Publications-Theses

題名 動態系統與生育率及死亡率的估計
Using dynamic system to model fertility and mortality rates
作者 李玢
貢獻者 余清祥
李玢
關鍵詞 微分方程
動態系統
生育率
死亡率
數值分析
Differential equation
Dynamic system
Fertility model
Mortality model
Numerical analysis
日期 2008
上傳時間 18-Sep-2009 20:11:24 (UTC+8)
摘要 人口統計學家在傳統上習慣將人口的種種變化視為時間的函數,皆試圖以決定型(deterministic)的函數來刻劃,例如:1825年Gompertz提出的死力法則、1838年Verhulst以羅吉斯函數描述人口成長。近年則傾向於逐項(item-by-item)分析各種可能因素,例如:1992年Lee-Carter提出的死亡率模型、目前英國實務上使用的Renshaw與Haberman(2003)提出改善Lee-Carter模型的Reduction Factor模型、加入世代(Cohort)因素的Age-Period-Cohort模型等。但台灣地區近年來生育率與死亡率皆不斷下降,且有隨著時間而變化加劇的傾向,使得以往使用的模型不易捕捉變化。
本文以另一個角度思考生育與死亡變化,將台灣人口視為一隨時間變化的動態系統,使用微分方程來刻劃,找出此動態系統的背後所隱含的規則。人口動態系統的變化,主要來源是出生、死亡與遷移,在建模的過程中,我們先各別針對其中一項,在其他條件不變的情況下,以常微分方程建模,之後再同時考慮各項變動,以偏微分方程建模,找出台灣人口變化的模型。在本文中,我們先介紹使用微分方程模型分別配適與估計出生與死亡。
由台灣地區人口統計資料顯示,不論總生育率或各年齡組的死亡率都有逐漸下降的趨勢,但是每年之間的震盪很大,因此我們提出「二次逼近法」,從出生或死亡對時間的變化率與曲度來估計生育率與死亡率,對於此種震盪幅度較大的資料,可以得到頗精確的估計。唯在連續幾年資料呈現近似線性上升或下降處,非線性的模型容易出現較大的估計誤差,針對此問題我們也提出一些可能的修正方法,以降低整體的模型誤差率。
Conventionally the change of population is considered as a function of time and described by using deterministic functions. The well-known examples are Gompertz law of mortality (1825) and Verhulst’s logistic growth model (1838). Recently demographers favor stochastic models when analyzing factors in an item-by-item fashion. Since 1992, Lee-Carter model is a most commonly used stochastic model in demographic studies. But empirical studies indicate that the rapid declines in both fertility and mortality rates are against the assumptions of Lee-Carter model.

In this study we treat Taiwan population as a dynamic system which changes over time and characterize it by differential equations. Since the changes are from birth, death and migration, we first separately build models using ordinary differential equations. Afterwards the model of Taiwan population can be built by using partial differential equations considering the three main factors simultaneously.

Total fertility and age-specific mortality rates in Taiwan decline over time but with shakes between years. Consequently we propose‘parabola approximation method’and apply it to velocity and acceleration of birth or death to solve the differential equations of Taiwan fertility and mortality. Empirical study shows the method allows us to get accurate estimates of mortality and fertility when the data change a lot in a short period of time. But we found the model may over-fit the data at some time point where the function does not seem to be very continuous.
