# Leslie matrices and women population in the United States of America.

ABSTRACT

This research tests the accuracy of the Leslie matrix, which is a discrete age-structured method that uses fertility and survival rates, as a tool for predicting women population. Based on available data for the year 2000, we have constructed a Leslie matrix that predicts female population in the United States for every five years from the years 2000 to 2020. To test the accuracy of this method, we compare the aforementioned obtained projected data for the year 2010 with the actual data for women population in the United States obtained by the 2010 U.S. Census.

Key words: Leslie matrices, population.

INTRODUCTION

In "Essay on the Principle of Population, Thomas Malthus was the first one to offer a scientific explanation and variables influencing long-term population projections (1). Such pioneering work prompted passing of Census Act 1800, the first national census in Britain that has been collected every decade since. In any projection, dimensions of population are the first to be considered. The first complex dimension accounts for age and sex of the species observed. In demography, a key dimension of population dynamics is the distinction between men and women. To clarify, the population growth almost completely correlates to the fertility rates of females. Thus, just by observing female population and the growth within this subgroup, it is possible to predict population growth.

METHODS AND DATA

In 1945, P. H. Leslie constructed a matrix as a means to project population of different species. The Leslie matrix requires three different categories of data divided into equal age groups: survival rates for each age group of females [s.sub.i], fertility rates for each age group of female population [F.sub.i], and the initial female population for each age group [p.sub.i]. The matrix is of dimension nxn, where n is number of age groups. Data is distributed and calculated as follows: fertility data occupies the first row of the matrix while survival rates occupy the first sub-diagonal. To project population data, the t-th power of the Leslie matrix is multiplied by the initial population vector [x.sub.0], where t is the number of years between the initial and final year, i.e.,

[x.sub.t] = [L.sup.t][x.sub.0],

where L is the matrix

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For all l[less than or equal to]i[less than or equal to]n, let [l.sub.f]=[s.sub.0][s.sub.1] ... [s.sub.i-1] and [m.sub.i] = [F.sub.i-1]/[s.sub.i-1], i.e., [l.sub.i] is the fraction of females surviving from birth to age i and [m.sub.i] is the number of females born, on average, to a female of age i. If [lambda] is a complex eigenvalue of the matrix L, then [lambda] satisfies the discrete Euler-Lotka equation

1 = [n summation over (i=1)][[lambda].sup.-i][l.sub.i][m.sub.i].

For further information, see (2), (3) and (4).

We used Census-based data for the year 2000 (see (5)) and applied the Leslie matrix model to project population for the next two consecutive 5-year intervals by using fertility and survival rates from (6). To compare the effectiveness of the Leslie matrix, we also used simulated data provided in (7), which uses previous projections and modified data to predict population. Moreover, we also analyze the spectrum (i.e., the set of eigenvalues) of each of the Leslie matrices obtained by using both sets of data, with the goal to compare our results with the conclusions of the Perron-Frobenius Theorem (see e.g. (3)), which claims that any matrix with nonnegative real entries has a unique largest real eigenvalue. The Leslie matrices, their eigenvalues and the predictions for each set of data were obtained by using the software MATHEMATICA 8.

RESULTS

Projections for the year 2010 obtained by using the Leslie matrix constructed from fertility and survival rates in Table I versus the data provided for female population in (5) is shown in Figure 1. To better compare actual versus projected values, Figure 2 shows how unreliable the data is by presenting the percent error per age group. In comparison, simulated data published in (7) gives better results, as shown in Table II. This data has been modified to the anticipated projections and thus result in better accuracy. In Figures 3 and 4, the complex eigenvalues of the Leslie Matrices corresponding to each set of data are shown. We can see that the Leslie matrices corresponding to the data in Table I and Table II, respectively, have a real eigenvalue that is larger than the others, which satisfies the conclusion of the Perron-Frobenius Theorem. However, the magnitudes of the eigenvalues corresponding to the Leslie matrix obtained from Table II are very similar as shown in Figure 4, whereas the eigenvalues of the Leslie matrix obtained from Table I are significatively different as shown in Figure 3.

