forecasting introduction quantitative models time series
DESCRIPTION
FORECASTING Introduction to Quantitative FORECASTINGTRANSCRIPT
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FORECASTING
Introduction Quantitative ModelsTime Series
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Forecasting Models
Forecasting
Quantitative Qualitative
Causal Model
Time series
Expert Judgment
Trend
Stationary
Trend
Trend + Seasonality
Delphi Method
Grassroots
Market Research
Jury Exec. Opinion
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FORECASTING
Introduction toQuantitative
FORECASTING
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Quantitative Forecasts
Quantitative forecasting models possess two important and attractive features:They are expressed in mathematical
notation. Thus, they establish an unambiguous record of how the forecast is made.
With the use of spreadsheets and computers, quantitative models can be based on an amazing quantity of data.
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Quantitative Forecasts
Two types of quantitative forecasting models :Causal modelsTime-Series models
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Quantitative Forecasts
- Forecasting based on data and models
Causal Models:
Time Series Models:
PricePopulationAdvertising
……
CausalModel
Year 2000 Sales
Sales1999
Sales1998
Sales1997
…………………
Time Series Model
Year 2000 Sales
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FORECASTING
Time Series Forecasting
FORECASTING
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Forecasting Models
Forecasting
Quantitative Qualitative
Causal Model
Time series
Expert Judgment
Trend
Stationary
Trend
Trend + Seasonality
Delphi Method
Grassroots
Market Research
Jury Exec. Opinion
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Time Series Forecasting Models
Time-series forecasting models produce forecasts by extrapolating the historical behavior of the values of a particular single variable of interest.
Time-series data are historical data in chronological order, with only one value per time period.
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Components of a Time Series
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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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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
Mean Absolute Deviation - MAD Average of the absolute errors
Evaluation of Forecasting Model
nError
nForecast) - (Actual
BIAS
n|Error|
nForecast - Actual|
MAD |
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Mean Square Error - MSE
Standard error Square Root of MSE
Mean Absolute Percentage Error - MAPE
Calculate the % error using the absolute error, then average the results
n(Error)
nForecast) - (Actual
MSE22
nActual
|Forecast - Actual|
MAPE
%100*
Evaluation of Forecasting Model
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Time Series: Stationary Models Stationary Model Assumptions
Assumes item forecasted will stay steady over time (constant mean; random variation only)
Techniques will smooth out short-term irregularitiesThe forecast is revised only when new data becomes
available.
Stationary Model TypesNaïve ForecastMoving Average/Weighted Moving AverageExponential Smoothing
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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 Average
For 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 period (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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Naïve Forecast Wallace Garden SupplyForecasting
PeriodActual Value
Naïve Forecast Error
Absolute Error
Percent Error
Squared Error
January 10 N/AFebruary 12 10 2 2 16.67% 4.0March 16 12 4 4 25.00% 16.0April 13 16 -3 3 23.08% 9.0May 17 13 4 4 23.53% 16.0June 19 17 2 2 10.53% 4.0July 15 19 -4 4 26.67% 16.0August 20 15 5 5 25.00% 25.0September 22 20 2 2 9.09% 4.0October 19 22 -3 3 15.79% 9.0November 21 19 2 2 9.52% 4.0December 19 21 -2 2 10.53% 4.0
0.818 3 17.76% 10.091BIAS MAD MAPE MSE
Standard Error (Square Root of MSE) = 3.176619
Storage Shed Sales
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Wallace Garden - Naive Forecast
0
5
10
15
20
25
February March April May June July August September October November December
Period
Shed
s Actual Value
Naïve Forecast
Naïve Forecast
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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
nY
Y
nT
Tit
t
1
nntY2tY1tY
ˆ=tF
n periodsn previousin valuesactual of Sumˆ
tF
