Hi again k9queenga,
I have embedded my responses with the questions you have provided by
adding the 2004 column (where each row in the tables represents
quarters in the given year):
1) Let's invent data where there is no seasonality but there is a
trend. Suupose these are order for a certain brand of computers. You
have been asked to give a forecast for 2004
yr.2001 yr. 2002 yr. 2003 yr. 2004 (forecasted)

1900 2100 2300 2500
1950 2150 2350 2550
2000 2200 2400 2600
2050 2250 2450 2650
Notes: The trend here is fairly clear..the orders are increasing by 50
for each subsequent period. So we can carry that pattern through for
2004.
2)Now invent data where we have no trend but pronounced seasonality
yr. 2001 yr. 2002 yr. 2003 yr. 2004 (forecasted)

1900 1900 1900 1900
2100 2100 2100 2100
2000 2000 2000 2000
3000 3000 3000 3000
Notes: This time we are only considering the same period in previous
years to forecast. In all cases, the 2001/2/3 values for a season are
the same, so we can carry them over to 2004.
3)Now invent data where we have a trend and seasonality, but the trend
is always a constant and the seasonality is always the same.
yr. 2001 yr. 2002 yr. 2003 yr. 2004 (forecasted)

1800 2000 2200 2400
2000 2200 2400 2600
2100 2300 2500 2700
3500 3700 3900 4100
Notes: This problem can be solved by considering the seasonality alone
 notice that the values for a given quarter in 2001/2/3 are
increasing by a constant 200. We can continue this constant trend to
2004. Another way of looking at this is the difference between
quarters in the same period. For example, the difference between Q1
and Q2 is always the same, etc.
4)Now invent data where we have a trend and seasonality but there is
some variation in both.
yr. 2001 yr.2002 yr. 2003 yr. 2004 (forecasted)

1900 2250 2325 2426.36
1950 2060 2100 2319.30
2000 2340 2560 2542.78
2050 2325 2765 2648.53
Notes: This is the only question that doesn't have a clear pattern and
contains elements of both trend and seasonality. This is a perfect
application of Winter's model:
"Winters' Model for Seasonality"
http://www.cba.uh.edu/~ekao/D6360S02LEC11A.pdf
(slides 2628)
We need to first define three smoothing constants that relate to the
forecast, trend, and seasonality (you can play around with these to
weight the result of one component of the forecast more heavily
compared to the others). For this forecast, I will use all three
(alpha, beta, and sigma) equal to 0.2 and the number of periods is 4.
Sameperiod next year forecast = F(t+n) = (F(t) + T(t+ 1))*S(n)
Where n is the period number (in your case this is the quarter) and t
is the current period. Note that this only works after we have a full
year of actual data (since the forecast needs to go back to the same
period last year). Continuing this through years 2002 and 2003 we can
forecast 2004 figures incorporating trend (which includes previous
forecasts) and seasonality.
Hopefully this has helped you understand the different methods of
forecasting  keep in mind that in real life the results are not a
perfect fits as in questions 1, 2, and 3 :)
Cheers!
answerguruga 
Clarification of Answer by
answerguruga
on
09 Nov 2003 11:43 PST
Hi again,
The reason as to how I arrived to answers #13 is based purely on the
"hint" given in the questions:
1. "there is no seasonality but there is a trend"
You already have all the actual values for the first 12 periods, but
the forecast is still done to factor in trends throughout that period.
Since we already have the actual values we can verify how close the
forecast was and adjust our value accordingly for the next period if
it is above or below. This is known as exponential smoothing. Since
the trend is perfectly linear (each value is exactly 50 higher than
the previous period), we just continue the linear trend.
Formula: Forecast of next period = (Forecast current period) +
(smoothing factor)*(Acutal value current period  Forecasted value
current period)
Where the smoothing factor is between 0 and 1
2. "no trend but pronounced seasonality"
This means that seasons are the only factor that matters in this case
 we forecast in a manner similar to #1 but only consider the previous
period in the same season. Again, there is a linear relationship (no
growth by season) so our forecast would be an exact reflection of the
previous season.
Formula: Forecast of next period = (Forecast same season last year) +
(smoothing factor)*(Acutal same season last year  Forecasted value
same season last year)
3. "we have a trend and seasonality, but the trend is always a
constant and the seasonality is always the same"
This time there is a linear trend for each season (across the rows of
the table)  there is an increase of 200 between each given period and
the same period in the previous year. The difference between the
periods within a year are always the same as well. That means there
are two ways of looking at the problem; either one will yield the
correct forecast.
Formula: Forecast of next period = (Forecast same season last year) +
(constant factor)
Where the constant factors are (incidentally all the same here):
Q1: Y2002  Y2001 = 200
Q2: Y2002  Y2001 = 200
Q3: Y2002  Y2001 = 200
Q4: Y2002  Y2001 = 200
4. "doesn't have a clear pattern and contains elements of both trend
and seasonality"
I already gave a description of Winter's model, which is the
appropriate way to approach this type of problem  I did the
calculations in excel and the result was as follows:
Period Actual Forecast Trend Seasonality Winters
1 1900 1900 1900 1
2 1950 1900 1900 1
3 2000 1960 1912 1
4 2050 2008 1921.6 1
5 2250 2058.4 1931.68 1.018616401 2055.600493
6 2060 2288.32 1977.664 0.980044749 2063.575951
7 2340 2014.336 1922.8672 1.032334625 2072.598902
8 2325 2405.1328 2001.02656 0.993336518 2178.847637
9 2325 2308.97344 1981.794688 1.016281319 2182.268245
10 2100 2328.205312 1985.641062 0.964432261 2053.795678
11 2560 2054.358938 1930.871788 1.075093866 2207.481945
12 2765 2661.128212 2052.225642 1.002475812 2375.007091
13 2661.128212 2785.774358 2077.154872 1.004076294 2426.35747
14 2785.774358 2636.198983 2047.239797 0.982893615 2319.303037
15 2636.198983 2815.689432 2083.137886 1.047325785 2542.775887
16 2815.689432 2600.300894 2040.060179 1.018547081 2648.528886
(the formatting for the table may get skewed...sorry)
The formulas for these are taken from the slides referenced in the
original answer  I haven't posted them here because the special
characters don't transfer well.
The reason I didn't include the formulas in the original question for
#13 is because this question is using special (easy) cases to teach
you that you don't always need a formula. It was more of a conceptual
question that was trying to get you to "see" the patterns rather than
just plugging them into a formula. I hope that the formulas above help
you understand how I arrived at the answers.
answerguruga
