Graph-based Virtual Sensing from Sparse and Partial Multivariate Observations
Authors: Giovanni De Felice
Andrea Cini
Daniele Zambon
Vladimir Gusev
Cesare Alippi
Venue: ICLR 2024 poster (2024)
Keywords: Spatio-temporal data
time series
virtual sensing
imputation
graph neural networks
deep learning
Primary area: representation learning for computer vision, audio, language, and other modalities
TL;DR: We present a novel framework for sparse multivariate virtual sensing that leverages dependencies between the target variable and available covariates.

Abstract: Virtual sensing techniques allow for inferring signals at new unmonitored locations by exploiting spatio-temporal measurements coming from physical sensors at different locations. However, as the sensor coverage becomes sparse due to costs or other constraints, physical proximity cannot be used to support interpolation. In this paper, we overcome this challenge by leveraging dependencies between the target variable and a set of correlated variables (covariates) that can frequently be associated with each location of interest. From this viewpoint, covariates provide partial observability, and the problem consists of inferring values for unobserved channels by exploiting observations at other locations to learn how such variables can correlate. We introduce a novel graph-based methodology to exploit such relationships and design a graph deep learning architecture, named GgNet, implementing the framework. The proposed approach relies on propagating information over a nested graph structure that is used to learn dependencies between variables as well as locations. GgNet is extensively evaluated under different virtual sensing scenarios, demonstrating higher reconstruction accuracy compared to the state-of-the-art.

Published as a conference paper at ICLR 2024
GRAPH-BASED VIRTUAL SENSING FROM SPARSE AND PARTIAL MULTIVARIATE OBSERVATIONS
Giovanni De Felice',, Andrea Cini", Daniele Zambon', Vladimir V. Gusev', Cesare Alippi
Universiity of Liverpool
The Swiss Al Lab IDSIA Università della Svizzera italiana
Politecnico di Milano
ABSTRACT
Virtual sensing techniques allow for inferring signals at new unmonitored loca- tions by exploiting spatio-temporal measurements coming from physical sensors at different locations. However, as the sensor coverage becomes sparse due to costs or other constraints, physical proximity cannot be used to support interpo- lation. In this paper, we overcome this challenge by leveraging dependencies between the target variable and a set of correlated variables (covariates) that can frequently be associated with each location of interest. From this viewpoint, co- variates provide partial observability, and the problem consists of inferring values for unobserved channels by exploiting observations other locations to learn how such variables can correlate. We introduce a novel graph-base based methodology to exploit such relationships and design a graph deep learning architecture, named GgNet, implementing the framework. The proposed approach relies on propa- gating information over a nested graph structure that is used to learn dependen- cies between variables as well as locations. GgNet is extensively evaluated under different virtual sensing scenarios, demonstrating higher reconstruction accuracy compared to the state-of-the-art.
INTRODUCTION
Spatio-temporal data analysis plays a significant role in applied and fundamental domains where systems evolve over both space and time. These include, for example, environmental science, urban planning, and epidemiology (Cressie & Wikle 2015, Wang al. 2020). In practice, the acquisi- tion of spatio-tempore scenarios. Sensor and communication failures, for instance, can result in partial even complete data loss at certain locations (Little & Rubin, 2019). Furthermore, the deployment of physical sen- sors incur high costs that often limit the number of monitored locations. Nonetheless, inferring the values of observables at new target locations holds a significant value for analysis and, even- tually, decision-maie makinn (Zhang et al., 2022; Hu et al., 2023). To address these challenges, virtual sensing (Liu et al. 2009) (also known as ppatio-tempore kriging) (Stein 1999) has emerged as a viable solution. Virtual sensing consists of reconstructing complete signals at unobserved locations by utilizing data gathered from monitored locations during the same time frame (Paepae et al.) [202]] Brunello et al. 202T]. As an example, virtual sensing techniques can be used to estimate solar irra- diance at particular location by exploiting neighboring sensors (Jayaaardene & Venayagamoorthy 2016), allowing, for instance, to plan for the installation of a new PV power plant. Most works rely on spatial correlations and on interpolating observations at neighboring locations (Appleby et al. 2020 Wu et al. 2021a b Zheng et al. 2023). However, as the sensor coverage becomes sparser due to costs and/or other (e.g., ecological) concerns, the number of neighbors from which extract use- ful information becomes limited or even null. This makes the spatial position less informative and limits the effectiveness of existing approaches. One such prominent example is material weathering testing (De Felice et al. 2022), where such sparsity, together with the total lack of any historical data regarding the variable infer the target location. makes virtual sensing bound fail. Despite be- ing quite common in the real world, this scenario of sparse data collection is still mostly overlooked in the scientific literature.
data is inevitably affected by all of the typical issues arising in real-world
Cooreeponndence gede-fed fel ce@l iverpool aac.uk.

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Published as a conference paper at ICLR 2024
In this work, we focus the multivariate setting and show how leveraging correlations be- tween variables can mitigate the challenges posed by sparse data scenarios. We consider sparsely distributed sensors, each of which is associated with multivariate time series (MTS). MTS can be partly unobserved, i.e., completely lack some of the channels. In such settings, the problem becomes that of reconstructing missing channels (target variables) from the avail- able ones (covariates) at the same location and from observations at related sensors (Fig. D This scenario typical of many real-world applications: in fact, variables correlated to the reconstruction target are often available at each location, e.g., satellite weather data can be used as covariates to infer photovoltaic energy production or material degradation. The approach developed in this paper follows from two observations. First, the mutual dependencies between the target variables and the covariates can be learned from the dynamics observed at locations where the target vari- able is being monitored. Such dependencies can then be lever- aged at the target location, where covariates provide partial observability, to infer the missing variable. Second, the tar- get variables can be reconstructed by integrating information from sensors that, although physically distant, can be consid- ered close (similar) in a certain functional latent space. Such similarities can be learned from the data. To solve the task while taking advantage of the above considerations, we design a novel graph-based framework allowing for modeling dependencies between variables and prop- agating information across locations through graph convolutional layers (Bronstein et al. 2021). Graph neural networks (GNNs) (Scarselli et al., 2008 Bacciu et al. 2020) indeed allow for ex- ploiting dependencies as architectural biases in deep learning architectures and have recently found widespread success in time series processing (Li et al., 2018 Cini al.) [223a Jin al. 2023) In particular, we propose a novel graph deep learning method, named Graph-graph Network (GgNet), designed to propagate information through nested graph structure encompassing both sensors and channels. Relying on such a structure, the architecture of the resulting model is biased toward prop- agating representations by exploiting learned dependencies among both variables and locations. Summarizing, our novel contributions are as follows:
?
Time
Figure Sparse multivariate vir- tual sensing: available data (yel- low), missing data (white), and predictions (red).
we introduce a general methodology for performing multivariate virtual sensing, specifi- cally designed to handle scenarios where available sensors do not provide adequate spatial coverage (Sec. 4.1;
we propose a nested graph representation allowing for effectively modeling dependencies within multivariate spatio-temporal data (Sec. 4.2
we design GgNet, the first spatio-temporal graph neural network explicitly tailored for multivariate virtual sensing from sparse observations (Sec. 4.3):
we carry out an extensive empirical evaluation by exploring different use cases and assess- ing the performance of the proposed method against the state-of -fht-e-rt (Sec. 5).
The proposed framework constitutes a first attempt at tackling a challenging setting for deep virtual sensing methods and arguably constitutes key contribution to the methodological advancement of the field, as well as a powerful tool practical applications.
PROBLEM FORMULATION
This section introduces the notation and formalizes the multivariate virtual sensing problem, with focus on settings where sensors provide a sparse coverage of the area of interest.
Notation We indicate scalar variables and indices as lowercase k. constants uppercase K, vec- tors as bold lowercase x, matrices as bold uppercase X, and higher order tensors and sets with calligraphic X. We use the notation x and X to indicate univariate and multivariate time series, respectively. Association with the n-th location is indicated with the notation 2/n]]. Measurements at time step t are indicated with subscript 24; channel (variable) d of a multivariate observation is indicated with superseript v.
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Multivariate spatio-temporal data Spatio-temporal data refers to a collection of temporal obser- vations coming from N distinct spatial locations with coordinates [q[n]]m- in a generic domain Ω, e.g,, the one induced by the geographic placement of sensors. For each spatial location n, consider multivariate time series (MTS) X[n] RT where 7 [n]] and D are, respectively, the number of time steps at location n. and the number of channels, i.e., observable variables. If observations are synchronous and an equal number of time steps are recorded across locations, the set of mea- surements can be represented as 3-D tensor X' E IR N Nx7xD Otherwise, the tensor representation can usually be obtained by padding or interpolating the missing observations (Lepot et al. 2017).
