Missing data detection and imputation for urban ANPR system using an iterative tensor decomposition approach

Transportation Research Part C 107 (2019) 337–355

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Transportation Research Part C journal homepage: www.elsevier.com/locate/trc

Missing data detection and imputation for urban ANPR system using an iterative tensor decomposition approach

T

Han Zhanga, Peng Chena, Jianfeng Zhengb, , Jingqing Zhub, Guizhen Yua, Yunpeng Wanga, Henry X. Liub,c ⁎

a

School of Transportation Science and Engineering, Beihang University, Beijing, China DiDi Chuxing, LLC, China c Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, United States b

ARTICLE INFO

ABSTRACT

Keywords: Automatic number plate recognition (ANPR) system Intelligent transportation system Iterative tensor decomposition Missing data detection and imputation Traffic volume data

The automatic number plate recognition (ANPR) system has been widely implemented as an important part of intelligent transportation system (ITS). However, similar to other traffic monitoring devices, missing data is a common and critical problem in the ANPR system. To solve the missing data problem, numerous tensor-based methods have been proposed in previous studies. Most of them, however, assume that where and when missing data occur in the dataset are known. This would be impractical, because missing data may occur randomly. In this study, we propose a novel tensor-based algorithm, specifically, an iterative tensor decomposition (ITD) approach, that utilizes multidimensional inherent correlation of traffic data to detect and impute missing data in the ANPR system. The proposed algorithm is tested with a real-world ANPR system dataset. The experimental results show that missing data from the ANPR system can be classified into three cases, i.e., no missing, random elements missing, and extreme missing. The proposed ITD can accurately detect and correct missing data under different missing cases. Furthermore, ITD is also compared with other state-of-the-art methods and the results show that ITD outperforms the existing methods.

1. Introduction Owing to the increasing traffic demands, traffic congestion has become a worldwide problem, which can no longer to be simply addressed by constructing new infrastructure or restricting travel demand for economic and environmental reasons (Kerner, 2009). To alleviate traffic congestion, the intelligent transportation system (ITS), which aims to enhance the efficiency of existing transportation system (Zhang et al., 2011), is a more desirable and cost-effective alternative. With the improvement of data collection technology, traffic data collected from multiple sources such as loop detectors, GPS, high-definition cameras, and video sensors are the key input for the ITS (Zhang et al., 2011; Ran et al., 2012). Sufficient and high-quality traffic data become increasingly important and critical for ITS applications such as traffic network planning, route planning, and driver assistance systems (Wang et al., 2013; Wang et al., 2015). Recently, automatic number plate recognition (ANPR) system has become increasingly popular and become an important part of the ITS. The ANPR system uses image processing techniques to collect vehicle id and/or vehicle speed information (Sulaiman et al., 2013). Thus far, ANPR systems have been widely deployed, especially in China. To capture information of vehicles, the ANPR system

⁎

Corresponding author. E-mail address: [email protected] (J. Zheng).

https://doi.org/10.1016/j.trc.2019.08.013 Received 29 October 2018; Received in revised form 13 August 2019; Accepted 14 August 2019 Available online 27 August 2019 0968-090X/ © 2019 Elsevier Ltd. All rights reserved.

