A prediction model for traffic emission based on interval-valued intuitionistic fuzzy sets and case-based reasoning theory

Abstract

Prediction of traffic emission is very important to management actions for traffic emission reduction. To overcome the shortcomings of ordinary prediction methods in previous studies, a novel model for the prediction of traffic emission is proposed for the combination of interval-valued intuitionistic fuzzy sets and case-based reasoning theory. Based on analysis of the factors that affect the traffic emission, a characteristic factor matrix of the source cases was constructed. Then, an interval-valued intuitionistic fuzzy set was introduced in order to describe the uncertainty of the case, and the source case that was the most similar to the target case, was picked out by calculating the similarities between the source cases and the target case, therefore leading to the improvement in prediction accuracy. Finally, a case study was conducted to evaluate the effectiveness of the constructed prediction model.

Keywords

Traffic emission prediction case-based reasoning interval-valued intuitionistic fuzzy set

1 Introduction

Because of the sharp increase in car ownership over the past years, automotive exhaust emission has become a primary source of urban environmental pollution, causing great harm to public health. There are multiple factors that affect road traffic emission, including road alignment, traffic flow and weather conditions. Moreover, part of them in dynamic uncertain environments lead to that traffic emission varies under different external environmental conditions [12]. However, accurate prediction of road traffic emission is crucial in the design of new strategies to control traffic emission.

Currently, international scholarship has mainly focused on the formation mechanisms and control techniques of traffic emission, while neglecting systematic investigation of a prediction model [13, 20]. Few studies mainly predict traffic emission based on the simulation model. Elkafoury et al. presented a VISSIM based microscopic traffic emission prediction model [1]. Smit and Mcbroom used microscopic simulation models to predict traffic emissions [14]. Besides, Smit and Mcbroom used overseas emissions models to predict traffic emissions in urban areas [15]. However, factors affecting transport emissions are so many that the simulation model can’t completely simulate the actual traffic and road conditions, resulting in some differences appear between predicted and actual value.

Alternatively, transport emissions would be treated as a black box, and mathematical methods could be used to directly predict the value, by analyzing the statistical laws of historical and real-time data. The following mature research methods can be adopted as a reference for the prediction of traffic emission:

Time series methods such as exponential smoothing and neural networks, which are mainly applicable to the prediction of evolution rules under relatively stable environments. Nagendra and Khare used artificial neural network to predict vehicular exhaust emission [16]. Boyd presented an exponential smoothing model for predicting traffic emission [6].

Causal forecasting methods [3 , 17] such as support vector machines, structural equations and regression, which have been mainly used for the prediction of emergencies under wartime conditions. Dunea et al. presented an integrated model of support vector machines and forward neural networks to improve air pollution forecasting. Momani et al. studied an assessment of automobile emissions based on structural equations [17]. Dong et al. analyzed of light-duty vehicle emission using regression model [5].

In the second type of prediction methods, some scholars have proposed the use of fuzzy set theory [2] and case-based reasoning to deal with the suddenness, incompleteness and uncertainty of random interference factors [4 , 18]. Specifically, the inaccuracy of the cases is depicted using fuzzy mathematics while the most similar source case can be determined based on the similarity between the cases, thus greatly enhancing accuracy and reliability in prediction. At present, this method is seldom used in forecasting traffic emission, but is widely used in other areas. Liu and Wang et al. put forward a prediction model of emergency supply requirements based on intuitionistic fuzzy case-based reasoning [4, 11]. Ji et al. investigated a management method for traffic congestion based on case-based reasoning [18]. Similarly, based on the case-based reasoning method, Wang et al. constructed a prediction model of the resource requirements for unconventional emergencies [19].

As above mentioned, type-2 fuzzy sets [10] and the intuitionstic fuzzy sets [7, 8] have been successfully incorporated into case-based reasoning models, while fewer prediction models have been constructed based on interval-valued intuitionistic fuzzy sets. As an extension of fuzzy sets, intuitionistic fuzzy sets can consider both membership degree and non-membership degree simultaneously, which makes this method more convenient and simple in dealing with problems of ambiguity and uncertainty. The interval-valued intuitionistic fuzzy set is a further extension of the intuitionistic fuzzy set. Using the interval-valued intuitionistic fuzzy set method, both membership degrees and non-membership degree of the elements in a nonempty set can be taken into account. Therefore, the interval-valued intuitionistic fuzzy set exhibits stronger expression ability than type-2 fuzzy sets and intuitionistic fuzzy sets in dealing with uncertain information.

