436 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Using (2), a = y N ∑ = 290 5 = 58 and b = xy x ∑ ∑ 2 = 34 10 = 3.4 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 × (–2) = 51.2 1975 –1 58 + 3.4 × (–1) = 54.6 1976 0 58 + 3.4 × 0 = 58.0 1977 1 58 + 3.4 × 1 = 64.4 1978 2 58 + 3.4 × 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 = y N ∑ and b = xy x ∑ ∑ 2 Calculations for fitting the straight-line trend 2 2 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 734 0 168 210 Year Value y x x xy YxXY −− −− −− −− ∑= ∑ = ∑ = Using (2), 734 91.75 8 y a N === ∑ , and 2 210 1.25 168 xy b x === ∑ ∑ TIME SERIES AND FORECASTING 437 ∴ From (1), the required equation of the straight-line trend is 91.25 1.25yx=+ Year X Trend Value (y = 91.75 + 1.25x) 1961 –7 91.75 + 1.25 × –7 = 83.0 1962 –5 91.75 + 1.25 × –5 = 85.5 1963 –3 91.75 + 1.25 × –3 = 88.0 1964 –1 91.75 + 1.25 × -1 = 90.5 1965 1 91.75 + 1.25 × 1 = 93.0 1966 3 91.75 + 1.25 × 3 = 95.5 1967 5 91.75 + 1.25 × 5 = 98.0 1968 7 91.75 + 1.25 × 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 different times of day or whatever. Seasonal variations are measured through their indices called the seasonal indices. The measurement of seasonal variations requires determining the seasonal component s t which indicates how a times series from quarter to quarter, month to month, or week to week, etc. through out a year. A series of numbers showing relative values of a variable during the quarters or months or weeks etc. of the year is called seasonal index for the variable. If Rs. x be the average quarterly (or monthly) sales in a year and I be the Seasonal index of that quarter (or month), then Sale for the Qaurter or Month =1% of 1 = 100 xx× . The following methods are commonly used for measuring seasonal variations. (a) Method of Averages (Quarterly, Monthly or Weekly) (b) Moving Average Method (c) Ratio to Trend Method (d) Link Relative Method (=) Method of Averages: This method is used when trend and cyclical fluctuations, if any, have little effect on the time series. 438 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES If quarterly data are given, first find quarterly totals for each quarter and the averages for the four-quarter of the years. To find these averages, we divide the quarterly totals by the number of the years for which the data are given. Then we find grand average of the 4 quarterly averages. Grand average +++ = 123 4 4 xxxx G If we use multiplicative identity, then the seasonal indices are the 4 quarterly averages expressed as percentages of the grand average G. i.e., 312 4 100, 100, 100, 100 xxx x GGGG ×××× Similarly, if we use additive model, then seasonal variations for the 4 quarters are 1 234 ,,, xGxGxGxG−−−− When monthly or weekly data are given, we find monthly (or weekly) averages for the 12 months or (52 weeks). Example 9. Calculate Seasonal indices for each quarter from the following percentages of wholesale price indices to their moving averages. Year Quarter I II III IV 1996 – – 11.0 11.0 1997 12.5 13.5 15.5 14.5 1998 16.8 15.2 13.1 15.3 1999 11.2 11.0 12.4 13.2 2000 10.5 13.3 – – Sol. Calculation for Seasonal Indices Year Quarter I II III IV 1996 – – 11.0 11.0 1997 12.5 13.5 15.5 14.5 1998 16.8 15.2 13.1 15.3 1999 11.2 11.0 12.4 13.2 2000 10.5 13.3 – – Total 51.0 53.0 52.0 54.0 Averages 12.75 13.25 13.0 13.5 Grand Average (G) 12.75 13.25 13.0 13.5 4 +++ = 52.5 13.125 4 == TIME SERIES AND FORECASTING 439 Using Multiplicative model, Seasonal index () 100 () i Average x Grand Average G =× ∴ Seasonal indices for the first, second, third, and fourth quarters are respectively 12.75 100 97.14 13.125 ×= 13.25 100 100.95 13.125 ×= 13.0 100 