Forecasting
Dr/Mohamed Sameh
Eng /Omar Elharoun
What is forecasting?
A planning tool that helps management in its attempts to cope with the uncertainty of the
future, relying mainly on data from the past and present and analysis of trends.
• How much will the economy grow over the next year?
1
• What about interest rates?
2
• What will be the hot new products?
3
• Underestimating leads to
many lost sales and unhappy
customers
• Overestimating demand also
is very costly due to
unneeded production or
storage capacity, forced price
reductions, lost opportunities
to market more profitable
goods
1.Sales
Forecasting
• Forecasts of the need for
spare parts
2.The Need
for Spare
Parts
Why forecasting?
• How much will the nation’s
gross domestic product grow
next quarter? Next year?
• What is the forecast for the
rate of inflation? The
unemployment rate?
3.Economic
Trends
4.Staffing
Needs
•
Providing too few agents to
answer the telephone leads to
unhappy customers, lost calls,
and perhaps lost business
• Too many agents cause
excessive personnel costs
Forecasting methods
1.Judgmental forecasting method
Based on experience (when no data are available for employing a statistical forecasting method).
1.Manager’s
opinion
involves
a single
manager using
his or her best
judgment to
make the
forecast.
2.Executives’
opinion
A small group
of high-level
managers who
pool their best
judgment to
collectively
make the
forecast.
3.Sales force
4.Consumer
market
survey
5.Delphi
method
Each
salesperson
provides an
estimate of
what sales will
be in his or her
region.
surveying
customers
regarding their
future
purchasing
plans and how
they would
respond to
various new
features in
products.
A panel of
experts in
different
locations who
independently
fill out a series
of
questionnaires.
Forecasting methods
2.Statistical forecasting methods
1.Constant level
2.Constant level plus
seasonal effects
3.Linear trend
Statistical forecasting
methods
1.Forecasting methods for a constant level model
a)Last-Value Forecasting Method
𝑭𝒕+𝟏 = 𝒙𝒕
 𝑭𝒕+𝟏 ----- The forecast at time t +1
 𝒙𝒕 ----- The sales of a particular product at time t
This method is called the naive method , because statisticians consider it naive
to use just a sample size of one when additional relevant data are available.
This method is used when ;
 The change is so fast and it does not depend on historical data.
 The time t is very small.
1.Forecasting methods for a constant level model
b)Averaging Forecasting Method
This method uses all the data points in the time series and simply averages these points
𝑭𝒕+𝟏 =
𝒙𝒊
𝒕
σ𝒊=𝟏
𝒕
This method is used for;
 Young processes because of a natural reluctance to use very old data
 Recent companies with no ancient history
Disadvantages;
 Using very old data that may no longer be relevant
 Gives the same weights to all observations
1.Forecasting methods for a constant level model
c)Moving-Average Forecasting Method
This method averages the data for only the last n periods as the forecast for the next period
𝒕
𝑭𝒕+𝟏 =
෍
𝒊=𝒕−𝒏+𝟏
𝒙𝒊
𝒏
This method is used for;
 Old companies with almost steady sales for years
Advantages;
 Combines the advantages of the last value and averaging methods by using only recent
history and using multiple observations.
Disadvantages;
 Gives the same weights to all observations
1.Forecasting methods for a constant level model
d)Exponential Smoothing Forecasting Method
This forecast is just a weighted sum of the last observation 𝒙𝒕 and the preceding forecast
𝑭𝒕 for the period just ended.
𝑭𝒕+𝟏 = 𝜶𝒙𝒕 + (𝟏 − 𝜶)𝑭𝒕
 𝜶 𝟎 < 𝜶 < 𝟏 − is called the smoothing constant
Another alternative forms for the exponential smoothing technique is given by;
𝑭𝒕+𝟏 = 𝜶𝒙𝒕 + 𝜶(𝟏 − 𝜶)𝒙𝒕−𝟏 + 𝜶(𝟏 − 𝜶)𝟐 𝒙𝒕−𝟐 + ⋯
𝑭𝒕+𝟏 = 𝑭𝒕 + 𝜶 𝒙𝒕 − 𝑭𝒕
 𝒙𝒕 − 𝑭𝒕 ----- The forecasting error
In this form, it becomes evident that exponential smoothing gives the
most weight to 𝒙𝒕 and decreasing weights to earlier observations.
