Interpretation: The best smoothing constant needs to be determined by evaluating the MSE .
Concept Introduction: Single Exponential Smoothing is a method which computes the weighted average of previous sales data to forecast the future sales value.
Explanation of Solution
Following formula can be used to calculate the forecasted values:
The Mean Square Error for various values of a is as follows:
At a = 0.1
Period | Observation | Forecast | Error | Squared Error |
1 | 1623 | 1623 | ||
2 | 1533 | 1623 | 90 | 8100 |
3 | 1455 | 1614 | 159 | 25281 |
4 | 1386 | 1598.1 | 212.1 | 44986.41 |
5 | 1209 | 1576.89 | 367.89 | 135343.0521 |
6 | 1348 | 1540.101 | 192.101 | 36902.7942 |
7 | 1581 | 1520.8909 | 60.1091 | 3613.103903 |
8 | 1332 | 1526.90181 | 194.9018 | 37986.71554 |
9 | 1245 | 1507.411629 | 262.4116 | 68859.86303 |
10 | 1521 | 1481.170466 | 39.8295 | 1586.38907 |
11 | 1421 | 1485.153419 | 64.15342 | 4115.661232 |
12 | 1502 | 1478.738078 | 23.2619 | 541.1159916 |
13 | 1656 | 1481.06427 | 174.396 | 30413.96482 |
14 | 1614 | 1498.557843 | 115.442 | 13326.85536 |
15 | 1332 | 1510.102059 | 178.1021 | 31720.34325 |
16 | 1492.291853 | SUM | 442777.2685 | |
MSE | 34059.79 |
At a = 0.2
Period | Observation | Forecast | Error | Squared Error |
1 | 1623 | 1623 | ||
2 | 1533 | 1623 | 90 | 8100 |
3 | 1455 | 1605 | 150 | 22500 |
4 | 1386 | 1575 | 212.1 | 44986.41 |
5 | 1209 | 1537.2 | 367.89 | 135343.0521 |
6 | 1348 | 1471.56 | 123.56 | 15267.0736 |
7 | 1581 | 1446.848 | 134.152 | 17996.7591 |
8 | 1332 | 1473.6784 | 141.6784 | 20072.76903 |
9 | 1245 | 1445.34272 | 200.3427 | 40137.20546 |
10 | 1521 | 1405.274176 | 39.8295 | 1586.38907 |
11 | 1421 | 1428.419341 | 7.419341 | 55.04661791 |
12 | 1502 | 1426.935473 | 23.2619 | 541.1159916 |
13 | 1656 | 1441.948378 | 174.396 | 30413.96482 |
14 | 1614 | 1484.758702 | 115.442 | 13326.85536 |
15 | 1332 | 1510.606962 | 178.607 | 31900.44687 |
16 | 1474.88557 | SUM | 382227.088 | |
MSE | 29402.08 |
At a = 0.3
Period | Observation | Forecast | Error | Squared Error |
1 | 1623 | 1623 | ||
2 | 1533 | 1623 | 90 | 8100 |
3 | 1455 | 1596 | 141 | 19881 |
4 | 1386 | 1553.7 | 212.1 | 44986.41 |
5 | 1209 | 1503.39 | 367.89 | 135343.0521 |
6 | 1348 | 1415.073 | 67.073 | 4498.787329 |
7 | 1581 | 1394.9511 | 186.0489 | 34614.19319 |
8 | 1332 | 1450.76577 | 118.7658 | 14105.30812 |
9 | 1245 | 1415.136039 | 170.136 | 28946.27177 |
10 | 1521 | 1364.095227 | 39.8295 | 1586.38907 |
11 | 1421 | 1411.166659 | 9.83334 | 96.69457556 |
12 | 1502 | 1414.116661 | 23.2619 | 541.1159916 |
13 | 1656 | 1440.481663 | 174.396 | 30413.96482 |
14 | 1614 | 1505.137164 | 115.442 | 13326.85536 |