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Moving Averages
Wallace Garden SupplyForecasting
PeriodActual Value Three-Month Moving Averages
January 10February 12March 16April 13 10 + 12 + 16 / 3 = 12.67May 17 12 + 16 + 13 / 3 = 13.67June 19 16 + 13 + 17 / 3 = 15.33July 15 13 + 17 + 19 / 3 = 16.33August 20 17 + 19 + 15 / 3 = 17.00September 22 19 + 15 + 20 / 3 = 18.00October 19 15 + 20 + 22 / 3 = 19.00November 21 20 + 22 + 19 / 3 = 20.33December 19 22 + 19 + 21 / 3 = 20.67
Storage Shed Sales
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Moving Averages Forecast
Wallace Garden SupplyForecasting 3 period moving average
Input Data Forecast Error Analysis
Period Actual Value Forecast ErrorAbsolute
errorSquared
errorAbsolute % error
Month 1 10Month 2 12Month 3 16Month 4 13 12.667 0.333 0.333 0.111 2.56%Month 5 17 13.667 3.333 3.333 11.111 19.61%Month 6 19 15.333 3.667 3.667 13.444 19.30%Month 7 15 16.333 -1.333 1.333 1.778 8.89%Month 8 20 17.000 3.000 3.000 9.000 15.00%Month 9 22 18.000 4.000 4.000 16.000 18.18%Month 10 19 19.000 0.000 0.000 0.000 0.00%Month 11 21 20.333 0.667 0.667 0.444 3.17%Month 12 19 20.667 -1.667 1.667 2.778 8.77%
Average 1.333 2.000 6.074 10.61%Next period 19.667 BIAS MAD MSE MAPE
Actual Value - Forecast
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Moving Average Forecast
Three Period Moving Average
0
5
10
15
20
25
1 2 3 4 5 6 7 8 9 10 11 12
Time
Valu
e Actual Value
Forecast
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Stability vs. Responsiveness
Should I use a 3-period moving average or a 5-period moving average? The larger the “n” the more stable the forecast. A 3-period model will be more responsive to
change. We don’t want to chase outliers. But we don’t want to take forever to correct for a
real change. We must balance stability with responsiveness.
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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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Weighted Moving Average Model
Assumes data from some periods are more important than data from other periods (e.g. earlier periods).
Uses weights to place more emphasis on some periods and less on others
Historical values of the time series are assigned different weights when performing the forecast
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Weighted Moving Average Model
Weighted n-Period Moving Average
Typically weights are decreasing: w1>w2>…>wn
Sum of the weights = wi = 1Flexible weights reflect relative importance
of each previous observation in forecasting
1ntYnw2tY2w1tY1 w=tF
iw
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Weighted MA: An Illustration
Month Weight DataAugust 17% 130September 33% 110October 50% 90
November forecast:
FNov = (0.50)(90)+(0.33)(110)+(0.17)(130)
= 103.4
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Weighted Moving AverageWallace Garden SupplyForecasting
PeriodActual Value Weights Three-Month Weighted Moving Averages
January 10 0.222February 12 0.593March 16 0.185April 13 2.2 + 7.1 + 3 / 1 = 12.298May 17 2.7 + 9.5 + 2.4 / 1 = 14.556June 19 3.5 + 7.7 + 3.2 / 1 = 14.407July 15 2.9 + 10 + 3.5 / 1 = 16.484August 20 3.8 + 11 + 2.8 / 1 = 17.814September 22 4.2 + 8.9 + 3.7 / 1 = 16.815October 19 3.3 + 12 + 4.1 / 1 = 19.262November 21 4.4 + 13 + 3.5 / 1 = 21.000December 19 4.9 + 11 + 3.9 / 1 = 20.036
Next period 20.185
Sum of weights = 1.000
Storage Shed Sales
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Weighted Moving AverageWallace Garden Supply Forecasting 3 period weighted moving average
Input Data Forecast Error Analysis
Period Actual value Weights Forecast ErrorAbsolute
errorSquared
errorAbsolute % error
Month 1 10 0.222Month 2 12 0.593Month 3 16 0.185Month 4 13 12.298 0.702 0.702 0.492 5.40%Month 5 17 14.556 2.444 2.444 5.971 14.37%Month 6 19 14.407 4.593 4.593 21.093 24.17%Month 7 15 16.484 -1.484 1.484 2.202 9.89%Month 8 20 17.814 2.186 2.186 4.776 10.93%Month 9 22 16.815 5.185 5.185 26.889 23.57%Month 10 19 19.262 -0.262 0.262 0.069 1.38%Month 11 21 21.000 0.000 0.000 0.000 0.00%Month 12 19 20.036 -1.036 1.036 1.074 5.45%
Average 1.988 6.952 6.952 10.57%Next period 20.185 BIAS MAD MSE MAPE
Sum of weights = 1.000
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Operational Problems With Moving Averages
The operational shortcoming of simple moving average models is that if n observations are to be included in the moving average, then (n-1) pieces of past data must be brought forward to be combined with the current (the nth) observation
All this data must be stored in some way, in order to calculate the forecast.
