LSTOD: Latent Spatial-Temporal Origin-Destination prediction model and its applications in ride-sharing platforms
Authors: Fan Zhou
Haibo Zhou
Hongtu Zhu
Venue:  (2020)
Keywords: Origin-Destination Flow
Spatial Adjacent Convolution Network
Periodically Shift Attention Mechanism

Abstract: Origin-Destination (OD) flow data is an important instrument in transportation studies. Precise prediction of customer demands from each original location to a destination given a series of previous snapshots helps  ride-sharing platforms to better understand their market mechanism. However, most existing prediction methods ignore the network structure of OD flow data and fail to utilize the topological dependencies among related OD pairs. In this paper, we propose a latent spatial-temporal origin-destination (LSTOD) model, with a novel convolutional neural network (CNN) filter to learn the spatial features of OD pairs from a graph perspective and an attention structure to capture their long-term periodicity. Experiments on a real customer request dataset with available OD information from a ride-sharing platform demonstrate the advantage of LSTOD in achieving at least 6.5% improvement in prediction accuracy over the second best model. 

Under review as a conference paper at ICLR 2020
LSTOD: LATENT
SPATIAL-TEMPORAL ORIGIN- DESTINATION PREDICTION MODEL AND ITS APPLI- CATIONS IN
PLATFORMS
Anonymous authors
Paper under double-bling review
ABSTRACT
Origin-Destination (OD) flow data is an important instrument transportation studies. Precise prediction of customer demands from each original location to a destination given a series of previous snapshots helps ride-sharing platforms to better understand their market mechanism. However, most existing prediction methods ignore the network structure of OD flow data and fail to utilize the topo- logical dependencies among related OD pairs. In this paper, spatial-temporal origin-destination (LSTOD) model, with a novel convolutional neural network (CNN) filter to learn the spatial features of OD pairs from a graph perspective and an attention structure to capture their long-term periodicity. Ex- periments on real customer request dataset with available OD information from a ride-sharing platform demonstrate the advantage of LSTOD in achieving at least 7.94% improvement in prediction accuracy over the second best model.
propose a latent
INTRODUCTION
Spatial-temporal prediction of large-scale network-based OD flow data plays an important role in traffic flow control, urban routes planning, infrastructure construction, and the policy design of ride- sharing platforms, among others. On ride-sharing platforms, customers keep sending requests with origins and destinations at each moment. Knowing the exact original location and destination of each future trip allows platforms to prepare sufficient supplies in advance to optimize resource uti- lization and improve users' experience. Given the destinations of prospective demands, platforms can predict the number of drivers transferring from busy to idle status. Prediction of dynamic de- mand flow data helps ride-sharing platforms to design better order dispatch and fleet management policies for achieving the demand-supply equilibrium as well as decreased passenger waiting times and increased driver serving rates.
Many efforts have been devoted to developing traffic flow prediction models in the past few decades. Before the rise of deep learning, traditional statistical and machine learning approaches dominate this field (Li et al. 2012 Lippi et al. 2013] Moreira-Matiass al., 2013 Shekhar & Williams 2007] Idé & Sugiyamal 2011 Zheng & Ni 2013). Most of these models are linear and thus ignore some important non- lon-linear correlations among the OD flows. Some other methods (Kwon & Murphy 2000 Yang et al. 2013)) further use additional manually extracted external features, but they fail to automatically extract the spatial representation of OD data. Moreover, they roughly combine the spatial and temporal features when fitting the prediction model instead of dynamically modelling their interactions.
The development of deep learning technologies brings a significant improvement of OD flow predic- tion by extracting non-linear latent structures that cannot easily covered by feature engineering. (Xingjian et al.) 2015) Ke et al.) 2017; Zhou et al.) 2018). Zhhang et al.] (2016) 2017) modeled the whole city are as an entire image and employed residual neural network to capture tempo- ral closeness. Ma et al. (2017) and Yu et al. (2017) also learned traffic as images but they used LSTM instead to obtain the temporal dependency. Yao et al. (2018b) proposed a Deep Multi- View Spatial-Temporal Network (DMVST-Net) framework to model both spatial and temporal relation- ships. However, using standard convolution filters suffers from the problem that some OD flows covered by receptive field of regular CNNs are not spatially important. Graph-based neural net-

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works (GNN) (Kipf & Welling, 2016 Defferarrd et al. 2016 Veličković al. 2017) proved to
be powerful tools in modelling ppatial-empoln network structures ( Yuu et al. 2018 Li et al. 2017). However, none of these frameworks
directly applicable here since both the historical observa-
tions and responses to predict are vertex-level variables. On the contrary, the OD flows we discuss in this paper are generated in the edge space by our definition.
