3. Trend Component
• a long-term increase or decrease in the datawhich
might not be linear. Sometimes the trend might
changedirection as time increases
Back
4. Seasonal Component
• exists when a series exhibits regularfluctuations based
on the season (e.g. every month/quarter/year).
Seasonality is always of a fixed and known period.
6. Trend Projection
• We can calculate trend projection with the method of least squares, the formula for the trend projection is shown below :
• Tt = b0 + b1t
• Where the symbol mean :
• Tt : Trend forecast for the time period t
• b1 : slope of the trend line
• b0 : trend line projection for the time 0
• And we also have to find b0 too, and the formula is down below :
• Where the symbol mean :
• Yt = observed value of the time series at time period t
• = average time period for the n observations
• = average of the observed values for Yt
7. • but we have to find the b1 first, and we can find
with the formula below:
8. • And we also have to find b0 too, and the formula is down
below :
Yt = observed value of the time series at time period t
= average time period for the n observations
= average of the observed values for Yt
9. Example
• The number of MONSTA X show appeared in korean
music bank in each of the last 12 month is listed on the
table below. Forecast the number of monsta x show’s will
perform in the next year on the first month (January) using
the least square methodThe number of MONSTA X show
appeared in korean music bank in each of the last 12
month is listed on the table below. Forecast the number of
monsta x show’s will perform in the next year on the first
month (January) using the least square method
10. MONTH MX SHOWS
January 23
February 30
March 29
April 40
May 55
June 42
July 65
August 78
September 73
October 68
November 70
December 80
11. MONTH t Yt tYt t2
January 1 23 23 1
February 2 30 60 4
March 3 29 87 9
April 4 40 160 16
May 5 55 275 25
June 6 42 252 36
July 7 65 455 49
August 8 78 624 64
September 9 73 657 81
October 10 68 680 100
November 11 70 770 121
December 12 80 960 144
SUM 78 653 5003 650
12. • = 78 / 12 = 6.5
• = 653 / 12 = 54.4166667 using round up formula in excel
54.42
• b1 = (12)(5003)-(78)(653)/(12)(650)-(78)^ = 5.3041958
• using round up formula in excel 5.31
• b0 = - b1 = 54.42 – 5.31(6.5) = 19.905 with roundup to 19.91
• T13 = 19.91 + (5.31)(13) = 88.94
Y
Y
13. Monsta x appear in music bank
• forecast for January (month 13) using a three-period
(n=3) weighted moving average with weight of .5, .3, .2
for the newest to oldest data, respectively. Then,
compare this month 13 weighted moving average
forecast with the month 13 trend projection forecast.
14. Three-month weighted moving average
• The forecast for January will be the weighted average of
the preceding three months: October, November,
December
• F13 = .2 Y oct + .3Y nov + .5Y dec
• = .2(68)+.3(70)+.5(80) = 74.6
16. Note : The meaning of the symbols
• Tt : Trend forecast for the time period t
• b1 : slope of the trend line
• b0 : trend line projection for the time 0
17. Trend and Seasonal
• A table below will explain Evelyn’s Bakery Shop weekly
sales during four different seasons; 1.) Mother’s Day 2.)
Christmas Day, 3.) New Year, and 4.) Valentine’s DaySeason
Year Mother’s Day Christmas Day New Year Valentine
’s Day
2015 1999 3867 5346 3097
2016 2005 2345 7777 1089
2017 1857 8907 8234 2011
2018 1230 6754 1089 6435
18. First Step : Calculate CMA (Centered Moving Average)
There are three distinct seasons in each year.
Hence, take a three-season moving average to
eliminate seasonal and irregular factors.
• 1st CMA: 1999+3867+5346+3097/4 = 3577,25
• 2nd CMA: 3867+5346+3097+2005/4 =
3578,75
• 3rd CMA: 5346+3097+2005+2345/4 = 3198,25
• 4th CMA: 3097+2005+2345+7777/4= 3806
• Etc.
