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Forecasting Quantitative - Time Series.ppt

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The document discusses various quantitative time series forecasting models including causal models and time series models. It describes stationary time series models including the naïve model, moving average models, and exponential smoothing. It explains that moving average models reduce random variation by averaging past data, and that exponential smoothing requires less data storage than moving averages as it applies a smoothing constant to weight the most recent period.

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FORECASTING Introduction Quantitative Models Time Series
Forecasting Models Forecasting Quantitative Qualitative Causal Model Time series Expert Judgment Trend Stationary Trend Trend + Seasonality Delphi Method Grassroots Market Research Jury Exec. Opinion
FORECASTING Introduction to Quantitative FORECASTING
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.
Quantitative Forecasts  Two types of quantitative forecasting models :  Causal models  Time-Series models
Quantitative Forecasts - Forecasting based on data and models  Causal Models:  Time Series Models: Price Population Advertising …… Causal Model Year 2000 Sales Sales1999 Sales1998 Sales1997 ………………… Time Series Model Year 2000 Sales
FORECASTING Time Series Forecasting FORECASTING
Forecasting Models Forecasting Quantitative Qualitative Causal Model Time series Expert Judgment Trend Stationary Trend Trend + Seasonality Delphi Method Grassroots Market Research Jury Exec. Opinion
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.
Components of a Time Series
11 Time Series Components Time Trend Random movement Time Cycle Time Seasonal pattern Demand Time Trend with seasonal pattern
12 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
13  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 n Error n Forecast) - (Actual BIAS     n | Error | n Forecast - Actual | MAD     |
14  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) n Forecast) - (Actual MSE 2 2     n Actual | Forecast - Actual | MAPE   % 100 * Evaluation of Forecasting Model
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 irregularities  The forecast is revised only when new data becomes available.  Stationary Model Types  Naïve Forecast  Moving Average/Weighted Moving Average  Exponential Smoothing
16 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.
17 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
Naïve Forecast Wallace Garden Supply Forecasting Period Actual Value Naïve Forecast Error Absolute Error Percent Error Squared Error January 10 N/A February 12 10 2 2 16.67% 4.0 March 16 12 4 4 25.00% 16.0 April 13 16 -3 3 23.08% 9.0 May 17 13 4 4 23.53% 16.0 June 19 17 2 2 10.53% 4.0 July 15 19 -4 4 26.67% 16.0 August 20 15 5 5 25.00% 25.0 September 22 20 2 2 9.09% 4.0 October 19 22 -3 3 15.79% 9.0 November 21 19 2 2 9.52% 4.0 December 19 21 -2 2 10.53% 4.0 0.818 3 17.76% 10.091 BIAS MAD MAPE MSE Standard Error (Square Root of MSE) = 3.176619 Storage Shed Sales
Wallace Garden - Naive Forecast 0 5 10 15 20 25 February March April May June July August September October November December Period Sheds Actual Value Naïve Forecast Naïve Forecast
20 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 n Y Y n T T i t t               1 n n t Y 2 t Y 1 t Y ˆ = t F n periods n previous in values actual of Sum ˆ t F 
Moving Averages Wallace Garden Supply Forecasting Period Actual Value Three-Month Moving Averages January 10 February 12 March 16 April 13 10 + 12 + 16 / 3 = 12.67 May 17 12 + 16 + 13 / 3 = 13.67 June 19 16 + 13 + 17 / 3 = 15.33 July 15 13 + 17 + 19 / 3 = 16.33 August 20 17 + 19 + 15 / 3 = 17.00 September 22 19 + 15 + 20 / 3 = 18.00 October 19 15 + 20 + 22 / 3 = 19.00 November 21 20 + 22 + 19 / 3 = 20.33 December 19 22 + 19 + 21 / 3 = 20.67 Storage Shed Sales
Moving Averages Forecast Wallace Garden Supply Forecasting 3 period moving average Input Data Forecast Error Analysis Period Actual Value Forecast Error Absolute error Squared error Absolute % error Month 1 10 Month 2 12 Month 3 16 Month 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
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 Value Actual Value Forecast
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.
