Life expectancy forecasts are obtained from the forecast agespecific death rates using standard life table methods (Preston et al. 2001) [16] . To obtain prediction intervals for life expectancies, we simulate a large number of future death rates, as described in Hyndman and Booth (2008) [5] , and obtain the life expectancy for each. Then the prediction intervals are constructed from percentiles of these simulated life expectancies. ...
... We split our age-and sex-specific data into a training sample (including data from years 1 to (n − 30)) and a testing sample (including data from years (n − 29) to n), where n represents the total number of years in the data. Following the early work by Hyndman and Booth (2008), we implement an expanding window approach as it allows us to assess the forecast accuracy among methods for different forecast horizons. With the initial training sample, we produce one-to 30-year-ahead forecasts, and determine the forecast errors by comparing the forecasts with actual out-of-sample data. ...
... The prediction intervals for age-specific mortality are obtained from (8), whereas the prediction intervals for life expectancy are obtained from the percentiles of simulated life expectancies obtained from simulated forecast mortality rates as described by Hyndman and Booth (2008). ...
A robust multilevel functional data method is proposed to forecast age-specific mortality rate and life expectancy for two or more populations in developed countries with high-quality vital registration systems. It uses a robust multilevel functional principal component analysis of aggregate and population-specific data to extract the common trend and population-specific residual trend among populations. This method is applied to age- and sex-specific mortality rate and life expectancy for the United Kingdom from 1922 to 2011, and its forecast accuracy is then further compared with standard multilevel functional data method. For forecasting both age-specific mortality and life expectancy, the robust multilevel functional data method produces more accurate point and interval forecasts than the standard multilevel functional data method in the presence of outliers.
... Research has moved in many directions since then as described by Basellini, Camarda, and Booth (2023) in a valuable paper in a special journal issue on the topic. One direction is the use of functional data models by Hyndman and Booth (2008). Another direction is "coherent" forecasting in which a group of countries is modeled and forecasted together, drawing strength from common patterns while preserving key differences (Li and Lee 2005), an approach explored further and critiqued by Booth (2020). ...
... We can do the same for fertility. We could also do this for in and out migration as well, for example using the methods in Cohen et al. (2008), or for net migration, for example using the methods in Azose and Raftery (2015) or Hyndman and Booth (2008). More typically values are assumed for migration and are held constant across the random scenarios. ...
The long human lifespan enables long run forecasts of population size and age distribution. New methods include biodemographic research on upper limits to life expectancy and incorporation of early experiences affecting later life mortality such as smoking, obesity, and childhood health shocks. Some fertility forecasts incorporate education and quantum‐tempo insights. Statistical time series and Bayesian methods generate probabilistic forecasts. Yet recent decades have brought surprising changes in the economy, natural environment, and vital rates. In these changing circumstances we need new methods and the increasing use of probabilistic models and Bayesian methods incorporating outside information. The increasing use of microsimulation combined with aggregate forecasting methods is a very promising development enabling more detailed and heterogeneous forecasts. Some new uses of stochastic forecasts are interesting in themselves. Probabilistic mortality forecasts are used in finance and insurance, and a new Longevity Swap industry has been built on them. Random sample paths used to generate stochastic population forecasts can stress‐test public pension designs for fiscal stability and intergenerational equity. Population forecasting a few decades ago was a dull backwater of demographic research, but now it is increasingly important and is full of intellectual and technical challenges.
... Ndirangu & Oduto (2011) noted that overcrowding lecture rooms and inadequate teaching resources are likely to impact negatively on teachers and pupils" performance. Herman (2009) argues that employees may perform differently put in different situations hence different results. ...
Measuring performance is a key strategy for organizational success. If performance of an employee is not quantified, one cannot be certain that there is any value gained from employee. Performance contracting is a concept that is used in Kenya to manage performance of employees. The contracting parties freely negotiate performance agreement which clearly specifies the intentions, obligations and responsibilities of the contracting parties. Performance contracts are gauged against an agreed set of indicators that provide a clear framework for establishing accountability. In Kenya it is a requirement for all civil servants to sign performance contracts. To date, the secondary school teachers have not yet fully accepted to sign performance contracts. The study, therefore, sort to establish the extent to which, performance indicators influence acceptance of performance contracting by secondary school teachers in Kenya, a case of Meru Central District. To achieve this objective, the study adopted descriptive research design. The study covered all the twenty one (21) public secondary schools in Meru Central District. The target population was all the government employed teachers within Meru Central District. A sample of thirty percent (30%) of teachers was picked from each of the 21 schools by use of simple random sampling. Primary data was collected by use of structured questionnaires and existing relevant documents were perused for secondary data. The data was analysed using descriptive and inferential statistics. This was done by use of the Statistical Package for Social Sciences (SPSS). Results were presented in charts, graphs and tables. The study findings indicated that performance indicators influenced acceptance of performance contracting by the secondary school teachers. The study concludes there is a positive relationship between performance indicators and acceptance of performance contracting by secondary school teachers.Key Words: Performance Contracting, Performance indicators, Secondary Schools
... The authors noted that "the approach is a natural extension of methods for mortality and fertility forecasting that have evolved over the last two decades" but did not specify which age groups could be modeled when studying mortality. However, already in their work [19], the author of the method includes individuals aged 0 to 100 years in the model. A number of authors, using the Hyndman and Ullah method, also included younger, older, and elderly groups in the model [20][21][22][23]. ...
Age-specific mortality forecasting in Kazakhstan plays a crucial role in public health planning and healthcare management. By predicting mortality rates across different age groups, policymakers, healthcare providers, and researchers can make informed decisions that improve health outcomes and allocate resources more effectively. We analyzed Kazakhstan’s annual mortality data from 1991 to 2023. The Lee–Carter model and its extensions were used to predict mortality. But they did not give satisfactory results for predicting mortality. Including external socio-economic factors in the model did not improve the forecasting accuracy. The accuracy of the forecast increased with a separate analysis of the subpopulations of children and adults. This was because, since 1991 in the children subpopulation there has been a pronounced linear downward trend, while in the adult subpopulation the global trend in mortality dynamics is nonlinear. As a result, it is possible to make forecasts for 7 years with a high degree of accuracy (error < 10%) and forecast for the 8th, 9th, and 10th years with a “good” degree of accuracy (error 10–20%). In 2024–2033, a further mortality decline is expected in most age groups. Only in groups over 80 years old is a slight increase in mortality predicted in the coming year, but then a downward trend will be observed again.
... In demography, mortality and fertility rates are given as a function of age (e.g. Erbas et al., 2007;Hyndman and Ullah, 2007;Hyndman and Booth, 2008), while in geophysical sciences, magnometers record the strength and direction of the magnetic field every five seconds. Due to the wide range of applications, functional time series and the development of techniques that allow to relax the i.i.d. ...
The literature on time series of functional data has focused on processes of which the probabilistic law is either constant over time or constant up to its second-order structure. Especially for long stretches of data it is desirable to be able to weaken this assumption. This paper introduces a framework that will enable meaningful statistical inference of functional data of which the dynamics change over time. We put forward the concept of local stationarity in the functional setting and establish a class of processes that have a functional time-varying spectral representation. Subsequently, we derive conditions that allow for fundamental results from nonstationary multivariate time series to carry over to the function space. In particular, time-varying functional ARMA processes are investigated and shown to be functional locally stationary according to the proposed definition. As a side-result, we establish a Cram\'er representation for an important class of weakly stationary functional processes. Important in our context is the notion of a time-varying spectral density operator of which the properties are studied and uniqueness is derived. Finally, we provide a consistent nonparametric estimator of this operator and show it is asymptotically Gaussian using a weaker tightness criterion than what is usually deemed necessary.
... Functional data analysis has received increasing amounts of attention in demographic forecasting (see, e.g., Cairns et al. 2011, D'Amato et al. 2011, Carfora et al. 2017) since its first application to demographic modeling and forecasting by Hyndman and Ullah (2007). While Hyndman and Shang (2009) and Hyndman et al. (2013) apply functional time series analysis to age-specific mortality or fertility rates, Hyndman and Booth (2008) extended this method to the modeling and forecasting of age-and sex-specific population sizes. We treat a time series of age-specific populations as a functional time series. ...
The age pension aims to assist eligible elderly Australians who meet specific age and residency criteria in maintaining basic living standards. In designing efficient pension systems, government policymakers seek to satisfy the expectations of the overall aging population in Australia. However, the population’s unique demographic characteristics at the state and territory level are often overlooked due to the lack of available data. We use the Hamilton-Perry model, which requires minimum input, to model and forecast the evolution of age-specific populations at the state and territory level. We also integrate the obtained sub-national demographic information to determine sustainable pension ages up to 2051. We also investigate pension welfare distribution in all states and territories to identify the disadvantaged residents under the current pension system. Using the sub-national mortality data for Australia from 1971 to 2021 obtained from AHMD (2023), we implement the Hamilton-Perry model with the help of functional time series forecasting techniques. With the forecasts of age-specific population sizes for each state and territory, we compute the old age dependency ratio to determine the nationwide sustainable pension age.
... Beyaztas & Shang, 2019;Beyaztas & Yaseen, 2019;R. J. Hyndman & Booth, 2008;Shang, 2013;Wanger-Muns et al., 2018). ...