99.95 13.125 ×= 13.5 100 102.86 13.125 ×= Example 10. Compute the Seasonal Index for the following data: Qaurters Year I II III IV 2001 75 60 54 59 2002 86 65 63 80 2003 90 72 66 85 2004 100 78 72 93 Sol. Let Cyclical Fluctuations and Trend are absent in the given data Year Quarters I II III IV 2001 75 60 54 59 2002 86 65 63 80 2003 90 72 66 85 2004 100 78 72 93 Total 351 275 255 317 Averages x i bg 87.75 68.75 63.75 79.25 (Total/4) Grand Average (G) = 87.75 68.75 63.75 79.25 4 +++ = 299.50 4 = 74.875 440 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Now, using Multiplicative model, Seasonal index Average x Grand Average G i () () ×100 Hence Seasonal indices for the 1st, 2nd, 3rd and 4th quarters are respectively 1st = 87 75 74 875 100 . . × = 117.20 2nd = 68 75 74 875 100 . . × = 91.82 3rd = 63 75 74 875 100 . . × = 85.14 4th = 79 25 74 875 100 . . × = 105.84 Hence the sum of Seasonal indices is 400. Similarly, we can obtain Seasonal indices using additive model. Using additive model, Seasonal index = Average N E ) – Grand average (G) Therefore 1st, 2nd, 3rd and 4th quarters are respectively 12.875, – 6.125, –11.125, 4.375 Hence the sum of seasonal indices for the four quarters is 12.875 + 4.375 – 11.125 − 6.125 = 0. Ans. (>) Moving Average Method (or ratio to moving averages): This is a improved method over method of averages and is widely used for measuring seasonal variation. According to this method, if monthly data are given, we find 12-month centred moving averages, if quarterly data are given, we find 4 quarter centred moving averages and so on. This represent trend and then eliminate the effect of trend by using either additive model or multiplicative model. Case 1: If multiplicative model is used, then express the original data as percentage of the corresponding moving averages expressed as percentage that is ratio to moving averages expressed as percentage. These percentages for corresponding months or quarters are then averaged by the method of averages, gives the required seasonal indices. This method is known as Ratio to Moving Average Method. Example 11. Calculate Seasonal indices by the ratio to moving average method from the following data. Iron Prices (In Rupees Per Kg.) Quarter Year 2001 2002 2003 2004 Quarter 1 75 86 90 100 Quarter 2 60 65 72 78 Quarter 3 54 63 66 72 Quarter 4 59 80 85 93 TIME SERIES AND FORECASTING 441 Calculation of Ratio to moving averages Year/Quarter Prices 4 Quarter 2 Point 4 Quarter Ratio to moving Moving Average Average total total (2-pt. Moving ÷ 8) 2001 Q 1 75 Q 2 60 Q 3 54 248 Q 4 59 259 507 63.375 54 100 85.21 63.375 ×= 2002 Q 1 86 264 523 65.375 59 100 90.25 65.375 ×= Q 2 65 273 537 67.125 86 100 128.12 67.125 ×= Q 3 63 294 567 70.875 65 100 91.71 70.875 ×= Q 4 80 298 592 74.000 63 100 85.14 74.000 ×= 2003 Q 1 90 305 603 75.375 80 100 106.41 75.375 ×= Q 2 72 308 613 76.625 90 100 117.46 76.625 ×= Q 3 66 313 621 77.625 72 100 92.75 77.625 ×= Q 4 85 323 636 79.500 66 100 83.02 79.500 ×= 2004 Q 1 100 329 652 81.500 85 100 104.29 81.500 ×= Q 2 78 335 664 83.000 100 100 120.48 83.000 ×= Q 3 72 343 678 84.750 78 100 92.04 84.750 ×= Q 4 93 442 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Calculation for Seasonal Indices Quarter Year Q 1 Q 2 Q 3 Q 4 2001 85.21 90.25 2002 128.12 91.71 85.14 106.14 2003 117.46 92.75 83.02 104.29 2004 120.48 92.04 Total 366.06 276.50 253.37 300.68 Averages 122.02 92.17 84.46 100.23 Grand Average (G) 122.02 92.17 84.46 100.23 99.72 4 +++ == 3 Seasonal indicies for 4 quarters are respectively Q 1 122.02 100 122.36 99.72 =×= Q 2 92.17 100 92.43 99.72 =×= Q 3 84.46 100 84.70 99.72 =×= Q 4 100.23 100 100.51 99.72 =×= Hence sum of seasonal