Moving average method maths ppt

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<p>Moving Average Method &amp; Least Square Method</p> <p>Moving Average Method&amp;Method Of Least SquaresBy:-HEEMA SUMANT&amp; ABHISHEKMoving average method </p> <p> A quantitative method of forecasting or smoothing a time series by averaging each successive group (no. of observations) of data values. </p> <p>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.</p> <p>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 : </p> <p> 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.</p> <p>Goals : Smooth out the short-term fluctuations.</p> <p> Identify the long-term trend. </p> <p>MERITS Of Moving average method</p> <p> 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 eliminatedThis method is used for determining seasonal, cyclic and irregular variations beside the trend values.</p> <p> LIMITATIONS Of Moving average method</p> <p> 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.Moving Average ExampleYear Units Moving1994 2 1995 5 3 1996 2 31997 2 3.671998 7 51999 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.</p> <p>Avg.Moving Average Example SolutionYear Response Moving Avg.1994 2 1995 5 3 1996 2 31997 2 3.671998 7 51999 6 94 95 96 97 98 99 8 6 4 2 0SalesL = 3No MA for 2 years Calculation of moving average based on period</p> <p>When period is odd- example:- Calculate the 3-yearly moving averages of the data given below:yrs1980198119821983198419851986198719881989Sales (million of rupees) 3 4 8 6 7 11 9 10 14 12 yearSales,(millions of rupees)3-yearly totals3-yearly moving averages(trends)1980198119821983198419851986198719881989 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) 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.</p> <p> 1980198119821983198419851986198719881989123yearssales4681012Actual lineTrend line(1981,5)(1983,7)(1984,8)(1988,12)(1988,14)(1985,11)Calculation of moving average based on period</p> <p>When period is even:-</p> <p>Example :- Compute 4-yearly moving averages from the following data:year19911992199319941995199619971998Annual sale(Rs in crores)3643433444543424Year (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 361992 43 156 391993 43(39+41)/2=80/2=40 164 411994 34(41+43.75)/2=84.75/2= 42.375 175 43.751995 44(43.75+41.50)/2=42.625 166 41.501996 54(41.50+39)/2=40.25 156 391997 341998 24</p> <p> salesyearActual lineTrend lineMethod of least squaresThis 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.</p> <p>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 seriesequation of straight line trend: Y=a+bX</p> <p> b = XY - X2 -n 2 </p> <p> a= -b </p> <p> 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 </p> <p>When =0 then b= XY X2 a= =Y/N</p> <p>MERITSThis 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.</p> <p>LIMITATIONSIt 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.Example:-Determine trend line:-Year:</p> <p>20002001200220032004Sales(in Rs 000):</p> <p>3556798040CalculationYear (x)Sales (Y)XX2XY200020012002200320043556798040-2-101241014-70-5608080TOTALY=290X=0X2 =10XY=34CalculationNow a = = Y/ N =290/ 5=58 and b = XY/X =34/10 =3.4Substituting these values in equation of trend line which is Y=58+3.4X ,with 2002=0</p> <p>Year (x) X=x-2002Trend values (Y=58+3.4X)20002001200220032004 -2 -1 0 1 258+3.4(-2)=51.258+3.4(-1)=54.658+3.4(0)=58.058+3.4(1)=61.458+3.4(2)=64.8</p> <p>A problem involving all four component of a time seriesfirm 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 IV1991 16 21 9 181992 15 20 10 181993 17 24 13 221994 17 25 11 211995 18 26 14 25Solution 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.Calculating the Seasonal Indexes1. 