LSTOD: L ATENT S PATIAL -T EMPORAL
O RIGIN -
D ESTINATION PREDICTION MODEL AND ITS APPLICATIONS IN RIDE - SHARING PLATFORMS
Anonymous authors
Paper under double-blind review
A BSTRACT
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 7.94% improvement in prediction accuracy over the second best model.
I NTRODUCTION
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 ridesharing 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 utilization 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 demand 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-Matias et al., 2013; Shekhar & Williams, 2007;
Id´e & Sugiyama, 2011; Zheng & Ni, 2013).
Most of these models are linear and thus ignore some important non-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 prediction by extracting non-linear latent structures that cannot be easily covered by feature engineering.
(Xingjian et al., 2015; Ke et al., 2017; Zhou et al., 2018).
Zhang et al.
(2016; 2017) modeled the whole city are as an entire image and employed residual neural network to capture temporal 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 relationships.
However, using standard convolution ﬁlters suffers from the problem that some OD ﬂows covered by a receptive ﬁeld of regular CNNs are not spatially important.
Graph-based neural net- works (GNN) (Kipf & Welling, 2016; Defferrard et al., 2016; Veliˇckovi´c et al., 2017) are proved to be powerful tools in modelling spatial-temporal network structures (Yu et al., 2018; Li et al., 2017).
However, none of these frameworks are directly applicable here since both the historical observations 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 Spatial-Temporal Origin-Destination (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 information, we introduce a 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 different regions of the ﬂow map, can be spatially correlated and topologically connected.
Moreover, for most ride-sharing platforms, a passenger is more likely to 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., 2018) and integrate it with VACN by using the sandwich-structured STconv 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 periodicity.
Finally, the combined short-term and long-term representations are fed into the ﬁnal prediction layer to complete the architecture.
Our contributions are summarized as follow:
• 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 a novel VACN architecture to capture the graph-based semantic connections and functional similarities among correlated OD ﬂows by modeling each OD ﬂow map as an 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 LSTOD outperforms many state-of-the-art methods in OD ﬂow prediction, with 7.94% to 15.14% improvement of testing RMSE.
D EFINITIONS AND P ROBLEM S TATEMENT
For a given urban area, we observe a 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 = { v 1 , v 2 , . . .
, v N } denote the vertex set with v i being the i -th sub-region.
The shape of each grid v
i could be either rectangles, hexagons or irregular sub-regions.
We deﬁne the dynamic OD ﬂow maps as { O d,t } , where d = 1 , . . .
, D and t = 1 , . .
, T represent the day and time indexes, respectively.
For each snapshot O d,t = ( o ijd,t ) ∈ R N × N , the edge weight o ijd,t at row i and column j denotes the ﬂow amount from node v i to node v j at time t of day d .
A larger edge weight o ijd,t is equivalent to a strong connection between nodes v i and v j .
The O d,t s’ are asymmetric since all the included OD ﬂows are directed.
Speciﬁcally, o ijd,t =
if there is no demand from v i to v j within the t -th time interval of day d .
The goal of our prediction problem is to predict the snapshot O d,t + j ∈ R N × N in the future time window ( t + 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 p 1 timestamps from t + 1 − p 1 to t , denoted by X
= { O d,t +1 − p 1 , O d,t +1 − p 1 +1 , . .
, O d,t } .
The long-term input data is made up of q time series { O d − ϕ,t
+ j − ( p 2 − 1) / 2 , . . .
, O d − ϕ,t + j +( p 2 − 1) / 2 } of length p 2 for each previous day ( d − ϕ ) with the predicted time index ( t + j ) in the middle for ϕ = 1 , . . .
, q .
We let X
=
{ O d − q,t + j − ( p 2 − 1) / 2 , . .
, O d − q,t + j +( p 2 − 1) / 2 , . .
O d − 1 ,t + j − ( p 2 − 1) / 2 , . .
, O d − 1 ,t + j +( p 2 − 1) / 2 } denote the entire long-term data.
Increasing p 1 and p 2 leads to higher prediction accuracy, but more training time.
We reformulate the set of short-term OD networks X 1 into a 4D tensor X 1 ∈ R N × N × p 1 × 1 and concatenate the long-term snapshots X 2 into a 5D tensor X
=
( X 2 ,d − 1 , . . .
