The Consequent Model Is Mathematically Formulated

Published Date: 02 Nov 2017

It can be assumed that the (concurrent) application of roadside and in-vehicle cooperative traffic management measures has an influence on complexity of driving conditions, possibly with an effect on driving behavior (e.g., route choice, lane changing and car-following behavior) and ultimately on the efficiency of these cooperative traffic management measures.

In order to ex ante assess the efficiency of cooperative traffic management measures, microscopic traffic flow simulation software packages (e.g., PARAMICS, VISSIM, CORSIM, OpenTraffic) are often used. Mathematical models of driving behavior (i.e., route choice, car-following and lane changing models) form the core of these simulation tools. However, we conjecture that the current models are fundamentally inadequate to determine the efficiency of cooperative traffic management measures, as these models do not take the influence of complexity of driving conditions on driving behavior (through for example human factors) into account. For instance, current mathematical models of car-following behavior can be qualified as mechanistic, as they do not account for changes in perceptual thresholds. It has been shown that these thresholds in perception are substantially influenced by the driving conditions.In this context, in Hoogendoorn et al., we took some first steps towards the incorporation of complexity of driving conditions in mathematical models of driving behavior by including task demands and driver capability in the Intelligent Driver Model.

In order to adequately determine the influence of complexity of driving conditions on driving behavior and ultimately on the efficiency of cooperative traffic management measures, an empirically underpinned quantification of complexity of the driving conditions is needed. Therefore in this contribution, we take some first steps towards the development of a quantification of complexity of driving conditions. In this context we propose to develop this quantification through complex systems theory using the notion of fuzzy entropy.

In this contribution we qualify driving conditions as a complex system, i.e., the traffic system. In this context, the next section discusses the characteristics of complex systems. Furthermore in this section we propose a taxonomy of the traffic system and provide a brief introduction into the notion of entropy. In the following section we provide an introduction into fuzzy entropy, followed by a case study using driving simulator data. This contribution finishes with a discussion section and recommendations for future research.

In the previous section we stated that the complexity of driving conditions has an influence on driving behavior and ultimately on the efficiency of cooperative traffic management measures. In order to quantify this influence, a quantified measure of complexity is needed. In order to achieve this, we propose to analyze the complexity of driving conditions through complex systems theory using the notion of (fuzzy) entropy. However, first we have to establish whether driving conditions can be qualified as a complex system. Therefore in this section we start with a brief description of the characteristics of complex systems.

According to Cilliers, in a complex system the interaction between elements is such that the nature of the whole cannot be determined by analysis of some subset (emergence). However, the most important characteristic of complex systems is self-organization. Lemke states in this context that self-organization in complex systems is the result of interactions with the environment and not purely an internal or autonomous process.

Other characteristics of complex systems have also been formulated in literature. In this sense, Cilliers formulates five additional characteristics of complex systems:

a large number of elements whose interaction defies analysis through traditional mathematical means;

dynamic interactions between elements, involving the transfer of information or energy;

redundancy that permits some subset of the system to carry out the function of the whole;

localized autonomy and lack of information sharing between all elements;

non-linear interactions between elements, which makes it possible for small perturbations to have large effects;

The fact that complex systems are characterized by non-linear interactions between elements, means that the causal connections are not proportional. The proportionality is violated where in non-linear systems feedback plays a role in the emergence of a new state of (dis)order. A related phenomenon, derived from chaos theory, is that complex systems are very sensitive to the initial conditions. This implies that a small difference in the initial situation may result substantially different outcomes.

In the ensuing of this contribution, we will refer to the driving conditions as the traffic system. Furthermore, we conjecture that this traffic system can indeed be qualified as a complex system. For instance, one of the main characteristics of traffic is self-organization. In this context, it has been proposed that there is a density range where homogeneous states of traffic flow due either to an instability or another kind of transition cannot exist and leads to a stop-and-go wave.

Also the other characteristics of complex systems can easily be applied to the traffic system, such as the non-linearity of interactions between elements (e.g., small changes in the behavior of one road user can have large implications for the system as a whole) as well as the dynamic character of the interactions involving the transfer of information or energy (e.g., information from roadside and in-vehicle cooperative systems is transferred to the road users).