Among several hydrological processes, river flow is an essential parameter that is vital for different water resources engineering activities. Although several methodologies have been adopted over the literature for modeling river flow, the limitation still exists in modeling the river flow time series curve. In this research, a functional quantile autoregressive of order one model was developed to characterize the entire conditional distribution of the river flow time series curve. Based on the functional principal component analysis, the regression parameter function was estimated using a multivariate quantile regression framework. For this purpose, hourly scale river flow collected from three rivers in Australia (Mary River, Lockyer Valley, and Albert River) were used to evaluate the finite‐sample performance of the proposed methodology. A series of Monte‐Carlo experiments and historical data sets were examined at three stations. Further, uncertainty analysis was adopted for the methodology evaluation. Compared with the existing methods, the proposed model provides more robust forecasts for outlying observations, non‐Gaussian and heavy‐tailed error distribution, and heteroskedasticity. Also, the proposed model has the merit of predicting the intervals of future realizations of river flow time series at the central and non‐central locations. The results confirmed the potential for predicting the river flow time series curve with a high level of accuracy in comparison with the benchmark existing functional time series methods.
... (2) Statistic Calculation: After obtaining the bootstrapped principal component scores and errors in the model residuals, one can then obtain the bootstrapped survival function S(x). We take the first six principal components as suggested by Hyndman and Booth (2008) and express ...
We introduce the function principal component regression (FPCR) forecasting method to model and forecast age-specific survival functions observed over time. The age distribution of survival functions is an example of constrained data whose values lie within a unit interval. Because of the constraint, such data do not reside in a linear vector space. A natural way to deal with such a constraint is through an invertible logit transformation that maps constrained onto unconstrained data in a linear space. With a time series of unconstrained data, we apply a functional time-series forecasting method to produce point and interval forecasts. The forecasts are then converted back to the original scale via the inverse logit transformation. Using the age- and sex-specific survival functions for Australia, we investigate the point and interval forecast accuracies for various horizons. We conclude that the functional principal component regression (FPCR) provides better forecast accuracy than the Lee–Carter (LC) method. Therefore, we apply FPCR to calculate annuity pricing and compare it with the market annuity price.
... They can be chosen based off a threshold function that measures the proportion of variance explained by each principal component, or by minimizing the integrated squared forecast error (Hyndman and Ullah, 2007). However, Hyndman and Booth (2008) noted that the HU models are relatively insensitive to the choice of K as long as it is sufficiently large, and that setting K = 6 is more than adequate to capture the essential features of the data. For consistency across all models, we adopt K = 6 for both the HU model variants and the proposed HUts model. ...
This study introduces an innovative methodology for mortality forecasting, which integrates signature-based methods within the functional data framework of the Hyndman-Ullah (HU) model. This new approach, termed the Hyndman-Ullah with truncated signatures (HUts) model, aims to enhance the accuracy and robustness of mortality predictions. By utilizing signature regression, the HUts model is able to capture complex, nonlinear dependencies in mortality data which enhances forecasting accuracy across various demographic conditions. The model is applied to mortality data from 12 countries, comparing its forecasting performance against variants of the HU models across multiple forecast horizons. Our findings indicate that overall the HUts model not only provides more precise point forecasts but also shows robustness against data irregularities, such as those observed in countries with historical outliers. The integration of signature-based methods enables the HUts model to capture complex patterns in mortality data, making it a powerful tool for actuaries and demographers. Prediction intervals are also constructed with bootstrapping methods
... The second sample, called the test set, comprises the values between 2009 and 2018. In alignment with previous research [18,20,33,95] and recommendations from the Human Mortality Database [53], we utilized data beginning from 1950. This decision was based on the understanding that post-1950 data exhibits relative stability, a condition likely attributable to the cessation of the Second World War, where mortality data were not informative. ...
Forecasting mortality is challenging. In general, mortality rate forecasting exercises have been based on the supposition that predictors’ residuals are random noise. However, issues regarding model selection, misspecification, or the dynamic behavior of the temporal phenomenon lead to biased or underperformed single models. Residual series might present temporal patterns that can still be used to improve the forecasting system. This paper proposes a new recursive direct multi-step Hybrid System for Mortality Forecasting (HyS-MF) that combines the Autoregressive Integrated Moving Average (ARIMA) with Neural Basis Expansion for Time Series Forecasting (N-BEATS). HyS-MF employs (i) ARIMA to model and forecast the mortality rate time series with a recursive approach and (ii) N-BEATS with the direct multi-step approach to learn and forecast the residuals of the linear predictor. The final output is generated by summing ARIMA with the N-BEATS forecasts in each time horizon. HyS-MF achieved an average Mean Absolute Percentage Error (MAPE) less than 1.34% considering all prediction horizons, beating statistical techniques, machine learning, deep learning models, and hybrid systems considering 101 different time series from the French population mortality rate.
... Research from [14] employing a non-parametric approach to the original Lee-Carter model to facilitate smooth age functions and robust modeling of age-specific fertility rates. This functional data methodology has found application in forecasting fertility trends in Australia [15] and Malaysia [16][17]. While [18] have explored both models using Malaysian mortality data, a comparative analysis between the functional data method and the original Lee-Carter model concerning Malaysian fertility data remains unexplored. ...
Global fertility has been experiencing a significant decline, reaching towards the replacement ratio. This trend, coupled with increasing life expectancies, has led to the emergence of an ageing population. In this study, we delve into an analysis of fertility patterns among Malaysian women, considering both their childbearing age and ethnic groups. A comprehensive 63-year fertility dataset, from 1958 to 2020, were obtained from the Department of Statistics Malaysia. These data were fitted into the Lee-Carter model and its modified version, which is the functional data model. The models were evaluated using the out-sample forecast error measures. Results indicate that the third-order functional data model able to capture most of variation present in the actual data, consequently outperforming the Lee-Carter model in forecasting fertility rates among Chinese and Indian populations. The 20-year forecasts reveal a noteworthy shift in maternal ages of the highest births to older ages suggesting a trend towards delayed pregnancies among women. It is predicted that the Malay total fertility rates will likely fall to below the replacement level reaching 1.71 in 2040 whereas Chinese and Indian total fertility rates will substantially decrease to the lowest level in history below 1.0 which are 0.54 and 0.70 respectively. The evolution in Malaysian fertility rates is an alarming fact as, together with low mortality rates, it may impact the Malaysian population structure in future. Proactive policy measures are urgently needed to address these demographic shifts.
... For these methods to work effectively, multiple observations on the migration indicators are needed. Hyndman and Booth (2008) forecasted age patterns of net migration using functional data analysis, a statistical approach for analysing data that are in the form of curves. Shang and others (2016) showed how functional data analysis may be used to forecast age patterns of mortality, fertility, immigration, and emigration using the same data as Wiśniowski and others (2015). ...
This technical paper proposes a methodology for inferring the age and sex profiles of net migration. This approach enhances the capability of the Population Division of the United Nations Department of Economic and Social Affairs (United Nations Population Division) to estimate and project populations for the World Population Prospects (WPP). The age and sex profiles of net migration are crucial inputs to demographic accounting models for population estimation and projection. However, most countries in the world do not directly measure migration, and residual estimation methods for inferring patterns have proven inadequate owing to errors in population measures, births, and deaths. Recognizing that net migration lacks consistent patterns across different ages and sexes, this study introduces a novel strategy: estimating immigration and emigration flows by age and sex—categories that demonstrate regular patterns–and using differences from these flows to estimate net international migration by age and sex. Empirical validations using data from Sweden and the Republic of Korea have yielded promising results, prompting the extension of the method to estimate age- and sex-specific net migration patterns for countries lacking migration data.
... We forecast the size and age and sex composition of the Portuguese population using the standard cohort-component method stochastically modelling the components of demographic change. Forecasts of age-specific fertility and net migration rates are generated using the functional demographic data modelling approach (Hyndman and Booth, 2008), calibrated to data provided by Statistics Portugal from 1960 to 2019. Net migration is estimated using the demographic growth-balance equation. ...
Continuous longevity improvements and population ageing have led countries to modify national public pension schemes by increasing standard and early retirement ages in a discretionary, scheduled, or automatic way, and making it harder for people to retire prematurely. To this end, countries have adopted alternative retirement age strategies, but our analyses show that the measures taken are often poorly designed and consequently misaligned with the pension scheme's ultimate goals. This paper discusses how to implement automatic indexation of the retirement age to life expectancy developments while respecting the principles of intergenerational actuarial fairness and neutrality among generations of the respective policy scheme design. With stable demographic conditions, we show in policy designs in which extended working lives translate into additional pension entitlements, the pension age must be automatically updated to keep the period in retirement constant. Alternatively, policy designs that pursue a fixed replacement rate are consistent with retirement age policies targeting a constant balance between active years in the workforce and years in retirement. Under conditions of population ageing, the statutory pension age will have to increase at a faster rate to meet the intergenerational equity criteria. The empirical strategy employed a Bayesian Model Ensemble approach to stochastic mortality modelling to address model risk and generate forecasts of intergenerationally and actuarially fair pension ages for 23 countries from 2000 to 2050. The findings show that the pension age increases needed to accommodate the effect of longevity developments on pay-as-you-go equilibrium and to reinstate equity between generations are sizeable and well beyond those employed and/or legislated in most countries. A new wave of pension reforms may be at the doorsteps.
... FDA methods for modelling and forecasting data across a range of health and demographic issues have significant advantages for better understanding trends, risk factor relationships, and the effectiveness of preventive measures (Erbas et al. 2007;Hyndman and Ullah 2007;Ullah and Finch 2013). This approach has become popular in demographic applications, especially in forecasting mortality, fertility, and migration (Erbas et al. 2007;Hyndman and Ullah 2007;Hyndman and Booth 2008;Hyndman and Shang 2009;Ullah and Finch 2010). Recent studies have used FDA to explore changes in age-specific mortality over time in low-mortality countries (Léger and Mazzuco 2021) and developing countries (Shair et al. 2019) as well as to analyze cause-specific mortality trends across countries (Stefanucci and Mazzuco 2022), successfully enhancing the accuracy of the analyses and providing innovative insights. ...