25 Smoothing Effect of MA Model Longer-period moving averages (larger n) react to actual changes more slowly
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
27 Weighted 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         1 n t Y n w 2 t Y 2 w 1 t Y 1 w = t F i w 
28 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
Weighted Moving Average Wallace Garden Supply Forecasting Period Actual Value Weights Three-Month Weighted Moving Averages January 10 0.222 February 12 0.593 March 16 0.185 April 13 2.2 + 7.1 + 3 / 1 = 12.298 May 17 2.7 + 9.5 + 2.4 / 1 = 14.556 June 19 3.5 + 7.7 + 3.2 / 1 = 14.407 July 15 2.9 + 10 + 3.5 / 1 = 16.484 August 20 3.8 + 11 + 2.8 / 1 = 17.814 September 22 4.2 + 8.9 + 3.7 / 1 = 16.815 October 19 3.3 + 12 + 4.1 / 1 = 19.262 November 21 4.4 + 13 + 3.5 / 1 = 21.000 December 19 4.9 + 11 + 3.9 / 1 = 20.036 Next period 20.185 Sum of weights = 1.000 Storage Shed Sales
Weighted Moving Average Wallace Garden Supply Forecasting 3 period weighted moving average Input Data Forecast Error Analysis Period Actual value Weights Forecast Error Absolute error Squared error Absolute % error Month 1 10 0.222 Month 2 12 0.593 Month 3 16 0.185 Month 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
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.
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: ) Ŷ )( - (1 ) (Y Ŷ = t F 1 - T 1 - T T     ) Ŷ - (Y Ŷ Ŷ = t F 1 - T 1 - T 1 - T T    value actual period Last : Y forecast period Last : ˆ F T period for Forecast : ˆ F Constant Smoothing : 1 - T 1 1 - T T    T T Y Y 
Exponential Smoothing Data Period Actual Value(Yt) Ŷt-1 α Yt-1 Ŷt-1 Ŷt January 10 = 10 0.1 February 12 10 + 0.1 *( 10 - 10 ) = 10.000 March 16 10 + 0.1 *( 12 - 10 ) = 10.200 April 13 10.2 + 0.1 *( 16 - 10.2 ) = 10.780 May 17 10.78 + 0.1 *( 13 - 10.78 ) = 11.002 June 19 11.002 + 0.1 *( 17 - 11.002 ) = 11.602 July 15 11.602 + 0.1 *( 19 - 11.602 ) = 12.342 August 20 12.342 + 0.1 *( 15 - 12.342 ) = 12.607 September 22 12.607 + 0.1 *( 20 - 12.607 ) = 13.347 October 19 13.347 + 0.1 *( 22 - 13.347 ) = 14.212 November 21 14.212 + 0.1 *( 19 - 14.212 ) = 14.691 December 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.)
Exponential Smoothing (Alpha = .419) Wallace Garden Supply Forecasting Exponential smoothing Input Data Forecast Error Analysis Period Actual value Forecast Error Absolute error Squared error Absolute % error Month 1 10 10.000 Month 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
Exponential Smoothing Exponential Smoothing 0 5 10 15 20 25 J a n u a r y F e b r u a r y M a r c h A p r i l M a y J u n e J u l y A u g u s t S e p t e m b e r O c t o b e r N o v e m b e r D e c e m b e r Sheds Actual value Forecast
36 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
Evaluating the Performance of Forecasting Techniques  Several forecasting methods have been presented.  Which one of these forecasting methods gives the “best” forecast?