15 | 1332 | 1537.796015 | 205.796 | 42351.99973 |
16 | 1476.05721 | SUM | 378792.0421 | |
MSE | 29137.84 |
At a = 0.4
Period | Observation | Forecast | Error | Squared Error |
1 | 1623 | 1623 | ||
2 | 1533 | 1623 | 90 | 8100 |
3 | 1455 | 1587 | 132 | 17424 |
4 | 1386 | 1534.2 | 212.1 | 44986.41 |
5 | 1209 | 1474.92 | 367.89 | 135343.0521 |
6 | 1348 | 1368.552 | 20.552 | 422.384704 |
7 | 1581 | 1360.3312 | 220.6688 | 48694.71929 |
8 | 1332 | 1448.59872 | 116.5987 | 13595.26151 |
9 | 1245 | 1401.959232 | 156.9592 | 24636.20051 |
10 | 1521 | 1339.175539 | 39.8295 | 1586.38907 |
11 | 1421 | 1411.905324 | 9.83334 | 96.69457556 |
12 | 1502 | 1415.543194 | 23.2619 | 541.1159916 |
13 | 1656 | 1450.125916 | 174.396 | 30413.96482 |
14 | 1614 | 1532.47555 | 115.442 | 13326.85536 |
15 | 1332 | 1565.08533 | 233.0853 | 54328.77103 |
16 | 1471.851198 | SUM | 393495.819 | |
MSE | 30268.909 |
At a = 0.5
Period | Observation | Forecast | Error | Squared Error |
1 | 1623 | 1623 | ||
2 | 1533 | 1623 | 90 | 8100 |
3 | 1455 | 1578 | 123 | 15129 |
4 | 1386 | 1516.5 | 212.1 | 44986.41 |
5 | 1209 | 1451.25 | 367.89 | 135343.0521 |
6 | 1348 | 1330.125 | 17.875 | 319.515625 |
7 | 1581 | 1339.0625 | 241.9375 | 58533.75391 |
8 | 1332 | 1460.03125 | 128.0313 | 16392.00098 |
9 | 1245 | 1396.015625 | 151.0156 | 22805.71899 |
10 | 1521 | 1320.507813 | 39.8295 | 1586.38907 |
11 | 1421 | 1420.753906 | 9.83334 | 96.69457556 |
12 | 1502 | 1420.876953 | 23.2619 | 541.1159916 |
13 | 1656 | 1461.438477 | 174.396 | 30413.96482 |
14 | 1614 | 1558.719238 | 115.442 | 13326.85536 |
15 | 1332 | 1586.359619 | 254.3596 | 64698.81585 |
16 | 1459.17981 | SUM | 412273.2873 | |
MSE | 31713.329 |
At a = 0.6
Period | Observation | Forecast | Error | Squared Error |
1 | 1623 | 1623 | ||
2 | 1533 | 1623 | 90 | 8100 |
3 | 1455 | 1569 | 114 | 12996 |
4 | 1386 | 1500.6 | 212.1 | 44986.41 |
5 | 1209 | 1431.84 | 367.89 | 135343.0521 |
6 | 1348 | 1298.136 | 17.875 | 319.515625 |
7 | 1581 | 1328.0544 | 252.9456 | 63981.47656 |
8 | 1332 | 1479.82176 | 147.8218 | 21851.27273 |
9 | 1245 | 1391.128704 | 146.1287 | 21353.59813 |
10 | 1521 | 1303.451482 | 39.8295 | 1586.38907 |
11 | 1421 | 1433.980593 | 9.83334 | 96.69457556 |
12 | 1502 | 1426.192237 | 23.2619 | 541.1159916 |
13 | 1656 | 1471.676895 | 174.396 | 30413.96482 |
14 | 1614 | 1582.270758 | 115.442 | 13326.85536 |
15 | 1332 | 1601.308303 | 269.3083 | 72526.96216 |
16 | 1439.723321 | SUM | 427423.3071 | |
MSE | 32878.71 |
At a = 0.7
Period | Observation | Forecast | Error | Squared Error |
1 | 1623 | 1623 | ||
2 | 1533 | 1623 | 90 | 8100 |