19. Second Step: Center the CMAs on integer-valued periods
• The first centered moving average computed in step 1 (11986,25) will be centered on
season 2 of year 1. Note that the moving averages from step 1 center themselves on
integer-valued periods because n is an odd number.
Year Season Dollar Sales (Yt) Moving Average
2015 1 1999
2 3867 3577,25
3 5346 3578,75
4 3097 3198,25
2016 1 2005 3806
2 2345 3304
3 7777 3267
4 1089 4907,5
2017 1 1857 5021,75
2 8907 5252,25
3 8234 5095,5
4 2011 4557,25
2018 1 1230 2771
2 6754 3881,5
3 1089
4 6435
20. Third Step: Determine the seasonal and irregular factors
(St It )
Isolate the trend and cyclical components. For each period t, this is given by:
St It = Yt /(Moving Average for period t )
Year Season Dollar Sales (Yt) Moving Average StIt
2015 1 1999
2 3867 3577,25 1,08 (3867/3577,25)
3 5346 3578,75 1,49
4 3097 3198,25 0,96
2016 1 2005 3806 0,52
2 2345 3304 0,7
3 7777 3267 2,38
4 1089 4907,5 0,22
2017 1 1857 5021,75 0,36
2 8907 5252,25 1,69
3 8234 5095,5 1,61
4 2011 4557,25 0,44
2018 1 1230 2771 0,44
2 6754 3881,5 1,74
3 1089
4 6435
21. Fourth Step: Determine the average seasonal factors
Averaging all St It values corresponding to that
season:
Season 1: 0,52+0,36+0,44/3 = 0,44
Season 2: 1,08 + 0,70+0,36+1,74/4=2,57
Season 3: 1,49+2,38+1,61/3= 1,82
Season 4: 0,96+0,22+0,44/3= 1,32
0,44+2,57+1,82+1,32= 6,15
22. Fifth Step: Scale the seasonal factors (St )
Year Season Dollar Sales (Yt) Moving Average StIt ScaledSt
2015 1 1999 0,28
2 3867 3577,25 1,08 (3867/3577,25) 1,67
3 5346 3578,75 1,49 1,18
4 3097 3198,25 0,96 0,86
2016 1 2005 3806 0,52 0,28
2 2345 3304 0,7 1,67
3 7777 3267 2,38 1,18
4 1089 4907,5 0,22 0,86
2017 1 1857 5021,75 0,36 0,28
2 8907 5252,25 1,69 1,67
3 8234 5095,5 1,61 1,18
4 2011 4557,25 0,44 0,86
2018 1 1230 2771 0,44 0,28
2 6754 3881,5 1,74 1,67
3 1089 1,18
4 6435 0,86
Average the seasonal factors = (0,44+2,57+1,82+1,32)/4 = 1,53. Then, divide each seasonal factor by the average of
the seasonal factors.
Season 1: 0,44/1,53= 0,28
Season 2: 2,57/1,53= 1,67
Season 3: 1,82/1,53= 1,18
Season 4: 1,32/1,53= 0,86
0,28+1,67+1,18+0,86= 3,99
23. Seventh Step: Determine a trend line of the deseasonalized
data
• Using the least squares method for t = 1, 2, ..., 16, gives:
• Tt = 1580.11 + 33.96t
24. Eight Step: Determine the deseasonalized predictions
• Substitute t = 17, 18, 19, and 20 into the least squares
equation:
• T17 = 1580.11 + (33.96)(17) = 2157,43
• T18 = 1580.11 + (33.96)(18) = 2191,39
• T19 = 1580.11 + (33.96)(19) = 2225,35
• T20 = 1580.11 + (33.96)(20) = 2259,31
25. Ninth Step: Take into account the seasonality
• Multiply each deseasonalized prediction by its seasonal
factor to give the following forecasts for year 5
• Season 1: (0,28)(2157,43) = 604,08
• Season 2: (1.67)(2191,39) = 3659,62
• Season 3: ( 1,18)(2225,35) = 2625,91
• Season 4: (0,86)(2259,31) = 1943