Univariate virtual sensing Consider a set of N sensors positioned at distinct locations in spatial domain Q where subset of N < N univariate time series [X'n] E RT N
are observed, while the
remaining are missing. In line with previous works (Cao et al. 2018 Cini et al. 2022), we model data availability with binary mask m[n] E {0, 1). which indicates if the n-th location is observed
(m [n] = 1) or missing (m[n) = 0). The task of univariate virtual sensing consists of estimating p[[[n]] q(zi), X), where time series X[n] is unavailable (i.e., m[n] 0), and the position q[z]] of the associated sensor is given together with the observation set X = [[[m),, q(zl)) m[n] = 1) n==
Note that spatial coordinates are assumed to be available everywhere.
Multivariate virtual sensing The problem can easily be extended to the multivariate case. Here, consider a set of multivariate sensors [Xn] E RRT'DJ,_ y where individual channels are missing at some locations. We can denote channel availability by extending the previous binary mask to mm[n] (0, 1), which indicates if the d-th channel is available at the n-th location. The observation
ser now consists of all the observed channels at all locations X' = [m), q(z]) [x] = 1) The task of multivariate virtual sensing (MVS) then consists of modeling:
p(x [m] q(m), X)
for every pair (n, d) such that mn O. Note that virtual sensing is, priori, much more chal- lenging task than the ordinary missing data imputation problem, due to the absence of any historical data to infer both the scale and the dynamics of the missing signal. When addressing this problem, the sensors' spatial proximity plays a pivotal role. If the sensors densely cover space, it is possible, in both the univariate and multivariate settings, to leverage the spatial proximity to available observa- tions. However, the problem is far more challenging in sparse settings: if the problem is univariate, no observations are available at the target or any neighboring location. Conversely, the multivariate settings might allow for dealing with the virtual sensing problem by exploiting dependencies on variables other than the target one.
3 RELATED METHODS
Many approaches to kriging and virtual sensing have been proposed in geospatial analysis and ma- chine learning literature (Cressie & Wikle 2015). Notably, among existing methods, some rely on Gaussian processes (Luttinen & Ilin (0122) and tensor dccomposition (Bahadori et al. [014). More related to approach, Cini et al. (2022) and Marisca et al. (2022) introduced imputation methods based on GNNs and showed that such methods can perform virtual sensing as well. Wu et al. (021a))
and Zheng al. (202)) directly tackle the virtual sensing problem exploiting inductive GNNs. In-
deed, GNNs are becoming popular in time series imputation and reconstruction (Ye et al. 2021
Kuppannagari al. 2021: Chen al., 2022 Jin et al. 2023). However, all these approaches focus on univariate settings. As consequence, they can only rely on the spatial proximity of the available sensors to perform the reconstruction and/or do not target sparser settings. Besides virtual sensing, deep learning methods have been widely applied in the context of MTS imputation by relying on a diverse range of model architectures, e.g., autoregressive RNNs (Cao al. 2008). attention- models (Shukla & Marlin 2021 Du et al. 2023). generative adversarial networks (Yoon et al.) 2018a; Luo et al.J 2019) and diffusion models (Tashiro et al. 2021). Although such approaches can in principle be used to perform virtual sensing in the multivariate case, they cannot take spatial correlations directly into account. To the best of our knowledge, the proposed GgNet is the first graph-based neural network architecture designed to tackle MVS in sparse scenarios by exploiting learned relationships across both observable variables and locations. We refer to Appendix (A for a more thorough discussion of the relevant literature for spatio-temporal kriging and MTS imputation.
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4 METHODOLOGY
Here, we introduce a general framework for solving the MVS problems formalized in Sec. 2 re- gardless of the spatial coverage offered by the available sensors. We begin by presenting concep- tualization of the inference problem as dealt with in our framework. Then, we introduce the details of the proposed novel architecture, aligned with such conceptualization.
4.1 CONCEPTUALIZATION
We identify two fundamental types of dependencies characterizing the MVS task, i.e., the rela- tionships between channels the same location (intra location) and among observations different locations (inter location). We model such dependencies by considering location-channel (1, d) pairs and defining the sets
2 [n|
8
v/ 8 d A -
m [v] 0
(target set)
T
[v]]
x ||| 5 = d A n
(observed target set)
[V x [v] 1 m }
(intra-location covariate set)
C c\( =
(inter-location covariate set)
where \n indicates all indices except n. Sets [n]] ""In,.cCd and C [\]] encompass, respec- tively, the target variable at the target location, the target variables at all other locations except n, the observed variables at the target location, and the observed variables at all other locations except By considering the above sets, we distinguish between the types of dependencies that allow for reconstructing missing channel [n] in the target set.
T-Y: Relations between observed targets at other locations [vz) and the target ["]...
C-+Y: Relations between observed covariates the target location (c)d(z)) and the target x [m].
The subtask of learning from T-+Y dependencies is similar to learning how to exploit observations at related locations typical of univariate virtual sensing. To weigh the importance different obser- vations in the observed target w.r.t. the target, notion of similarity between locations is required. Kriging methods address it by relying on spatial proximity (dense scenario), e.g., as derived from the sensor positions {q[=]}, weigh the reciprocal importance of observations at different locations. In contrast, with the sparse settings in mind, we model T-+Y dependencies in a data-driven fash- ion, learning similarity score for each pair of locations. In particular, as detailed in Sec. 442,, we associate a representation (node embedding) (Cini et al.) (2023a) to each location and learn a score function taking as input pairs of such representations.
the same time, these are used as additional learned covariates to tailor C-+Y in modeling location-specific dynamics. Such framework allows for the extension of virtual sensing to generic MTS datasets, as it does not rely on pre-defined spatial relationships such as physical proximity. In the following, we design model that directly accounts for both types dependencies with architectural inductive biases that align the processing with the modeling of intra and inter-location dependencies.
4.2 NESTED GRAPH STRUCTURE
To take advantage of both T-Y and C-+Y relations, we learn a nested graph structure composed of an inter-location graph and an intra-location graph (Fig. 2 The inter-location graph connects different spatial locations, while the intra-location one explicits dependencies among channels.
Inter-location graph G:
We associate each sensor location node of graph G, which we call inter-location graph, and model the association between different sensors edges of G, thus accounting for the T-+Y modeling.
As physical proximity cannot be exploited in sparse settings, we learn the graph topology (N N adjacency matrix Ag) from the data to account for dependencies among sensors that can far apart in space. In particular, following previous works local representations (Cini al. 2023a),
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Observed dharnel
Time
Figure 2: Dataset collection of multivariate time series (left) and its nested graph representation (right). Each location is represented as node in the inter-location graph (G), capturing generic relations between locations. Within each node in G, smaller intra-location graph (g) models de- pendencies between the channels.
associate each n-th node in G with a learnable (static) embedding ea(n] E RRA. These are learned as parameters, jointly with the network weights, and are used estimate relationships relevant to the downstream MVS task. The weighted adjacency matrix is then obtained from the similarities between embeddings. In particular, we model Ag
Ag Softmar MLP1 (Ec) MLP2
where the matrix Eg E RR G contains the learnable node embeddings for all nodes in G. Convo- lutions on such graph directly account for T-Y dependencies by propagating information across locations. Similarly, we learn an adjacency matrix for location dependencies.
Intra-location graph g: We identify each of the D channels, at every n-th sensor, with node of second graph, which we name intra-location graph g. accounting for the relationships between channels (C-Y).
Similarly inter-location case, we associate each d-th node in g with learnable embedding cy Id]. We learn adjacency matrix Ag R of g as a function D D matrix & of free parameters, treated as edge scores. As the physical quantities considered at each location are the same, we assume to be the same in all locations.
Note that, in this paper, for simplicity, we do not consider graphs G and g that change over time and do not constrain G and g to be discrete binary objects. That being said, more advanced graph learning methods (e.g., Kipf et al. 2018 Niculae et al. 2023) Cini et al. 2023b) can be incorporated into the framework and constitute possible extension for future works. Note that the node embed- dings eg [n] need be learned for every location, which makes the model intrinsically transductive. Despite that, the fine-tuning new node embeddings can be done efficiently (Cini et al. 2023a).
4.3 GRAPH-GRAPH NETWORK
In this section, present the Graph-graph Network (GgNet), proposed architecture relying the nested graph representations provided by G and g. The model is composed of several blocks, as detailed below and depicted in Fig.
Input encoder To account for specifies along the temporal, spatial and channel dimensions, we obtain a hidden representation for each time step f, location n, and channel d given the input data X' and the learned node embeddings Eg, E., similarly to Marisca et al. (2022)
h [n] m[[n]] MLP MLP (x4[[) + MLP MLP (e..:::2(eG(])))pp ey(d)
(3)
where [n] R are the resulting encoded representations, while MLP ene.1 and MLP cue.2 two distinct multilayer perceptions. Note that representations are obtained by exploiting observations wherever available and relying exclusively node embeddings for the targets.