This may become a problem when a company needs to forecast the demand for thousands of individual products on an item-by-item basis.
The next weighting scheme addresses this problem.
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Exponential Smoothing
• Moving average technique that requires little record keeping of past data.• Uses a smoothing constant α with a value between 0 and 1. (Usual range 0.1
to 0.3)• Applies alpha to most recent period, and applies one minus alpha distributed
to previous values• α = The weight assigned to the latest period (smoothing constant)• Forecast = α(Actual value in period t-1) + (1- α)(Forecast in period t-1)
• Can also be forecast for period t-1 plus α times the difference between the actual value and forecast in period t-1:
)Y)( -(1 )(YY=tF 1-T1-TT
)Y-(Y YY=tF 1-T1-T1-TT
valueactual periodLast :Yforecast periodLast :ˆF
T periodfor Forecast :ˆF
Constant Smoothing :
1-T
11-T
T
T
T
Y
Y
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Exponential Smoothing Data
PeriodActual
Value(Yt) Ŷt-1 α Yt-1 Ŷt-1 Ŷt
January 10 = 10 0.1February 12 10 + 0.1 *( 10 - 10 ) = 10.000March 16 10 + 0.1 *( 12 - 10 ) = 10.200April 13 10.2 + 0.1 *( 16 - 10.2 ) = 10.780May 17 10.78 + 0.1 *( 13 - 10.78 ) = 11.002June 19 11.002 + 0.1 *( 17 - 11.002 ) = 11.602July 15 11.602 + 0.1 *( 19 - 11.602 ) = 12.342August 20 12.342 + 0.1 *( 15 - 12.342 ) = 12.607September 22 12.607 + 0.1 *( 20 - 12.607 ) = 13.347October 19 13.347 + 0.1 *( 22 - 13.347 ) = 14.212November 21 14.212 + 0.1 *( 19 - 14.212 ) = 14.691December 19 14.691 + 0.1 *( 21 - 14.691 ) = 15.322
Storage Shed Sales
Class Exercise: What is the forecast for January of the following year? How about March? Find the Bias, Mad & MAPE. (Note: α equals 0.1.)
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Exponential Smoothing (Alpha = .419)
Wallace Garden SupplyForecasting Exponential smoothing
Input Data Forecast Error Analysis
Period Actual value Forecast ErrorAbsolute
errorSquared
errorAbsolute % error
Month 1 10 10.000Month 2 12 10.000 2.000 2.000 4.000 16.67%Month 3 16 10.838 5.162 5.162 26.649 32.26%Month 4 13 13.000 0.000 0.000 0.000 0.00%Month 5 17 13.000 4.000 4.000 16.000 23.53%Month 6 19 14.675 4.325 4.325 18.702 22.76%Month 7 15 16.487 -1.487 1.487 2.211 9.91%Month 8 20 15.864 4.136 4.136 17.106 20.68%Month 9 22 17.596 4.404 4.404 19.391 20.02%Month 10 19 19.441 -0.441 0.441 0.194 2.32%Month 11 21 19.256 1.744 1.744 3.041 8.30%Month 12 19 19.987 -0.987 0.987 0.973 5.19%
Average 2.608 9.842 14.70%Alpha 0.419 MAD MSE MAPE
Next period 19.573
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Exponential Smoothing
Exponential Smoothing
0
5
10
15
20
25
Janu
ary
Februa
ryMar
chApr
ilMay
June Ju
ly
Augus
t
Septem
ber
Octobe
r
Novem
ber
Decem
ber
Shed
s Actual value
Forecast
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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 Suitable for relatively stable time series
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Evaluating the Performance of Forecasting Techniques
Several forecasting methods have been presented.
Which one of these forecasting methods gives the “best” forecast?
