A Choice of Representative Sections for the Estimation of the Transport Network Loading

. The problem of a choice of representative sections for transport network loading management is considered in the following article. To solve the problem, the use of Principal component analysis and the procedures of cluster analysis are offered. The example of the choice of representative sections at network traffic control is presented.


Introduction
Taking into account the quantitative growth of individual cars we can see that one of the characteristics of Russian cities are transport jams. They affect not only on availability of transport infrastructure, but also reduce efficiency of work of emergency services. In its turn it causes a potential threat of serious emergency situations and untimely help. Because of the bad organization of city transport system and appearance of jams a real economic damage exists. [7][8][9][10] In the Russian Federation temporarily-dependent traffic light control with beforehand calculated signal schedules are traditionally used. [4][5][6][7]11]. In such situations when network loading comes nearer to the saturation state, any fluctuations in a traffic flow lead to spontaneous appearance of a transport jam. The key moment of solution of the given problem is the usage of modern methods of adaptive control of a city motion.
In most of known researches the attention is paid to computational methods of signal schedules, thus the attention is not practically paid to the supply with information of operation of control systems [9,12,13].

Problem statement
It is impossible to arrange with transport detectors all allowed directions of motion in the controlled area. Usually it is necessary to select representative subsets of points, information of which can give an objective estimate of a transport situation in all area of control [13][14][15].
There are, at least, three basic types of basic preconditions causing possibility of transition from the big number р of initial indicators of analyzed transport system condition to the essentially smallest number р' of the most informative sections. It is, first of all: • Duplication of information caused by strong correlation of intensity on adjacent sites; • Small variability of received data for sections, intensity on which is poorly changed; • Aggregation possibility (the simple or weighed summation) on some sections. Technically, the task of passage with the least losses in self-descriptiveness to a new gang of sections of a network, in which measuring will be processed z (1) , z (2) … z (pʹ) can be presented as follows [2,3].
Let Z = Z (X) be some r-dimensional vector function of initial variables, x (1) , x (2) … x (p) (р'<<p) and let Iр'(Z(X)) be a definitely given measure of self-descriptiveness р 'dimensional system of indications Z(X)=(z (1) . Concrete select of a functional Iр'(Z(X)) depends on specificity of the following actual task and is based on one of the possible criteria: • The measure of auto self-descriptiveness aimed at maximum maintenance of information, containing in the initial array { } =1, ̅̅̅̅̅ concerning the initial indications; • The measure of external self-descriptiveness aimed at maximum "squeezing" the information from { } =1, ̅̅̅̅̅ , containing in this array concerning some external indexes.
The task is in definition of such feature set ̃ discovered in class F of admissible conversions of initial indexes x (2) Both of variants of a concrete definition of this setting (defining the concrete select of standard of self-descriptiveness Iр'(Z(X)) and a class of admissible conversions) leads to the concrete method of lowering of dimensionality.

Task solution
The method of principal component analysis (PCA) belongs to the methods, which allow to reduce dimensionality. The method provides usage of various linear orthogonal normalized combinations of initial indexes as a class of admissible conversions F: where As a self-descriptiveness standard р '-dimensional system of indexes (z (1) (X)…,z (р') (X)) the expression is used: where D is a character of dispersible evaluation. Following the general optimization setting of the task of size lowering and supposing a parsed as X r-dimensional aleatory variable measure with a vector of average values where L is a matrix and its lines satisfy to the orthogonality requirement: Gained in this way variables         , is a such normalized centered linear combination of these indexes, which among all other possesses the greatest variance, and the k principal component is a such normalized centered linear combination of indexes, which is not correlated with k-1 previous principal components and, among all other, possesses the greatest variance.

Method application
Now we consider finding of an amount and a dislocation of controlled areas which can allow us to have the objective information on network loading. Thus, intensity of motion for cuts, where transport detectors were not set, can be defined as a function of received measuring.
Outcomes of evaluations are the percentage shares of the general variance, explained by principal components for the given observations of a traffic flow, represented in the figure 1. As we can see, for an explanation of 99.4196 % of the general variance in the considered network fragment there is enough usage of 4 representative cuts.
The concrete definition of representative cuts for installing transport detectors is fulfilled by clusterization of given measures and pointing out 4 clusters based on the amount of principal components (figure 2). As the near metric of objects the Euclidean distance was used. The building-ups of the hierarchical tree of clusters were made on algorithm of a centroid, which uses the "barycentres" distance of groups. As we see, clusters №1 and 3 contain one direction of motion, clusters № 2 and 4 contain 6 and 3 units accordingly. Let's consider procedure of select of representative cut for the cluster №2 in detail. Intensity of motion for cluster directions has similar dynamics (figure 3). As the cluster centre we select the direction having the minimum aggregate distance to other units of the cluster (table 2). As we see from results of calculations for the cluster №2, it is expedient to select a direction №7 as the representative cut. The correlation analysis has shown high values of correlation coefficient of the traffic flow dynamics in a cluster with representative cut (table 3) that testifies the effectiveness of the offered method.