(15,18,12,17,13)
𝑭𝟔 = 𝒙𝟓 = 𝟏𝟑
𝑭𝒕+𝟏 = 𝒙𝒕
σ𝟓𝒕=𝟏 𝒙𝒕 𝟏𝟓 + 𝟏𝟖 + 𝟏𝟐 + 𝟏𝟕 + 𝟏𝟑
𝑭𝟔 =
=
= 𝟏𝟓
𝟓
𝟓
𝒕
𝑭𝒕+𝟏 =
෍
𝒊=𝒕−𝒏+𝟏
𝒙𝒊
𝒏
σ𝟓𝒕=𝟑 𝒙𝒕 𝟏𝟐 + 𝟏𝟕 + 𝟏𝟑
𝑭𝟔 =
=
= 𝟏𝟒
𝟑
𝟑
 The averaging method seems to be the best, since all five months of data are relevant
in determining the forecast of sales for the next month.
𝒕
𝑭𝒕+𝟏 =
෍
𝒊=𝒕−𝒏+𝟏
𝒙𝒕−𝟑 = 𝟖𝟎𝟓
𝒙𝒊
𝒏
Forecast for the last month
Forecast For this month
𝑭𝒕 = 𝟕𝟖𝟐
𝒙𝒕−𝟏 + 𝒙𝒕−𝟐 + 𝒙𝒕−𝟑
𝑭𝒕 =
𝟑
𝒙𝒕 + 𝒙𝒕−𝟏 + 𝒙𝒕−𝟐
𝑭𝒕+𝟏 =
𝟑
𝒙𝒕 = 𝟕𝟗𝟑
𝒙𝒕−𝟏 + 𝒙𝒕−𝟐 = 𝟑 ∗ 𝟕𝟖𝟐 − 𝟖𝟎𝟓 = 𝟏𝟓𝟒𝟏
𝑭𝒕+𝟏
𝟏𝟓𝟒𝟏 + 𝟕𝟗𝟑
=
= 𝟕𝟕𝟖
𝟑
𝑭𝒕+𝟏 = 𝑭𝒕 + 𝜶 𝒙𝒕 − 𝑭𝒕
Or
𝑭𝒕+𝟏 = 𝜶𝒙𝒕 + (𝟏 − 𝜶)𝑭𝒕
𝑭𝑱𝒖𝒏𝒆 = 𝑭𝑴𝒂𝒚 + 𝜶 𝒙𝑴𝒂𝒚 − 𝑭𝑴𝒂𝒚
𝟑𝟖𝟎 = 𝟑𝟗𝟎 + 𝟎. 𝟓 𝒙𝑴𝒂𝒚 − 𝟑𝟗𝟎
𝒙𝑴𝒂𝒚 = 𝟑𝟕𝟎
𝑭𝑴𝒂𝒚 = 𝑭𝑨𝒑𝒓𝒊𝒍 + 𝜶 𝒙𝑨𝒑𝒓𝒊𝒍 − 𝑭𝑨𝒑𝒓𝒊𝒍
𝟑𝟗𝟎 = 𝟑𝟖𝟎 + 𝟎. 𝟓 𝒙𝑨𝒑𝒓𝒊𝒍 − 𝟑𝟖𝟎
𝒙𝑨𝒑𝒓𝒊𝒍 = 𝟒𝟎𝟎
𝑭𝒕+𝟏 = 𝜶𝒙𝒕 + (𝟏 − 𝜶)𝑭𝒕
𝒙𝑴𝒂𝒚 = 𝟑𝟕𝟎
𝑭𝑴𝒂𝒓𝒄𝒉 = 𝜶𝒙𝑭𝒆𝒃 + (𝟏 − 𝜶)𝑭𝑭𝒆𝒃
𝒙𝑨𝒑𝒓𝒊𝒍 = 𝟒𝟎𝟎
𝑭𝑭𝒆𝒃 = 𝜶𝒙𝑱𝒂𝒏 + (𝟏 − 𝜶)𝑭𝑱𝒂𝒏
𝑭𝑴𝒂𝒓𝒄𝒉 = 𝜶𝒙𝑭𝒆𝒃 + 𝜶 𝟏 − 𝜶 𝒙𝑱𝒂𝒏 + (𝟏 − 𝜶)𝟐 𝑭𝑱𝒂𝒏
After change
𝒙ƴ 𝑭𝒆𝒃 = 𝒙𝑭𝒆𝒃
𝑭ሖ 𝑱𝒂𝒏 = 𝑭𝑱𝒂𝒏
ሖ
𝑭ሖ 𝑴𝒂𝒓𝒄𝒉 = 𝜶𝒙ƴ 𝑭𝒆𝒃 + 𝜶 𝟏 − 𝜶 𝒙ƴ 𝑱𝒂𝒏 + (𝟏 − 𝜶)𝟐 𝑭𝑱𝒂𝒏
𝑭ሖ 𝑴𝒂𝒓𝒄𝒉 = 𝑭𝑴𝒂𝒓𝒄𝒉 + 𝜶 𝟏 − 𝜶 𝒙ƴ 𝑱𝒂𝒏 − 𝒙𝑱𝒂𝒏 = 𝟒𝟎𝟎 + 𝟎. 𝟐𝟓 ∗ 𝟑𝟐 = 𝟒𝟎𝟖
𝑭ሖ 𝑨𝒑𝒓𝒊𝒍 = 𝜶𝒙ƴ 𝑴𝒂𝒓𝒄𝒉 + 𝟏 − 𝜶 𝑭ሖ 𝑴𝒂𝒓𝒄𝒉 = 𝟑𝟖𝟒
𝑭ሖ 𝑴𝒂𝒚 = 𝜶𝒙ƴ 𝑨𝒑𝒓𝒊𝒍 + 𝟏 − 𝜶 𝑭ሖ 𝑨𝒑𝒓𝒊𝒍 = 𝟑𝟗𝟐
𝑭ሖ 𝑱𝒖𝒏𝒆 = 𝜶𝒙ƴ 𝑴𝒂𝒚 + 𝟏 − 𝜶 𝑭ሖ 𝑴𝒂𝒚 = 𝟑𝟖𝟏
2.Incorporating seasonal effects into forecasting
It is fairly common for a time series to have a seasonal pattern with higher or lower
