修勻與小區域人口之研究 - 政大學術集成

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(1)立. 政治大. ‧ 國. 學 ‧. A Study of Smoothing Methods for Small Area Population. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v.

(2) R. 立. 政治大. n. al. er. io. sit. y. Nat. 100. ‧. ‧ 國. 學. 98. Natalie. Ch. engchi. i Un. v.

(3) (Longevity Risk). ‧. (Block. Age-Period-Cohort. (Standard Mortality Ratio). er. io. sit. y. Lee-Carter. Nat. Bootstrap). (Smoothing, Graduation). 學. ‧ 國. 立. 政治大. n. a(Coherence) iv l C n hengchi U. i.

(4) Abstract The population size plays a very important role in statistical estimation, and it is difficult to derive a reliable estimation for small areas. The estimation is even more difficult if the geographic and social attributes within the small areas vary widely. However, although the population aging and longevity risk are common phenomenon in the world, the problem is not the same for different countries. The aim of this study is to explore the population projection and mortality models for small areas, with the consideration of the small area’s distinguishing characteristic.. 政治大 directions: data quality, population size, number of base years, and projection horizon. 立. The difficulties for small area population projection can be attributed into four. ‧ 國. 學. The data quality is beyond the discussion of this study and the main focus shall be laid on the other three issues. The smoothing methods and coherent models will be applied. ‧. to improve the stability and accuracy of small area estimation. In the study, the block. sit. y. Nat. bootstrap and the smoothing methods are combined to project the population to the. io. al. er. small areas in Taiwan. Besides, the Lee-Cater and the age-period-cohort model are. iv n C We found that the smoothinghmethods i U the fluctuation of estimation e n g can c hreduce n. extended by the smoothing and coherent methods.. and projection in general, and the improvement is especially noticeable for areas with smaller population sizes. To obtain a reliable population projection for small areas, we suggest using at least fifteen-year of historical data for projection and a projection horizon not more than twenty years. Also, for developing mortality models for small areas, we found that the smoothing methods have similar effects than those methods using more complicated models, such as the coherent models.. Key Words: Small Area Population Projection; Mortality Models; Smoothing Methods; Standard Mortality Ratio; Longevity Risk. ii.

(5) ................................................................................................................ 1 .................................................................................................... 1 ........................................................................................ 5 ...................................................................................................... 10 .......................................................................................... 10 .............................................................................................. 12 .................................................................................................. 17 .................................................................................. 22. 政治大. .................................................................................................. 22 .................................................................................................. 23. 立. ...................................................................................................... 29. ‧ 國. 學. .......................................................................... 29 .......................................................................... 52. ‧. .............................................................................. 64. Nat. sit. y. .................................................................................................. 67. er. io. .......................................................................................................... 67. n. 70 a.......................................................................................... v i l C n ...................................................................................................................... 72 U hengchi .......................................................................................... 76 .............................................................. 77 Lee-Carter. .............................................................. 81 .................................................................................. 84. iii.

(6) 3-1. 12. ........................................................................ 22. 3-2 3-3. .................................................................................... 23 MAPE. ................................................................................. 28. 4-1. 7. 4-2. 1991. 4-3. 2009. 7. .............................................. 32 SMR. ................................................ 33. .......................................................................... 34. 4-4. .................................................................... 36. 4-5. 15. ............................................................................ 50. 4-6. 15. ............................................................................ 51 治政大. 4-9. Lee-Carter. ........................................................................... 57. 4-11. APC. 4-12. APC. 4-13. APC. (Testing). .............................................................. 60. .................................................................................. 62. y. ...................................................................... 64. al. ........................................................................ 64. io. (Training). n. (Testing). 3-1. Lee-Carter. 3-2. APC. ‧. Lee-Carter. .............................................................. 60. Nat. 4-10. (Training). sit. Lee-Carter. ............................................................................... 53. er. 4-8. 立. ‧ 國. Lee-Carter. 學. 4-7. Ch. e nsg c h i x. sx. iv n U ........................................... 83. .................................................... 83. 4-1. Lee-Carter. ............................................................... 88. 4-2. Lee-Carter. ............................................................... 88. 4-3. Lee-Carter. ............................................................... 88. 4-4. Lee-Carter. ............................................................... 88. 4-5. Lee-Carter. ............................................................... 89. 4-6. Lee-Carter. ............................................................... 89. 4-7. Lee-Carter. ............................................................... 89. 4-8. Lee-Carter. ............................................................... 89. iv.

(7) 1-1. .......................................................................... 1. 1-2. 2010. 65. ...................................................... 2. 1-3. .............................................. 3. 1-4. 1991. 2009. Reduction Factor........................... 4. 1-5. Lee-Carter. kt ......................................................................... 8. 1-6. Lee-Carter. bx ......................................................................... 8. 2-1. ............................................................ 15. sx. 4-1. 33 政治 .................................................... 大 ............................................... 34. 12. 4-3. 立. /. ‧ 國. .................................................... 35. 4-5. io. .................................................................................... 39. y. .................................................................................... 40 41 a......................................................................................... v i l C n s 1.0 ......................... 42 U hengchi. n. a. ........................................................ 38. sit. 4-8. ....................................................... 38. sx. Nat. 4-7. 1.0. ‧. 4-6. sx. 學. 4-4. 4-9. .............................................................................. 31. er. 4-2. 7. 4-10. x. sx. 4-11. .......................... 43. 4-12. .......................................................... 44. 4-13. .......................................................................... 44. 4-14. .................................................................................. 46. 4-15. 1980. 1994. .............................................. 48. 4-16. /. ................................................. 51. 4-17. /. ................................................. 52. 4-18. Lee-Carter. ............................................. 53. 4-19. Lee-Carter. ............................................. 55. v.

(8) 4-20. Lee-Carter. ............................................. 55. 4-21. Lee-Carter. ..................................................................... 56. 4-22. 1980. 4-23. APC. 1994. .............................. 59 ...................................................... 63. 1-1. 2008. ............................................................ 76. 2-1. sx. 2-2. sx. 1.0. ....................................... 77 ........................................ 78. 2-3. Partial SMR. sx. 1.0. ................... 79. 2-4. Whittaker. sx. 1.0. ................... 79. 2-5. 立. 2-6. Partial SMR 治 s 政大s Whittaker x. .................... 80. x. .................... 80. 5. ............................................. 81. 3-2. Lee-Carter. 10. ........................................... 81. 3-3. Lee-Carter. 5. ............................................. 82. 3-4. Lee-Carter. 10. ........................................... 82. sit. y. ‧. 4-4. al. er. ............................................................................ 84. n. 4-3. io. 4-2. Nat. 4-1. ‧ 國. Lee-Carter. 學. 3-1. ............................................................................ 84. iv n C h............................................................................ 85 engchi U ............................................................................ 85. 4-5. 1980~1994. .......................................... 86. 4-6. 1980~1994. .......................................... 86. 4-7. 1980~1994. .......................................... 86. 4-8. 1980~1994. .......................................... 87. 4-9. 1980~1994. .......................................... 87. vi.

(9) 2010 2000. 65. 10.7%. 10. 0.2%. 5. 2.1 2010. 1-1. 立. 20%. 14%. (Super-aged). 學 ‧. ‧ 國. 2025. 2060 治政 2017 大. n. er. io. sit. y. Nat. al. Ch. engchi. 1-1. 1. i Un. v. (Aged).

(10) 2010 1-2. 8.2% 15.7%. (Small Area Estimation). 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. 1-2. Ch. engchi 2010. i Un. v. 65. (Cohort Component Method). 2.

(11) 1990. 2009. 2010. (Block Bootstrap) 95%. 2019. 5. (Prediction Interval). 1-3. 260 30. 10. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. 1-3. (Longevity Risk). 3.

(12) Rao (2003). 1-4 1991. 1. 2009. (Reduction Factor, RF). RF RF. 學. ‧ 國. 立. 政治大 12. ‧. n. er. io. sit. y. Nat. al. Ch. engchi. 1-4. 1991. i Un. 2009. v. Reduction Factor. 1991. 1. q x ,t. RF. q x ,0 4. x. t.

(13) 立. ‧ 國. 學. Smith (1987). 政治大 (Population Size). (Projection Horizon). ‧. (Base Period). sit. y. Nat. io. n. al. er. Cannan (1895). Ch. engchi. i Un. v. 10. 30 5.

(14) 1990. 16. 12. 政 12 治大. 立2010. 30. 11. Nat. y. ‧. ‧ 國. 學 0. 2,000. n. er. io. al. sit. 0. Ch. engchi. i Un. v. Smith and Sincich (1990) (Baseline Period). 5 20. 6. 10. 10. 20.

(15) (Smith, 1987). (Projection Horizon). 12. 政治大. 立. ‧ 國. 學 ‧. Lee-Carter. io. n. al. x. er. bx. sit. y. Nat. Lee-Carter. Ch q x ,t q x ,t −1. n engchi U. iv. kt kt. = exp[bx (kt − k t −1 )]. (1.1). ax ax bx. kt. kt 1980. (Random Walk with Drift, RWD) 2009. kt. Lee-Carter 1980. 1994. kt. 1-5 RWD. 7.

