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- 1. Problem 1: Fit an equation of the form y = a + b x + c x2 to the data given below. X 1 2 3 4 5 Y 25 28 33 39 46 Answer: X Y x x2 x3 x4 xY Yx2 Trend Values 1 25 -2 4 -8 16 -50 100 24.88 2 28 -1 1 -1 1 -28 28 28.26 3 33 0 0 0 0 0 0 32.92 4 39 1 1 1 1 39 39 38.86 5 46 2 4 8 16 92 184 46.08 171 10 0 34 53 351 Fitting of a Second Degree (Parabolic) Trend: ∑y = na + b∑x + c∑x2 171 = 5a+0b+10c …..(i) ∑xy = a∑x + b∑x2 + c∑x3 53=0a+10b+0c …..(ii)
- 2. ∑x2y = a ∑x2 + b∑x3 + c∑x4 351=10a+0b+34c …..(iii) By (ii) b = 5.3; Solving (i) and (iii) [Multiply (i) by 2 and deduct that from (iii)] we get c = o.64 (14c = 9) and a = 32.92 (171-10*0.64=5a) Therefore the equation is: y = 32.92 + 5.3 x + 0.64 x2 Problem 2: Fit an equation of the form y = A. Bx to the data given below x 1 2 3 4 5 y 1.6 4.5 13.8 40.2 125 Answer: x y Y= log y Yx x2 Trend Values 1 1.6 0.2041 0.2041 1 1.6 2 4.5 0.6532 1.3064 4 4.6 3 13.8 1.1399 3.4197 9 13.8 4 40.2 1.6042 6.4168 16 41.1 5 125 2.0969 10.4845 25 122.3 15 5.6983 21.8315 Fitting of a Exponential Curve: y = A. Bx …..(i) Taking Logarithm we get: log y = log A+ x log B Y = a + bx …..(ii); Y = log y; a = log A; b = log B …..(iii) Equation (ii) can be written as: ∑Y = na + b∑x 5.6983 = 5a + 15b …..(iv)
- 3. ∑xY = a∑x + b∑x2 21.8315 = 15a+55b …..(v) By solving (iv) & (v) we get b = 0.4737 & a = -0.2814 Take Antilog we get A = 0.5231; B = 2.977; Therefore the trend equation is: y = 0.5231*(2.977)x METHOD OF MOVING AVERAGES This is the simple and flexible method of measuring trend. Moving Average is an averaging process that smoothens out the fluctuations and ups & downs in the given data. The Moving Average of period ‘m’ is a series of successive averages of m overlapping values at a time, starting with 1st, 2nd, 3rd value and so on. Problem 3: Calculate 5 yearly Moving Average from the data given below: 10; 14; 18; 22; 26; 30; 34; 38; 42; 46 Answer: Year Values 5 yearly Moving Total Average 1 10 2 14 3 18 90 18 4 22 110 22 5 26 130 26 6 30 150 30 7 34 170 34 8 38 190 38 9 42 210 42 10 46 11 50
- 4. Problem 4: Calculate 4 yearly Moving Average from the following data: 37.4; 31.1; 38.7; 39.5; 47.9; 42.6 Answer: Year Production 4 yearly Moving 2 PeriodCentered Total Average Moving Moving Total Average 1991 37.4 1992 31.1 146.7 36.675 1993 38.7 75.975 37.99 157.2 39.300 1994 39.5 81.475 40.74 168.7 42.175 1995 47.9 1996 42.6 SEASONAL VARIATIONS The variations due to such forces which operate in a regular periodic manner with period less than one year. The objectives of studying this is as follows: o To isolate seasonal variations: To determine the effect of seasonal swings on the values of a given phenomenon.
- 5. o To eliminate them: To determine the value of the phenomenon if there were no seasonal ups & downs. Methods: o Method of “Simple Averages” o “Ratio to Trend” Method o “Ratio to Moving Averages” Method o “Link Relative” Method SIMPLE AVERAGES This is the simplest method of measuring the seasonal variations in a time series and involves the following steps: o Arrange the data by years & months o Compute the average for the months o Compute the overall average o Obtain seasonal Indices for different months Problem 5: Compute the seasonal index from the data given: Quarter 1990 1991 1992 1993 1994 1995 I 3.5 3.5 3.5 4.0 4.1 4.2 II 3.9 4.1 3.9 4.6 4.4 4.6 III 3.4 3.7 3.7 3.8 4.2 4.3 IV 3.6 4.8 4.0 4.5 4.5 4.7
- 6. Answer: Year I Qtr. II Qtr. III Qtr. IV Qtr. 1990 3.5 3.9 3.4 3.6 1991 3.5 4.1 3.7 4.8 1992 3.5 3.9 3.7 4.0 1993 4.0 4.6 3.8 4.5 1994 4.1 4.4 4.2 4.5 1995 4.2 4.6 4.3 4.7 TOTAL 22.8 25.5 23.1 26.1 A.M. 3.8 4.25 3.85 4.35 Seasonal 93.6 104.7 94.8 107.1 Index X = 4.06 {(3.8+4.25+3.85+4.35)/4} o {(3.8/4.06)*100}=93.6 o {(4.25/4.06)*100}=104.7 o {(3.85/4.06)*100}=94.8 o {(4.35/4.06)*100}=107.1 RATIO TO TREND This is a method which is an improvement over the previous method. This is on the assumption that seasonal fluctuations for any season are a constant factor of the trend. This involves the following steps: o Compute the trend values by the appropriate method o Assuming multiplicative model, trend is eliminated.