, X 2 ,d − q ∈ R q × N × N × p 2 × 1 .
Each X 2 ,d − ϕ ∈ R N × N × p 2 × 1 for day d − ϕ is a 4D tensor for ϕ = 1 , . . .
, q .
Therefore, we can ﬁnally deﬁne our latent prediction problem as follows:
[Equation]
where F ( · , · ) represents the LSTOD model, which captures the network structures of OD ﬂow data as well as the temporal dependencies in multiple scales.
A notation table is attached in the appendix.
LSTOD F RAMEWORK
In this section, we describe the details of our proposed LSTOD prediction model.
See Figure 1 for the architecture of LSTOD.
The four major novelties and functionalities of LSTOD model include
• an end-to-end framework LSTOD constructed by all kinds of CNN modules to 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 can not be covered by traditional CNNs; • a 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 convolutional ST-Conv blocks to efﬁciently utilize the long-term information by measuring its similarities with short-term data.
3. S PATIAL A DJACENT C ONVOLUTION N ETWORK
Before introducing the detailed architecture of VACN, we want to discuss why directly applying standard CNNs to the OD ﬂow map O d,t may disregard the connections between neighboring OD ﬂows in the graph space ﬁrst.
Figure 2 demonstrates that it fails to capture enough semantic information using the real-world example of ride demands.
For the OD ﬂow starting from v 1 to v 2 , as illustrated in the upper sub-ﬁgure, the most related OD ﬂows should be those with either origin or destination being v 1 or v 2 in the past few timestamps.
A certain part of the travel requests from v 1 to v 2 can be matched with some historical ﬁnished trips from a third-party location to V 1 by the same group of people, for example a trip from v 3 to v 1 .
However, as the lower-left sub-ﬁgure illustrates, some of the OD ﬂows covered by a single CNN ﬁlter (the green square) such as the four corners of the kernel window may be topologically far away from the target one in the graph.
More generally, let’s consider a target
OD ﬂow o ijd,t in a map O d,t .
Most of the components covered by a standard CNN of size q × q with o ijd,t in the middle such as o ( i +1)
( j +1)
d,t are less correlated than some out of the kernel window, for example o ki d,t when | k − j | > ( q + 1) / 2 .
Moreover, if we change the order of the N vertexes in O d,t , then the network structure is unchanged, but a different set of
OD ﬂows is covered by the receptive ﬁeld of the same size with the center being o ijd,t .
As shown in the lower right sub-ﬁgure of Figure 2, OD ﬂows with either origin or destination being v i or v j , covered by the red and yellow kernel windows, are considered to be the most related ones for o ijd,t in row i and column j .
To better understand the differences between our proposed VACN over graph edges with those vertex-based convolutions such as GCN (Kipf & Welling, 2016) or GAT (Veliˇckovi´c et al., 2017), we introduce the concept of line graphs L ( G ) (Godsil & Royle, 2013).
Each node in L ( G ) corresponds 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 a joint vertex.
L ( G ) contains 12 [ N 3 − N ( N − 1)] while there are only 12 N ( N − 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 d,t = { a ijd,t } ∈ R N × N × m 0 consist of m 0 feature maps as the input with each a ijd,t ∈ R m 0 representing a feature vector at the edge from v i to v j .
The learned representation of each target edge is deﬁned as 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 v i to v j is deﬁned as follows:
[Equation]
where [ R n 1 , R n 2 , C n 1 , C n 2 ] = W n ∈ R 4 m 1 × m 0 such that [ W 1 , . . . , W N ] ∈ R 4 m 1 × m 0 ×
N are 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 at v j or end at v i .
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
R n 1 , R n 2 , C n 1 or C n 2 .
We denote the overall output of a multi-layer VACN as Y d,t ∈ R N × N × m s at time ( d, t ) .
Figure 4 in the appendix more clearly illustrates the architecture of VACN.
3. T EMPORAL G ATED CNN
We use temporal gated CNN (TGCNN) (Yu et al., 2018) instead of RNN-based 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 a potential long temporal sequence length make RNNs notoriously difﬁcult to 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 N 2 OD ﬂows along the time axis and the other being a gated linear unit (GLU) as the gate mechanism.
