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# Time series mnr

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### Time series mnr

1. 1. Quantitative Methods for Business PGDMA-624 Time Series Forecasting Note: Adapted from “Quantitative Methods for Business by Anderson et all.
2. 2. 20-7-2010 2 Quantitative Approaches to Forecasting  Quantitative methods are based on an analysis of historical data concerning one or more time series.  A time series is a set of observations measured at successive points in time or over successive periods of time.  If the historical data used are restricted to past values of the series that we are trying to forecast, the procedure is called a time series method.  If the historical data used involve other time series that are believed to be related to the time series that we are trying to forecast, the procedure is called a causal method.
3. 3. 20-7-2010 3 Time Series Methods  Three time series methods are:  smoothing  trend projection  trend projection adjusted for seasonal influence  Two primary methods: causal models and time series methods Causal Models (Regression Models) Let Y be the quantity to be forecasted and (X1, X2, . . . , Xn) are n variables that have predictive power for Y. A causal model is Y = f (X1, X2, . . . , Xn). A typical relationship is a linear one: Y = a0 + a1X1 + . . . + an Xn
4. 4. 20-7-2010 4 Time Series Methods  A time series is just collection of past values of the variable being predicted. Also known as naïve methods. Goal is to isolate patterns in past data. (See Figures on following pages) Components of Time series  Trend  Seasonality  Cycles  Irregular Component or Randomness
5. 5. 20-7-2010 5 Components of a Time Series  The trend component accounts for the gradual shifting of the time series over a long period of time.  Any regular pattern of sequences of values above and below the trend line is attributable to the cyclical component of the series.  The seasonal component of the series accounts for regular patterns of variability within certain time periods, such as over a year.  The irregular component of the series is caused by short-term, unanticipated and non-recurring factors that affect the values of the time series. One cannot attempt to predict its impact on the time series in advance.
6. 6. 20-7-2010 6
7. 7. 20-7-2010 7 Notation Conventions  Let Y1, Y2, . . . Yn, . . . be the past values of the series to be predicted (demands?). If we are making a forecast during period t (for the future), assume we have observed Yt , Yt-1 etc.  Let Ft = forecast made in period t Models of Time Series Additive Model: Y= S+T+C+I Multiplicative Model : Y =S.T.C.I
8. 8. 20-7-2010 8 Evaluation of Forecasts  The forecast error in period t, et, is the difference between the forecast for demand in period t and the actual value of demand in t.  For one step ahead forecast: et = Yt – Ft  To evaluate Forecasting accuracy we develop a chart of Forecasting errors using: Mean Square error =MSE = (1/n) Σ ei 2 Root Mean Square error = RMSE = √MSE Mean absolute Deviation: MAD = (1/n) Σ | e i |
9. 9. 20-7-2010 9 Forecasting for Stationary Series  A stationary time series has the form: Dt = µ + εt where µ is a constant and εt is a random variable with mean 0 and var σ2  Stationary series indicate stable processes without observable trends  Two common methods for forecasting stationary series are moving averages and exponential smoothing. Time Series with irregular (Random) component
10. 10. 20-7-2010 10 Week Sales (1000 of gallons) 1 17 2 21 3 19 4 23 5 18 6 16 7 20 8 18 9 22 10 20 11 15 12 22 Example Sales (1000 of gallons) 0 5 10 15 20 25 1 2 3 4 5 6 7 8 9 10 11 12 Week Sales(1000sofones)
11. 11. 20-7-2010 11 Moving Averages  In words: the arithmetic average of the n most recent observations. For a one-step- ahead forecast: Ft = (1/N) (Y t - 1 + Y t - 2 + . . . + Y t - n )
12. 12. 20-7-2010 12 Summary of Moving Averages  Advantages of Moving Average Method  Easily understood  Easily computed  Provides stable forecasts  Disadvantages of Moving Average Method  Requires saving lots of past data points: at least the N periods used in the moving average computation  Lags behind a trend  Ignores complex relationships in data
13. 13. 20-7-2010 13 What about Weighted Moving Averages?  This method looks at past data and tries to logically attach importance to certain data over other data  Weighting factors must add to one  Can weight recent higher than older or specific data above others Selecting length of moving averages
14. 14. 20-7-2010 14 Exponential Smoothing Method A type of weighted moving average that applies declining weights to past data. 1. New Forecast = α (most recent observation) + (1 - α) (last forecast) where 0 < α < 1 and generally is small for stability of forecasts ( around .1 to .2)
15. 15. 20-7-2010 15 Exponential Smoothing (cont.) In symbols: Ft+1 = α Yt + (1 - α ) Ft = α Yt + (1 - α ) (α Yt-1 + (1 - α ) Yt-1) = α Yt + (1 - α )(α )Yt-1 + (1 - α)2 (α )Yt - 2 + . . .  Hence the method applies a set of exponentially declining weights to past data. It is easy to show that the sum of the weights is exactly one.