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with high-definition cameras will take an image when a vehicle passes through a monitoring station. These images, which include useful traffic data, are stored by traffic management, to extract traffic information such as vehicle volume count, traffic density, and vehicle speed. The extracted data from ANPR systems is then further used for ITS applications. Unfortunately, similar to traffic data from other sources (Xu et al., 2010; Smith et al., 2003; Ni et al., 2005; Sun, 2010), the problem of missing data also exists in the ANPR system. This missing data problem could be caused by device malfunctions, transmission distortion, and/or loss of vehicles’ identification from images (Faouzi et al., 2011; Patel et al., 2013). Because the ITS is highly sensitive to data quality, the traffic data collected from the ANPR system are restricted from being used as input without improvement of the missing data. For example, Chen et al. (2012) showed that missing data significantly influences performance of traffic prediction. The traffic control system fails to generate appropriate traffic management strategies if insufficient traffic flow data is provided (Carlson et al., 2010). Consequently, it is necessary to develop a method to solve the missing data problem in the ANPR system. In the last two decades, numerous imputation methods have been developed to tackle missing data problem. Generally, existing methods can be categorized into three groups: vector-based (one dimension) algorithms, matrix-based (two dimensions) algorithms, and tensor-based (three or more dimensions) algorithms. The traditional vector-based algorithms usually develop a set of traffic vectors from historical data and then apply simple interpolation or extrapolation algorithms to recover missing data (Tan et al., 2013). The frequently used vector-based methods are historical imputation methods and spline (including linear)/regression imputation methods. The historical imputation method means to fill a missing data point with a known data point collected on the same site at the same daily time interval but from a neighboring day (as close in temporal distance as possible) (Schafer, 1997; Chen and Shao, 2000; Allison, 2001). The spline (including linear)/regression imputation methods recover the missing values by applying mathematical interpolation algorithms according to surrounding known data points collected during the same day (Qu et al., 2009). These methods model the traffic data as a vector pattern, which depends heavily on surrounding missing data. Such methods cover little spatial–temporal information, and would fail to work when missing ratio is high. Different from the vector-based approaches, the matrix-based imputation methods impute missing traffic data by applying twomode coherence of traffic data simultaneously, and thereby better take into account spatial–temporal information. The most popular matrix-based methods are principal component analysis (PCA)-based methods. Qu et al. (2008) proposed a Bayesian Principal Component Analysis (BPCA) based imputing method, which provides an appropriate fusion of all the recorded information, to handle traffic flow volume data incompleteness. Qu et al. (2009) proposed a Probabilistic Principal Component Analysis (PPCA) based imputation method accounting for periodicity, local predictability, and statistical information of traffic flow volume data. Tan et al. (2014b) developed a Robust Principal Component Analysis (RPCA) based imputation method that not only utilizes the temporal correlation, but also considers the physical limitation (i.e., road capacity) to avoid the possibility of generating negative and overcapacity values of traffic data. Li et al. (2013) improved performance of missing data imputing by fusing the information of multiple points and compared PPCA- and kernel probabilistic principle component analysis (KPPCA) based imputing methods. However, matrix-based models are limited in that only two modes or diversities are considered. In fact, significant inherent correlations exist in traffic data. For example, the traffic data temporal correlations contain the relations from day to day, hour to hour, and so forth. In addition, the significant spatial correlations also exist in data from detectors at nearby locations. These correlations are difficult to be accounted for by the matrix-based methods. Recently, tensor-based approaches have been introduced to resolve traffic data missing problem. Tensor decomposition originated with Hitchcock’s work in 1927 (Hitchcock, 1927a,b), and it received great attention after Tucker’s work (Tucker, 1966) which made tensor-based methods more practical. Another typical tensor-based method, CP (CANDECOMP/PARAFAC) was proposed by Carroll and Chang (1970). In fact, the CP model can be viewed as a special case of the Tucker model, where the core tensor is super-diagonal and the mode rank along each mode is equal (Kolda and Bader, 2009). Recent studies have demonstrated that tensor decomposition can significantly contribute to estimating the missing traffic data. Chen et al. (2013) incorporated factor priors into a novel tensor completion method called simultaneous tensor decomposition and completion, which completes missing data entries while exploiting the factorization structure. Tan et al. (2013) proposed a Tucker decomposition-based imputation method (TDI) to impute the missing traffic values. A tensor pattern is introduced to model traffic data for the first time. Based on this, the Tucker weighted optimization (Tucker-Wopt) was proposed by using a nonlinear optimization technique on a Grassmann manifold Tan et al. (2014a). Zhao et al. (2015) proposed a fully Bayesian CP factorization which can naturally handle incomplete and noisy tensor data. Based on the nuclearnorm based method-HaLRTC, Ran et al. (2016) and Tan et al. (2016) proposed an algorithm using the spatial information to improve the performance for missing data imputation. Goulart et al. (2017) proposed a tensor completion algorithm that can iteratively adjust the thresholding of the Tucker model’s core. Wu et al. (2018) proposed a CP tensor factorization approach to fuse the l2 -norm constraint, sparseness (l1 norm), manifold, smooth information, and low-rank properties simultaneously to complete and analyze the sparsely observed. Because tensor decomposition can conveniently take into account multimode correlations (Ishteva, 2009; Acar et al., 2011), it can be applied in many fields of traffic data mining. Han and Moutarde (2016) achieved long-term prediction of large-scale traffic by using non-negative tensor factorization. Tan et al. (2016) designed a short-term traffic flow prediction approach by using dynamic tensor completion (DTC). To characterize the collective mobility patterns from high-dimensional datasets, Sun and Axhausen (2016) built a multiway probabilistic factorization model based on tensor decomposition. Tang et al. (2018) proposed a tensor-based Bayesian probabilistic model to estimate urban personalized travel time based on big and sparse GPS trajectories collected from taxicabs. The tensor decomposition techniques were utilized to discover the spatial–temporal patterns and underlying structure from incomplete data (Chen et al., 2018). Although numerous missing data imputation methods based on tensor decomposition have been developed, the problem is still 338

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Table 1 Summary of selected literature on experimental dataset and missing cases of experiment. Literature

Year

Traffic state

Missing cases

Tan et al. Tan et al. Ran et al. Goulart et al. Chen et al. Chen et al. Chen et al.

2013 2014 2016 2017 2018 2019b 2019a

Volume Volume Volume Speed Speed Speed Speed

Random missing, extreme missing and missing under adverse weather Random and structured missing Random and extreme missing Systematic missing Element-like random missing and fiber-like random missing Random missing and fiber missing Random missing and non-random missing

open and has not yet been fully tapped with the following challenges. As shown in Table 1, the experimental datasets of most existing tensor-based imputation methods are complete. The entries of complete dataset are manually discarded to generate different missing cases for the experiment. That is, it is assumed that missing data occur at known locations and times, which may not be practical for real-world traffic data. In reality, missing data could occur completely at random, and may or may not be related to neighboring data points (Qu et al., 2009). Therefore, a notable obstacle exists between practical traffic missing data problems and existing tensor-based imputation methods. The missing cases assumed in the existing literature are shown in Table 1, but the types of missing cases existing in the ANPR system are still uncertain. To resolve these issues, we propose an iterative tensor decomposition (ITD) method to deal with the missing data problem in the ANPR system. This method first utilizes multidimensional inherent correlations of traffic volumes to identify occurrence of missing data. Then, missing data cases of the ANPR system are categorized. Finally, the missing data are corrected by estimated values. To test the performance, experiments are conducted based on real-world ANPR data. The experiment results show that missing cases in the ANPR system can be classified into three typical cases, i.e., no missing, random elements missing, and extreme missing. To further evaluate the performance of the proposed algorithm, different scenarios corresponding to different missing cases are simulated. The results show that the ITD can accurately detect and correct missing data. Comparison results also show that our proposed algorithm outperforms the other state-of-the-art methods with higher accuracy and robustness. The rest of the paper is organized as follows. Section 2 introduces the notations used in this paper and gives a brief introduction of the Tucker decomposition. Section 3 presents the methodology proposed in this paper. The experimental results are shown in Section 4, followed by conclusions and future work in Section 5. 2. Notations and Tucker decomposition In this section, we introduce notations used in our paper, which are referred from (Kolda and Bader, 2009). Calligraphic letters (A , B , ...) are used to represent tensors, italic capitals (A, B, ...) for matrices, and italic lower-case letters (a, b, ...) for vectors and scalars. For example, x i, j, k represents the element (i, j, k ) of a third-order tensor X . Matricization, also known as unfolding or flattening, is the process of rearranging an N-order tensor into a matrix. For example, I1× I2× × IN and its mode-n matricization are defined as follows: entry transformation between a N-order tensor X N