- 7. o Arrange values according to the years, months or quarters o These seasonal indices are adjusted to the total of 1200 for monthly data or 400 for quarterly data. Problem 6: Using “Ration to Trend” method, determine seasonal index. Year I Quarter II Quarter III Quarter IV Quarter 1 68 60 61 63 2 70 58 56 60 3 68 63 68 67 4 65 56 56 62 5 60 55 55 58 Answer: Year Total Average x x2 xy Trend values 1 252 63.0 -2 4 -126 64.3 2 244 61.0 -1 1 -61 62.85 3 266 66.5 0 0 0 61.4 4 242 60.5 1 1 60.5 59.95 5 224 56.0 2 4 112 58.5 307 0 10 -14.5 Fitting of Linear Trend: y = a + b x To find a & b: ∑y = n a + b∑ x 307 = a*5 + b*0 a = 61.4 ∑xy = a ∑x + b ∑x2 -14.5 = a*0 + b*10 b = -1.45 Therefore the equation will be given by: y = 61.4 -1.45x Quarterly values will be: increment of (-1.45/2 = -0.36) Between II & III quarter: - 0.36/2 = -0.18
- 8. Trend Values Trend Eliminated Values Year I Quarter II Quarter III Quarter IV Quarter I Quarter II Quarter III Quarter IV Quarter 1 64.84 64.48 64.12 63.76 104.9 93.05 95.13 98.81 2 63.39 63.03 62.67 62.61 110.4 92.02 89.36 96.29 3 61.94 61.58 61.22 60.86 109.8 102.3 111.1 110.1 4 60.50 60.14 59.78 59.42 107.4 98.10 93.68 104.3 5 59.06 58.70 58.34 57.98 101.6 93.70 87.42 100.03 Total 534.1 479.2 476.7 509.6 Average 106.8 95.84 95.33 101.9 Adjusted Seasonal Indices 106.9 95.9 95.4 101.9 Sum of the averages: 106.8 + 95.84 + 95.33 + 101.9 = 399.90 Trend Eliminated Values are: (Given Value for that Quarter / Trend Value for that Quarter)* 100 Therefore the Correction Factor is: 400/ 399.90 RATIO TO MOVING AVERAGES This is a method which is an improvement over the previous method. This is a widely used measure which involves the following steps: o Obtain 12-month (4-quarter) moving average values. o Express the original values as a percentage of centered moving average. o Arrange these according to the years/months/quarter o These indices should be 1200 or 400. Problem 7:
- 9. Calculate the seasonal indices. 1991 I Quarter 68 II Quarter 62 III Quarter 61 63.125 IV Quarter 63 62.250 1992 I Quarter 65 62.375 II Quarter 58 62.750 III Quarter 66 62.875 IV Quarter 61 63.875 1993 I Quarter 68 64.125 II Quarter 63 64.500 III Quarter 63 IV Quarter 67 Answer: Ratio to Moving Averages: (61/63.125)*100 = 96.63; (63/62.250)*100 = 101.20; ….. and so on. Trend Eliminated Values I Quarter II Quarter III Quarter IV Quarter Year 1991 - - 96.63 101.20 1992 104.21 92.43 104.97 95.50 1993 106.04 97.67 - - Total 210.25 190.1 201.6 196.7 Averages 105.13 95.05 100.80 98.35 399.33 Adjusted Seasonal Indices 105.31 95.21 100.97 98.52 LINK RELATIVES This is the value of the given phenomenon in any season expressed as a percentage of its value in the preceding season. This involves the following steps: o Convert the original data into link relatives.
- 10. o Average these link relatives for each month. o Convert Link Relatives into Chain relatives. o Obtain CR for the first month o Obtain Corrected Chain relatives. Problem 8: Wheat Prices (10 Kgs.) Year 1990 1991 1992 1993 Quarter I Qtr. (Jan- Mar) 75 86 90 100 II Qtr. (Apr – June) 60 65 72 78 III Qtr. (Jul – Sept.) 54 63 66 72 IV Qtr. (Oct. – Dec.) 59 80 85 93 Answer: Note: Link Relatives for any month = (Current Month’s Value / Previous Month’s Value) * 100 Chain Relative for any month = (Link Relative of that month * Chain Relative of the preceding month) / 100 New CR for the First Quarter: (LR of I Qtr. * CR of last Qtr.)/100 (123.303 * 89.81) / 100 =112.54 d = ¼(New CR of first Qtr. -100) = ¼(112.54 – 100) = 3.135
- 11. Adjusted CR: 78.395 – 3.135 = 75.26; 72.69 – 6.27 = 66.42; 89.81 – 9.405 = 80.41 Year I Quarter II Quarter III Quarter IV Quarter 1990 - 80 90 109.26 1991 145.76 75.58 96.92 126.98 1992 112.5 80 91.67 128.79 1993 117.65 78 92.31 129.17 Total 375.91 313.58 370.90 494.20 Average 125.3 78.395 92.725 123.55 Chain Relative 100 78.395 72.69 89.81 Adjusted CR 100 75.26 66.42 80.41 322.09 Seasonal Indices 124.2 93.47 82.49 99.87 400 CYCLICAL VARIATIONS This is an approximate or crude method of measuring cyclical variations, which consists of estimating trend, seasonal components and then eliminating their effect from the given Time Series. RANDOM VARIATIONS These can not be estimated accurately, we can not obtain an estimate the variance of random components.

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