The aim of this paper is to introduce a hierarchical Latent (LSTOD) prediction model to jointly extract the complex spatial-temporal features of OD data by using some well-designed CNN-based architectures. Instead of modelling the dynamic OD networks as a sequence of images and applying standard convolution filters to capture their spatial informa- tion, we introduce novel Vertex Adjacent Convolution Network (VACN) that uses an irregular convolution filter to cover the most related OD flows that share common vertecies with the target one. The OD flows connected by common starting and/or ending vertexes, which may fall into dif- ferent regions of the flow map, can be spatially correlated and topologically connected. Moreover, for most ride-sharing platforms, passenger is more likely send a new request from the location where his/her last trip ends in. To learn such sequential dependency, we introduce a temporal gated CNN (TGCNN) Yu et al. (018)) and integrate it with VACN by using the sandwich-structure ST- conv block in order to collectively catch the evolutionary mechanism of dynamic OD flow systems. A periodically shifted attention mechanism is also used to capture the shift in the long-term period- icity. Finally, the combined short-term and long-term representations are fed into the final prediction layer to complete the architecture. Our contributions are summarized follow:
Origin-Destinanl Destination
To the best of our knowledge, it is the first time that we propose purely convolutional structures to learn both short-term and long-term spatio-temporal features simultaneously from dynamic origin-destination flow data.
We propose novel VACN architecture to capture the graph-based semantic connections and functional similarities among correlated OD flows by modeling each OD flow map as adjacency matrix.
We design a periodically shift attention mechanism to model the long-term periodicity when using convolutional architecture TGCNN in learning temporal features.
Experimental results on two real customer demand data sets obtained from a ride-sharing platform demonstrate that LSOO outperforms many state-of-the-ard methods in OD flow prediction, with 7.94% to 15.14% improvement of testing RMSF.
2 DEFINITIONS AND PROBLEM STATEMENT
For given urban area, we observe sequence of adjacency matrices representing the OD flow maps defined on a fixed vertex set V. which indicates the N selected sub-regions from this area. We let V {V1, ..,..., J denote the vertex set with e being the i-th sub-region. The shape of cach grid vn could be either rectangles, hexagons or irregular sub-regions. We define the dynamic OD flow maps (Oa,t), where l...., D and t 1,..., represent the day and time indexes, respectively. For each snapshot Ou, = (o%,) E RNN the edge weight off, at row i and column j denotes the flow amount from node U; node Vy at time 1 of day d. larger edge weight of, is equivalent to strong connection between nodes U; and dy. The O are asymmetric since all the included OD flows are directed. Specifically, off = 0 if there is no demand from v; to U within the t-th time interval of day d.
The goal of our prediction problem is predict the snapshot O8æ4ÿ E NNN in the fu- ture time window (L + j) of day d given previously observed data, including both short-term and long-term historical information. The short-term input data consists of the last P1 times- tamps from t + 1 - P1 to t. denoted by X1 = 10a=-1-p::00--1110 Os. long-term input data is made up of q time series (Oa-p.t -11- Od-p.ttit(p--- 11/23 of length p2 for each previous day (d Φ)) with the predicted time index (1 + j) in the middle for 40 = l,....g. We let X2
Ou.h The
(o-e: Oa-1,t+ (..++j((pp-)) (pa-1)/2..... Oa-I.ttitop 11/2] denote the entire long-term data. Increasing Pl and P2 leads to higher prediction accuracy, but more training time.
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We reformulate the set of short-term OD networks X1 into a 4D tensor X₁ R NxNxpix and con- catenate the long-term smapshots X2 into a 5D tensor X₂ = (X2,d-1:.... X2 E XXxNxXxxxxx Each X X2.d-p E RN Xp2x1 for day d - yp is a 4D tensor for = =,, ,. Therefore, we can finally define our latent prediction problem as follows:
0d1 = F(X, X2):
where .). represents the LSTOD model, which captures the network structures of OD flow data as well as the temporal dependencies multiple scales. notation table is attached in the appendix.
3 LSTOD FRAMEWORK
In this section, we describe the details of our proposed LSTOD prediction model. See Figure for the architecture of LSTOD. The four major novelties and functionalities of LSTOD model include
end-to-end framework LSTOD constructed by all kinds of CNN modules process dynamic OD flow maps and build spatio-temporal prediction models;
a novel multi-layer architecture VACN to extract the network patterns of OD flow maps by propagating through edge connections, which not be covered by traditional CNNs;
special module ST-Conv block used to combine VACN and gated temporal convolutions to coherently learning the essential spatio-temporal representations;
a periodically shifted attention mechanism which is well designed for the purely convo- lutional ST-Com blocks to efficiently utilize the long-term information by measuring its similarities with short-term data.
Day al
O
blikk
1
K
19g
Garey
T
Figure 2: An example to show how standard CNN fails capture the network structure of OD flows
Figure 1: The architecture of LSTOD consisting of (CI) ST-
Conv blocks, (C2) TGCNN, (C3) VACN, and (C4) attention.
3.1 SPATIAL ADIACENT CONVOLUTION NETWORK
Before introducing the detailed architecture of VACN, we want to discuss why directly applying standard CNNs the OD flow map O may disregard the connections between neighboring OD flows in the graph space first. Figure Satureedeemoo demonstrates that it fails to capture enough semantic infor- mation using the real-world example of ride demands. For the OD flow starting from en to 12, as illustrated in the upper sub-figure, the most related OD flows should be those with either origin or destination being in or U2 in the past few timestamps. A certain part of the travel requests from en to U2 can be matched with some historical finished trips from a third-party location V1 by the same group of people, for example a trip from Vg to f1. However, as the lower-left sub-figure illustrates, some of the OD flows covered by single CNN filter (the green square) such as the four corners of the kernel window may be topologically far away from the target one the graph.
o More generally, let's consider a target OD flow og, in a map O Most of the components covered
((+1)(11) are less correlated than by a standard CNN of size x with of in the middle such as o some out of the kernel window, for example of, when lk jl (q 1)/2. Moreover, if change
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the order of the N vertexes in Od,,, then the network structure is unchanged, but different set of OD flows is covered by the receptive field of the same size with the center being of As shown in the lower right sub-figure of Figure 2 OD flows with either origin or destination being vs vj covered by the red and yellow kernel windows, are considered to be the most related ones for o in row i and column j.