(16) bx. 0. 1-6. bx. Female 2. 2. Male Observed Predicted. Observed Predicted. 1. 1. 政治大 0 kt -1. 2000. 2005. 2010. 1980. y. -1 -2. -2. 1995. 1985. 1990. io. 1-5. Lee-Carter. n. al. Ch. engchi 1.0. Male 1.0. 2000. 2005. 2010. Year. i Un. kt. v. Female. 0.0. 0.2. 0.4. 0.6. 0.8. bx. -0.2. -0.2. 0.0. 0.2. bx. 0.4. 0.6. 0.8. bx. er. Year. bx. 1995. sit. 1990. ‧. 1985. Nat. 1980. 學. ‧ 國. kt. 0. 立. 1~4. 20~24. 40~44. 60~64. 80~84. 1~4. Age group. 1-6. 20~24. 40~44 Age group. Lee-Carter. 8. bx. 60~64. 80~84.

(17) 立. 政治大. ‧. ‧ 國. 學. Smith and Sincich (1990). n. al. Li and Lee (2005) Carter. APC. Ch. v ni. Coherent. engchi U. 9. er. io. sit. y. Nat. APC. Lee-Carter. Lee-.

(18) 學. ‧ 國. 立. 政治大. Cannan (1895). (Cohort Component Method). ‧ y. Nat. n. er. io. al. sit. (Cohen,1986; Smith, 1987). Ch. engchi. i Un. v. (Stochastic Forecast Method) (Random Scenario Method). (ex Post Method). (2008). (Block Bootstrap). 10.

(19) (Bootstrap). Efron (1979) Hall. (1985). Künsch (1989). Denton et al. (2005) (Canada). (Uniform). (Linear Weighted). 立. 政治大. 95%. n. al. er. io. sit. 2.5. y. ‧. ‧ 國. 學. Nat. 97.5. Ch. engchi. Y = [Y1 , Y2 ,L, Yn ]T. i Un. v. n. ∆ log Y = [log Y2 − log Y1 , log Y3 − log Y2 L , log Yn − log Yn −1 ]T .  Y  Y   Y  = log 2 , log 3 ,L , log n   Y2   Yn −1    Y1  n-1. b. 11. T. n-b. (2.1).

(20) Rao (2003). 立. 學. ‧ 國. Cohen (1986). 政治大 (Sweden). Smith and Sincich (1990). ‧. (SHIFT). sit. y. Nat. io. al. n. Kanaroglou et al. (2009). (Multi-regional Migration Model). er. Rogers (1995). (Ontario). Smith (1987). i n C U hengchi (Hamilton). v. (2010). Gompertz (1825) 12. (Force of Mortality).

(21) Heligman and Pollard (1980). (Cohort). Lee-Carter. 立. ‧. ax. io. t. n. al. y. kt. x. ε x,t. er. x. (2.2). t. Nat. bx. log(m x ,t ) = a x + bx k t + ε x ,t. x. m x ,t. 學. ‧ 國. Lee and Carter (1992). 政治大. sit. (Period). Ch. ki Un t. eb n g c h i0. v. x. ∑. x. bx = 1. ∑k t. t. =0. Lee and Carter. kt. Koissi et al. (2006). Lawson and Hanson (1974) SVD). (Singular Value Decomposition,. (Principal Component Analysis, PCA). (Maximum Likelihood Estimation, MLE). SVD. kt. (Weighted Least Square, WLS). 13.

(22) (RWD). (Li et al., 2004). SVD. ax T. min ∑ [log(m x ,t ) − a x − bx k t ]2. (2.3). t =1. bx kt. ax. 0. T. ax = [log(m x ,t ) − a x ] V. T. PCA. x ,t. ) (2.4). T UΣVT. kt. 政治 Σ 大 (Singular Value). SVD. Lee-Carter. ‧. SVD. al. er. io. sit. y. Nat. APC. (Age-Period-Cohort Model). n. APC. t =1. 學. (2008) SVD. 立. ‧ 國. U. bx. SVD. ∑ log(m. (Age). (Period). i n C(Cohort) U hengchi. v. APC 2-1 = −. (Non-identification problem) APC log m x ,t = µ + α x + β t + γ t − x + ε x ,t m x ,t. t. x. αx. µ. 14. (2.5). βt. γ t−x.

(23) (Relative Risk). (Sequential Method). (Intrinsic Estimator). (2006). Yang et al. (2007). 立. APC. (2008). 政治大. ‧. ‧ 國. 學. n. al. er. io. sit. y. Nat 2-1. Ch. i Un. i e n gYang c h(2008). 15. v.

(24) Bayesian Bayesian. Pedroza (2006). Lee-Carter. Cairns et al. (2011). Bayesian. APC (Markov Chain Monte Carlo, MCMC) (Posterior distribution). Njenga and Sherris (2009). 立. 學. (2010) (Loading). (Score). er. io. sit. y. Nat. al. iv n C HMD(Human Database) h e nMortality gchi U. n. Coherent Lee-Cater Li and Lee (2005). (Coherence). ‧. ‧ 國. VAR. 治政 VECM (Vector 大 Error‐Correction Model). 15. Lee-Carter. Bx. SVD. Kt. ax. log(mx ,t ) − a x − Bx K t. (2.6). bx. SVD. 16. kt.

(25) log(m x ,t ) = a x + B x K t + bx k t + ε x ,t Kt. kt. Kt. Li and Lee (2005). (2.7). RWD. kt. (First Order Autocorrelation). (Smoothing). 政治大. 立. ‧ 國. (Fit). ‧. (Smoothness). 學. 1997. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Whittaker Whittaker. Whittaker (1923). Whittaker-Henderson. Henderson (1924, 1925). Whittaker. n. n -z. M = F + hS = ∑ w x ( v x − u x ) + h ∑ ( ∆ z v x ) 2 2. x =1. F. S. (2.8). x =1. ux. x. 17. vx. x.

(26) wx. x. x. Z. (Difference). h. vx. (2.8). ∆Z. Z Whittaker. v v v = (W + hk z′ k z ) −1Wu. (2.9). k z = Dn− z +1 L Dn−1 ⋅ Dn. (2.10). − 1 1 0 0  0 −1 1 0 D= M M M M  0 L L L. 0 0  M  1. L L L L O O. 政治L −大 1. 立. (2.11). h 2. 3. h. 學. Z. 4. Partial SMR. sit. y. Nat. (Standard Mortality Ratio,. io. er. Partial SMR. al. n. SMR) SMR. (1997). ‧. ‧ 國. n. ni C h ∑d U i SMR =e n g c h. v. x. x. ∑e. (2.12). x. x. dx. ex. x. x. SMR. SMR 1.0. SMR. 1.0 x. dx. (2.12). ∑d SMR = ∑n ⋅u x. x. x. x. 18. * x. (2.13).

(27) nx. u *x. x. x. SMR Lee (2003). SMR. Partial SMR  d x × hˆ 2 × log(d x / ex ) + (1 − d x / ∑ d x ) × log(SMR)   vx = u*x × exp 2 ˆ   d x × h + (1 − d x / ∑ d x )  . vx. x. x. (2.14). SMR. 政治大立Lee (2003). hˆ 2. h2. (. ). ‧ 國. (2.15). ‧. (Heterogeneity). Nat. sit. 0. n. er. io. SMR × u *x. al. Whittaker Partial SMR. Ch. y. hˆ 2. 學. 2   ˆh 2 = max ∑ (d x − e x × SMR) − ∑ d x , 0  2 2   SMR × ∑ e x  . engchi. SMR. i Un. v. SMR SMR Whittaker SMR. rx = 0 SMR. x. rx > 2 × SMR. SMR. Whittaker. rx. Hyndman et al. (2011) 19. rx = u x / u *x.

(28) p x = u x × u *x. (Functional Time Series Models) Product-Ratio. (Sweden) Whittaker Product-. Ratio. Bayesian (Bayesian). (Prior Information) (Posterior). Jones (1967). ‧ 國. v v v u | t ~ N (t , B ). ‧. v m. sit. y. (2.16). A. n. al. er. io. v u. v v v v v = m + ( I + BA−1 ) −1 (u − m). Nat. I. v t ~ N (m, A). 學. Bayesian. 立v. Kimeldorf and. 政治大. Axy =. Ch. i Un. v. h 0i ≤ r < 1 Axx eAn yy rg c x− y. B. (2.17) (Binomial. Distribution). (Spline) Spline P-Spline (Penalized Likelihood Estimation). 20.

(29) Currie et al. (2004) P-Spline 2-d P-Splines (Twodimensional P-Splines) Ugarte et al. (2009) (Spain). (Vitoria). 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 21. i Un. v.

(30) 12. 1990. 5. 9. 90. io. 95. n. al. 94. 100. 99. 100. sit. 4. y. Nat. 0. 20. er. 2009. 12. ‧. 1990. 16. 學. ‧ 國. 立. 政治大. Ch. 100. engchi. i Un 1. v. 3-1. 3-1 1990 1992 1998. 1991 1997 2009. 0 90+ 0 100+ 0 100+. Gompertz B > 0. 12. Gompertz C. 1. 0 85+ 0 95+ 0 100+. x. Gompertz 22. µ x = BC x.

(31) (Weighted Least Square, WLS). Yue (2002). 1950. 2009. 60 3-2. 1979. 2009. 31. 3-2 0 0 0. 立. 1991 1994 1997 2009. 70+ 80+ 85+. 0 0 0. 政治大 0 90+ 0 95+ 0 95+ 0 100+. 學. 1992 1995 1998. 1957 1969 1974. 70+ 80+ 85+. 0 85+ 0 90+ 0 95+ 0 100+. ‧. ‧ 國. 1950 1958 1970 1975. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. (Moving Average). v. Whittaker. (Copas and Haberman, 1983; Ramlan-Hansen, 1983) Bayesian. 23. Partial SMR. (Kernel).

