Confidential 1 DCPs in Forecasting Edward Kambour, Senior Scientist Roxy Cramer, Scientist.
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Transcript of Confidential 1 DCPs in Forecasting Edward Kambour, Senior Scientist Roxy Cramer, Scientist.
Confidential
1
DCPs in Forecasting
Edward Kambour, Senior Scientist
Roxy Cramer, Scientist
Confidential
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
0
1
2
3
4
5
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0 1 2 3 4 5
Days Prior
Tota
l Boo
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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
05
101520253035
0 2 4 6 8 10
Days Prior
To
tal
Bo
oki
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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
0
5
10
15
20
25
30
35
0 2 4 6 8 10Days Prior
To
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Bo
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10 DCP Booking Curve10 DCP Booking Curve
Booking Curve
0
5
10
15
20
25
30
35
0 2 4 6 8 10Days Prior
To
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Bo
ok
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2 DCP Booking Curve2 DCP Booking Curve
Booking Curve
0
5
10
15
20
25
30
35
0 2 4 6 8 10Days Prior
To
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ok
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2 DCP Booking Curve2 DCP Booking Curve
Booking Curve
0
5
10
15
20
25
30
35
0 2 4 6 8 10Days Prior
To
tal B
oo
kin
gs
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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
Bo
oki
ng
s
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Real Fitted Booking CurveReal Fitted Booking Curve
Aggregated Booking Curve
Days Prior to Departure
Bo
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Real Booking CurvesReal Booking Curves
Aggregated Booking Curve
Days Prior to Departure
Bo
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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.