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PDF (DX094490.pdf) - White Rose Etheses Online

PDF (DX094490.pdf) - White Rose Etheses Online

as following: 84 "Let y,

as following: 84 "Let y, y 2 , ..., y be uncorrelated, observable random variables such that y 1 = a +x. + e 1 where a and are unknown parameters, x 1 are observable mathematical (non- random) variables, and e 1 are uncorrelated, unobservable 2 2. random variables with mean 0 and variance a , where a is not a function of a, , or x111. Two points of interest arise concerning the use of such a linear model with the type of data involved in the current study. The first is the definition of dependent and independent variables, the second is the distribution of the variables. From a purely explanatory point of view it would seem obvious to define as dependent variable the number of vehicles entering while the size of the gap is defined as the independent variable. However, from the point of view of errors due to observational mistakes, it is very unlikely that any should be present in the counting of the number of entries associated with each gap. On the other hand such errors are much more likely in the estimation of the size of the gaps. Furthermore, the number of entering vehicles is a step function while the distribution of the gap lengths accepted by a given number of vehicles is markedly skew i.e. there are more smaller such gaps than longer ones. Thus the normality assumption of linear regression models is violated. This does not allow the full benefits of the linear regression method to be exploited. However, it does not invalidate the use of a linear model. It points to the possibility of introducing modifications to

85 produce reasonable predictions, accepting the fact that the least squares regression assumptions will not be met. Thus, the term "least squares regression" will not be used, instead the model will be referred to as "simple linear". This is in line with the arguments of Cooper and Wennell (1978). It is termed "simple" to distinguish it from the two-line model proposed by Armitage and McDonald. The justification of such a model will be dependent on its ability to successfully analyze data and provide values for the gap-acceptance parameters which are as near to their true values as it is possible to determine. In order to arrive at the best model, data with known gap-acceptance parameters have been produced using computer simulation. These data are analyzed using the available linear models. This way the model producing the best results can be chosen. 4.5.2 Simulated Data The data for the checking were produced using a computer program simulating a continually saturated single- lane entry to a roundabout. The program is given in Appendix 2. It assumes a shifted negative exponential distribution for the circulating flow. It allows changes in the values of the critical gap, e, the move-up time, , the circulating flow and the minimum headway, T. Values of the gap-acceptance parameters a and were constant throughout the simulation for the initial runs, although later work allowed variation in these parameters (see section 4.5.8). The period of the simulation can be extended indefinitely; however, the pseudo- random generating subroutine has a cycle of 16384; therefore

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