(32) (Block Size) Bühlmann (2002). Politis and Romano. (1994). (Empirical Criteria-Based Methods). 政治大. 立. ‧ 國. 學. f(Y) g (∆ log f (Y )). g. Nat. io. sit. f. n. al. er. 1,000. ‧. f ( g (∆ log Y )). y. g(Y) Y. Whittaker. Ch. engchi. i Un. v. 0 Partial SMR. SMR Whittaker Bayesian. Whittaker x. rx = u x / u *x. u *x x. ux 「 rx = 0. rx > 2 × SMR 」 rx. SMR. 24.

(33) Whittaker. Whittaker Z. h. 3. Bayesian Bayesian. Partial SMR Whittaker. x. Axx. A. 0. (2.17). x. 0. 0. Whittaker. SMR. rx. (2.16). r. 立. 政治大. 0.9. 學 ‧. ‧ 國. Bayesian. n. er. io. sit. y. Nat. al. Lee-Carter. Ch. engchi. i Un. v. Lee-Carter. Currie (2006). P-Splines. Lee-Carter. 2-d P-Splines. 25. APC. 0.

(34) Partial SMR. Whittaker. 1. SVD. bx. Lee-Carter. Bx. Kt. kt. bx. 2.. SMR. ∑b × n R= ×n 政 ∑ B治 x. x. x. x. x. (3.1). x. rx =. 學. bx Bx. R. Partial SMR. sit. n. al. er. io. bx. Whittaker. y. Nat i. Partial SMR. (3.2). ‧. rx. ‧ 國. 立. 大. Ch. Partial SMR. engchi. i Un. v. hˆ 2 SMR. (Weighted Geometric Mean).  w log(d x / e x ) + log( SMR)   v x = u *x × exp x wx + 1  . bx. (3.3). wx SMR. Partial SMR. (Weighted Geometric. Mean). 26.

(35) ii. Whittaker rx. Z. Whittaker. bx. h. 3. kt. 3.. i. Partial SMR (Weighted Geometric Mean). bx ii. Whittaker. kt. 立. 政治大. SVD. Whittaker. Whittaker. ‧. ‧ 國. 學. Coherent APC. sit. Li and Lee (2005). al. n β t′ 、 γ t−′ x. Lee-Carter. er. io APC. y. Nat Coherent LC. 0. Ch. engchi. i Un. v. β t 、 γ t− x. Coherent APC log m x ,t = µ + α x + β t′ + γ t′− x + β t + γ t − x + ε x ,t. β t′ 、 γ t′− x. (3.4). Li and Lee (2005). βt 、 γ t−x AR(1). 27.

(36) (Data Mining). -. (Training-testing). (In-sample) (Out-sample) MAPE (Mean Absolute Percentage Error). MAPE =. Y. Yˆ. 20%. 50%. MAPE. 政治大. 3-3. MAPE. < 10%. 10% 20%. 3-3. 學. 20% 50%. ‧. ‧ 國. (3.5). Lewis (1982). 立. io. sit. y. Nat. n. al. er. MAPE. 1 n Yt − Yˆt × 100% ∑ n t =1 Yt. Ch. engchi. 28. i Un. v. > 50%.

(37) Lee-Carter APC. 立. 政治大. ‧. ‧ 國. 學. n. al. er. io. sit. y. Nat. Bayesian. 12. Partial SMR. Ch. engchi. Whittaker. Whittaker. 29. i Un. v.

(38) 12. 2008. 23. 0 SMR. Bayesian. Whittaker. Partial SMR. Whittaker 0. 2008. 17. 0. 4. 5. 9. 80. 84. 85. 2008. 治政 1990 2009 大. Lee-Carter. 立. ‧ 國 sx. 1.0. 3.. sx. 1.2. 4.. sx. 5.. sx. ux. 2008. x. al. er. 4-1. sit. y. ‧. 2.. 1,000. n. 0.8. io. sx. nx. Nat. 1.. sx. 學. nx × u x. sx. 2008. Ch. engchi U. v 80% i n. 120%. 30.

(39) sx. 7.. sx. 2.0. 6.. 上上下下先下下先上上(V 型) 先上上先下下(倒V型) 1.5 Sx. 1.0. 1.0. 政治大. ‧ 國. 20~24. 40~44. 60~64. 80~84. 學. 0~4. 0.5. 0.8. 立. 0.6. 0~4. 20~24. 40~44. Age Group. ‧. sx. 4-1. 7. io. sit. y. Nat. 4-1. Age Group. n. al. er. Sx. 1.2. 1.4. Sx=0.8 Sx=1.0 Sx=1.2. Ch. engchi. i Un. v. SMR. Partial SMR. SMR. Partial SMR SMR. Whittaker. (Exploratory Data Analysis, EDA) Partial SMR Whittaker. 31. 60~64. 80~84.

(40) 4-1. 7. MAPE (%). 80%. 100%. 120%. 40.8. 36.7. 33.1. 42.5. 33.4. 35.2. 39.6. Whittaker. 31.7. 29.6. 27.9. 32.9. 28.4. 31.7. 29.8. Bayesian. 12.6. 5.6. 10.7. 27.2. 30.6. 28.0. 29.1. Partial SMR. 4.3. 3.8. 3.6. 46.7. 25.6. 25.5. 18.5. 12.7. 12.7. 12.0. 32.8. 22.3. 13.5. 16.0. Whittaker. V. MAPE. 政治大. 立. ‧. ‧ 國. 學. n. al. er. io. sit. y. Nat 12 1990. 2009. 12. 40% 80%. 120%. Ch. 12. engchi. i Un. v. SMR. 4-2 4-2. 12 Partial SMR. 32. V.

(41) 4-2. 1991. 2009. SMR. SMR ( ). 0.793. 0.853. 0.869. 0.988. 0.989. 1.005. SMR ( ). 0.818. 0.884. 0.915. 0.986. 1.000. 1.006. SMR ( ). 1.042. 1.057. 1.133. 1.167. 1.193. 1.200. SMR ( ). 1.061. 1.034. 1.080. 1.169. 1.175. 1.153. 1.5. Nat. n. al. er. 0.5. io. sit. y. ‧. 1.0. Ratio. ‧ 國. 1.5 Ratio. 學. 1.0. 大大大 , 松松大,中中大中松大 , 信信大 ,文松大 ,內內大 , 士士大北北大 , 南南大 ,萬萬大 ,大大大. 政治大. 立. 0.5. Female. 2.0. 2.0. Male. 大大大 ,松松大, 中中大中松大 ,信信大 ,文松大 , 內內大 ,士士大北北大 ,南南大 ,萬萬大 , 大大大. 0~4. 20~24. Ch. 40~44 60~64 Age Group (year-old). 4-2. 1990. 0~4. engchi. 2009. v. 40~44 60~64 Age Group (year-old). Lee-Cater. sx. 7. 2004. (Data Mining). i Un. 20~24. 12. 1990. sx. 80~84. 2005. -. (Training-Testing). 1,000. 2009. 1,000 MAPE. 33. 80~84.

(42) 4-3. V. 4-1. 4-3 Whittaker. 20%. Partial SMR. 4-3. Partial SMR Whittaker. 80%. 100% 120%. 42.4. 41.1. 36.6. 39.4. 36.3. 39.4. 36.2. 16.9. 16.8. 16.3. 37.5. 30.0. 27.5. 22.9. 27.7. 23.6. 24.4. 30.1. 28.1. 23.6. 22.7. 23.3. V. 16.1. 15.7. 15.1. 36.0. 21.1. 20.2. 20.1. 24.6. 28.5. ‧. Ch. engchi. i Un. v. 0.6. 0.8. 1.0. 1.2. n. al. er. io. Partial SMR Mortality Ratio Graduation. sit. y. MAPE. Nat Ratio. MAPE (%). 學. Whittaker. 1.4. 7. 治政 24.2 23.2 21.9 大 28.8 立. ‧ 國. Partial SMR. 10%. 80%. 100%. 120%. Increase Decrease V shape Inverted V Scenario. 4-3. /. 34. V.

(43) 1990. 2009. 1990 4-2. 2004. 2005. 4-2. 2009. 1.0. Partial SMR 4-4 4-4 Partial SMR. 2%. Whittaker. 5%. 1,000 1,000 政治大. 立. 1,000. ‧. ‧ 國. 學 sit i Un. v. 40 20 0. 0. 20. 40. No graduation Partial SMR Mortality ratio. er. 100 MAPE (%). engchi. 60. Ch. 80. al. 80. 100. y. 140 120. No graduation Partial SMR Mortality ratio. n. MAPE (%). Forecast -> Graduation. 60. io. 120. Nat. 140. Graduation -> Forecast. 0~4. 20~24. 40~44. 60~64. 80~84. Age Group (year-old). 0~4. 20~24. 40~44 Age Group (year-old). 4-4. 35. 60~64. 80~84.

(44) 4-4. MAPE (%) Partial SMR. Whittaker. →. 24.7. 30.1. →. 26.3. 25.4. 51.1. 政治大. 立. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. 1950. 2009. Ch. engchi. Lee-Carter 1980. 2009. v. 1950. 1979. 30. 30. (Jump-off Population). 1. sx ＝1.0：. i Un. 150. 10. 25. sx. 50. ＝1.0 2. sx. 10. 36. 25. 50.

