Explainable Spatio-Temporal Forecasting with Shape Functions
Authors: Xianbin Cao
Vy Vo
Tingjin Chu
Guoqi Qian
Mingming Gong
Venue: NeurIPS 2022 Submitted (2022)
Keywords: Spatio-temporal Forecasting
Shape Function
Spatial Weight Matrix

Abstract: Spatio-temporal modelling and forecasting are challenging due to their complicated spatial dependence, temporal dynamics, and scenarios. Many statistical models, such as Spatial Auto-regression Model (SAR) and Spatial Dynamic Panel Data Model (SDPD), are restricted by a pre-specified spatial weight matrix and thus are limited to reflect its flexibility. Graph-based or convolution-based methods can learn more flexible representations, but they fail to show the exact interactions between locations due to the lack of explainability. This paper proposes a spatial regression model with shape functions to address the limitations of existing methods. Our method learns the shape functions by incorporating shape constraints, which are able to capture spatial variability or distance-based effects over distance. Therefore, our approach enjoys a learnable spatial weight matrix with a distance-based explanation. We demonstrate our method's efficiency and forecasting performance on synthetic and real data.

Explainable Spatio-Temporee Forecasting with Shape Functions
Anonymous Author(s)
Affiliation Address email
Abstract
Spatio-tempone modeling and forecasting are challenging due to their complicated spatial dependence. temporal dynamics. and scenarios. Many statistical models, such as Spatial Auto-regression Model (SAR) and Spatial Dynamic Panel Data Model (SDPD), are restricted by a pre-specified spatial weight matrix and thus are limited to reflect its flexibility. Graph-based or convolution-basee methods can learn more flexible representations, but they fail to show the exact interactions between locations due to the lack of explainability. This paper proposes spatial gression model with shape functions to address the limitations of existing methods. Our method learns the shape functions by incorporating shape constraints, which are able to capture spatial variability or distance- based effects over distance. There- fore, our approach enjoys learnable spatial weight matrix with distance-based explanation. We demonstrate our method's 's efficiency and forecasting performance synthetic and real data.
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Introduction
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Spatio-temporal data is widely observed in many areas, such as transportation 33
climatology
(), and environmental research(19). The popularity of spatio-temporal data brings varieties of tasks for researchers, and one of the key tasks forecasting. Spatio-temporal data has some inherent characteristics, namely, spatial dependence and temporal dynamics, which need be considered for modeling and forecasting.
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Spatial dependence means that the observations at different locations not independent, and observations at closer locations often have a stronger correlation. In the statistics community, extensive research has been conducted model spatial dependence, and various spatial models have been proposed. For example, in the spatial autoregressive (SAR) models, the spatial dependence is modeled product of an unknown parameter and pre-specified spatial weight matrix [ 11] (2). Combined with the panel data, various types of spatial panel data models have been used to analyze spatio-temporal data (35) 13 7 22). One limitation of the autoregressive models is that the elements of the spatial weight matrix are pre-specified, such as an inverse distance. Although these pre- specified spatial weight matrices are applied capture decreased distance-based effects, they fail capture complex distance relations in real-world applications.
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Researchers in the computer science community have developed various methods modeling spatio- temporal data using deep neural networks. Various neural network architectures have been proposed and applied to spatio-temporal forecasting, for example, Dpatio-temporal LSTM (3I), fully connected
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39 gated graph architecture (20). Convolutional LSTM (23) and etc. One advantage of these methods that they can incorporate unstructured data and rely high-performance computing platform
35 to learn complicated representations for spatio-temporal problems. However, a critical limitation of
36 these methods is that they fail to explain how the spatial interaction works explicitly. The lack of
Submitted to 36th Conference on Neural Information Processing Systems (NeurIPS 2022). Do not distribute.

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interpretability restricts its reliability and deep insights into the underlying spatio-temporal process. The explanation can be obtained if we can estimate the coefficient matrix that intuitively explains spatio-temporal interactions.
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In this paper, we propose Explainable
Forecasting (ESTF) model, which utilizes a spatial autoregressive model with shape functions to address the current limitations. Our method extends the vector autoregressive (VAR) model (24) by incorporating distance information into the temporal coefficient matrix using shape functions (3) The shape constraints are designed to be
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consistent with the common fact that observations from neighbours have stronger spatial dependence versus pongdiitance pairs. It is known as Tobler's ' First Law, which is "Everything is related to everything else, but near things are more related than distant things" [8). Unlike the pre-specified spatial weight matrix, this coefficient matrix is learnable and is thus more flexible in capturing real-world complex spatial relations. Moreover, the shape functions are represented as a combination of basis functions, and thus smaller number of parameters needs to be estimated. Finally, ESTF can
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We conduct experiments on both simulated and real data, and the results demonstrate that our method achieves better forecast accuracy and is computationally efficient and more explainable.
