COMPARISON BETWEEN OPTIMAL SEARCHING ALGORITHM VERSUS KNIFE-EDGE CUTTING METHOD FOR CONVERSION OF LENGTH DISTRIBUTION TO AGE COMPOSITION

SCRS/003/077 Col. Vol. Sci. Pap. ICCAT, 56(4): 1417-147 (004) COMPARISON BTWN OPTIMAL SARCHING ALGORITHM VRSUS KNIF-DG CUTTING MTHOD FOR CONVRSION OF LNGTH DISTRIBUTION TO AG COMPOSITION Yin Chang 1 and Shean-Ya Yeh SUMMARY An Optimal Searching Algorithm (OSA) for converting length distribution into age composition was provided and compared to the commonly used Knife-edge Cutting Method (KCM) of the same task. Simulative populations, which composed of individual fork length with age associated, were used in this study. The results so far obtained indicated that: (1) The KCM consistently under-estimate the most abundant age class. () The KCM also tends to over-estimate the age classes between the most dominant classes. (3) We suggested the minimum sampling rate for applying OSA was 0.5%. (4) Above the minimum sampling rate we suggested, the resultant age composition obtained by the OSA has much smaller least square value (as compares to the known age composition) than those obtained by using KCM. RÉSUMÉ Un algorithme de recherche optimal (OSA) conçu pour convertir la distribution de taille en composition démographique a été fourni et comparé à la méthode de découpage en arête vive (KCM) de la même tâche. La présente étude a utilisé des populations simulées, composées de longueur-fourche individuelle associée à l âge. Les résultats obtenus à ce our ont indiqué que : (1) La KCM sous-estime invariablement la classe d âge la plus abondante. () La KCM a également tendance à surestimer les classes d âge parmi les classes dominantes. (3) Nous avons suggéré que le taux d échantillonnage minimum pour appliquer l algorithme de recherche optimal était de 0,5%. (4) Au-delà du taux d échantillonnage minimum que nous avons suggéré, la composition démographique résultante obtenue par l OSA a des valeurs de moindres carrés bien plus faibles (par rapport à la composition démographique connue) que celles obtenues en utilisant la KCM. RSUMN Se proporcionó un algoritmo óptimo de búsqueda (OSA) para convertir la distribución por tallas en composición por edades y se comparó con el método de separación de edades filo de cuchillo (KCM) de la misma tarea comúnmente utilizado. n este estudio se utilizaron poblaciones simuladas, compuestas de longitud a horquilla individual asociada con la edad. Los resultados obtenidos hasta el momento indicaban que: (1) l KCM subestima constantemente la clase de edad más abundante; () el KCM tiende también a sobreestimar las clases de edad en lo que se refiere a las clases dominantes; (3)sugerimos que la tasa mínima de muestreo para aplicar el OSA sea de 0,5%; (4) por encima de la tasa mínima de muestreo sugerida, la composición por edad resultante obtenida por el OSA presenta unos valores de cuadrados mínimos muy inferiores (si se compara con la composición por edades conocida) a los obtenidos utilizando el KCM. KY WORDS Size-age conversion, Optimal Searching Algorithm, Knife-edge Cutting Method 1 Institute of Oceanography National Taiwan University. -mail: d8541004@ntu.edu.tw Institute of Oceanography National Taiwan University. -mail: sheanya@ntu.edu.tw 1417