(45) sx Partial SMR. Whittaker. 1,000 10. sx. 12. 1.0. 150. 4-5. 30. 政治大. 4-5. 150. 40% 8. 學. ‧ 國. 立15. 10. 15. ‧. 150. n. al. er. io. s x = 1.0. 4-5. 20%. sx. sit. y. Nat. SMR. 15. 10. Ch. engchi U 4-6. 10 v i n. Partial. 15. Partial SMR. Whittaker 10 1.0. 37. 15. sx.

(46) 120. No graduation 100,000 250,000 500,000 1,500,000. 60 0. 20. 40. MAPE (%). 80. 100. Population Population Population Population. 5. 10. 15. 20. Block size. 120. Whittaker Ratio. 100,000 250,000 500,000 1,500,000 (No graduation). Population 100,000 Population 250,000 Population 500,000 Population 1,500,000 (No graduation). MAPE (%) 40 60 80. 15. 5. 20. 10. Block size. sx. 4-5. ‧. 120. Nat. er. al. n 60. 100,000 250,000 500,000 1,500,000. sit. 100. Population Population Population Population. 80. io. Ch. engchi. i Un. v. 0. 20. 40. 20. 1.0. No graduation. MAPE (%). 15 Block size. y. 10. 學. 5. ‧ 國. 0. 0. 20. 立. 政治大 20. 60 40. MAPE (%). 80. 100. Population Population Population Population. 100. 120. Partial SMR. 5. 10. 15. 20. Block size. Whittaker Ratio. 80. 100 120. Population 100,000 Population 250,000 Population 500,000 Population 1,500,000 (No graduation). 0. 20. 20. 40. 60. 100,000 250,000 500,000 1,500,000 (No graduation). 0. MAPE (%). 100. Population Population Population Population. MAPE (%) 40 60 80. 120. Partial SMR. 5. 10. 15. 5. 20. 10. 15 Block size. Block size. sx. 4-6. 38. 20.

(47) 80 60 0. 20. 40. MAPE (%). 100. 120. Population 100,000 Population 250,000 Population 500,000 Population 1,500,000. 0. 5. 10. 15. 20. 25. 30. Projection year. 4-7. 政治10 s大1.0. (. 立. 1.0. sit. y. 10 10. n. al. iv n C (Discrete Distribution) h eUniform ngchi U. B ~ Discrete Uniform[b ± a] B. b. 4-5. (4.1). b. 10. 15. b=7. a=2. a 4-8. 50. b. B. b. a b = 10. a=3. 1.0. er. io. B. sx. ‧. ‧ 國. 學. sx. Nat. 4-7. ). x. b = 15. a=5. 39. b.

(48) a 4-9. b. a. 15. 40. 政治大. 20. ‧. ‧ 國. 30. 立. 學. 5. 10. n. al. Ch. 4-8. 15. 20. er. io. 0. y. Nat. 10. MAPE (%). 50. Fixed 7 Fixed 10 Fixed 15 Random 7 Random 10 Random 15. i Un. Projection year. engchi. (. 50. 40. sit. 60. a. sx. 25. 30. v. 1.0. ).

(49) 100. No graduation. 40. 60. 100,000 250,000 500,000 1,500,000. 0. 20. MAPE (%). 80. Population Population Population Population. 0. 2. 4 6 parameter a. 8. 10. 政治大. 60. 0. 4 6 parameter a. 8. 10. 2. 4 6 parameter a. ( B ~ DU[15 ± a] ). a. 4-9. 0. sx. 1.0. sit. n. er. io. al. Ch. 10. engchi. i Un. v. 1,000. 5. 3. 10. 20. 15. 25. 7 30. [15 ± 5] 4-10. 10. y. Nat. 15. 8. ‧. ‧ 國. 20. 40 20. 2. Population 100,000 Population 250,000 Population 500,000 Population 1,500,000 (No graduation). 學. 0. 100. 100,000 250,000 500,000 1,500,000 (No graduation). 立. 0. MAPE (%). 80. Population Population Population Population. Whittaker Ratio. M A P E (% ) 40 60 80. 100. Partial SMR. sx. 1.0. 4-11 2-3. 41. 2-6.

(50) 150. 10 15. 25. 10. 20. 100 80. Population 250,000. 20. 政治大 30. Population 500,000. 15 20 Baseline Period (years). 25. 30. Population 1,500,000. sit er. engchi. i Un. v. 0. 0. Ch. 20. n. al. MAPE (%) 40 60. io. 20. 10. y. Nat. MAPE (%) 40 60. 5. 80. 80. 25. ‧. ‧ 國. 15 20 Baseline Period (years). 100. 10. 學. 5. 0. 0. 立. 100. 20. MAPE (%) 40 60. 100. Projection horizon = 10 Projection horizon = 20 Projection horizon = 30. MAPE (%) 40 60. Population 100,000. 80. Smith and Sincich (1990). 5. 10. 15 20 Baseline Period (years). 25. 30. 5. 10. 15 20 Baseline Period (years). sx. 4-10. 42. 1.0. 25. 30.

(51) 200. 100. Population 250,000. 0. 0. 25. 50. MAPE (%) 50 75. MAPE (%) 100 150. 125. 250. Population 100,000 Projection horizon = 10 Projection horizon = 20 Projection horizon = 30. 10. 15 20 Baseline Period (years). 125. 5. 25. 30. 5. 10. 15 20 Baseline Period (years). Population 1,500,000. MAPE (%) 25. 100 MAPE (%) 50 75. 政治大 0. 15 20 Baseline Period (years). 25. 30. 5. 學. 10. ‧ 國. 0. 25. 立 5. 10. 15 20 Baseline Period (years). 4-12. sit. y. al. er. 20. n. 1.0. 30. ‧. io. 50. Nat. 10. 25. sx. 4-11. sx. 30. 50. Population 500,000. 25. Ch. engchi. 10. i Un. 10. v. 4-12 Partial SMR 4-13. 20 10. sx. Partial SMR. 1.0. sx Whittaker. 20. 10. 40% 15. 43. 50.

(52) 70% 10 5 15. 15 15. 立. 政治大. 學 ‧. ‧ 國 io. sit. y. Nat. 4-12. n. al. er. 10. 10. Ch. engchi. 4-13. i Un. v. 20. 44. 20.

(53) 1,000. [15 ± 5]. 25. 2. ‧ 國. 50. 10. ‧ 20. io. sit. y. Partial SMR. n. al. er. Nat. 1.0. Ch. i U e n g c h20%. sx 20%. 20. 學. ( MAPE < 20% ). sx. 20%. 3. 政治大 50. 立. 20. 4-14. v ni. 20 10 10. 10. 45.

(54) Population 100,000. Population 500,000. 60. MAPE (%). 80. 100. No graduation, Sx = 1.0 No graduation, Sx is increasing Partial SMR, Sx = 1.0 Whittaker Ratio, Sx is increasing. 0. 20. 40. 60 40 20 0. 5. 10. 15. 20. Projection year. 25. 25. 30. 0. 5. 10. 15. 政治大立4-14 Discrete Uniform [15 ± 5]. 學 ‧. io. sit. y. Nat. n. al. Ch. engchi. i Un. v. 4-5. 4-6. sx Partial SMR. 1.0 4-12. 10. 15 50. 20. Projection year. er. 0. ‧ 國. MAPE (%). 80. 100. No graduation, Sx = 1.0 No graduation, Sx is increasing Partial SMR, Sx = 1.0 Whittaker Ratio, Sx is increasing. 10. 20. 4-14 20%. 46. 25. 30.

(55) 4-12 15. 10. 10. 10. 5. 50. 10. 5. 4-14. 4-11. 12. 政治大. 立. ‧ 國. 學. 100. 10. al. 100. er. 20. n. 100. io. 20. y. Nat 15. ‧. 20. 50. sit. 4-10. Ch. e n g c h20i. i Un. v. 1980 20 10 (Training). 50. 2009. 100 1980 15. 1994. 15 1995. 47. 2009. 15.

(56) 20. 15. (Testing). 10 4-15. 1980. 1994 SMR. 1. 1.04. 1.03. sx. 1.0. Partial SMR. sx. V. Partial SMR. 4-3 治政大. Whittaker. 立. V. 學. ‧ 國. Whittaker. Partial SMR. Partial SMR 1.39. 1.37. ‧. SMR. 2.0. n. er. io. sit. y. Nat. al. CMaleh. engchi. v. Male Female 1 Male SMR 2 Female SMR. 1 2. 0.5. 1.0. 1.0. 1 2. R atio 1.5. R atio. 1.5. 2.0. Female 1 Male SMR 2 Female SMR. i Un. 0~4. 20~24. 40~44. 60~64. 80~84. 4-15. 1980. 0~4. 20~24. 40~44 Age Group. Age Group. 1994. 48. 60~64. 80~84.

(57) 4-5 0. 4-6 14. 15. 64. 65. 0 MAPE. 6. 50% 4-1. 政治大. 學. Partial SMR Partial SMR. ‧. ‧ 國. 64. 7 6. 立. 15. 20%. 20%. 4-4. 14. sit. y. Nat. 4-2. io. n. al. er. (Bias). Ch. e n g4-5c h i. i Un. v. Whittaker SMR. Partial Partial SMR Partial SMR. V Whittaker 6. 15. Partial SMR. 64. Partial SMR 4-4. 49. 4Whittaker.

