# Mathematical and Graphical Methods

These are simple or direct methods since they operate with past population records and take no account of the components of change. Although the two types are somewhat interchangeable because mathematical methods can be plotted and graphical data can be described mathematically. Both methods are generally based on an idealized model of exponential growth followed by linear growth and finally decreasing growth.

These include:

· Linear growth

· Exponential growth

· Decreasing growth

· Correlation method

· Component method

## Linear growth

Steady growth can be represented as:

DP / Dt = (Pb – Po) / (tb – to) = K1

where:

DP = change in population

Dt = change in time

Pb = base population (start of projection)

Po = initial population (in the applicable linear growth period)

tb = base year (start of projection)

to = initial year (earliest year in the applicable linear growth period)

K1 = growth rate = slope of line

A projection assuming linear growth can be calculated using the formula:

Pf = Pb + K1t

where:

Pf = future population

t = tf – tb = # of years projected into the future

tf = future year (end of projection)

## Exponential growth

First-order growth can be represented as:

DP / Dt = K2P, or ln (Pb / Po) = K2(tb – to)

where:

DP = change in population

Dt = change in time

K2 = growth coefficient

P = population at a given year

Pb = base population (start of projection)

Po = initial population (in the applicable exponential growth period)

tb = base year (start of projection)

to = initial year (earliest year in the applicable exponential growth period)

A projection assuming exponential, or geometric, growth can be calculated using the formula:

Pf = Pbe

^{K}2^{t}
where:

Pf = future population

K2 = ln (Pb / Po) / (tb - to)

t = tf – tb = # of years projected into the future

tf = future year (end of projection)

## Decreasing growth

Decelerating growth is assumed to asymptotically approach a saturation population, that is, the maximum population predicted for the geographic area of interest. The saturation population may be based on practical limitations such as the maximum number of dwellings under the zoning restrictions or other constraints.

Decreasing growth can be represented as:

DP / Dt = K3(S – P)

where:

DP = change in population

Dt = change in time

K3 = growth coefficient

S = saturation population

P = population at a given year

As P approaches S, DP / Dt approaches zero, and a projection assuming decreasing growth can be calculated using the formula:

Pf = S – (S – Pb)e

^{-K}3^{t}
where:

Pf = future population

Pb = base population (start of projection)

K3 = -ln [(S – Pb) / (S – Po)] / (tb - to)

Po = initial population (in the applicable decelerating growth period)

t = tf – tb = # of years projected into the future

tf = future year (end of projection)

tb = base year (start of projection)

to = initial year (earliest year in the applicable decelerating growth period)

## Correlation method

The correlation method uses a separate population projection for a similar community or region as a basis for determining a growth rate for the community or region of interest, assuming that the ratio of base populations is representative of the ratio of the projected populations. This method can be expressed mathematically as:

Pf2 / Pf1 = Pb2 / Pb1 = K4

where:

Pf2 = future population of community 2

Pf1 = future population of community 1

Pb2 = base population of community 2

Pb1 = base population of community 1

K4 = population ratio

## Component method

This method, as suggested by its name, involves a summation of population growth components such as:

· Births

· Deaths

· Migration

· Employment opportunities

· Available housing

· Other data reflecting potential changes in the population

While this method may require significantly more effort than the previous methods, due to the amount of data needed, it may result in a more accurate projection if the data is accurate.

The component method can be represented by the following formula:

Pf = Pb + (F1 + F2 + … Fn)

where:

Pf = future population

Pb = base population

F1, F2, Fn = population changes expected during the selected time period due to specific factors

(births, deaths, etc.)

While the above techniques may utilize plots of data, they are based on mathematical approaches. A purely graphical technique that could be called the superposition method may also be used. It involves an approach described below.

# The Employment Method

Given a series of past values of the activity rate, i.e., economically active population / persons in working age groups =

E / W

And the ratio, Persons in working age groups / total population =

W / P

It is possible by using graphical or mathematical methods such as those just described to produce future values for these ratios. Then, given forecasts of total employment, population may be estimated, since

E/W X W/P = E/P

A range or employment forecasts (made for different assumptions about the economic ‘climate’) will yield a range of population forecasts. If regression techniques are used to project E/W and W/P, then the application of calculated estimating errors will of itself produce ranges of values for E/P.

The reliability of this method is certainly no greater than those already discussed and should not be used for long-range forecasting.

# Ratio Methods

This family of methods rests on the assumption that changes in any geographical area are a function of those experienced in (successively) wider areas. Thus, the population of a city is held to be a function of that of the region, which itself is a function of that of the nation, and so on.

The requirements for such projections are time-series of populations for the areas to be used in the analysis and a forecast or set of forecasts for the largest area. In ratio methods the population of the second is plotted against that of the parent area (the nation), thus:

A curve is fitted to the points thus obtained and, by least squares, correlation, graphical or other method is extrapolated to intersect the projected value for the parent area at a given forecast date. Clearly, if a range had been given for the parent-area forecast this would have resulted in a range for the region.

In the second step down the process is repeated using data for the study area and the region.

Again the curve is fitted and extrapolated to intersect the (derived) forecast for the parent area.

This method has the great benefits of simplicity and the use of readily available data. However, this does not directly examine the components of population change which are subsumed in the central assumption, i.e. that there are certain forces at work in nations, regions and sub-regions which make for patterns and order in the proportionate share which the latter have in the former. Further, it is assumed that these relationships change but slowly over time.