DISCUSSION

Based on our results, we believe Leslie matrix projections are not reliable for the age groups 0-10 and age groups over 85 years old, but are reliable for the rest of the age groups. This claim is supported by our percent error calculations, which show maximum error in projection to be 8%, whereas at extremes it exceeds well above 50%. Furthermore, by the behavior of eigenvalues, we can see that available data is not enough to accurately predict the population. In these age categories, we believe this is caused by inaccurate data in fertility rates in the 0-10 and over 60 age groups. While the rates may be very small, they are far from insignificant and result in such extreme deviations of projections from accurate data. Simulated data used in (7), give more accurate projections, which leads to conclusion that not all data is true and/or available for age groups under 10 years old. When data for these age groups is manipulated, the results are closer to the true value, which is confirmed the data in Table II. Moreover, we have concluded that in order to give better predictions using manipulated data, the available information has to be manipulated in such a way that the corresponding Leslie matrix has complex eigenvalues whose magnitudes are very close to the magnitude of the unique largest real eigenvalue.

REFERENCES

(1.) Lutz W and Samir K: Dimensions of global population projections: What do we know about future population trends and structures? Philos. Trans. R. Soc. Lond. B Biol. Sci., 365(1554): 2779-2791, 2010. URL: http://www.cdc.gov/nchs/data/nvsr/nvsr54/nvsr54_14.pdf [cited August 21, 2012].

(2.) Caswell H: "Matrix Population Models: Construction, Analysis, and Interpretation." Sinauer Associates, 2nd edition, 2001.

(3.) Kot M: "Elements of Mathematical Ecology." Cambridge University Press, p377, p386, 2001.

(4.) Leon S J: "Linear Algebra with Applications." Prentice Hall, 8th edition, 2010.

(5.) Interim State Population Projections, 2005. Technical report, U.S. Census Bureau, Population Division, January 2012. URL: http://www.census.gov/population/www/projections/files/Summary-TabC3.pdf [cited August 21, 2012].

(6.) Sandy S F and Bindra K: Vital statistics of California, Table 1-5 2003. URL: http://s3.amazonaws.com/zanran_storage/www.cdph.ca.gov/ContentPages/34385493.pdf#page=20 [cited August 21, 2012].

(7.) Child Trends Data Bank: Birth and fertility birth and fertility rates. Technical report, March 2012. URL: http://www.childtrendsdatabank.org/sites/default/files/79_Birth_Rate.pdf [cited August 21, 2012].

Brittney Nelson, Denise T. Reid, Antonija Tangar and Jose A. Velez-Marulanda *

Valdosta State University, Valdosta, GA 31698

* Corresponding author

javelezmarulanda@valdosta.edu

Fax: (229) 219-1257

This research tests the accuracy of the Leslie matrix, which is a discrete age-structured method that uses fertility and survival rates, as a tool for predicting women population. Based on available data for the year 2000, we have constructed a Leslie matrix that predicts female population in the United States for every five years from the years 2000 to 2020. To test the accuracy of this method, we compare the aforementioned obtained projected data for the year 2010 with the actual data for women population in the United States obtained by the 2010 U.S. Census.

Key words: Leslie matrices, population.

INTRODUCTION

In "Essay on the Principle of Population, Thomas Malthus was the first one to offer a scientific explanation and variables influencing long-term population projections (1). Such pioneering work prompted passing of Census Act 1800, the first national census in Britain that has been collected every decade since. In any projection, dimensions of population are the first to be considered. The first complex dimension accounts for age and sex of the species observed. In demography, a key dimension of population dynamics is the distinction between men and women. To clarify, the population growth almost completely correlates to the fertility rates of females. Thus, just by observing female population and the growth within this subgroup, it is possible to predict population growth.