By employing VACN at each of r successive timeslots, we can build a 4D tensor Y =
( Y d,t ) ∈ R N × N × r × m s
which is then fed into the TGCNN operator:
[Equation]
where G ∗ γ represents the TGCNN kernel and γ includes all the related parameters to learn.
2 m t 3D convolutional kernels of size (1 × 1 × K ) with zero paddings map the input Y into a single output [ Y 1 Y 2 ]
∈ R N × N × ( r − K +1)
× 2 m t , which is split in half to obtain Y 1 and Y 2 with the same number of feature channels.
(cid:12) here denotes the element-wise Hadamard product.
3. ST-C ONV BLOCKS
We use the spatial-temporal convolutional block (ST-conv block) to jointly capture the spatialtemporal features of OD ﬂow data, which has a ‘sandwich’-structure architecture with a multi-layer
VACN operator in the middle connecting the two TGCNNs.
The use of ST-Conv blocks have two major advantages.
First, the block can be stacked or extended based on the dimension and characteristic 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 l -th block Z l − 1 ∈ R N × N × r l − 1 × c l − 1 ( Z 0 is the original the OD ﬂow data with c 0 = 1), the output is computed as follows:
[Equation]
where G 1 ∗ γ l 1 and G 0 ∗ γ l 0 are two TGCNN kernels and S ∗ θ l is a multi-layer VACN operator being applied to each timestamp.
The G 1 and G 2 operators from all the stacked ST-Conv blocks employ the same kernel sizes, which are (1 × 1 × K 1 ) and (1 × 1 × K 2 ) , respectively.
Thus, we have r l = r l − 1 − ( K 1 + K 2 − 2) .
After applying ( r 0 − 1) /
( K 0 + K 1 − 2) ST-Conv blocks to the input
Z 0 , the temporal length is reduced from r 0 to 1 .
When input being the short-term OD ﬂow data Z 0 = X 1 ∈ R N × N × p 1 × 1 , we use L 0 =
( p 1 − 1) /
( K 0 ST + K 1 ST − 2) blocks to obtain the spatial-temporal representations
Z ST = f ( Z L 0 ) ∈
R N × N × c ST .
f here squeezes the captured 4D output into a 3D tensor by dropping the temporal axis.
The kernel sizes of each G 1 and G 0 are (1 × 1 × K ST 0 ) and (1 × 1 × K ST 1 ) , respectively.
The detailed propagation of the l -th ST-Conv block is deﬁned as
[Equation]
3. P ERIODICALLY S HIFTED A TTENTION M ECHANISM
In addition to capturing the the spatial-temporal features from short-term OD ﬂow data X 1 , 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 travelling schedule and the city’s trafﬁc conditions.
Directly applying ST-Con blocks to 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 p 2 time intervals for each day d − ϕ when predicting the time window ( d, t + 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 et al., 2015; Yao et al., 2018a; Liang et al., 2018) in spatial-temporal prediction problems, we propose a modiﬁed periodically shifted attention to work for the CNN-based ST-Conv blocks here.
Different from Yao et al.
(2018a) that measures the similarity between hidden units of LSTMs, the attention here is built on the intermediate outputs of TGCNNs where the concatenations are then fed into a new set of ST-Conv blocks.
For each day ( d − ϕ ) , we apply a series of L 1 ST-Conv blocks to the day-level p 2 -length sequential OD ﬂow data X 2 ,d − ϕ and reduce the temporal length from p 2 to n 0 LT .
Each block contains two TGCNN layers with the same kernel size 1 × 1 × K 0 LT , such that
[Equation] [Equation]
and the propagation rule of the l -th ST-Conv blocks is deﬁned as:
[Equation]
7) with Z nd − ϕ and Z n +1 d − ϕ representing the input and output of the l -the block.
We denote the day-level representations of day ( d − ϕ ) captured by the L 1 ST-Conv blocks above as ˜ Z
d − ϕ where each ˜ z ijd − ϕ,φ represents the φ -th element along the time axis for the OD ﬂow from v i to v j .
On the other hand, we let z ij ST be the learned short-term representation at the OD ﬂow from v i to v j .