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 1 100 2 110 3 90 4 80 100.00 -20.00 20.00 25.00 98.00 -18.00 18.00 22.50 5 105 93.33 11.67 11.67 11.11 89.00 16.00 16.00 15.24 6 115 91.67 23.33 23.33 20.29 94.50 20.50 20.50 17.83 18.33 18.80 18.17 18.52 BIAS MAD MAPE BIAS MAD MAPE
MAD for the moving average technique: MAD for the weighted moving average technique: Performance Measures – MAD for the Sample Example 33 . 18 3 33 . 23 67 . 11 20         n | Error | n | Forecast - Actual | MAD 17 . 18 3 50 . 20 16 18         n | Error | n | Forecast - Actual | MAD
MAPE for the moving average technique: MAPE for the weighted moving average technique: Performance Measures – MAPE for the Sample Example 80 . 18 115 33 . 23 105 67 . 11 80 20 % 100 *       3 n Actual | Forecast - Actual | MAPE 52 . 18 115 50 . 20 105 16 80 18 % 100 *       3 n Actual | Forecast - Actual | MAPE
 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
Excel Solver EXCEL SOLVER EXAMPLE
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
44 Time Series Components Time Trend Random movement Time Cycle Time Seasonal pattern Demand Time Trend with seasonal pattern
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
Linear Trend Analysis Midwestern Manufacturing Sales Scatter Diagram Actual value (or) Y Period number (or) X 74 1995 79 1996 80 1997 90 1998 105 1999 142 2000 122 2001 Sales(in units) vs. Time 0 20 40 60 80 100 120 140 160 1994 1995 1996 1997 1998 1999 2000 2001 2002
Least Squares for Linear Regression Midwestern Manufacturing Least Squares Method Time Values of Dependent Variables Objective: Minimize the squared deviations!
Trend Analysis - Least Squares Method For Linear Regression Y ^ ] X n - XY [ _ _ Y         _ 2 2 X n - 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,…
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
50 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 X b n Y a n X X n Y X Y X b i i i i i i i i                             2 2 ) ( / Regression formula is an optional learning objective
51  Find the regression line with Excel  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 Curve Fitting: Simple Linear Regression
Linear Trend Data & Error Analysis Midwestern Manufacturing Company Forecasting Linear trend analysis Input Data Forecast Error Analysis Period Actual value (or) Y Period number (or) X Forecast Error Absolute error Squared error Absolute % 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 MAPE Slope 10.536 Next period 141.000 8 Enter the actual values in cells shaded YELLOW. Enter new time period at the bottom to forecast
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 Value Actual values Linear (Actual values)
54 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.
55 Pattern-based forecasting – Seasonal Deseasonalize Forecast Reseasonalize Actual data Deseasonalized data Example (SI + Regression)
56 Pattern-based forecasting – Seasonal Deseasonalization  Deaseasonalized Data = Reseasonalization  Reseasonalized Forecast = (Deseasonalized Forecast )* (Seasonal Index) Index Seasonal Value Actual
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.
Forecasting Seasonal Data: Calculating Seasonal Indexes Eichler Supplies Year Month Demand Average Demand Ratio Seasonal Index 1 January 80 94 0.851 0.957 February 75 94 0.798 0.851 March 80 94 0.851 0.904 April 90 94 0.957 1.064 May 115 94 1.223 1.309 June 110 94 1.170 1.223 July 100 94 1.064 1.117 August 90 94 0.957 1.064 September 85 94 0.904 0.957 October 75 94 0.798 0.851 November 75 94 0.798 0.851 December 80 94 0.851 0.851 2 January 100 94 1.064 0.957 February 85 94 0.904 0.851 March 90 94 0.957 0.904 April 110 94 1.170 1.064 May 131 94 1.394 1.309 June 120 94 1.277 1.223 July 110 94 1.170 1.117 August 110 94 1.170 1.064 September 95 94 1.011 0.957 October 85 94 0.904 0.851 November 85 94 0.904 0.851 December 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 units Forecast demand Year 3 May: 100 X 1.309 = 131 units
Forecasting Seasonal Data: Calculating Seasonal Indexes Eichler Supplies Year Month Demand Average Demand Ratio Seasonal Index 1 January 80 94 0.851 0.957 February 75 94 0.798 0.851 March 80 94 0.851 0.904 April 90 94 0.957 1.064 May 115 94 1.223 1.309 June 110 94 1.170 1.223 July 100 94 1.064 1.117 August 90 94 0.957 1.064 September 85 94 0.904 0.957 October 75 94 0.798 0.851 November 75 94 0.798 0.851 December 80 94 0.851 0.851 2 January 100 94 1.064 0.957 February 85 94 0.904 0.851 March 90 94 0.957 0.904 April 110 94 1.170 1.064 May 131 94 1.394 1.309 June 120 94 1.277 1.223 July 110 94 1.170 1.117 August 110 94 1.170 1.064 September 95 94 1.011 0.957 October 85 94 0.904 0.851 November 85 94 0.904 0.851 December 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 . 1 94 100 January : Y2 851 . 94 80 January : Y1     957 . 2 064 . 1 851 . 2 Ratio January : Y2 Ratio January : Y1 January for Index Seasonal     
Seasonal Forecasting  Seasonal Forecasting Example
61 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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Forecasting Quantitative - Time Series.ppt

  • 2. Forecasting Models Forecasting Quantitative Qualitative Causal Model Time series Expert Judgment Trend Stationary Trend Trend + Seasonality Delphi Method Grassroots Market Research Jury Exec. Opinion
  • 4. 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.