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Performance Measures – Sample Example• Find the forecasts and the errors for each forecasting
technique applied to the following stationary time series.
3-Period Moving Average 3-Period Weighted Moving Avg. (.5,.3,.2)Period Sales Forecast Error A. Error P. Error Forecast Error A. Error P. Error
1 1002 1103 904 80 100.00 -20.00 20.00 25.00 98.00 -18.00 18.00 22.505 105 93.33 11.67 11.67 11.11 89.00 16.00 16.00 15.246 115 91.67 23.33 23.33 20.29 94.50 20.50 20.50 17.83
18.33 18.80 18.17 18.52BIAS MAD MAPE BIAS MAD MAPE
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MAD for the moving average technique:
MAD for the weighted moving average technique:
Performance Measures – MAD for the Sample Example
33.183
33.2367.1120
n|Error|
n|Forecast - Actual|
MAD
17.183
50.201618
n|Error|
n|Forecast - Actual|
MAD
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MAPE for the moving average technique:
MAPE for the weighted moving average technique:
Performance Measures – MAPE for the Sample Example
80.1811533.23
10567.11
8020%100*
3nActual
|Forecast - Actual|
MAPE
52.1811550.20
10516
8018%100*
3nActual
|Forecast - Actual|
MAPE
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Use the performance measures to select a good set of values for each model parameter. For the moving average:
the number of periods (n). For the weighted moving average:
The number of periods (n), The weights (wi).
For the exponential smoothing: The exponential smoothing factor ().
Excel Solver can be used to determine the values of the model parameters.
Performance Measures –Selecting Model Parameters
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Excel Solver
EXCEL SOLVER EXAMPLE
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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 & Seasonality
Trend analysis Technique that fits a trend equation (or curve) to a series of
historical data points Projects the equation into the future for medium and long
term forecasts. Typically do not want to forecast into the future more than half the number of time periods used to generate the forecast
Seasonality analysis Adjustment to time series data due to variations at certain
periods. Adjust with seasonal index - ratio of average value of the
item in a season to the overall annual average value. Examples: demand for coal in winter months; demand for
soft drinks in the summer and over major holidays
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Linear Trend AnalysisMidwestern Manufacturing Sales
Scatter Diagram
Actual value (or)
Y
Period number (or) X
74 199579 199680 199790 1998
105 1999142 2000122 2001
Sales(in units) vs. Time
0
20
40
60
80
100
120
140
160
1994 1995 1996 1997 1998 1999 2000 2001 2002
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Least Squares for Linear RegressionMidwestern Manufacturing
Least Squares Method
Time
Valu
es o
f Dep
ende
nt V
aria
bles
Objective: Minimize the squared deviations!
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Trend Analysis - Least Squares Method For Linear Regression
Y^
]Xn - XY[ __
Y
_
22 Xn -X
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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bt a Y^
Where
Y^
= predicted value of the dependent variable (demand)
a = Y- intercept
b = Slope of the regression line
t = independent variable (time period = 1, 2, 3, ….)
Trend Analysis - Least Squares Method For Linear Regression
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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:
nX
bnY
a
nX
XnYX
YXb
ii
ii
iiii
22 )(
/
Regression formula is an optional learning objective
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Find the regression line with ExcelUse Excel’s Tools | Data Analysis | Regression
Curve Fitting: Multiple RegressionTwo or more independent variables are used to
predict the dependent variable: Y = b0 + b1X1 + b2X2 + … + bpXp
Use Excel’s Tools | Data Analysis | Regression
Curve Fitting: Simple Linear Regression
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Linear Trend Data & Error AnalysisMidwestern Manufacturing CompanyForecasting Linear trend analysis
Input Data Forecast Error Analysis
PeriodActual value
(or) YPeriod number
(or) X Forecast ErrorAbsolute
errorSquared
errorAbsolute % error
Year 1 74 1 67.250 6.750 6.750 45.563 9.12%Year 2 79 2 77.786 1.214 1.214 1.474 1.54%Year 3 80 3 88.321 -8.321 8.321 69.246 10.40%Year 4 90 4 98.857 -8.857 8.857 78.449 9.84%Year 5 105 5 109.393 -4.393 4.393 19.297 4.18%Year 6 142 6 119.929 22.071 22.071 487.148 15.54%Year 7 122 7 130.464 -8.464 8.464 71.644 6.94%
Average 8.582 110.403 8.22%Intercept 56.714 MAD MSE MAPESlope 10.536
Next period 141.000 8
Enter the actual values in cells shaded YELLOW. Enter new time period at the bottom to forecast
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Least Squares Graph
Trend Analysis
y = 10.536x + 56.714
0
20
40
60
80
100
120
140
160
1 2 3 4 5 6 7
Time
Valu
e
Actual values Linear (Actual values)
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Pattern-based forecasting – Seasonal
The methods we have learned (Heuristic methods and Regression) are not suitable for data that has pronounced fluctuations.
