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Forecasting Population
Websters definitions: Projection
an estimate of future possibilities based on a current trend Estimate
a rough or approximate calculation; a numerical value obtained from a
statistical sample and assigned to a population parameter
Forecast
to calculate or predict (some future event or condition) usually as a result ofstudy and analysis of available pertinent data
Definitions
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Forecasting: Need
Short-term
How much population would be available for next growth rate (interest)?
Working on proposals without knowing what to expect could result in a waste oftime
Long-term
Planning for long-lasting population assets
Borrowing is usually a long-term commitment
Growth rate accrue in the future
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Forecasting Considerations
Forecaster must understand the local governments population growth
system
How is it administered Tax base structure
Other population sources
Understand the factors that have affected past population growth
Structure/definition
Administrationo Are all people posted?
o All people growth at the same way?
Must have adequate and timely data
o No data is better than false data
Use graphs of variables against time to visualize changes
Are changes small
Are changes large
Are changes seasonal
Are any patterns evident and are these congruent with regional or national
population grpwth pattern?
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Forecasting process should be transparent Identify assumptions
Define assumptions
Avoids under or over-forecasting Individual population sources need to be forecast separately Different population sources respond to different economic and policy factors
Monitor and revise forecasts Review initial forecast to determine source of error and enhance forecasting for
future years
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Forecasting Methods
1. Simplistic Trend extrapolation or projection using historical data
Most common local government population estimation tool2. Multiple regression
Use of IVs to predict populations
3. Econometric Complex multivariate technique using composite measures to estimate populations
4. Microsimulation
Estimates based on sample of relevant data
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Simplistic Models
Assumption
o Past trends will continue
o No major legislative or tax change expected
Future population
o Extrapolated from historical data or previous forecasting
Constant increments
population increased by 5000 person for the past 5 years
Constant percentage change population increased by 5% for the past 5 years
Simple average compounded growth r = (Y / X)^1/n - 1
Linear (R = a + bt) time trends
Nonlinear ( lnR = a + bt)
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Decomposition to Time Series
Breaks the time series into trend, cycle, seasonal (for monthly orquarterly forecasts), and irregular (or residual) components
o The method adjust for four basic elements that contribute to the behavior of aseries over time
S = seasonal factor Regular fluctuations; driven by weather and propriety
T = the adjustment for trend
Long-run pattern of growth or decline C = cycle
Periodic fluctuations around the trend level
I = the irregular or residual influence Erratic change that follows no pattern
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8
Decomposition Model
Rt = (St ) (Tt ) (Ct ) (It )
Sequence for each of filters is as shown in the equation
o R = population to be forecasto S = seasonal
extracted by using a centered seasonal moving average
o T = trend
Adjusted by linear regression against time of the seasonally adjusted data
o
C = cycle Identified by removing the trend from the deseosonalized data
o I = the irregular or residual influence
Isolated by removing the cyclical component from the series
o t = time of the data (historic or forecast)
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Multiple Regression
Estimates population as a function of one or more IVs
o Each equation used to estimate a population source is independent of the others
o Estimates for the independent variables are generated independent of the
regression equation
o The equation with the best goodness of fit is selected
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The Regression Model
The mathematical equation for a straight line is used to predict the valueof the dependent variable (Y) on the basis of the independent variable(X):
Y = a + b1X1 + b2X2 + biXi + e
a is called the Y-intercept. It is the expected value of Y when X=0. This is the base-line amount because it is what Y should be before we take the level of X into
account.b is called the slope (or regression coefficient) for X. This represents the amount that
Y changes for each change of one unit in X
e is called the error term or disturbance term. The difference between actual andpredicted values.
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Econometric Models
Uses a system of simultaneously interdependent equations to predict
population
o The equations are linked by theoretical and empirical relationships
o These models while preferred by economist because of their theoretical soundness,
are in practice not much more accurate than multiple regression models
o Better in predicting macroeconomic variables
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Microsimulation Models
A statistical sample of tax data is used to forecast population from a taxsource
o How the sample is drawn and its updating is criticalo Economic activity expected in the budget year is included in the analysis
o More applicable to estimate how population would be affected by proposed policychanges
o Also useful for regular forecasting
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Factors Influencing the Choice of Forecasting Method
Resources--Money
--Personnel
--Time
Can we afford it?
Face Validity--Availability of Data
--Quality of Data
Are the Inputs Good?
Plausibility
Do the Outputs Make Sense?