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Related work
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Statistical models Several works focus on temporal dynamics when considering spatio-temporal
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forecasting problems. The classical time series models, such as VAR, and ARIMA models, are applied to spatio-temporal process modeling (1 38). Besides, a spatial weight matrix is also introduced to the ARIMA model to capture spatial dependence (28). The non-stationarity, particularly unit-root non- stationarity, is mainly modeled by ARIMA or Co-integration models. In addition, spatial regression models or panel data are classical models in econometries and can also be applied to model spatio- temporal problems. These models, for example, spatial auto-regression models, take spatial weight matrix into consideration and estimate parameters in the framework of regression. However, the common characteristics of these models need a pre-specified spatial weight m35 (). Elements in the matrices are generally an inverse distance of corresponding locations. Meanwhile, these spatial models focus statistical inference on the scalar parameters placed before the spatial weight 25). Although there are many choices for the spatial weight matrix, such as inverse distance, adjacency relationships, and K-nearest neighbors, there is a lack of research on estimating the spatial weight matrix. The pre-specified spatial weight matrix restricts models' application and fails to capture more complicated underlying spatial dependence. Some researchers developed sparse spatio-temporal model that can estimate a sparse spatial weight matrix (17). The strict sparse setting also restricts the wide application of the spatial weight matrix.
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71 Graph-based methods Graph-based methods are widely applied for a non-Euclidean domain. Some types of spatio-temporal data, for example, traffic flow data brain network data, can be 73 represented as graphs. The graph structures well model the complicated spatial dependence. Thus, the definition or pre-specified graphs structure is normally required when developing a graph-based
75 model. Related works can be found in (30 [4). The common typical method is GraphCNN, which is
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apply a convolutional transformation to the neighbors of each node (29 34). The graph convolution can capture patterns and features in the spatial domain. Graph-based methods have been proposed and widely applied to lots of real cases. Traffic flow data modeling and forecasting a popular topic in this area (30) 20). Other topics, for example, climate sensor data (16), video (10) and etc, are also applied by variant graph- based models. RNN or LSTM combined with graphs, i.c., a sequence of graphs, are also considered in spatio-temporal forecasting problems (10).
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CNNebased methods Unlike graph-based methods, CNN-based methods are more suitable for modeling spatio-temporal data collected in regular grid locations. It applies filters to find relationships between neighboring inputs. Although some works (32) applied convolution neural networks to model non-grid traffic data, it is more common to see CNN-based methods process grid structures, e.g., images, video rather than a general domain. As some spatio-temporal data are collected from a regular grid in the Fuclidean space (29), they thus can be viewed as a kind of special image. The CNN 88 structure combined with RNN or LST has been developed to make forecasting for spatio-temporal data, for example, diffusion convolutional RNN (15), Convolutional LSTM networks (23) 36)and etc.
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90 3 Proposed method
3.1 Problem formulation and notation
92 We use a n x 1 vector X, = (Xit: x ", xut) to denote observations time 4, where n the number of locations. At each location i, Si (c c² is the coordinates of the location d. The distance between location S and Sj is de (d + (dl)), where dij = | s51 and c cg Our goal is to make forecasting for spatio-temporal data: given training data set X XT, we would like to make forecasting for the next h, ,, ,XTth-
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The stationary spatio-temporal model with shape functions
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We first consider the stationary case. To model the spatio-temporal stationary process, we consider
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the following model
X, x, Σ W
(1)
k=1
where Wk is spatial weight matrix for capturing the spatial dependence at lag k; and et is white noise. Moreover, we assume the (i,j)th element of Wk. w depends on the distance diy. That is, depends on function fi
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For spatio-temporal data, the spatial dependence, represented by wi between locations decreases as the distance between two locations increases. In other words, there is a shape constraint for the function fú (d), such a decreasing function. In order to estimate the shape function, we model (d) as a linear combination of basis functions ge(d), i = 1, 2, . m. More specifically, the shape function fa (d)) is linear combination of basis functions and coefficients with positive value (d) = ((,)
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.... + "m (4), where 01,k; Om, are parameters to be estimated.