values at certain times of the year than others, so the forecasting methods presented
in the preceding section should not be applied directly.
The seasonal factor is calculated to be
𝑺𝑭 =
𝒂𝒗𝒆𝒓𝒂𝒈𝒆 𝒇𝒐𝒓 𝒕𝒉𝒆 𝒑𝒆𝒓𝒊𝒐𝒅
𝒐𝒗𝒆𝒓𝒂𝒍𝒍 𝒂𝒗𝒆𝒓𝒂𝒈𝒆
The outline of the general procedure
 Use the following formula to seasonally adjust each value in the time series:
𝑺𝒆𝒂𝒔𝒐𝒏𝒂𝒍𝒍𝒚 𝒂𝒅𝒋𝒖𝒔𝒕𝒆𝒅 𝒗𝒂𝒍𝒖𝒆 =
𝐚𝐜𝐭𝐮𝐚𝐥 𝐯𝐚𝐥𝐮𝐞
𝐬𝐞𝐚𝐬𝐨𝐧𝐚𝐥 𝐟𝐚𝐜𝐭𝐨𝐫
 Select a time series forecasting method and apply it to the seasonally adjusted time
series to obtain a forecast of the next seasonally adjusted value.
 Multiply this forecast by the corresponding seasonal factor to obtain a forecast of
the next actual value.
𝑺𝒆𝒂𝒔𝒐𝒏𝒂𝒍 𝒇𝒂𝒄𝒕𝒐𝒓 =
𝐨𝐯𝐞𝐫𝐚𝐥𝐥 𝐚𝐯𝐞𝐫𝐚𝐠𝐞 =
𝐚𝐯𝐞𝐫𝐚𝐠𝐞 𝐟𝐨𝐫 𝐭𝐡𝐞 𝐩𝐞𝐫𝐢𝐨𝐝
𝐨𝐯𝐞𝐫𝐚𝐥𝐥 𝐚𝐯𝐞𝐫𝐚𝐠𝐞
𝟐𝟑 + 𝟐𝟐 + 𝟑𝟏 + 𝟐𝟔 + 𝟏𝟗 + 𝟐𝟏 + 𝟐𝟕 + 𝟐𝟒 + 𝟐𝟏 + 𝟐𝟔 + 𝟑𝟐 + 𝟐𝟖
= 𝟐𝟓
𝟏𝟐
Quarter
Three-year average
Seasonal Factor
1
(23+19+21)/3 =21
21/25 = 0.84
2
(22+21+26)/3=23
0.92
3
(31+27+32)/3=30
1.2
4
(26+24+28)/3=26
1.04
𝑺𝒆𝒂𝒔𝒐𝒏𝒂𝒍𝒍𝒚 𝒂𝒅𝒋𝒖𝒔𝒕𝒆𝒅 𝒗𝒂𝒍𝒖𝒆 =
𝒙ƴ 𝑸𝟒)𝟑
𝒙𝑸𝟒)𝟑
𝟐𝟖
=
=
= 𝟐𝟕
𝑺𝑭𝑸𝟒 𝟏. 𝟎𝟒
𝑭𝑸𝟏)𝟒 = 𝒙𝑸𝟒)𝟑 ∗ 𝑺𝑭𝑸𝟏
= 𝟐𝟕 ∗ 𝟎. 𝟖𝟒 = 𝟐𝟑
𝒂𝒄𝒕𝒖𝒂𝒍 𝒗𝒂𝒍𝒖𝒆
𝒔𝒆𝒂𝒔𝒐𝒏𝒂𝒍 𝒇𝒂𝒄𝒕𝒐𝒓
Quarter
Three-year
Average
Seasonal
Factor
1
21
0.84
2
23
0.92
3
30
1.2
4
26
1.04
Quarter
Three-year
Average
Seasonal Factor
Forecasted value for the
fourth year
1
21
0.84
23
2
23
0.92
(23/0.84)x0.92=25
3
30
1.2
(25/0.92)x1.2=33
4
26
1.04
(33/1.2)x1.04=29