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layers
layers
(T-M))
1C-
X M
Input Encoder
Temporal
Convolution
g Convolution
Readout
Convolution
E
Figure 3: Overview of GgNet. Temporal convolutions (blue) encode temporal patterns, G- convolutions (green) propagate information across the inter location graph and model dependencies between locations; g-convolutions (purple) propagate information across the intra-location graph and model dependencies between channels. Starred modules refer to channel-wise operations.
Temporal processing Stacks of Temporal Convolutions (TC) (Bai et al. 2018 are used to encode temporal information for each location and channel
Hd [n] = TC (Hr [m)
where [n] RTTx As channels may present largely heterogeneous temporal features, we use a different set of weights (filters) for processing data in each channel. Temporal convolutions are highly parallelizable making them preferable, in this setting, over, e.g., recurrent neural networks. Furthermore, we symmetrically pad the input sequence to avoid duplicating the architecture to en- code forward and backward dynamics and exploit exponentially increasing dilation rates to capture both short and rongerange temporal dependencies.
Inter-location (spatial) processing Spatial information propagated across locations (T+Y) by means of Graph Convolutions (GC) (Bronstein et al. 2021) over the inter-location graph (G- convolution Fig.
H = GC
(H. m Eg. Ac)
(5)
where H. E R NxH and the same convolution is performed independently on each channel and each time step. Specifics of each channel are accounted for via the conditioning on the embeddings in Eg. Note that G-convolution is synchronous operation that assumes that the time frames of each time series are aligned.
Intra-location (channel-wise) processing As the next step. information is propagated across channels (C-Y) by performing GCs over the intra-location graph (g-convolution Fig.
H [n] GC H. [n], m. [n], Eg, Ag)
(6)
for all time steps and locations, and where H,(n) [n] E DDH g-convolutions allow for inferring observations at unavailable channels by modeling dependencies among targets and covariates. Note that weights shared across locations, as local dynamics accounted for by conditioning on node embeddings Eg.
Multiple T-, G-, and g-convolution blocks can be stacked, allowing for designing deep architectures. We refer to Appendix E.2 for an extensive ablation study assessing the impact of each component.
Readout Finally, readout composed of d MLPs maps representations predictions at all target locations and time steps
x [n]
MLPd A (h][m), mf(n], ea[n], eyld))
Note that the GgNet implementation considered here is just of many possible instantiations of the framework. The model can be extended, e.g., by including more sophisticated graph processing and learning modules.

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4.4 MODEL TRAINING
For a generic MVS task, the reconstruction error can be defined as
C(R,X. En Ea m [n) l(x
[n])
B=1
(8)
Σn Ex [m]
where [n] and x= [n] are, respectively, the predicted and true val- ues, and -) is a loss function between time series, e.g., the mean absolute error or the mean squared error. The model is trained and evaluated by simulating the presence of missing channels partition- ing the available N x D channels into training, validation, and test channels (Fig. 4. simulating a transductive learning setting. To bias the training toward the virtual sensing task, mask out fraction of the available channels for each training batch and train the model to reconstruct the input data, giving higher weight to the masked out observations. We have found this strategy to be crucial for models like GgNet that do not rely forecasting as surrogate objective. tional small fraction of data points, at random, in the training data to enforce robustness to random missing values (Cini et al. 2022 Du al. 2023).
==
Train
Val.
Test
Figure 4: Training and eval- uation splits for MVS.
Finally, we mask out addi-
EXPERIMENTAL EVALUATION
In this section, we evaluate the proposed method in several MVS tasks. Experiments conducted across three different datasets, allowing for the investigation of settings characterized by (i) different temporal resolutions and (ii) different degrees of sparsity, as well as (iii) assessing the limitations of
the different approaches. Alongside GgNet, we provide results for the following baselines: KNN, i.e., a simple baseline averaging the nearest geographical neighbors. BRITS (Cao et al., 2018) as
representative of autoregressive RNN global model SAITS (Du et al. 2023) as representative of transformer- based global models unaware of spatial relations, GRIN (Cini et al., (022) as repre- sentative of graph-based imputation models, leveraging spatial correlations. Moreover, we consider a variation of the standard GRIN model, named GRIN,,, m. modified to better deal with multivariate data at each location; specifically, we add self-loops in GRIN's Spatial Decoder for conditioning on covariates at the same time step of the reconstruction target. To assess the impact of some of the introduced design choices, also compare 4 progressively more advanced recurrent architec- tures: 1) RNN, a standard global recurrent neural network; 2) RNN its bidirectional extension; 3) RNNEm.. which, besides being bidirectional, incorporates local component by concatenating learnable node embeddings to the input observations; 4) RNNG. which adds a convolution opera- tion over an inter-location graph on top of the previous architecture. For each model, we consider different hyperparameter configurations, with increasing complexity; reported results correspond to the berf-performing model, given the reconstruction accuracy on the validation set. Regarding the preprocessing steps, we follow the common practice of standardizing the data, ensuring that, across locations and time, each of the channels has zero mean and unit variance. Missing val- ues, if present, are masked out. Details about the baselines can be found in Appendix B while the complete experimental setup is described Appendix C
5.1 MISSING-CHANNEL RECONSTRUCTION CLIMATIC DATA
In the first experiment, address reconstructing missing at-random (MaR) channels in multivari- ate spatio-temporal datasets. We build two datasets with different temporal resolutions of climatic variables from the NASA Langley Research Center POWER Project. More information about the data and APIs is available in Appendix D To emulate a reduced spatial coverage, we collect data in correspondence with the 235 national capitals. For each location, we collect multiple correlated variables with daily and hourly resolutions. Within this setting, we aim to reconstruct MaR climatic variables at different locations from the available data. Among all N channels, we use 70% for training, 10% for validation, and the remaining 20% for testing.
With the term "global" we refer to models whose parameters are shared in all locations, in contrast to "local" ones that have spcetic parameters (Montero-Manso Hyndman 2021 Cini et al. 2023a
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Table 1: Climatic dataset: channel-wise MRE (%) for daily data and average accuracy for daily (D)
and hourly (H) data. Results are averaged across locations and 5 random seeds. The berf-performing method is in bold, the second-best is underlined.
CR.
TEMP, MEAN
TEMP RANGE
TEMP:
WIND SPEED
REL. HUM.
PREC.
TEMP.
CLOUDS
IRR.
IRR.
Ava (D)
AvG
MAX
DEW
SHORT
KNN
15.6-2.6 42.1 12.4+1.5 33.7+4.6
12.3
102.8
23.9+2.9
34.0.o
34.0
18.0
6.0
36.2+1.9 30.1+1.0
BRITS
4.020.6
17.522.
3.3
32.1
5.5
80.5g2.4
8.4g1.1
29.9ga.9
20.921.
3.4
26.3go.8 20.5g..5
GRIN
4.2g07
18.6
4.3
30.7+2.
5.6
77.3_s.6
8.721.
29.
29.1
15.3
3.0g0.3
19.8
23.3
GRIN
2.9
12.9
2.8
30.71s.9
3.6.0.9
69.9
6.4
20.5+a.9
11.7+0.6
3.6go.4
16.52...
22. 22.7
SATTS
24
11.2
2.3
26.9
3.3.1.0
06.0.22
20.6ea.7
14.2#0.9
2.8 =0.3
15.5go..
22.2
GeNET
2.1
9.620.7
2.0
23.9 2
2.720.7
60.619.1
4.2g0.8
16.5
9.2g0.
2.9
13.4
20.4.006
% IMP.
12.5%
14.3%
13.0%
11.1%
18.1%
8.1%
16.0%
19.9%
21.4%
-3.6%
13.5%
8.1%
RNN
5.6
21.940.7
5.8
32.721.2
6.9
83.821.8
10.8
35.521.
21.8
3.8
22.9
28.8
B:D.
4.2
19.1
4.7
31.1g0.8
5.9
78.0=26
8.6
31.4ga.6
19.0
3.6go.4
20.6
25.7
EMB.
4.120.5
17.5
4.3gm.4
30.2ga.0
5.5
77.121.
8.521.
30.2go.4
17.5 3.620.3
19.8
26.220.9
3.6.0.3
16,5+1s
3.940.4 27.6+1.6
5.2+0.4
74.7=1.
7.3+0.9
27.7=0.s 14.0 3.0
18.4+0.6
23.2+1.1
Road Town (British Virgin Islands)
TSNE projection GgNet's learned embeddings
Truhh
11.
GRIN PMRE 12.4%)
GATTS ONRE 364)
40
30
Wind Speed (m/s)
9.
10
-10
-20
-30
-10
1906
Peb 1906
Mar 1996
Apr 2995
May 1906
1906
-30
-20
20
Figure 5: (Left) Reconstruction of wind speed (poorly correlated with the other channels) in Road Town (which can exploit observations at nearby Caribbean capitals). (Right) t-SNE representation of the node embeddings learned by GgNet; colours refer to the Köppen-Geigee climate classification.