To better understand the differences between our proposed VACN over graph edges with those bortex-based convolutions such as GCN (Kipf Welling 2016) or GAT (Veličković et al. 2017). introduce the concept of line graphs L(G) (Godsil & Royle 2013). Each node in L(G) corre- sponds to an edge in G, while each individual edge in L(G) can be mapped to a pair of edges in G that connect to joint vertex. L(G') contains [ - (N - 1)] while there are only |NN - 1) for
G. Thus, applying GNNs over L(G) is memory intensive. Our edge-based convolution operations VACN is much more efficient.
Formally, every VACN layer takes a 3D snapshot A.,, = (a E RNxNXm, consist of mo feature
maps as the input with each a² E R representing feature vector the edge from 2 to V3- The learned representation of each target edge is defined the weighted sum of those from the same row or column in the adjacency matrix, and those from the row or column in the transposed adjacency matrix. The output of the layer-wise VACN propagation for the OD flow from o; to vy defined as follows:
F
R ai + CTa ni + Rial + C2 CTa E R!!
such that [W] ...., w E [emm XN the weights to learn. F()) represents an elementwise activation function, such as ReLU(x) = max(0,x)). The first part of 2 works by summing up the feature values of OD flows having either the same origin or destination with the target OD flow. The second part covers another set of OD flows that either start by or end at U;. Similar to standard CNNs, different OD flows are treated with unequal importance by VACN and edges more related to the target one are assigned higher weights RR, CT or Cg. We denote the overall output of a multi-layer VACN as Yd, E NNN X xm, at time (d, t). Figure 4 in the appendix more clearly illustrates the architecture of VACN.
where [R] R2, Cf C2 W E
3.2 TEMPORAL GATED CNN
We use temporal gated CNN (TGCNN) (Yu et al. 2018) instead of RNNNbased architectures such as LSTMs to capture the temporal representation, which makes our LSTOD a pure convolutional architecture. RNNs suffer from the problem of lower training efficiency, gradient instability, and time-consuming convergence. Moreover, the high dimension of the spatial representations captured by VACN and potential long temporal sequence length make RNNs notoriously difficult train. The CNNs is more flexible in handling various data structures and allows parallel computations to increase training speed.
TGCNN consists of two parts including one being a 3D convolution kernel applied to the spatial representations of all the 2 OD flows along the time axis and the other being gated linear unit (GLU) as the gate mechanism. By employing VACN at each of r successive timeslots, we can build a 4D tensor Y (Ya,)) E R. XNNxxm, which is then fed into the TGCNN operator:
G Y= Y, (Y2) E (NXNN(--K1)xxm,
where G+.,, represents the TGCNN kernel and 7 includes all the related parameters to learn. 2m, 3D convolutional kernels of size (1 x 1 x K) with zero paddings map the input into a single output which is split in half to obtain Y, and 22 with the same number of feature channels. C here denotes the element-wise Hadamard product.
[Y, Y2) E R (+xNx((=++)))xe
3.3 ST-CONV BLOCKS
We use the spatial-temporal convolutional block (ST-conv block) jointly capture the spatial- temporal features of OD flow data, which has 'sandwich' architecture with a multi-layer VACN operator in the middle connecting the two TGCNNs. The use of ST-Conv blocks have two
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major advantages. First, the block can be stacked extended based on the dimension and character- istic of the spatio-temporal input data. Second, a temporal operation is applied before extracting the spatial information, which greatly reduces its computation complexity and memory consumption.
Both the input and output of each individual ST-Conv block are 4D tensors. For the input of the 1-th block Z'-1 E R Xx--1Xx------- (Z" is the original the OD flow data with co 1), the output is computed as follows:
z' G1 $r1 [S sgi {Go $76 Z1-
where G1 f and Go are two TGCNN kernels and S*0: is a multi-layer VACN operator being applied to each timestamp. The G₁ and G2 operators from all the stacked ST-Conv blocks employ the same kernel sizes, which are (1 x 1 x K1) and (1
K2), respectively. Thus, we have 72-1 (K1 - K2 2). After applying (ro- 1)/ + Ki 2) ST-Conv blocks to the input
z", the temporal length is reduced from To to 1.
