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Module 4. Forecasting
MGS3100
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Forecasting
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• Casual Models:
Causal Model
Year 2000 Sales
Price PopulationAdvertising
……
• Time Series Models:
Time Series Model
Year 2000 Sales
Sales1999 Sales1998
Sales1997……
--Forecasting based on data and models
Quantitative Forecasting
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Causal forecasting
• RegressionFind a straight line that fits the data best.
y = Intercept + slope * x (= b0 + b1x)
Slope = change in y / change in x
0
2
4
6
8
10
12
10 11 12 13 14 15 16 17 18 19 20
Best line!
Intercept
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Causal Forecasting Models
• Curve Fitting: Simple Linear Regression– One Independent Variable (X) is used to predict one
Dependent Variable (Y): Y = a + b X– Given n observations (Xi, Yi), we can fit a line to the
overall pattern of these data points. The Least Squares Method in statistics can give us the best a and b in the sense of minimizing (Yi - a - bXi)2:
n
Xb
n
Ya
n
XX
n
YXYXb
ii
ii
iiii
22 )(
/
Regression formula is an optional learning objective
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• Curve Fitting: Simple Linear Regression – Find the regression line with Excel
• Use Function:
a = INTERCEPT(Y range; X range)
b = SLOPE(Y range; X range) • Use Solver• Use Excel’s Tools | Data Analysis | Regression
• Curve Fitting: Multiple Regression– Two or more independent variables are used to
predict the dependent variable:
Y = b0 + b1X1 + b2X2 + … + bpXp
– Use Excel’s Tools | Data Analysis | Regression
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Time Series Forecasting Process
Look at the data (Scatter Plot)
Forecast using one or more techniques
Evaluate the technique and pick the best one.
Observations from the scatter Plot Techniques to try Ways to evaluate
Data is reasonably stationary (no trend or seasonality)
Heuristics - Averaging methods Naive Moving Averages Simple Exponential Smoothing
MAD MAPE Standard Error BIAS
Data shows a consistent trend
Regression Linear Non-linear Regressions (not covered in this course)
MAD MAPE Standard Error BIAS R-Squared
Data shows both a trend and a seasonal pattern
Classical decomposition Find Seasonal Index Use regression analyses to find the trend component
MAD MAPE Standard Error BIAS R-Squared
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• BIAS - The arithmetic mean of the errors
– n is the number of forecast errors– Excel: =AVERAGE(error range)
• Mean Absolute Deviation - MAD
– No direct Excel function to calculate MAD
Evaluation of Forecasting Model
n
Error
n
Forecast) - (Actual BIAS
n
|Error|
n
Forecast - Actual| MAD
|
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• Mean Square Error - MSE
– Excel: =SUMSQ(error range)/COUNT(error range)
– Standard error is square root of MSE
• Mean Absolute Percentage Error - MAPE
• R2 - only for curve fitting model such as regression• In general, the lower the error measure (BIAS, MAD,
MSE) or the higher the R2, the better the forecasting model
n
(Error)
n
Forecast) - (Actual MSE
22
nActual
|Forecast - Actual|
MAPE
%100*
Evaluation of Forecasting Model
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Stationary data forecasting
• Naïve I sold 10 units yesterday, so I think I will sell 10 units
today.
• n-period moving averageFor the past n days, I sold 12 units on average.
Therefore, I think I will sell 12 units today.
• Exponential smoothing I predicted to sell 10 units at the beginning of yesterday;
At the end of yesterday, I found out I sold in fact 8 units. So, I will adjust the forecast of 10 (yesterday’s forecast) by adding adjusted error (α * error). This will compensate over (under) forecast of yesterday.
