Introduction to Interpolation

The time series data which is recorded after regular or irregular interval of time consists of values of a phenomenon at a certain point in time or the values of some hypothetical function corresponding to a few values of independent variable are not sufficient to provide information regarding the values of the same phenomenon in between the specified intervals. For example the production of wheat in different years, the export of cotton goods during the last ten years, yearly enrolment at the university or it may be the values of logarithms to a specified base corresponding to natural numbers, the demand of a product at different levels of prices etc., for a period within the specified time not mentioned in the series or the values of $\log X$, the demand etc., corresponding to the values of $X$ or $P$ from within the specified values. The process and technique of estimation of such values in termed as interpolation.
This unit is restricted to the case where these related variables follow approximately a linear trend. The formulae are developed on the assumption of linearity. A number of different are available for interpolation, mostly by Newton. We shall confine here only to a few of them satisfying the need of BBA, Economics and MBA students. Only an application of these formulae shall be explained with help of a few worked examples. The derivation of these formulae is beyond the scope.
In addition to linearity the data should satisfy two more assumptions for the better application of these formulae:

• No sudden jumps are present in the data from one time period to another; this implies that the data are in the form of continuous or smooth curves.
• The data has a uniform rate of change; this assumption is infecting equivalent to the assumption of linearity.