When input being the short-term OD flow data Z = X1 E RN*NXxIXI, we use Lo = (pl 1)/(K3 + KST 2) blocks obtain the spatial-temporal representations ZST s(Z n(Z RNxNxeer f here squeezes the captured 4D output into a 3D tensor by dropping the temporal
axis. The kernel sizes of each G₁ and Gu are (1x 1 x KÿT)) and (1 (1x1x. K;T), respectively. The
detailed propagation of the 1-th ST-Conv block defined as
z' = G1 S5gp
[S *0'er (Go #0 Z'-
(5)
3.4 PERIODICALLY SHIFTED ATTENTION MECHANISM
In addition to capturing the the spatial-temporal features from short-term OD flow data X1, we also take into account the long-term temporal periodicity due to the potential day-wise cycling patterns insides the OD flow data, decided by customer's 's travelling schedule and the city's 's traffic conditions. Directly applying ST-Con blocks an extremely long OD sequence which covers previous few days or weeks is computationally expensive. Only a small set of timestamps from each previous day is necessary to capture the long-term periodicity. As mentioned, we pick p2 time intervals for each day d - &0 when predicting the time window (d, + j) considering the non-strict long-term periodicity.
This slight time shifting may be caused by unstable traffic peaks, holidays and extreme weather conditions among different days.
Inspired by the recently widely used attention mechanisms (Xu al. 2015 Yao et al. 2018a Liang et al., 2018) in spatial-temporal prediction problems, propose modified periodically shifted attention to work for the CNN-based ST-Conv blocks here. Different from Yao et al. (2018a) Sa) that measures the similarity between hidden units of LSTMs, the attention here is built on the intermedi- ate outputs of TGCNNS where the concatenations are then fed into a new set of ST-Conv blocks. For each day (d :p), we apply series of L1 ST-Conv blocks the day-level pe-length sequential OD flow data X p and reduce the temporal length from P2 to n².. Each block contains two TGCNN layers with the same kernel size 1 x x K2r- such that
nLr = Li (2K2) - 2)
(6)
and the propagation rule of the 1-th ST-Conv blocks is defined as:
Zy- = G1 * > [S *0t
(Go * 2er 1
with - and
+1+1 representing the input and output of the L-the block.
We denote the day-level representations of day (d - up) captured by the L, ST ST-Conv blocks above as Za-0 where each زt =y/ represents the ø-th element along the time axis for the OD flow from Ua to uy. On the other hand, we let 2sS be the learned short-term representation at the OD flow from U s/
to uy. Subsequently, a re-weighted day-level output 25 2اا can be obtained by summing up 2d-p.0" 's
using weights مy- which measure their similarities with the short-term representation zgT
1/
(8)
d 2, d-
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where 3 Pd-2. quantifies the similarity between d-ppo and ij score(2 (222221-0)))) p,0 p. 2TT), which is defined as:
2ST based function
exp z1 d-p.d 2gT
81-p00
(9)
Eg exp(score (2 p.00 22T)
is defined as Moreover, score 2 2ST)
Jzz tanh
(aanh(Wi5) (Wi + W22ST + ba);
(10)
where W1, W2 and Ug are learned projection matrices, b, is the added bias term. By assuming that Za-p (#2-p) denotes the day-level output for all the N entries after re-weighting, can build
a new day-wise time series Zia of length g by concatenating all the Za- 's along an additional axis in the third dimension as
Lrr po
(11)
to build a new 4D day-wise time series Z) and finally apply another set of ST-Conv blocks to it to obtain the long-tern teretermm apatial-tempore representations, which is denoted by Z,T E RNXNXxer CLT the number of feature channels.
We concatenate the short-term and long-term spatial-temporal representations ZST and ZLT together along the feature axis as Z = ZsT @ ZuT E It NxCC where C = CsT + CLT. Then, Z is modified to 2D tensor Z R by flattening the first two dimensions while keeping the third one. We apply a fully connected layer to the C feature channels together with an element-wise non-linear sigmoid function get the final predictions for all the N2 OD flows.
3.5 DATA PROCESSING AND TRAINING
We normalize the original OD flow data in the training set to (0, 1) by Max-Min normalization and use 'sigmoid' activation for the final prediction layer to ensure that all predictions fall into (0, 1). The upper and lower bounds saved and used to denormalize the predictions of testing data get the actual flow volumes.
We use La loss to build the objective loss during the training. The model is optimized via Back- propagation Through Time (BPTT) and Adam (Kingma & Ba, 2014). The whole architecture of our model is realized using Tensorflow (Abadi et al. (016)) and Keras (Chollet et al. 2015). All experiments were run on a cluster with one NVIDIA 12G-memory Titan GPU.
4 EXPERIMENT
In this section, we compare the proposed LSTOD model with some state-of-the- -f--th--rtrrrr art approaches for latent traffic flow predictions. All compared methods are classified into traditional statistical meth- ods and deep-learning based approaches. We use the demand flow data collected by ride-sharing platform to examine the finite sample performance of OD flow predictions for each method.
4.1 DATASET DESCRIPTION
We employ a large-scale demand dataset obtained from large-scale ride-sharing platform to do all the experiments. The dataset contains all customer requests received by the platform from 04/01/2018 06/30/2018 two big cities A and B. Within each urban area, N 50 hexagonal regions with the largest customer demands are selected to build the in total 22 2500 OD flows. Since one-layer VACN has a computation complexity ((N) at each of the N2 entries (globally O(N*)), the memory consumption highly increases N gets bigger. Considering the computation efficiency and storage limitation, we choose N = 50 here which can cover more than 80% total demands and thus satisfy the operation requirement of the ride-sharing platform.
