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7/30/2019 Pop Fore Cast

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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,


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:



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



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