Subsequently, a re-weighted day-level output z ijd − ϕ can be obtained by summing up ˜
z
ijd − ϕ,φ ’s using weights β ijd − ϕ,φ which measure their similarities with the short-term representation z ijST :
[Equation]
where β ijd − ϕ,φ quantiﬁes the similarity between ˜
z ijd − ϕ,φ and z ijST based on a score function score (˜ z ijd − ϕ,φ , z ijST ) , which is deﬁned as:
[Equation]
Moreover, score (˜ z ijd − ϕ,φ , z ijST ) is deﬁned as
[Equation]
where W 1 , W 2 and v φ are learned projection matrices.
b s is the added bias term.
By assuming that
Z d − ϕ
=
( z ijd − ϕ ) denotes the day-level output for all the N 2 entries after re-weighting, we can build a new day-wise time series Z 0 LT of length q by concatenating all the Z d − ϕ ’s along an additional axis in the third dimension as
[Equation]
to build a new 4D day-wise time series Z 0 LT . and ﬁnally apply another set of ST-Conv blocks to it to obtain the long-term spatial-temporal representations, which is denoted by Z LT ∈ R N × N × c LT .
c LT is the number of feature channels.
We concatenate the short-term and long-term spatial-temporal representations Z ST and Z LT together along the feature axis as Z = Z ST ⊕ Z LT ∈ R N × N ×C , where C =
c
ST + c LT .
Then, Z is modiﬁed to a 2D tensor Z ∈ R N 2 ×C 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 to get the ﬁnal predictions for all the N 2 OD ﬂows.
3. D ATA P ROCESSING AND T RAINING
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 are saved and used to denormalize the predictions of testing data to get the actual ﬂow volumes.
We use L 2 loss to build the objective loss during the training.
The model is optimized via Backpropagation Through Time (BPTT) and Adam (Kingma & Ba, 2014).
The whole architecture of our model is realized using Tensorﬂow (Abadi et al., 2016) and Keras (Chollet et al., 2015).
All experiments were run on a cluster with one NVIDIA 12G-memory Titan GPU.
E XPERIMENT
In this section, we compare the proposed LSTOD model with some state-of-the-art approaches for latent trafﬁc ﬂow predictions.
All compared methods are classiﬁed into traditional statistical methods and deep-learning based approaches.
We use the demand ﬂow data collected by a ride-sharing platform to examine the ﬁnite sample performance of OD ﬂow predictions for each method.
4. D ATASET D ESCRIPTION
We employ a large-scale demand dataset obtained from a large-scale ride-sharing platform to do all the experiments.
The dataset contains all customer requests received by the platform from 04/01/2018 to 06/30/2018 in 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 N 2 = 2500 OD ﬂows.
Since one-layer VACN has a computation complexity O ( N ) at each of the N 2 entries (globally
O ( N 3 ) ), the memory consumption highly increases as N gets bigger.
Considering the computation efﬁciency and storage limitation, we choose N = 50 here which can cover more than 80% of 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 06/17/2017 to 06/30/2017 (14 days) serves as the testing 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 v i to v j is the cumulative number of customer requests.
We make predictions for all the 50 2 OD ﬂows in the incoming 1st, 2nd, and 3rd 30 minutes (i.e. t + 1 , t + 2 , t + 3 ) by each compared method, given the historical data with varied ( p 1 , p 2 ) combinations.
For those model settings incorporating long-term information, we trace back q =days to capture the time periodicity.
We use Rooted Mean Square Error to evaluate the performance of each method:
[Equation]
o ijd,t and (cid:98) o ijd,t are the true value and prediction at the OD ﬂow from vertex v i to vertex v j at time ( d, t ) , respectively.
T 0 is the set containing all the predicted time points in the testing data.
4. C OMPARED M ETHODS
All state-of-the-art methods to be compared are listed as follows, some of which are 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 integrated moving average (ARIMA) , (iii) Support Vector Machine Regression (SVMR) , (iv) Latent Space Model for Road Networks (LSM-RN)
(Deng et al., 2016), (v) Dense + BiLSTM (DLSTM) (Altch´e & de La Fortelle, 2017) and (vi) Spatiotemporal Recurrent Convolutional
Networks (SRCN)
(Yu et al., 2017).
We only consider latent models in this paper, that is, no external covariates are allowed, while only the historical OD ﬂow data is used to extract the hidden spatial-temporal features.