We split the whole dataset into two parts. The data from 04/01/2018 to 06/16/2018 is used for model training, while the other part from 06/17/2011 06/30/2017 (14 days) serves the testing

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set. The first two and half months of OD flow data is further divided in half to the training and validation sets. The size ratio between the two sets is around 4:1. We let 30 min be the length of each timestamp and the value of the OD flow from V4 to by is the cumulative number of customer requests. We make predictions for all the 50 OD flows in the incoming 1st, 2nd, and 3rd 30 minutes (i.e. + t + 2, f + 3) by each compared method, given the historical data with varied (p1.22) combinations. For those model settings incorporating long-term information, trace back q days to capture the time periodicity. We use Rooted Mean Square Error to evaluate the performance of each method:
N
N
RMSE
Σ
(12)
N *
(d.t)cTa
od and 02, are the true value and prediction at the OD flow from vertex V/ to vertex vy at time (d, I), respectively. To is the set containing all the predicted time points in the testing data.
4.2 COMPARED METHODS
All state-of-the-art methods to be compared are listed follows, some of which modified to work for the OD flow data: (i) Historical average (HA): HA predicts the demand amount at each OD flow by the average value of the same day in previous 4 weeks. (ii) Autoregressive inte- grated moving average (ARIMA), (iii) Support Vector Machine Regression (SVMR), (iv) La- tent Space Model for Road Networks (LSM-RN) (Deng al., 2016), (v) Dense + BiLSTM (DLSTM) (Atcché & de La Fortelle 2017) and (vi) Spatiotemporal Recurrent Convolutional Networks (SRCN) (Yu et al., (017)) We only consider latent models in this paper, that is, no ex- ternal covariates are allowed, while only the historical OD flow data is used to extract the hidden spatial-temporal features.
4.3 PREPROCESSING AND PARAMETERS
We tune the hyperparameters of each compared model to obtain the optimal prediction performance.
Specifically, we get (p" d" -²") (3, 0,3) for ARIMA and k- 15, 7" =2- X = 10 for LSM- RN. The optimal kernel size of the spatial based CNN kernel is 11 x 11 in SRCN model.
For fair comparison, we set the length of short-term OD flow sequence be p1 9 (i.e., previous 4.5 hours), = 3 for long-term data which covers the three most recent days, and the length of each day-level time series P2 predicted time index). More analysis of how variant (p1. P2) combinations may affect the prediction performance of L.TOD will studied latter.
capture the periodicity shifting (one hour before and after the
A two-layer architecture is used by all the deep-learning based methods to extract the spatial patterns inside the OD flow data (L for both short-term and long-term VACN). We set the filter size of all deep learning layers in both spatial and temporal space to be 64, including the VACNs and TGCNNs in our LSTOD model with CST = CLT = 64.
4.4 RESULTS
Comparison with state-of-the-art methods. Table 1 summarizes the finite sample performance for all the competitive methods and our LSTOD model in terms of the prediction RMSE on the testing data of city A. We compute the mean, variance, 25% quantile and 75% quantile of the 14 day-wise RMSE on the testing set. LSTOD outperforms all other methods on the testing data with the lowest average day-wise RMSE (2.41/2.55/2.66), achieving (8.02%/7. 14%./92444444 improvement over the second best method 'SRCN', In general, deep-learning based models perform more stably than traditional methods with smaller variance and narrower confidence intervals. Both 'ARIMA' and 'LSM-RN' perform poorly, even much worse than HA, indicating that they cannot capture enough short-term spatial-temporal features to get the evolution trend of OD flow data. Among the deep learning models, LSTOD can more efficiently control the estimation variance compared to all the others. This demonstrates the advantages of using spatial-temporal architecture and long-term periodicity mechanism in modelling the dynamic evolution of OD flow networks. The improvement becomes more significant when the time scale increases since the contributions of
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Table I: Comparison with -tate-of-the-art methods for City A
City A
Method
30 min
60 min
90 min
HA
3.61(3.2443.7700.4))
ARIMA
5.11(4.36/5.2//122)
5.37(4.62/6.1/1..2)
5.8765.09/6.65/1399
LSVR
3.5502.55/4.51/2.09)
3.8392.64//..4422995)
4.65(3.42/5.78/3224)
LSM-RN
5.60(4.6118872/1.866 6.26(5.417.19/1197)
6.7065.71/7.63/2.11
DLSTM
3.03(2.51/3.5000552 3.5202.88/3.0300.51
3.9103.33/4
SRCN
2.621..09/3.2/10.54) 2.7712.14333500.72)
1L12.18/3.60/00990
LSTOD
2/112.16/2.69001117))
2.55(2.277/.68001199
26712.35/3.01/.2555)
long-term periodicity are more emphasized as the short-term signals getting weaker. The LSTOD performs even better on city B compared to the baseline methods since the long-term periodical pattern in city may be more sigmificant compared with that in city A. Detailed results about City B are summarized in Table 3 of the appendix.