參考文獻 中文部分
中華民國內政部統計資訊網,http://www.moi.gov.tw/stat/
內政部(1949~2005),中華民國台閩地區人口統計,內政部編印。
閰守誠(1997),中國人口史,臺北:文津出版社。
楊靜利與李大正(2007),台灣出生與死亡資料之編製與調整:1905-1943與1951-1997,2007年台灣人口學會學術研討會論文
王郁萍與余清祥(2007),台灣地區死亡率APC模型之研究,2007年台灣人口學會學術研討會論文
余清祥(2008),高齡死亡率模型的實證研究,2008年台灣人口學會學術研討會論文
余清祥與曾奕翔(2005),Lee-Carter模型分析:台灣地區死亡率推估之研究,2005年台灣人口學會學術研討會論文
郭孟坤與余清祥(2008),電腦模擬、隨機方法與人口推估的實證研究,人口學刊,第36期,67-98頁
許鳴遠(2006),台灣人口死亡率模型之探討: Reduction Factor模型的實證研究,國立政治大學風險管理與保險研究所碩士論文
賴思帆與余清祥(2006),臺灣與各國生育率模型之實證與模擬比較,人口學刊,第33期,33-59頁
賴思帆 (2005),生育率模型與臺灣各縣市生育率之實證研究,國立政治大學統計研究所碩士論文
余清祥與藍銘偉(2003),台灣地區生育率模型之研究,人口學刊,第27期,105-131頁
李芯柔 (2008),電腦模擬在生育、死亡、遷移及人口推估之應用,國立政治大學統計研究所碩士論文
王德睦、劉一龍與李大正(2005),台灣存活曲線的矩形化與死亡率壓縮,2005年台灣人口學會學術研討會論文
英文部分
Bogue, Donald J. (1969), Principles of Demography, John Wiley and Sons, Inc., New York.
Brouhns, N., Denuit, M, and Vermunt, J.K. (2002), A Poisson Log-bilinear Regression approach to the Construction of Projected Lifetables, Insurance: Mathematics and Economics, Vol. 31, pp. 373-393.
Brown, R.L. (1991), Introduction to the Mathematics of Demography, 2nd Edition, Axtex Publications, Winsted, CT.
Cairns et al (2009), A Quantitative Comparison of Stochastic Mortality Models Using Data from England and Wales and the United States, North American Actuarial Journal, Vol.13(1), pp. 1-35(Reprint).
Caselli, G, and Lopez, A.D. (1996), Health and Mortality Among Elderly Populations, Clarendon Press, Oxford.
Duan, R., and Li, M.R., and Yang, T.(2008), Propagation of Singularities in the Solutions to the Boltzmann Equation near Equilibrium, Mathematical Models and Methods in Applied Sciences, Vol.18(7), pp.1093-1114
Edelstein-Keshet, L. (2005), Mathematical Models in Biology, SIAM, Philadelphia.
Gompertz, B. (1825), On the Nature of the Function Expressive of the Law of Human Mortality and On a New Mode of Determining Life Contingencies, Philosophical Transactions of the Royal Society of London, 115:513-585.
Guillard, A.(1855), Eléments de Statistique Humaine Ou Démographie Comparée, Guillaumin et Cie., Paris, 376 p.
Hinde, A.(1998), Demographic Methods, Arnold Publishers, London.
Holford, T.R. (1983), The Estimation of Age, Period and Cohort Effects for Vital Rates, Biometrics, 39:311-324.
Huang, H., Yue, C.J., and Yang, S.S. (2008), An Empirical Study of Mortality Models in Taiwan, APRIA, Vol. 3(1), pp. 150-164.
Kammeyer, Kenneth C.W. (1971), An Introduction to Population, Chandler Pub. Co., San Francisco.
Kannisto, V. (2000), Measuring the Compression of Mortality, Demographic Research 3, Article 6.(www. demographic-research.org/Volumes/Vol3/6)
Lotka, A.J. (1956), Elements of Mathematical Biology(formerly published under the title Elements of Physical Biology 1925), Dover Publications, New York.
Lee, R.D., and Carter, L.R.(1992), Modeling and Forecasting U.S. Mortality, Journal of the American Statistical Association, Vol.87(419), pp. 659-671.
Lewis, C. D. (1982), Industrial and Business Forecasting Methods, Butterworths, London.
Li, M.R. (2008), Estimates for the Life-Span of the Solutions for Semilinear Wave Equations, Communications on Pure and Applied Analysis. Vol.7(2), pp. 417-432.
Li, M.R. (to appear), On the Blow-up Time and Blow-up Rate of Positive Solutions of Semi-linear Wave Equations □ u - = 0 in 1-dimensional Space, submitted to Communications on Pure and Applied Analysis (CPAA).
Malthus, T.R. (1826), An Essay on the Principle of Population, Cambridge University Press.
Marshall, G. (1998), A Dictionary of Sociology, http://www.encyclopedia.com
Pitchford, J.D. (1974), Population in Economic Growth, North Holland/American
Elsevier.