1 Introduction Fisheries biologists have utilized the information from commercial catches in their research for a long time. Usually, the information of commercial catches is presented as size-frequency distributions. We need to convert the Catch-at-size data to Catch-at-age data if any age-structured stock assessment model is intending to be applied. For estimating age composition from Fork Length () distribution, statistical methods could be more accurate than the deterministic methods of dissecting a mixed distribution (Fournier et al., 1998). In this study, both Optimal Searching Algorithm (OSA) and Knife-edge Cut Method (KCM) were used to convert simulative distribution (the individual age of was known) to age composition to compare the accuracy of the two methods. Materials and Methods.1 The simulative distribution The Figure 1 is the flow chart of this paper. In this paper, we first set two age compositions (in number), the first one was a unimodal distribution and the second one was a bimodal distribution. The numbers of the two settings were 50,000. Figure was the histograms of the age composition settings. The next step was to generate the simulative (with age associated). In this procedure, we first assumed that the distribution of an age class was normal distributed. According to the growth equation provided by Lee & Yeh (1993), the mean at age can therefore be calculated by the growth equation: i 0.1454 (i+ 0.647) = 14.8 (1 e ) eq1. Where: i : The mean at age i. i : age. Based on the original ring reading data of South Atlantic albacore spines (Lee et al., 1993), the standard deviation of at age ( ) ranged about from 3 to 5 cm (Table 1). It is therefore select to be 3.0, 4.0 and 5.0cm to generate the simulative associated with age i by the Inverse of the Normal Cumulative Distribution Function (INCDF): 1 i = F (P i, ) = { : F(P i, ) = P} eq. Where: P is probability, i is the simulative associated with age i and (t i ) i 1 P = ( i i, ) = e dt eq3. π The result i was the solution of the integral equation above where you supply the desired P. In this study, the P is generated by the evenly distributed random number greater in the Microsoft Visual Basic. The whole of i was our simulative population. i Throughout randomized the sequence of i in the simulative population and then random sampled the by different sampling rate (0.%, 0.5%, 1.0%,.0% and 5.0%) to constitute the sample, and the age composition associated to the sample was the TRU age composition. We transferred the sample into frequency distribution and re-converted it into Age compositions by KCM and OSA, which we will expatiate as follows.. The Knife-edge Cut Method The frequency distributions of sample were re-converted into age compositions by KCM first. The Cut Point of age i and i+1 ( CP i,i + 1 ) can be calculated by: CP ( 0.1454((i+ 0.5) + 0.674)) = 14.8(1 e ). eq4. i,i+ 1 1418

The 19 Cut Points were listed in Table. We use the symbols and LB to express the Upper Boundary and Lower Boundary of class. If there is no Cut Point within the class and CPi 1,i LB and CPi,i + 1, then the corresponding age of class was i. On the other hands, if there is one Cut Point, CP i,i + 1, within the class, the frequency of class need to divide into age i and i+1 as: CPi,i + 1 LB ) = f ( ) eq5. LB CPi,i + 1 + 1) = f ( ) eq6. LB Where ) and + 1) are the frequencies need to divide into age i and i+1, the f ( ) is the frequency of class. And if there are two Cut Points, CP i,i + 1 and CP i + 1,i +, within the class, the frequency of class need to divide into age i, i+1 and i+ as: CPi,i + 1 LB ) = f ( ) eq7. LB CPi + 1,i+ CPi,i + 1 + 1) = f ( ) eq8. LB CPi + 1,i+ + ) = f ( ) eq9. LB Where ), + 1) and + ) are the frequencies need to divide into age i, i+1 and i+..3 Optimal Searching Algorithm In the OSA method, we first assumed the distribution of each age is normal distributed and the constant for all age. So the theoretical frequency of OSA can be written in the form of: fˆ ( 1 i age ( ) 1 ) e = π MAX ) = eq10. i 1 is Where: was the class value of class, was the mean of age class i, fˆ( ) i was the theoretical frequency of class (in percentage), ) was the estimated frequency of age class i (in percentage), MAX age was the maximum age class, was the estimated, i was the age class index and was the class index. The algorithm of OSA was to search a better age composition, ), and for getting a newer theoretical distribution, which is one step closer to the observed distribution. In so doing, the obective function, OBJ (),of OSA can thus be written as follow: OBJ() N = 1 [ ] fˆ( ) f ( ) = eq11. min OBJ( ) eq1. We used the Microsoft xcel software with a "Solver.xla" Add-In to perform this task. The value of obective function mentioned above will be used as the criteria for continuation of searching. 1419

The algorithm of the Microsoft xcel's "Solver.xla" Add-In is based on the Generalized Reduced Gradient (GRG) algorithm (Abadie et al., 1969). The settings of Solver.xla Add-In for optimal age composition searching are listed in Table 3..4 Compare The age compositions estimated by OSA and KCM were compared with the TRU age composition by Least Square (LS) as: MAX age LS = [ T ) )] eq13 i= 1 Where: T ) was the TRU frequency of age i and ) was the stimated frequency (by OSA or KCM) of age i. 3 Results 3.1 The Simulative Population The simulative distributions generated from the age composition settings with different were listed in Figure 3. The left part of the Figure 3 was the age composition setting 1 and its associated simulative distributions for =3, 4 and 5cm. The age composition setting 1 was unimodal distribution and the associated simulative distributions were unimodal distribution, too. The mode of the age composition setting 1 was age 6 and the modes of the simulative distributions of =3, 4 and 5cm were 90cm, 88cm and 86cm. The right part of the Figure 3 was the age composition setting and its associated simulative distributions for =3, 4 and 5cm. The age composition setting was a bimodal distribution and the associated simulative distributions were bimodal distribution, too. The modes of the age composition setting were age 4 and 9, and the modes of the 3 simulative distributions of different settings were 7 and 110cm. 3. The Comparison between the TRU and stimated age composition The TRU age compositions and the age compositions estimated by OSA and KCM were list in Table 4a and Table 4b (The Table 4a was the unimodal cases and the Table 4b was the bimodal cases). The TRU and the estimated by OSA were also listed in the tables. The modes of the TRU age compositions in Table 4a were in the class of age 6, almost all modes of age compositions estimated by OSA or KCM were in the same class, too. But it was needed to take notice of the KCM, because the KCM was consistently under-estimate the frequency of the most abundant age class. The modes of the TRU age compositions in Table 4b were in the classes of age 4 and 9, almost all modes of age compositions estimated by OSA and KCM were in the same classes, too. It was also needed to take notice of the KCM, because not only the KCM was still consistently under-estimate the