Comparison with variants of LSTOD. Table 2 shows the finite sample performance of our pro- posed model LSTOD and its different variants based on the demand data from city A. We can see that the complete LSTOD model outperforms the short-term model and the one without using at- tention mechanisms in terms of smaller means and variances, and narrower confidence intervals. It indicates that the attention design we use can capture the shift of the day-wise periodicity and extract more seasonal patterns to improve prediction accuracy. The left sub-plot of Figure compares the predictions by each model against the true values at two selected OD flows on the last testing days in the 60-minute scales. Two abnormal change points are marked by black circles. The short-term model fails in this case because it ignores the long-term information. The complete LSTOD model outperforms the one without using attention mechanisms since it can better catch the shift of the periodicity. The right sub-plot visualizes the distribution curves of the day-wise RMSES on the 14 testing days by each of the three compared models. The lighter tail of the red curve demonstrates that the complete LSTOD is more predictive and stable especially for those unusual cases. We do some more experiments to show how different hyperparameter configurations influence the model performance. For more details, please refer to Section E of the appendix.
VACN VS standard local CNN. In this experiment, we will show that our proposed VACN out- performs standard CNNs in capturing the hidden network structure of the OD flow data. Given the model setting that N = 50 sub-regions of city A are used to build the dynamic OD flow matrices, the number of pixels being covered by VACN at each single snapshot is 50 x 4 200. For fair comparison, the largest receptive filed of standard CNN should be no bigger than a 15 x 15 window, which includes 225 elements each time. We consider five different kernel sizes including 5 x 5, 8 x 8, 11 x 11, 14 x 14, and 15 x 15. We replace VCAN in our model by standard CNN in or- der to fairly compare its performance. All hyper-parameters are fixed but only the kernel size of CNNs being changed. Moreover, we only consider the bascline short-term mode of LSTOD model while ignoring the long-term information. As Figure 4 illlsstraess standard CNN achieves the best performance with the smallest RMSE = 2.64 on testing data with the filter size being 11 x 11, which is still higher than that using VACN with RMSE = 2.54. Specifically, RMSE increases when the receptive field is larger than 11 x 11 since the spatial correlations among the most related OD flows (sharing common origin or destination nodes) are smoothed with the increase in the filter size ((8 2 /6 > (14 x 2 1)/196). This experiment shows that treating the dynamic demand matrix as an image, and applying standard CNN filters does not capture enough spatial correlations among related OD flows without considering their topological connections from the perspective of graphs. For more details, please refer to Figure
As we mentioned above, one-layer VACN has global computation complexity O(N3) at each timestamp. For standard CNN, O(N3) executions are still needed by applying local CNN filters to N windows with each of the N entries in the middle. Therefore, when the side length of the CNN filter is in the order O(VN), the total cost of our VACN and standard CNN are in the same order.
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RMSE
Method
30 min
60 min
90 min
VACN TGCNN (short-term) LSTOD (no attention)
2.5202.21/22.000.20
2.7442.44/2.8100.344)
2.8102.3333.5500445
2.6202.3112..000.228) 2.5602.27//..8801199)
2.70(2.34/3. 12/00.3511 2667(2.35/3.0100.55))
LSTOD (complete)
2.4102.16//2.6900111)
Table Evaluation of LSTOD and its variants
2.73
2,67
RMSE
255
Local CNN Size
Figure RMSE on testing data with spect to ACN and standard CNN using different kernel sizes.
Figure 3: Day-wise RMSE comparison between varied STOD models
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Jiangchuan Zheng and Lionel M Ni. Time-dependent trajectory regression on road networks via multi-task learning. In Twenty- Seventh AAAI Conference on Artificial Intelligence, 2013.
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Under review as a conference paper at ICLR 2020
A
NOTATION TABLE
O
OD flow maps of day d, time
Odtt
$
Flow amount from vertex to oj at day d, time
X1, X2
Short-term and long-term spatial-temporal OD flow data
2
long-term spatial-tempont OD flow data of day
Z
Output of n-th ST-Conv block of day d - +
d=p
Zo-4
features of day (d ip) captured by ST-Conv blocks @-th element along the time axis of Za-Ø from v/ to 1y
di
Za-p
day-level output of day d -
ZLT
long-term spatial-temports representations
Z
Zst ① ZLT, combined ppatial-tempon representations
B ILLUSTRATION OF VACN
1
Σndt R ads
Vn
C2 de
Vi
Σ" n=1
C
nf d
R2 dd
n=1
Vm
Figure 5: Working mechanism of spatial adjacent convolution network (VACN) for a target OD flow
from v; to Uj
C TRAINING DETAILS
Batch normalization is used in the VACN component. The batch size in experiment was to 10, corresponding to 10 randomly sampled timestamps and all the 502 OD flows in each snapshot. The initial liearning rate the deep bearning-based methods where the training process is terminated when the RMSE over validation has been improved for 10 successive epochs. The maximal number of epochs allowed is 100.
be 10-4 with decay rate 10-6 We early stopping for all
D TABLE3: COMPARISON WITH STATE-OF-THE-ART METHODS FOR CITY B
COMPARISON OF DIFFERENT HYPERPARAMETER CONFIGURATIONS
In this section, we want to explore how some important hyperparameters of input OD flow data, for example P1 and p2, may affect the performance of our L.TOO model.