(X(n) )in, j = x i1, i2, … , iN ,

U

where j = 1 +

k 1

(i k

1) Jk with Jk =

Im m=1 m n

k=1 k n

The mode-n products of a tensor are denoted as ×n . For example, the mode-n product of a tensor X J × In is denoted as X ×n U . Elementwise, we have

(1) I1× I2×

× IN

with a matrix

In

(X ×n U )i1

in 1 jin + 1

iN

=

x i1 i2

iN ujin

(2)

in = 1

and the size of result is I1 × ×In 1 × J × In + 1 × ×IN . The Hadamard product is the elementwise product. Given two tensors X , Y with the same size I1 × I2 × product is defined as:

X

Y =

x11 y11 x12 y12 x21 y21 x22 y22

x1J y1J x2J y2J

xI 1 yI 1 xI 2 yI 2

xIJ yIJ

×IN , their Hadamard

(3)

If a is a constant, then the Hadamard product is defined as:

a

Y=

ay11 ay12 ay21 ay22

ay1J ay2J

ayI 1 ayI 2

ayIJ

(4) 339

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Fig. 1. Illustration of tucker decomposition for a third-order tensor.

The Kronecker product of matrices A

A

B=

a11 B a12 B a21 B a22 B

a1J B a2J B

aI 1 B aI 2 B

aIJ B

I ×J

and B

K ×L

is denoted by A

B , which result is a matrix of size (IK × JL) :

(5)

The Tucker decomposition was first introduced by Tucker (1963) and then refined by following articles by Levin (1963) and Tucker (1964, 1966). It has been shown to be a good analytical method for dealing with multidimensional data. Fig. 1 shows an example of Tucker decomposition for a third-order tensor, and the Tucker decomposition decomposes a given third-order tensor n2 × r 2 n3 × r 3 n1× r 1 V n1× n2 × n3 r 1× r 2 × r 3 X into a core tensor G and three factor matrices U , , and W . The factor matrices are r 1× r 2 × r 3 usually orthogonal and can be viewed as principal components in each mode. The entries of the core tensor G show the level of interaction of different components (Kolda and Bader, 2009). 3. Methodology In this study, the objective is to first detect, and then correct, the missing data in the ANPR system. The framework of our algorithm is shown in Fig. 2. The first step is to construct an incomplete tensor based on observed ANPR data. Here, we extract traffic volumes from the ANPR system data. In step 2 and 3, we refer to the methods proposed by Chen et al. (2018). We obtain the initialization for the tensor decomposition by using the truncated singular value decomposition (SVD) method in step 2. In step 3, we get the first recovered tensor by using the SVD-combined tensor decomposition (STD) method. To detect and correct the missing data more accurately, we propose a new algorithm, named ITD, in step 4. 3.1. Tensor construction In a traffic ANPR system, the HD camera captures snapshots once moving vehicles pass the detecting station, and identifies each vehicle based on their plate number. Based on vehicle ids, we can obtain traffic volume easily. Since traffic volumes depend on many factors, including location, date, and time period (Tan et al., 2013), therefore the dataset is naturally a high-dimensional one. In this n1× n2 × n3 regard, we construct a three-dimensional incomplete tensor X , where n1 represents the number of ANPR stations, n2 represents the number of days, n3 represents number of time slices (each time slice is 30 min) of a day, and its entries represent the number of passing vehicles at detecting station during 30 min. Our goal is to detect entries of missing data within the incomplete tensor X and to impute these missing entries. 3.2. Initialization by truncated singular value decomposition (SVD) As mentioned in Section 2, Tucker decomposition decomposes a given tensor X n2 × r 2 n3 × r 3 n1× r 1 V factor matrices U , , and W , as shown in following function:

n1× n2 × n3

into a core tensor G

r 1× r 2 × r 3

and (6)

X = G ×1 U ×2 V ×3 W

Because the size of the core tensor is critical for the results of the Tucker decomposition (Chen et al., 2015), we use the truncated SVD method proposed by Chen et al. (2018). The truncated SVD is a sensitivity-driven rank selection method that considers the ratio threshold of singular values of each unfolding. As mentioned in Section 2, an N-order tensor can be unfolded into a mode-n matrix. The SVDs of each unfolding of incomplete tensor X are formulated by the following functions: 340

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Fig. 2. Framework of the proposed algorithm.

X(1) = U

(1)

XT

X(2) = V

(2)

YT

(3)

X(3) = W

ZT

(7) (q )

= diag ( 1(q), 2(q), ..., where U, V, and W are the leading singular vectors, The size of the core tensor can be determined by the following functions: where pm(q) 1 < p

rq = m , m

pm(q) =

i=1 i

(q) nq )

q = 1, 2, 3 are singular-value diagonal matrices.

pm(q)

(q ) i (q) i

(8)

where the parameter p is a ratio threshold which decides the size of the core tensor and the factor matrices. Because we obtain the rank - (r1,r2,r3), truncated factor matrices can be obtained:

U (T

= Ur 1

n1× r 1

V (T SVD)

SVD)

= Vr 2

n2 × r 2

W (T

= Wr 3

SVD)

n3× r3

(9)

The mode-1 unfolding of core tensor G (T (T G(1)

U (T

SVD)

= U (T

SVD) T X (1)

( W (T

SVD)

SVD)

V (T

is SVD) )

(10)

Then, we can obtain the core tensor G (T SVD) by folding SVD ), W (T SVD), V (T SVD) as the initialization of the next step.

(T SVD) G(1) .