4. P REPROCESSING AND P ARAMETERS
We tune the hyperparameters of each compared model to obtain the optimal prediction performance.
Speciﬁcally, we get ( p ∗ , d ∗ , q ∗ ) =
(3 , 0 , 3) for ARIMA and k ∗ = 15 , γ ∗ = 2 − 5 , λ ∗ = 10 for LSMRN.
The optimal kernel size of the spatial based CNN kernel is 11 × 11 in SRCN model.
For fair comparison, we set the length of short-term OD ﬂow sequence to be p 1 = 9 (i.e., previous 4.5 hours), q = 3 for long-term data which covers the three most recent days, and the length of each day-level time series p 2 = 5 to capture the periodicity shifting (one hour before and after the predicted time index).
More analysis of how variant ( p 1 , p 2 ) combinations may affect the prediction performance of LSTOD will be studied latter.
A two-layer architecture is used by all the deep-learning based methods to extract the spatial patterns inside the OD ﬂow data (L = 2 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 c ST = c LT = 64 .
4. R ESULTS
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 theday-wise RMSE on the testing set.
LSTOD outperforms all other methods on the testing data with the lowest average day-wise RMSE (2.41/2.55/2.67), achieving (8.02%/7.94%/8.24%) 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 our 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
[Table]
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 B may be more signiﬁcant 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 proposed 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 3 compares the predictions by each model against the true values at two selected OD ﬂows on the last 3 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 thetesting 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 outperforms 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 × 4 = 200 .
For fair comparison, the largest receptive ﬁled of standard CNN should be no bigger than a 15 × 15 window, which includes 225 elements each time.
We consider ﬁve different kernel sizes including 5 × 5 , 8 × 8 , 11 × 11 , 14 × 14 , and 15 × 15 .
We replace VCAN in our model by standard CNN in order to fairly compare its performance.
All hyper-parameters are ﬁxed but only the kernel size of
CNNs being changed.
Moreover, we only consider the baseline short-term mode of LSTOD model while ignoring the long-term information.
As Figure 4 illustrates, standard CNN achieves the best performance with the smallest RMSE = 2 .
64 on testing data with the ﬁlter size being 11 × 11 , which is still higher than that using VACN with RMSE = 2 . 54 .
Speciﬁcally, RMSE increases when the receptive ﬁeld is larger than 11 × 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 − 1) / 64 > (14 × 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 4.
As we mentioned above, one-layer VACN has a global computation complexity O ( N 3 ) at each timestamp.
For standard CNN, O ( N 2 ) executions are still needed by applying local CNN ﬁlters to
N 2 windows with each of the N 2 entries in the middle.
Therefore, when the side length of the CNN ﬁlter is in the order O ( √ N ) , the total cost of our VACN and standard CNN are in the same order.
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B I LLUSTRATION OF VACN
C T RAINING D ETAILS
Batch normalization is used in the VACN component.
The batch size in our experiment was set to 10, corresponding to 10 randomly sampled timestamps and all the 50 2 OD ﬂows in each snapshot.
The initial liearning rate is set to be 10 − 4 with a decay rate 10 − 6 .
We use early stopping for all the deep learning-based methods where the training process is terminated when the RMSE over validation set has not been improved for 10 successive epochs.
The maximal number of epochs allowed is 100.
D T ABLE 3: C OMPARISON WITH S TATE - OF - THE - ART METHODS FOR C ITY B E C OMPARISON OF DIFFERENT HYPERPARAMETER CONFIGURATIONS
In this section, we want to explore how some important hyperparameters of input OD ﬂow data, for example p 1 and p 2 , may affect the performance of our LSTOD model.
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Figure 6 (b) compares RMSE on testing data by STOD model with different data settings.
Varied combinations of the short-term sequence length p 1 and the long-term day-level sequence length pare studied.
We can see that the best performance is achieved as ( p 1 , p 2 ) =
(7 , 5) with RMSE = 2.41.
Speciﬁcally, settings with different p 1 ’s under p 2 = 5 consistently outperform those under p 2 = 7 .
It may demonstrate that the shift can usually be captured within a short time range, while a longer time sequence may smooth the signiﬁcance.
Table 4 provides the detailed prediction results for each data setting.
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