This work analyses the contribution of ages and causes of death to gender gap in life expectancy in 20 European and non-European countries between 1959 and 2015, using Functional Data Analysis. Data were retrieved from the WHO Mortality Database and from the Human Mortality Database. We propose a Functional Principal Component Analysis of the age profiles of cause-specific contributions, to identify the main components of the distribution of the age-specific contributions according to causes of death, and to summarize them with few components. Our findings show that the narrowing gender gap in life expectancy was mainly driven by decreasing differences in cardiovascular diseases. Additionally, the study reveals that the age cause contributions act almost entirely on only two dimensions: level (extent of the cause-specific contribution to the overall mortality gender gap) and age pattern (location of the curves across ages). Notably, in the last period, it is not the "quantum" of the cause-specific contributions that matters, but the "timing", i.e. location across the age spectrum. Moreover, our results show that in the most recent period the gender gap in life expectancy is affected by composition of the causes of death more than it was in previous periods. We emphasise that Functional Data Analysis could prove useful to deepen our understanding of complex demographic phenomena.
... In contrast, new AI methods mainly focus on short-term forecasting. internal migration, long-term, [29] ; international migration, long-term, [30]; international migration, long-term, [31]. [32]; international migration, long-term, [33]; international migration, long-term, [34]; international migration, long-term, [35]; international migration, long-term, [36]; international migration, long-term, [37]. ...
As a fundamental, overall, and strategic issue facing human society, human migration is a key factor affecting the development of countries and cities given constantly changing population numbers. The fuzziness of the spatiotemporal attributes of human migration limits the pool of open-source data for human migration prediction, leading to a relative lag in human migration prediction algorithm research. This study expands the definition of human migration research, reviews the progress of research into human migration prediction, and classifies and compares human migration algorithms based on open-source data. It also explores the critical uncertainty factors restricting the development of human migration prediction. Given the effect of human migration prediction, in combination with artificial intelligence and big data technology, the paper concludes with specific suggestions and countermeasures aimed at enhancing human migration prediction research results to serve economic and social development and national strategy.
... Ramsay and Silverman (2005) and Tsay (2010) provide background for functional data analysis and time series analysis respectively, while (Hörmann and Kokoszka 2012) provides background for functional time series. Hyndman and Shahid Ullah (2007) and Hyndman and Booth (2008) propose a forecasting approach for functional time series and apply it to demographic data. Hyndman and Shang (2009) proposes a weighted functional approach which assigns more weight to recent observations and yields an improvement in forecast accuracy. ...
A functional time series approach is proposed for investigating spatial correlation in daily maximum temperature forecast errors for 111 cities spread across the U.S. The modelling of spatial correlation is most fruitful for longer forecast horizons, and becomes less relevant as the forecast horizon shrinks towards zero. For 6-day-ahead forecasts, the functional approach uncovers interpretable regional spatial effects, and captures the higher variance observed in inland cities versus coastal cities, as well as the higher variance observed in mountain and midwest states. The functional approach also naturally handles missing data through modelling a continuum, and can be implemented efficiently by exploiting the sparsity induced by a B-spline basis. The temporal dependence in the data is modeled through temporal dependence in functional basis coefficients. Independent first order autoregressions with generalized autoregressive conditional heteroskedasticity [AR(1)+GARCH(1,1)] and Student-t innovations work well to capture the persistence of basis coefficients over time and the seasonal heteroskedasticity reflecting higher variance in winter. Through exploiting autocorrelation in the basis coefficients, the functional time series approach also yields a method for improving weather forecasts and uncertainty quantification. The resulting method corrects for bias in the weather forecasts, while reducing the error variance.
... Lee and Miller (2001) then adopted 1950 in order to better meet the assumption of fixed b x (see 'Fixed b x '). While (Hyndman & Ullah, 2007) used data from 1899 to 2001 (and robust estimation), (Hyndman & Booth, 2008) 7 Except that slight differences in individual k t values may result from SVDs based on different fitting periods. adopted a starting year of 1950 in order to ''avoid difficulties with war years, the 1918 Spanish influenza pandemic, and structural change over the course of the twentieth century'' (p. ...
The introduction of the Lee–Carter (LC) method marked a breakthrough in mortality forecasting, providing a simple yet powerful data-driven stochastic approach. The method has the merit of capturing the dynamics of mortality change by a single time index that is almost invariably linear. This thirtieth anniversary review of its 1992 publication examines the LC method and the large body of research that it has since spawned. We first describe the method and present a 30-year ex post evaluation of the original LC forecast for U.S. mortality. We then review the most prominent extensions of the LC method in relation to the limitations that they sought to address. With a focus on the efficacy of the various extensions, we review existing evaluations and comparisons. To conclude, we juxtapose the two main statistical approaches used, discuss further issues, and identify several potential avenues for future research.
... The statistical analysis of functional time series has become increasingly important to many scientific fields including climatology (Shang and Hyndman, 2011), finance (Kokoszka and Zhang, 2012), geophysics (Hörmann and Kokoszka, 2012), demography (Hyndman and Booth, 2008), manufacturing (Woodall, 2007), and environmental modeling (Fortuna et al., 2020). A functional time series is a sequence of functions (i.e., infinite objects), observed over time. ...
Detecting changepoints in functional data has become an important problem as interest in monitory of climatologies and other various processing monitoring situations has increased, where the data is functional in nature. The observed data often contains variability in amplitude (y-axis) and phase (x-axis). If not accounted for properly, incorrect changepoints can be detected, as well as underlying mean functions at those changes will be incorrect. In this paper, an elastic functional changepoint method is developed which properly accounts for these types of variability. Additionally, the method can detect amplitude and phase changepoints which current methods in the literature do not, as they focus solely on the amplitude changepoint. This method can easily be implemented using the functions directly, or to ease the computational burden can be computed using functional principal component analysis. We apply the method to both simulated data and real data sets to show its efficiency in handling data with phase variation with both amplitude and phase changepoints.
... This is a problem of the fitting period selection; many studies prefer to choose a shorter period to avoid data volatility. For example, Hyndman and Booth [30] used 1950 as the starting year to avoid the difficulties of the war years and the 1918 Spanish influenza pandemic. Other related studies are by Tuljapurkar et al. [31] and Lee and Miller [32]. ...
The Lee–Carter model could be considered as one of the most important mortality prediction models among stochastic models in the field of mortality. With the recent developments of machine learning and deep learning, many studies have applied deep learning approaches to time series mortality rate predictions, but most of them only focus on a comparison between the Long Short-Term Memory and the traditional models. In this study, three different recurrent neural networks, Long Short-Term Memory, Bidirectional Long Short-Term Memory, and Gated Recurrent Unit, are proposed for the task of mortality rate prediction. Different from the standard country level mortality rate comparison, this study compares the three deep learning models and the classic Lee–Carter model on nine divisions’ yearly mortality data by gender from 1966 to 2015 in the United States. With the out-of-sample testing, we found that the Gated Recurrent Unit model showed better average MAE and RMSE values than the Lee–Carter model on 72.2% (13/18) and 67.7% (12/18) of the database, respectively, while the same measure for the Long Short-Term Memory model and Bidirectional Long Short-Term Memory model are 50%/38.9% (MAE/RMSE) and 61.1%/61.1% (MAE/RMSE), respectively. If we consider forecasting accuracy, computing expense, and interpretability, the Lee–Carter model with ARIMA exhibits the best overall performance, but the recurrent neural networks could also be good candidates for mortality forecasting for divisions in the United States.
... • Discrete vs. continuous. A discrete model in time usually describes a system whose time-evolution can be considered as a result of individual and separate events occurring in the system, i.e. molecular reaction-diffusion process (Erban and Chapman, 2009), population systems (Hyndman and Booth, 2008), epidemiology (Bansal et al., 2007). ...
Collision-exchange processes play a prominent role in a variety of natural systems where system members interact to engender the change of quantities, material transfer and information exchange over a population, driving a macroscopic evolution of system state in time. Mathematical modelling provides a useful approach to the quantitative characterisation of collision-exchange processes in order to assist the identification of change of system state. The established mathematical model that involves state variables and model parameters is expected to be identified based on the relevant experimental observations. Although collision-exchange processes have been extensively studied in many systems, especially in particulate systems, by formulating models based on discrete element methods, these models still suffer from several limitations, in particular the significant computational intensity required by simulations that restrict the further research into the models, leading to the difficulty using these models in model-based tasks, including design of experiments and optimisation. This project focuses on the investigation of a stochastic modelling approach for collision-exchange processes and the development of identification strategies for stochastic models. The work addresses the following challenges: i) the development of a stochastic model to simulate the collision-exchange process and predict the dynamical evolution of system state within tractable computational time; ii) the design and execution of experiments in an industrial seed coating process for the verification of the established stochastic model; iii) the development of parameter estimation and model-based design of experiments techniques suitable for stochastic models, i.e. the model outputs with uncertainty. The work presented in the Thesis facilitates an alternative modelling approach for collision-exchange processes, providing a systematic methodology for the identification and optimisation of stochastic systems with higher accuracy in prediction and less computational intensity.
... Bishai and Opuni (Bishai, Opuni 2009) highlighted the importance of selection of an appropriate method of data transformation for the research involving the comparison of various demographic indices. Similarly, H. Booth (Hyndman and Booth 2008), followed by H.L. Shang (Shang 2015), argued that a correctly chosen transformation is prerequisite for correct modelling and predicting various demographic parameters and occurrences. ...