Daily climatic data Considering a daily resolution, we collect 10 climatic variables (listed in Ap- pendix D encompassing 30 years, which results in N = 235, T = 10958, D = 10. Tab. reports the Mean Relative Error (MRE) for each test channel, grouped by variable and averaged across lo- cations. All results are consistent across different metrics as shown in Appendix E.8 From the results, we first observe that GgNet outperforms models relying exclusively on pre-defined spatial relations (GRIN), demonstrating superior performance in handling sparse settings. Notably, GRIN underperforms especially in correspondence with remote locations, where no neighboring sensor is available (Appendix EX The GRIN,, variant improves over the results of GRIN, but nonetheless underperforms w.r.t. Ggt SAITS performs reasonably well on this dataset, ranking second in most of the setups. In particular, SAITS can account for intra-location covariates, but cannot take advantage of inter-location relations. This is particularly evident for missing channels that are hard to infer from the other variables the same location (e.g., wind speed, precipitations, clouds and shortwave irrudiation), but can be reconstructed from spatially close observations, as in the exam- ple shown in Fig. (left). Tab. Is lower section reports results for the RNN model variants. The notable performance improvement obtained by the bidirectional variant demonstrates the impor- tance of accounting for both past and future values. Node embeddings yield modest yet consistent improvements. Additionally, the introduction of the inter-location graph G results in a substantial performance gain, supporting the importance of inter-location relations. Alongside MVS results, we provide in Fig. 5 (right) t-SNE visualization of the node embeddings (Eg, Sec. 4.2) as learned by GgNet w.r.t each location. Each embedding is coloured according to the Köppen- Geiger climate classification (Peel et al.) 2007) that divides the globe into different climates, with similar colours corresponding to similar climates. Note that the model does not have access, at any point, to either these climate labels to the geographical coordinates of any location. The correspondence between
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Table Photovoltaic dataset: MAE (W*h). For each N, results are averaged across 5 sampling of the locations and 5 different runs for each set (25 runs in total). The best-performing method is in bold, the second-bor underlined.
NICESTINS
10
20
30
50
70
100
150
200
GRIN
581
170
536
52
533
49
514 36
531 34
524 28
523 1 40
528 43
GRIN...
497 + 117
458 49
446 + 42
424 +: 23
428 +: 21
423 + 13
417 13
418 - 12
SAITS
466 + 68
451 + 46
448 - 37
427
+ 29
436 + 24
420 + 16
417 = 14
416 - 14
GGNET
452 67
455
439
32
415 + 26
410 26
391 + 18
367
357 16
1MP.
3.0%
-0.9%
1.6%
2.1%
4.2%
6.9%
12.0%
14.2%
such labels and the formed clusters serves as a qualitative assessment of the capability of GgNet to capture similarities the observed dynamics reflecting the climate each location. Appendix B reports several additional analyses.
Hourly climatic data Then, we address the same MaR task in the case of climatic variables sampled with hourly resolution. Data cover one year, which results in N = 235, 1' = 8760, D = 7. Differently from daily data, hourly data are more difficult to predict, being characterized by high- frequency components both temporally and spatially. As a consequence, exploiting inter-location dependencies can help in providing better conditioning and hence lead to more accurate predictions. Average MRE is reported in the last column of Tab. [ while channel-wise results are shown in Ap- pendix E.8 Graph-based models effectively exploit the additional information to obtain a relatively good reconstruction accuracy. In Ggtet, accounting for the interplay between spatial aggregation (T-Y) and global modeling of the relations across channels (C-Y) arguably results in superior average performance.
5.2 EXTENDING SPATIAL COVERAGE: PHOTOVOLTAIC ENERGY PRODUCTION
We consider the problem of predicting specific target variable (whose direct observation is ex- pensive) at new target locations by exploiting a set of available covariates. This constitutes an interesting case for our multivariate virtual sensing task. For example, renewable energy pro- duction or material degradation can be inferred from widely available satellite climatic data (De Fe- lice al.) 2022). In particular, we consider as target variable the daily photovoltaic power (PV) production simulated over continental North America by Hu et al. (2022). To simulate sparse scenario, we randomly sample few locations from the provided dense grid. As covariates, the same 10 climatic variables from Sec. 5.1 collected in correspondence with the sampled loca- tions. For both target and covariates, we consider 1 full year of daily observations, which results in T= 365, D = 11. By varying the number of locations between 10 and 200, we aim to infer the PV production at new target locations from the climate at such locations. We use 70% of the sensors for training. 10% for validation, and 20% for testing. Tab. rrpport the reconstruction accuracy in terms of the Mean Absolute Error (MAE). As more locations are considered, results show a progressive increment of the relative improvement of GgNet over SAITS. This could be related to the superior capability of GgNet to exploit spatial information.
CONCLUSION
We presented a novel framework for virtual sensing from sparse multivariate spatio-temporal obser- vations. In doing so, we introduced novel methodology to exploit dependencies between covariates and relations across locations. In this context, we proposed GgNet, a graph deep learning architec- ture leveraging a nested graph structure to account for such relations. The relational structure is learned end-to-end to maximize reconstruction accuracy. Compared to state-of-the-art, GgNet can achieve superior performance in settings with poor sensor coverage, where other methods fail. As future directions, we believe it would be interesting to explore the application of a similar method- ology to collections of asynchronous time series, typically found in healthcare applications. Finally, GgNet is limited, in its current form, to transductive learning settings and its extension to inductive learning would broaden the applicability of the proposed method.

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REPRODUCIBILITY STATEMENT
Python code reproduce the experiments is available online at https://githui. com/ gdefe/ tafte//gnett httg//ggnnttttaaaltgggg wirtual-sensingg
A script to download climatic data from the database in correspondence with the world capitals is included in the provided code. PV datasets at differ- ent spatial densities, together with the corresponding climatic variables, are provided alongside the code. Complete climatic and photovoltaic datasets are publicly available online through the refer- ences provided in Appendix() D The sampling of locations (photovoltaic experiment) and simulation of missing channels (both climatic and photovoltaic experiment) are controlled by a fixed seed for reproducibility purposes. Hyperparameters for all models are provided in Appendix C
ACKNOWLEDGMENTS
G.D.F. acknowledges the Beckers Group for funding this research. V.V.G. thanks Leverhulme Trust for support via the Leverhulme Research Centre for Functional Materials Design and EPSRC under grant number EP/VO26887. This research was partly funded by the Swiss National Science Founda- tion under grant 204061: High- Order Relations and Dynamics in Graph Neural Nerworks. Climate data was obtained from the National Aeronautics and Space Administration (NASA) Langley Re- search Center (LaRCC) Prediction of Worldwide Energy Resource (POWER) Project funded through the NASA Earth Science/Applied Science Program. The authors wish to thank the anonymous viewers whose comments considerably improved the final version the paper.
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APPENDIX
A RELATED METHODS
Methods that can perform virtual sensing can be broadly categorized into spatio-temporal kriging methods and generic MTS imputation approaches. Kriging methods exploit the density of the net- work and infer the missing sensors by interpolating neighboring observations. Among those, ordi- nary kriging (Stein 1999) is a linear interpolation procedure that assigns optimal weights for neigh- boring data points based on a variogram model. Similarly, cokriging (Goovaartss, 1998] Cressie & Wikle| 2015) also exploits correlation with auxiliary covariate information (such as elevation, soil type, or temperature) and estimates them simultaneously with the variable of interest. However, these are linear spatial interpolation methods, therefore, they fail to capture any temporal patterns, as well as nonlinear dependencies within the data. Multi-task Gaussian Processes leverage flexi- ble kernel structures to capture apatio-temporal correlations (Bonilla et al.. [007]] Luttinen & Ilim 2012]. In addressing scalability issues when dealing with extensive datasets, matrix/tensor com- pletion methods capture global consistency in the data by imposing a low-rank assumption, well as local consistency by diverse key regularization structures (Bahadori et al. 2014; Takeuchi et al.
2017 Lei et al. 2022. Wu et al., 2022). More related to our approach, imputation methods based on GNNs have demonstrated great effectiveness in modeling spatial dependencies between time se- ries (Ye et al. 202T Kuppannagari et al. 2021 Roth & Liebig 2022 Kong et al., 2023 Jin al.]
2023]. Notably, GRIN (Cini et al.] 2022) accounts for both temporal and spatial correlation within the data by combining autoregressive modeling with message-passing operations: SPIN (Marisca et al. 2022) introduces a spatiotemporal attention mechanism to weigh discrete points in time and space; IGNNK (Wu et al., 2021a) tackle inductive virtual sensing by specializing
graph neural networks, later extended in SATCN (Wu et al. 2021b) by allowing multiple transfor- mations of the spatial information to contribute to the adjacency matrix; INCREASE (Zheng et al.] 2023) also considers different heterogeneous spatial relations. However, they all assume that the spatial information characterizing the placement of each sample is available to the model; either as a given graph structure or as geographical information to compute one. In sparse settings, or when such information is missing, the main limitation of these methods is deriving an appropriate adjacency matrix. This problem is even exacerbated in the multivariate case. As evidence, MTS sim- ilarity is a challenging problem on its own (Mikalsen et al. [0188] De Felice et al.) 2023). Adaptive methods (Bai al., 2020). such as AGRN (Chen al.) 20022.. address this by learning the adjacency matrix together with the network weights. Despite this, existing GNN-based approaches only target dense and univariate sensor networks and do not explicitly model the relations between channels in individual locations.