(58) 4-16. 4-17 6. 6. Partial SMR. Whittaker. Lee-Carter. 立. 政治大. ‧. ‧ 國. 學. n. al. er. io. sit. y. 50. Nat. 100. Ch. engchi. 4-5. i Un. v. 15. MAPE (%). Partial SMR Whittaker. 0-14 37.6 24.5 28.8. 15-64 8.4 5.5 8.7. 65 + 6.9 5.9 6.6. 43.5 26.7 27.8. Partial SMR Whittaker. 0-14 19.1 18.5 17.2. 15-64 5.1 4.3 4.9. 65 + 11.7 12.1 11.1. 43.9 27.2 36.0. MAPE 50.

(59) 4-6. 15. MAPE (%). Partial SMR Whittaker. 0-14 50.0 26.4 27.0. 15-64 22.6 17.8 14.6. 65 + 18.4 16.9 19.2. 70.0 45.6 44.0. Partial SMR Whittaker. 0-14 66.4 30.6 31.7. 15-64 26.5 8.9 16.6. 65 + 16.8 14.8 14.1. 102.4 48.4 57.8. MAPE. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. 4-16. Ch. engchi /. 51. i Un. v.

(60) 4-17. 立. sit. y. Nat. Lee-Carter. ‧. ‧ 國. 學. Lee-Carter. 政 /治大. Li and Lee (2005). Lee-Carter. er. io. n. LC aCoherent iv l C n Lee-Carter h e n g c h i U APC. Lee-. Carter. Lee-Carter. 1950. 2009. (Training) 1980. 1950 (Testing). 1,000. 1979. 30 Lee-Carter 4-7. 52.

(61) 4-7. Lee-Carter. MAPE (%). 10. 25. 50. 150. 10. 16.3. 11.4. 7.9. 6.6. 5.7. 20. 20.2. 15.6. 13.1. 12.3. 11.7. 30. 23.4. 18.4. 16.1. 15.4. 15.0. 30. 5. 15 10. MAPE (%). 20. 25. Projection horizon = 10 Projection horizon = 20 Projection horizon = 30. 300. 4-18. 550. 800. 1,050. 1,300. 1,500. Population (in thousands). ‧. ‧ 國. 50. 學. 0. 5. 立. 政治大. Lee-Carter. Lee-Carter. n. al. er. io. sit. y. Nat. 5. 20. Ch 20%. Lee-Carter. engchi. i Un. v. 5. 4-18. 5 150 (Training). 30. Lee-Carter. sx. 1.0. 10. 30. sx. 1,000. (Training Error). (Testing. Error) 4-19. 53. 4-20.

(62) Coherent LC 4-19 Coherent LC Lee-Carter. Lee-Carter. Lee-Carter. Partial SMR. Coherent LC. Whittaker. Lee-Carter. kt AR(1). sx. Coherent LC Whittaker. 1.0. 政治大. 立. Lee-Carter. Coherent LC. ‧. ‧ 國. 學. AR(1). y. Nat. Whittaker Whittaker. io. n. al. er. Partial SMR. sit. Coherent LC. sx. Partial SMR. Coherent LC. Ch. engchi U. v ni. 4-20. Partial SMR. sx. Partial SMR. 4-19 Coherent LC Lee-Carter Coherent LC. Bx. Kt. bx. kt. 54.

(63) 1.0 0.8. Ratio. 0.4 0.2 0.0. 0.0. 0.2. 0.4. 0.6. Ratio 0.6 0.8. 1.0. 1.2. Population 100,000 (Training error). 1.2. Population 50,000 (Training error). 1958. 1962. 1966. 1970. 1974. 1978. 1950. 1954. 1958. 1962. 1974. Population 50,000 (Testing error). Population 100,000 (Testing error). 1978. 1.0. 1.0. 0.8. Ratio. 0.4 0.0. ‧ 國. 1992. 0.2. 0.4 0.2 0.0. 1988. 1996. 2000. 2004. 2008. 1980. 1984. 1988. 1992. Lee-Carter. Nat. 2004. 2008. y. sit er. 1.0. 1.2. i Un. v. 0.0. 0.0. 0.2. 0.4. 0.6 0.4. engchi. 0.2. Ch. Ratio. n. al. 0.8. io. 1.0. 2000. Population 100,000 (Training error). 0.6. 1.2. Population 50,000 (Training error). 0.8. 1996 Year. ‧. 4-19. Original Model Partial SMR, Sx is increasing Whittaker Ratio, Sx is increasing Coherent LC, Sx is increasing. Partial SMR, Sx = 1.0 Whittaker Ratio, Sx = 1.0 Coherent LC, Sx = 1.0. 學. 1984. 0.6. Ratio 0.6 0.8. 政治大. Year. 1958. 1962. 1966. 1970. 1974. 1978. 1950. 1954. 1958. 1962. 1966. 1970. 1974. Year. Year. Population 50,000 (Testing error). Population 100,000 (Testing error). 1978. 1.0 0.8 0.4 0.2 1980. 1984. 1988. 1992. 1996. 2000. 2004. 2008. Year. 4-20. Original Model Partial SMR, Sx is increasing Whittaker Ratio, Sx is increasing Coherent LC, Sx is increasing. Partial SMR, Sx = 1.0 Whittaker Ratio, Sx = 1.0 Coherent LC, Sx = 1.0. 0.0. 0.0. 0.2. 0.4. 0.6. Ratio. 0.8. 1.0. 1.2. 1954. 1.2. 1950. 0.6. Ratio. 1970. Year. 立 1980. Ratio. 1966. Year. 1.2. 1954. 1.2. 1950. 1980. 1984. 1988. 1992. 1996 Year. Lee-Carter 55. 2000. 2004. 2008.

(64) Lee-Carter. 4-21. sx. 5. 1950. Lee-Carter. 1980. kt. 2009. Whittaker. 1979 Lee-Carter. Lee-Carter Coherent LC. 政治大. No Graduation. -4 -6 -8. Mortality rate (log scale). -2. 80~84. io. y. sit. 60~64. -10. -4 -10. -8. -6. ‧ 國 40~44. ‧. 20~24. Nat. 0~4. 1950~1959 1960~1969 1970~1979 1980~1989 1990~1999 2000~2009. 學. Mortality rate (log scale). -2. 立. 0~4. 20~24. n. al. er. Age group. 20~24. 40~44. 60~64. -4 0~4. 20~24. 40~44 Age group. Lee-Carter 5. 56. 80~84. Real Mortality. -10. 80~84. 60~64. 1950~1959 1960~1969 1970~1979 1980~1989 1990~1999 2000~2009. Age group. 4-21. v. -6. -6 -8 0~4. i Un. 40~44. 1950~1959 1960~1969 1970~1979 1980~1989 1990~1999 2000~2009. Age group. -8. Mortality rate (log scale). -2. engchi. -4. Ch. 1950~1959 1960~1969 1970~1979 1980~1989 1990~1999 2000~2009. -10. Mortality rate (log scale). -2. Coherent LC. Whittaker Ratio. sx. 60~64. 80~84.

(65) Lee-Carter. Coherent LC. Coherent LC Lee-Carter. -. Partial SMR. Whittaker Ratio. Coherent. LC Coherent LC. bx k t. AR(1) AR(1). Coherent LC. Coherent LC. AR(1) 4-8. LC. Coherent. 政治大. Lee-Carter. 立. LC. Lee-Carter. ‧ 國. Lee-Carter. Lee-Carter. n. al. 7.1% 9.6%. Partial SMR + Lee-Carter. Ch. e3.2% ngchi. MAPE. er. io. (Training Error). sit. y. ‧. Nat. 4-8. Lee-Carter Lee-Carter Coherent LC. Coherent. 學. Lee-Carter. sx =1.0. 10. v. Whittaker Ratio + Lee-Carter. Coherent LC. 5.1% 8.5%. 7.2% 7.1%. i Un. 7.2% (Testing Error). Lee-Carter. Partial SMR + Lee-Carter. Whittaker Ratio + Lee-Carter. Coherent LC. 19.1% 20.1%. 15.6% 17.1%. 16.8% 17.5%. 17.3% 18.3%. Lee-Carter Coherent LC 10. sx =1.0. 57.

(66) Lee-Carter. Lee-Carter 10 5. 20 2. 10 1980. 2009. 2009. 1980 1994. (Testing). (Training) 1995. 立. ‧ 國. ‧ sit. n. 3. 2009. er. io. 2. 14. v. SMR. i n C1.39 U h e n g c1.37 hi. 1.38. 1.26. 1.09. 96,210 2009. 0. y. Nat. al. 0.91 1.11. 4-22. 學. 3. V. 1.01. kt. Lee-Carter 治政大. 232,497 14.3% 58. 340,964. 530,824.

(67) Ratio. 1.5. Ratio. 2.0. Male Female Male SMR 2 Female SMR. 1. 2.0. Taitung County Male Female Male SMR 2 Female SMR. 1.5. Penghu County. 1. 1. 2. 1.0. 1.0. 1 2. 20~24. 40~44. 60~64. 0~4. 20~24. 40~44. 60~64. Age Group. Hualien County. Nantou County. Ratio. 1.5. 1. 80~84. Male Female Male SMR 2 Female SMR. 2.0. 2.0. Male Female Male SMR 2 Female SMR 1. 1. 1. 2. 政治大 1.0. 1.0. 20~24. 40~44 Age Group. 1980. 60~64. 80~84. 0~4. 20~24. 1994. 60~64. ‧. 1980. 1994. sit. io. al. n Lee-Carter. 80~84. y. Nat 4-9. 15. 40~44 Age Group. 學. 4-22. 立. 2. er. 0~4. ‧ 國. Ratio. 80~84. Age Group. 1.5. 0~4. C hLee-Carter U n i engchi 20%. v. Coherent LC. 4-10 Coherent LC Partial SMR. 59.