341

We use the core tensor G (T

SVD)

and factor matrices

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3.3. Tensor decomposition combined truncated SVD Our proposed algorithm can be viewed as a non-constraint optimization problem based on tensor decomposition. In this paper, we use the same objective function as (Chen et al., 2018), which can be expressed as follows:

1 2

min L =

(X

B

2 F

G ×1 U ×2 V ×3 W )

+

2

( G

2 F

+ U

2 F

+ V

2 F

+ W

2 F)

(11)

where is the parameter of regularization term. The first part of the right-hand side (RHS) of Eq. (11) is to minimize the squared n1× n2 × n3 errors, the other is an additive regularization. Here, B is a binary tensor in which all values are 1 by default and will be updated in the next step. To solve the optimization problem, we use the gradient descent method. With respect to the decision variables G , U , W , V in Eq. (11), we have: L G L U L V L W

=

E ×1 U T ×2 V T ×3 W T + G

=

E(1) (W

T V ) G(1) + U

=

E(2) (W

T U ) G(2) + V

=

E(3) (V

T U ) G(3) + W

where E = B

(12)

G ×1 U ×2 V ×3 W ) . Then, the updates of the decision variables G , U , W , V can be formulated as:

(X

G U

(1

) G + ·E ×1 U T ×2 V T ×3 W T

(1

) U + E(1) (W

T V ) G(1)

V

(1

) V + E(2) (W

T U ) G(2)

W

(1

) W + E(3) (V

T U ) G(3)

(13)

where is the learning rate of gradient descent method. For details on how to obtain Eqs. (12) and (13), interested readers are referred to Chen et al. (2018). Tensor decomposition can have a better performance if an optimal initialization is provided (Acar et al., 2011), and thus we use G (T SVD), U (T SVD), W (T SVD) and V (T SVD) from Section 3.2 as an optimal initialization. The pseudo-code of the STD is given in Algorithm 1. Algorithm 1: Tensor decomposition combined truncated SVD Input: incomplete tensor X , default binary tensor B Output: recovered tensor X 1. Set learning rate , regularization parameter , tolerance , and maximum iteration count kmax 2. Initialize G (0) = G (T

SVD ),

U (0) = U (T

SVD ),

W (0) = W (T

SVD),

V (0) = V (T

SVD )

3. X (0) = G (0) ×1 U (0) ×2 V (0) ×3 W (0) 4. For k = 1, 2, ...,kmax do: 5. E (k ) = B

X (k

(X

6. Update G (k )

1) )

U (k )

(1

) U (k

1)

+ E (k ) (1) (W (k

1)

V (k

1) T × W (k 1) T 3 1) ) G (k 1) T (1)

V (k )

(1

) V (k

1)

+ E (k ) (2) (W (k

1)

U (k

1) ) G (k 1) T (2)

(1

) W (k 1)

W (k )

) G (k

(1

1)

+ ·E (k ) ×1 U (k

+

E (k )

(3)

1) T

(V (k 1)

×2 V (k

(k 1) T U (k 1) ) G (3)

7. X (k ) = G (k ) ×1 U (k ) ×2 V (k ) ×3 W (k ) 8. If X (k )

X (k

1) 2 F

<

then break

9. X (k 1) = X (k ) 10. End for 11. X = X (k )

3.4. Iterative tensor decomposition In this section, we proposed the ITD procedure for detecting and correcting missing data. Because we obtain the recovered tensor X from Algorithm 1, we first calculate the mean absolute percentage error (MAPE) between every entry of the incomplete tensor X and the recovered tensor X . Then, a new space is defined as below:

= {(i, j, k )|MAPE(x i, j, k , x i, j, k )

MAPE(a,b) =

|a

b| a

(14)

q}

× 100%

(15) 342

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where q is a predefined threshold. The entries of the incomplete tensor X belonging to space are treated as missing entries, and vice versa. A default binary tensor in which all values are 1 is used in Section 3.3. Here, we update the binary tensor B by removing its entries that belong to space

0, (i , j, k ) 1, (i , j, k )

bi, j, k =

(16)

We use the updated binary tensor B as the input of Algorithm 1. Then a new recovered tensor X is obtained from the output of Algorithm 1. The steps above are iterated until the difference in the missing-detecting rate (MDR) reaches a threshold. Here, we define the MDR as:

MDR =

| | n1 × n2 × n3

(17)

At each iteration, the first item of the RHS objective function as shown in Eq. (11) is adjusted owing to the update of the binary tensor. Not only the missing entries but the entries whose estimation errors are higher than the threshold are included in space . Hence, the gradient descent method will be more efficient and the performance of recovery will be more accurate. After iteration, the final results of recovery are stored in the recovered tensor X . The detecting results are stored in the binary tensor B and space , i.e., if (i, j, k ) then x i, j, k is detected as missing data. The pseudo-code of the ITD is given in Algorithm 2. Algorithm 2: Iterative tensor decomposition Input: incomplete tensor X , recovered tensor X Output: recovered tensor X , binary tensor B 1. Set the threshold q, and maximum iteration count nmax 2. Build space

(0)

by Eq. (14), calculate MDR(0)

3. Update binary tensor B (0) by Eq. (16) 4. For n = 1, 2, ...,n max do:

5. Execute Algorithm 1 with the X and B (n) as the input

6. Obtain new recovered tensor X 7. Update

(n)

8. If MDR(n) 9. End for 10. X = X

MDR(0) < (n)

(n )

from output of Algorithm 1

and B (n) , calculate MDR(n) then break

, B = B (n)