Renshaw, A.E., and Haberman, S. (2003), On the Forecasting of Mortality Reduction Factors, Insurance: Mathematics and Economics, Vol. 32, pp. 379-401.
Renshaw, A.E., and Haberman, S. (2003), Lee-Carter Mortality Forcasting with Age-specific Enhancement, Insurance: Mathematics and Economics, Vol. 33, pp. 255-272.
Shieh, T.H., and Li, M.R.(2009), Numerical Treatment of Contact Discontinuity with Multi-gases, Journal of Computational and Applied Mathematics, Vol. 230(2), pp. 656-673.
Shieh, T.H. et al, (2009), Analysis on Numerical Results with Different Exhaust Holes, International Communications in Heat and Mass Transfer, Vol.36(4), pp. 342-345.
Turchin, P. (2001), Does Population Ecology Have General Laws?, Oikos 94:17-26.
United Nations Statistics Division, http://unstats.un.org/unsd/demographic
United Nations (1958), Multilingual Demographic Dictionary, English Section, Department of Economic and Social Affairs, Population Studies, No. 29(United Nations publication, Sales No. E.58.XIII.4).
United Nations (1998), Principles and Recommendations for Population and Housing
Censuses Rev. 1., Statistics Division, Series M, No. 67, Rev. 1 (United Nations
Publication, Sales No. E.98.XVII.8).
Verhulst, P.F.(1838), Notice Sur la Loi Que la Population Poursuit Dans Son Accroissement, Correspondance Mathématique et Physique, Vol.10, pp. 113-121.
Yue, C.J. (2002), Oldest-Old Mortality Rates and the Gompertz Law: A Theoretical and Empirical Study Based on Four Countries. Journal of Population Studies, Vol.24, pp. 33-57.
描述 碩士
國立政治大學
統計研究所
95354003
97
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0953540031
資料類型 thesis
dc.contributor.advisor 余清祥zh_TW
dc.contributor.author (Authors) 李玢zh_TW
dc.creator (作者) 李玢zh_TW
dc.date (日期) 2008en_US
dc.date.accessioned 18-Sep-2009 20:11:24 (UTC+8)-
dc.date.available 18-Sep-2009 20:11:24 (UTC+8)-
dc.date.issued (上傳時間) 18-Sep-2009 20:11:24 (UTC+8)-
dc.identifier (Other Identifiers) G0953540031en_US
dc.identifier.uri (URI) https://nccur.lib.nccu.edu.tw/handle/140.119/36932-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 統計研究所zh_TW
dc.description (描述) 95354003zh_TW
dc.description (描述) 97zh_TW
dc.description.abstract (摘要) 人口統計學家在傳統上習慣將人口的種種變化視為時間的函數,皆試圖以決定型(deterministic)的函數來刻劃,例如:1825年Gompertz提出的死力法則、1838年Verhulst以羅吉斯函數描述人口成長。近年則傾向於逐項(item-by-item)分析各種可能因素,例如:1992年Lee-Carter提出的死亡率模型、目前英國實務上使用的Renshaw與Haberman(2003)提出改善Lee-Carter模型的Reduction Factor模型、加入世代(Cohort)因素的Age-Period-Cohort模型等。但台灣地區近年來生育率與死亡率皆不斷下降,且有隨著時間而變化加劇的傾向,使得以往使用的模型不易捕捉變化。
本文以另一個角度思考生育與死亡變化,將台灣人口視為一隨時間變化的動態系統,使用微分方程來刻劃,找出此動態系統的背後所隱含的規則。人口動態系統的變化,主要來源是出生、死亡與遷移,在建模的過程中,我們先各別針對其中一項,在其他條件不變的情況下,以常微分方程建模,之後再同時考慮各項變動,以偏微分方程建模,找出台灣人口變化的模型。在本文中,我們先介紹使用微分方程模型分別配適與估計出生與死亡。
由台灣地區人口統計資料顯示,不論總生育率或各年齡組的死亡率都有逐漸下降的趨勢,但是每年之間的震盪很大,因此我們提出「二次逼近法」,從出生或死亡對時間的變化率與曲度來估計生育率與死亡率,對於此種震盪幅度較大的資料,可以得到頗精確的估計。唯在連續幾年資料呈現近似線性上升或下降處,非線性的模型容易出現較大的估計誤差,針對此問題我們也提出一些可能的修正方法,以降低整體的模型誤差率。
zh_TW
dc.description.abstract (摘要) Conventionally the change of population is considered as a function of time and described by using deterministic functions. The well-known examples are Gompertz law of mortality (1825) and Verhulst’s logistic growth model (1838). Recently demographers favor stochastic models when analyzing factors in an item-by-item fashion. Since 1992, Lee-Carter model is a most commonly used stochastic model in demographic studies. But empirical studies indicate that the rapid declines in both fertility and mortality rates are against the assumptions of Lee-Carter model.

