Moving average method maths ppt

ppt on maths

1. Moving Average Method & Method Of Least Squares By:- HEEMA SUMANT & ABHISHEK

2. Moving average method A quantitative method of forecasting or smoothing a time series by averaging each successive group (no. of observations) of data values. term MOVING is used because it is obtained by summing and averaging the values from a given no of periods, each time deleting the oldest value and adding a new value.

3. For applying the method of moving averages the period of moving averages has to be selected This period can be 3- yearly moving averages 5yr moving averages 4yr moving averages etc. For ex:- 3-yearly moving averages can be calculated from the data : a, b, c, d, e, f can be computed as :

4. If the moving average is an odd no of values e.g., 3 years, there is no problem of centring it. Because the moving total for 3 years average will be centred besides the 2nd year and for 5 years average be centred besides 3rd year. But if the moving average is an even no, e.g., 4 years moving average, then the average of 1st 4 figures will be placed between 2nd and 3rd year. This process is called centering of the averages. In case of even period of moving averages, the trend values are obtained after centering the averages a second time.

5. Goals : – Smooth out the short-term fluctuations. Identify the long-term trend.

6. MERITS Of Moving average method simple method. flexible method. OBJECTIVE :- If the period of moving averages coincides with the period of cyclic fluctuations in the data , such fluctuations are automatically eliminated This method is used for determining seasonal, cyclic and irregular variations beside the trend values.

7. LIMITATIONS Of Moving average method No trend values for some year. M.A is not represented by mathematical function - not helpful in forecasting and predicting. The selection of the period of moving average is a difficult task. In case of non-linear trend the values obtained by this method are biased in one or the other direction.  

8. Moving Average Example Year Units Moving 1994 2 1995 5 3 1996 2 3 1997 2 3.67 1998 7 5 1999 6 John is a building contractor with a record of a total of 24 single family homes constructed over a 6-year period. Provide John with a 3-year moving average graph. Avg.

9. Moving Average Example Solution Year Response Moving Avg. 1994 2 1995 5 3 1996 2 3 1997 2 3.67 1998 7 5 1999 6 94 95 96 97 98 99 8 6 4 2 Sales L = 3 No MA for 2 years

10. Calculation of moving average based on period When period is odd- example:- Calculate the 3-yearly moving averages of the data given below: yrs 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 Sales (million of rupees) 3 4 8 6 7 11 9 10 14 12

11. year Sales,(millions of rupees) 3-yearly totals 3-yearly moving averages(trends) 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 3 4 8 6 7 1 1 9 1 0 1 4 1 2 1 5 1 8 2 1 2 4 2 7 3 0 3 3 3 6 5=(1 5/3) 6=(1 8/3) 7=(2 1/3) 8=(2 4/3) 9=(2 7/3) 1 0=(3 0/3) 1 1=(3 3/3) 1 2=(3 6/3)

12. In Figure, 3-yrs MA plotted on graph fall on a straight line, and the cyclic f luctuation have been smoothed out. The straight Line is the required trend line. 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1 2 3 years sales 4 6 8 10 12 Actual line Trend line (1981,5) (1983,7) (1984,8) (1988,12) (1988,14) (1985,11)

13. Calculation of moving average based on period When period is even:- Example :- Compute 4-yearly moving averages from the following data: year 1991 1992 1993 1994 1995 1996 1997 1998 Annual sale (Rs in crores ) 36 43 43 34 44 54 34 24

14. Year (1) Annual sales (Rs in crores) (2) 4-yearly moving total (T) (3) 4-yearly moving averages (A) (3)/4 {4} 4-yearly centred moving averages OR (trend values) (5) 1991 36 1992 43 156 39 1993 43 ( 39+41)/2=80/2=40 164 41 1994 34 (41+43.75)/2=84.75/2= 42.375 175 43.75 1995 44 (43.75+41.50)/2=42.625 166 41.50 1996 54 (41.50+39)/2=40.25 156 39 1997 34 1998 24

15. sales year Actual line Trend line

16. Method of least squares This is the best method for obtaining trend values. It provides a convenient basis for obtaining the line of best fit in a series. Line of the best fit is a line from which the sum of the deviations of various points on its either side is zero. The sum of the squares of the deviation of various points from the line of best fit is the least. – That is why this method is known as method of least squares.