However, in order to be able to analyze traffic system complexity, a taxonomy of the driving conditions is needed. Therefore in the next subsection, we introduce a crude structure of driving conditions.

In order to be able to quantify traffic system complexity, we need to define the factors possibly contributing to traffic system complexity. In Fig. \ref{structure} the proposed taxonomy of the traffic system is displayed. Following this figure, we propose four main factors, namely:

Driver characteristics;

Road design;

Environmental factors;

Characteristics of the vehicle;

These four factors consist of subfactors, which in turn have their own subfactors. For instance, driver characteristics consist of static and dynamic driver characteristics. Subfactors of static driver characterstics are age and driving experience, while subfactors of dynamic driver characteristics are situation awareness and driver workload.

In this contribution as subfactors of road design we propose alignment, surface condition and other road design factors. Furthermore we propose that the third main factor, i.e., environmental factors consists of the traffic condition (i.e., traffic intensity and interaction with other road users), urban versus rural environments and the presence of traffic management measures (traditional and cooperative). The fourth main factor, characteristics of the vehicle, consists of braking and acceleration capabilities of the vehicle, the present speed, but also in-vehicle systems such as traditional navigation systems and cooperative traffic management measures.

However, it must be stated here that the factors included in this taxonomy of the traffic system must be viewed as an example. Many other subfactors may me included.

In the previous section, we argued that driving conditions can be qualified as a complex system. In this contribution we propose to quantify traffic system complexity through the notion of (fuzzy) entropy. In the present section we provide a brief introduction into entropy, in which we will focus on Shannon's entropy.

In general, entropy can be defined as a measure of the unpredictability of the content of information. In the Shannon entropy was formulated as follows:

H(X)=E[I(X)]=E[- \ln (P(X))]

In this equation H denotes the entropy of a discrete random variable X, with values x_1,...,x_n along with probability mass function P(X). Furthermore, E denotes the expected value operator and I the information content. Assuming a finite sample, we get:

H(X)&=&\sum_i P(x_i)I(x_i)\\

&=&-\sum_i P(x_i)\log_b P(x_i)\\

We will show the workings of entropy displayed in Eq. 2 through a brief example. Suppose we have a set of n possible dynamic maximum speed limits with outcomes x_1,...,x_n, generated by an in-vehicle system (Intelligent Speed Adaptation System). Furthermore suppose, that these outcomes have an equal probability, i.e., p(x_i)=\frac{1}{n}. In this case, the level of uncertainty would become:

u=\log_b

Now, let b=2. This would mean that when a driver wants to specify one of the n values, log_2(n) bits would be required.

However, entropy in the traffic system is not only determined by the presence of in-vehicle traffic management measures, but by numerous other factors as proposed in the previous section. When we have to deal with two or more independent factors related to uncertainty, the characteristic of additivity is of high importance. This characteristic is clearly captured by the logarithm.

In the previous example we assumed a set of of n possible dynamic maximum speed limits with outcomes x_1,...,x_n generated by an in-vehicle system. In a more complicated example let us add a second source of uncertainty, i.e., dynamic route advice communicated through a roadside traffic management system, with $m$ possible outcomes y_1,...,y_m. When combining route choice advice with dynamic maximum speed limits, we would get mn possible outcomes $x_iy_i:i=1,...,n,j=1,...,m From the aforementioned the uncertainty would become:

u=\log_b(nm)=\log_b(n)+\log_b(m)

Again, this would mean that when a driver wants to specify the n value as well as the m value, log_2{(n)+\og_2(m) bits would be required.

Now let us return to the example in which we only have one source of uncertainty: the dynamic maximum speed limits provided through the in-vehicle system. From the fact that we assumed that the probability of each possibly outcome is $\frac{1}{n}$, it follows that:

u_i&=&\log_b \left( \frac{1}{p(x_i)} \right)\\

&=&-\log_b(p(x_i))\\

&=&-\log_b \left (\frac{1}{n} \right)

Furthermore, in case of random continuous variables, it is assumed that $u_i=-\log_b p(x_i)$. According to Shannon this is called a 'surprise'. In other words: the lower the probability, the higher the uncertainty ('surprise'). The average uncertainty can then be calculated through the following equation, in which the average $\bar{u}$ is used as the definition of entropy:

\bar{u}=\sum_{i=1}^n p(x_i)u_i=-\sum_{i=1}^n p(x_i) \log_b(p(x_i))

In this section we briefly described the notion of entropy through Shannon's entropy. However, we assumed that the different factors constituting traffic system complexity are independent. In reality, this is however not the case. We therefore need to use more advanced methods to analyze traffic system complexity. To this end, in the next section we provide insight into a special type of entropy, namely fuzzy entropy.