If data seasonalized: Deseasonalize the data. 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
Deseasonalization
Deaseasonalized Data =
Reseasonalization Reseasonalized Forecast = (Deseasonalized Forecast )* (Seasonal Index)
Index SeasonalValue Actual
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Forecasting Seasonal Data With Trend
1. Calculate the seasonal indices.2. Calculate “deseasonalized” trend by divide the
actual value (Y) by the seasonal index for that period.
3. Find the trend line, and extend the trend line into the desired forecast period.
4. Now that we have the Seasonal Indices and Trend line, we can reseasonalize the data and generate the “seasonalized” forecast by multiplying the trend line values in the forecast period by the appropriate seasonal indices for each time period.
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Forecasting Seasonal Data: Calculating Seasonal IndexesEichler Supplies
Year Month DemandAverage Demand Ratio
Seasonal Index
1 January 80 94 0.851 0.957February 75 94 0.798 0.851
March 80 94 0.851 0.904April 90 94 0.957 1.064May 115 94 1.223 1.309June 110 94 1.170 1.223July 100 94 1.064 1.117
August 90 94 0.957 1.064September 85 94 0.904 0.957
October 75 94 0.798 0.851November 75 94 0.798 0.851December 80 94 0.851 0.851
2 January 100 94 1.064 0.957February 85 94 0.904 0.851
March 90 94 0.957 0.904April 110 94 1.170 1.064May 131 94 1.394 1.309June 120 94 1.277 1.223July 110 94 1.170 1.117
August 110 94 1.170 1.064September 95 94 1.011 0.957
October 85 94 0.904 0.851November 85 94 0.904 0.851December 80 94 0.851 0.851
Seasonal Index – ratio of the average value of the item in a season to the overall average annual value.
Example: average of year 1 January ratio to year 2 January ratio. (0.851 + 1.064)/2 = 0.957
Ratio = Demand / Average Demand
If Year 3 average monthly demand is expected to be 100 units.Forecast demand Year 3 January: 100 X 0.957 = 96 unitsForecast demand Year 3 May: 100 X 1.309 = 131 units
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Forecasting Seasonal Data: Calculating Seasonal IndexesEichler Supplies
Year Month DemandAverage Demand Ratio
Seasonal Index
1 January 80 94 0.851 0.957February 75 94 0.798 0.851
March 80 94 0.851 0.904April 90 94 0.957 1.064May 115 94 1.223 1.309June 110 94 1.170 1.223July 100 94 1.064 1.117
August 90 94 0.957 1.064September 85 94 0.904 0.957
October 75 94 0.798 0.851November 75 94 0.798 0.851December 80 94 0.851 0.851
2 January 100 94 1.064 0.957February 85 94 0.904 0.851
March 90 94 0.957 0.904April 110 94 1.170 1.064May 131 94 1.394 1.309June 120 94 1.277 1.223July 110 94 1.170 1.117
August 110 94 1.170 1.064September 95 94 1.011 0.957
October 85 94 0.904 0.851November 85 94 0.904 0.851December 80 94 0.851 0.851
1. Take average of all Demand Values (Average Demand Column)
2. Get Ratio of “Actual Value: Average Value” for each period (Ratio Column)
3. Average the ratio for corresponding periods to get seasonal index
064.194
100 January :Y2
851.9480 January :Y1
957.2
064.1851.2
RatioJanuary :Y2 RatioJanuary :Y1January for Index Seasonal
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Seasonal Forecasting
Seasonal Forecasting Example
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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?