4. Numerical experiments In this section, based on real-world ANPR system data, we experimentally investigate the performance of the proposed algorithm. 4.1. Experiment preparation In the experiment, we use real-world ANPR systems’ data, which are obtained from the traffic management department of a city in China for analysis. The dataset contains 237 ANPR systems from December 1st 2017 to December 31th 2017 (i.e., 31 days), as shown in Fig. 3. During each day, the data from 6:00 to 22:00 (where most traffic jams take place) are extracted and aggregated with 30-min time windows. In such case, each ANPR system should have 32 vol values every day. Then, the dataset is structured into an in237 × 31 × 32 complete tensor denoted by X . The size of the incomplete tensor is X . The parameters used in this study are set as shown in Table 2. Because the size of the core tensor is crucial for the final result of tensor decomposition, the ratio threshold of T-SVD is set as p = 0.70 by referring to Chen et al. (2018). How to determine the value of other parameters are discussed in Appendix A. 4.2. Categories of missing data cases In this section, we analyze categories of missing data cases existing in the ANPR systems. First, the incomplete tensor X is imputed by the ITD method. The input of the ITD is the incomplete tensor X and the default binary tensor B that has the same size as X and all values are set as 1. Then, the recovered tensor X and binary tensor B are obtained from the output of the ITD. The recovered tensor X stores all the estimated values and the detection results are stored in the binary tensor B . The detection results are then analyzed. From the binary tensor B , all the entries of missing data (i.e., the entries of B with 0 value) of the incomplete tensor X detected by the ITD can be obtained. The analysis results indicate that missing data cases in the ANPR system can be categorized into three cases: no missing, random elements missing, and extreme missing. These missing data cases are similar to assumption of missing cases in previous studies. Fig. 4 343

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Fig. 3. Locations of the ANPR systems. Table 2 Parameter set values in the experiment. Parameter

Description

Value

kmax nmax

Maximum number of iterations in Algorithm 1 Maximum number of iterations in Algorithm 2 Tolerance Learning rate for gradient descent method Regularization parameter in Eq. (11) MAPE threshold in Eq. (14) Stopping threshold for missing-detecting rate Ratio threshold of the T-SVD

103 30 10−3 2e−11 1e−1 50% 5e−3 0.7

q p

shows examples of three missing cases. The time-series of original and estimated values are plotted. The X-axis is the time of day, which lasts for 31 days. The Y-axis represents the traffic volumes per 30 min of an ANPR system. The black and red curves represent original and estimated volumes, respectively. The line at the bottom of the figure is the detection line. It is generated according to the values of the binary tensor B . If the value is 0 (i.e., the corresponding entry is detected as a missing entry), the line will be red. Otherwise, the line will be green. The three missing cases are further described as below. Case 1. No missing. For case 1, the chosen ANPR system’s location is 200 m west of the intersection of Jingshi Rd. and Hongshan Rd. As shown in Fig. 4(a), the estimated curves match with the original curves closely. Meanwhile, no missing entries are detected by the proposed algorithm (i.e., the detection line does not have any red segments). In this case, the ANPR system can be regarded as working in a good condition. Case 2. Random elements missing. For case 2, the location of the selected ANPR system is 280 m west of the intersection of Heping Rd. and Shanda Rd. As shown in Fig. 4(b), some entries of the original data are detected as missing entries. These entries are completely independent of each other and appear as isolated points (e.g., the missing entries on December 21st and 22nd). The traffic volumes at these missing entries dramatically decrease, which appears without a regular pattern. Hence, the random elements missing case can be described by missing points that are isolated and independent of each other, whose values dramatically decrease. Case 3. Extreme missing. For case 3, the location of the selected ANPR system is the intersection of Lvyou Rd. and Jiangshuiquan Rd. As shown in Fig. 4(c), the original data on December 15th are totally missing. Traffic volumes at these entries shown an unreasonable descent to 0. The extreme (also named structured) missing case is defined to data that are all missing for one or several days, and the associated dates could be either independent or sequential. This case could be caused by a device malfunction that lasts for more than one day. To further evaluate the efficiency and effectiveness of the proposed algorithm, analysis is conducted to quantify: (1) the accuracy of detection of missing data and (2) the accuracy of estimation of missing data. A complete dataset without missing data is necessary to further investigate our proposed algorithm. Since it is difficult to collect such large-scale actual traffic volumes, we select data from the ANPR system that in case 1 as our investigation dataset. The dataset in case 1 has an approximately complete dataset, and traffic 344

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Fig. 4. Demonstration of three missing data cases.

volumes in this case have a similar distribution without unreasonable fluctuations, all indicating good quality of data (Tan et al., 2013). Thus, we resampled the dataset in case 1 to simulate the rest 2 missing cases. The detailed performance of the proposed algorithm is shown as follows. 4.3. Performance in different missing cases To further investigate the performance of the proposed algorithm, we create two missing data scenarios corresponding to cases 2 and 3 in this section. All ANPR systems in case 1 are selected as our complete traffic volume dataset . To simulate the random elements missing scenario, the traffic volumes of the ANPR systems belonging to are randomly removed following a missing ratio . The process can be defined as follows; (18)

x i, j, k = Bernouli ( ) × x i, j, k , i

where x i, j, k represents a value of incomplete tensor X at entry (i, j, k), i.e., the ith ANPR system, jth day, and kth time slice. If the 345

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outcome of the Bernoulli distribution is 0, the value at the corresponding entry will be discarded. To simulate the extreme missing scenario, we randomly remove traffic volumes of an ANPR system of several days following a missing ratio :

X(i, j,:) = Bernouli ( )

X(i, j,:),

(19)

i

where X(i, j,:) represents all traffic volumes of the jth day values for the ith ANPR system, and * is the elementwise product, which is defined in Section 2. If the outcome of the Bernoulli distribution is 0, the values for the corresponding day are discarded. A space 1 including all discarded entries is defined as: 1

= {(i , j, k )|if i

(20)

and x i, j, k = 0}

Then, the incomplete tensor X combined with randomly discarded entries is set as the input of ITD. The output of the ITD is the recovered tensor X and binary tensor B . Then, the estimated traffic volumes, which are obtained from recovered tensor X , at the ANPR systems belonging to are compared to the complete traffic volumes to test accuracy of the estimation. As mentioned in Section 3.4, the space can be obtained from the binary tensor B ; then, space and space 1 are utilized to test the detection accuracy of missing data. To test the accuracy of the estimation, three criteria are used, including MAPE, mean absolute error (MAE) and root-mean-square error (RMSE). If x¯i and x i are the ith estimated value and actual value, respectively, then the criteria can be defined as:

MAPE = MAE =

RMSE =

1 N

xi

N

1 N

N i=1

1 N

x¯i xi

i=1

|x i

x¯i |

(x i

x¯i )2

× 100

(21) (22)

N

(23)

i=1

To test the detection accuracy, there are three criteria, i.e., detection rate, correction rate, and error rate are used. If | 1| is the number of random discarded entries, | | is the number of missing entries detected by ITD, nc is the number of correct detecting entries and (i, j, k ) (i.e., {(i , j, k )|if (i, j, k ) and ne represents the number of error detecting entries (i.e., 1} ), {(i , j, k )|if (i, j, k ) and (i, j, k ) 1} ), then the criteria can be defined as:

detecting rate = correct rate = error rate =

nc |

1|

× 100

(24)

nc × 100 | |

(25)

ne × 100 | |

(26)

4.3.1. Case of random elements missing To simulate this case, we randomly discard the traffic volumes in our complete traffic volume dataset , using a missing ratio that

Fig. 5. Heatmap of random elements missing under 80% missing ratio and its reconstruction. 346

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Fig. 6. Demonstration of random elements missing with 80% missing ratio.

is set from 10% to 90%. The results are then analyzed. We first select an ANPR system under 80% missing ratio as an example to demonstrate the performance of the ITD. Fig. 5(a) shows a heatmap of the complete data after randomly discarding entries (represented by black dots). No clear distribution and pattern are found for such a missing ratio. The reconstruction of the volumes by the ITD is shown as Fig. 5(b). We can see that the morning peak hours (7:00–8:00) and evening peak hours (16:30–19:00) of a weekday have been discovered. The different traffic patterns between weekdays and weekends (such as December 2nd, December 3rd, December 9th, December 10th, December16th, December 17th, December 23rd, December 24th, December 30th, and December 31st) have also been discovered. Therefore, it shows that the proposed algorithm can still distinguish different traffic patterns and estimate missing data according to different traffic patterns, even under an extreme missing ratio. The time-series of actual volumes, estimated volumes, and missing volumes under 80% missing ratio are plotted in Fig. 6. The green, red, and black curves represent the actual, estimated, and missing volumes under 80% missing ratio, respectively. From the black curves, it can be found that most of the volumes are discarded under such an extreme missing ratio. The red curves are generated by the ITD with the black curves as input. The line at the bottom of the figure represents the detection results. The Table 3 Cases of detection results. Cases

Color

Condition

1: 2: 3: 4:

Green Red Yellow Purple

(i , (i , (i , (i ,

No missing Correct detecting Error detecting Miss missing

347

j, j, j, j,

k) k) k) k)

and and and and

(i , (i , (i , (i ,

j, j, j, j,

k) k) k) k)

1 1 1 1

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Table 4 Performance of the proposed algorithm with scenario of segments missing under different missing ratios. Measures

10%

20%

30%

40%

50%

60%

70%

80%

90%

Detection rate Correction rate Error rate MAPE MAE (vehs) RMSE (vehs)

100% 99.17% 0.83% 7.30% 26.7 34.8

100% 99.46% 0.54% 7.35% 26.9 35.1

100% 99.77% 0.23% 7.54% 27.7 36.5

100% 99.73% 0.27% 7.76% 28.6 37.3

100% 99.88% 0.12% 7.97% 29.6 38.8

100% 99.87% 0.13% 8.87% 33.7 44.9

100% 99.98% 0.02% 9.56% 38.1 50.3

100% 99.98% 0.02% 11.04% 48.1 69.4

99.0% 98.65% 1.35% 19.98% 82.2 126.0

detection results include four cases, as shown in Table 3. From Fig. 6(a), we can see that few yellow or purple segments exist in the detection line. Consequently, it shows that our proposed algorithm can accurately detect the missing entries. To further demonstrate the estimation accuracy, the actual volumes and estimated volumes from December 11th, 2017 to December 17th, 2017 are selected as an example in Fig. 6(b). The residual area represents the estimation performance. The smaller residual area represents the more accurate estimation. It can be seen that the red curves fit the green curves very well, indicating that the estimation is accurate. Moreover, the ITD can capture the different traffic patterns accurately (i.e., weekdays from December 11th to December 15th and weekend from December 16th to December 17th). Table 4 further illustrates the performance of the proposed algorithm as the missing ratio ranges from 10% to 90%. The detection rate measures whether the proposed algorithm can detect missing entries, while the metrics of correct and error rate quantify the detection accuracy. It can be seen that the detection rate retains a high percentage, even when the missing ratio reaches 90%. The error detection rate remains at a low percentage. The error rate under low missing ratio may be higher than under high missing ratio. That is because the number of error detections does not change significantly, but the number of detected missing entries increases when the missing ratio ascends. These three measures demonstrate that the proposed algorithm can accurately detect missing entries, even under high missing ratio. As the missing ratio increases from 10% to 90%, the measures of MAPE, MAE and RMSE also increase. However, three measures remain stable when the missing ratio is less than 50%. The estimation errors increase when the missing ratio is higher than 50%, especially from 80% to 90%. Therefore, ITD can provide accurate data recovery when the missing ratio is less than 80%. The performance of ITD degrades quickly and becomes unstable when the missing ratio is higher than 90%. 4.3.2. Case of extreme missing As shown in case 4 of Section 4.2, the ANPR system observations may be lost for a whole day. Because the tensor model can utilize both temporal and spatial information, missing data in such a case can also be detected and recovered. To investigate the performance of the proposed algorithm in this case, we simulate this case by randomly discarding data of several days. The missing days are set from 1 to 21 days (corresponding to missing ratio ranging from approximately 3–70%) in this section. Fig. 7 is an example of 15 days missing of an ANPR system from December 1st 2017 to December 31st 2017. We can see that there are 5 continuous missing days from December 7th to December 11th from Fig. 7(a), likely due to device malfunctions or data transmission problems. Nevertheless, the ITD still works to impute the data in this situation, as shown in Fig. 7(b). The proposed algorithm can estimate traffic volumes according to different traffic patterns (i.e., December 7th, 8th, and 11th in the weekday pattern and December 9th and 10th in the weekend pattern). There is no yellow or purple segment in the detection line, which means the missing entries are accurately detected. From Fig. 8, we can see that the estimated curves fit actual curves well. This demonstrates that the ITD works well in the extreme missing cases and can accurately impute missing data.