METHODS AND DATA

In 1945, P. H. Leslie constructed a matrix as a means to project population of different species. The Leslie matrix requires three different categories of data divided into equal age groups: survival rates for each age group of females [s.sub.i], fertility rates for each age group of female population [F.sub.i], and the initial female population for each age group [p.sub.i]. The matrix is of dimension nxn, where n is number of age groups. Data is distributed and calculated as follows: fertility data occupies the first row of the matrix while survival rates occupy the first sub-diagonal. To project population data, the t-th power of the Leslie matrix is multiplied by the initial population vector [x.sub.0], where t is the number of years between the initial and final year, i.e.,

[x.sub.t] = [L.sup.t][x.sub.0],

where L is the matrix

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For all l[less than or equal to]i[less than or equal to]n, let [l.sub.f]=[s.sub.0][s.sub.1] ... [s.sub.i-1] and [m.sub.i] = [F.sub.i-1]/[s.sub.i-1], i.e., [l.sub.i] is the fraction of females surviving from birth to age i and [m.sub.i] is the number of females born, on average, to a female of age i. If [lambda] is a complex eigenvalue of the matrix L, then [lambda] satisfies the discrete Euler-Lotka equation

1 = [n summation over (i=1)][[lambda].sup.-i][l.sub.i][m.sub.i].

For further information, see (2), (3) and (4).

We used Census-based data for the year 2000 (see (5)) and applied the Leslie matrix model to project population for the next two consecutive 5-year intervals by using fertility and survival rates from (6). To compare the effectiveness of the Leslie matrix, we also used simulated data provided in (7), which uses previous projections and modified data to predict population. Moreover, we also analyze the spectrum (i.e., the set of eigenvalues) of each of the Leslie matrices obtained by using both sets of data, with the goal to compare our results with the conclusions of the Perron-Frobenius Theorem (see e.g. (3)), which claims that any matrix with nonnegative real entries has a unique largest real eigenvalue. The Leslie matrices, their eigenvalues and the predictions for each set of data were obtained by using the software MATHEMATICA 8.

RESULTS

Projections for the year 2010 obtained by using the Leslie matrix constructed from fertility and survival rates in Table I versus the data provided for female population in (5) is shown in Figure 1. To better compare actual versus projected values, Figure 2 shows how unreliable the data is by presenting the percent error per age group. In comparison, simulated data published in (7) gives better results, as shown in Table II. This data has been modified to the anticipated projections and thus result in better accuracy. In Figures 3 and 4, the complex eigenvalues of the Leslie Matrices corresponding to each set of data are shown. We can see that the Leslie matrices corresponding to the data in Table I and Table II, respectively, have a real eigenvalue that is larger than the others, which satisfies the conclusion of the Perron-Frobenius Theorem. However, the magnitudes of the eigenvalues corresponding to the Leslie matrix obtained from Table II are very similar as shown in Figure 4, whereas the eigenvalues of the Leslie matrix obtained from Table I are significatively different as shown in Figure 3.

