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DCPs in Forecasting

Edward Kambour, Senior Scientist

Roxy Cramer, Scientist

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Forecasting BackgroundForecasting 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

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DCP ForecastingDCP 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

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Small Numbers ProblemSmall 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

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Small Numbers (cont.)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

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Example 1Example 1

5 Day Booking period Constant Poisson arrival rate

1 per day Examine forecast accuracy

5 DCPs Single DCP

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Example 1 Booking CurveExample 1 Booking Curve

Booking Curve

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Days Prior

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Example 1: ForecastingExample 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

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Example 1: Forecast AccuracyExample 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

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Example 1: SimulationExample 1: Simulation

5 historical departure dates

Arrival Date 0 to 1 1 to 2 2 to 3 3 to 4 4 to 51 1 0 1 1 32 1 0 0 1 13 1 0 1 2 34 0 0 1 0 05 2 0 3 3 3

5 DCP 1 0 1.2 1.4 2Single DCP 1.12 1.12 1.12 1.12 1.12

Days Out

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Example 1: Simulation Forecast Errors

Example 1: Simulation Forecast Errors

Single DCP MSE = 0.0144, MAE = 0.12

5 DCPs MSE = 0.44, MAE = 0.52

5 DCP 1 0 1.2 1.4 2Single DCP 1.12 1.12 1.12 1.12 1.12Truth 1 1 1 1 1

5 DCP Error 0 -1 0.2 0.4 1Single Error 0.12 0.12 0.12 0.12 0.12

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Example 2Example 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

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Example 2 Booking CurveExample 2 Booking Curve

Booking Curve

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0 2 4 6 8 10

Days Prior

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Example 2: ForecastingExample 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

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Example 2: Forecast AccuracyExample 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

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Example 2: SimulationExample 2: Simulation

5 historical departure dates

Date 0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-101 4 7 1 5 1 0 0 0 2 12 11 6 9 4 10 1 0 0 1 23 5 4 8 6 7 1 1 2 0 34 7 9 7 5 2 0 2 0 0 25 6 10 2 5 1 0 2 1 0 0

10 DCP 6.6 7.2 5.4 5 4.2 0.4 1 0.6 0.6 1.62 DCP 5.68 5.68 5.68 5.68 5.68 0.84 0.84 0.84 0.84 0.84

Single DCP 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26

Days Out

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Example 2: Simulation Forecast Errors

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

10 DCP 6.6 7.2 5.4 5 4.2 0.4 1 0.6 0.6 1.62 DCP 5.68 5.68 5.68 5.68 5.68 0.84 0.84 0.84 0.84 0.84

Single DCP 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26 3.26Truth 5 5 5 5 5 1 1 1 1 1

10 DCP Error 1.6 2.2 0.4 0 -0.8 -0.6 0 -0.4 -0.4 0.62 DCP Error 0.68 0.68 0.68 0.68 0.68 -0.16 -0.16 -0.16 -0.16 -0.16Single Error -1.74 -1.74 -1.74 -1.74 -1.74 2.26 2.26 2.26 2.26 2.26

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10 DCP Booking Curve10 DCP Booking Curve

Booking Curve

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10 DCP Booking Curve10 DCP Booking Curve

Booking Curve

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2 DCP Booking Curve2 DCP Booking Curve

Booking Curve

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2 DCP Booking Curve2 DCP Booking Curve

Booking Curve

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Finding the Best DCP Structure

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

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DCP Selection AlgorithmDCP 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

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Real Data Booking CurveReal Data Booking Curve

Aggregated Booking Curve

Days Prior to Departure

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Real Fitted Booking CurveReal Fitted Booking Curve

Aggregated Booking Curve

Days Prior to Departure

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Real Booking CurvesReal Booking Curves

Aggregated Booking Curve

Days Prior to Departure

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ConsiderationsConsiderations

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

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RobustnessRobustness

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

DCP structures

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ConclusionConclusion

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

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Technical ReferencesTechnical 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.