Fig. 7. Heatmap of extreme missing with 15 days missing and its reconstruction. 348

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Fig. 8. Demonstration of extreme missing with 15 days missing. Table 5 Performance of the proposed algorithm with scenario of days missing under different missing days. Measures

1

3

6

9

12

15

18

21

Detection rate Correct rate Error rate MAPE MAE (vehs) RMSE (vehs)

100% 97.24% 2.76% 7.45% 27.6 35.9

100% 98.59% 1.41% 7.52% 27.8 36.2

100% 99.23% 0.77% 7.94% 30.4 40.9

100% 99.77% 0.23% 8.11% 32.1 44.7

98.7% 99.74% 0.26% 10.66% 51.9 80.9

98.3% 99.84% 0.16% 12.39% 66.0 105.5

95.3% 99.90% 0.10% 18.64% 100.2 150.0

91.0% 99.60% 0.40% 26.76% 132.3 188.2

Fig. 9. Experimental results under Type 1 nonrecurrent event with 50% missing ratio. 349

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Fig. 10. Experimental results under Type 2 nonrecurrent event with 50% missing ratio. Table 6 Hyperparameters setting for SPC. Parameter

Description

Value

p

Type of smoothness constraint (1 for TV and 2 for QV) Smoothness parameter vector Signal to distortion ratio Stopping threshold

1 [0.5, 0.5, 0] 25 dB 0.01

SDR

Table 5 shows the performance of the proposed algorithm under different missing days. The changing tendency of detection measures (detection rate and error detecting rate) and accuracy measures (MAPE and MAE) is similar to that in the scenarios of random elements missing. However, the data reconstruction in the scenarios of extreme missing is more difficult than in random elements missing, as also reflected in previous studies (Qu et al., 2009; Ishteva, 2009). The performance of the proposed algorithm dramatically declines when the number of missing days is larger than 15 (corresponding to approximately 50% of the missing ratio). 4.3.3. Case of nonrecurrent traffic events Traffic flow contains both recurrent pattern and nonrecurrent pattern. As stated in Yang et al. (2014), the nonrecurrent traffic event includes incident, extreme weather and social activities. During the nonrecurrent traffic event, traffic volumes may vary significantly. Hence, it is useful to investigate the performance of the proposed algorithm during nonrecurrent traffic event. Since such nonrecurrent event data are difficult to obtain in practice, we considered two types of nonrecurrent traffic events by referring to Yang et al. (2014) in order to investigate the performance of the proposed algorithm on missing data detection and correction during nonrecurrent traffic event. Type 1 nonrecurrent event is associated with extreme weather such as rainstorm and snowfall, which will lead to reduction of traffic volumes in a large-scale network. Here it is assumed that extreme weather occurs during 7 AM–9 AM and 6 PM–8 PM for two days (i.e., December 11th and 13th). During the occurrence of traffic event, the traffic volumes of all road networks decrease by 40–60% in random compared to normal conditions. Type 2 nonrecurrent event is related to social activities that occasionally occur, which result in unusual increase of traffic volumes. Here it is assumed that a social activity was held at a certain place during 2PM to 5PM for three consecutive days (i.e., December 6th, 7th and 8th) and only traffic volumes from the ANPR system around the place were affected. During the social activity, the affected traffic volumes from the ANPR system increase by 40–60% in random compared to normal conditions. For the results of analysis, Fig. 9 presents the experimental results of the proposed algorithm under Type 1 nonrecurrent traffic event in random elements missing case with 50% missing ratio. As illustrated in blue shadow areas in Fig. 9, traffic volumes reduced sharply during the periods of nonrecurrent event, while the estimated curves (in red) fit the actual ones (in green) closely. MAPEs, MAEs and RMSEs are 7.79%, 38.9 vehs and 52.8 vehs, respectively. One possible explanation is that, though getting affected by the noncurrent event, traffic flow data from most of the ANPR systems may share similar changing tendency. It assists the proposed algorithm in achieving the desirable performance for missing data imputation in such cases. Besides, there is no yellow or purple segments existing in the bottom detection line as shown in Fig. 9. It represents that the proposed algorithm can accurately detect the missing entries. Fig. 10 shows the experimental results of the proposed algorithm under Type 2 nonrecurrent traffic event in random elements missing case with 50% missing ratio. It is assumed that traffic volumes from one certain ANPR system increased due to the occurrence of a social activity nearby, as illustrated in blue shadow areas in Fig. 10. In such case the estimated curves are generally below the actual ones. It can be attributed to that sudden increase of traffic volumes under nonrecurrent event essentially differs from normal conditions. Furthermore, the data from the affected ANPR system may not have significant spatial and temporal correlations with those from downstream and upstream ANPR systems, leading to a deteriorated performance of the proposed algorithm for missing data imputation. Nevertheless, the estimation errors in terms of MAPEs, MAEs and RMSEs are 14.4%, 75.2 vehs and 102.9 vehs, respectively, and still acceptable. In addition, the bottom detection line, as shown in Fig. 10, indicates that the proposed algorithm can still accurately detect the missing entries in such case. 350

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Fig. 11. Comparison of different methods.