Based on the results of related research, this article combined the interval-valued intuitionistic fuzzy set and case-based reasoning in order to propose a novel road traffic pollution emission prediction model. By monitoring the parameter changes on the intelligent transformation platform, the causal relationship between the external environment and the target can be rapidly analyzed in accordance with the related information of the most similar source case. By this method, the traffic pollution emission of the target case can be predicted.

2 The concept and definition of interval-valued intuitionistic fuzzy sets

As stated above, there are multiple factors that affect road traffic pollution emission, and the decision-making environment’s effects on characteristic factors of the emissions should be taken into account. When large amounts of historical statistical data are unavailable, the uncertainty of the effecting factors can be described by an interval-valued intuitionistic fuzzy set in combination with the experts’ empirical knowledge. This method is simple, scientific and easy to operate. The concept and operation of the interval-valued intuitionistic fuzzy set and the formula for measuring the similarity between two interval-valued intuitionistic fuzzy sets will be introduced below [7, 9]

Definition 1. Assuming a discourse domain U, for any x ∈ U, if A is a intuitionistic fuzzy set in U, i.e., A ={ 〈 x, M_A (x) , N_A (x) 〉 |x ∈ U }, the following conditions should be satisfied:

$M_{A} (x) = [t_{A}^{-} (x), t_{A}^{+} (x)], N_{A} (x) = [f_{A}^{-} (x), f_{A}^{+} (x)], P_{A} (x) = [t_{A}^{-} (x), t_{A}^{+} (x)], N_{A} (x) = [f_{A}^{-} (x), f_{A}^{+} (x)], P_{A} (x) = [π_{A}^{+} (x) = 1 - t_{A}^{-} (x) - f_{A}^{-} (x), π_{A}^{-} (x) = 1 - t_{A}^{+} (x) - f_{A}^{+} (x)],$ in which M_A (x), N_A (x) and P_A (x) denote the true membership, the false membership and the uncertainty range;

M_A (x) ⊆ [0, 1] , N_A (x) ⊆ [0, 1] , P_A (x) ⊆ [0, 1];

$Sup M_{A} (x) + Sup N_{A} (x) = t_{A}^{+} (x) + f_{A}^{+} (x) \leq 1 .$

Definition 2. Assuming a discourse domain U, for any x ∈ U, if A is a single-parameter interval-valued intuitionistic fuzzy set expanded based on an intuitionistic fuzzy set, the following expression should be satisfied:

P_A (x) = [0, π_A (x)] , M_A (x) = [t_A (x) , t_A (x) + λ_A (x) π_A (x)] , N_A (x) = [f_A (x) , f_A (x) + (1 - M_A (x) λ_A (x)) π_A (x)] (0 ≤ λ_A (x) ≤1), in which, N_A (x) and P_A (x) denote the true membership, the false membership and the uncertainty range.

Definition 3. Assuming three intuitionistic fuzzy sets in the discourse domain U, A, B and C, if A ⊆ B ⊆ C, N (A, C) ≤ N (A, B) ∩ N (B, C) can be satisfied, where N(A,B) is the similarity between intuitionistic fuzzy sets A and B, and N(A,B) can be written as:

$\begin{matrix} N (A, B) & = & \frac{\sum_{i = 1}^{m} [t_{A} (x_{i}) \land t_{B} (x_{i})]}{\sum_{i = 1}^{m} [t_{A} (x_{i}) \lor t_{B} (x_{i})]} \\ \land \frac{\sum_{i = 1}^{m} [f_{A} (x_{i}) \land f_{B} (x_{i})]}{\sum_{i = 1}^{m} [f_{A} (x_{i}) \lor f_{B} (x_{i})]} \end{matrix}$ (1)

Assuming that the weight of x_i in the discourse domain U is represented as w_i, w_i ≥ 0 and $\sum_{i = 1}^{m} w_{i} = 1$ , Equation (1) can be rewritten as:

$\begin{matrix} N (A, B) & = & \frac{\sum_{i = 1}^{m} w_{i} [t_{A} (x_{i}) \land t_{B} (x_{i})]}{\sum_{i = 1}^{m} w_{i} [t_{A} (x_{i}) \lor t_{B} (x_{i})]} \\ \land \frac{\sum_{i = 1}^{m} w_{i} [f_{A} (x_{i}) \land f_{B} (x_{i})]}{\sum_{i = 1}^{m} w_{i} [f_{A} (x_{i}) \lor f_{B} (x_{i})]} \end{matrix}$ (2)

It can therefore be concluded that if A and B are two interval-valued intuitionistic fuzzy sets in the discourse domain U, they can be transformed into intuitionistic fuzzy sets, and the similarity between A and B can be calculated according to Equations (1) and (2).

3 Prediction model of road traffic pollution emission based on interval-valued intuitionistic fuzzy sets and case-based reasoning theory

3.1 Effecting factors of road traffic pollution emission

Several factors can affect traffic pollution emission. These factors are mainly dependent on the traffic conditions at the monitoring point. In the present study, seven effecting factors were selected in total, including road alignment, traffic flow and weather and etc.. As shown in Table 1, some factors are static while some factors are dynamic. Data for these dynamic factors should be acquired through the intelligent transport system (ITS). These factors can interfere with each other, thereby increasing the variability of traffic pollution emission.

3.2 Construction of the prediction model for road traffic pollution emission

In the case library of road traffic pollution, n source cases, denoted as C ={ c₁ , c₂, ·· · , c_n }, were included, and m characteristic factors, H ={ h₁ , h₂, ·· · , h_n }, were involved. The weights of these m characteristic factors are denoted as w = (w₁, w₂, ·· · , w_m), $\sum_{j = 1}^{m} w_{j} = 1$ .

Using the experiences and knowledge of decision makers for reference, an interval-valued intuitionistic fuzzy matrix of the characteristic factors of all source cases in the case library was constructed and then transformed into an intuitionistic fuzzy matrix. Then, the similarities between the target case and all of the historical cases were calculated. By this method, the road pollution emission of the target case was predicted. This method also offers greatly improved prediction accuracy and reliability. The specific procedure is described below.

(1) Construction of the interval-valued intuitionistic fuzzy matrix of all factors that affect the traffic pollution emission

Based on the historical traffic pollution emission data collected by ITS as well as the decision makers’ fuzzy experiences and judgments, the true membership and false membership of the i-th source case c_i to the j-th characteristic factor are expressed as $t_{c_{i}} (h_{j}) = [t_{c_{i}}^{-} (h_{j}), t_{c_{i}}^{+} (h_{j})]$ and $f_{c_{i}} (h_{j}) = [f_{c_{i}}^{-} (h_{j}), f_{c_{i}}^{+} (h_{j})]$ , respectively. Thus, the set of interval-valued intuitionistic fuzzy characteristic vectors corresponding to any a source case c_i can be written as: $\begin{matrix} V_{C_{i}} & = & {< t_{c_{i}} (h_{1}), f_{c_{i}} (h_{2}) >, < t_{c_{i}} (h_{2}), f_{c_{i}} (h_{2}) > \\ \cdot \cdot \cdot < t_{c_{i}} (h_{m}), f_{c_{i}} (h_{m}) >} . \end{matrix}$

Therefore, the interval-valued intuitionistic fuzzy matrix of the characteristic factors of all source cases in the case library can be described as: $A = (\begin{matrix} < t_{c_{1}} (h_{1}), f_{c_{1}} (h_{1}) > \cdot \cdot \cdot & < t_{c_{1}} (h_{m}), f_{c_{1}} (h_{1}) > \\ \cdot & \cdot \\ \cdot & \cdot \\ \cdot & \cdot \\ < t_{c_{n}} (h_{1}), f_{c_{1}} (h_{1}) > \cdot \cdot \cdot & < t_{c_{n}} (h_{m}), f_{c_{n}} (h_{m}) > \end{matrix})$ (3)

Similarly, the vector set of the characteristic factors of the target case T can be written as: V_T = {< t_T (h₁) , f_T (h₂) > , < t_T (h₂) , f_T (h₂) > ·· · < t_T (h_m) , f_T (h_m) >}. V_T is also an interval-valued intuitionistic fuzzy characteristic vector set.