In this study we treat Taiwan population as a dynamic system which changes over time and characterize it by differential equations. Since the changes are from birth, death and migration, we first separately build models using ordinary differential equations. Afterwards the model of Taiwan population can be built by using partial differential equations considering the three main factors simultaneously.

Total fertility and age-specific mortality rates in Taiwan decline over time but with shakes between years. Consequently we propose‘parabola approximation method’and apply it to velocity and acceleration of birth or death to solve the differential equations of Taiwan fertility and mortality. Empirical study shows the method allows us to get accurate estimates of mortality and fertility when the data change a lot in a short period of time. But we found the model may over-fit the data at some time point where the function does not seem to be very continuous.
en_US
dc.description.tableofcontents 第一章 前言……………………………………………………………1
第一節 研究動機與目的………………………………………………………………………1
第二節 研究範圍與研究架構…………………………………………………………………3
第二章 文獻探討與模型介紹…………………………………………4
第一節 文獻探討………………………………………………………………………………4
第二節 相關模型………………………………………………………………………………6
第三章 非線性微分方程與動態系統…………………………………10
第一節 非線性微分方程與動態系統…………………………………………………………10
第二節 非線性微分方程二次逼近法求解……………………………………………………11
第三節 非線性微分方程在人口問題的應用…………………………………………………17
第四節 非線性微分方程模型的特色…………………………………………………………19
第四章 台灣地區資料實證結果………………………………………21
第一節 台灣出生人口數的模型配適情形……………………………………………………21
第二節 台灣年齡別死亡率的模型配適情形…………………………………………………23
第五章 結論與建議……………………………………………………25
第一節 結論……………………………………………………………………………………25
第二節 建議……………………………………………………………………………………27

參考文獻………………………………………………………………29
附錄……………………………………………………………………34
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dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0953540031en_US
dc.subject (關鍵詞) 微分方程zh_TW
dc.subject (關鍵詞) 動態系統zh_TW
dc.subject (關鍵詞) 生育率zh_TW
dc.subject (關鍵詞) 死亡率zh_TW
dc.subject (關鍵詞) 數值分析zh_TW
dc.subject (關鍵詞) Differential equationen_US
dc.subject (關鍵詞) Dynamic systemen_US
dc.subject (關鍵詞) Fertility modelen_US
dc.subject (關鍵詞) Mortality modelen_US
dc.subject (關鍵詞) Numerical analysisen_US
dc.title (題名) 動態系統與生育率及死亡率的估計zh_TW
dc.title (題名) Using dynamic system to model fertility and mortality ratesen_US
dc.type (資料類型) thesisen
dc.relation.reference (參考文獻) 中文部分zh_TW
dc.relation.reference (參考文獻) 中華民國內政部統計資訊網,http://www.moi.gov.tw/stat/zh_TW
dc.relation.reference (參考文獻) 內政部(1949~2005),中華民國台閩地區人口統計,內政部編印。zh_TW
dc.relation.reference (參考文獻) 閰守誠(1997),中國人口史,臺北:文津出版社。zh_TW
dc.relation.reference (參考文獻) 楊靜利與李大正(2007),台灣出生與死亡資料之編製與調整:1905-1943與1951-1997,2007年台灣人口學會學術研討會論文zh_TW
dc.relation.reference (參考文獻) 王郁萍與余清祥(2007),台灣地區死亡率APC模型之研究,2007年台灣人口學會學術研討會論文zh_TW
