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PDF (DX094490.pdf) - White Rose Etheses Online

PDF (DX094490.pdf) - White Rose Etheses Online

88 Armitage and

88 Armitage and McDonald's two-line least squares method involves the abstraction of all or most of, the rejected gaps since the value of L, the intercept, is not known in advance, while the majority of these rejected gaps, the ones less than L, will not in fact be considered in the analysis. 4.5.4 Extreme Values The distribution of the circulating flow used in the simulation was shifted negative exponential (see section 4.3. for a more detailed description). This distribution allows the occasional large gap to be present even though in reality such gaps are often more common than suggested by the theoretical distribution. When the simulated data were divided in groups, the frequency of large gaps per group was very small; often no such gap was present. Furthermore, it was difficult to define a consistent way of determining the lower limit of these extreme values. For example, the simulated gap distribution based on a circulating flow of 0.44 veh/s had only a few gaps allowing 4 vehicles to enter, less allowing 5 vehicles, and none allowing more than 5 vehicles. The 20 groups into which these gaps were divided were analyzed with and without the gaps allowing 4 or 5 vehicles to enter. The results are given in Table 4.3 which also includes the results of analyzing the same groups but reversing the definition of dependent and independent variables (see section 4.6). As can be seen from the table, some groups did not have gaps of length that were large enough to be excluded, and therefore, no gap-acceptance values are shown under the heading "highest values excluded". The

89 exclusion of top values decreased the values of c, and increased the value of at all groups. The effect on the mean value over all 20 groups is shown on Table 4.4. In the case of assuming the number of entries as the dependent variable, exclusion decreased the accuracy of the prediction of the mean but also reduced the standard deviation. In the case of the gap size as dependent variable, the prediction was improved, giving the best results of the four sets. However, the criterion for excluding extreme values was not satisfactory, as it could not be explicitly defined. The effect of exclusion of large gaps for data collected in the field would be very uncertain as the total nunther of gaps would be very small compared to the simulated data. It was decided therefore to develop other procedures for ensuring reasonable predictions without resorting to exclusion of the extreme values. 4.5.5 The Use of Weights In general, weights are introduced into a least squares analysis to counterbalance distributions of data which overrepresent certain parts of the range, since the latter may introduce inaccuracies in the parameters of the analysis. The distribution of the gaps is of a type that more smaller gaps are present than larger. This would occur with either a negative exponential or a shifted negative exponential distribution assumed for the circulating flow. This over- representation of the smaller gaps would be observed not only over the whole range but also each value of the step function describing variable N would exhibit a similar distribution, for example there should be more smaller gaps accepted by two

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