Mortality crises are periods of unusually high mortality resulted from a combination of epidemic episodes, climatic phenomena, historical events and sociopolitical factors. The most pronounced setback in the methodology applied to analyse mortality rates of historical populations is the inability to establish their size. Reference publications do not provide unambiguous measures of the intensity and scale of mortality crisis periods. This problem was approached with the use of the Standardised Demographic Dynamics Rate (SDDR) whose value provides information about the condition of a population, disregarding the size of the group. Demographic crises were indicated and identified among the population living in the 19th century in central Poland in the rural parish. The analysis was based on data obtained from parish registers, made use of the measure expressing the ratio of the number of births to the number of deaths, without using the size of the group. Results obtained from the analysis of data were set against the information about events causing a sudden growth in mortality derived from the widely-accessible literature. Value of the Standardised Demographic Dynamics Rate (SDDR) provides information aboutthe condition of a population, disregarding the size of the group. Nevertheless, only by combining the statistically obtained data with the information derived from written records it is possible to attempt to answer the question of the possible root cause of a demographic crisis.
... If we turn out our attention to Europe, interesting is also the approach of [68], as they apply functional data models and time-series methods to forecast the components of change in mortality, fertility and net international migration and use them in forecasting the population of France [69]. Such a probabilistic population forecast is compared with the official population projections for the country, which are based on traditional deterministic scenarios. ...
The transition from a demographic regime of high mortality and high fertility to one with low mortality and low fertility is universal and comes along with the process of socio-economic modernization. The Spanish total fertility rate has decreased to below replacement levels in the last decades. The decline has persisted since the 1960s and is diverse across the country. Based on that diversity, the use of population forecasts, not only at national but at regional levels, for planning purposes (governments and private sector) with large horizons has become a must to provide essential services. Using a Bayesian hierarchical model we constructed probabilistic fertility forecasts for Spain at the regional level. Although this approach is already issued by the United Nations little research has been done focusing on the Spanish subnational level. Our objective is to disaggregate the national projections of the total fertility rate for Spain into regional forecasts. The results of this research will show the model fitting, first to the national level and then using a multifaceted and continuous evolution of fertility over time, at the regional level, to check its convergence.
... The best conditions for this to occur would be in the scenario where population data and the corresponding subnational units are relatively fine in spatial resolution and there are numerous time points upon which to fit the logistic growth/decay curves and the natural cubic splines. Ideally, the projected population numbers would be independently derived from demographic models Hyndman and Booth, 2008), utilising fertility and mortality as inputs as opposed to the simple, yet global in coverage, exponentially extrapolated population data used here . Many of these demographic models produce "scenarios" with "low" ...
Since 1950, the world’s population has shifted from being largely rural to majority urbanised. This trend of increasing urbanisation of population and increasing land use transitions promoting the growth of settlements and the built-environment, are expected to continue in future decades, particularly in low- and middle-income countries. These trends are accompanied by rapidly shifting subnational demographics and spatial distributions of populations, even within urbanised areas. Accurate and timely data is required to develop adaptive strategies for these shifting trends and minimising potential negative impacts. While multi-temporal, high-resolution datasets of built-settlement extent have become globally available, there remain gaps in their coverage and globally consistent methods of predicting future built-settlement expansion at regular intervals have not kept pace with these new data.
This thesis develops and validates a country-specific yet globally applicable means of annually interpolating built-settlement extents and projecting built- settlement extents into the near future using relative changes in subnational population and lights at night radiance. Additionally, I demonstrate the utility of this modelling framework within a global population modelling context across a period of 13 years. This thesis improves upon previous urban growth modelling approaches by demonstrating that relative changes in population can be sufficient, in and of themselves and as causal proxies for changes in economics, for accurately predicting areas undergoing built-settlement expansion across time and space. Additionally, this thesis validates its predictions at the pixel level, something not done by previous global urban and settlement modelling approaches. By addressing the limits that exist within current global urban modelling approaches, such as large or specific data requirements and subjective assumptions of growth factors/parameters, the modelling frameworks presented in this thesis allows for more consistent, frequent, and accurate built-settlement predictions. By extension, these accurate, time-specific built-settlement predictions allow for better, time-specific population mapping across the globe. Improved knowing of where and when built-settlement appeared allows for further investigations into arable land use consumption in relation to population dynamics, temporally fine-scale changes in population distributions across space in relation to climate change stresses, built-settlement expansion and greenhouse gas emissions, and trends in built-settlement expansion in relation to sea level rise, to name a few.
... Functional data analysis (FDA) encompasses a variety of techniques including forecasting [21,22,23], regression analysis [24,25,26], non-parametric modeling [27,28], clustering [29,30,31], smoothing [32,33,34], and data reduction [24,35,36,33]. In our particular application, FDA is used for data reduction, i.e., to diffuse the infinitely dimensional physiological functional data into features that can be used as input for machine learning models. ...
... • using more than one principal component for forecasting. Hyndman and Booth (2008) used functional data models with time series coefficients to model age-specific mortality and fertility rates. Hyndman et al. (2013) proposed a method for nondivergent or coherent forecasting of mortality rates for two or more subpopulations, based on functional principal components models of simple and interpretable functions of rates. ...
In recent years, the studies of demographic forecasts have grown significantly. One of the goals of demography is to statistically analyze and predict mortality and fertility rates without relying on subjective opinions of experts. Therefore, to identify the characteristics of the mortality dynamics of a population, many models were developed since the introduction of the famous model proposed by Lee and Carter (1992). Many research available in the literature tend to focus on the time series perspective of forecasting mortality rates. Lack of studies from the spatial framework sparked our interest in investigating the mortality rates from the spatial framework. The extension of the Lee-Carter (1992) model by incorporating the idea of functional data analysis (FDA) inspired the first part of this thesis where the FDA concept was applied to the spatial demographic analysis framework.We investigate the existence of spatial autocorrelation in mortality data of neighbouring countries. A functional spatial principal component method is proposed to reveal spatial patterns by directly considering spatial information. A functional Moran’s I statistic is introduced. This statistic aids in determining the spatial autocorrelation in functional data through the implementation of the spatio-functional PCA. This functional Moran’s I statistic is the first of its kind in the functional data framework.The second part of this thesis investigates the impact of the VigilanS system (program to prevent suicide reattempts in France) on suicide recidivism where the data from this system (patient’s age, sex, address, history of suicide attempts, hospital stay etc.) are mapped on the map of the Nord-Pas-de-Calais region while constructing spatial prediction models. The risks of suicide attempts are mapped with the help of spatial probit models. We propose a partially linear probit model for spatially dependent data. This model has not been investigated in the literature from a theoretical point of view and this part fills that gap by addressing a spatial autoregressive error (SAE) model where the spatial dependence structure is integrated in a disturbance term of the studied model. A semi-parametric estimation method is obtained by combining the generalized method of moments approach and the weighted likelihood method. We examined the use of this spatial probit regression model as well as other existing models in the literature to study the suicide relapses of patients involved in the VigilanS system. This thesis highlights the importance of spatial statistics in analyzing demographic and suicide problems. It is also interesting to see how functional data can be used as a tool in the field of demography especially in capturing spatial autocorrelation in mortality rates where space is of concern. Furthermore, the use of spatial regression models to study suicidal relapses sheds light on the impact of neighbouring locations on suicide cases.
Background
Estimates of the future prevalence of childhood cancer are informative for health system planning but are underutilized. We describe the development of a pediatric oncology microsimulation model for prevalence (POSIM-Prev) and illustrate its application to produce projections of incidence, survival, and limited-duration prevalence of childhood cancer in Ontario, Canada, until 2040.
Methods
POSIM-Prev is a population-based, open-cohort, discrete-time microsimulation model. The model population was updated annually from 1970 to 2040 to account for births, deaths, net migration, and incident cases of childhood cancer. Prevalent individuals were followed until death, emigration, or the last year of simulation. Median population-based outcomes with 95% credible intervals (CrIs) were generated using Monte Carlo simulation. The methodology to derive model inputs included generalized additive modeling of cancer incidence, parametric survival modeling, and stochastic population forecasting. Individual-level data from provincial cancer registries for years 1970 to 2019 informed cancer-related model inputs and internal validation.
Results
The number of children (aged 0–14 y) diagnosed with cancer in Ontario is projected to rise from 414 (95% CrI: 353–486) in 2020 to 561 (95% CrI: 481–653) in 2039. The 5-y overall survival rate for 2030–2034 is estimated to reach 90% (95% CrI: 88%–92%). By 2040, 24,088 (95% CrI: 22,764–25,648) individuals with a history of childhood cancer (diagnosed in Ontario or elsewhere) are projected to reside in the province. The model accurately reproduced historical trends in incidence, survival, and prevalence when validated.
Conclusions
The rising incidence and prevalence of childhood cancer will create increased demand for both acute cancer care and long-term follow-up services in Ontario. The POSIM-Prev model can be used to support long-range health system planning and future health technology assessments in jurisdictions that have access to similar model inputs.
Highlights
This article describes the development of a population-based, discrete-time microsimulation model that can simulate incident and prevalent cases of childhood cancer in Ontario, Canada, until 2040. Use of an open cohort framework allowed for estimation of the potential impact of net migration on childhood cancer prevalence. In addition to supporting long-term health system planning, this model can be used in future health technology assessments, by providing a demographic profile of incident and prevalent cases for model conceptualization and budget impact purposes. This modeling framework is adaptable to other jurisdictions and disease areas where individual-level data for incidence and survival are available.