Needs of the Users
--Geographic Detail--Demographic Detail
--Temporal Detail
Are User Needs Satisfied?
Model Complexity--Ease of Application
--Ease of Explanation
Can we do this?
Can we explain
what we did?
Political AcceptabilityAre the Outputs Acceptable?
Forecast AccuracyIs the Forecast Accurate?
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Simplistic Models1- Arithmetic increase method :
In this method, the rate of growth of population is assumed to be constant. This
method gives too low an estimate, and can be adopted for forecasting populations oflarge cities which have achieved saturation conditions.
Validity: The method valid only if approximately equal incremental increases have
occurred between recent censuses.
kdt
dp
t
to
pt
po
kdtdp
tkppot
dp/dt : rate of change of population
Pt : population at some time in the future
po: present or initial population
t : period of the projection in decades
k : population growth rate (constant)
Population Projection
Arithmetic increase method
Time (decade)
population
slopet
pk
slopettppko
ot
Note: decade = 10 years
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2- Uniform percentage of increase:
Assumption: This method assumes uniform rate of increase, that is the rate
of increase is proportional to population).
pkdt
dp1
tpt
podtkp
dp
01
tkppot
1lnln
)(lnln 1oot
ttkpp
dp/dt : rate of change of population
Pt : population at some time in the future
Po : present or initial population
: period of the projection in years
k : population growth rate
n : number of years
t
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3- Logistic method : ( Saturation method )
This method has an S-shape combining a geometric rate of growth at low population
with a declining growth rate as the city approaches some limiting population.
A logistic projection can be based on the equation:
tba
satt
e
pp
1
2
12
2
2
121 )(2
ppp
ppppppp
o
oosat
2
2
ln p
ppa
sat
)(
)(ln
1
1
1
osat
sato
ppp
ppp
nb
pt : population at some time in the future
po: base population
psat: population at saturation level
p1 , p2 : population at two time periods
n : time interval between succeeding censuses
: no. of years after base yeart
population
Population Projection
Logistic method
Time (year)
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4- Declining growth method :
This technique, like the logistic method, assumes that the city has some limiting
saturation population, and that its rate of growth is a function of its population deficit:)(2 ppk
dt
dpsat
osat
sat
pp
pp
nk
ln1
2
)1)(( 2 tkosatot epppp
may be determined from
successive censuses and the
equation:2
k
then,
pt : population at some time in the future
po: base population
psat: population at saturation level
p , po : are populations recorded n years apart
: no. of years after base yeart
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5- Curvilinear method (Comparative graphical extention method) :
This technique, involves the graphical projection of the past population growth curve,
continuing whatever trends the historical data indicate. This method includes
comparison of the projected growth to the recorded growth of other cities of largersize. The cities chosen for the comparison should be as similar as possible to the city
being studied.:
STUDY CITY - A BC
DE
A
YEARS
POPU
LATION
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6- Incremental increase method : (Method of varying increment)
In this technique, the average of the increase in the population is taken as per
arithmetic method and to this, is added the average of the net incremental increase,
one for every future decade whose population figure is to be estimated. In thismethod, a progressive increasing or decreasing rate rather than constant rate is
adopted. Mathematically the hypothesis may be expressed as:
pt : population at some time in the future
po: present or initial population
k : rate of increase for each decade
a : rate of change in increase for each decade
n : period of projection in decades
a
nn
knpp ot .2
)1(
.
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7- Geometric increase method :
This method assumes that the percentage of increase in population from decade to
decade is constant. This method gives high results, as the percentage increase
gradually drops when the growth of the cities reach the saturation point. This methodis useful for cities which have unlimited scope for expansion and where a constant
rate of growth is anticipated. The formula of this estimation is :
)1( kn
ot
pp
pt : population at some time in the future
po: present or initial population
k : average percentage increase (geometric mean)
n : period of projection in decade
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Example :
The population of a town as per the senses records are given below for the years 1945
to 2005. Assuming that the scheme of water supply will commence to function from
2010, it is required to estimate the population after 30 years, i.e. in 2040 and also, theintermediate population i.e. 15 years after 2010.