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The constraint decrease needs parameters non-negative and thus each parameters squared. The spatial weight matrix can take the value of decreased shape function directly. The element of W
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is w = fx(de)) The details of the shape function and the corresponding basis functions can be found in Section 3.4
The parameters shape functions can be estimated from the neural network illustrated in Figure The neural network can be trained from the following criterion:
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P
ΣIX 1 Σ X
Σ
Wv Wr
min (WE
k=1
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3.3
The non-stationary spatio-temporal model with time-variant shape functions
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The static spatial weight matrix Wk can reflect spatial dependence and thus can be applied to stationary scenarios. Next, consider the nonstationary case. Therefore, we extend the stationary model to non-stationary cases. The spatial weight matrices only reflect static relationships across time lags in the static model. Unlike these settings, we change spatial weight matrices be time-variant.
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The spatial weight matrices formed by time-variant shape functions can thus capture non-stationary dynamic spatial dependence. The ota-stationary model has the form below,
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Σ W k=1
X,
where E is white noise, and W. W.k relies on shape function (d). Similar with stationary settings, the time-variant shape functions are still represented as a linear combination of basis functions ( = 1.2.... m. The coefficients are therefore time-variant. The shape function time has the form below (i, = af,,,01 (d) +... + mm.1999m(d Unlike stationary setting, the coefficients
of nonstationary setting, {a,,, ) depend on the time f. The non-stationary model can be trained from the criterion by minimizing
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min (W.+))
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The networks for the stationary model as well as the non con-stationary model are presented in the Figure
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The neural network for the stationary spatio-temporal process (left) and non-stationary spatio-temporal process (right).
Figure 1:
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3.4
The basis functions for shape functions
The shape functions are integrated into our model to obtain distance-based explanations in stationary and non-stationary scenarios. The motivation of the proposed shape functions is that as the distance between two observations increases, the effects between these two locations decreases. These distance- based effects can be reflected in spatial weight matrix W and each element in the matrix can measure how the corresponding locations interact. The shape function is represented a linear combination of basis functions. The basis functions, satisfying shape constraint, rely on the corresponding definition of basis functions.
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Definition of basis functions for various shape constraints. We list the definition basis functions for increased and decreased shape (3). The distance quantile among {di diggt21>' diy di at quantile level ()1, 42, 9m is denoted by {d(1), d(2). ..., d(m)) where Q < q2 < < Qm S
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1 and {g1, 42, ... 4m } ={ mm 1}. Here, we can set the number of m << n², and thus, the number of parameters is significantly reduced.
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For the constraint of monotone decreasing function, the basis function is defined as g, (d))) = 1 (d The basis function for the shape function with the constraint of concave decrease defined as (d) = (d(a) d)
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1 (de sd) and convex decrease is defined as gi(d) (d() d) (d ) for 1 < i < m. Figure 2 shows the definition of basis functions for monotone decreased and increased
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shape functions, respectively. We only present four basis functions for each shape and each of them
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is related to four quantile levels. The dashed lines indicate the turning points for each basis function
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and they equal one or zero at the beginning and turn to zero or one at turning points.
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Figure 2: The basis functions for decreased shape (left) and for increased shape (right). The arrows indicate domain of each basis functions.

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3.5
Model forecasting
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The stationary model requires fixed shape functions and related spatial weight matrix are time- invariant. Given training data set X2. ,r,, we can estimate spatial weight matrix
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WT. W2. W and make forecasting iteratively. That is XT+1= E WXXT+1-k; X7+2=
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W, W1XT+ 1+1 + Ex= WxXT+2-k; Xr+h= Σi=1 W
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The non-stationary model incorporate time variant spatial weight matrix We,. Given the training data set X1, x2.... ;,. XT, we can obtain corresponding shape functions fi f.. Sr... where denotes time lag. For lag p I, we can use (f:) to represent time-variant shape functions for convenience. We can make dynamic forecasts for the next windows. One simple forecasting method is to use Wr.k to make forecast for NT+h. that is
X = Σ WT.
k=1
The alternative method is to retrain the new forecast to obtain the latest shape functions as well as spatial weight matrix. Given long-term forecast window I,, we first make short-term forecast for steps
X = Σ WT
where h 2, ... and WT+1,k is estimated by training forecast value of XT+ We repeat the process until L steps in total have been predicted.