12

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Under review as a conference paper at ICLR 2020
Table 3: Comparison with State-of-the-art methods for City B
City
Method
30 min
60 min
90 min
HA
1.9502.78/4.58/2.211
ARIMA
4.8703.81/6.8311.22
5.1443.91/6.4/1.711)
5.3964.0446.65/2.155)
LSVR
1.21(3.19/5.3//1.98
4.87(3.5666.2/33.66)
5.24(3.77/6.83/3.68))
LSM-RN
5.9965.1277.05/2.003 6.7215.7077.742..14)
DLSTM
3.8663.344.37705559 4.04(3.52/460/00.0
190.81 3.17(2.50/3.87/1. .02)
SRCN
2.800.33/3.4/0059) 5.06(2.53/3.7000..8
LSTOD
2.56(2.25/2.8800220))
2.630..911..//02244)
2.990..22/..50003111
Figure (b) compares RMSE on testing data by STOD model with different data settings. Varied combinations of the short-term sequence length p1 and the long-term day-level sequence length P2 are studied. We can see that the best performance achieved as (p1,p2) (7,5) with RMSE 2.41. Specifically, settings with different pi 's under P2 consistently outperform those under P2 = 7. may demonstrate that the shift can usually be captured within a short time range, while longer time sequence may smooth the significance. Table provides the detailed prediction results for each data setting.
Table 4: Comparison of STOD under different P2 combinations
P2
(K LT. K Kir)
Pi
(K ST, K ST)
RMSE
(2, 2)
2.45
(2,3) 3)
2.41
5
(2,2)
9
(3,3)
2.42
11
(3,4)
2.43
13
(4,4)
2.43
(2,2)
2.45
(2,3)
2.44
7
(3, 2)
9
(3,3)
2.44
11
(3,4)
2.44
13
(4,4)
2.49
250
P2 5
P2
2.40
242
2.40
5
7
9
11
Pi length
Figure 6: RMSE on testing data with respect to STOD with different and P2 combinations.
13


Reviews:
[{"review_id": "SkeWVbvntB", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper proposes a  deep learning architecture for forecasting Origin-Destination (OD) flow. The model integrates several existing modules including spatiotemporal graph convolution and periodically shifted attention mechanism. Experiments on a large-scale ride-sharing demand dataset demonstrate improved forecasting accuracy.\n\n+ The spatiotemporal forecasting problem is an important task in the context of ride-sharing platforms.\n+ The description of the model is clear and the real-world dataset experiments are interesting.\n\n- The novelty of the model is rather limited. Spatiotemporal graph convolution has been previously proposed in [Yu et al. 2018] and later studies. Periodically shifted attention is a slight modification of the earlier work.\n- The mathematical notations are messy and redundant. For example, the superscript index i, j can be removed and write all equations in matrix form. The subscript index can be replaced with a range operator. In Eqn (2), r_1 and c_2 are non-identifiable. \n- The experiments are not convincing. The paper is missing the following important baselines:\n\nBing Yu, Haoteng Yin, and Zhanxing Zhu. Spatio-temporal graph convolutional networks: A deep learning framework for traffic forecasting. In Proceedings of the 27th International Joint Conference on Artificial Intelligence, 3634-3640. AAAI Press, 2018.\n\nYaguang Li, Rose Yu, Cyrus Shahabi, Yan Liu. \"Diffusion Convolutional Recurrent Neural Network: Data-Driven Traffic Forecasting.\" International Conference on Learning Representations (ICLR), 2018.\n\nThe authors should provide confidence intervals and visualizations for the predictions.\n\n\n", "extra_scores": {"experience_assessment": "I have published in this field for several years.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I carefully checked the derivations and theory.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly."}, "extra_text": {}}, {"review_id": "S1eFS7t3FB", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Summary: This paper proposes a latent spatial-temporal origin-destination model to address the OD flow prediction problem. \n\nThe main contributions are summarized as follows:\n\n1.\tThe authors propose a purely convolutional framework to learn both short-term and long-term spatio-temporal features simultaneously from dynamic origin-destination flow data.\n2.\tThe authors propose a novel SACN architecture to capture the relevance of OD flows by modeling each OD flow map as an adjacency matrix.\n3.\tThe authors design a periodically shift attention mechanism to model the long-term periodicity.\n4.\tThe results demonstrate that the proposed model outperforms state-of-the-art methods in OD flow prediction, with 6.5% to 15.0% improvement of testing RMSE.\n\nHowever, my major concerns are as follows:\n\n1. On Page 4, the authors explain the reasons why they use TGCNN instead of RNN-based architectures to capture the temporal representation. However, it would be more convincing if quantitative analysis or empirical results are provided.\n\n2. On Page 4, readers might be confused with the symbols in the formulation (3) that are not defined clearly.\n\n3. On Page 7, the experiment only considers one metric, i.e., RMSE. The efficiency comparisons between the proposed model and the baselines are missing.", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper."}, "extra_text": {}}, {"review_id": "H1lEfW8aFB", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "\nReview Summary\n--------------\nWhile I think there are some interesting innovations here, I don't think this is ready for ICLR. I have concerns that the evaluation doesn't focus enough on application-relevant scenarios (why exclude cells that are not in the top 50? why not use 7-day windows to capture day-of-week effects?), and that the evaluations don't quantify uncertainty (and thus the claimed improvements due to attention may not be significant). The presentation quality of the method details in Sec. 3 needs a significant rewrite to improve clarity. I do think the non-standard convolution operator is a nice idea.\n\nPaper Summary\n-------------\nThis paper addresses the forecasting of origin-destination demand data, with a focus on ride-sharing transportation applications. We assume an urban area has been divided up into N cells (known in advance). The problem is to forecast demand for *directed* rides from cell i to cell j, using historical demand data.