In the previous section we introduced the notion of entropy through the Shannon entropy . One of the assumptions made in the aforementioned was that the different factors (sources of uncertainty) are independent. In the traffic system this is clearly not the case. In this contribution, we therefore propose to quantufy traffic system complexity using fuzzy entropy. In this subsection we will start with providing a brief introduction into neuro fuzzy logic modeling.

Fuzzy entropy was inspired by Shannon's entropy. However, in order to obtain an estimate of fuzzy entropy, we need to represent the traffic system as a fuzzy inference system. A fuzzy inference system consists of a set of r rules, for instance:

\text{If } x_1 \text{ is } A_1^1 \text{ and } x_2 \text{ is } A_2^1 \text{ ... } \text{ and } \\

x_n \text{ is } a_n^1 \text{ then } y^1=f^1(x_1,x_2,...,x_n)\\

...\\

\text{If } x_1 \text{ is } A_1^r \text{ and } x_2 \text{ is } A_2^r \text{ ... } \text{ and } \\

x_n \text{ is } a_n^r \text{ then } y^r=f^r(x_1,x_2,...,x_n)

Here A^i_j is a fuzzy set on the $j$th premise variable defined by a membership function, i.e., $\mu^i_j=\Re^n \rightarrow [0,1]$. In Eq. \ref{fuzzy1} also a consequent can be observed. This consequent is a function f^ with i=1,...,r of the input vector [x_1,x_2,...,x_n]/

Through the fuzzy sets A^i_j the input is divided into smaller regions where the mapping is approximated by the models $f^i$. A weighted mean is used to recombine all the local representations in a global approximation:

In Eq. \ref{fuzzy2}, $\mu^i$ represents the degree to which the $i$th rule is fulfilled. However, when the consequent is a linear model, the system can be used to return a local linear approximation of a generic point of the input domain. Suppose we have the input $\hat x=[\hat x_1, \hat x_2,...,\hat x_m]$. In this case Eq. \ref{fuzzy2} will return a linear approximation of $f_{lin}(\hat x)$:

However, in the aforementioned traditional approach to fuzzy systems, the membership functions and models are fixed according to prior knowledge (expert opinions). However, when this knowledge is (not yet) available, the components (given a certain data set) can be represented in a parametric form and the parameters tuned through a learning procedure. In this case the fuzzy system turns into a neuro fuzzy approximator .

In this context, we firstly aimed at finding a suitable number of rules and a proper partition of the input space. With regard to the structure of the neuro fuzzy logic model of the traffic system, it may be assumed that there are a lot of different structure / parameter combinations possible. In this context, we therefore aimed at finding the solution which provides the best performance in terms of generalization. To this end we adopted an incremental approach where different architectures with different levels of model complexity are assessed and cross-validated. The initialization of the architecture is provided by a hyper-ellipsoidal fuzzy clustering procedure \cite{Babuska1997}. In \cite{Babuska1997}, it is proposed to cluster the data in the input-output domain through which a set of hyper-ellipsoids is obtained. This set can be regarded as a coarse representation of the input - output mapping. To achieve this, we gradually increased the number of local models. Next we compared the different model structures in relation to their performance $J_{CV}$ using a K-fold cross validation

Methods for initializing the parameters of a neuro fuzzy system were derived from the procedure described in . In the present contribution, we used the eigenvectors of the scatter matrix to initialize the parameters of the consequent functions $f^i$. Furthermore we projected the cluster centers on the input domain to initialize the centers of the antecedents and adopted the scatter matrix in order to compute the width of the membership functions.

