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C-1 A p p e n d i x C The purpose of this appendix is to supplement the information provided in Chapter 6 with more background information on some key issues related to the evaluation of uncertainty with respect to DDFSs. In particular, the concept of confidence intervals will be explained in more detail, along with the distribution of uncertainty between annual forecasts and DDFS forecasts. C.1 Confidence Intervals Confidence intervals are described in detail in ACRP Report 76: Addressing Uncertainty about Future Airport Activity Levels in Airport Decision Making http://onlinepubs.trb.org/onlinepubs/ acrp/acrp_rpt_076.pdf. In general, they provide a method for describing and quantifying how an actual future value of an activity measure, such as aircraft operations, is likely to deviate from the predicted value. By measuring historical variations in activity from the long-term average, the most likely distributions of activity around the long-term average can be measured and applied to forecast values. These distributions are often described as confidence intervals. A confidence interval represents the probability that an actual activity measure will fall within a given range. For example, if there is a forecast of 85 peak hour operations and there is a 90 per- cent confidence interval of plus or minus five operations, it means that there is a 90 percent chance that forecast peak hour operations will be between 80 and 90 operations. Another way of describing the confidence interval in this example is that there is at least a 95 percent chance that peak hour operations will be 80 operations or higher (five percent chance that peak hour operations will be lower than 80), and a five percent chance that peak hour operations will be higher than 90 operations (95 percent chance that they will be lower than 90). Exhibit C.1 provides an example of confidence intervals associated with a normal probability distribution curve (sometimes described as a bell curve). The mean represents the projected activity level, and the area beneath the curve represents the expected distribution of outcomes (actual activity levels). As shown, outcomes are most likely to cluster near the mean, but outliers should also be expected but much less frequently. The vertical bars representing the 90 percent confidence level encompass 90 percent of the graph (measured in area). The x-axis in Exhibit C.1 represents the standard deviation. The standard deviation is the square root of the variance of the population around the mean. When only a sample of the population is being evaluated, which is usually the case in forecasting, an additional adjustment is made for degrees of freedom, which serves to slightly increase the effective standard deviation. (The degrees of freedom are equal to the sample size minus one. When calculating the variance of a sample and applying it to the population, the sum of the deviations in the sample is divided by (n -â1) rather than (n), where (n) represents the sample size.) The part of the curve that is encompassed by 1 standard deviation on either side accounts for approximately 68 percent of Evaluation of DDFS Uncertainty
C-2 Guidebook for preparing and Using Airport design day Flight Schedules occurrences, the part encompassed by 2 standard deviations accounts for about 95 percent of occur- rences, and the part encompassed by 3 standard deviations accounts for 99.7 percent of occurrences. When the spread is equal to approximately 1.25 standard deviations, 90 percent of the area under the curve is encompassed. The distribution shown in Exhibit C.1 is a normal distribution which means that deviations around the mean are symmetric. This is not always the case. For example, if one were to estimate the distribution of average seats per aircraft in the world commercial passenger fleet, the aver- age would be about 150 seats per aircraft. The low extreme would be about 19 seats per aircraft for small turboprops but the high range would be an Airbus A380 that would have 500 seats or more. Therefore, the outliers on the upper end of the distribution are more extreme than those on the lower end, and the probability distribution becomes skewed. Normal distributions are typically assumed as a matter of analytical convenience, but they sometimes break down with real world data, especially at high or low extremes. C.2 Share of Uncertainty between Annual and DDFS Forecasts A critical question when evaluating uncertainty is how much of the uncertainty is directly attributable to the DDFS and how much is attributable to the annual forecasts upon which it is based. One way of examining this issue is a Monte Carlo analysis. Monte Carlo analysis is briefly described in Chapter 8. ACRP Report 76: Addressing Uncertainty about Future Airport Activity Levels in Airport Decision Making http://onlinepubs.trb.org/onlinepubs/acrp/acrp_rpt_076.pdf also describes Monte Carlo analysis in more detail and how to use Monte Carlo simulation to generate probability distributions for annual forecast outputs. In a Monte Carlo analysis, probability distributions are identified for forecast input factors and forecast parameters. Using these probability distributions, the inputs and parameters are randomized and integrated within the forecast equations to generate multiple forecast out- comes. The forecast outcomes are then aggregated to provide a general forecast probability -4 .0 0 -3 .7 5 -3 .5 0 -3 .2 5 -3 .0 0 -2 .7 5 -2 .5 0 -2 .2 5 -2 .0 0 -1 .7 5 -1 .5 0 -1 .2 5 -1 .0 0 -0 .7 5 -0 .5 0 -0 .2 5 0. 00 0. 25 0. 50 0. 75 1. 00 1. 25 1. 50 1. 75 2. 00 2. 25 2. 50 2. 75 3. 00 3. 25 3. 50 3. 75 4. 00 Standard Deviations 90 % Confidence Interval Mean Exhibit C.1. Normal probability distribution curves and confidence intervals.
evaluation of ddFS Uncertainty C-3 distribution that incorporates all the probability distributions associated with the inputs and parameters. Separate Monte Carlo tests were performed on an example large-hub airport to estimate confi- dence intervals based on DDFS forecast factors by themselves, based on annual forecast factors by themselves, and based on annual and DDFS factors in combination. Peak hour originations were evaluated. Exhibit C.2 shows the confidence intervals based solely on variations in DDFS factors, namely the peak hour percentage. For the purpose of this test, no variation was assumed in the annual forecast factors. As shown, the distributions around the mean are very symmetric and relatively small. Exhibit C.3 is similar to Exhibit C.2 except that is holds the peak hour percentage constant and allows the uncertainty in the annual forecast factors to generate the confidence intervals. In this instance, the annual forecast factors were projected regional income and average air fares, along with a random Black Swan variable representing infrequent disruptive events. In comparison with Exhibit C.2, the confidence intervals are much broader, indicating that there is much more uncertainty associated with the annual forecasts than with the DDFS fore- cast. Also, both the mean and the median of the Monte Carlo distribution are lower than the base forecast. This is because disruptive events represented by the Black Swan variable almost always have a negative impact on aviation activity. Typically, base planning forecasts do not include a future Black Swan variable. Exhibit C.4 is similar to Exhibits C.2 and C.3 except that it combines annual and DDFS uncertainty in a single Monte Carlo test. Although the confidence intervals are slightly wider in Exhibit C.2. Monte Carlo example: peak hour uncertainty independent of annual forecasting uncertainty.
Exhibit C.3. Monte Carlo example: peak hour uncertainty based on annual forecasting uncertainty. Exhibit C.4. Monte Carlo example: peak hour uncertainty based on annual and ddfs forecasting uncertainty.
evaluation of ddFS Uncertainty C-5 Exhibit C.4 than in Exhibit C.3, the differences are not readily detectable when looking at the graphs. It should be noted that the confidence intervals associated with DDFS uncertainty are not additive to the confidence intervals associated with annual forecasting uncertainty. Within a Monte Carlo framework, there are many instances when a negative deviation in an annual forecast factor will be offset by a positive deviation in a DDFS forecast factor and vice versa. In general, adding uncertainty to a system will generate broader confidence intervals, but the increase will not be linear. Ideally, the analysis would be performed for a large sample of airports. However, this test does suggest that it is more important to accurately assess the uncertainty associated with the annual forecasts than with the DDFS forecasts. Exceptions would be instances in which planning was based on annual activity levels. Since the annual activity level would essentially be defined as constant, an analysis of DDFS uncertainty would be appropriate to capture variations that occur even when annual activity levels do not change.