\n\nThe paper's first contribution is the design of a convolutional neural net architecture that uses non-rectangular receptive fields more suitable to origin-destination data. Instead of assuming that cells in a given 2D euclidean neighborhood are correlated, it assumes that all journeys that overlap with the one of the cells in the current origin-destination pair are similar. They call this the \"SACN\" architecture.\n\nThe second contribution is developing an \"ST-Conv\" block that combines the above SACN convolution with gated temporal convolutions, so that both space and time can be summarized via convolution operators. A periodically-shifting attention mechanism is further used to measure long-term data's similarities to short-term data and weight accordingly.\n\nOverall, these two ideas (SACN convolutions and ST-Conv blocks with attention) are combined into what they call their Latent Spatio-Temporal Origin-Destination or \"LSTOD\" architecture, which is claimed to be the first to use both short-term and long-term features in prediction.\n\nEvaluation examines demand for ride-sharing in two major cities, A and B (presumably masked because they are proprietary). RMSE error comparing to several classic (e.g. ARIMA) and deep feature learning (e.g. LSTMs and SRCN) baselines, with primary results in Table 1. A few side experiments examine superiority over standard convolutions (Fig 3) and experiments without the attention and long-term bits of the model (Table 2). \n\n\nNovelty & Significance\n-----------------------\nThe current paper is quite niche when it comes to significance; I think it may be a bit too specialized for most ICLR readers. Solving this kind of forecasting problem seems important to ride-sharing applications, but is highly specialized for origin-destination demand data. Would be nice to see the paper attempt to connect to other problems beyond ride-sharing (maybe animal migration? maybe package logistics?).\n\nThat said, I think there's sufficient novelty. The architectural design contributions here do appear new to me (though I don't follow this kind of data closely). \n\n\nMethod Concerns\n------------------\n\n## M1: Complexity analysis missing, scaling could be a problem\n\nWhen comparing the current square receptive field architecture for CNNs with the proposed SACN, I felt there was an opportunity to clarify how both scale with N and other key problem size parameters in terms of number of parameters or execution time. I'm concerned that it will be non-scalable given the O(N) cost in terms of number of parameters and the O(N^2) cost of runtime (you need to execute Eq. 2 for each of the N^2 entries at each layer). I'd like to see some careful breakdown of this compared to other approaches. \n\nExperimental Concerns\n---------------------\n\n## E1: Why not use a 7-day past history?\n\nShouldn't ride share demand have a day-of-week trend? Why wouldn't we use last Sunday's demand to predict this Sunday's demand? I find it quite odd that for the Historical Average baseline, only the last 5 days (rather than last 7) are used, and for the presented method, only the last 3 days are used. I'd be happy to be proven wrong, but I'd guess including a 7-day window would lead to noticeably better results.\n\n## E2: Lack of error bars / uncertainty quantification\n\nA natural question is, can we reliably tell that the difference between (for example) RMSE 2.49 and 2.54 is significant and not noise?  I'd like to see some attempt at quantifying the uncertainty for measurements in Table 1 (perhaps taking each full day in test set as its own \"mini\" test set that produces one RMSE score, then reporting average as well as 2.5th and 97.5th percentiles or something across each day). Without this, I think the claim that attention is useful here is unproven, since the change in performance is so small (less than 0.1 RMSE). \n\n## E3: Why focus only on the N=50 most common cells?\n\nI would think to really assess demand forecasting, you want to know when to task drivers to visit less-common cells. The focus on the cells with only the top 80% of journeys means that 1 of every 5 rides would not be covered by this prediction system. I wonder if part of the reason is that using N much larger than 50 is problematic due to the scaling mentioned earlier.\n\n\nPresentation Concerns\n---------------------\n\n## P1: Need to simplify notation\n\nI found the descriptions of the neural net architecture throughout Sec. 3 quite hard to parse. I think there's an overreliance on math notation and there could be more simple description of high level intent and motivation.\n\nFor example, Eq. 2 could be simplified to avoid the channel \"m\" and layer \"l\" notation and thus let the reader focus on what matters, which is how any part of input A that involves the origin node i or the destination node j is included in the weighted sum.  Words can be used around this to clarify this same operation can happen across channels and layers. \n\nSimilarly, Eq. 3 and Eq. 4 could perhaps be replaced by a good diagram. Currently, Eq. 3 both P and Q are undefined and quite confusing.\n\n## P2: Name of the convolution operation\n\nI'm not sure \"SACN\" is the best name. The receptive field of a standard CNN looks much more \"spatially adjacent\" to me (a compact rectangle surrounding the target cell). Instead, you might call it \"topologically adjacent\" or even something like \"vertex adjacent\". You want to emphasize that you are getting strength by finding all journeys whose start or end overlaps with one of the endpoints of the current journey.", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "N/A"}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper proposes a  deep learning architecture for forecasting Origin-Destination (OD) flow. The model integrates several existing modules including spatiotemporal graph convolution and periodically shifted attention mechanism. 

The reviewers agree that the paper is not written well, and the experiments are also not executed well. Overall, we recommend rejection.
Decision: Reject