This study introduces an innovative methodology for mortality forecasting, which integrates signature-based methods within the functional data framework of the Hyndman–Ullah (HU) model. This new approach, termed the Hyndman–Ullah with truncated signatures (HUts) model, aims to enhance the accuracy and robustness of mortality predictions. By utilizing signature regression, the HUts model is able to capture complex, nonlinear dependencies in mortality data which enhances forecasting accuracy across various demographic conditions. The model is applied to mortality data from 12 countries, comparing its forecasting performance against variants of the HU models across multiple forecast horizons. Our findings indicate that overall the HUts model not only provides more precise point forecasts but also shows robustness against data irregularities, such as those observed in countries with historical outliers. The integration of signature-based methods enables the HUts model to capture complex patterns in mortality data, making it a powerful tool for actuaries and demographers. Prediction intervals are also constructed with bootstrapping methods.
The subject of battery sources and electrical energy accumulators is currently very topical. Moreover, the possibilities of reusing already discarded sources are being explored as so-called “second-life batteries”. This article is concerned with studying and modelling the behaviour of a battery in an electric aircraft in operation — the voltage during discharge. Outcomes from extensive experiments on real long-term stored batteries have provided statistically robust sets of data on both long-term stored and new batteries; some of the data, however, are truncated. A modern approach that neglects the truncated issues and is based on functional data analysis and modified with a specific time series is used to model the process. This suggested model is much more accurate than the model used previously as it can effectively process truncated data. It also allows a certain degree of generalization. The aim is to determine the probability density of the time when the battery reaches the critical value, including the numerical statistics, for both stored and new batteries. The results are compared using the specific statistical Kullback–Leibler divergence approach to determine the degree of difference. The proposed model applies to similar issues where battery voltage is modelled in a time domain while the data form is truncated. It is proved, however, that further use of the stored batteries does not disrupt the safe and reliable operation of an electric airplane in terms of their functionality.
Purpose
Forecasting population movement trends is crucial for implementing effective policies to regulate labor force growth and understand demographic changes. Combining migration theory expertise and neural network technology can bring a fresh perspective to international migration forecasting research.
Design/methodology/approach
This study proposes a conditional generative adversarial neural network model incorporating the migration knowledge – conditional generative adversarial network (MK-CGAN). By using the migration knowledge to design the parameters, MK-CGAN can effectively address the limited data problem, thereby enhancing the accuracy of migration forecasts.
Findings
The model was tested by forecasting migration flows between different countries and had good generalizability and validity. The results are robust as the proposed solutions can achieve lesser mean absolute error, mean squared error, root mean square error, mean absolute percentage error and R ² values, reaching 0.9855 compared to long short-term memory (LSTM), gated recurrent unit, generative adversarial network (GAN) and the traditional gravity model.
Originality/value
This study is significant because it demonstrates a highly effective technique for predicting international migration using conditional GANs. By incorporating migration knowledge into our models, we can achieve prediction accuracy, gaining valuable insights into the differences between various model characteristics. We used SHapley Additive exPlanations to enhance our understanding of these differences and provide clear and concise explanations for our model predictions. The results demonstrated the theoretical significance and practical value of the MK-CGAN model in predicting international migration.
As a fundamental, holistic, and strategic issue facing human society, human migration is a key factor affecting the development of countries and cities, given the constantly changing population numbers. The fuzziness of the spatiotemporal attributes of human migration limits the pool of open-source data for human migration prediction, leading to a relative lag in human migration prediction algorithm research. This study expands the definition of human migration research, reviews the progress of research into human migration prediction, and classifies and compares human migration algorithms based on open-source data. It also explores the critical uncertainty factors restricting the development of human migration prediction. Based on the analysis, there is no “best” migration prediction model, and data are key to forecasting human migration. Social media’s popularity and its increase in data have enabled the application of artificial intelligence in population migration prediction, which has attracted the attention of researchers and government administrators. Future research will aim to incorporate uncertainty into the predictive analysis framework, and explore the characteristics of population migration behaviors and their interactions. The integration of machine-learning and traditional data-driven models will provide a breakthrough for this purpose.
En los últimos años se ha propuesto una importante cantidad de métodos estadísticos demográficos. La gran mayoría han sido desarrollados con la finalidad de pronosticar las componentes demográficas y/o medidas derivadas a partir de la suposición de un modelo subyacente. El presente trabajo pretende realizar un ejercicio comparativo integral a través de la estimación y pronóstico de la fecundidad a partir de tres propuestas —métodos clásicos de pronóstico, tales como los modelos ARIMA y los suavizados exponenciales, modelos para datos funcionales (MDF) y modelos jerárquicos bayesianos (MJB)—, como un primer paso hacia el estudio de las proyecciones de población derivadas de cada una de ellas, empleando datos de la Argentina. El ejercicio tiene como horizonte final la estimación de la mortalidad y la fecundidad a través de los tres métodos mencionados para luego integrarlos en proyecciones de población.
Detecting changepoints in functional data has become an important problem as interest in monitoring of climate phenomenon has increased, where the data is functional in nature. The observed data often contains both amplitude (‐axis) and phase (‐axis) variability. If not accounted for properly, true changepoints may be undetected, and the estimated underlying mean change functions will be incorrect. In this article, an elastic functional changepoint method is developed which properly accounts for these types of variability. The method can detect amplitude and phase changepoints which current methods in the literature do not, as they focus solely on the amplitude changepoint. This method can easily be implemented using the functions directly or can be computed via functional principal component analysis to ease the computational burden. We apply the method and its nonelastic competitors to both simulated data and observed data to show its efficiency in handling data with phase variation with both amplitude and phase changepoints. We use the method to evaluate potential changes in stratospheric temperature due to the eruption of Mt. Pinatubo in the Philippines in June 1991. Using an epidemic changepoint model, we find evidence of a increase in stratospheric temperature during a period that contains the immediate aftermath of Mt. Pinatubo, with most detected changepoints occurring in the tropics as expected.
En los últimos años se ha propuesto una importante cantidad de métodos estadísticos demográficos. La gran mayoría han sido desarrollados con la finalidad de pronosticar las componentes demográficas y/o medidas derivadas a partir de la suposición de un modelo subyacente. El presente trabajo pretende realizar un ejercicio comparativo integral a través de la estimación y pronóstico de la fecundidad a partir de tres propuestas —métodos clásicos de pronóstico,tales como los modelos ARIMAy los suavizados exponenciales, modelos para datos funcionales(MDF) y modelos jerárquicos bayesianos (MJB)—, como un primer paso hacia el estudio de las proyecciones de población derivadas de cada una de ellas,empleando datos de la
255Argentina. El ejercicio tiene como horizonte final la estimación de la mortalidad y la fecundidad a través de los tres métodos mencionados para luego integrarlos en proyecciones de población.
In water resource management daily flow to the reservoir is most important factor, in this study the daily flow to the reservoir as well as upstream has been forecast by
different models. Autoregressive Integrated Moving Average (ARIMA) and
Autoregressive Moving Average (ARMA) models has been used to forecast daily flow of different dams and linked canals in Pakistan. Application in practical science ARIMA model plays an important role. Both ARIMA and ARMA model has been used to compare the capability of autoregressive forecast of daily dam reservoir inflow. Forecast accuracy of Tarbela Dam reservoir inflow has been increased if we increase the number of parameter in ARMA and ARIMA models. To forecast Dam reservoir inflow polynomial for ARMA and ARIMA models was derived up to four and six parameter. Root mean square error concludes that ARIMA model can be used to forecast the level of water for different rivers in Pakistan with less error than ARMA model.
Stochastic volatility, or variability that is well approximated as a random process, is widespread in modern finance. While understanding that volatility is essential for sound decision making, the structural and data constraints associated with complex financial instruments limit the applicability of classical volatility modeling. This article investigates stochastic volatility in functional time series with the goal of accurately modeling option surfaces. We begin by introducing a functional analogue of the familiar stochastic volatility models employed in univariate and multivariate time series analysis. We then describe how that functional specification can be reduced to a finite dimensional vector time series model and discuss a strategy for Bayesian inference. Finally, we present a detailed application of the functional stochastic volatility model to daily SPX option surfaces. We find that the functional stochastic volatility model, by accounting for the heteroscedasticity endemic to option surface data, leads to improved quantile estimates. More specifically, we demonstrate through backtesting that Value-at-Risk estimates from the proposed functional stochastic volatility model exhibit correct coverage more consistently than those of a constant volatility model.
With the development of the economy, the global fertility rate has generally decreased, and even led to negative population growth in some developed countries. In recent years, China, where family planning is implemented, has also gradually shown such a trend. The decline of fertility rate limits the further development of the society. In this paper, based on the SIR infectious disease model, we improve the SIR fertility structure model applicable to the prediction of female fertility structure based on the state transformation process existing in the female fertility stage, and forecast the female fertility structure, female population, and total population of China from 2018–2027 with the data related to the female population in 2017. Compared with the prediction results of the three classical models of Grey model (1, 1), BP Neural Network model and Logistic model, the SIR fertility structure and population prediction model in this paper can predict the female fertility structure in the short term in the future and deduce the population size accordingly, which has better prediction accuracy and the total population accuracy rate can reach 99.945%.The model can provide reference and basis for population prediction under the condition of low fertility level.
We propose an alternative to k-nearest neighbors for functional data whereby the approximating neighboring curves are piecewise functions built from a functional sample. Using a locally defined distance function that satisfies stabilization criteria, we establish pointwise and global approximation results in function spaces when the number of data curves is large. We exploit this feature to develop the asymptotic theory when a finite number of curves is observed at time-points given by an i.i.d. sample whose cardinality increases up to infinity. We use these results to investigate the problem of estimating unobserved segments of a partially observed functional data sample as well as to study the problem of functional classification and outlier detection. For such problems our methods are competitive with and sometimes superior to benchmark predictions in the field. The R package localFDA provides routines for computing the localization processes and the estimators proposed in this article.