indices is 400. Ans. Case 2: If the additive model is used, to eliminate trend subtract the moving averages from the original data and also obtain deviations from trend. Now apply the method of averages to these deviations to obtain required seasonal variations. Example 12. Obtain Seasonal Fluctuation from the following time series using moving averages method. Quaterly output for 4 years Year I II III IV 1998 65 58 56 61 1999 68 63 63 67 2000 70 59 56 52 2001 60 55 51 58 TIME SERIES AND FORECASTING 443 Sol. Calculation of moving averages and deviations from trend Year/Quarter Output 4-Quarter 2-Point 4-Quarter Deviation from Moving total Moving total Moving Average Trend 1998 Q 1 65 Q 2 58 Q 3 56 240 Q 4 61 243 483 483/8 = 60.38 56 – 60.38 = –4.38 1999 Q 1 68 248 491 491/8 = 61.38 61 – 61.38 = –0.38 Q 2 63 255 503 503/8 = 62.88 68 – 62.88 = 5.12 Q 3 63 261 516 516/8 = 64.50 63 – 64.50 = –1.50 Q 4 67 263 524 524/8 = 65.25 63 – 65.50 = –2.50 2000 Q 1 70 259 522 522/8 = 65.25 67 – 65.25 = 1.75 Q 2 59 252 511 511/8 = 63.88 70 – 63.88 = 6.12 Q 3 56 237 489 489/8 = 61.13 59 – 61.13 = –2.13 Q 4 52 227 464 464/8 = 58.00 56 – 58.00 = –2.00 2001 Q 1 60 223 450 450/8 = 56.25 52 – 56.25 = –4.25 Q 2 55 218 441 441/8 = 55.13 60 – 55.13 = 4.87 Q 3 51 224 442 442/8 = 55.25 55 – 55.25 = –0.25 Q 4 58 Calculate the Seasonal Fluctuations Deviation from Trend Year Quarter-I Quarter-II Quarter-III Quarter-IV 1998 –4.38 –0.38 1999 5.12 –1.50 –2.50 1.75 2000 6.12 –2.13 –2.00 –4.25 2001 4.87 –0.25 Total 16.11 –3.88 –8.88 –2.88 Average i x 5.37 –1.29 2.96 –0.96 Grand Average (G) = 5.37 + (–1.29) + (2.96) + (–0.96) = 0.16 ÷ 4 = 0.04 Therefore the seasonal functions are () i xG − i.e., 5.37 – 0.04 = 5.33 –1.29 – 0.04 = –1.33 2.96 – 0.04 = 2.92 –0.96 – 0.04 = –1.00 respectively (? ) Ratio to Trend Method: In this method trend values are first determined by the method of least squares fitting a mathematical curve and the given data are expressed as percentage of the corresponding trend values. Using the multiplicative identity these percentages are then averaged by the method of averages. R S | T | U V | W | U V | W | 444 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES (@) Link Relative Method: According to this method for given data for each quarter or m089onth are expressed as percentage of data for the preceding quarter or month. These percentages are known as Link relatives. The link relative for the first quarter (or 1st month) of the year cannot be determined. An appropriate average (Arithmetic Mean or Median) of the link relatives for each quarter (or month) is then found. From these average link relatives, we find the chain relative with respect to 1st quarter (or 1st month) for which the chain relative is taken as 100. If Q 1 , Q 2 , Q 3 , Q 4 denotes 4 quarters respectively and chain relative represents by C.R., or link relative represents by L.R. then C.R. for Q 2 = (Average L.R. for Q 2 × C.R. for Q 1 ) ÷ 100 C.R. for Q 3 = (Average L.R. for Q 3 × C.R. for Q 2 ) ÷ 100 C.R. for Q 4 = (Average L.R. for Q 4 × C.R. for Q 3 ) ÷ 100 and 2nd C.R. for Q 1 = (Average L.R. for Q 1 × C.R. for Q 4 ) ÷ 100 Generally, the 2nd C.R. for Q 1 will be either higher or lower than the first C.R. 100 for Q 1 depending on the presence of an increase or decrease in trend. If d = 2nd C.R. for Q 1 –100, i.e., the difference between 1st and 2nd C.R. for Q 1 then we subtract (1 ÷ 4) (2 ÷ 4)d, (3 ÷ 4) (4 ÷ 4)d From the chain relatives for Q 2 , Q 3 and Q 4 and the 2nd C.R. for Q 1 respectively to obtain the adjusted chain relatives. These adjusted chain relatives expressed as percentages of their A.M. gives the required Seasonal indices. Example 13. Calculate