(68) Coherent LC. Lee-Carter. 4-10 Lee-Carter. 272.6%. 4-9. Lee-Carter. 25.3 30.4. Whittaker Ratio. 24.8. Coherent LC. 27.4. 32.6. 12.0. 18.2. MAPE (%). 10.2. 21.8 12.8 政14.9 治大 11.7 20.2 12.6 立30.9 29.1 14.7 16.6 10.1 31.0. MAPE. Lee-Carter. (Testing). 13.7. 18.2. 12.8. 16.1. 17.1. 10.1. 15.1. 15.0. 11.9. 12.9. MAPE (%). 272.6. n. al. Ch. Partial SMR. 39.9. 44.2. Whittaker Ratio. 47.2. 58.7. Coherent LC. 36.7. 38.6. 39.6. 50.6. 32.8. 32.6. e32.5 hi n g c38.6 31.4. 32.7. er. io. sit. y. Nat 69.6. 10.8. ‧. 4-10. 17.4. 學. ‧ 國. Partial SMR. (Training). 50.8. 78.1. 40.7. 55.2. 35.8. 27.8. 30.6. 33.6. 46.4. 25.4. 40.2. 50.1. 33.5. 26.8. 30.8. 33.8. i Un. v. MAPE. 0 15. 64. 65. 14 4-1. 8. Coherent LC. 4-22. 60. 4-.

(69) 4-7 64 Lee-Carter. Lee-Carter. 立. 政治大. 學 ‧. ‧ 國 io. sit. y. Nat. Coherent LC. n. al. er. 15. APC (2008). Ch. engchi APC. i Un. v. Lee-Carter. -. Lee-Carter Lee-Carter. APC. Lee-Carter. APC Coherent APC. 1950. 61. 2009.

(70) -. 1,000. APC. APC. Coherent. APC. 23. APC 4-. APC 4-19. Lee-Carter. APC. Lee-Carter. s x = 1.0. Whittaker. Partial SMR. Coherent APC. Lee-Cater. 政治 APC 大. 立. AR(1). Coherent APC Coherent APC. ‧. 4-11. 學. ‧ 國. APC. Coherent. y. al. n. 3-2. sit. io. Partial SMR. Whittaker. er. APC. Nat. Coherent APC. Ch 4-11. n engchi U. iv. sx. APC. MAPE (Training Error). APC Coherent APC. APC. Partial SMR + APC. Whittaker Ratio + APC. Coherent APC. 21.0% 23.9%. 5.1% 10.7%. 5.6% 10.9%. 21.1% 23.8%. (Testing Error) Partial SMR Whittaker Ratio + APC + APC. APC APC Coherent APC. 45.9% 48.8% 10. 36.4% 38.4%. sx =1.0 62. 36.5% 38.5%. Coherent APC 42.4% 45.7%.

(71) 1.0 0.8. Ratio. 0.4 0.2 0.0 1954. 1958. 1962. 1966. 1970. 1974. 1978. 1950. 1954. 1958. 1962. 1966. 1970. 1974. Year. Population 50,000 (Testing error). Population 100,000 (Testing error). 1978. 1992. 1996. 2000. 0.6 0.4 0.2. 2004. 2008. 1980. 1984. Year. 4-23. 1996. 2000. 2004. Year. ‧ sit. Lee-Carter. n. a4-9 l C h. Lee-Carter. APC. v. APC. er. io. Whittaker. 1992. APC. Nat. 4-12. 1988. y. 1988. 0.8. 1.0 1984. Original Model Partial SMR, Sx is increasing Whittaker Ratio, Sx is increasing Coherent APC, Sx is increasing. Partial SMR, Sx = 1.0 Whittaker Ratio, Sx = 1.0 Coherent APC, Sx = 1.0. 學. 1980. ‧ 國. 0.0. 0.2. 立. 政治大 0.0. 0.4. 0.6. Ratio. 0.8. 1.0. 1.2. Year. 1.2. 1950. Ratio. 0.6. 0.8 0.6 0.0. 0.2. 0.4. Ratio. 1.0. 1.2. Population 100,000 (Training error). 1.2. Population 50,000 (Training error). i Un. (2008). engchi. Coherent APC APC 4-13. 50%. APC. Lee-Carter. APC. 50% Lee-Carter. Lee-Carter. Coherent APC Coherent LC. 63. 4-10. 2008.

(72) 4-12. APC. (Training). MAPE (%). 41.2. 43.0. 34.7. 34.9. 32.9. 30.8. 29.8. 33.6. Partial SMR. 40.8. 38.7. 34.1. 33.5. 31.9. 29.3. 28.5. 32.6. Whittaker Ratio. 30.6. 32.4. 21.4. 23.6. 19.5. 20.4. 20.3. 22.5. Coherent APC. 43.6. 44.9. 38.0. 36.7. 36.2. 30.7. 31.0. 34.7. MAPE. 4-13. 50.8 46.4. (Testing). MAPE (%). 政治大 56.1 39.4 67.5 立62.2 52.9 50.1 35.6 59.5. Whittaker Ratio. 44.1. 49.0. 39.1. 33.9. 46.6. Coherent APC. 42.2. 60.0. 47.8. 36.9. 64.6. 39.7. 59.2. 47.4. 36.2. 55.2. 44.3. 28.1. 44.8. 36.9. 39.7. 54.7. 42.9. 學. ‧ 國. Partial SMR. APC. ‧. MAPE. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. (Credibility Theory). Estimate = Z × [Observation] + (1 − Z) × [Other Information] , 0 ≤ Z ≤ 1 Z. (Credibility) Z 64. (4.2) (1 − Z).

(73) (Other Information) (2010). HMD. (Moving Average) Whittaker (Kernel). 政治大 Partial SMR Whittaker 立. ‧ 國. ‧. Whittaker. io. Whittaker. n. al. Ch. SMR. y. SMR. SMR. sit. Nat. sx. sx. Whittaker Partial SMR. s x = 1.0. sx. sx. 學. Partial SMR. engchi. er. Bayesian. i Un. v. Partial SMR. s x = 1.0. Partial SMR 4-1. sx. Partial SMR. 65.

(74) Partial SMR Whittaker. kt. Lee-Carter. 政治大 Li and Lee (2005). 立. Coherent LC. ‧ 國. ‧. Product. sit. y. al. n. Product-Ratio. Ch. APC Ratio. engchi. 66. er. Bayesian. io. (2011). Product-Ratio. 學. Hyndman et al. (2011). Nat. Cairns et al. (2011). APC. i Un. v. Whittaker. Hyndman et al..

(75) (Smoothing, Graduation). 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 67. i Un. v.

(76) 12 -. 政治大. 立. sx. 12. Whittaker. ‧. sx. 1.0. 學. ‧ 國. Partial SMR. n. er. io. sit. y. Nat. al. 15. Ch. e n g c h10i U. v ni. 15. 5 Smith and Sincich (1990). 1-3 B. B ~ Discrete Uniform[b ± a]. 68. a.

(77) 20. 15 20 12. Whittaker. 立. ‧ 國. 學. 100. ‧. n. al. er. io. sit. y. Nat. 100. Ch. engchi. Currie (2006) P-Spline. Partial SMR. 政治大. Lee-Carter. i Un. v. (CMI) APC 69.

(78) BIC (Bayesian Information Criterion). Lee-Carter. Li and Lee (2005). APC. Coherent LC. Coherent APC Coherent. Hyndman et al. (2011). kt. (Random Walk with Drift). Order Autocorrelation). 立. (First. 政治大. (Autoregressive. Integrated Moving Average, ARIMA). (Autoregressive. ‧ 國. 學. Fractionally Integrated Moving Average, ARFIMA). ‧. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. 15. (Multi-regional Migration Model). Partial SMR Whittaker. Whittaker. Kt. Coherent LC. SMR. Whittaker 70. 49.

(79) (1997) Partial SMR. Coherent LC. SMR. Product-Ratio. APC. 治政 PBF (Postcode Best Fit) 大立 (Bates, 2008) ‧. ‧ 國. 學. n. al. er. io. sit. y. Nat. Ugarte et al. (2009). Ch. engchi. 71. i Un. v.

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(84) Male. -4 -6. Mortality (log scale). -2. Raw Data Whittaker. -10. -8. -2 -4 -6 -8 -10. Mortality (log scale). Female. Raw Data Whittaker. 0~4. 20~24. 40~44 Age Group. 60~64. 80~84. 0~4. Male -4 Mortality (log scale). 60~64. 80~84. -6 -8 -10. 60~64. sit er. Mortality (log scale). y. -4. 40~44 Age Group. engchi -10. Ch. -6. -4 -6 -8 -10. n. al. 80~84. 0~4. i Un. v. 20~24. 40~44 Age Group. 60~64. 80~84. 60~64. 80~84. Female Raw Data Mortality Ratio Standard (Taipei). -6 -8 -10. -10. -8. -6. Mortality (log scale). -4. Raw Data Mortality Ratio Standard (Taipei). -4. 40~44 Age Group. Raw Data Partial SMR Standard (Taipei). Male. Mortality (log scale). 20~24. Female. io 20~24. 0~4. ‧. ‧ 國. 80~84. -8. -4 -8 -10. Mortality (log scale). -6. 60~64. Nat. Mortality (log scale). 40~44 Age Group. Raw Data Partial SMR Standard (Taipei). 0~4. 80~84. Raw Data Bayesian Prior (Taipei). 學. Male. 60~64. 政治大. 立 20~24. 40~44 Age Group. Female. Raw Data Bayesian Prior (Taipei). 0~4. 20~24. 0~4. 20~24. 40~44 Age Group. 60~64. 80~84. 1-1. 0~4. 2008. 76. 20~24. 40~44 Age Group.