  • 5. Quantitative Forecasts  Two types of quantitative forecasting models :  Causal models  Time-Series models
  • 6. Quantitative Forecasts - Forecasting based on data and models  Causal Models:  Time Series Models: Price Population Advertising …… Causal Model Year 2000 Sales Sales1999 Sales1998 Sales1997 ………………… Time Series Model Year 2000 Sales
  • 8. Forecasting Models Forecasting Quantitative Qualitative Causal Model Time series Expert Judgment Trend Stationary Trend Trend + Seasonality Delphi Method Grassroots Market Research Jury Exec. Opinion
  • 9. 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.
  • 10. Components of a Time Series
  • 12. 12 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
  • 13. 13  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 n Error n Forecast) - (Actual BIAS     n | Error | n Forecast - Actual | MAD     |
  • 14. 14  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) n Forecast) - (Actual MSE 2 2     n Actual | Forecast - Actual | MAPE   % 100 * Evaluation of Forecasting Model
  • 15. 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 irregularities  The forecast is revised only when new data becomes available.  Stationary Model Types  Naïve Forecast  Moving Average/Weighted Moving Average  Exponential Smoothing
  • 16. 16 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.
  • 17. 17 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
  • 18. Naïve Forecast Wallace Garden Supply Forecasting Period Actual Value Naïve Forecast Error Absolute Error Percent Error Squared Error January 10 N/A February 12 10 2 2 16.67% 4.0 March 16 12 4 4 25.00% 16.0 April 13 16 -3 3 23.08% 9.0 May 17 13 4 4 23.53% 16.0 June 19 17 2 2 10.53% 4.0 July 15 19 -4 4 26.67% 16.0 August 20 15 5 5 25.00% 25.0 September 22 20 2 2 9.09% 4.0 October 19 22 -3 3 15.79% 9.0 November 21 19 2 2 9.52% 4.0 December 19 21 -2 2 10.53% 4.0 0.818 3 17.76% 10.091 BIAS MAD MAPE MSE Standard Error (Square Root of MSE) = 3.176619 Storage Shed Sales
  • 19. Wallace Garden - Naive Forecast 0 5 10 15 20 25 February March April May June July August September October November December Period Sheds Actual Value Naïve Forecast Naïve Forecast
  • 20. 20 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 n Y Y n T T i t t               1 n n t Y 2 t Y 1 t Y ˆ = t F n periods n previous in values actual of Sum ˆ t F 
  • 21. Moving Averages Wallace Garden Supply Forecasting Period Actual Value Three-Month Moving Averages January 10 February 12 March 16 April 13 10 + 12 + 16 / 3 = 12.67 May 17 12 + 16 + 13 / 3 = 13.67 June 19 16 + 13 + 17 / 3 = 15.33 July 15 13 + 17 + 19 / 3 = 16.33 August 20 17 + 19 + 15 / 3 = 17.00 September 22 19 + 15 + 20 / 3 = 18.00 October 19 15 + 20 + 22 / 3 = 19.00 November 21 20 + 22 + 19 / 3 = 20.33 December 19 22 + 19 + 21 / 3 = 20.67 Storage Shed Sales
  • 22. Moving Averages Forecast Wallace Garden Supply Forecasting 3 period moving average Input Data Forecast Error Analysis Period Actual Value Forecast Error Absolute error Squared error Absolute % error Month 1 10 Month 2 12 Month 3 16 Month 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
  • 23. 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 Value Actual Value Forecast
  • 24. 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.