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We summarize the whole process of our model when making spatio-temporal forecasts.
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Step
Given the observation (X+)T= and its coordinates, calculate all distance pairs among all locations, denoted by {di ... di
Step2 2 Calculate { m. ".,,,, 11 quantile levels and obtain corresponding distance quantile value {d(t), d(2).. .,, dd()))
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Step3 Determine the shape constraints and construct corresponding basis functions. Specify the time lag p.
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Step Train the model according to the illustration of Figure
Experiment
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In order to assess our model in stationary and non-stationary scenarios, we synthesize data. Then, we apply our model to make some comparisons. On the one hand, we need evaluate how the estimated shape functions look and assess their similarity and accuracy. On the other hand, our model can make spatio-temporal forecasting after estimating for spatial weight matrix. The basic idea for completing the two goals is to set up the expected shape function and compare estimated parameters with real one. Next, we assess the forecasting performance with baseline models. Codes and data for replicating our experiments are anonymously published at https / /anonymoos. 4open. science////TVAR--FL
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4.1
Simulation for stationary model
Here, we synthesize 100 stationary spatio-temporal data sets, The spatial domain consists of 30 locations and their coordinates can be found at STVAR-F16E/ For each location, we observe 500 values. The observation is generated from the stationary model X, Σi=1 W + Ez, where e, is randomly generated from the standard normal distribution. The next step is to construct random spatial weight matrices for each synthesized data set. The shape functions are set to be decreasing, and we set them as a logarithmic function:
4open.science//1
+ 1) + loo(lo00))
167 where a is randomly generated from uniform distribution 10.05,0.06) but kept to be fixed for each simulated data set. We use d - 1 to avoid zero value. This setting can make the real shape function
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decrease and make it equal to zero when d = 169. The stationary model can iteratively generate the X, given initial value Xo. where X. randomly generated from a uniform distribution with bounds [-0.01,0.01]. The time lags are set as p
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Estimation for shape functions. In Figure 3
Stationary Simulation
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the estimated shape function is presented in red, while the real shape function is presented in blue.
Actual 5F Estimated
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It can be seen that the estimated shape function
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can capture the trend of the real shape function.
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Training details. The first 300 steps are used as training data, saving the last 200 steps for eval- uation. We train all models for 100 epochs with of 0.01. The process involves parallel training across 10 CPUs. We select 100 quantile levels, and thus
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Adam optimizer (5) and learning
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Distance
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100 basis functions (i(d) were generated as the
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inputs for the model.
Figure 3: The sample of estimated shape function.
Distances are shown every 20th quantile.
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Assessment for forecasting. We assess the fore- casting performance for the stationary model with baseline models. As introduced in the liter-
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ature review, the baseline models are selected from the VAR model(21), the spatial panel dPta
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model that applied pre-specified spatial weigh matrix (28), graph-based models (20; 37 and convolution-based models (15)
(3). The enror metrics are mean absolute error and root mean squared /
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+I
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error defined by No respectively. The Table I shows the six baseline models with the proposed model. As totally we have 100 synthesised data sets, x and X denote i th variable in j th data sets. = 30 is the number of locations and N 100 is the number of synthesised data. We conducted one-step forecasting for the next 200 observations.
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Compared with baseline models, the proposed model performs better under the metric MAE and RMSE The proposed method outperforms the closest competing method, DC-RNN, by 10%.
4.2
Experiments for non-stationary model
We conduct simulation for the non non-stationary model with time lag p 1 and synthesize 100 data sets using a similar approach to the stationary model simulation. The initial value Xo and e, are generated from uniform and normal distribution respectively. The locations of observations same as those in the stationary model simulation. In order to construct W., the time-varying shape functions are created under the decreased constraint. The shape function time t is constructed as
the
Re(-log(de + 1) + (170)),)),
where controls the level of value at each time l. éz is generated from normal distribution. Xo is generated from a uniform distribution with bound [-0.001,0.001].
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Shape functions settings and estimation. The shape functions are set as time-variant, as they can simulate the non-stationary process across time. We specified ag 0 from uniform distribution
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[1 x 10-4.2 222 x 10 and then make an interpolation from Qg to C5000 The total length for every E O = 0.0001(1 - ))= 0.001 7. where T = 500. This setting guarantee that shape functions vary
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location is 500 and we set aisu = 10 x ag- For example, generally og 0.0001, we have
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from lower level to higher level. The larger a is, the more larger distance-based based effects they have. Thus, the corresponding spatial weight matrix consists of dynamic shape functions and can reflect the non-stationary dependence among each site. We present the estimated shape functions in Figure and compare them with the real ones.