frequencies of the dominant age classes 4 and 9, but also the KCM trended to over-estimate the frequencies between the two dominant age classes. The LS of OSA and KCM were listed in the Table 5. In this table we can find: (1) Within the same sample, the LS of OSA was less than the LS of KCM when the sampling rate was equal or higher than 0.5%. () The minimum LS of OSA was 1.4, occurred in Unimodal setting, =3.0cm and sampling rate = 5.0%. This minimum LS was also the smallest LS in the Table 5. (3) The maximum LS of OSA was 3465.8, occurred in Unimodal setting, =4.0cm and sampling rate = 0.%. This maximum LS was also the largest LS in the Table 5. (4) The minimum LS of KCM was 1., occurred in Unimodal setting, =3.0cm and sampling rate = 0.5%. The maximum LS of KCM was 6.6, occurred in Bimodal setting, =5.0cm and sampling rate = 1.0%. 140

4 Disscussion We had mentioned that the frequencies of dominant age classes estimated by KCM were constantly less than the TRU settings. This may cause underestimated of the dominant age class when applying the KCM. Besides, the KCM also tended to over-estimate the age classes between the abundant one. This may cause overestimated these age classes. We should very careful when applying KCM of converting frequencies into age composition because there is always bias. The population number was 50,000 and the sampling rates used in the study were 0.% (sample size=100), 0.5% (sample size=50), 1.0% (sample size=500),.0% (sample size=1000) and 5.0% (sample size=1000). We have found when the sampling rates below 0.5%, most age compositions estimated by OSA were poor than KCM. On the other way, when the sampling rates above 0.5%, all the age compositions estimated by OSA were better. So our suggestion of the minimum sampling rates for applying OSA was 0.5%. Besides age composition, OSA can also estimate. On the other hand, the KCM was deterministic method, that is, it did not consider the term of the errors when converting the mixed distribution into age composition. For estimating age-group parameters from size-frequency data conventional statistical methods can be more accurate than the commonly used KCM of dissecting a mixed distribution (Doubleday, 198). Nowadays, the mostly common used statistical method is MULTIFAN (Fournier, et al., 1990). MULTIFAN is a likelihood-based method for analyzing a time series of size-frequency data for a fishery. The maor difference between MULTIFAN and OSA is that the OSA can transfer any single size-frequency data into age composition. If a time series of size-frequency data is not available or a fixed time size-frequency is intending to be transferred into age composition, the OSA introduced in this study provide a convenient way for size-age converting. Acknowledgments We are indebted to the Council of Agriculture for the financial support. References ABADI, J. and Carpentier, J. 1969. Generalization of the Wolfe reduced gradient method to the case of nonlinear constraints. Symposium of the Institute of Mathematics and Its Applications, University of Keele, ngland, 1968, dited by Fletcher, R., Academic Press, New York. 37-47. DOLDAY, W. G. 1976. A least square approach to analyzing catch at age data. Int. Comm. Northwest Atl. Fish. Res. Bull. 1: 69-18. FOURNIR, D. A. and Archibald, C. 198. A general theory for analyzing catch at age data. Can. J. Fish. Aquat. Sci. 39: 1195-107. FOURNIR, D. A., Sibert, J. R., Makowski, J. and Hampton. J. 1990. MULTIFAN a likelihood-based