Transcript Logistic Curve Method (Saturation Method)

POPULATION
FORECASTING LOGISTIC
CURVE
METHOD
Page 1
GROUP MEMBERS
Kush Poorunsing
Aman Sahadeo
Arshaad Jeedaran
Nevin Sunassee
Pamben Moonsamy
Kishan Joorawon
Page 2
Population Forecasting
• Important process in the design of waste
water treatment plants
• Factors affecting change in population:
 Fertility/ Birth rate
 Death rate
 Migration
Page 3
Logistic Method
Saturation population
• Follows some logical mathematical
relationship.
• has an S-shape combining a geometric
rate of growth at low population with a
declining growth rate as the population
approaches some limiting value.
Population
Decreasing
rate
Increasing
rate
Low rate
Years
Page 4
Equation - 1
• 𝑃𝑡 =
𝑃𝑠𝑎𝑡
1+𝑒 (𝑎+𝑏∆𝑡)
………………. (1)
• Where
Pt : population at some time in the future
Psat: population at saturation level
t: number of years after base year
a, b: data constants
Page 5
Equation - 2
• 𝑃𝑠𝑎𝑡 =
2𝑃0 𝑃1 𝑃2 − 𝑃12 (𝑃0 + 𝑃2 )
𝑃0 𝑃2 − 𝑃12
……………….. (2)
• Where
Psat: population at saturation level
Po: base population
P1 , P2 : population at time periods
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Equation - 3
• 𝑎 = ln(
• 𝑏=
𝑃𝑠𝑎𝑡 − 𝑃0
𝑃0
) …………………. (3)
1
𝑃0 (𝑃𝑠𝑎𝑡 − 𝑃1 )
ln
………………(4)
𝑛
𝑃1 (𝑃𝑠𝑎𝑡 − 𝑃0 )
• Where
n: time interval between successive
censuses
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Example
Census year
Population
1851
495
1861
693
1901
3162
1911
4829
1921
6584
1931
8202
1944
11885
1952
13333
1962
18335
1972
24769
1983
33082
1990
34204
2000
35779
• The data for population
censuses from
1851 to 2000 for
1871
1108
Island of Rodrigues
Estimate the
1881 is given below.1431
expected population
1891 in 2011.
2068
Page 8
STEP 1
Census year
Population
1851
495
1861
693
1871
1108
1921
6584
1931
8202
1944
11885
1952
13333
1962
18335
1972
24769
1983
33082
1990
34204
2000
35779
1881
1431
• Three successive
years, represented
by to,
2068
t1, and t2 are1891
chosen
such
that
they
are
1901
3162
equidistant from
each other.
1911
4829
t0
t1
t2
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Logistic Curve Plot
40000
35000
30000
Population
25000
20000
15000
10000
5000
0
1840
1860
1880
1900
1920
1940
Census year
1960
1980
2000
2020
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STEP 2
Census year
Population
1851
495
1861
693
1871
1108
1911
4829
1921
6584
1931
8202
1944
11885
1952
13333
1962
18335
1972
24769
1983
33082
1990
34204
2000
35779
• The number1881
of years from 1431
to to t1 and t1 to
2068
t2 is denoted1891
by
n.
1901
3162
t0
n=28
t1
n=28
t2
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STEP 3
Census year
Population
1851
495
1861
693
1871
1108
1921
6584
1931
8202
1944
11885
1952
13333
1962
18335
1972
24769
1983
33082
1990
34204
2000
35779
1881 figures corresponding
1431
• The population
to to,
2068
t1, and t2 are1891
P
,
P
and
P
respectively
o
1
2
1901
3162
through which
is to pass.
1911the logistic curve
4829
t0
n=28
t1
n=28
t2
P0
P1
P2
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CALCULATIONS - 1
• n=28
• P0= 11885, P1= 24769 and P2= 35779
• From equation 2, 𝑃𝑠𝑎𝑡 =
• ∴ 𝑃𝑠𝑎𝑡 =
2𝑃0 𝑃1 𝑃2 − 𝑃12 (𝑃0 + 𝑃2 )
𝑃0 𝑃2 − 𝑃12
2(11885)(24769)(35779) −247692 (11885 +35779)
(11885)(35779) − 247692
= 43431.1
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CALCULATIONS - 2
• From equation 3, 𝑎 = ln(
𝑃𝑠𝑎𝑡 − 𝑃0
𝑃0
)
43431.3 −11885
ln(
)
11885
• ∴𝑎=
= 0.9761
• From equation 4, 𝑏 =
• ∴𝑏=
1
𝑃0 (𝑃𝑠𝑎𝑡 − 𝑃1 )
ln
𝑛
𝑃1 (𝑃𝑠𝑎𝑡 − 𝑃0 )
1
11885(43431.3 −24769)
ln
28
24769(43431.3 −11885)
= −0.04497
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CALCULATIONS - 3
• Replacing the values obtained in equation
1, the expected population for 2011 is:
• ∆𝑡 = 2011-1944 = 67 years
• 𝑃𝑡 =
43431.3
1+𝑒 (0.9761−0.04497(67))
= 38420.8
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CONCLUSION
• Based on census reports, the population
of Rodrigues in 2011 was 37,922
(Wikipedia)
• Estimated population with logistic method
is 38420.8
• 1.31% deviation from actual value
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Advantages and Disadvantages
• Advantages
 Longer projection period compared to simple
geometric procedures
 Gives a better approximation than other methods.
• Disadvantages
 Mechanical process
 Cannot predict decreasing population
 Demographic and socioeconomic changes not
considered
 Requires a large number of observations covering a
long period
Page 17
References
• Punmia, B.C, Arun, K.J., 2005. Water Supply
Engineering. 2nd ed. New Delhi: Laxmi
Publications
• Annual Digest of Statistics 2011 [Online].
Available from:
http://www.gov.mu/portal/goc/cso/file/AnnualDig
estofStatistics2011.pdf
• Rodrigues [Online]. Available from:
http://en.wikipedia.org/wiki/Rodrigues
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Thank You For Your
Attention
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