17. Method of least squares Least squares, also used in regression analysis, determines the unique trend line forecast which minimizes the mean squares of deviations. The independent variable is the time period and the dependent variable is the actual observed value in the time series equation of straight line trend: Ŷ=a+bX

18. b = ∑XY - ∑X 2 -n 2 a= -b Where, a = Y-intercept b = slope of the best-fitting estimating line. X = value of independent variable Y = value of dependent variable = mean of the values of the independent variable = mean of the values of the dependent variable

19. When =0 then b= ∑XY ∑X 2 a= =∑Y/N

20. MERITS This method gives the trend values for the entire time period. This method can be used to forecast future trend because trend line establishes a functional relationship between the values and the time. This is a completely objective method.

21. LIMITATIONS It requires some amount of calculations and may appear tedious and complicated for some. Future forecasts made by this method are based only on trend values; seasonal, cyclical or irregular variations are ignored. If even a single item is added to the series a new equation has to be formed.

22. Example:- Determine trend line:- Year : 2000 2001 2002 2003 2004 Sales(in Rs ‘000): 35 56 79 80 40

23. Calculation Year (x) Sales (Y) X X 2 XY 2000 2001 2002 2003 2004 35 56 79 80 40 -2 -1 1 2 4 1 1 4 -70 -56 80 80 TOTAL ∑Y=290 ∑X=0 ∑X 2 =10 ∑XY=34

24. Calculation Now a = =∑ Y/ N =290/ 5=58 and b = ∑XY/∑X =34/10 =3.4 Substituting these values in equation of trend line which is Y=58+3.4X ,with 2002=0 Year (x) X =x-2002 Trend values (Y=58+3.4X) 2000 2001 2002 2003 2004 -2 -1 1 2 58+3.4×(-2)=51.2 58+3.4×(-1)=54.6 58+3.4×(0)=58.0 58+3.4×(1)=61.4 58+3.4×(2)=64.8

25. A problem involving all four component of a time series firm that specializes in producing recreational equipment . To forecast future sale firm has collected the information. Given time series -1)trend 2)cyclic 3)seasonal Quarterly sales Sales per quarter(× $10,000) Year I II III IV 1991 16 21 9 18 1992 15 20 10 18 1993 17 24 13 22 1994 17 25 11 21 1995 18 26 14 25

26. Solution Procedure for describing information in time series consist of four stages:- 1. finding seasonal indices- using moving average method 2. Deseasonalized the given data. 3. Developing the trend line. 4.Finding the cyclical variation around the trend line.

27. Calculating the Seasonal Indexes 1. Compute a series of n -period centered moving averages, where n is the number of seasons in the time series. 2. If n is an even number, compute a series of 2-period centered moving averages. 3. Divide each time series observation by the corresponding centered moving average to identify the seasonal-irregular effect in the time series. 4. For each of the n seasons, average all the computed seasonal-irregular values for that season to eliminate the irregular influence and obtain an estimate of the seasonal influence, called the seasonal index , for that season.

28. Deseasonalizing the Time Series The purpose of finding seasonal indexes is to remove the seasonal effects from the time series. This process is called deseasonalizing the time series. By dividing each time series observation by the corresponding seasonal index, the result is a deseasonalized time series. With deseasonalized data, relevant comparisons can be made between observations in successive periods.

29. Calculation of 4-Qr centered moving average : Year (1) Quarter (2) Actual sales (3) Step 1: 4-Qr moving total (4) Step 3: 4-Qr centered moving average (6) Step:4 % of actual to moving averages (7)={(3)×100}÷(6) 1991 16 21 9 18 64 63 15.875 15.625 56.7 115.2 1992 15 20 10 18 62 63 63 65 15.625 15.750 16.000 16.750 96.0 127.0 62.5 107.5 1993 17 24 13 22 69 72 76 76 17.625 18.500 19.000 19.125 96.5 129.7 68.4 115.0

30. year (1) quarter (2) Actual sales (3) Step 1: 4-Qr moving total (4) Step 2: 4-Qr moving average (5)=(4)÷4 Step 3: 4-Qr centered moving average (6) Step:4 % of actual to moving averages (7)={(3)×100}÷(6) 1994 17 25 11 21 77 75 74 75 19.25 18.75 18.50 18.75 19.000 18.625 18.625 18.875 89.5 134.2 59.1 111.3 1995 18 26 14 25 76 79 83 19.00 19.75 20.75 19.375 20.250 92.9 128.4