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Figure 4: The sample of estimated shape function for the 120 testing time steps. Distances are shown every 40th quantile.
Training details. Similar the stationary simulation, the train-test split is 300 -
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size of 500. However, we train all models for 100 epochs with Adam optimizer (5) at a learning rate of 0.001. We train models in parallel across 10 CPUs.
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Forecasting performance. The forecasting performance is assessed by the same metrics used in the previous simulation for the stationary case. We made one-step forecast by our model. As for the baseline models, we adjusted their published code accordingly. The results show that the proposed model can still capture non-stationary processes compared with baseline models. The proposed method outperforms the other competing methods. The error metric shown in Table
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Table 1: The error metrics with baseline models for simulation.
Stationary Simulation
Non-stationary Simulation
Methods
MAE
RMSE
MAE
RMSE
VAR
2.9611 + 1.8573
3.2588 + 1.8077
2.4426 + 1. 1.2285
2.7676 + 1.2015
SPM
1.8850 0.6348
1.8671 0.6778
2.1918 + 0.7350
2.2161 + 0.6876
DC-RNN
0.8960 + 0.0370
1.1168 0.0426
0.9017 0.0358
1.1328 = 0.0463
FC-GAGA
2.5425 + 0.2965
3.1066 0.3633
1.0270 = 0.0080
1.2939 = 0.0120
GMAN
1.6806 0.1491
1.9293 + 0.1483
1.5714 + 0.1104
1.8608 0.1155
ConvLSTM
2.9495 = 0.2980
3.2509 = 0.2887
2.2478 = 0.2295
2.5469 + 0.2324
ESTF
0.7997 + 0.0015
1.0017 + 0.0016
0.8075 + 0.0016
1.0112 + 0.0020
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4.3
Real case studies
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Air quality data. We apply our model air quality data, which records air quality in California over
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2021
The daily mean of PM 2.5 is recorded across 172 sites.
We obtain the first 200 steps for training and perform forecasting for the next 165 steps. All models
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are trained for 100 epochs using Adam optimizer (5), at a learning rate of 0.01 and batch size of
50. We present the estimated time-variant shape functions in supplemental file. The value of shape
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functions decays to zero at around 5.926, which 80% quantile in the sample of distance pairs. In
other words, the distance-based effects decay to zero at distance equal or larger than 5.926. Our
https///www.ppgggoootttoooo airquuuuooooooruuiiittttatiaaat dat./lloddddt//ooloaddddd daily
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model has ideal performance with low time consummation compared with baseline models. We put detailed forecasting results of simulation and real cases supplemental file.
MAE
RMSE
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CPU time (S)
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Error Value
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CPU times
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Quantile Value
Figure 6: The significant distance-based effect Figure 5: Comparing efficiency vs. performance among 30 locations.
trade-off at different quantile values.
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The result is shown in Table 2 The ESTF performs best in terms of RMSE, while the DC-RNN method performs best in terms of MAE. For the computational time, the ESTF method is significantly faster than most machine learning methods, and only takes around 1 /00 time of DC-RNN, In Figure MAE, RMSE, and time are presented with different numbers of m. As m increases, the computational time increases while both MAE and RMSE decrease. There a significant increase in the forecasting performance when m increases from 10 to 50. For m 50, the forecasting performance does not increase much as m increases.
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One key advantage of the ESTF method is that we can make an explicit distance- explanation for our dataset. Figure shows the distance-based effects at time 9. We only present the effects using& threshold to obtain more concise visualization. The estimated shape function fa ranges from to 9.8 and we set 5 as the threshold. The red line indicates the value of the shape function larger than 7, while the gray line indicates the value between and 7. Figure[Jshow how any two locations interact and measure the distance-based effects quantitatively. For example, air quality monitoring sites around the Greater Los Angeles(red circle Figure area have a strong spatial interaction with each other, such node 7 and node 8.
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4.4
SO₂ data
Texas the second largest manufacturing state in the USA and prediction for SO₂ is critical task for researchers. The data records daily SO2 at 31 locations in 2021. More detailed spatial information 246 can be found in the supplemental file. The numeric result is listed in Table
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Table 2: The metrics with baseline models for real case study. Clock time (in seconds) for real case study is recorded when training each model for 100 epochs on a single CPU.