  • 25. 25 Smoothing Effect of MA Model Longer-period moving averages (larger n) react to actual changes more slowly
  • 26. 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
  • 27. 27 Weighted 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         1 n t Y n w 2 t Y 2 w 1 t Y 1 w = t F i w 
  • 28. 28 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
  • 29. Weighted Moving Average Wallace Garden Supply Forecasting Period Actual Value Weights Three-Month Weighted Moving Averages January 10 0.222 February 12 0.593 March 16 0.185 April 13 2.2 + 7.1 + 3 / 1 = 12.298 May 17 2.7 + 9.5 + 2.4 / 1 = 14.556 June 19 3.5 + 7.7 + 3.2 / 1 = 14.407 July 15 2.9 + 10 + 3.5 / 1 = 16.484 August 20 3.8 + 11 + 2.8 / 1 = 17.814 September 22 4.2 + 8.9 + 3.7 / 1 = 16.815 October 19 3.3 + 12 + 4.1 / 1 = 19.262 November 21 4.4 + 13 + 3.5 / 1 = 21.000 December 19 4.9 + 11 + 3.9 / 1 = 20.036 Next period 20.185 Sum of weights = 1.000 Storage Shed Sales
  • 30. Weighted Moving Average Wallace Garden Supply Forecasting 3 period weighted moving average Input Data Forecast Error Analysis Period Actual value Weights Forecast Error Absolute error Squared error Absolute % error Month 1 10 0.222 Month 2 12 0.593 Month 3 16 0.185 Month 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
  • 31. 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.
  • 32. 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: ) Ŷ )( - (1 ) (Y Ŷ = t F 1 - T 1 - T T     ) Ŷ - (Y Ŷ Ŷ = t F 1 - T 1 - T 1 - T T    value actual period Last : Y forecast period Last : ˆ F T period for Forecast : ˆ F Constant Smoothing : 1 - T 1 1 - T T    T T Y Y 
  • 33. Exponential Smoothing Data Period Actual Value(Yt) Ŷt-1 α Yt-1 Ŷt-1 Ŷt January 10 = 10 0.1 February 12 10 + 0.1 *( 10 - 10 ) = 10.000 March 16 10 + 0.1 *( 12 - 10 ) = 10.200 April 13 10.2 + 0.1 *( 16 - 10.2 ) = 10.780 May 17 10.78 + 0.1 *( 13 - 10.78 ) = 11.002 June 19 11.002 + 0.1 *( 17 - 11.002 ) = 11.602 July 15 11.602 + 0.1 *( 19 - 11.602 ) = 12.342 August 20 12.342 + 0.1 *( 15 - 12.342 ) = 12.607 September 22 12.607 + 0.1 *( 20 - 12.607 ) = 13.347 October 19 13.347 + 0.1 *( 22 - 13.347 ) = 14.212 November 21 14.212 + 0.1 *( 19 - 14.212 ) = 14.691 December 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.)
  • 34. Exponential Smoothing (Alpha = .419) Wallace Garden Supply Forecasting Exponential smoothing Input Data Forecast Error Analysis Period Actual value Forecast Error Absolute error Squared error Absolute % error Month 1 10 10.000 Month 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
  • 36. 36 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
  • 37. Evaluating the Performance of Forecasting Techniques  Several forecasting methods have been presented.  Which one of these forecasting methods gives the “best” forecast?
  • 38. 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 1 100 2 110 3 90 4 80 100.00 -20.00 20.00 25.00 98.00 -18.00 18.00 22.50 5 105 93.33 11.67 11.67 11.11 89.00 16.00 16.00 15.24 6 115 91.67 23.33 23.33 20.29 94.50 20.50 20.50 17.83 18.33 18.80 18.17 18.52 BIAS MAD MAPE BIAS MAD MAPE
  • 39. MAD for the moving average technique: MAD for the weighted moving average technique: Performance Measures – MAD for the Sample Example 33 . 18 3 33 . 23 67 . 11 20         n | Error | n | Forecast - Actual | MAD 17 . 18 3 50 . 20 16 18         n | Error | n | Forecast - Actual | MAD
  • 40. MAPE for the moving average technique: MAPE for the weighted moving average technique: Performance Measures – MAPE for the Sample Example 80 . 18 115 33 . 23 105 67 . 11 80 20 % 100 *       3 n Actual | Forecast - Actual | MAPE 52 . 18 115 50 . 20 105 16 80 18 % 100 *       3 n Actual | Forecast - Actual | MAPE
  • 41.  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
  • 43. 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
  • 45. 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
  • 46. Linear Trend Analysis Midwestern Manufacturing Sales Scatter Diagram Actual value (or) Y Period number (or) X 74 1995 79 1996 80 1997 90 1998 105 1999 142 2000 122 2001 Sales(in units) vs. Time 0 20 40 60 80 100 120 140 160 1994 1995 1996 1997 1998 1999 2000 2001 2002
  • 47. Least Squares for Linear Regression Midwestern Manufacturing Least Squares Method Time Values of Dependent Variables Objective: Minimize the squared deviations!