On the other hand, recent advancements in generic MTS imputation have led to models that can be adapted to perform virtual sensing, Among these, the most widely adopted are deep autoregressive methods based on recurrent neural networks (Che et al. 2018 Yoon et al. 2018b), e.g. BRITS (Cao) et al. 2018) is bidirectional model that also takes advantage of the relationships between dimen- sions. Moreover, imputation can be performed by modeling the underlying data distribution, gen- erally employing generative adversarial neural networks (Yoon al.; [20T8a] Luo et al., (019). In recent times, imputation techniques have been proposed that rely on attention mechanisms (Ma) cet al. 2019 Shukla & Marlin 2021; Du et al., 2023) and diffusion models (Tashiro et al. (022)). However, these models cannot take spatial correlations directly into account. Indeed, these methods learn single model shared across locations and take into account single MTS at time. One could still exploit such approaches by considering all the available data as single MTS, but the result- ing model would not be appropriate for virtual sensing, as the same variables at different locations would be modeled as entirely different channels.
B BASELINE DETAILS
imputes missing test channels by averaging the corresponding train channels in the k- nearest geographical neighbors. The number of neighbors & E 2,, 5, 10) is selected based on the best performance on the validation set. This baseline has access to the geographical coordinates, with which neighbors are identified.
KNN:
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global model where data at different locations X[n] RTx are considered as different training instances to train a shared model. Specifically, this model imputes missing values from a linear transfommation of the hidden state of global Recurrent Neural Network (RNN). Formally, the following operations are sequentially iterated for every time step f. in the considered temporal window:
RNN:
X. W Wout h +
(9)
X₁ m. ④ X2 + m, ④ &t h GRU X mm]: h,-1)
(10)
(11)
where the notation [ni] is implicit for all inputs, masks and hidden states.
RNNBID: this model replicates the previous RNN model both in the forward direction and in the backward direction. Imputations performed by Eq. Sq. discarded and the final imputation is obtained with separate MLP acting on the aggregated hidden representations:
x MIP Fwed
h
band
(12)
RNNemb: global models, it is not possible for the two previous models to specialize the mod- eling on specifics of each location. this extension, we incorporate local components into RNNBD by concatenating the node embeddings e[n] to all input time steps before the state update and readout. Formally, the model sequentially iterates, on both directions, for every time step t, Eq. 9 Eq. and:
[x[r]] = GRU [x [a] [x [[e[x]]] h--1
(13)
where we have here explicit the notation [n]]. Finally, as before, we aggregate hidden representations to output the final result:
N [n] MLP(
[h/ hh [ri] || brod (r]] [[][]]])
(14)
we further build here on the previous model by introducing synchronous message-passing operations between locations. The resulting architecture mimics time-then-space models that are typical the spatio-temporal graph processing literature (Gao & Ribeiro [022]. First, inputs are
RNNG:
processed by the RNNemb. as above, up Eq. 13 then, a GC layer performs convolution operation over the inter-location graph G:
H AG H, skip +
(15)
where H. H wdd
H Howd and AG is obtained from Eq. 2 Finally, RNN and GNN states are concatenated and fed to an MLP readout to produce the final imputation:
& [n] MLP( [ fwd [r]]
band [n]] [m] || [[m]]]
(16)
BRITS: global bidirectional RNN model for generic MTS imputation from Cao al. (018 The 2D form is recovered by processing one MTS at time, i.e., in practice, flattening the location
dimension along the batch dimension.
SAITS:
global self-attention-ben based MTS imputation model from Du al. 2023 The 2D form
recovered by processing one MTS at a time, i.e., in practice, flattening the location dimension along the batch dimension.
GRIN: graph-based model for spatio-temporal data imputation from Cini et al. 2022 This baseline has access the geographical coordinates. from which we compute the adjacency matrix with thresholded Gaussian kernel (Shuman et al., 2013):
dist(i,j)² g²
Ag = A3s = exp
if dist(i. otherwise
(17)
0
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with -)) being the Haversine distance, o² the distances variance. and d custom threshold. For all datasets with a number of nodes N 2 70, we set to the required value to obtain an average of 10 edges per node, e.g.. ó(climatic datasets) 3500 km. For smaller datasets, we fix 6 to the corresponding value at = 70. The GRIN,, variant adds self-loops to the original Spatial Decoder of GRIN.
C DETAILED EXPERIMENTAL SETTING
Shared settings For GgNet and all baselines, adopt the Adam optimizer (Kingma & Ba 2015) with a learning rate Ir = 0.001 paired with a cosine annealing learning rate scheduler. All models are trained for maximum of 500 epochs, with 30 epochs patience for early stopping. All batches have a size set to 32 and consider temporal windows of tw = 24 time steps. As for the settings that are specific virtual sensing: we set, for all methods, the probability of randomly masking training points to Puithen-points = 0.05, the probability of randomly masking entire channels to -chsnmeens 0.3 and the weight of masked points during loss calculation to Wyhhsten 5. Different values Warhiten E {3,55, 5, 10} led to a negligible difference in performance. A brief study on the robustness under changes the Tuithen-channdd mask can found Appendix [. Ex- periments are conducted within the Python (Van Rossum al. 1995) library Torch Spatiotempo- ral (Cini & Marisca 2022), which is also used to implement all methods. Experiments are tracked using Weight and Biases (Biewald 2020). Code to reproduce the experiments is available online
Hyperparameter settings on climatic dataset In the climatic data, all models are evaluated under different hyperparameter configurations, with increasing complexity. Separately for the daily and hourly datasets, the best-performing configuration of parameters on the validation set across the training epochs used testing. For GgNet, this resulted in the same set of parameters for both datasets: hidden size d, = 128 for all convolutions and MLPs; location node embedding size he.G 16, channel node embedding size le.g 8; 2 blocks, each of which is composed of 3 layers of temporal convolution (with filter length kemmp. fiter = 3 centered at each time step t). layer of G-convolution and layer of g-convolution; "elu" activation functions for the encoding MLPs, all graph convolutions and decoding MIPs; "Tanh" activation functions for the MI.Ps transforming the embeddings (Fq.2). Hyperparameter choices for all baselines are available within the configurations file in the provided code. RNN variants are set to the same hidden size h 64 for comparability.
Hyperparameter settings on photovoltaic dataset the photovoltaic dataset, to assess perfor- mance scaling with the number of locations, set common hidden size of h 64 to all models and leave all other parameters to the best-performing setting in the climatic daily experiment. For all models, also set dropout value of dr 0.1 in all experiments up N 70, and dr afterward.
C.I LOSS FUNCTIONS
GgNet loss function As our loss function ), we employ the sum of three quantile losses, (also referred to as "pinball loss') (Koenker Bassett Jr 1978), with quantile levels selected as follows: q-a 0.159, 9r- 0.5, and a+o 0.841. The median quantile aligns with the Mean Absolute Error (MAE), while q-a and g+o allow for the concurrent estimation of uncertaintics. We believe this is particularly relevant to virtual sensing, not all channels in all locations can be equally reconstructed from the available data. As in (Cini et al., 2023a), further embedding regularizations can be considered in addition to the training objective.
Baseline loss functions RNN and RNN use MAE loss; RNN and RNNG use the same three- quantiles loss GgNet; BRITS, SAITS and GRIN use same loss functions as the authors provide in the respective code implementations.
httpss///githbb..cmmggeeeee
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D DATASETS DETAILS
Climatic dataset: We obtain daily and hourly climatic data from the POWER Project's Daily and Hourly 2.3.5 version 2023/02/26. This online database contains data about surface solar energy fluxes and other meteorological quantities obtained through satellite systems and further reanalysis. For radiation data, spatial resolution is 1.0 (latitude) by 1.0" (longitude); for meteorological data, this is 1/2" (latitude) by 5/8" (longitude). For each point on this global-scale grid, data are pro- vided in time-series format with customizable temporal resolution. Further information, together with data and API, is available at the project website h For daily data, we select the following 10 variables: mean temperature (C), temperature range (C), maximum temperature (C), wind speed m//s, relative humidity (%), precipitation (mm//day, dew/frost point (C), cloud amount (%), all- sky surface shortwave irradiance (W/v) //mm and all-sky surface longwave irradiance (W/m²). Daily data extend for 30 years (1991-01-01 to 2022-12-31). For hourly data, we reduce the dimensionality to 7 variables, as maximum temperature, temperature range and cloud amount are not available at this resolution. Hourly data extend for year (2022-01-01 2022-12-31).