In the parametric estimation the best set of parameters was searched for by minimizing the sum-of-squares cost function $J_M$ dependent solely on the training data set. As the model proposed in this paper is a linear model, the minimization procedure was decomposed into a least squares problem to estimate the linear parameters of the consequent models $f^i and a non-linear minimization (Levemberg-Marquant) to find the parameters of the membership functions $A^i_j$ \cite{Bontempi1997} (see also Eq. \ref{fuzzy1}). In this context in this paper we used triangular shaped membership functions of the antecedents. Mathematically, these membership functions can be formulated as follows:

The consequent model is mathematically formulated as follows:

According to De Luca and Termini entropy equals fuzziness and entropy equals information. Let $E:F(s^x) \rightarrow [0,1]$, where $E$ is a fuzzy set. Motivated by the previously discussed Shannon entropy } De Luca and Termini \cite{Deluca1972} propsed a parametrized entropy measure $E_k(A)=D_k(A)+D_k(A^c)$, in which $k>0$ and $D_k(A)=-k \sum_i m_A(x_i) \log m_A(x_i)$.

However, in this contribution we will use the similarity measure as proposed by Tanimoto (see also \cite{Kosko1986}). In this context, the following formulation of fuzzy entropy can be proposed :

Therefore we propose that the level of entropy of the traffic system can be approximated by the ratio of sigma counted underlap versus the sigma counted overlap of the neuro fuzzy sets.

In this section we briefly described the notion of neuro fuzzy modeling and fuzzy entropy. In the next section, through a case study, we will show the workings of the proposed method using data derived from a driving simulator experiment with experimental condition with a variying degree of complexity of the driving conditions.

\caption{Estimated membership functions of the neighborhood variables Age, Experience, following distance $\Delta x$, relative speed $\Delta v$ and speed $v$. The graphs on the left represent the normal driving condition, while the graphs on the right represent the complex driving condition.}

In this section we provide a simple case study of using fuzzy entropy as an quantification of complexity traffic system complexity. In this context we aim to quantify the added complexity to the traffic system of a change in the road design (normal lanes versus narrow lanes with concrete barriers). We start with a brief presentation of the data collection method, followed by the method used to process the data. In the final subsection, we provide a brief overview of the results of this case study.

In this contribution, we used the data described in In this driving simulator experiment all participants participated in a control condition as well as in an experimental condition, rendering up a complete within-subjects design. Driving behavior, represented by speed $v$, acceleration $a$, relative speed $\Delta v$ and spacing $s$ were measured through registered behavior in the driving simulator at a sampling rate of 10 samples per second during both conditions.

For the purpose of the experiment, a driving environment was developed consisting of three segments. The first segment consisted of a short test drive through a suburban area to accustom participants to driving in a driving simulator and also to investigate whether the participants were prone to simulator sickness.

The other two segments were used in the experiment. These test trials took place on a virtual motorway with three lanes in the same direction. The length of the three segments combined was 9.45 km. In the control condition normal driving conditions with a medium density were simulated, while in the experimental condition narrow lanes and roadside concrete barriers were applied aimed at increasing the complexity of the driving conditions. Traffic with a medium intensity was simulated. The behavior of the other vehicles was derived from a pilot study and consisted of larger values of spacing $s$ as well as a reduction in speed $s$.

The research population consisted of 25 employees and students of Delft University of Technology (16 male and 9 female participants). The age of the participants varied from 22 to 54 years with a mean age of 29.68 years ($SD=6.93$). Driving experience varied from 1 to 35 years with a mean of 9.6 years ($SD=7.50$).

In the aforementioned we proposed a structure of the traffic system (see Fig. 1). We assumed that the traffic system consists of four different factors, namely: driver characteristics, road design, environmental factors and characteristics of the vehicle. In this case study we assume that implementing narrow lanes with conrete barriers (road design) adds to the complexity of driving conditions, reflected in an increase in the quantified measure of traffic system complexity.

We therefore created two separate input - output matrices; one for the condition with normal driving lanes and one for the condition with narrow lanes and concrete barriers. As a representation of driver characteristics we chose to add driving experience and age to the matrices, while as a representation of environmental factors we chose to add following distance $\Delta x$ and relative speed $\Delta v$ (interactions with other road users). As a representation of characteristics of the vehicle we chose to include only speed $v$. We assumed that the other factors displayed in Fig. 1 were constant and were therefore excluded from the analyses. As output of the traffic system we chose acceleration $a$. The input and output variables were all scaled from -1 to 1.