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Naïve Model
• The simplest time series forecasting model
• Idea: “what happened last time (last year,
last month, yesterday) will happen again
this time”• Naïve Model:
– Algebraic: Ft = Yt-1
• Yt-1 : actual value in period t-1
• Ft : forecast for period t
– Spreadsheet: B3: = A2; Copy down
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Moving Average Model• Simple n-Period Moving Average
• Issues of MA Model – Naïve model is a special case of MA with n = 1 – Idea is to reduce random variation or smooth data– All previous n observations are treated equally (equal
weights)– Suitable for relatively stable time series with no trend or
seasonal pattern
nnt
Y2t
Y1t
Y=
n
periodsn previousin valuesactual of Sumt
F
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Smoothing Effect of MA Model
Longer-period moving averages (larger n) react to actual changes more slowly
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Moving Average Model
• Weighted n-Period Moving Average
– Typically weights are decreasing:
w1>w2>…>wn
– Sum of the weights = wi = 1
– Flexible weights reflect relative importance of each previous observation in forecasting
– Optimal weights can be found via Solver
ntY
nw
2tY
2w
1tY
1 w=
tF
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Weighted MA: An Illustration
Month Weight Data
August 17% 130
September 33% 110
October 50% 90
November forecast:
FNov = (0.50)(90)+(0.33)(110)+(0.17)(130)
= 103.4
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Exponential Smoothing
• Concept is simple!– Make a forecast, any forecast– Compare it to the actual– Next forecast is
• Previous forecast plus an adjustment• Adjustment is fraction of previous forecast error
– Essentially• Not really forecast as a function of time• Instead, forecast as a function of previous actual and
forecasted value
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Simple Exponential Smoothing• A special type of weighted moving average
– Include all past observations– Use a unique set of weights that weight recent observations
much more heavily than very old observations:
( )
( )
( )
1
1
1
2
3
weight
Decreasing weights given to older observations
0 1
Today
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Simple ES: The Model
New forecast = weighted sum of last period actual value and last
period forecast – : Smoothing constant
– Ft : Forecast for period t
– Ft-1: Last period forecast
– Yt-1: Last period actual value
321
32
21
)1()1(
)1()1(
tttt
tttt
YaYYF
YYYF
11 )1( ttt FYF
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Simple Exponential Smoothing
• Properties of Simple Exponential Smoothing
– Widely used and successful model
– Requires very little data
– Larger , more responsive forecast; Smaller
, smoother forecast (See Table 13.2)
– “best” can be found by Solver
– Suitable for relatively stable time series
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Time Series Components
• Trend– persistent upward or downward pattern in a time series
• Seasonal – Variation dependent on the time of year – Each year shows same pattern
• Cyclical – up & down movement repeating over long time frame– Each year does not show same pattern
• Noise or random fluctuations – follow no specific pattern – short duration and non-repeating
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Time Series Components
Time
Trend
Randommovement
Time
Cycle
Time
Seasonalpattern
Dem
and
Time
Trend with seasonal pattern
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Trend Model• Curve fitting method used for time series
data (also called time series regression model)
• Useful when the time series has a clear trend• Can not capture seasonal patterns • Linear Trend Model: Yt = a + bt
– t is time index for each period, t = 1, 2, 3,…
0
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10
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Pattern-based forecasting - Trend
• Regression – Recall Independent Variable X, which is now time variable – e.g., days, months, quarters, years etc. Find a straight line that fits the data best.
y = Intercept + slope * x (= b0 + b1x)
Slope = change in y / change in x
0
2
4
6
8
10
12
10 11 12 13 14 15 16 17 18 19 20
Best line!
Intercept
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Pattern-based forecasting – Seasonal
• Once data turn out to be seasonal, deseasonalize the data.– The methods we have learned (Heuristic methods and
Regression) is not suitable for data that has pronounced fluctuations.
• Make forecast based on the deseasonalized data• Reseasonalize the forecast
– Good forecast should mimic reality. Therefore, it is needed to give seasonality back.
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Pattern-based forecasting – Seasonal
Deseasonalize
Forecast
Reseasonalize
Actual data Deseasonalized data
Example (SI + Regression)
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Pattern-based forecasting – Seasonal
• DeseasonalizationDeseasonalized data = Actual / SI
• ReseasonalizationReseasonalized forecast
= deseasonalized forecast * SI
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Seasonal Index
• What’s an index?– Ratio– SI = ratio between actual and average demand
• Suppose– SI for quarter demand is 1.20
• What’s that mean?• Use it to forecast demand for next fall
– So, where did the 1.20 come from?!
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Calculating Seasonal Indices
• Quick and dirty method of calculating SI– For each year, calculate average demand– Divide each demand by its yearly average
• This creates a ratio and hence a raw index• For each quarter, there will be as many raw indices
as there are years– Average the raw indices for each of the quarters– The result will be four values, one SI per quarter
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Classical decomposition
• Start by calculating seasonal indices• Then, deseasonalize the demand
– Divide actual demand values by their SI valuesy ’ = y / SI
– Results in transformed data (new time series)– Seasonal effect removed
• Forecast– Regression if deseasonalized data is trendy– Heuristics methods if deseasonalized data is stationary
• Reseasonalize with SI
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Causal or Time series?
• What are the difference?
• Which one to use?
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Can you…
• describe general forecasting process?• compare and contrast trend, seasonality and
cyclicality?• describe the forecasting method when data is
stationary?• describe the forecasting method when data
shows trend?• describe the forecasting method when data
shows seasonality?