Photovoltaic dataset: We collect photovoltaic data from Hu et al. 2022 which simulate photo- voltaic power production densely over the Continental North America. Simulations extend over the entire year 2019, with a temporal resolution of one hour, which we average across days to obtain daily data. The spatial resolution of the mesh grid is 12 km, resulting in 56, 66,776 points, from which we randomly sample few locations to recreate a sparse scenario. From the 13 simulated photo- voltaic panels, we select module 00. Finally, we take the average across the 21 forecast members of the provided ensemble.
E ADDITIONAL ANALYSIS
E.1 SCALABILITY AND COMPUTATION TIME
The asymptotic computational complexity of GgNet originates from the sum of the complexities of its components. The time and space complexity are both O(NTTD) + *(NTDD). As it is often safe to assume that D < N2, the overall complexity reduces O(NTTD) The main bottleneck in GgNet is due to the inter-locations graph which makes message-passing operations scale quadratically w.r.t. the number of locations. If scalability to large N is a concern, existing sparse graph learning methods can be considered (Niculae et al. [023 Cini et al. (023b). However, note that scenarios that involve significantly more sensors than the selected datasets would not likely be sparse and different set of techniques should be used.
We also report here the computational times of different methods on the daily climatic dataset (N = 235, T 10958, D 10) under the best hyperparameter setting. It takes approximately (on average) 2h for BRITS to complete one training. 7h for SAITS, 16h for GgNet, 14h for GRIN. Approximate time per epoch are: 40s for BRITS, 4m 40s for SAITS, 6m for GgNet, 8m 30s for GRIN. The temporal modeling in GgNet is based on temporal convolutions, which makes faster than the popular graph-based representative GRIN, which uses a recurrent model instead. As for BRITS and SAITS, these are faster than GgNet as they do not account for spatial dependencies. Timings are taken on a machine equipped with an NVIDIA A100 GPU.
E.2 ABLATION STUDY
The GgNet architecture is composed of the following principal constituents: 3 layers of temporal convolutions, which we indicate with 3T, and one pairs of alternate spatial and channel convolutions: G-g. The sequence is iterated twice to allow for deeper processing, resulting in the following layers' pattern: 2(3T - G - g). We provide in Tab. 3 an ablation study by removing individual components; model's hidden size is h 64, without residual connections and variable embeddings (Ey). Considerably worse performances are obtained by removing the g-convolution, i.e., 2(3T supporting the importance of modeling the relations with the covariates in reconstructing the missing channel in a sparse setting. Removing the temporal convolution, i.c., 2(G - g). forces the model exchange only instantaneous information across locations and channels, missing the temporal
hhttps://pwwer lars. masa.gov.
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Table 3: Ablation study. Results averaged across 3 runs. 2(3T- G- represents the architecture for which results are presented in the main text of the paper.
TEMP
TEMP:
TEMP:
WIND
REI-
PREC:
TEMP.
CLouns
IRR.
IRR. LONG
AvG (D)
MEAN
RANGE
SPEED
HUM.
SHORT
2(3T G)
9.2 25.9
8.411.
33.1
8.5 83.0
14
33.1
17.7 a.
23.
2(G 9)
4.8.0.8 18.512.4
3.911.
28.012.4
5.610.a
19.1
7.6113
20.611.9
11.311.2
3.2
17.3
2(3T g)
3.120.2 11.920.7
2.5.0.3
29.1
2.840.5
68.0g3.0
4.920.2
20.820.4
14.121.0
3.4g0.3
16.1
2(3T G 9) 2.6 10.9 2.6 25.0 2.8 61.7.1.4
4.400.s 17.340.4 9.2a0.8 3.2 14.0
context. Removing the G-convolution, i..., 2(3T 9), negates the model from exploiting information from other locations at inference. This variant has access to the same information as other models that do not consider spatial information, e.g., SAITS.
E.3 GEOGRAPHICALLY ISOLATED LOCATIONS
In this section, we qualitatively discuss the disadvantages of graph-based approaches in reconstruct- ing virtual sensors in geographically sparse locations. This is done, in Fig. 6 (left), by illustrative example on the daily climatic dataset. The plot compares the reconstruction of the maximum tem-
Port-aux-Françall (south Indian Ocean)
GgNat's iib locaticnn adjacency meb
TDPa HDNI
EG
SAFTS (MR2-126 (70
GRSV (PRC- 33..4%
TESK A4922
3.2
Ggfeet (NRE-10 394)
Temp. Max
ChaTt3
Jul 18 1909
Sep
MAAREE
ML
Figure 6: (Left) Reconstruction of maximum temperature (well correlated with other available vari- ables) in Port-aux- Port-aux-Français, which cannot exploit observations at nearby locations geographically isolated in the south Indian Ocean, (Right) colourmap visualization of the GgNet's learned intra- location graph (g), representation of the mutual dependencies between climatic channels.
perature channel, with different models, in Fort-aux-Français a geographically isolated location in the south Indian Ocean. For such a remote location, no information can inferred from nearby points, making T + Y dependencies ineffective in performing virtual sensing. As consequence, graph-based methods, e.g., GRIN, heavily underperform, as they rely on geographical proximity to build the graph and model spatial dependencies. On the contrary, GgNet or global recurrent mod- els, e.g.. SAITS, can effectively infer the missing channel by leveraging. at the target location, the learned dependencies between the target and the available covariates (C + Y dependencies). With the intent of enhancing this effect, we deliberately choose maximum temperature as target variable, which is highly correlated with other variables in the dataset.
E.4 LEARNED DEPENDENCIES BETWEEN VARIABLES
In GgNet, relationships between channels are accounted for by means of the intra-location graph g. In particular, we learn the edge scores of its adjacency matrix from D D free parameter matrix, normalized with a softmax function. In Fig. 6 (right) we provide a colourmap visualization of the leamed weighted (dense and bidirected) adjacency matrix. Interestingly, after processing the daily climatic dataset, several meaningful correlations have been captured: e.g., temperatures correlate with each other; lemperature range affects precipitation and wind speed; relative humidity corre-
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Published as a conference paper at ICLR 2024
lates with temperatures, precipitation and wind speed: precipitation correlate with clouds; longwave irradiation correlates with clouds and shortwave irradiation.
E.5 LOCATION-WISE ANALYSIS
Tables throughout the paper present results in a channel-wise fashion, i.c., for each channel d, per- formance metries are averaged across test locations. In this additional analysis, we present, for the daily climatic dataset, results averaged across variables, i.c., location wise results. With this anal-
SAITS
GgNet
0.3
MRE
0.25
0.2
0.15
0.1
0.2
0.3
0.4
0.5
0.6
0.2
Fraction training channels
Figure 7: (left) location-wise analysis of GgNet performance over SAITS for the daily climatic dataset. (right) Robustness of GgNet to the fraction of masked training channels on the daily climatic dataset.
ysis, we aim to demonstrate that the improvement of GgNet over SAITS not restricted to a few critical locations but is actually spread across the considered locations. As evidenced by Fig, (left), GgNet improves over SAITS for 2/3 of the considered target locations.
E.6 ROBUSTNESS TESTS
With reference to Sec. 4.4 the training of GgNet and the adopted baselines is guided by means of a specific mask. In partieular, we mask all timestamps in a batch) and train the models to reconstruct such masked portions of data. This is core aspect, it pushes the training toward learning the virtual-sensing task. In this regard, investigate the accuracy of GgNet on the daily climatic dataset for different values of the fraction of masked training channels. The study is reported in Fig. 7 (right); the model's 's layer organization is 4T 2(G-g), model's 's hidden size is h = 64, without residual connections and variable embeddings (E Results show a rapid improvement as soon as small fraction used, followed by robust performances.
fraction of the available training channels (entirely for
E.7 UNCERTAINTY ESTIMATES
Virtual sensing models often employed guide decision-making at unsampled locations. In this regard, is important to pair the estimate for the virtual channel with a prediction of the corre- sponding uncertainty. In fact, channels some locations may be, in principle, more challenging reconstruct than others. In GgNet, uncertainty estimates are obtained by employing the sum of three quantile losses (Appendix C.I). Fig. X explores four different scenarios where the reconstruction is gradually more challenging. The extent of the uncertainty estimates is consistent with the dominant factors assessing the quality the predictions, i.e., the correlation between the target and the available covariates and the presence of similar latent representations in the training set.
