# TREND EXTRAPOLATION

TREND EXTRAPOLATION FOR TECHNOLOGY FORECASTING

This is one of the simplest and most commonly used forecasting techniques. This method uses historical data on selected technological parameters for projecting the future trends. It thus implies that the historical trends are a resultant of a number of forces which will continue to behave as in the past unless there are strong reasons to the contrary. A single function or an attribute of a technological performance or a technical characteristic is used for projection into the future. It is generally assumed that it is possible to quantify the changes in the performance of a technology based on either a straight line trend extrapolation or an exponential trend extrapolation on continuation of past attributes. If there has been some regular rate of change in the past of functional capability, it is quite likely that the rate of progression will continue at least in the near future.

However, there are two instances where trend extrapolation treatment should not be pursued, namely :

a) in cases where there are known natural limits such as 100% efficiency achievement in a steam engine or for that matter any conversion of energy, and

b) cases where it is known that conditions governing a specific trend in the past have changed.

With these brief introductory remarks, let us consider how the techniques of trend extrapolation could be made use of in practice.

LINEAR TREND EXTRAPOLATION

Straight line extrapolation method uses projection of a parameter assuming a linear growth trend.

The linear trend could be represented by

yi = αxi+β, where

yi = Value of parameter estimated in the ith year.

xi = Value of the it” year, α and β are constants to be estimated.

α and β could be, estimated by the method of sum of squares and minimising them from the projected trend extrapolation. Therefore, sum of squares, S (α, β) are obtained by

where n = number of observations.

On minimising the estimates of, α and β, we get

Which give rise to two normal equations:

Solving the equations 3 and 4 simultaneously we obtain

EXTRAPOLATION USING EXPONENTIAL TREND

It has been found by empirical study that many technologies do grow exponentially though there is no sound theoretical basis for it. When the exact behaviour of a technology is unknown and there is no reason to suspect departure from an exponential growth, the forecaster may be justified empirically to assume that the technology in question would grow exponentially. The extrapolation using exponential trend is thus suitable to deal with the growth of particular technological capability, or production trends etc. when they are plotted against time. A linear regression of the logarithm of the parameter against time, or alternatively, a straight line on a semi-log plot, is equivalent to exponential growth of functional capability.

The exponential growth curve could be assumed to be

yi = abxi, where

yi Value of parameter to be estimated

xi = Value of the i`th year a, b are constants to be estimated.

Taking logarithm on both sides,

In yi = Ina + xi lnb

= A + B xi ,where A = Ina B = Inb

which is a straight line.

Now, applying the sum of squares method and minimising it, as before, we get two normal equations, viz :

We can easily solve these two equations from a set of observed data and find the estimates of A and B; and from those the estimates of ‘a’ and `b’.

A word of caution is necessary here for pursuing the exercise of curve fitting for trend extrapolation. You may realise that some of the trends will not follow a straight line pattern or an exponential pattern to describe the trend and its underlying direction. In such cases a parabolic trend [y = as + alx + a2x2] or a polynomial trend [y = ao + aix + a2x2….at,xp] may be applied to identify the underlying trend. Regression analysis can be performed by general method described earlier by finding out the normal equations by minimising the sum of squares. You are advised to work out the normal equations by yourself as an exercise.

APPLICATIONS

Extrapolation of past data can be used for predicting future trends and thus finds application in-predicting future performance characteristics of a technology, production level of an item/product or for that matter any parameter that is amenable to such treatment. The crucial aspect of this methodology is identification of an appropriate index/parameter.

In general the trend projection method consists of the following steps :

Step 1: selecting an appropriate parameter to describe the attribute of interest to the forecaster,

Step 2 : collecting past data of this parameter for a reasonable period,

Step 3 : plotting the data graphically to determine if straightline/ exponential/ parabolic etc. can best describe the trend, .

Step 4 : fitting an appropriate curve as described earlier and using the mathematical equation to project events in the near future.

Trend extrapolation has been used extensively for forecasting technological as well as non technological parameters. Routine and mechanical use of this technique has been responsible for many of the serious forecasting errors, since extrapolations cannot predict the trend if underlying causes are drastically changed at a point of time. Extrapolation can be used only for short range forecasting.