method for estimating growth parameters and age composition from multiple length frequency data sets illustrated using data for southern bluefin tuna (Thunnus maccoyii). Can. J. Fish. Aquat. Sci. 47: 301-317. FOURNIR, D. A., Hampton, J. and Sibert, J. R. 1998. MULTIFAN-CL: a length-based, age-structured model for fisheries stock assessment, with application to South Pacific albacore, Thunnus alalunga. Can. J. Fish. Aquat. Sci. 55: 105-116. L, L.K. and Yeh, S. Y. 1993. Studies on the age and growth of South Atlantic albacore (Thunnus alalunga) specimens collected from Taiwanese longliners. ICCAT Col. Vol. Sci. Pap. XL():354-360. 141

Table 1. The size. observed by Lee & Yeh (1993) using traditional ring reading method, the N was the sample Age 3 4 5 6 7 8 9 10 11 1 (cm) 3.31 3.51.87.89 5.03 4.77 4.45 3.94.67 3.08.35 N 6 3 4 13 1 40 8 83 58 7 6 Table. The Cut Point of KCM. Age 1 & & 3 3 & 4 4 & 5 5 & 6 6 & 7 7 & 8 8 & 9 9 & 10 10 & 11 Cut point (cm) 38.6 5.6 64.7 75. 84.3 9.1 98.9 104.8 109.9 114.3 Age 11 & 1 1 & 13 13 & 14 14 & 15 15 & 16 16 & 17 17 & 18 18 & 19 19 & 0 Cut point (cm) 118.0 11.3 14. 16.6 18.7 130.6 13. 133.5 134.7 Table 3. The setting of the Microsoft xcel's "Solver.xla" Add-In for optimal ) searching. Items Setting The maximum calculation time: 3000 seconds The maximum number of iterations: 3000 The precision: 0.00001 The integer tolerance value: % Assume Linear Model: False Assume Non-Negative: False Use Automatic Scaling: True The type of estimates: Quadratic The type of derivatives: Central The type of search: Conugate Gradient The convergence value: 0.00001 Show Iteration Results: False and 14

Table 4a. The comparison between the TRU age composition and the age composition estimated by OSA and KCM. (Unimodal Distribution) 1 0.0 0.1 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.1 3 1.0 1.6 1.0 0.8 1.4 1.8.4.9.6 1.8 1.9.1 1.5 1.7 1.9 4 14.0 17.0 14. 16.8 13.7 15.0 15.0 14.7 13.5 15.6 16.5 15. 15. 15. 14.4 5 33.0 41.1 36. 35.6 35.0 34.4 9.6 9.7 9.1 3.8 3. 31.1 31.7 3.3 30.1 6 43.0 34.6 33.9 35.6 36.1 33.5 43.8 43.1 38.4 39.5 38.5 34.4 40.5 39.6 35.5 7 8.0.3 11.7 10.4 9.8 11. 6.8 7.7 1.9 7.1 7.4 1.4 7.9 8.0 13.6 8 1.0 3.3.9 0.8 3.7 4.0. 1.7.9.3. 3.0.5.3 3. 9 0.0 0.0 0.1 0.0 0.0 0.0 0. 0.1 0.6 0.5 0.7 1.1 0.4 0.7 0.9 10 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0. 0.3 0.4 0. 0.0 0. 11 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 13 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.0 0.0 0.0 14 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 15 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 16 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 17 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.0 0.0 0.0 18 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 19 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 LS 183.4 111.3 19.6 1. 1.9 68.8.5 59. 1.4 60.6 3.0.8 NA 3.0.7 NA 3.0.9 NA 3.0 3.1 NA 3.0 3.1 NA 1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.7 0.0 0.0 0.0 0.0 0. 0.5 0.4 0.1 0.1 0.1 0. 0. 0. 3 4.0 11.9 3.7 0.4 0.5 1.1 1.8 3.5.5 1.8 1.4.3 1.6 1.7.3 4 15.0 10.6 14.5 16.0 13.5 1.6 16.4 17. 14.6 17.4 17.1 16. 16.5 16.8 15.4 5 30.0 68.3 9.9 9. 5.8 7.5 33.0 38. 31.7 8.7 8.3 8.8 33. 33.4 31.3 6 46.0.4 38. 41.6 4.0 36.1 40.0 3.6 3.9 41.3 4.4 34.4 39.1 37.1 3.3 7 3.0 6.1 10.6 10.0 13.6 16. 7.0 6.7 14.9 9.1 8.6 14.3 6.8 8.6 14.1 8.0 0.0.0.0 3.4 5. 1.0 0.0.1 1.4 1.9 3.. 1.1 3.1 9 0.0 0.0 1.1 0.8 0.4 0.6 0.0 0.4 0.3 0. 0.1 0.5 0. 