Table I. Results using data from (6). Actual Estimated Population Population Age Fertility Survival 2005 2005 2010 2015 Group rates rates [F.sub.i] [s.sub.i] 5 0 0.993164 9.36507 4.39275 4.19878 4.11921 10 0.000001 0.998622 10.0262 9.30105 4.36273 4.17008 15 0.00005 0.99927 10.0079 10.0124 9.28823 4.35671 20 0.0009 0.998303 9.82889 10.0006 10.0051 9.28145 25 0.0485 0.996493 9.27619 9.81221 9.9836 9.98812 30 0.1123 0.997464 9.58258 9.24366 9.77779 9.94859 35 0.1214 0.996375 10.1886 9.55827 9.22021 9.753 40 0.0941 0.995024 11.388 10.1517 9.52363 9.18679 45 0.0404 0.993177 11.3128 11.3313 10.1012 9.47624 50 0.0079 0.990092 10.2029 11.2356 11.254 10.0323 55 0.0005 0.984612 8.97782 10.1018 11.1243 11.1425 60 0.0005 0.975703 6.96051 8.83967 9.94636 10.9531 65 0.0003 0.960272 5.66882 6.79139 8.6249 9.70469 70 0.0002 0.936197 5.13318 5.44361 6.52158 8.28225 75 0.0001 0.905493 4.95453 4.80567 5.09629 6.10548 80 0.00001 0.860207 4.37136 4.48629 4.3515 4.61466 85 0 0.789186 3.11047 3.76027 3.85914 3.74319 90 0 0.681708 1.91332 2.45474 2.96755 3.04558 95 0 0.537671 0.830206 1.30432 1.67342 2.02301 100 0 0.462403 0.228669 0.446378 0.701297 0.899747 105 0 0.236645 0.040397 0.105737 0.206406 0.324282 Actual Population Age Fertility Survival 2020 2010 Group rates rates [F.sub.i] [s.sub.i] 5 0 0.993164 4.13708 9.88194 10 0.000001 0.998622 4.09105 9.95902 15 0.00005 0.99927 4.16433 10.0973 20 0.0009 0.998303 4.35353 10.7367 25 0.0485 0.996493 9.2657 10.5718 30 0.1123 0.997464 9.9531 10.4663 35 0.1214 0.996375 9.92336 9.9656 40 0.0941 0.995024 9.71764 10.1376 45 0.0404 0.993177 9.14108 10.497 50 0.0079 0.990092 9.41158 11.4995 55 0.0005 0.984612 9.93285 11.3649 60 0.0005 0.975703 10.971 10.1412 65 0.0003 0.960272 10.6869 8.74042 70 0.0002 0.936197 9.31915 6.58272 75 0.0001 0.905493 7.75381 5.03419 80 0.00001 0.860207 5.52847 4.13541 85 0 0.789186 3.96956 3.44895 90 0 0.681708 2.95407 2.34659 95 0 0.537671 2.0762 1.02398 100 0 0.462403 1.08771 0.288981 105 0 0.236645 0.416046 0.044202 Table II. Results using data from (7). Actual Estimated Population Population Age Fertility Survival 2005 2005 2010 2015 Group rates rates [F.sub.i] [s.sub.i] 5 0 0.993164 9.36507 9.15632 9.59477 9.61736 10 0 0.998622 10.0262 9.30105 9.09373 9.52918 15 0.0009 0.99927 10.0079 10.0124 9.28823 9.0812 20 0.0477 0.998303 9.82889 10.0006 10.0051 9.28145 25 0.1097 0.996493 9.27619 9.81221 9.9836 9.98812 30 0.1135 0.997464 9.58258 9.24366 9.77779 9.94859 35 0.0912 0.996375 10.1886 9.55827 9.22021 9.753 40 0.0397 0.995024 11.388 10.1517 9.52363 9.18679 45 0.008 0.993177 11.3128 11.3313 10.1012 9.47624 50 0.5 0.990092 10.2029 11.2356 11.254 10.0323 55 0 0.984612 8.97782 10.1018 11.1243 11.1425 60 0 0.975703 6.96051 8.83967 9.94636 10.9531 65 0 0.960272 5.66882 6.79139 8.6249 9.70469 70 0 0.936197 5.13318 5.44361 6.52158 8.28225 75 0 0.905493 4.95453 4.80567 5.09629 6.10548 80 0 0.860207 4.37136 4.48629 4.3515 4.61466 85 0 0.789186 3.11047 3.76027 3.85914 3.74319 90 0 0.681708 1.91332 2.45474 2.96755 3.04558 95 0 0.537671 0.830206 1.30432 1.67342 2.02301 100 0 0.462403 0.228669 0.446378 0.701297 0.899747 105 0 0.236645 0.040397 0.105737 0.206406 0.324282 Actual Population Age Fertility Survival 2020 2010 Group rates rates [F.sub.i] [s.sub.i] 5 0 0.993164 9.02188 9.88194 10 0 0.998622 9.55161 9.95902 15 0.0009 0.99927 9.51605 10.0973 20 0.0477 0.998303 9.07457 10.7367 25 0.1097 0.996493 9.2657 10.5718 30 0.1135 0.997464 9.9531 10.4663 35 0.0912 0.996375 9.92336 9.9656 40 0.0397 0.995024 9.71764 10.1376 45 0.008 0.993177 9.14108 10.497 50 0.5 0.990092 9.41158 11.4995 55 0 0.984612 9.93285 11.3649 60 0 0.975703 10.971 10.1412 65 0 0.960272 10.6869 8.74042 70 0 0.936197 9.31915 6.58272 75 0 0.905493 7.75381 5.03419 80 0 0.860207 5.52847 4.13541 85 0 0.789186 3.96956 3.44895 90 0 0.681708 2.95407 2.34659 95 0 0.537671 2.0762 1.02398 100 0 0.462403 1.08771 0.288981 105 0 0.236645 0.416046 0.044202