3 | 1455 | 1560 | 105 | 11025 |
4 | 1386 | 1486.5 | 212.1 | 44986.41 |
5 | 1209 | 1416.15 | 367.89 | 135343.0521 |
6 | 1348 | 1271.145 | 17.875 | 319.515625 |
7 | 1581 | 1324.9435 | 256.0565 | 65564.93119 |
8 | 1332 | 1504.18305 | 172.1831 | 29647.00271 |
9 | 1245 | 1383.654915 | 138.6549 | 19225.18545 |
10 | 1521 | 1286.596475 | 39.8295 | 1586.38907 |
11 | 1421 | 1450.678942 | 9.83334 | 96.69457556 |
12 | 1502 | 1429.903683 | 23.2619 | 541.1159916 |
13 | 1656 | 1480.371105 | 174.396 | 30413.96482 |
14 | 1614 | 1603.311331 | 115.442 | 13326.85536 |
15 | 1332 | 1610.793399 | 278.7934 | 77725.75957 |
16 | 1415.63802 | SUM | 437901.8765 | |
MSE | 33684.75 |
At a = 0.8
Period | Observation | Forecast | Error | Squared Error |
1 | 1623 | 1623 | ||
2 | 1533 | 1623 | 90 | 8100 |
3 | 1455 | 1551 | 96 | 9216 |
4 | 1386 | 1474.2 | 212.1 | 44986.41 |
5 | 1209 | 1403.64 | 367.89 | 135343.0521 |
6 | 1348 | 1247.928 | 17.875 | 319.515625 |
7 | 1581 | 1327.9856 | 253.0144 | 64016.28661 |
8 | 1332 | 1530.39712 | 198.3971 | 39361.41722 |
9 | 1245 | 1371.679424 | 126.6794 | 16047.67646 |
10 | 1521 | 1270.335885 | 39.8295 | 1586.38907 |
11 | 1421 | 1470.867177 | 9.83334 | 96.69457556 |
12 | 1502 | 1430.973435 | 23.2619 | 541.1159916 |
13 | 1656 | 1487.794687 | 174.396 | 30413.96482 |
14 | 1614 | 1622.358937 | 115.442 | 13326.85536 |
15 | 1332 | 1615.671787 | 283.6718 | 80469.68301 |
16 | 1388.734357 | SUM | 443825.0609 | |
MSE | 34140.38 |
At a = 0.9
Period | Observation | Forecast | Error | Squared Error |
1 | 1623 | 1623 | ||
2 | 1533 | 1623 | 90 | 8100 |
3 | 1455 | 1542 | 87 | 7569 |
4 | 1386 | 1463.7 | 212.1 | 44986.41 |
5 | 1209 | 1393.77 | 367.89 | 135343.0521 |
6 | 1348 | 1227.477 | 17.875 | 319.515625 |
7 | 1581 | 1335.9477 | 245.0523 | 60050.62974 |
8 | 1332 | 1556.49477 | 224.4948 | 50397.90176 |
9 | 1245 | 1354.449477 | 109.4495 | 11979.18802 |
10 | 1521 | 1255.944948 | 39.8295 | 1586.38907 |
11 | 1421 | 1494.494495 | 9.83334 | 96.69457556 |
12 | 1502 | 1428.349449 | 23.2619 | 541.1159916 |
13 | 1656 | 1494.634945 | 174.396 | 30413.96482 |
14 | 1614 | 1639.863494 | 115.442 | 13326.85536 |
15 | 1332 | 1616.586349 | 284.5863 | 80989.39029 |
16 | 1360.458635 | SUM | 445700.1073 | |
MSE | 34284.623 |
After evaluating the MSE of all
After comparing the MSEs of Exponential smoothing method performed in this question and moving average method in previous problem, The moving average method seems better option based on the given data.
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Chapter 9 Solutions
OM6 ONLINE-LMS INTEGRATED ACCESS
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