Average houses sold per season =100/4=25
Quarter
Average value
Seasonal Factor
Actual Forecasted
value
1
25
0.84
25x0.84=21
2
25
0.92
25x0.92=23
3
25
1.2
25x1.2=30
4
25
1.04
25x1.04=26
3. Exponential smoothing method for a linear trend model
Denote the slope of the linear trend by B, where the slope is called the trend factor. The
model is represented by
𝑿𝒊 = 𝑨 + 𝑩𝒊 + 𝒆𝒊 ,
for 𝒊 = 𝟏, 𝟐, …
 𝑿𝒊 --------The random variable that is observed at time i
 A ---------is a constant
 B ---------The trend factor
 𝒆𝒊 --------The random error occurring at time i
Adapting Exponential Smoothing to the linear trend model
This is done by also using exponential smoothing to estimate this trend factor.
𝑭𝒕+𝟏 = 𝜶𝒙𝒕 + (𝟏 − 𝜶)𝑭𝒕 + 𝑻𝒕+𝟏
𝑻𝒕+𝟏 = 𝜷𝑳𝒕+𝟏 + (𝟏 − 𝜷)𝑻𝒕
𝑳𝒕+𝟏 = 𝜶 𝒙𝒕 − 𝒙𝒕−𝟏 + (𝟏 − 𝜶) 𝑭𝒕 − 𝑭𝒕−𝟏
 𝑻𝒕+𝟏 = Exponential smoothing estimate of the trend factor at time t+1
 𝑳𝒕+𝟏 = latest trend at time t+1 based on the last two values 𝒙𝒕 &𝒙𝒕−𝟏 and the
last two forecasts 𝑭𝒕 &𝑭𝒕−𝟏
 𝜷 = The trend smoothing constant which, like 𝜶, must be between 0 and 1.
Given
𝜷 = 𝟎. 𝟐
𝜶 = 𝟎. 𝟏
𝒙𝒕−𝟏 = 𝟒𝟔𝟓𝟓
𝒙𝒕 = 𝟒𝟗𝟑𝟓
𝑭𝒕−𝟏 = 𝟒𝟕𝟐𝟎
𝑭𝒕 = 𝟒𝟗𝟕𝟓
𝑻𝒕 = 𝟐𝟒𝟎
𝑳𝒕+𝟏 = 𝜶 𝒙𝒕 − 𝒙𝒕−𝟏 + 𝟏 − 𝜶 𝑭𝒕 − 𝑭𝒕−𝟏
= 𝟎. 𝟏 ∗ 𝟒𝟗𝟑𝟓 − 𝟒𝟔𝟓𝟓 + 𝟎. 𝟗 𝟒𝟗𝟕𝟓 − 𝟒𝟕𝟐𝟎 = 𝟐𝟓𝟕. 𝟓
𝑻𝒕+𝟏 = 𝜷𝑳𝒕+𝟏 + 𝟏 − 𝜷 𝑻𝒕
= 𝟎. 𝟐 ∗ 𝟐𝟓𝟕. 𝟓 + 𝟎. 𝟖 ∗ 𝟐𝟒𝟎 = 𝟐𝟒𝟑. 𝟓
𝑭𝒕+𝟏 = 𝜶𝒙𝒕 + 𝟏 − 𝜶 𝑭𝒕 + 𝑻𝒕+𝟏
= 𝟎. 𝟏 ∗ 𝟒𝟗𝟑𝟓 + 𝟎. 𝟗 ∗ 𝟒𝟗𝟕𝟓 + 𝟐𝟒𝟑. 𝟓 = 𝟓𝟐𝟏𝟓