Methods
Air quality data Training time (6)
SO₂ data
MAE
RMSE
Inference Time
MAE
RMSE
Training Time (8)
Inference Time (s)
VAR
16.9844
22.3410
3.56
0.04
6,2705
9.1388
3.330
0.016
SPM
8.4547
13.8262
0.31
0.03
7.1453
9.1086
0.143
0.027
DC-RNN
4.7157
9.3873
203
1.211
3,5094
6.8681
264.215
1.366
FC-GAGA
7,8671
18.1870
181
2.759
4.5976
7.7528
169.425
2,889
GMAN
12.5268
17.3817
1.823
172.016
1,581
ConvL.STM
12,6292
17.91499
53
1.940
4.1445
8.0688
96.233
1.656
ESTF
5.2237
9.2169
1.625
4.2966
6.8307
31,050
1.868
Similar conclusions be drawn in SO₂ data that of air quality data. The ESTF model performs best under the RMSE metric, while DC-RNN is best in the MAE metric. In terms of training time,
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the proposed ESTF method costs around 10% of that of DC-RNN. For the inference time, ESTF and DC-RNN are comparable.
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RMSE
CPU
Emor Value
CPU times ($)
330
25
Quantile Value
Figure 8: The significant distance-based effect Figure 7: Comparing efficiency vs. performanceamong all 31 locations 90
trade-off at different quantile values.
The efficiency analysis and performance at different quantiles are shown in Figure 7 Together with Figure we can see that the increasing number basis functions does not have much improvement when the number of basis functions is larger than 50, while the training time increases as the number of basis functions increases. The spatial distribution time t is presented Figure & where coordinates are denoted by latitude and longitude. Two significant clusters, representing Houston and Dallas respectively, have the strongest distance-based effect. It quantitatively shows how these neighbors affect each other. Counties around Dallas-Fort Worth metropolitan area show strong interaction, which should be noted by environmental policy-makeen More detailed results are presented in the supplemental file.
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Discussion
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This paper applies learnable shape functions to capture distance-based effects. It can model dynamic spatial dependence for stationary and non-stationary spatio-temporal data based on their distance. The model does not have the limitations of classical statistical spatial models and provides a more explanatory model than usual deep learning methods. Furthermore, some spatio-temporal data, such as temperature for sea surface and air quality monitoring data, usually viewed as collected from the continuous field, more suitable for the proposed models since these kinds of data follow the basic rule that variability between two locations is significantly affected by their distance. However, some spatio-temporal data, such as traffic flow or some biology data, do follow the rule. As result, the spatial dependence may rely on road structure or biological mechanisms instead of distance. It is worth researching such data by considering graph structure when estimating spatial weight matrix. In addition, we can develop
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causal inference based on the ESTF model. Grander causal analysis can be done by fitting the first-order VAR model (24). The estimation of the coefficients matrix of the VAR model attracts researchers' interest as it can be treated as a causal transition matrix. the causal inference community, lots of work have been conducted on the VAR model 8 19. However, there is a lack of research on causal inference under the spatio-temporal process.
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The quantitative distance-base based effects in ESTF can be further researched and extended to develop a spatio-temporal causal model.
278 References
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[1] ANSELIN, L. Spatial econometrics: methods and models, vol. 4. Springer Science & Business Media, 1988.
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[2] CASTRUCCIO, S., AND GENTON, G. Principles for statistical inference on big spatio- temporal data from climate models. Statistics & Probability Letters 136 (2018), 92-96.

