DCPs in Forecasting Edward Kambour, Senior Scientist Roxy Cramer, Scientist

1. DCPs in ForecastingEdward Kambour, Senior ScientistRoxy Cramer, Scientist 1

2. Forecasting Background • The booking period is broken down into intervals during which the underlying demand process is stable • Handles heterogeneity in the arrival rates • Addresses the small numbers problem • Signal to noise • Sample sizes

3. DCP Forecasting • Aggregate all transactions that occur during an interval of the booking process • Use historical aggregated bookings to forecast the arrival rate during the DCP • Forecast the arrival rate for any given day in the interval by breaking up the DCP forecast • Assume constant arrival rate

4. Small Numbers Problem • Signal to noise • Finer granularity implies a lower signal to noise ratio • For Poisson data, the SNR = sqrt(mean) • Problematic for detecting demand shifts, seasonal trends, and holiday effects • Aggregating to the DCP level increases the signal to noise ratio

5. Small Numbers (cont.) • Sample Size • Aggregating m different days into a DCP increases the sample size by a factor of m • Using a 10 day DCP results in having 10 observations per departure date • Leads to superior forecast accuracy because we use more information about the demand process

6. Example 1 • 5 Day Booking period • Constant Poisson arrival rate • 1 per day • Examine forecast accuracy • 5 DCPs • Single DCP

7. Example 1 Booking Curve

8. Example 1: Forecasting • Suppose we have observations for n departure dates • Forecast the number of bookings between 4 and 5 days out • Single DCP: constant arrival rate • Average number of bookings over all the days out • 5 DCPs • Average number of bookings between days 4 and 5

9. Example 1: Forecast Accuracy • Both estimators are unbiased • Single DCP estimate is based on a sample size of 5n • Variance = 1/(5n), MSE = 1/(5n) • 5 DCP estimate is based on a sample size of n • Variance = 1/n, MSE = 1/n • The Single DCP estimate is more accurate

10. Example 1: Simulation • 5 historical departure dates

11. Example 1: Simulation Forecast Errors • Single DCP • MSE = 0.0144, MAE = 0.12 • 5 DCPs • MSE = 0.44, MAE = 0.52

12. Example 2 • 10 Day Booking period • Constant Poisson arrival rate over the first 5 days and the last 5 days • 1 per day in the first 5 • 5 per day in the last 5 • Examine forecast accuracy • 10 DCPs • 2 DCPs • Single DCP

13. Example 2 Booking Curve

14. Example 2: Forecasting • Suppose we have observations for n departure dates • Forecast the number of bookings on between 4 and 5 days out • Single DCP: constant arrival rate • Average number of bookings over all the days out • 2 DCPs: constant arrival rate from 5-10 and 0-5 days out • Average number of bookings from 0-5 days out • 10 DCPs • Average number of bookings between days 4 and 5

15. Example 2: Forecast Accuracy • 10 DCPs and 2 DCPs are unbiased • Single DCP will overestimate for 5-10 days out and underestimate for 0-5 days out (Absolute Bias = 2) • Single DCP, sample size of 10n • Variance = 3/(10n), MSE = 3/(10n) + 4 • 2 DCP, sample size of 5n • Variance = 1/n, MSE = 1/n • 10 DCP estimate is based on a sample size of n • Variance = 5/n, MSE = 5/n • The 2 DCP estimate is most accurate

16. Example 2: Simulation • 5 historical departure dates

17. Example 2: Simulation Forecast Errors • Single DCP: MSE = 4.07, MAE = 2 • 10 DCPs: MSE = 0.92, MAE = 0.7 • 2 DCPs: MSE = 0.24, MAE = 0.42

18. 10 DCP Booking Curve

19. 10 DCP Booking Curve

20. 2 DCP Booking Curve

21. 2 DCP Booking Curve

22. Finding the Best DCP Structure • Gather data for numerous departure dates • Fit every possible every possible DCP structure and select the one that has the smallest Mean Squared Error (MSE) • The structure with the smallest MSE will generally be the one with the fewest DCPs and negligible bias. • Recall that the MSE = Variance + Bias2

23. DCP Selection Algorithm • Configure the DCP question into a multiple linear regression with indicator predictors • Utilize the change point regression methodology from McLaren (2000) • Minimizes the estimated Expected MSE (risk), Eubank (1988) • Utilizes a mixture of Backward Elimination, Draper (1981), and Regression by Leaps and Bounds, Furnival (1974) • Extend the method to partition the MSE into its variance and squared bias components

24. Real Data Booking Curve

25. Real Fitted Booking Curve

26. Real Booking Curves

27. Considerations • Business rules and requirements • Application specific requirements • Concerns about the proportion of demand in each DCP • Don’t want to “put all the eggs in one basket” • Day of Week issues • Long haul versus short haul

28. Robustness • Yields a mathematical starting point • Finds best “sub-optimal” structures • Quantifies the effect of using different DCP structures

29. Conclusion • The number of DCPs is important • Too many leads to low SNR and high forecast error • Too few leads to biased forecasts, and hence high forecast error • Want constant arrival rate throughout a DCP interval • Examine historical booking curves • Keep in mind the randomness involved

30. Technical References Draper, N. and Smith, H. (1981) Applied Regression Analysis. Wiley, New York. Eubank, R. L. (1988) Spline Smoothing and Nonparametric Regression. Marcel Dekker, Inc., New York. Furnival, G. M. and Wilson, R. W. (1974). Regression by Leaps and Bounds. Technometrics, 16, 499-511. McLaren, C. E., Kambour, E. L., McLachlan, G. J. Lukaski, H. C., Li X., Brittenham, G. E., and McLaren, G. D. (2000). Patient-specific Analysis of Sequential Haematologial Data by Multiple Linear Regression and Mixture Modelling. Statistics in Medicine, 19, 83-98.