(2) Transformation into an intuitionistic fuzzy matrix of the factors that affect traffic pollution emission

According to Definition 2, by introducing the risk attitude of the decision maker λ (λ ∈ [0, 1]), the interval-valued intuitionistic fuzzy matrices of the traffic pollution emission sources and the target case can be transformed into intuitionistic fuzzy matrices: $t_{c_{i}} (h_{j}) = t_{c_{i}}^{-} (h_{j}) + λ [t_{c_{i}}^{+} (h_{j}) - t_{c_{i}}^{-} (h_{j})]$ and $f_{c_{i}} (h_{j}) = f_{c_{i}}^{-} (h_{j}) + λ [f_{c_{i}}^{+} (h_{j}) - f_{c_{i}}^{-} (h_{j})]$ . Whenλ → 0, $t_{c_{i}} (h_{j}) \to t_{c_{i}}^{-} (h_{j})$ ; when λ → 1, $t_{c_{i}} (h_{j}) \to t_{c_{i}}^{+} (h_{j})$ .

Additionally, because the characteristic factors of traffic pollution emission differ in their measuring units, they should be standardized in order to solve their incommensurability.

(3) Determination of the weight coefficients of the characteristic factors of traffic pollution emission cases

For the i-th source case c_i, assuming that the value of the j-th characteristic factor h_j is a_ij and the corresponding traffic pollution emission is denoted as k_i, the correlation coefficient between the j-th characteristic factor and the traffic pollution emission k_i of the i-th source case c_i can be defined as: $r_{j} = \frac{\sum_{i = 1}^{n} (a_{i j} - {\bar{a}}_{j}) (k_{i} - \bar{k})}{\sqrt[]{\sum_{i = 1}^{n} {(a_{i j} - \bar{a_{j}})}^{2}} \sqrt[]{\sum_{i = 1}^{n} {(k_{i} - \bar{k})}^{2}}}$ (4) where $\bar{a_{j}} = \frac{1}{n} \sum_{i = 1}^{n} a_{ij}$ and $\bar{k} = \frac{1}{n} \sum_{i = 1}^{n} k_{i}$ .

Clearly, the greater the value of r_j, the more significant the effects of h_j on k_i. The weight coefficient of the characteristic factor h_j can be calculated by: $w_{j} = \frac{| r_{j} |}{\sum_{j = 1}^{m} | r_{j} |}$ (5)

(4) Calculation of Similarity between the source case and the target case

Given the weight vectors of the characteristic factors w = (w₁, w₂, ·· · w_m), the similarity N (c_i, T) between the i-th source case c_i and the target case T can be calculated by:

$\begin{matrix} N (c_{i}, T) \\ = \frac{\sum_{j = 1}^{m} w_{j} [(t_{C_{i}} (h_{j}) + λ_{c_{i}} (h_{j}) π_{c_{i}} (h_{j})) \land t_{T} (h_{j})]}{\sum_{j = 1}^{m} w_{j} [(t_{c_{i}} (h_{j}) + λ_{c_{i}} (h_{j}) π_{c_{i}} (h_{j})) \lor t_{T} (h_{j})]} \\ \land \frac{\sum_{j = 1}^{m} w_{j} [(f_{c_{i}} (h_{j}) + (1 - λ_{c_{i}} (h_{j})) π_{c_{i}} (h_{j})) \land f_{T} (h_{j})]}{\sum_{j = 1}^{m} w_{j} [(f_{c_{i}} (h_{j}) + (1 - λ_{c_{i}} (h_{j})) π_{c_{i}} (h_{j})) \lor f_{T} (h_{j})]} \end{matrix}$ (6)

(5) Determination of the most similar source case of the target case

Then, the threshold value α can be set, where if N (c_i, T) ≥ α, the source case c_i is similar to the target case T. The source case c_p with the best similarity $(p = max_{i} N (c_{i}, T))$ is the most similar source case to the target case.