E.8
OTHER METRICS
Results provided in the main text in terms of the most relevant metric for the task at hand. This is MRE for the climatic dataset, as different channels are expressed in different units, and MAE for the photovoltaic dataset, power output is the only variable being reconstructed. Here, also define
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Published as a conference paper at ICLR 2024
Andorm Vella
large
Prodicted
Prodicted
Truth
Truth
1996: Neat. (0)
-20
Vienna
Yaren
Predicted Truth
Predicted Truth
Tutts Parga (C)
10
clubs (N)
2966
2000
Time
Figure 8: Daily climatic dataset: GgNet predictions for different channels at different locations, uncertainty estimates appear consistent with the difficulty of the reconstruction.
a custom metric, Variance Rescaled Error (VRE), tailored to this task:
i [n] - =M[n]
VRE(X* (n))
(18)
T
σd [n]
where [nnn] and x [n] are the predicted and true d-th channel n-th location, [n[n] is the standard deviation along the temporal axis of the true signal. Results are consistent across different metrics. We support this statement by reporting below the testing accuracy, for all experiments, in terms of the other metrics. Specifically, Tab. 6 contains the channel-wist results for the hourly climatic dataset that were omitted from Tab. to respect the space constraints.
Table 4: Daily climatic dataset: channel-wise MAE. Results are averaged across locations and random seeds. The best- persorrerfomiggggg method is in bold, the second-best is underlined.
Cn.
TeMm:
ToMe:
TEME:
WIND
REL.
Purc:
TOME:
CLOUDS
IsR.
Ing.
KNN
3.21
2.84
2.974
1.47=0.19
9.00=a..0
2.
3.49
19.53-o.ss
3..46+1.22
21.87+aa.3
BRITS
0.82+o.00 1.19
0.78+0.12
1.40.0.04
4.04
2.30+o.21
1.22#0.15
17.16
42.
42.41+2.18
12.49+1.00
GRIN
0.86 1.26
1.04
1.33
4.1520.45
2.20
1.26
16.70_o.32
31.14
14.30 20.92
GRIN mM
0.59 0.88
0.67
L3320.12
2.6420.65
2.00+0.19
0.93
11.79
0.39
23.70+n.94
13.13:+122
SAITS
0.49 20.07
0.76
0.5610.10
1.72000
2.42
1.8840.0
0.73
20.14
11.86
28. 1..5
10.07
GGNKT
0..07
0.65
0.48
20.00
1.04
1.98
1.73
0.62
9.48
20.35
18.73
10.394g.s1
RNN
1.1610.10 1.4020.13
1.40a0.12
1.42.0.05
5.08
2.3940.20
1.57
20.38_0.24
11.15
13.97
RNN.
0.87 1.30
1.1240.30
1.35
4.34
2.2240.16
1.25
18.06_0.1
38.591u.9t
13.13
RNN
0.85 1.1920.12
1.03
1.31g007
4.01
2.20
1.23
17.36_o.2
35.48
13. 18.18
RNNG
0.74+0.04 1.12#0.12
0.94
1.20+0.05
3. 3.800
2.13+0.20
1.06
15.90-0.33
28.47+1.00
13.04#111
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Published as a conference paper at ICLR 2024
Table 5: Daily climatic dataset: channel-wise VRE (%). Results are averaged across locations and 5 random seeds. The best-performing method is in bold, the second-best underlined.
CH.
TEMP: MEAN
TEMP: RANGE
TEMP: MAX
WIND SPEED
REL. RUM.
PREC.
TEMP. DEW
CLOUDS
IRR. SHORT
[RR. LONG
AVo (D)
KNN
105.8 16..3
213.3+s5.e
90.9.7.4
116.4=26.a
102.3+s.4
4 56.8+4.2
100.8=10.0
72.1+2.
59.7+3.7
95.9
101.4=9.a
BRITS
27.5=s.1
75.1+11.2
23.4+a.4
104.0=13.2
45.2+6.6
41.6+122
30.9+4.1
63.7+s.4
07.2+4.4
53.9
53.2+1.9
GRIN
29.7=3.6
75.8+12.2
31.0+2.5
99.4+1a.7
48.2+4.1
39.9+1.4
36.1+2.1
61.5+18
50.4.3.
64.3a9.5
53.6+a.2
GRIN
23.3=s.1
35.6+7.5
21.966.
101.8=18.6
30.1
37.0+2.
27.4a..1
13.6
39.6
61.810..
44.5+2.7
SAITS
16.8
44.8
16.5,2.9
88.8
26.0
(8877
20.1
44.0+2.1
47.5
38.341.5 45.2
GGNET
14.6
35.8
14.4
20.5
22.1
30.921.8
17.3
35.2
31.323.
48.9.
33.1
RNN
37.2
89.6g5.4
39.3
105.2:12.0
56.824.3
44.0ga.
42.0ga.
74.7211
69.1
60.0g7.
61.8g1.1
RNN
28.923.4
74.829.5
32.7aa.3
10100111.
48.7a4.2
40.92aa
33.32a.
66.4a1
61.42s8
57.3
54.520.8
RNN
26.924.1
63.213.
29.7a3.
99.821277
45.823.9
40.0g1.4
33.323.
63.8
57.324.
58.4
51.82n.
RNNo
24.5ga.a
60.217.5
28.0gn.3
93.12m
43.622.4
38.62a0
29.0ga.1
58.511.
46.223
58.6210.5
48.0ga.0
Table 6: Hourly climatic dataset: channel-wise MRE (%). Results are averaged across locations and 5 random seeds. The best-performing method is in bold, the second-best underlined.
CH.
TEMP: MEAN
WIND SPEED
REL. HUM.
PREC.
TEMP:
IRR. SHORT
IRR. LONG
AvG (H)
DEW
KNN
15.9
38.025.2
14.6
133.6_7.s
23.1
21 21.421.9 6.7
36.2+1.9
BRITS
7.4.2.5 34.621.
6.5
85.4
9.111.5
36.512.
4.610.4
26.31.0.
GRIN
6.5 35.5=1.
5.3
86.9+2.3
8.641.2
16.2#1.0
4.4
23.3+0.7
GRIN
6.0.2.22
34.5
4.8
85.5
8.0g14
16.1
4.3
22.74:0.4
SAITS
4.7.1.s 29.040.8
4.3.0.8
82.8
6.6,1.2
24.441.6
3.7
22.2+0.5
GGNET
29.2 4.5
4.3
78.4
5.720.9
17.4+3.s
3.6
20.4
RNN
8.0.2.7 38.9
6.6
90.811.3
9.711.
43.213.3
4.5
28.810.8
RNN
5.9-2.0
36.5
5.3
87.1
7.7+12
33.3+2.0
4.1
25.7+0.5
RNN cma
6.642.8
36.4.0.9
5.6+1.1
86.8+1.1
8,2+1.5
35.643.2
4.5+0.4
26.2+0.9
RNNG
5.722.
32.2=2.7
5.120.9
83.023.0
7.1210
25.1
4.0
23.2
Table 7: Hourly climatic dataset: channel-wise MAE. Results are averaged across locations and random seeds. The best perst-eerfomin method is in bold, the second-best is underlined.
REL. NUM.
PREC.
Temp. Dew
MRAN
SPEID
SHORT
LONG
KNN
3.24::.22 1.6420.25
10.59
0.16.0.01
3.54 43.52+4.19
24.54+1.55
BRITS
1.49+0.44 1.4940.11
4.68
0.10
1.39
74.13=44.00
16.84
GRIN
1.3220.40 1.53g0.00
3.84
0.10
1.31
20.00
32.91
15.99
GRIN m
1.23
1.4940.16
3.4720.00
0.10
10.00
1.22
32.74
15.6020.90
SAITS
0.96
1.25
3..1
0.10
20.01
1.00
49.67-9.34
13.31
10.77
GGNET
0.91 1.26
3.11
0.09
=0.01
0.87
15
35.28
13.1440.40
RNN
1.61 1.68
4.7640.50
0. 20.00
1.49
87.8626.70
4..2
RNN
1.194037 1.57
3.82
0.10
1.1810.1
67.75
15.00
RNN cmb
1.34 1.57
4.09
0.10
1.25
72.50
16.29 1.30
RNNG
1.1420-43 1.39
3.72
0.10
1.09
50.9925.50
14.71
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Published as a conference paper at ICLR 2024
Table 8: Hourly climatic dataset: channel-wise VRE (%). Results averaged across locations and random seeds. The best-performing method is in bold, the second-best underlined.
Cu.
TuMP: MEAN
WIND
KEL..
Pauc:
TEMe: DEW
Iast SHOKT
InR. LONG
AvG
KNN
77.4=5.2
96.9
83.9+11.9
49.1
106.9-21.2
16.3+1.4
83.5
73.4=4.1
BRITS
35.9
83.324.4
36.1
29.0
38.425.0
26.821.5
59.226.5
44.1
GRIN
37.8 85.0
32.6
29.2
42.7
12.2.0.s
59.1
42.7a1.1
GRIN
40.1
82.8
30.0
29.2+0.4
40.0
12.1
58.2+7.4
41.8
SAITS
24.3
69.9
25.5
28.3
28.1
18.321.3
46.5g1.1
34.4g1.
GGNET
24.0=6.0
70.6
26.3
27.3+1.4
25.3
12.