DISCUSSION

Based on our results, we believe Leslie matrix projections are not reliable for the age groups 0-10 and age groups over 85 years old, but are reliable for the rest of the age groups. This claim is supported by our percent error calculations, which show maximum error in projection to be 8%, whereas at extremes it exceeds well above 50%. Furthermore, by the behavior of eigenvalues, we can see that available data is not enough to accurately predict the population. In these age categories, we believe this is caused by inaccurate data in fertility rates in the 0-10 and over 60 age groups. While the rates may be very small, they are far from insignificant and result in such extreme deviations of projections from accurate data. Simulated data used in (7), give more accurate projections, which leads to conclusion that not all data is true and/or available for age groups under 10 years old. When data for these age groups is manipulated, the results are closer to the true value, which is confirmed the data in Table II. Moreover, we have concluded that in order to give better predictions using manipulated data, the available information has to be manipulated in such a way that the corresponding Leslie matrix has complex eigenvalues whose magnitudes are very close to the magnitude of the unique largest real eigenvalue.

REFERENCES

(1.) Lutz W and Samir K: Dimensions of global population projections: What do we know about future population trends and structures? Philos. Trans. R. Soc. Lond. B Biol. Sci., 365(1554): 2779-2791, 2010. URL: http://www.cdc.gov/nchs/data/nvsr/nvsr54/nvsr54_14.pdf [cited August 21, 2012].

(2.) Caswell H: "Matrix Population Models: Construction, Analysis, and Interpretation." Sinauer Associates, 2nd edition, 2001.

(3.) Kot M: "Elements of Mathematical Ecology." Cambridge University Press, p377, p386, 2001.

(4.) Leon S J: "Linear Algebra with Applications." Prentice Hall, 8th edition, 2010.

(5.) Interim State Population Projections, 2005. Technical report, U.S. Census Bureau, Population Division, January 2012. URL: http://www.census.gov/population/www/projections/files/Summary-TabC3.pdf [cited August 21, 2012].

(6.) Sandy S F and Bindra K: Vital statistics of California, Table 1-5 2003. URL: http://s3.amazonaws.com/zanran_storage/www.cdph.ca.gov/ContentPages/34385493.pdf#page=20 [cited August 21, 2012].

(7.) Child Trends Data Bank: Birth and fertility birth and fertility rates. Technical report, March 2012. URL: http://www.childtrendsdatabank.org/sites/default/files/79_Birth_Rate.pdf [cited August 21, 2012].

Brittney Nelson, Denise T. Reid, Antonija Tangar and Jose A. Velez-Marulanda *

Valdosta State University, Valdosta, GA 31698

* Corresponding author

javelezmarulanda@valdosta.edu

Fax: (229) 219-1257

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Author: | Nelson, Brittney; Reid, Denise T.; Tangar, Antonija; Velez-Marulanda, Jose A. |
---|---|

Publication: | Georgia Journal of Science |

Article Type: | Report |

Date: | Sep 22, 2013 |

Words: | 2233 |

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