dc.relation.reference (參考文獻) 余清祥(2008),高齡死亡率模型的實證研究,2008年台灣人口學會學術研討會論文zh_TW
dc.relation.reference (參考文獻) 余清祥與曾奕翔(2005),Lee-Carter模型分析:台灣地區死亡率推估之研究,2005年台灣人口學會學術研討會論文zh_TW
dc.relation.reference (參考文獻) 郭孟坤與余清祥(2008),電腦模擬、隨機方法與人口推估的實證研究,人口學刊,第36期,67-98頁zh_TW
dc.relation.reference (參考文獻) 許鳴遠(2006),台灣人口死亡率模型之探討: Reduction Factor模型的實證研究,國立政治大學風險管理與保險研究所碩士論文zh_TW
dc.relation.reference (參考文獻) 賴思帆與余清祥(2006),臺灣與各國生育率模型之實證與模擬比較,人口學刊,第33期,33-59頁zh_TW
dc.relation.reference (參考文獻) 賴思帆 (2005),生育率模型與臺灣各縣市生育率之實證研究,國立政治大學統計研究所碩士論文zh_TW
dc.relation.reference (參考文獻) 余清祥與藍銘偉(2003),台灣地區生育率模型之研究,人口學刊,第27期,105-131頁zh_TW
dc.relation.reference (參考文獻) 李芯柔 (2008),電腦模擬在生育、死亡、遷移及人口推估之應用,國立政治大學統計研究所碩士論文zh_TW
dc.relation.reference (參考文獻) 王德睦、劉一龍與李大正(2005),台灣存活曲線的矩形化與死亡率壓縮,2005年台灣人口學會學術研討會論文zh_TW
dc.relation.reference (參考文獻) 英文部分zh_TW
dc.relation.reference (參考文獻) Bogue, Donald J. (1969), Principles of Demography, John Wiley and Sons, Inc., New York.zh_TW
dc.relation.reference (參考文獻) Brouhns, N., Denuit, M, and Vermunt, J.K. (2002), A Poisson Log-bilinear Regression approach to the Construction of Projected Lifetables, Insurance: Mathematics and Economics, Vol. 31, pp. 373-393.zh_TW
dc.relation.reference (參考文獻) Brown, R.L. (1991), Introduction to the Mathematics of Demography, 2nd Edition, Axtex Publications, Winsted, CT.zh_TW
dc.relation.reference (參考文獻) Cairns et al (2009), A Quantitative Comparison of Stochastic Mortality Models Using Data from England and Wales and the United States, North American Actuarial Journal, Vol.13(1), pp. 1-35(Reprint).zh_TW
dc.relation.reference (參考文獻) Caselli, G, and Lopez, A.D. (1996), Health and Mortality Among Elderly Populations, Clarendon Press, Oxford.zh_TW
dc.relation.reference (參考文獻) Duan, R., and Li, M.R., and Yang, T.(2008), Propagation of Singularities in the Solutions to the Boltzmann Equation near Equilibrium, Mathematical Models and Methods in Applied Sciences, Vol.18(7), pp.1093-1114zh_TW
dc.relation.reference (參考文獻) Edelstein-Keshet, L. (2005), Mathematical Models in Biology, SIAM, Philadelphia.zh_TW
dc.relation.reference (參考文獻) Gompertz, B. (1825), On the Nature of the Function Expressive of the Law of Human Mortality and On a New Mode of Determining Life Contingencies, Philosophical Transactions of the Royal Society of London, 115:513-585.zh_TW
dc.relation.reference (參考文獻) Guillard, A.(1855), Eléments de Statistique Humaine Ou Démographie Comparée, Guillaumin et Cie., Paris, 376 p.zh_TW
dc.relation.reference (參考文獻) Hinde, A.(1998), Demographic Methods, Arnold Publishers, London.zh_TW
dc.relation.reference (參考文獻) Holford, T.R. (1983), The Estimation of Age, Period and Cohort Effects for Vital Rates, Biometrics, 39:311-324.zh_TW
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