(85) Projection year. Projection year. 政治大. Projection year. 立. Projection year. ‧. ‧ 國. 學 er. io. sit. y. Nat. n. a lProjection year iv n Ch engchi U. Projection year. 2-1 10. 50 Partial SMR. sx. 1.0. 77. Whittaker.

(86) Projection year. Projection year. Projection year. 立. Projection year. 政治大. ‧. ‧ 國. 學 er. io. sit. y. Nat Projection year. n. al. Ch. engchi. i Un. v. 2-2 10. 50 Partial SMR. sx. 78. Whittaker. Projection year.

(87) Population 100,000. Population 250,000 50 MAPE (%) 25. 0. 0. MAPE (%) 25. 50. Projection horizon = 10 Projection horizon = 20 Projection horizon = 30. 5. 10. 15 20 Baseline Period (years). 25. 30. 5. 10. 15 20 Baseline Period (years). 25. 30. MAPE (%) 25. 50. Population 500,000. 5. 10. sx. 1.0. y. sit i Un. v. 0. 25. engchi. 0. 25. Ch. Population 250,000. er. 100. n. al. MAPE (%) 50 75. io. 100. 30. 125. Population 100,000. Projection horizon = 10 Projection horizon = 20 Projection horizon = 30. MAPE (%) 50 75. 25. Partial SMR. Nat. 125. 15 20 Baseline Period (years). ‧. ‧ 國. 0. 立. 學. 2-3. 政治大. 25. 30. 5. 125. 15 20 Baseline Period (years). 10. 15 20 Baseline Period (years). MAPE (%). MAPE (%) 50 75. 100. Population 500,000. 25. 10. 0. 5. 5. 2-4. 10. 15 20 Baseline Period (years). Whittaker. 79. 25. 30. sx. 1.0. 25. 30.

(88) 125. 125. Population 100,000. 25. MAPE (%) 50 75. 100. Population 250,000. 0. 0. 25. MAPE (%) 50 75. 100. Projection horizon = 10 Projection horizon = 20 Projection horizon = 30. 10. 15 20 Baseline Period (years). 25. 30. 5. 125. 5. 10. 15 20 Baseline Period (years). 25. 75 50. 政治大. 立 5. 10. 15 20 Baseline Period (years). sx. y. sit. 125 MAPE (%) 50 75. n. i Un. v. 0. 0. 25. engchi. 25. Ch. Population 250,000. er. io. al. 100. 125. 30. ‧. Nat. Population 100,000. 100. 25. Partial SMR. Projection horizon = 10 Projection horizon = 20 Projection horizon = 30. MAPE (%) 50 75. 學. 2-5. ‧ 國. 0. 25. MAPE (%). 100. Population 500,000. 25. 30. 5. 125. 15 20 Baseline Period (years). 10. 15 20 Baseline Period (years). 100. Population 500,000. 25 0. 10. MAPE (%) 50 75. 5. 5. 2-6. 10. 15 20 Baseline Period (years). Whittaker. 80. 25. 30. sx. 25. 30. 30.

(89) 50 40 30. MAPE (%). 0. 0. 20~24. 40~44. 60~64. 80~84. 20~24. 40~44. Testing error, Sx is increasing 50. Testing error, Sx = 1.0. 40 30 20 0. ‧ 國. 10. 20 10 0. 學. 20~24. MAPE (%). 40 30. 政治大. 立 40~44. 60~64. 80~84. 0~4. 20~24. 40~44. Lee-Carter. sit. y. Nat. er. 50 40 MAPE (%). No Graduation Weighted Geometric Mean Whittaker Ratio Coherent LC. i Un. v. 0. 0. 10. 20. engchi. 10. Ch. 30. 40. n. al. 20. io. 50. Training error, Sx is increasing. No Graduation Weighted Geometric Mean Whittaker Ratio Coherent LC. 30. 80~84. 5. →. Training error, Sx = 1.0. 20~24. 40~44. 60~64. 80~84. 0~4. 20~24. 40~44. 60~64. Age group. Testing error, Sx = 1.0. Testing error, Sx is increasing. 30. 40. No Graduation Weighted Geometric Mean Whittaker Ratio Coherent LC. 10. MAPE (%). 80~84. 0. 0. 10. 20. 30. 40. No Graduation Weighted Geometric Mean Whittaker Ratio Coherent LC. 50. Age group. 50. 0~4. 20. MAPE (%). 60~64. Age group. ‧. 3-1. 80~84. No Graduation Weighted Geometric Mean Whittaker Ratio Coherent LC. Age group. MAPE (%). 60~64. Age group. No Graduation Weighted Geometric Mean Whittaker Ratio Coherent LC. 0~4. 0~4. Age group. 50. 0~4. MAPE (%). No Graduation Weighted Geometric Mean Whittaker Ratio Coherent LC. 10. 20. 30. 40. No Graduation Weighted Geometric Mean Whittaker Ratio Coherent LC. 10. MAPE (%). Training error, Sx is increasing. 20. 50. Training error, Sx = 1.0. 0~4. 20~24. 40~44. 60~64. 80~84. 0~4. Age group. 3-2. Lee-Carter. 20~24. 40~44 Age group. →. 81. 10. 60~64. 80~84.

(90) 50 40 30. MAPE (%). 0. 0. 20~24. 40~44. 60~64. 80~84. 0~4. 20~24. 40~44. Testing error, Sx = 1.0. Testing error, Sx is increasing 50. Age group. 30. 40. No Graduation Weighted Geometric Mean Whittaker Ratio Coherent LC. 10. 10. 政治大. 40~44. 0. 立. 0. 20~24. 60~64. 80~84. 0~4. 20~24. 40~44. 3-3. Lee-Carter. ‧. Training error, Sx is increasing. sit. 30. er. i Un. No Graduation Weighted Geometric Mean Whittaker Ratio Coherent LC. v. 0. 10. engchi. 0. Ch. y. 50 40. n. MAPE (%). 40 30. io. al. 20. 50. Nat. No Graduation Weighted Geometric Mean Whittaker Ratio Coherent LC. 20. 80~84. 5. →. Training error, Sx = 1.0. 10. 60~64. Age group. 學. ‧ 國. Age group. 20~24. 40~44. 60~64. 80~84. 0~4. 20~24. 40~44. 60~64. Age group. Testing error, Sx = 1.0. Testing error, Sx is increasing 50. Age group. No Graduation Weighted Geometric Mean Whittaker Ratio Coherent LC. 40 MAPE (%). 80~84. 10 0. 0. 10. 20. 30. 40. No Graduation Weighted Geometric Mean Whittaker Ratio Coherent LC. 30. 50. 0~4. 20. MAPE (%). 80~84. 20. MAPE (%). 20. 30. 40. No Graduation Weighted Geometric Mean Whittaker Ratio Coherent LC. 0~4. MAPE (%). 60~64. Age group. 50. 0~4. MAPE (%). No Graduation Weighted Geometric Mean Whittaker Ratio Coherent LC. 10. 20. 30. 40. No Graduation Weighted Geometric Mean Whittaker Ratio Coherent LC. 10. MAPE (%). Training error, Sx is increasing. 20. 50. Training error, Sx = 1.0. 0~4. 20~24. 40~44. 60~64. 80~84. 0~4. Age group. 3-4. Lee-Carter. 20~24. 40~44 Age group. →. 82. 10. 60~64. 80~84.

(91) 3-1. Lee-Carter. MAPE. (Training Error) Lee-Carter. Partial SMR + Lee-Carter. Whittaker Ratio + Lee-Carter. Coherent LC. 8.4% 10.8%. 12.2% 15.4%. 6.2% 9.7%. 7.9% 7.8%. Lee-Carter Coherent LC. (Testing Error) Lee-Carter. Partial SMR + Lee-Carter. Whittaker Ratio + Lee-Carter. Coherent LC. 19.0% 21.9%. 19.3% 22.3%. 15.5% 17.2%. 17.6% 18.6%. sx. 10. 立. 學. ‧ 國. MAPE. Whittaker Ratio + APC. 21.1% 25.2%. 11.8% 16.2%. 5.4% 11.0%. io. n. al. Partial SMR + APC. Whittaker Ratio + APC. i v35.9% n C h 40.6% e n g c h i U 38.4% 43.4% sx. 83. 21.1% 25.0%. sit. (Testing Error). Coherent APC. y. ‧. Partial SMR + APC. 46.5% 49.6% 10. APC. APC. APC APC Coherent APC. 3-2. (Training Error). Nat. APC Coherent APC. 政治大. er. Lee-Carter Coherent LC. Coherent APC 44.0% 47.2%.