There is a growing concern about the implications of environmental activities on the profitability of upstream companies in Nigeria. In view of the individual importance of the environment and the continued survival of business organizations, attention is focused on the interplay of sustainability of the environment and profitability of corporate organizations. The study specifically assessed the effect of environmental conservation cost on return on assets and net profit after tax of upstream companies in Nigeria. The study adopted the ex-post factor research design and data were sourced from the companies’ annual audited financial reports for the period 2009 to 2018. The multiple regression technique was applied in estimating the study’s parameters. Findings showed that environmental conservation cost has positive and insignificant effect on return on assets and profit after tax of upstream companies in Nigeria. The study therefore concluded that green cost accounting has positive and insignificant effect on financial performance of selected upstream companies in Nigeria. Based on the findings, it is therefore recommended that Nigerian upstream Companies should ensure proper management of environmental conservation costto enhance their profitability
The chain-ladder model is the most widely used technique for property and casualty insurance to estimate unpaid claims, including incurred but not reported (IBNR) claims. Inspired by the reserving method, we first apply a distribution-free method (the chain-ladder model) and its variant and a distributional method (the lognormal model) to project future mortality rates. Next, to simulate mortality rates for more applications, we also propose corresponding stochastic versions associated with both the lognormal model and the variant of the chain-ladder model. Finally, we demonstrate numerical illustrations with mortality data from the Human Mortality Database for both genders of the US, the UK, and Japan. To compare the forecasting performances of the proposed three models and the other five models (the Lee-Carter model, the Renshaw-Haberman model, the Cairns-Blake-Dowd model, the M6 and M7 models), we adopt mean absolute error (MAE), root mean square error (RMSE), and mean absolute percentage error (MAPE) as metrics. Numerical illustrations show that the proposed three models overall outperforms the other five models.
The existing literature on ex-post errors observed for historical population forecasts made by statistical agencies in 16 industrialized countries is reviewed. The amount and type of data in these studies (total population size, age structure, population growth rate, components of change), the level of detail (numbers of births and deaths, crude rates, total fertility rates, life expectancies), and the period covered vary considerably among the countries. Attention is given to forecast accuracy for total population size, population growth rates, fertility, mortality, the age structure, and the dependency ratio. Among the issues covered are relative errors for fertility and mortality, and common patterns across the countries. On the basis of a data set from Norway (covering the period 1969-1989, forecasts made between 1969 and 1987) and one from the Netherlands (for the period 1950-1986, forecasts made between 1950 and 1980) we investigate a possible correlation between forecast errors for fertility and mortality, and a possible reduction in forecast errors over time. The article concludes with suggestions for including results from ex-post evaluations in official population forecasts.
The Lee-Carter (LC) method of mortality forecasting is well known and widely used. Two recent variants are the Lee-Miller (LM) variant and the Booth-Maindonald-Smith (BMS) variant. Both aim to improve the performance of the method. These two variants and the original Lee-Carter method are evaluated using data for twenty populations for 1900-2001, with the fitting period ending in 1985 and the forecast period beginning in 1986. Forecast errors are compared and decomposed, and uncertainty is examined. For these short-term forecasts, the two variants are generally more accurate than the LC method with narrower prediction intervals; and BMS marginally outperforms LM on these criteria overall. Further evaluation using different fitting periods is required.
Fertility forecasting is the weak point of stochastic population forecasts. Changing trends account for large forecasting errors even a few years ahead. On the other hand, fertility trends have been shown to be common to different European countries. This paper explores the possibility of improving forecasts by jointly modelling total fertility rate (TFR) trends within relatively homogeneous clusters of countries. We propose different varieties of non-stationary dynamic factor models applied to Southern European countries. The forecasting performance of the common factor models is compared to alternative univariate and multivariate forecasting methods using data for the period 1950–2000. Joint forecasts show forecasting gains in terms of root mean square error of prediction (RMSE), particularly for longer forecast horizons. This corroborates the convenience of modelling fertility jointly for population forecasting.
This paper implements Wilmoth’s [Computational methods for fitting and extrapolating the Lee–Carter model of mortality change, Technical report, Department of Demography, University of California, Berkeley] and Alho’s [North American Actuarial Journal 4 (2000) 91] recommendation for improving the Lee–Carter approach to the forecasting of demographic components. Specifically, the original method is embedded in a Poisson regression model, which is perfectly suited for age–sex-specific mortality rates. This model is fitted for each sex to a set of age-specific Belgian death rates. A time-varying index of mortality is forecasted in an ARIMA framework. These forecasts are used to generate projected age-specific mortality rates, life expectancies and life annuities net single premiums. Finally, a Brass-type relational model is proposed to adapt the projections to the annuitants population, allowing for estimating the cost of adverse selection in the Belgian whole life annuity market.
We compare the short- to medium-term accuracy of five variants or extensions of the
Lee-Carter method for mortality forecasting. These include the original Lee-Carter, the
Lee-Miller and Booth-Maindonald-Smith variants, and the more flexible Hyndman-Ullah
and De Jong-Tickle extensions. These methods are compared by applying them to sexspecific
populations of 10 developed countries using data for 1986–2000 for evaluation.
All variants and extensions are more accurate than the original Lee-Carter method for
forecasting log death rates, by up to 61%. However, accuracy in log death rates does not
necessarily translate into accuracy in life expectancy. There are no significant differences
among the five methods in forecast accuracy for life expectancy.
This report is part of a state-commissioned project undertaken after California enacted Senate Bill 910 (Vasconcellos, Statutes of 1999, Chapter 948), mandating the Secretary of Health and Human Services to develop a plan to address the impending demographic, economic, and social changes triggered by the state s aging and increasingly diverse population. The University of California was asked to conduct the data analyses and provide background information needed to formulate this plan. CPRC has been coordinating this effort over the past three years, drawing on research experts from UC and other institutions. A faculty working group, chaired by Professor Andrew Scharlach (School of Social Welfare, Berkeley), helped guide the project. The authors wish to acknowledge Andrew Scharlach s substantial contributions to the substance and structure of this report, as well as his many valuable editorial suggestions at all stages of the writing. The authors of this report were charged with developing a composite demographic profile of Californians that would provide (1) a snapshot of aging demographics; (2) a summary of key variables, projections, and their degree of certainty; and (3) an estimation of service needs of elderly Californians in the context of the state budget and changing demographics. A forthcoming report by other authors (Planning for a Comprehensive Database on Aging Californians) assesses data the state already collects and where the gaps are, what questions the state needs to ask to project needs for policy planning purposes, and how the questions can be answered by enhancing and linking data that currently exist and selectively collecting new data.
We analyse empirical errors observed in historical population forecasts produced by statistical agencies in 14 European countries since 1950. The focus is on forecasts for three demographic variables: fertility (Total Fertility Rate - TFR), mortality (life expectancy at birth), and migration (net migration). We inspect forecast bias and forecast accuracy in the historical forecasts, as well as the distribution of the errors. Finally, we analyse for each of the three variables correlation patterns in forecast errors across countries and, for mortality, the correlation between errors for men and women. In the second part of the report we use time series model to construct prediction intervals to 2050 for the TFR, the life expectancy for men and women, and net migration in 18 European countries. GARCH models are used for fertility and mortality, while net migration is modelled as an autoregressive process
The first concern of this work is the development of approximations to the distributions of crude mortality rates, age-specific mortality rates, age-standardized rates, standardized mortality ratios, and the like for the case of a closed population or period study. It is found that assuming Poisson birthtimes and independent lifetimes implies that the number of deaths and the corresponding midyear population have a bivariate Poisson distribution. The Lexis diagram is seen to make direct use of the result. It is suggested that in a variety of cases, it will be satisfactory to approximate the distribution of the number of deaths given the population size, by a Poisson with mean proportional to the population size. It is further suggested that situations in which explanatory variables are present may be modelled via a doubly stochastic Poisson distribution for the number of deaths, with mean proportional to the population size and an exponential function of a linear combination of the explanatories. Such a model is fit to mortality data for Canadian females classified by age and year. A dynamic variant of the model is further fit to the time series of total female deaths alone by year. The models with extra-Poisson variation are found to lead to substantially improved fits.
Human lifespan has increased enormously this century. But we remain uncertain about the forces that reduce mortality, and about the cost implications of ageing populations and their associated social burden. The poor understanding of the factors driving mortality decline, and the difficulty of forecasting mortality are due in part to the pronounced irregularity of annual to decadal mortality change. Here we examine mortality over five decades in the G7 countries (Canada, France, Germany, Italy, Japan, UK, US). In every country over this period, mortality at each age has declined exponentially at a roughly constant rate. This trend places a constraint on any theory of society-driven mortality decline, and provides a basis for stochastic mortality forecasting. We find that median forecasts of life expectancy are substantially larger than in existing official forecasts. In terms of the costs of ageing, we forecast values of the dependency ratio (that is, the ratio of people over 65 to working people) in 2050 that are between 6% (UK) and 40% (Japan) higher than official forecasts.
Traditional population forecasts made by statistical agencies do not quantify uncertainty. But demographers and statisticians have developed methods to calculate probabilistic forecasts.