16. 16. 20-7-2010 16 Exponential Smoothing (cont.) In symbols: Ft+1 = α Yt + (1 - α ) Ft = α Yt + (1 - α ) (α Yt-1 + (1 - α ) Yt-1) = α Yt + (1 - α )(α )Yt-1 + (1 - α)2 (α )Yt - 2 + . . .  Hence the method applies a set of exponentially declining weights to past data. It is easy to show that the sum of the weights is exactly one.
17. 17. 20-7-2010 17 Weights in Exponential Smoothing:
18. 18. 20-7-2010 18 Comparison of MA and ES  Similarities  Both methods are appropriate for stationary series  Both methods depend on a single parameter  Both methods lag behind a trend
19. 19. 20-7-2010 19 Comparison of MA and ES  Differences  ES carries all past history (forever!)  MA eliminates “bad” data after N periods  MA requires all N past data points to compute new forecast estimate while ES only requires last forecast and last observation of ‘demand’ to continue
20. 20. 20-7-2010 20 Forecasting with Trend and Seasonal Components  Steps of Multiplicative Time Series Model 1. Calculate the centered moving averages (CMAs). 2. Center the CMAs on integer-valued periods. 3. Determine the seasonal and irregular factors (StIt ). 4. Determine the average seasonal factors. 5. Scale the seasonal factors (St ). 6. Determine the deseasonalized data. 7. Determine a trend line of the deseasonalized data. 8. Determine the deseasonalized predictions. 9. Take into account the seasonality.
21. 21. 20-7-2010 21 Year Quarter Sales(1000s) 1 1 4.8 2 4.1 3 6 4 6.5 2 1 5.8 2 5.2 3 6.8 4 7.4 3 1 6 2 5.6 3 7.5 4 7.8 4 1 6.3 2 5.9 3 8 4 8.4 Trend and Seasonal Components Sales(1000s) 0 1 2 3 4 5 6 7 8 9 Y1 Q1 Y1 Q2 Y1 Q3 y1 Q4 Y2 Q1 Y2 Q2 Y2 Q3 y2 Q4 Y3 Q1 Y3 Q2 Y3 Q3 y3 Q4 Y4 Q1 Y4 Q2 Y4 Q3 y4 Q4 Year Sales(000s)
22. 22. 20-7-2010 22 Calculating Seasonal Indices De seasonalizing time series Estimating Trend Forecasting by adjusting seasonal variations Fore casting
23. 23. 20-7-2010 23 Using Regression for Times Series ForecastingUsing Regression for Times Series Forecasting  Regression Methods Can be Used When a TrendRegression Methods Can be Used When a Trend is Present.is Present.  Model: Dt = a + bt +Model: Dt = a + bt + tt..  If t is scaled to 1, 2, 3, . . . , -- it becomes aIf t is scaled to 1, 2, 3, . . . , -- it becomes a number i -- then the least squares estimates fornumber i -- then the least squares estimates for aa andand bb can be computed as follows: (n is thecan be computed as follows: (n is the number of observation we have)number of observation we have)