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.Deseasonalizing the Time SeriesThe 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.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 </p> <p> 1621</p> <p>9 1864</p> <p>63</p> <p>15.875</p> <p>15.62556.7</p> <p>115.21992 </p> <p> 15</p> <p>20</p> <p>10</p> <p>1862</p> <p>63</p> <p>63</p> <p>6515.625</p> <p>15.750</p> <p>16.000</p> <p>16.75096.0</p> <p>127.0</p> <p>62.5</p> <p>107.51993 </p> <p> 17</p> <p>24</p> <p>13</p> <p>2269</p> <p>72</p> <p>76</p> <p>76</p> <p>17.625</p> <p>18.500</p> <p>19.000</p> <p>19.12596.5</p> <p>129.7</p> <p>68.4</p> <p>115.0 year (1) quarter (2)Actual sales (3)Step 1: 4-Qr moving total (4)Step 2: 4-Qr moving average (5)=(4)4Step 3: 4-Qr centered moving average (6)Step:4 % of actual to moving averages (7)={(3)100}(6)1994 </p> <p> 17</p> <p>25</p> <p>11</p> <p>2177</p> <p>75</p> <p>74</p> <p>7519.25</p> <p>18.75</p> <p>18.50</p> <p>18.7519.000</p> <p>18.625</p> <p>18.625</p> <p>18.87589.5</p> <p>134.2</p> <p>59.1</p> <p>111.31995 </p> <p> 18</p> <p>26</p> <p>142576</p> <p>79</p> <p>83</p> <p>19.00</p> <p>19.75</p> <p>20.7519.375</p> <p>20.250</p> <p>92.9</p> <p>128.4</p> <p>Computing the seasonal indexYear I II III IV1991 - - 56.7 115.2 96.0 127.0 62.5 107.51993 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: 1882=94.45 II: 258.12=129.05 III: 121.62=60.80 IV: 226.32=113.15 397.45Quarter indices Adjusting factor = seasonal indices =400/397.45=1.0064 I 94.45 1.0064 = 95.1 II 129.05 1.0064 = 129.9III 60.80 1.0064 = 61.2IV 113.15 1.0064 = 113.9 sum of seasonal indices = 400.1Calculation 0f deaseasonalised time series valuesYear (1)Quarter (2)Actual sales (3)Seasonal index/100(4)Deseasonalized sales (5)=(3)(4)1991IIIIIIIV16219180.9511.2990.6121.13916.816.214.715.81992</p> <p>IIIIIIIV152010180.9511.2990.6121.13915.815.416.315.81993</p> <p>IIIIIIIV172413220.9511.2990.6121.13917.918.521.219.31994</p> <p>IIIIIIIV172511210.9511.2990.6121.13917.919.218.018.41995</p> <p>IIIIIIIV182614250.9511.2990.6121.13918.920.022.921.9Identifying the trend componentYear (1)Qr (2)Deseasonalized sales (3)Translating or coding the time variable(4)x (5)=(4)2xY (6)=(5)(3) x (7)=(5)1991</p> <p>IIIIIIIV 16.8 16.2 14.7 15.8-9.5-8.5-7.5-6.5-19-17-15-13 -319.2-275.4-220.5-205.43612892251691992</p> <p>IIIIIIIV 15.8 15.4 16.3 15.8-5.5-4.5-3.5-2.5-11-9-7-5-173.8-138.6-114.1-79.01218149251993</p> <p>MeanIII</p> <p>IIIIV 17.9 18.5</p> <p> 21.2 19.3-1.5-0.5 0*0.51.5-3-1</p> <p>13-53.7-18.5</p> <p>21.257.991</p> <p>191994IIIIIIIV 17.9 19.2 18.0 18.42.53.54.55.55791189.5134.4162.0202.42549811211995IIIIIIIV 18.9 20.0 22.9 21.9 Y=360.96.57.58.59.513151719245.7300.0389.3416.1169225189361We assign mean=0 ,b/w II&amp;III(1993) &amp; measure translated time,x,by 0.5 because periods is even b= XY X2 a= =Y/NIdentifying the cyclical variationYear (1)Quarter (2)</p> <p>Deseasonalised sales (3)Y=a+bx (4)(Y100)/Y percent of trend (5)1991</p> <p>IIIIIIIV16.816.214.715.818+0.16(-19)=14.96 18+0.16(-17)=15.28 18+0.16(-15)=15.60 18+0.16(-13)=15.92112.3106.094.299.21992IIIIIIIV15.815.416.315.818+0.16(-11)=16.24 18+0.16(-9)=16.56 18+0.16(-7)=16.88 18+0.16(-5)=17.2097.393.096.691.91993IIIIIIIV17.918.521.219.318+0.16(-3)=17.52 18+0.16(-1)=17.84 18+0.16(1)=18.16 18+0.16(3)=18.48102.2103.7116.7104.41994IIIIIIIV19.919.218.018.418+0.16(5)=18.80 18+0.16(7)=19....</p>

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