1.0 1.0 10 0.0 0.0 0.0 0.0 0.8 0.7 0.4 0.5 0.4 0.0 0.1 0.3 0.1 0.0 0. 11 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 13 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 14 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 15 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 16 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 17 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 18 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 19 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0 0.0 0.0 0.0 0.0 0.0 0.0 0. 0. 0.0 0.0 0.0 0.0 0.0 0.0 0.0 LS 3465.8 10.9 34.4 95.3 87.3 118.5.3 80.7 9.3 107.3 4.0 4.5 NA 4.0 3.3 NA 4.0 3.5 NA 4.0 3.9 NA 4.0 3.9 NA 1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0. 0.0 0.0 0.0 0.1 0.5 0.3 3 1.0.3.9.8.4 3.8 1.8.0.7 1.6.0 3. 1.6 0.3.4 4 9.0 6.1 5.9 14.4 13.9 16.4 14.8 14.8 16.0 17.4 14.5 15.3 15.7 16.5 15.7 5 31.0 9.4 31.4 31.6 3.5 30.0 31.6 31.3 8.8 30.4 9.4 9.3 3.1 30.0 9. 6 51.0 43.7 39. 36.8 35.4 7.3 41. 39.8 3.7 40.5 37.5 31.3 41.4 43.6 3.6 7 5.0 13. 14.5 8.4 10.8 14.5 8.4 6.6 13.7 7.7 13.5 15.0 6.5 5.6 14.0 8 1.0 0.0.0 5.6 0.0 4.7.0 5.5 4.4 1.9.4 4.5.1 3.0 4.1 9 0.0 4.6 3.0 0.0 5.0 3.1 0. 0.0 1. 0. 0.7 0.9 0. 0.3 1. 10.0 0.6 1.1 0.4 0.0 0. 0.0 0.0 0. 0.3 0.0 0.3 0. 0.0 0.3 11 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.1 1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 13 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 14 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 15 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 16 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0. 0.0 17 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 18 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 19 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 LS 158.8 51.5 65.6 146. 17.7 117.8 5. 154.1 13.5 146. 5.0 3.1 NA 5.0 4.5 NA 5.0 4.1 NA 5.0 4.8 NA 5.0 5.1 NA 143

Table 4b. The comparison between the TRU age composition and the age composition estimated by OSA and KCM. (Bimodal distribution) 1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.0 1.0.0 4.4 5.4 5.4 1.8.0.5 3.1 3. 3.4.7.6 3.1 4 4.0 0.0.0 6.4 4.6 3.9 7.4 7.0 4.7 5.9 5.7 3.9 4.0 4. 1.8 5 14.0 16.6 15.0 14.0 14.1 13.4 1. 11.9 13.0 11.6 11.9 1.7 13.7 13.4 14.0 6 4.0 4.4 4.6 8.0 8.3 9.0 5. 5.5 5.3 7.0 6.3 6.3 7.4 7.6 7.6 7 0.0 0.0 0.4 0.0 0.0 1.8 1. 0.4.3 0.9 1.8.9 0.6 0.3.8 8 13.0 1. 1. 8.8 8.8 9.7 7. 8.7 10.0 9.4 7. 9.1 8.1 7.9 8.4 9 3.0 18.9 17.1 4.8 4.7 17.3 4.4 6.0 19.7 6.4 6.0 0.0 4.8 1.1 17.5 10 16.0 15.1 14.5 10.4 5.3 10.6 17.0 1.3 13.4 13. 14.0 14.0 14.6 18. 15.3 11 4.0 10.0 8.8 1.6 8.8 6.3 1.4 3.6 5.5 1..5 5.1.7 3.7 6.4 1 0.0 1.5 3.5 1. 0.0.1 1.0 1.9.3 0.9 0.0 1.3 0.7 0.0 1.8 13 0.0 0.1 0.0 0.0 0.0 0. 0.4 0.0 0.3 0.0 1. 0.6 0.3 0.5 0.5 14 0.0 0.0 0.0 0.0 0.0 0. 0.4 0.1 0.5 0.1 0.0 0.3 0.0 0.3 0. 15 0.0 0.0 0.0 0.4 0.0 0.0 0. 0.6 0.3 0.1 0.0 0.0 0.0 0.0 0.1 16 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 17 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 18 0.0 0.0 0.0 0.0 0.0 0.0 0. 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 19 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.1 0.0 0.1 0.0 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0. 0.0 0. 0.1 0.0 0.0 0.0 LS 80.4 77.8 83.4 9.3 33.9 71.9 10.9 67.1 8.5 77.5 3.0. NA 3.0.8 NA 3.0.8 NA 3.0.8 NA 3.0 3.1 NA 1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.1 0.0 0.0 0.0 1.0 1.5 1.0 0.0 0.1 0.0 0. 0.7 0.6 0.0 0. 0. 0.0 0.0 0.1 3 5.0 4. 5.9 3.6 3.9 4.7 3.4.4 3.3.6 1.0.6.4.9 3.7 4 3.0 3.7 0.5 7.6 6.4 3.8 6. 