Unit 4: Interpolation and Extrapolation Meaning Let x0, x 1 , x 2 … x n be given values of X and y 0, y1, y2 … yn be the values of Y. If we want to estimate Y for any value of X between x 0 and x n it can be done by interpolation But if we want to estimate Y for any value of X outside the range of the given X series, we use the technique of Extrapolation Assumptions of Interpolation and Extrapolation 1. No sudden jumps in values of dependent variable from one period to another. 2. There is a sort of uniformity in rise or fall of values of dependent variable. 3. No consecutive missing values in series. Methods of Interpolation and Extrapolation Binomial Expansion Method It is applicable if: 1) Values of the independent variable should have a common difference. That is the values should be in arithmetic progression. 2) The value of X for which the value of Y is to be interpolated must be one of the values of X Newton’s Advancing Difference Method The formula is 𝑦𝑥 = 𝑦0 + 𝑥∆10 + 𝑥= 𝑥(𝑥 − 1) 2! 2 0 + 𝑥(𝑥 − 1)(𝑥 − 2) 3! 3 0+ 𝑥(𝑥 − 1)(𝑥 − 2)(𝑥 − 3) 4! 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑋 𝑡𝑜 𝑏𝑒 𝑖𝑛𝑡𝑒𝑟𝑝𝑜𝑙𝑎𝑡𝑒𝑑 − 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑋 𝑎𝑡 𝑜𝑟𝑖𝑔𝑖𝑛 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡𝑤𝑜 𝑎𝑑𝑗𝑜𝑖𝑛𝑖𝑛𝑔 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑋 𝑥= 𝑡ℎ𝑒 𝑦𝑒𝑎𝑟 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑝𝑜𝑙𝑎𝑡𝑖𝑜𝑛 − 𝑡ℎ𝑒 𝑦𝑒𝑎𝑟 𝑜𝑓 𝑜𝑟𝑖𝑔𝑖𝑛 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡𝑤𝑜 𝑎𝑑𝑗𝑜𝑖𝑛𝑖𝑛𝑔 𝑦𝑒𝑎𝑟𝑠 Conditions for using Newton’s Formula is that it must be used when the value to be interpolated is in the beginning of the data.