(6) Prediction of traffic pollution emission of the target case

As stated above, according to the traffic pollution emission k_i of the most similar source case c_p, the traffic pollution emission of the target case T can be predicted after appropriate adjustments.

4 Case study

In total, 15 source cases were included in the case library of a city’s road traffic pollution emission. Table 2 lists this collected data at different monitoring points dynamically tracked by ITS. Judging from the experience of the decision makers, the interval-valued intuitionistic fuzzy matrix of the source cases was constructed and shown in Table 3. Then, the traffic pollution emission of the target case can be predicted based on interval-valued intuitionistic fuzzy and case-based reasoning.

When λ = 0.6, the interval-valued intuitionistic fuzzy matrix of the effecting factors was transformed into the intuitionistic fuzzy matrix of the traffic pollution emission, and the correlation coefficients between seven effecting factors and the traffic pollution emission were calculated. The results are shown in Fig. 1, where the coefficients between the effecting factors (t₁, t₂, t₃, t₄, t₅, t₆ and t₇) and the traffic pollution emission are 0.03, –0.792, –0.328, 0.82, 0.974, 0.655 and –0.33, respectively. Therefore, t₂, t₃ and t₇ are negatively correlated with the traffic pollution emission while t₁, t₄, t₅ and t₆ exhibit positive correlations with the traffic pollution emission. As shown in Fig. 2, the weight coefficients of the effecting factors are 0.008, 0.202, 0.083, 0.209, 0.248, 0.167 and 0.084, respectively. This indicates that t₅ has a largest weight while t₁ has a smallest weight.

Then, the similarities between the traffic pollution emission source cases and the target case T were calculated. The results were 0.945, 0.9696, 0.715, 0.71, 0.6721, 0.61, 0.754, 0.723, 0.72, 0.627, 0.527, 0.734, 0.849, 0.586 and 0.68, respectively, as shown in Fig. 3. If α was set to 0.7, c₁, c₂, c₃, c₄, c₇, c₈, c₉, c₁₂ and c₁₃ can be regarded as the most similar cases. In similarity, c₂ > c₁ > c₁₃ > c₇ > c₁₂ > c₈ > c₉ > c₃ > c₄.

It can be concluded that c₂ is the most similar source case. Because the traffic pollution emission depends on the traffic volume, the road traffic pollution emission of the target case can be calculated with reference to the related data of source case c₂,i.e., $k = 985 \times \frac{588.6}{980} = 591.6 g / km .$

In addition, the effect of the value of λ on the prediction result of the target case was investigated. The results are shown in Fig. 4. As λ increased, the calculated similarities between the source cases and the target case exhibited some differences. However, the overall variation tendency exhibited no great changes, and only the similarities between some source cases and the target case show some slight adjustments. Therefore, it can be affirmed that the selection of the source case c₁₅ as the most similar case imposes almost no effect on the prediction of the traffic pollution emission of the target case.

5 Conclusions

How to accurately predict traffic emission is very important to management actions for traffic emission reduction. To overcome the shortcomings of ordinary prediction methods in previous studies, this paper presented a prediction model for traffic emission based on interval-valued intuitionistic fuzzy sets and case-based reasoning theory. The work described in this thesis is as follows:

In the present study, interval-valued intuitionistic fuzzy numbers were used to describe the uncertain characteristic factors of traffic emission. Where sample data is missing or incomplete, the method is still simple and easy to operate. Moreover, the prediction results are more accurate and close to the practical conditions.

An investigation into the variation in the similarity between the source cases and the target case through the parameter variations using the case-based reasoning method can provide theoretical guidance for the analysis of the effects of the variation of external factors on traffic emission results.

Compared with intuitionistic fuzzy sets, interval one could use different values of the parameters to describe uncertain factors of road traffic emissions, which reflect actual cases that were more similar cases. Hence, propose model could contribute to improve forecast accuracy pollution.

If the change of a characteristic factor of traffic emission imposes only slight effects on the calculated similarities between the source cases and the target case, this factor is weakly correlated with the emission. According to the results of the present study, traffic volume exhibits the strongest correlation with traffic emission, followed by traffic flow, road capacity, road type, speed, the composition of traffic and weather conditions, while the road alignment is most weakly correlated with traffic pollution emission.