9.0 3.7 4.7 7..3 5. 3.8 1.1 5 8.0 8.0 8.6 14.8 13.3 13.3 17.0 13.5 16.0 13.6 11.9 13.7 15.0 15.3 15.3 6 11.0 1.1 13.0 6.0 8.5 9.1 6.0 7.0 7.9 7.1 7.4 7.4 7.0 7.7 7.9 7.0 0.0 1.0 0.0 0.0.7 0.8 1.8 3.7 0. 1.4 3.7 0.6 0.4 3.5 8 7.0 14.5 11.6 8.4 5.4 8.8 7.6 5.8 7.3 7.9 5.0 8.5 8.0 7.9 9.5 9 6.0 16.5 15.6 4.8 7.9 17.5.0 19.1 14.4 6.0. 16. 5.5.3 16.4 10 1.0 11.1 11.7 11.6 10.5 11.7 14.0 15.8 1. 13.9 1.7 14.7 13.3 15.6 13.3 11.0.8 5. 1.6 0.0 4.0 1.6 4.8 7.8.4 0.0 6.9 1.9.8 5.9 1 1.0 5.4 5.0 0.0.4.4 1.0 0.0 1.9 1.4 0.7.3 0.5 0.. 13.0 0.0 0.0 0.4 0.0 0.3 0.0 0.0 0.8 0.0 1.1 0.7 0. 1.0 0.7 14 0.0 0.0 0.8 0.4 0.0 0.3 0.0 0.0 0. 0.0 0.0 0.5 0.1 0.0 0. 15 0.0 0.0 0. 0.8 1.3 0.8 0. 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.1 16 0.0 0. 0.0 0.0 0.3 0.4 0.0 0.3 0. 0.1 0.0 0.0 0.0 0.0 0.1 17 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 18 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 19 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.0 0.0 0.0 LS 177.1 173. 38.7 99.5 49.9 10.3 104.4 136.4 0. 133.0 4.0 3.1 NA 4.0 3.8 NA 4.0 4.1 NA 4.0 4.1 NA 4.0 3.8 NA 1 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.5 0.4 0.0 0. 0. 0.1 0.4 0.5 0.1 0.1 0.3 3.0 7.8 3.9.8 7.1 4.3 4.0 4.4 5.1 3.6.5 4.4 3.1. 4.7 4 17.0 11.6 1.5 3. 3.1 19.3 3.6 0.5 18.7 4. 3.0 0.3 4.6 1.7 18.8 5 13.0 14.0 1.3 14.8 1.7 13.6 13. 17.0 15.8 13.3 1.9 13. 13.6 17.4 15.4 6 4.0 0.6 5.4 9. 4.7 10.4 8.0 5.8 7.3 6.3 8.3 7.8 7.0 5.9 7. 7 0.0 6.9 5.9 1.6 6. 4.3 0.8.5 4.3 0.5 1.4 3.6 0.4 0.4 4.7 8 16.0 14.8 1.5 6.8 7.3 9.3 7.4 7.5 9.1 8.9 5.6 9.9 8.8 8.8 10. 9 5.0 8.6 1.1 0.8 0.4 1.7 7. 18.7 15.0 6.0 4.4 17.3 5.1 0.1 14.5 10 0.0 7. 14.5 17. 1.4 14.0 14.6.0 15.5 14. 17.8 1.6 14. 16. 1.1 11 1.0 6.7 6.6 3. 5.5 8.1 0.8 0.0 5.9. 0.0 6..1 6.9 7.1 1.0 0.0 3.4 0.4 0.0.9 0. 0.0 1.9 0.4 1.7. 0.6 0.0 3.0 13 0.0 0.0 0.6 0.0 0.0 0.7 0. 1.4 1.1 0.1 1.9 1.0 0. 0.0 0.9 14 0.0 1.1 0.4 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.1 0.6 0.1 0.0 0.6 15 0.0 0.7 0.9 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.3 0.0 0.0 0. 16 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 17 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0. 0.0 18 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 19 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 LS 340.9 15.8 93.4 139.9 161.6 6.6 43.8 18.0 77.1 08.4 5.0 3. NA 5.0 3.3 NA 5.0 3.8 NA 5.0 4.8 NA 5.0 5.0 NA 144

Table 5. Compare the LS of OSA and KCM. Unimodal Bimodal 0.% 0.5% 1.0%.0% 5.0% OSA KCM OSA KCM OSA KCM OSA KCM OSA KCM 3.0 183.4 111.3 19.6 1. 1.9 68.8.5 59. 1.4 60.6 4.0 3465.8 10.9 34.4 95.3 87.3 118.5.3 80.7 9.3 107.3 5.0 158.8 51.5 65.6 146. 17.7 117.8 5. 154.1 13.5 146. 3.0 80.4 77.8 83.4 9.3 33.9 71.9 10.9 67.1 8.5 77.5 4.0 177.1 173. 38.7 99.5 49.9 10.3 104.4 136.4 0. 133.0 5.0 340.9 15.8 93.4 139.9 161.6 6.6 43.8 18.0 77.1 08.4 Set Age Composition (in number) Generate (with age associated) by INCDF Cumulate all the generated The simulative population Randomization Procedure Random Sampling with Different Sampling Rate The sample (in frequency distribution) The TRU Age composition associated to the sample OSA Age Composition stimated by OSA Conclusion Compare by Least Square KCM Age Composition stimated by KCM Figure 1. The flow chart of this paper. 145

Figure. The histograms of age composition setting 1 (Unimodal) and (Bimodal). 146

Figure 3. The comparison between the age composition settings and their associated simulative distributions for different settings. 147