Recent patterns of fertility in Europe show marked differences between countries. Recent United Kingdom and Irish fertility curves show 'distortions' in terms of a 'bulge' in early age fertility, distinct from the smoother curves of other European countries. These patterns may not be adequately described by mathematical functions used by previous studies to model fertility curves. A mixture model with two component distributions may be more appropriate. The suitability of the simple and mixture Hadwiger functions is examined in relation to the fertility curves of a number of European countries. While the simple Hadwiger model fits recent period age-specific fertility distributions for some countries, others which display a 'bulge' in early age fertility distributions for some countries, others which display a 'bulg' in early age fertility require a mixture Hadwiger model. Some of the parameters of the Hadwiger models appear to be related to familiar demographic indices. The simple and mixture Hadwiger models appear useful in describing and comparing fertility patterns across European countries.
"This article presents and implements a new method for making stochastic population forecasts that provide consistent probability intervals. We blend mathematical demography and statistical time series methods to estimate stochastic models of fertility and mortality based on U.S. data back to 1900 and then use the theory of random-matrix products to forecast various demographic measures and their associated probability intervals to the year 2065. Our expected total population sizes agree quite closely with the Census medium projections, and our 95 percent probability intervals are close to the Census high and low scenarios. But Census intervals in 2065 for ages 65+ are nearly three times as broad as ours, and for 85+ are nearly twice as broad. In contrast, our intervals for the total dependency and youth dependency ratios are more than twice as broad as theirs, and our ratio for the elderly dependency ratio is 12 times as great as theirs. These items have major implications for policy, and these contrasting indications of uncertainty clearly show the limitations of the conventional scenario-based methods."
PIP
1 solution to the dimensionality problem raised by projection of individual age-specific fertility rates is the use of parametric curves to approximate the annual age-specific rates and a multivariate time series model to forecast the curve parameters. Such a method reduces the number of time series to be modeled for women 14-45 years of age from 32 to 40 (the number of curve parameters). In addition, the curves force even longterm fertility projections to exhibit the same smooth distribution across age as historical data. The data base used to illustrate this approach was age-specific fertility rates for US white women in 1921-84. An important advantage of this model is that it permits investigation of the interactions among the total fertility rate, the mean age of childbearing, and the standard deviation of age at childbearing. In the analysis of this particular data base, the contemporaneous relationship between the mean and standard deviation of age at childbearing was the only significant relationship. The addition of bias forecasts to the forecast gamma curve improves forecast accuracy, especially 1-2 years ahead. The most recent US Census Bureau projections have combined a time series model with longterm projections based on demographic judgment. These official projections yielded a slightly higher ultimate mean age and slightly lower standard deviation than those resulting from the model described in this paper.
A multivariate ARIMA model is combined with a Gamma curve to predict confidence intervals for age-specific birth rates by 1-year age groups. The method is applied to observed age-specific births in Norway between 1900 and 1995, and predictive intervals are computed for each year up to 2050. The predicted two-thirds confidence intervals for Total Fertility (TF) around 2010 agree well with TF errors in old population forecasts made by Statistics Norway. The method gives useful predictions for age-specific fertility up to the years 2020-30. For later years, the intervals become too wide. Methods that do not take into account estimation errors in the ARIMA model coefficients underestimate the uncertainty for future TF values. The findings suggest that the margin between high and low fertility variants in official population forecasts for many Western countries are too narrow.
In the analysis of data it is often assumed that observations y1, y2, …, yn are independently normally distributed with constant variance and with expectations specified by a model linear in a set of parameters θ. In this paper we make the less restrictive assumption that such a normal, homoscedastic, linear model is appropriate after some suitable transformation has been applied to the y's. Inferences about the transformation and about the parameters of the linear model are made by computing the likelihood function and the relevant posterior distribution. The contributions of normality, homoscedasticity and additivity to the transformation are separated. The relation of the present methods to earlier procedures for finding transformations is discussed. The methods are illustrated with examples.
Lee and Carter (LC) published a new statistical method for forecasting mortality in 1992. This paper examines its actual and hypothetical forecast errors, and compares them with Social Security forecast errors. Hypothetical historical projections suggest that LC tended to underproject gains, but by less than did Social Security. True eo was within the ex ante 95% probability interval 97% of the time overall, but intervals were too broad up to 40 years and too narrow after 50 years. Projections to 1998 made after 1945 always contain errors of less than two years. Hypothetical projections for France, Sweden, Japan, and Canada would have done well. Changing age patterns of mortality decline over the century pose problems for the method.
The paper discusses the problem of modeling demographic variables for the purpose of forecasting. It is argued that theory rarely provides a complete dynamic specification for the model. Two empirical model selection procedures, a time series approach (Haugh and Box 1977 and Granger and Newbold 1977) and a sequential testing procedure (Hendry and Mizon 1978), are applied to suggest final-form forecasting equations for an Australian births series, first nuptial confinements. The suggested models are then assessed by comparing their post-sample forecast performance with that of univariate ARMA type models of confinements, which are regarded as approximations to the confinements final equation model. This modeling strategy is contrasted with the method used to construct the Australian government's IMPACT demographic module.
The development of a ‘law of mortality’, a mathematical expression for the graduation of the age pattern of mortality, has been of interest since the development of the first life tables by John Graunt (1662) and Edmund Halley (1693). Although Abraham De Moivre proposed a very simple law as early as 1725 the best known early contribution is probably that of Benjamin Gompertz (1825). Since World War II mathematical formulae have been used to graduate sections of the English Life Tables, as well as assured lives mortality, and pensioner and annuitant mortality. Reviews of attempts at finding the ‘law of mortality’ have been given by Elston and Benjamin and Haycocks.
In an effort to provide probabilistic measures of the accuracy of population projections, stochastic models for population growth are defined from the classical discrete deterministic model by assuming respectively that (1) the deterministic model is subject to additive random errors; (2) the elements of the transition matrix represent probabilities, rather than rates; and (3) the transition matrices are random variables. The mean of each process is shown to reproduce the deterministic process, while the variance can be expressed as the weighted sum of one-step conditional variances. For the second model, these “innovation variances” will be small for large populations, while for the first and third models their size will depend on the observed variability of, respectively, prediction errors and vital rates. Since it is known empirically that both the latter are quite variable, these models could be expected to yield relatively high prediction variances, and this expectation is confirmed by a numerical example.The first model seems unsatisfactory as demographic theory, while the second does not account for the observed imprecision of population projections. The third model does, however, seem to provide a satisfactory method of estimating prediction variances of population projections.
Time series methods are used to make long-run forecasts, with confidence intervals, of age-specific mortality in the United States from 1990 to 2065. First, the logs of the age-specific death rates are modeled as a linear function of an unobserved period-specific intensity index, with parameters depending on age. This model is fit to the matrix of U.S. death rates, 1933 to 1987, using the singular value decomposition (SVD) method; it accounts for almost all the variance over time in age-specific death rates as a group. Whereas e0 has risen at a decreasing rate over the century and has decreasing variability, k(t) declines at a roughly constant rate and has roughly constant variability, facilitating forecasting. k(t), which indexes the intensity of mortality, is next modeled as a time series (specifically, a random walk with drift) and forecast. The method performs very well on within-sample forecasts, and the forecasts are insensitive to reductions in the length of the base period from 90 to 30 years; some instability appears for base periods of 10 or 20 years, however. Forecasts of age-specific rates are derived from the forecasts of k, and other life table variables are derived and presented. These imply an increase of 10.5 years in life expectancy to 86.05 in 2065 (sexes combined), with a confidence band of plus 3.9 or minus 5.6 years, including uncertainty concerning the estimated trend. Whereas 46% now survive to age 80, by 2065 46% will survive to age 90. Of the gains forecast for person-years lived over the life cycle from now until 2065, 74% will occur at age 65 and over. These life expectancy forecasts are substantially lower than direct time series forecasts of e0, and have far narrower confidence bands; however, they are substantially higher than the forecasts of the Social Security Administration's Office of the Actuary.
The literature on population forecasting reveals a continuing debate over the relative ex post accuracy of forecasts from simple and complex models. In this paper ex ante projections from simple models are used to evaluate the plausibility of point and interval forecasts from a complex cohort‐component model.This paper compares complex with simple forecasts along two dimensions: simplification of input schedules and of the projection models using those input schedules. Projections of vital rates that are the inputs for a cohort‐component projection may be provided by extrapolations of historical trends in age‐specific rates, relational methods that link future age‐specific rates to a general trend variable and a “standard”; schedule, or time series forecasts of the parameters of a functionally specified model schedule. This paper compares point and interval forecasts of fertility patterns that result from such alternative input specifications.Furthermore, the cohort‐component framework may be replaced by an aggregated model for population forecasting, for example, by a time series model of total population growth. This paper uses the point and interval forecasts from such an aggregate time series model to establish the plausibility of the complex cohort‐component projection.
The calculation of net immigration for the purpose of estimating the resident population in Australia is based on net permanent and long–term (12 months or more) movements into and out of the country. All international movements with duration of stay in Australia or travel abroad of less than 12 months (defined as short–term) are excluded.
However, changes between short–term and long–term/permanent status can occur when people extend or shorten their stay or travel. Because net immigration is a significant component of Australia’s population growth (accounting for 40–50 per cent of annual growth), adjusting for these changes in migration status is thought to result in better estimates of net immigration and the resident population.
The paper shows that adjusting for change of status can have a large impact on net immigration, particularly when the immigrant intake is small. Failure to adequately adjust for change of status can also lead to misleading conclusions about the relative contributions of net temporary and permanent movements to total net immigration. The effect on the resident population, however, is relatively small, being less than 1 per cent of the total population. The paper also addresses the question of how important it is for countries to adjust for change of migrant status in international migration statistics in the context of increasing international mobility.