seasonal indices by method of link relatives from the data given in Example 12. Sol. Calculate of Average Link Relatives Link Relatives Year/Quarter Quarter-I Quarter-II Quarter-III Quarter-IV 1998 89.23 96.55 108.93 1999 111.48 92.65 100.00 106.35 2000 104.48 84.29 94.92 92.86 2001 115.38 91.67 92.73 113.73 Total 331.34 357.84 384.20 421.87 Averages (A.M.) 110.45 89.46 96.05 105.47 The link relative (L.R.) for the 1st quarter Q 1 of the first year 1998 cannot be determined. For the other three quarters of 1998, L.R. for Q 2 = (58 ÷ 65) × 100, L.R. for Q 3 = (56 ÷ 58) × 100, L.R. for Q 4 = (61 ÷ 56) × 100, Similarly, we determine the other link relative. From the average link relatives obtained in the last row of the above table, we now find chain relatives, taking 100 as the chain relative (C.R.) for Q 1 . C.R. for Q 1 = 100 C.R. for Q 2 = (89.46 × 100) ÷ 100 = 89.46 TIME SERIES AND FORECASTING 445 C.R. for Q 3 = (96.05 × 89.46) ÷ 100 = 85.93 C.R. for Q 4 = (105.47 × 85.93) ÷ 100 = 90.63 2nd C.R. for Q 1 = (110.45 × 90.63) ÷ 100 = 100.10 ∴ d = 100.10 – 100 = 0.10; ∴ 1 0.025, 4 d = 2 0.050, 4 d = 3 0.075, 4 d = 4 0.10. 4 d = The adjusted chain relatives are respectively 100, 89.46 – 0.025, 85.93 – 0.05, 90.63 – 0.075, i.e. 100, 89.435, 85.88, 90.555 or 100, 89.44, 85.88, 90.56. A.M. of the adjusted chain relative = (100 + 89.44 + 85.88 + 90.58) ÷ 4 = 91.47 The required seasonal indices are 100 89.44 85.88 90.56 100, 100, 100, 100 91.47 91.47 91.47 91.47 ×××× i.e., 109.33, 97.78, 93.89, 99.00. Ans. 10.4.3 Analysis of Cyclical Fluctuation or Cyclic Variations To analyse Cyclical fluctuation, we first find trend (T) and seasonal Variation (S) by any suitable method i.e., by the method of moving averages or other method and then eliminate them from the original data by using additive or multiplicative identity. Irregular movement is removed by using moving average of appropriate period depending average duration of irregular movement, leaving only cyclical fluctuation. 10.4.4 Analysis of Irregular of Random Movements Irregular movements are obtained by eliminating trend (T), seasonal variation (S) and cyclical fluctuation (C) from the original data. Normally irregular movements are found to be of small magnitude. 10.5 IMPORTANCE OF TIME SERIES The time series analysis is of great importance not only to a businessman, scientist or economist, but also to people working in various disciplines in natural, social and physical sciences. Some of its uses are: 1. It enables us to study the past behaviour of the phenomenon under consideration, i.e., to determine the type and nature of the variations in the data. 2. The segregation and study of the various components is of paramount importance to a businessman in the planning of future operations and in the formulation of executive and policy decisions. 3. It enables us to predict or estimate or forecast the behaviour of the phenomenon in future, which is very essential for business planning. 4. It helps us to compare the changes in the values of different phenomenon at different times or places, etc. . series. 438 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES If quarterly data are given, first find quarterly totals for each quarter and the averages for the four-quarter of the years. To find. the type and nature of the variations in the data. 2. The segregation and study of the various components is of paramount importance to a businessman in the planning of future operations and in. the original data and also obtain deviations from trend. Now apply the method of averages to these deviations to obtain required seasonal variations. Example 12. Obtain Seasonal Fluctuation from