4.3.4. Compared with previous algorithm In this section, the performance of the proposed algorithm was compared with several state-of-art tensor-based methods, i.e., SVD-combined Tensor Decomposition (STD) proposed by Chen et al. (2018), the method proposed by Yang et al. (2016) and smooth PARAFAC tensor completion (SPC) proposed by Yokota et al. (2016)1. Furthermore, an iterative version of SPC was also developed for comparison purpose. For the basic inputs, the incomplete tensor X generated by the data of ANPR systems are the same for all the methods. Nevertheless, in contrast to ITD and the iterative version of SPC which use a default binary tensor as another input, a binary 237 × 31 × 32 tensor S where si, j, k = 1, (i , j, k ) 1 and si, j, k = 0, (i , j , k ) 1 is defined as another input of STD, Yang’s method and SPC (the hyperparameters setting for SPC is shown in Table 6). The performance of different methods was compared under different missing cases, i.e., random elements missing and extreme missing. To enable a fair comparison, the same random seed was utilized to generate missing data and the missing entries are the same for different methods. The results are shown in Fig. 11. Under the case of random elements missing, all the methods were examined under different missing ratios ranging from 10% to 90% and the MAPEs are provided as shown in Fig. 11(a). It can be found that ITD has the best performance when the missing ratio is smaller than 80% and the iterative version of SPC outperforms the other methods when the missing ratio is higher than 80%. SPC

1

The Matlab codes are available in https://sites.google.com/site/yokotatsuya/home/software. 351

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outperforms STD obviously when missing ratio is less than 60%, and the performances of SPC and STD are approximately comparable when missing ratio is higher than 60%. Meanwhile, the performance of both STD and SPC got improved when adopting an iterative version of the methodology as proposed in this study. Fig. 11(b) shows the performance of different methods under the case of extreme missing. Similarly, ITD has the best performance when missing days are less than 15 and the iterative version of SPC outperforms the other methods when missing days are more than 18. It is notable that the performance of almost all the methods declines dramatically when the missing days are greater than 18 days (out of 31 days), implying that missing data imputation under extreme missing cases are more difficult to conduct than under random elements missing cases. Overall, the comparison results show that the methods adjusted in an iterative version have better performance under different missing ratios. The main reason is that the iterative method such as ITD can adjust the objective function dynamically according to the update of the binary tensor, which helps to make the estimation more accurate and robust. 5. Conclusions and future work In this study, a novel tensor-based algorithm, i.e., ITD, was proposed to detect and correct missing data in the ANPR system. The iterative characteristics of the algorithm help to adjust the objective function dynamically according to the update of the binary tensor, making the estimation more accurate and robust., Real-world ANPR system data were utilized to investigate the performance of the proposed algorithm. The cases of random elements missing (missing ratio ranges from 10% to 90%) and extreme missing (missing days ranges from 1 to 21 days) were analyzed by conducting experimental studies. The results show that ITD outperforms several state-of-the-art tensor-based methods under different missing ratios. As shown in the experiment results, the critical point of random elements missing is higher than extreme missing, which indicates the reconstruction of the scenarios of extreme missing is more difficult than those of random elements missing. This deficiency is considered to be improved by investigating multi-dimensional nature of traffic data and utilizing advanced data fusion technology in our future work. Moreover, tensor-based approaches show a great potential in traffic prediction as demonstrated in Tan et al. (2016). To extend the ITD for traffic prediction is another future research direction. Acknowledgement The authors appreciate the National Natural Science Foundation of China (U1811463, 61873018 & U1564212) for support of this research. Appendix A The parameters k max and n max are determined with a tradeoff between computing efficiency and the performance of the proposed algorithm. It is expected that the larger these two values, the better the performance but the lower the efficiency. To examine the sensitivity of the algorithm performance to these two values, experiments under the random elements missing case with 10%, 50% and 80% missing ratios were conducted with varied values of these two parameters. To be specific, the parameter k max is the maximum number of iterations in Algorithm 1. Fig. A1 illustrates the changing tendency of the algorithm performance in terms of MAPEs and MAEs with the increase of k max while the other parameters were set the same as in Table 2. It can be found that MAPEs and MAEs decrease rapidly with an increase of k max and becomes less sensitive to the increase of k max when exceeding 103. Hence, the optimal value of the maximum number of iterations in Algorithm 1 k max is determined as 103. Similarly, the maximum number of iterations in Algorithm 2 n max is determined as 30 by making a balance between computing efficiency and algorithm performance as shown in Fig. A2.

Fig. A1. The performance of the proposed algorithm vs. the parameter k max . 352

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Fig. A2. The performance of the proposed algorithm vs. the parameter n max .

Fig. A3. The performance of the proposed algorithm vs. the parameter q.

q and are the parameters utilized in Algorithm 2. They were examined to be insensitive to the change of the dataset, thereby remaining the same in different experiments once determined. To determine the best values of these parameters, experiments under the random elements missing case with 10%, 50% and 80% missing ratios were conducted with varied values of the parameters. Fig. A3 shows the changing tendency of the algorithm performance in terms of MAPEs and MAEs versus a series of q while the other

Fig. A4. The performance of the proposed algorithm vs. the parameter . 353

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parameters were set the same as in Table 2. It can be found that the proposed algorithm achieves the best performance when q is around 50%. The parameter is the stopping threshold for missing-detecting rate in Algorithm 2. It was found that the performance of the proposed algorithm was not sensitive to as shown in Fig. A4. is determined as 5e−3 when the algorithm achieves the best performance. , , and p are the parameters utilized in Algorithm 1. How to determine these parameters has been introduced in Chen et al. (2018). Note that given different characteristics of the study datasets (i.e., traffic volume data in this study and traffic speed data in Chen et al. (2018)) the determined values of these parameters are somewhat different. Appendix B. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.trc.2019.08.013.

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