PopulationYear
401851945445221955
603951965
756141975
988861985
1242301995
1587902005
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Solution :
1- Arithmetic increase method: Increase in population from 1945 to 2005 , i.e.for 6 decades: 15880040185 = 118615 = total increment
Increase per decade = 118615 / no. of decade = 118615 / 6 = 19769
IncreasePopulationYear
------401851945
4452240185 = 433744522195515873
60395196515219756141975
23272988861985
253441242301995
345701588002005
118615Total
118615/6=19769Average
tkppot
capita
pp
,198338
)2)(19769(158800
)2)(19769(20052025
capita
pp
,227992
)5.3)(19769(158800
)5.3)(19769(20052040
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2- Geometric increase method :
Rate of growthIncreasePopula
tion
Year
------401851945
4337 / 40185 = 0.1084452240185 = 43374452219550.35615873603951965
0.25215219756141975
0.308232729888619850.256253441242301995
0.278345701588002005
)1( kn
ot
pp
2442.0278.0256.0308.0252.0356.0108.06 xxxxxk
capitapp ,245828)2442.01(2
20052025
capitapp ,341166)2442.01(5.3
20052040
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3- Incremental increase method : (Method of varying increment) :
Incremental
increase (a)
Increase
(k)
Populat
ion
Year
------4018519454452240185 = 4337445221955
11536+15873603951965
- 65415219756141975
8053+23272988861985
2072+253441242301995
9226+345701588002005
30233118615Total
604719769Average
ann
knpp ot .
2
)1(.
capitax
xann
knpp ,21647960472
32197692158800.
2
)1(.19952015
capitax
xann
knpp ,27561260472
5.45.3197695.3158800.
2
)1(.19952040
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4- Uniform percentage of increase:
Growth rate
(k1)
PopulationYear
------4018519450.014452219550.03603951965
0.022756141975
0.027988861985
0.0291242301995
0.0251588002005
0.143Total
0.024Average
capitaep
xp
tkpp
,256530
455.1220024.0158800lnln
lnln
455.12
2025
2025
120052025
)(lnln 1o
ot
ttkpp
t
ppk ot
lnln1
capitaep
xp
tkpp
,367692
815.1235024.0158800lnln
lnln
815.12
2040
2040
120052040
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5- Logistic method: PopulationYear
401851945
445221955
603951965756141975
988861985
1242301995
1588002005
ot
1t
2
t
30n
30n
o
p
2
p
1
p
tba
satt
e
pp
1
260053)75614()158800)(40185(
)15880040185()75614()158800)(75614)(40185(2)(22
2
2
12
2
2
121
ppp
ppppppp
o
oosat
45.0158800
158800260053lnln
2
2
p
ppa sat
027.0)40185260053(75614
)75614260053(40185ln
30
1
)(
)(ln
1
1
1
osat
sato
ppp
ppp
nb
capitae
px
,1896021
26005320)027.0(45.02025
capita
ep
x,208404
1
26005335)027.0(45.02040
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Example :
The population of a city A as per the senses records are given below for the years
1950 to 1990. Estimate the population of city A in year 2020 according to senses
records for cities B,D, and E that similar to city A.
City E
Pop.Year
300001914
350001924
420001934
470001944
510001954
530001964
580001974
620001984
680001994
715002004
City A
Pop.Year
320001950
360001960
400001970
450001980
510001990
City B
Pop.Year
250001910
320001920
380001930
430001940
510001950
590001960
690001970
800001980
930001990
1100002000
City C
Pop.Year
250001915
310001925
360001935
420001945
510001955
580001965
680001975
730001985
860001995
960002005
City D
Pop.Year
310001913
350001923
420001933
460001943
510001953
550001963
610001973
680001983
720001993
800002003
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City ECity DCity CCity BCity A
Pop.YearPop.YearPop.YearPop.YearPop.Year300001914310001913250001915250001910320001950
350001924350001923310001925320001920360001960
420001934420001933360001935380001930400001970
470001944460001943420001945430001940450001980
510001954510001953510001955510001950510001990
530001964550001963580001965590001960
580001974610001973680001975690001970
620001984680001983730001985800001980
680001994720001993860001995930001990
7150020048000020039600020051100002000
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capitaApop ,707504
)62000680007300080000().( 2020
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Problems :1- The recent population of a city is 30000 inhabitant. What is the predicted
population after 30 years if the population increases 4000 in 5 years .
2- The recent population of a city is 30000 inhabitant. What is the predictedpopulation after 30 years if the growth rate is 3.5% .
3- The population of a town as per the senses records are given below , estimate the
population of the town as on 2040 by all methods.
PopulationYear
580001957
650001967
730001977
810001987
950001997
1150002007