46.9.4.4
33.3+0.0
RNN
40.5
93.5
36.5142
31.2#0.9
44.3445
31.822
56.0s2.o
47.7=2.0
RNN
29.9
87.4g4.
30.5+5.
29.521.
33.3+a.1
24.8
51.7+2.2 41.0g1.
RNN cmna
35.6284
88,4ga
33.846.9
29.1a1.0
33.917
26.212.
57.221.8
43.4ga2
RNNG
32.9-11.1
78.0+4.6
31.0+a.6
27.8+1.6
33.7+6.0
18.5+1.
52.9.2.2
39 39.3
Table Photovoltaic dataset: MRE (%). For each N, results are averaged across sampling of the locations and 5 different runs for each set (25 runs in total). The best-performing method is in bold, second-best underlined.
NEACATINS
10
20
30
70
100
150
200
GRIN
20.1 +
18.8
18.6 +: 2.2
18.2
18.8
18.4
18.3
18.5 + 1.6
GRIN
4.7
2..0
1.5
0.8
14.6
SAITS
3.0
15.8 2:3
15.6 1.8
15.1 1.141.4
15.4 1.8
14.7
14.6 14.6
GGNET
15.6 +: 3.5
15.9
15.3 1.9
14.7 14..111.5 1.5
14.5 +: 1.3
13.7 +: 1.0
12.8 +: 0.7
12.5 + 0.7
Table Photovoltaic dataset: VRE (%). For each results are averaged across sampling of the locations and 5 different runs for each set (25 runs in total). The best best-performing method is in bold, second-best underlined.
NiccaTows
30
50
100
150
200
GRIN
40.7 + 11.9
38.2 4.3
37.8 3.6
37.4 3.5
37.0
36.9 : 2,8
37.4 3.0
GRIN...
31.8 7.1
32.8
31.7 3.3
29.8 1.8
30.3 1.7
30.0
29.5 - 1.2
29.8 -
SAITS
32.8 -
32.4
31.7
3..0
30.8 1.9
29.7 1.3
29.5
1.2
29.6
GGNET
31.8 4.4
32.6 4 3.4
31.2 2.6
29.2 2.0
29.0 = 2.0
27.8 -: 1.7
26.0 + 1.4
25.5 1.4
22


Reviews:
[{"review_id": "tdOR0mu9OF", "reviewer": "Reviewer_CZHK", "summary": "This paper proposes a solution to the multivariate spatial interpolation problem through a nested graph representation and graph convolutional networks. Real climate data is used to validate the performance of the proposed solution.", "questions": "1. Why Kriging would not work in this case is not adequately justified. The semi-variogram function can be selected from a variety of options depending on the assumptions. As long as the two locations are not farther than a threshold (the range) apart, they can be assumed to have correlations. \n\n2. Another traditional model that might solve this problem is Markov random field. Why not considering it in the baseline?\n/?？\n3. How to define the graph edges based on (static) embedding? What information do you used to learn it?", "limitations": "", "rating": 3, "confidence": 5, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The problem of spatial interpolation through deep learning is generally meaning for and important.\n2. The authors acknowledge traditional spatial interpolation methods such as Kriging\n3. Real-world dataset used for experiments.", "weaknesses": "1. The claim that Kriging or similar approach won't work due to sparse sensors lacks sufficient justification. The proposed solution can be solved by multivariate Kriging. The authors should at least show the results on such a well-known spatial solution and compare it with their solution.\n\n2. The graph of sensors is built by measuring similarities between node embeddings. However, it is unclear how to learn such embeddings and why it is not affected by sensor sparsity. If two sensors are too far away their values have no correlation. The edge defined in under such a case would be meaningless. \n\n3. There is no demonstration of a successful prediction of missing values on a spatial map.", "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": "", "extra_scores": {}}, {"review_id": "jjNP7ABgGq", "reviewer": "Reviewer_L9oj", "summary": "In this paper, the authors presented a novel graph-based framework for virtual sensing from sparse multivariate spatio-temporal observations, called GgNet. They leveraged a nested graph structure to account for dependencies between covariates and relations across locations and learned end-to-end to maximize reconstruction accuracy. The GgNet achieves superior performance in settings with poor sensor coverage, where other state of the art fail, and contributes to the methodological advancement of the field as well as a powerful tool in practical applications.", "questions": "①\tWhy is GgNet only applicable to transductive learning settings in its current form? What are the limitations? How to extend GgNet to support inductive learning?\n②\tHow does the efficiency of Ggnet compare to other models? What is the impact of missing data in the dataset used in the experiment on the experimental results? Can you offer a more detailed explanation of the results?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "①\tThe GgNet proposed in the article effectively addresses the limitations of traditional methods in the context of sparse sensor coverage by inferring signal values at unmonitored locations through learning dependencies between variables and positions.\n②\tThe article provides numerous details in the appendices to help readers gain a more comprehensive understanding of the framework. For instance, Appendix B and C offered a detailed description of the experimental settings and baseline parameter details.\n③\tThe author employed effective training method. They masked an additional small fraction of data points, at random, in the training channels, which can enforce robustness to random missing value.\n④\tThe experimental baseline selection is quite comprehensive and reasonable, including KNN, BRITS, SAITS, and GRIN, along with various progressively more advanced recurrent RNN. These choices in baselines contribute to a more comprehensive and reliable set of experimental results.", "weaknesses": "①\tThis article lacks a thorough analysis of the experimental results and a comprehensive interpretation of the data. It primarily presents the experimental outcomes in tables without providing detailed explanations.\n②\tThe article did not include an overview diagram of the overall model structure, which made it difficult for readers to comprehend the architecture of the model.\n③\tThe experiment evaluated accuracy-related metrics such as MAE and MRE but did not provide information on the computation time and performance on large datasets. As a result, readers cannot assess the efficiency of this model.\n④\tThe experimental training methods have the potential to enhance the robustness of the network, but in the end, no robustness test results for the model were provided.\n⑤\tThe experiment did not conduct tests of this framework with different data types of other domains, which makes it difficult to assess the comprehensiveness of its application in various fields.", "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": "", "extra_scores": {}}, {"review_id": "1w2AqSiD5I", "reviewer": "Reviewer_cynR", "summary": "The paper presents a novel framework for virtual sensing of missing multivariate spatio-temporal data in settings with limited sensor coverage. The authors introduce GgNet, a deep learning architecture capable of leveraging inter-location dependencies and relationships between covariates for accurate data reconstruction. \nThe contributions of the paper include: \n•\tThe introduction of a novel deep learning architecture designed for multivariate virtual sensing tasks, particularly in sparse scenarios. \n•\tNested Graph Structure: GgNet employs a nested graph structure to capture both spatial relationships between locations and dependencies between covariates, learning them end-to-end for improved reconstruction accuracy.", "questions": "•\tThe paper mentions the computational time for GgNet and other baselines but does not discuss scalability. How does GgNet's performance scale with more extensive sensor networks or larger datasets?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "•\tThe problem formulation of multivariate virtual sensing in sparse scenarios is itself an original and significant contribution, as it addresses a practical challenge across various domains. The introduction of GgNet leverages a nested graph structure to capture dependencies between covariates and relations across locations, is highly original. The combination of graph-based modeling and deep learning techniques to address the problem is innovative and distinguishes this work from prior methods in the field. \n•\tThe technical content, methodology, and experimental design are well-supported with empirical evidence. The thoroughness of the experimental evaluation, conducted across multiple datasets, with different degrees of sparsity and temporal resolutions, enhances the paper's quality. The extensive experimentation, use of real-world datasets, and evaluation further emphasize the paper's significance.", "weaknesses": "•\tThe paper, while generally well-structured, could benefit from more explicit and detailed explanations in certain technical aspects. For instance, the paper could provide a more comprehensive explanation of the nested graph structure used in GgNet, which is a key component of the proposed method.", "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": "", "extra_scores": {}}, {"review_id": "PPLseI8SNX", "reviewer": "Reviewer_MbBb", "summary": "This paper aims to address the data sparsity problem by using multivariate data. The authors proposed a model GgNet which can leverage the correlations/dependencies between the target variable/channel and the covariates to reconstruct the target values at certain locations. Extensive experiments are conducted.", "questions": "Please address the questions above.", "limitations": "", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper is well-presented and well-organized.\n2. The paper proposed a new model GgNet aiming to reconstruct the target values at certain locations.\n3. Extensive experiments are conducted to validate the proposed model, and the results seem promising.", "weaknesses": "1. This paper has a lot of assumptions, for example, it assumes the covariates are always available, and the mutual dependencies between target and covariates are invariant and can be leveraged at all locations. Here are some concerns: we cannot guarantee that those assumptions hold for all cases (at least, there should be more examples or references to support them), and if they do not hold, does the proposed method still work?\n2. It seems the method requires the latent representations of locations where the target values need to be reconstructed are close, so is it possible to deal with the cases where the latent representations of locations are remote?\n3. The model somewhat lacks novelty, it seems the final model just combines several existing models.", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)