DSc 3120 Generalized Modeling Techniques with Applications

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DSc 3120 Generalized Modeling Techniques with
Applications

• Part II. Forecasting

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Forecasting

• Why Forecasting?
• Characteristics of Forecasts
• Forecasts are usually wrong or seldom correct
• Aggregate forecasts are usually more accurate
• Less accurate further into the future
• Assumptions of Forecasting Models
• Information (data) about the past is available
• The pattern of the past will continue into the
  future.

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Qualitative Forecasting
--Forecasting based on experience, judgement,
and knowledge

• Sales force composites (field sales force)
• Consumer market survey (users expectations)
• Jury of executive
• The Delphi method

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Quantitative Forecasting
--Forecasting based on data and models

• Casual Models

Price Population Advertising
Causal Model
Year 2000 Sales

• Time Series Models

Sales1999 Sales1998 Sales1997
Time Series Model
Year 2000 Sales
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Overview of Forecasting Models
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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

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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 Excels 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 Excels Tools Data Analysis Regression

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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)
• 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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Time Series Model Building

• Historical data collection
• Data plotting (time series plot)
• Forecasting model building
• Evaluation and selection of model
• Forecasting with the final selected model

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Components of A Time Series

• Trend long term overall up or down movement
• Seasonality periodic pattern repeating every
  year
• Cycles up down movement repeating over long
  time frame
• Random Variations random movements follow no
  pattern

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Components of A Time Series
Cycle
Trend
Random movement
Time
Time
Seasonal pattern
Trend with seasonal pattern
Demand
Time
Time
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Types of Time Series Models

• Nonseasonal Model
• Trend
• Naïve
• Moving average
• Exponential
• Seasonal Model
• Time Series Decomposition

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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,

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Trend Model (Cont.)

• Nonlinear Trend Models
• Power Yt atb
• Quadratic Yt a bt ct2

Power
Quadratic
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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 observation 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 w1gtw2gtgtwn
• Sum of the weights ?wi 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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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

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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

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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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Holts Model Exponential Smoothing with Trend

• Ft Forecast for period t
• Lt Level term (intercept)
• Tt Trend term (slope)
• Yt-1 Last period actual value
• ? Smoothing constant for Level L
• ? Smoothing constant for Trend T

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Time Series Decomposition Model

• Basic Idea a time series is composed of several
  basic components Trend, Seasonality, Cycle, and
  Random Error
• The multiplicative decomposition model
• These components contribute to time series value
  in a multiplicative way

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Time Series Decomposition

• The basic model is
• Y Trend ? Cyclical ? Seasonal ? Error
• Since we cannot easily extract or predict
  cycles, we will assume that the trend component
  will capture cycles during the forecast period
• Since we have to live with error (cannot predict
  it), our model is simplified to
• Y Trend ? Seasonal

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I. Estimate Seasonal Component - Seasonal Index

• Step 1 Calculate 1-Year Moving Averages
• For quarterly data, use 4-period MA
• For monthly data, use 12-period MA
• Step 2 Calculate Centered Moving Averages
• Simple average of two adjacent MAs
• Step 3 Calculate Seasonal Ratio (SR)
• SR Y / CMA
• Step 4 Calculate Seasonal Index (SI)

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Calculate Seasonal Index (Steps 1-3)
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Calculate Seasonal Index (Step 4)

• ASR Average Seasonal Ratio
• GA Grand Average average of all ASRs
• The average of Seasonal Indices (SI) must be 1

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II. Estimate Trend Component

• Step 1 Remove seasonal effect
• Deseasonalized datat Yt / SIt
• Step 2 Fit a trend line to deseasonalized
• data using least squares method
• Step 3 Calculate the trend value for each
• period
• Note If the deseasonalized data look stable (no
  apparent trend), simple exponential smoothing may
  be used in Steps 2 and 3 to calculate the
  forecast (rather than trend) for each period.

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III. Forecast

• Combine seasonal and trend components
• Ft Trend Valuet ? Seasonal Indext
• This final step is also called reseasonalizing
• Trend Valuet is the trend estimate for the period
  t, based on the trend model fitted to the
  deseasonalized data