This study employs parameterized model schedules of the age profiles of fertility to summarize age-specific rates with a small number of parameters. Time series models of the parameters capture the temporal patterns of the age profiles, and use this information to project the profiles of future vital rates. The model schedules are estimated by application of nonlinear least squares to fertility rates for women aged 14 through 49, for each calendar year from 1917 through 1988. Time series models of the parameters provide a representation of the historical trends of fertility behavior and are employed to forecast fertility profiles. Forecast evaluation over a 10-year holdout sample demonstrates the viability of the methodology. The time series models are updated through 1988 to generate fertility forecasts to the year 2000. The forecasts, showing an increase in the mean age of childbearing and a 9% rise in the total fertility rate, are compared with projections from the U.S. Bureau of the Census.
The Lee-Carter method for mortality forecasting is outlined, discussed and improved utilizing standard time series approaches. The new framework, which integrates estimation and forecasting, delivers more robust results and permits more detailed insight into underlying mortality dynamics. An application to women's mortality data illustrates the methods.
Observed schedules of migration rates exhibit strong regularities in age patterns. These regularities may be captured and represented by a mathematical expression known as the multiexponential model migration schedule. Fitting this function to empirical data requires non-linear regression methods and often some experimentation with alternative initial estimates of the parameters. Simpler, linear methods of estimation are adequate for most applications. These may be carried out with hand calculators or simple spreadsheet-based calculations on the computer. Such methods are studied and appear to perform satisfactorily.
To develop high-low bounds for population projections, traditional demographic forecasts assume fertility is always high or always low. Such bounds are related to bounds on average annual fertility up to year t rather than to individual year bounds in time series forecasts, and so the autocorrelation of time series forecast errors is important in practical applications. This paper develops methods for using time series methods to make constrained long term forecasts of fertility. Specifically, age-time variations in fertility are modeled with a single time-varying parameter, or fertility index; upper and lower bounds on the total fertility rate are imposed by forecasting an inverse logistic transform of the fertility index; the long run level of the fertility forecast is also constrained to equal a prespecified level. The principal interest is in the variance and the autocorrelation structure of the forecast errors. Based on these for the USA we conclude: (1) the probability interval for average fertility up to time t begins to contract after about 50 years, but only very slightly: (2) the probability interval for average fertility up to year 2065 is about three-fifths as wide as that for single year fertility in 2065, but is still far wider than the band for official forecasts; (3) realizations of the simple ARMA (1,0,1) forecast model exhibit long fluctuations something like actual fertility in industrial nations; (4) the model of fertility age patterns fits poorly at older ages, but may be adequate for present purposes.
A new method is proposed for forecasting age-specific mortality and fertility rates observed over time. This approach allows for smooth functions of age, is robust for outlying years due to wars and epidemics, and provides a modelling framework that is easily adapted to allow for constraints and other information. Ideas from functional data analysis, nonparametric smoothing and robust statistics are combined to form a methodology that is widely applicable to any functional time series data observed discretely and possibly with error. The model is a generalization of the Lee–Carter (LC) model commonly used in mortality and fertility forecasting. The methodology is applied to French mortality data and Australian fertility data, and the forecasts obtained are shown to be superior to those from the LC method and several of its variants.
We provide a new approach to automatic forecasting based on an extended range of exponential smoothing methods. Each method in our taxonomy of exponential smoothing methods provides forecasts that are equivalent to forecasts from a state space model. This equivalence allows: (1) easy calculation of the likelihood, the AIC and other model selection criteria; (2) computation of prediction intervals for each method; and (3) random simulation from the underlying state space model. We demonstrate the methods by applying them to the data from the M-competition and the M3-competition. The method provides forecast accuracy comparable to the best methods in the competitions; it is particularly good for short forecast horizons with seasonal data.
Approaches and developments in demographic and population forecasting since 1980 are reviewed. Three approaches to forecasting demographic processes are extrapolation, expectation (individual-level birth expectations or population-level opinions of experts), and theory-based structural modelling involving exogenous variables. Models include 0–3 factors (age, period and cohort). Decomposition and disaggregation are also used in multistate models, including macrosimulation and microsimulation. Forecasting demographic change is difficult; accuracy depends on the particular situation or trends, but it is not clear when a method will perform best. Estimates of uncertainty (model-based ex ante error, expert-opinion-based ex ante error, and ex post error) differ; uncertainty estimation is highly uncertain. Probabilistic population forecasts are based on stochastic population renewal or random scenarios. The approaches to population forecasting, demographic process forecasting and error estimation are closely linked. Complementary methods that combine approaches are increasingly employed. The paper summarises developments, assesses progress and considers the future.
The propagation of error in stochastic cohort-component forecasts of population is discussed. The uncertainty of the forecasts is due to uncertain estimates of the jump-off population, and to errors in the forecasts of the vital rates: fertility, mortality, and migration. Empirically based (ex post) estimates of each source are presented and propagated first through a simplified analytical model of population growth. Being analytic, the model readily permits the assessment of the role of each component in the total error. Then, we consider numerical estimates based on the (ex ante) errors of an actual vector ARIMA forecast of the vital rates and propagate them through a forecast of the US female population. The results agree in broad outline with those of the analytical model. In particular, the uncertainty in the forecasts of fertility is shown to be so much higher than that in the other sources that the latter can be ignored in the propagation of error calculations for those cohorts that are born after the jump-off year of the forecast. This simplifies the propagation of error calculations considerably. However, both the uncertainty of the jump-off population, migration, and mortality needs to be considered in the propagation of error for those alive at the jump-off time of the forecast.
The highly acclaimed The Future Population of the World contains the most authoritative assessment available of the extent to which population is likely to grow over the next 50 to 100 years. The book provides a thorough analysis of all the components of population change and translates these factors into a series of projections for the population of the world's regions. This revised and updated version incorporates completely new scenario projections based on updating starting values and revised assumptions, plus several methodological improvements. It also contains the best currently available information on global trends in AIDS mortality and the first ever fully probabilistic world population projections. The projections, given up to 2100, add important additional features to those of the UN and the World Bank: they show the impacts of alternative assumptions for all three components (mortality and migration, as well as fertility); they explicitly take into account possible environmental limits to growth; and, for the first time, they define confidence levels for global populations. Combining methodological innovation with overviews of the most recent data and literature, this updated edition of The Future Population of the World is sure to conform its reputation as the most comprehensive and essential publication in the field. © International Institute for Applied Systems Analysis, 1996 All rights reserved.
The relationship between classical demographic deterministic forecasting models, stochastic structural econometric models and time series models is discussed. Final equation autoregressive moving average (ARMA) models for Australian total live-births are constructed. Particular attention is given to the problem of transforming the time series to stationarity (and Gaussianity) and the properties of the forecasts are analyzed. Final form transfer function models linking births to females in the reproductive age groups are also constructed and a comparison of actual forecast performance using the various models is made. Long-run future forecasts are generated and compared with available projections based on the deterministic cohort model after which some policy implications of the analysis are considered.
This article links parameterized model mortality schedules with time series methods to develop forecasts of U.S. mortality to the year 2000. The use of model mortality schedules permits a relatively concise representation of the history of mortality by age and sex from 1900 to 1985, and the use of modern time series methods to extend this history forward to the end of this century allows for a flexible modeling of trend and the accommodation of changes in long-run mortality patterns. This pilot study demonstrates that the proposed procedure produces medium-range forecasts of mortality that meet the standard tests of accuracy in forecast evaluation and that are sensible when evaluated against the comparable forecasts produced by the Social Security Administration.
Lee and Carter (LC) published a new statistical method for forecasting mortality in 1992. This paper examines its actual and hypothetical forecast errors, and compares them with Social Security forecast errors. Hypothetical historical projections suggest that LC tended to underproject gains, but by less than did Social Security. True e0 was within the ex ante 95% probability interval 97% of the time overall, but intervals were too broad up to 40 years and too narrow after 50 years. Projections to 1998 made after 1945 always contain errors of less than two years. Hypothetical projections for France, Sweden, Japan, and Canada would have done well. Changing age patterns of mortality decline over the century pose problems for the method.
Is human life expectancy approaching its limit? Many--including individuals planning their retirement and officials responsible for health and social policy--believe it is, but the evidence presented in the [Policy Forum][1] suggests otherwise. For 160 years, best-performance life expectancy has steadily increased by a quarter of a year per year, an extraordinary constancy of human achievement. Mortality experts have repeatedly asserted that life expectancy is close to an ultimate ceiling; these experts have repeatedly been proven wrong. The apparent leveling off of life expectancy in various countries is an artifact of laggards catching up and leaders falling behind.
[1]: http://www.sciencemag.org/cgi/content/full/296/5570/1029
"This paper describes a particular approach to stochastic population forecasting, which is implemented for the U.S.A. through 2065. Statistical time series methods are combined with demographic models to produce plausible long run forecasts of vital rates, with probability distributions. The resulting mortality forecasts imply gains in future life expectancy that are roughly twice as large as those forecast by the Office of the Social Security Actuary.... Resulting stochastic forecasts of the elderly population, elderly dependency ratios, and payroll tax rates for health, education and pensions are presented."
"This paper considers parametric graduation for mortality, fertility and migration with particular reference to the development of parameterized local and regional demographic projections. Parametric graduations facilitate comparisons of demographic schedules across many areas and across time points--a feature which can be used to advantage in making forecasts of the three demographic components and thus in setting the assumptions for projections. Particular methodological issues raised are the questions of parsimony in fit and...of overdispersion in relation to binomial or Poisson assumptions. The analysis is illustrated with cross-sectional material for the 32 London boroughs and with time series at the level of Greater London."