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A textbook of Computer Based Numerical and Statiscal Techniques part 46

A textbook of Computer Based Numerical and Statiscal Techniques part 46. By joining statistical analysis with computer-based numerical methods, this book bridges the gap between theory and practice with software-based examples, flow charts, and applications. Designed for engineering students as well as practicing engineers and scientists, the book has numerous examples with in-text solutions. | 436 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES J y Using 2 a -N 58 and b -J 3.4 5 J x2 10 From 1 the required equation of the best fitted straight-line is Y 58 3.4x. Year x Trend Values y 58 3.4x 1974 -2 58 3.4 x -2 51.2 1975 -1 58 3.4 x -1 54.6 1976 0 58 3.4 x 0 58.0 1977 1 58 3.4 x 1 64.4 1978 2 58 3.4 x 2 64.8 Example 8. Fit a straight-line trend equation by the method of least square and estimate the trend value. Year 1961 1962 1963 1964 1965 1966 1967 1968 Values 80 90 92 83 94 99 92 104 Sol. Here N Number of years 8 which is even Let the straight line trend equation by the method of least squares with the origin at the mid point of 1964 and 1965 and unit of x as 1 2 year be y a bx . 1 Then a and b are given by a Jy and b J N J x Calculations for fitting the straight-line trend Year Value y x x2 xy 1961 80 -7 49 -560 1962 90 -5 25 -450 1963 92 -3 9 -276 1964 83 -1 1 -83 1965 94 1 1 94 1966 99 3 9 297 1967 92 5 25 460 1968 104 7 49 728 Total J Y 734 0 J x2 168 J XY 210 Using 2 a J 734 91.75 and b -2 210 1.25 N 8 J x2 168 TIME SERIES AND FORECASTING 437 From 1 the required equation of the straight-line trend is y 91.25 1.25x Year X Trend Value y 91.75 1.25x 1961 -7 91.75 1.25 x -7 83.0 1962 -5 91.75 1.25 x -5 85.5 1963 -3 91.75 1.25 x -3 88.0 1964 -1 91.75 1.25 x -1 90.5 1965 1 91.75 1.25 x 1 93.0 1966 3 91.75 1.25 x 3 95.5 1967 5 91.75 1.25 x 5 98.0 1968 7 91.75 1.25 x 7 100.5 Note If the number of years is even there is no middle year and in this case the midpoint which is take as the origin lies midway between the two middle years. In example 8 the midpoint i.e. the origin lies midway between July 1 1964 and July 1 1965 which is January 1 1965 or December 31 1964 . To avoid fractions the units of x are taken as 1 2 year or 6 months . 10.4.2 Analysis of Seasonal Variation Seasonal variations are short term fluctuations in recorded values due to different circumstances which affect results at different times of the year on different days of the week at