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Reviews:
[{"review_id": "zmZsujkqNJZ", "reviewer": "Reviewer_JW2N", "summary": "This manuscript focuses on the patio-temporal forecasting problem and proposes a new method with shape functions toward a learnable and explainable model. The proposed method extends the statistical models via learnable bash shape functions, making it possible to capture the spatial variability and distance-based effects over distance. The proposed method is evaluated on both real and synthetic datasets.", "questions": "- It would be interesting to add more discussion about complex shape functions.\n\n- For the evaluation, it is also suggested to add the inference time beyond the training time, which is more important in real scenarios.\n", "limitations": "Yes", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Pros:\n\n- The idea of modeling the spatial weight matrix with learnable shape functions is novel and interesting. It generalizes the traditional statistical method by introducing more learnable components.\n\n- Experimental results on both synthetic and real datasets indeed show the effectiveness of the proposed method. It even performs better or is comparable with respect to some DNN-based methods under some scenarios.\n\nCons:\n\n- The proposed method is only evaluated on a single real-world dataset, making it hard to figure out if the method can consistently work well under real setups.\n\n- The authors provide several specifications of the relations between the distance and the weights. However, the authors did not discuss the advantage or disadvantages of each specification. On a given dataset, how should we choose among them?\n\n- According to section 3.4, the basis function for the shape constraints is simple. There is no discussion about some more complex functions and their influences on the performances. \n\nMinor issues:\n\n- \"modelling\" should be \"modeling\"\n\n- The citation format is a little strange with semicolons and parentheses, it is better to reformat them.\n\n- For better visualization, it is highly recommended to adopt vector graphics for all figures.", "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": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "c9s6M63--5-", "reviewer": "Reviewer_WyxW", "summary": "In this paper, the authors propose an interpretable spatio-temporal forecasting method by learning shape functions from data. The shape function is designed as a function of the distance between pairwise locations, which is then used to represent the vector autoregressive model coefficients. As a result, the proposed model can incorporate both spatial and temporal information. Furthermore, the interactions between different locations can be interpreted by the shape function outputs, and a graph can be used to visualize the interactions. Finally, the proposed method also achieves promising forecasting performance compared to deep learning methods.", "questions": "Questions:\nIn Eq (1), the authors only consider time lagged correlations. Is it possible to incorporate instantaneous relations at the same time step? The reason why I ask this question is because there might exist instantaneous relations in some applications.\n\nWhat is the motivation behind the choice of specifications in section 3.2? Are the authors trying to mimic some properties in the classical statistical methods?", "limitations": "The authors addressed the limitations.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength:\nThe work is well motivated by the challenge in spatio-temporal forecasting, i.e., the tradeoff between accuracy and interpretability. The proposed method uses shape functions to constrain the statistical relations between different locations and achieves good forecasting performance. Overall, I think the paper makes a good contribution to spatio-temporal forecasting.\n\nThe explanation of relations to previous works is clear. The proposed method is an extension of traditional methods by replacing the predefined spatial functions with learnable shape functions. The proposed method enjoys good forecasting accuracy as deep learning methods, while maintaining the interpretability of traditional methods.\n\nWeakness:\nThe main assumption of this paper is the correlation between two locations decreasing with distance. While this assumption may make sense in some applications, it would be better to give some discussions or case studies of this assumption.\n\nIn the nonstationary setting, the model in Eq (2) has a different weight matrix for each time step. It seems to be that this design suffers from the small sample problem, as the data at each time step is limited. This would cause fluctuations in the estimations across time. \n\nIn section 3.4, the authors give examples of monotonically increasing/deceasing functions. However, in the experiments, only monotonically decreasing functions are investigated. If increasing functions are not going to be used in the forecast model, it might be better not to mention it the method section.", "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": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "blQO5Bc2JTH", "reviewer": "Reviewer_tdLh", "summary": "The paper proposes a spatio-temporal prediction model that tries to incorporate a varying strength of spatial dependency over distance. The idea is to model spatial dependency weights as a shape constraint. There are some empirical evaluations on simulation datasets and real-world datasets. ", "questions": "1. The Gaussian process model is geostatistics also captures the effect of weakening spatial dependency with increasing distance. How does the proposed idea compare with the Gaussian process strategy? \n\n2. The proposed idea is under the category of traditional machine learning methods. It is suspicious that the proposed model can outperform deep learning models in prediction accuracy. How do you explain the results in Table 1?", "limitations": "The paper does have some interesting discussions on the limitations of the proposed idea, e.g., not capturing the non-Euclidean space like spatial networks, and the lack of causal interpretations. ", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength:\n\n* The paper solves an important problem, i.e., training explainable spatio-temporal forecasting model that can automatically learn the dependency structure.\n\n* The paper is overall well-written and easily understandable.\n\nWeakness:\n\n* Although the problem of learning spatio-temporal dependency structure is interesting, the proposed solution is not well-justified. The proposed method explicitly models spatial dependency as a function of distance. But the similar philosophy has already been explored in spatial statistics, such as the Gaussian process (it represents the covariance matrix of different sample locations based on distance between those locations). \n\n* The proposed model seems limited in model representation with a linear function. Currently, it is not convincing that such models can achieve better performance than recent deep learning models. ", "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": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Reject