Confidential 1 DCPs in Forecasting Edward Kambour, Senior Scientist Roxy Cramer, Scientist

Confidential 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

Confidential 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

Confidential 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

Confidential 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

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

Confidential Example 1 Booking Curve

Confidential 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

Confidential 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

Confidential Example 1: Simulation 5 historical departure dates

Confidential Example 1: Simulation Forecast Errors Single DCP MSE = , MAE = DCPs MSE = 0.44, MAE = 0.52

Confidential 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

Confidential Example 2 Booking Curve

Confidential 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

Confidential 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) 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

Confidential Example 2: Simulation 5 historical departure dates

Confidential Example 2: Simulation Forecast Errors Single DCP: MSE = 4.07, MAE = 2 10 DCPs: MSE = 0.92, MAE = DCPs: MSE = 0.24, MAE = 0.42

Confidential 10 DCP Booking Curve

Confidential 10 DCP Booking Curve

Confidential 2 DCP Booking Curve

Confidential 2 DCP Booking Curve

Confidential 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 + Bias 2

Confidential 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

Confidential Real Data Booking Curve

Confidential Real Fitted Booking Curve

Confidential Real Booking Curves

Confidential 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

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

Confidential 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

Confidential 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, 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,