Method AiBi and Logistics for the
projection of small areas: an
application to micro-Angicos - RN
Cristiane Silva Corrêa – CEDEPLAR/UFMG e UFRN
Luana Junqueira Dias Myrrha – CEDEPLAR/UFMG e UFRN
Moema Fígoli – CEDEPLAR/UFMG
Presented at the International Seminar on Population Estimates and Projections:
Methodologies, Innovations and Estimation of Target
Population applied to Public Policies
9-11 November 2011,Rio de Janeiro
Goals
1 Discuss two methods of projection of small
area: the logistic and AiBi
2 Illustrate the application of methods in the
population of micro-Angicos / RN, as the
larger area and the eight municipalities that
make up the large area.
Methods Logistic and AiBi
Projected population of smaller areas as a
function of a larger area.
Método AiBi
n
P(t )   Pi (t )
i 1
Pi (t )  ai P(t )  bi
Pi (t1 )  Pi (t0 )
ai 
P(t1 )  P(t0 )
bi  Pi (t0 )  ai P(t0 )
ai: the coefficient of proportionality of the increased population of smaller
area i relative to the increase of the population of the larger area;
bi:the linear coefficient of correction
Coefficient ai

The coefficient ai reports the percentage
growth of the larger area which the growth of
the smaller area was responsable, between
two periods
n
a
i 1
i
1
Method AiBi

Each population of smaller area i, at time t, is
a proportion of the population of the larger
area, corrected by a correction factor bi.
Coefficient bi


The coefficient bi is a correction factor that
adjusts the relative distribution of the
populations of small areas at time t, to the
distribution at the beginning of the projection.
bi can be less than 0.
n
b  0
i 1
i
Limits for bi:

Considering that ai can only assume
values ​between 0 and 1, we can define what
are the possible limits ​for bi:
lim ai 0 bi  lim ai 0 Pi (t )  ai P(t )  Pi (t )
lim ai 1 bi  lim ai 1 Pi (t )  ai P(t )  Pi (t )  P(t )
Method Limitation

It is not consistent when the population
growth of the larger area, and the smaller
areas, have opposite directions.
Solution to Method Limitation


Divide the larger area into two subgroups

smaller areas growing population

smaller areas that decrease in population.
Problem: result affected by the criterion
chosen to divide the population of the larger
area.
Another solution: estimate Φi
Pi (t )  ai P(t )  bi
Pi (t ) bi  ai P(t )

P (t )
PT (t )
i  ai  bi / P(t )

The relationship
between Pi (t) and P(t)
changes from linear to
hyperbolic.

No more modeled Pi(t)
but Φi

0< Φi<1.
New interpretation of the coefficients - bi

The sign of bi indicates the concavity of the
hyperbola which gives the relationship between Pi
(t) and P (t):



bi> 0, the hyperbola is concave and Φi (t) is related to P(t)
in decreasing order;
bi <0, the hyperbola is convex and the relationship between
Φi (t) and P(t) is positive.
The possibility of having negative and positive
relationships between population of small and large
areas is the main advantage of modeling Φi(t)
instead of modeling Pi(t) as the method AiBi.
New interpretation of the coefficients - ai

Ai: the limit of Φi
when P(t) tends to
infinity.
Limites para Φi

Analyzing
lim P ( t )0 i e lim P ( t ) i
Tabela 5 – Limites de Φi para cada combinação de valores de ai e bi.
ai
bi
>0
>0
<0
<0
<0
>0
<0
>0
Φi
Limite Inferior (L1 )
0
ai
0
0
Limite Superior (L2)
ai
1
0
1
Logistics function

The logistic function has the advantage of
establishing a limit to growth
 i (t )  L1 

L2  L1
1  exp{   (t1  t0 )}

1   L2   i (t1 ) 

   
ln

t1  t0    i (t1 )  L1 

 L   (0) 
i

  ln  2

(
0
)

L
1 
 i
Application of the Method




Micro Angicos - RN
8 municipalities: Afonso Bezerra, Angicos,
Wind River Caiçara Fernando Pedroza,
Garden Angicos, Plates, Black Stone, Pedro
Avelino.
Known data: Population census in 1991 and
2000.
Total population in 2010 estimated by the
exponential growth rate between 1991 and
2000.
Tabela 7 - Observed population and the population estimated by Ai Bi abd
Logistics method, by municipality ,Angicos - RN, 2010
Município
Afonso Bezerra
Angicos
Caiçara do Rio do Vento
Fernando Pedroza
Jardim de Angicos
Lajes
Pedra Preta
Pedro Avelino
Total
População
Recenseada 2010
10844
11549
3308
2854
2607
10381
2590
7171
51304
População Estimada
AiBi
11022
11731
3158
2414
2938
10224
3006
5219
49711
Fonte: Atlas do Desenvolvimento Humano do Brasil e Censo Demográfico, 2010.
Logístico
10457
11203
2694
2641
2510
8868
2709
8628
49711
Conclusion

For the micro-Angicos - RN, the logistic method
yielded better results.
Método
AiBi
Logístico
Soma dos Quadrados dos Erros
4.398.982
2.335.101
Referências:




ATLAS DE DESENVOLVIMENTO HUMANO NO BRASIL. Rio de
Janeiro, PNUD, IPEA, Fundação João Pinheiro, 2003.
FRIAS, Luiz Armando de M. Projeções da população residente e do
número de domicílios particulares ocupados por situação urbana e
rural, segundo as unidades da Federação no período 1985-2020 In:
WONG, Laura R; HAKKERT, Ralph; LIMA, Ricardo(Org) Futuro da
população brasileira: projeções, previsões e técnicas Embu, São Paulo:
ABEP, p148-172, 1987.
IBGE, Censo Demográfico 2010. Disponível em
<http://www.ibge.gov.br/home/>, acesso dia 03 de junho de 2011.
WALDVOGEL, B.C. Técnicas de projeção populacional para o
planejamento regional. Belo Horizonte, CEDEPLAR, 1998.
Cristiane Silva Corrêa
[email protected]