(92) 25-29 year-old. -6.4 -6.6. Mortality (log scale). -6.8. -8.0 -8.5 -9.0. Raw data (training) Raw data (testing) Partial SMR Whittaker Ratio. -7.0. -9.5. Mortality (log scale). -7.5. -6.2. 5-9 year-old. 1985. 1990. 1995. 2000. 2005. 2010. 1980. 1990. 45-49 year-old. 65-69 year-old. 2010. 2005. 2010. 2005. 2010. 2005. 2010. -3.8. 2005. 1990. -3.9 -4.0 -4.2. ‧ 國. -5.2. 1985. -4.1. Mortality (log scale). -5.1. -5.0. 政治大. 1995. 2000. 2005. 2010. 1980. 1985. 1990. 4-1. sit. y. Nat. er. -7.4. Mortality (log scale). -8.5. engchi. 1990. 1995. 2000. 2005. 2010. i Un. v. Raw data (training) Raw data (testing) Partial SMR Whittaker Ratio. -8.2. -9.0. Ch. -7.8. al. -7.0. 25-29 year-old. n. -8.0. io. 1980. 1985. 1990. 1995. 2000. Year. Year. 45-49 year-old. 65-69 year-old. -4.5 -4.7 -4.9. Mortality (log scale). -4.3. -6.2 -6.1 -6.0 -5.9 -5.8 -5.7. 1985. 2000. Year. 5-9 year-old. 1980. 1995. ‧. Year. Mortality (log scale). 2000. 學. 1980. Mortality (log scale). 1995 Year. 立. -5.3. Mortality (log scale). 1985. Year. -4.9. 1980. 1980. 1985. 1990. 1995. 2000. 2005. 2010. 1980. Year. 1985. 1990. 1995 Year. 4-2 84. 2000.

(93) 25-29 year-old. -5.8 -6.2. Mortality (log scale). -7.0 -7.5. Raw data (training) Raw data (testing) Partial SMR Whittaker Ratio. -6.6. -8.0. Mortality (log scale). -5.4. 5-9 year-old. 1985. 1990. 1995. 2000. 2005. 2010. 1980. 1990. 1995. 45-49 year-old. 65-69 year-old. 2005. 2010. 2005. 2010. 2005. 2010. 2005. 2010. Mortality (log scale). -4.4 -4.5. ‧ 國. 1990. -3.7 -4.1. -4.7. 1985. -3.9. -4.6. 政治大. 1995. 2000. 2005. 2010. 1980. 學. 1980. 1985. 1990. 1995. Year. ‧. 4-3. y. Nat. sit er. -6.5 -7.0. -8.0. engchi. -9.0. 1990. 1995. 2000. 2005. 2010. v. 1980. 1985. 1990. 1995. 2000. Year. Year. 45-49 year-old. 65-69 year-old. -4.0 -4.2 -4.4. Mortality (log scale). -4.8. -4.6. -5.2 -5.4 -5.6 -5.8. Mortality (log scale). -3.8. 1985. i Un. Raw data (training) Raw data (testing) Partial SMR Whittaker Ratio. -8.0. -8.5. Ch. -7.5. al. Mortality (log scale). -7.0. 25-29 year-old. n. -7.5. io. 1980. 2000. Year. 5-9 year-old. Mortality (log scale). 2000. Year. 立. -4.8. Mortality (log scale). 1985. Year. -3.5. 1980. 1980. 1985. 1990. 1995. 2000. 2005. 2010. Year. 1980. 1985. 1990. 1995 Year. 4-4 85. 2000.

(94) Male. Female. 1.5. 2.0. 1980~1982 1983~1985 1986~1988 1989~1991 1992~1994. 0.5. 1.0. Ratio. 1.5 0.5. 1.0. Ratio. 2.0. 1980~1982 1983~1985 1986~1988 1989~1991 1992~1994. 0~4. 20~24. 40~44. 60~64. 80~84. 0~4. 20~24. 40~44. Age Group. 4-5. 3.0 2.5 2.0. 60~64. Ch. 80~84. 0~4. er. sit. 1.0 0.5. al. n. 4-6. y. 1.5. Ratio. ‧ 國. 3.0 2.5 2.0 1.0 0.5. io 40~44. Age Group. v ni. 20~24. e n1980~1994 gchi U. Male. 60~64. 80~84. 1.5 1.0 0.5. 1.0. 1.5. Ratio. 2.0. 2.5. 3.0. 1980~1982 1983~1985 1986~1988 1989~1991 1992~1994. 2.0. 3.0. 1980~1982 1983~1985 1986~1988 1989~1991 1992~1994. 2.5. 40~44. Age Group. Female. 0.5. Ratio. 1980~1982 1983~1985 1986~1988 1989~1991 1992~1994. ‧. 1.5. 1980~1982 1983~1985 1986~1988 1989~1991 1992~1994. Female. 學. Ratio. 立. 政治大. Nat 20~24. 80~84. 1980~1994. Male. 0~4. 60~64. Age Group. 0~4. 20~24. 40~44. 60~64. 80~84. 0~4. Age Group. 4-7. 20~24. 40~44 Age Group. 1980~1994. 86. 60~64. 80~84.

(95) Male. Female. 1.0. 1.5. Ratio 2.0. 2.5. 3.0. 1980~1982 1983~1985 1986~1988 1989~1991 1992~1994. 0.5. 0.5. 1.0. 1.5. Ratio 2.0. 2.5. 3.0. 1980~1982 1983~1985 1986~1988 1989~1991 1992~1994. 0~4. 20~24. 40~44. 60~64. 80~84. 0~4. 20~24. Age Group. 40~44. 立. Female. 3.0. sit. engchi. i Un. v. 0.5. 0.5. Ch. 1.0. n. er. 1.5. io. 1.5. Ratio 2.0. Nat. al. 1980~1982 1983~1985 1986~1988 1989~1991 1992~1994. y. 2.5. 3.0. 1980~1982 1983~1985 1986~1988 1989~1991 1992~1994. ‧. 2.5. 學. ‧ 國. Male. 1.0. 80~84. 治政 1980~1994 大. 4-8. Ratio 2.0. 60~64. Age Group. 0~4. 20~24. 40~44. 60~64. 80~84. 0~4. Age Group. 4-9. 20~24. 40~44 Age Group. 1980~1994. 87. 60~64. 80~84.

(96) 4-1. Lee-Carter. 0-14 Partial SMR Whittaker Ratio Coherent LC. 15-64. 16.0 26.3 21.8 19.4. 65. 6.2 7.6 5.9 6.6. MAPE (%) +. 0-14. 9.0 7.9 7.1 6.3. 15-64. 25.6 25.4 23.9 17.7. 12.1 10.6 11.2 9.9. 65. +. 7.9 5.0 5.9 5.5. MAPE. 4-2. Lee-Carter. 0-14. 立. ‧ 國. 32.4 42.1 31.4. 10.0 19.5 6.2. 54.7 82.5 53.5. 65. +. 61.4. 29.6. 10.4 21.3 16.7. 22.6 33.0 10.5. MAPE. ‧. al. n. 7.4 10.7 11.9 11.0. 4.1 7.0 4.0 4.4. Ch. 65. +. 2.8 4.2 2.7 4.5. sit. 15-64. 0-14. er. io. 0-14. MAPE (%). y. Lee-Carter. Nat. 4-3. Partial SMR Whittaker Ratio Coherent LC. 4.7 19.4 4.8. 15-64. 學. 15. 65 + 0-14 治政 41.8 28.9 512.1 大. 15-64. 28.8 Partial SMR Whittaker Ratio Coherent LC. MAPE (%). engchi U. 15-64. v ni. 8.9 19.9 16.2 15.4. 6.8 10.0 4.9 3.4. 65. +. 3.2 6.2 4.3 4.2. MAPE. 4-4 0-14 Partial SMR Whittaker Ratio Coherent LC. 16.5 19.5 31.9 20.8. Lee-Carter 15-64. 65. 14.5 7.8 11.8 6.8. MAPE (%) +. 22.3 22.2 14.5 4.4. 15. 0-14. 15-64. 28.8 39.2 30.2 40.9 MAPE. 88. 31.0 8.3 18.8 3.9. 65. +. 25.9 17.5 26.1 9.3.

(97) 4-5. Lee-Carter. 0-14 Partial SMR Whittaker Ratio Coherent LC. 15-64. 6.8 9.8 12.0 6.3. 65. 2.5 6.1 3.0 2.7. MAPE (%) +. 4.8 4.3 4.1 3.4. 0-14. 15-64. 10.0 15.2 9.3 6.6. 9.0 9.1 4.2 5.4. 65. +. 6.7 5.9 5.1 3.8. MAPE. 4-6. Lee-Carter. 0-14. 15-64. 17.4. 立. ‧ 國. 15-64. 治 97.9 政 20.4 大38.6 6.9 15.7. 65. +. 11.0. 43.1. 40.0. 13.5 8.5. 16.8 7.5 8.8. 14.6 17.5 9.7. 15.7 4.1. 35.5 32.6. MAPE. Lee-Carter. MAPE (%). io. 15-64. n. al. 3.5 5.7 3.7 4.2. Ch. 65. +. sit. y. Nat. 0-14. 7.3 7.6 6.4 8.0. ‧. 4-7. Partial SMR Whittaker Ratio Coherent LC. 0-14. 0-14. 15-64. er. 15. 20.9 16.4 19.1. +. 學. Partial SMR Whittaker Ratio Coherent LC. 65. MAPE (%). 3.6 3.5 4.2 3.9. 7.3 6.4 7.8 7.0. n engchi U. 5.0 4.3 3.7 3.4. iv. 65. +. 3.8 2.7 3.1 2.8. MAPE. 4-8 0-14 Partial SMR Whittaker Ratio Coherent LC. 22.7 18.3 21.0 24.4. Lee-Carter 15-64. 65. 9.7 6.4 11.1 5.4. MAPE (%) +. 19.4 14.9 24.6 4.1. 15. 0-14. 15-64. 40.5 32.8 37.8 33.6 MAPE. 89. 23.8 8.0 13.5 8.4. 65. +. 43.2 24.0 23.2 17.7.

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