  • 48. Trend Analysis - Least Squares Method For Linear Regression Y ^ ] X n - XY [ _ _ Y         _ 2 2 X n - 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,…
  • 49. 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
  • 50. 50 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 X b n Y a n X X n Y X Y X b i i i i i i i i                             2 2 ) ( / Regression formula is an optional learning objective
  • 51. 51  Find the regression line with Excel  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 Curve Fitting: Simple Linear Regression
  • 52. Linear Trend Data & Error Analysis Midwestern Manufacturing Company Forecasting Linear trend analysis Input Data Forecast Error Analysis Period Actual value (or) Y Period number (or) X Forecast Error Absolute error Squared error Absolute % 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 MAPE Slope 10.536 Next period 141.000 8 Enter the actual values in cells shaded YELLOW. Enter new time period at the bottom to forecast
  • 53. 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 Value Actual values Linear (Actual values)
  • 54. 54 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.
  • 55. 55 Pattern-based forecasting – Seasonal Deseasonalize Forecast Reseasonalize Actual data Deseasonalized data Example (SI + Regression)
  • 56. 56 Pattern-based forecasting – Seasonal Deseasonalization  Deaseasonalized Data = Reseasonalization  Reseasonalized Forecast = (Deseasonalized Forecast )* (Seasonal Index) Index Seasonal Value Actual
  • 57. 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.
  • 58. Forecasting Seasonal Data: Calculating Seasonal Indexes Eichler Supplies Year Month Demand Average Demand Ratio Seasonal Index 1 January 80 94 0.851 0.957 February 75 94 0.798 0.851 March 80 94 0.851 0.904 April 90 94 0.957 1.064 May 115 94 1.223 1.309 June 110 94 1.170 1.223 July 100 94 1.064 1.117 August 90 94 0.957 1.064 September 85 94 0.904 0.957 October 75 94 0.798 0.851 November 75 94 0.798 0.851 December 80 94 0.851 0.851 2 January 100 94 1.064 0.957 February 85 94 0.904 0.851 March 90 94 0.957 0.904 April 110 94 1.170 1.064 May 131 94 1.394 1.309 June 120 94 1.277 1.223 July 110 94 1.170 1.117 August 110 94 1.170 1.064 September 95 94 1.011 0.957 October 85 94 0.904 0.851 November 85 94 0.904 0.851 December 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 units Forecast demand Year 3 May: 100 X 1.309 = 131 units
  • 59. Forecasting Seasonal Data: Calculating Seasonal Indexes Eichler Supplies Year Month Demand Average Demand Ratio Seasonal Index 1 January 80 94 0.851 0.957 February 75 94 0.798 0.851 March 80 94 0.851 0.904 April 90 94 0.957 1.064 May 115 94 1.223 1.309 June 110 94 1.170 1.223 July 100 94 1.064 1.117 August 90 94 0.957 1.064 September 85 94 0.904 0.957 October 75 94 0.798 0.851 November 75 94 0.798 0.851 December 80 94 0.851 0.851 2 January 100 94 1.064 0.957 February 85 94 0.904 0.851 March 90 94 0.957 0.904 April 110 94 1.170 1.064 May 131 94 1.394 1.309 June 120 94 1.277 1.223 July 110 94 1.170 1.117 August 110 94 1.170 1.064 September 95 94 1.011 0.957 October 85 94 0.904 0.851 November 85 94 0.904 0.851 December 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 . 1 94 100 January : Y2 851 . 94 80 January : Y1     957 . 2 064 . 1 851 . 2 Ratio January : Y2 Ratio January : Y1 January for Index Seasonal     
  • 60. Seasonal Forecasting  Seasonal Forecasting Example
  • 61. 61 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?