Forecasting

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```					Module 4. Forecasting

MGS3100

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Forecasting

Forecasting

Quantitative                               Qualitative

Causal Model                              Expert Judgment

Trend                     Delphi Method

Time series                                 Grassroots

Stationary

Trend

Trend + Seasonality                        2
Quantitative Forecasting
--Forecasting based on data and models
• Casual Models:
Price
Population
Causal             Year 2000
……

• Time Series Models:
Sales1999
Sales1998        Time Series           Year 2000
Sales1997          Model                 Sales
……

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Causal forecasting

• Regression
Ø Find a straight line that fits the data best.
Best line!

Intercept

Ø y = Intercept + slope * x (= b0 + b1x)
Ø Slope = change in y / change in x
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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 S(Yi - a - bXi)2:

Regression formula is an optional learning objective   5
• 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 = b 0 + b 1X 1 + b 2X 2 + … + b pX p
– Use Excel’s Tools | Data Analysis | Regression

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Time Series Forecasting Process

Look at the data                  Forecast using one or           Evaluate the technique
(Scatter Plot)                    more techniques                 and pick the best one.

Observations from the
Techniques to try               Ways to evaluate
scatter Plot

Heuristics - Averaging methods       MAD
Data is reasonably
 Naive                              MAPE
stationary
 Moving Averages                    Standard Error
(no trend or seasonality)
 Simple Exponential Smoothing       BIAS
Regression
 MAPE
Data shows a consistent        Linear
 Standard Error
trend                          Non-linear Regressions (not
 BIAS
covered in this course)
 R-Squared
Classical decomposition
 MAPE
Data shows both a trend and    Find Seasonal Index
 Standard Error
a seasonal pattern             Use regression analyses to find
 BIAS
the trend component
 R-Squared
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Evaluation of Forecasting Model
• 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

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Evaluation of Forecasting Model
• 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
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Stationary data forecasting

• Naïve
Ø I sold 10 units yesterday, so I think I will sell 10 units
today.

• n-period moving average
Ø For the past n days, I sold 12 units on average.
Therefore, I think I will sell 12 units today.

• Exponential smoothing
Ø 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)
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

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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 = åwi = 1
– Flexible weights reflect relative importance of
each previous observation in forecasting
– Optimal weights can be found via Solver

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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:

weight
Decreasing weights
given
to older observations

Today          17
Simple ES: The Model

New forecast = weighted sum of last period
actual value and last period forecast
– a:      Smoothing constant
– Ft :    Forecast for period t
– Ft-1:   Last period forecast
– Yt-1:   Last period actual value

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Simple Exponential Smoothing
• Properties of Simple Exponential Smoothing
– Widely used and successful model

– Requires very little data

– Larger a, more responsive forecast; Smaller a,
smoother forecast (See Table 13.2)

– “best” a 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
– short duration and non-repeating

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Time Series Components

Cycle
Trend

Random
movement

Time                               Time

Seasonal                        Trend with

Demand
pattern                         seasonal pattern

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Time                               Time
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,…

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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.
Ø Find a straight line that fits the data best.
Best line!

Intercept

Ø y = Intercept + slope * x (= b0 + b1x)
Ø Slope = change in y / change in x

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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
Example (SI + Regression)

Actual data                        Deseasonalized data

Deseasonalize

Forecast

Reseasonalize

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Pattern-based forecasting – Seasonal

• Deseasonalization
ØDeseasonalized data = Actual / SI

• Reseasonalization
ØReseasonalized 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 values
y ’ = 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?

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