31. Computing the seasonal index Year I II III IV 1991 - - 56.7 115.2 96.0 127.0 62.5 107.5 1993 96.5 129.7 68.4 115.0 1994 89.5 134.2 59.1 111.3 92.9 128.4 - - modified sum=188.9 258.1 121.6 226.3 modified mean: Qr I: 188÷2=94.45 II: 258.1÷2=129.05 III: 121.6÷2=60.80 IV: 226.3÷2=113.15 397.45 Quarter indices × Adjusting factor = seasonal indices =400/397.45=1.0064 I 94.45 1.0064 = 95.1 II 129.05 1.0064 = 129.9 III 60.80 1.0064 = 61.2 IV 113.15 1.0064 = 113.9 sum of seasonal indices = 400.1

32. Calculation 0f deaseasonalised time series values Year (1) Quarter (2) Actual sales (3) Seasonal index/100 (4) Deseasonalized sales (5)=(3)÷(4) 1991 I II III IV 16 21 9 18 0.951 1.299 0.612 1.139 16.8 16.2 14.7 15.8 1992 I II III IV 15 20 10 18 0.951 1.299 0.612 1.139 15.8 15.4 16.3 15.8 1993 I II III IV 17 24 13 22 0.951 1.299 0.612 1.139 17.9 18.5 21.2 19.3 1994 I II III IV 17 25 11 21 0.951 1.299 0.612 1.139 17.9 19.2 18.0 18.4 1995 I II III IV 18 26 14 25 0.951 1.299 0.612 1.139 18.9 20.0 22.9 21.9

33. Identifying the trend component Year (1) Qr (2) Deseasonalized sales (3) Translating or coding the time variable(4) x (5)=(4)×2 xY (6)=(5)×(3) x² (7)=(5)² 1991 I II III IV 16.8 16.2 14.7 15.8 -9.5 -8.5 -7.5 -6.5 -19 -17 -15 -13 -319.2 -275.4 -220.5 -205.4 361 289 225 169 1992 I II III IV 15.8 15.4 16.3 15.8 -5.5 -4.5 -3.5 -2.5 -11 -9 -7 -5 -173.8 -138.6 -114.1 -79.0 121 81 49 25 1993 Mean I II III IV 17.9 18.5 21.2 19.3 -1.5 -0.5 0* 0.5 1.5 -3 -1 1 3 -53.7 -18.5 21.2 57.9 9 1 1 9 1994 I II III IV 17.9 19.2 18.0 18.4 2.5 3.5 4.5 5.5 5 7 9 11 89.5 134.4 162.0 202.4 25 49 81 121 1995 I II III IV 18.9 20.0 22.9 21.9 ∑Y=360.9 6.5 7.5 8.5 9.5 13 15 17 19 245.7 300.0 389.3 416.1 169 225 189 361 We assign mean=0 ,b/w II&III(1993) & measure translated time,x,by 0.5 because periods is even

34. b= ∑XY ∑X 2 a= =∑Y/N

35. Identifying the cyclical variation Year (1) Quarter (2) Deseasonalised sales (3) Ŷ= a+bx (4) (Y×100)/Ŷ percent of trend (5) 1991 I II III IV 16.8 16.2 14.7 15.8 18+0.16(-19)=14.96 18+0.16(-17)=15.28 18+0.16(-15)=15.60 18+0.16(-13)=15.92 112.3 106.0 94.2 99.2 1992 I II III IV 15.8 15.4 16.3 15.8 18+0.16(-11)=16.24 18+0.16(-9)=16.56 18+0.16(-7)=16.88 18+0.16(-5)=17.20 97.3 93.0 96.6 91.9 1993 I II III IV 17.9 18.5 21.2 19.3 18+0.16(-3)=17.52 18+0.16(-1)=17.84 18+0.16(1)=18.16 18+0.16(3)=18.48 102.2 103.7 116.7 104.4 1994 I II III IV 19.9 19.2 18.0 18.4 18+0.16(5)=18.80 18+0.16(7)=19.12 18+0.16(9)=19.44 18+0.16(11)=19.76 95.2 100.4 92.6 93.1 1995 I II III IV 18.9 20.0 22.9 21.9 18+0.16(13)=20.08 18+0.16(15)=20.40 18+0.16(17)=20.72 18+0.16(19)=21.04 94.1 98.0 110.5 104.1

36. Graph of time series , trend line and 4-Qr centered moving average for sales data Time series (all Components) Ŷ=18+0.16X (trend only) 4-Qr centered moving average (both trend & cyclical component) sales X=0