Our study is of great importance to management actions for traffic emission reduction. However, it has the following disadvantages and future research work should take into consideration:

Ignore difference in each decision-maker‘s intuitionistic fuzzy ambiguities on traffic emission factors, resulting in predictions have one-sidedness.

Our mode can not handle more prediction target of traffic emission, and thus can not be widely applied.

Footnotes

Acknowledgments

This paper is funded by open fund for State Key Joint Laboratory of Environment Simulation and Pollution Control (14K07ESPCT); Jiangsu Construction Technology Program (2013ZD38).

References

Elkafoury

, Negm

A.M.

, Bady

M.F.

, et al., VISSIM Based Microscopic Traffic Emission Prediction Model, International Conference on Urban Planning and Transportation (2015), 34–51.

Sun

B.Z.

, Ma

W.M.

and Zhao

H.Y.

, A fuzzy rough set approach to emergency material demand prediction over two universes, Applied Mathematical Modeling37 (2013), 7062–7070.

Dunea

, Pohoata

and Iordache

, Using wavelet–feed forward neural networks to improve air pollution forecasting in urban environments, Environmental Monitoring & Assessment187(7) (2015), 1–16.

Liu

D.Y.

, Zhu

C.F.

and Ding

, Research on emergency supply demand forecast methods based on case reasoning, Journal of Lanzhou Jiaotong University32(4) (2013), 119–123.

Dong

, Song

, Wang

X.Y.

, et al., Comparative analysis of light-duty vehicle emission model, Journal of Zhongnan University43(10) (2012), 4123–4128.

Boyd

J.P.

, The Devil’s invention: Asymptotic, superasymptotic and hyperasymptotic serie, Acta Applicandae Mathematicae56(1) (2000), 1–98.

Atanassov

, Intuitionistic Fuzzy Sets: Theory and Applications, Physica-Verlag, Heidelberg, (1999), pp. 78–120.

Atanassov

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems20(2) (1986), 87–96.

Atanassov

and Gargov

, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems31 (1989), 343–349.

10.

Zadeh

L.A.

, The concept of linguistic variable and its application to approximate reasoning, Information Science8(2) (1975), 199–249.

11.

Wang

L.Y.

, Guo

Z.H.

, Zhang

Y.F.

, Zhang

Y.S.

and Zhang

, An emergency supplies demand prediction model based on intuitionist fuzzy case reasoning, Journal of China University of Mining & Technology44(4) (2015), 775–781.

12.

Wei

, Li

Y.L.

and Chen

, Linguistic multiple attribute group decision making model for traffic pollution control plan evaluation, Journal of Guangxi University(Nat Sci Ed)41(2) (2016), 23–32.

13.

Groot

, De Schutter

and Hellendoorn

, Integrated Model Predict ive Traffic and Emission Control Using a Piecewise-Affine Approach, IEEE Transactions on Intelligent Transportation Systems14(2) (2013), 587–598.

14.

Smit

and Mcbroom

, Use of Microscopic Simulation Models to Predict Traffic Emissions, Road & Transport Research18(2) (2009), 49–54.

15.

Smit

and Mcbroom

, Use of overseas emissions models to predict traffic emissions in urban areas: Technical note, Road & Transport Research18(3) (2009), 52–60.

16.

Nagendra

S.M.S.

and Khare

, Artificial neural network based line source models for vehicular exhaust emission predictions of an urban roadway, Transportation Research Part D Transport & Environment9(3) (2004), 199–208.

17.

Momani

T.M.A.

, Altaani

A.A.

, Nasser

A.D.A.

, et al., An assessment of automobile emissions in Irbid, Northwest Jordan, Electronic Journal of Applied Statistical Analysis4(1) (2011), 404–423.

18.

X.F.

and Liu

, Traffic congestion management method based on case-based reasoning, Journal of Southwest Jiaotong University44(3) (2009), 416–421.

19.

Wang

and Zhang

Y.M.

, Forecasting model of unconventional emergency incidents resource demand based on case-based reasoning, Journal of Xidian University: Social Science Edition20(4) (2010), 22–26.

20.

Pachiappan

and Govindarai

, Regression modeling of traffic noise pollution, Građ evinar65(12) (2013), 1089–1096.