DeformTime: capturing variable dependencies with deformable attention for time series forecasting
Authors: Yuxuan Shu
Vasileios Lampos
Venue: Accepted by TMLR ()

Abstract: In multivariable time series (MTS) forecasting, existing state-of-the-art deep learning approaches tend to focus on autoregressive formulations and often overlook the potential of using exogenous variables in enhancing the prediction of the target endogenous variable. To address this limitation, we present DeformTime, a neural network architecture that attempts to capture correlated temporal patterns from the input space, and hence, improve forecasting accuracy. It deploys two core operations performed by deformable attention blocks (DABs): learning dependencies across variables from different time steps (variable DAB), and preserving temporal dependencies in data from previous time steps (temporal DAB). Input data transformation is explicitly designed to enhance learning from the deformed series of information while passing through a DAB. We conduct extensive experiments on 6 MTS data sets, using previously established benchmarks as well as challenging infectious disease modelling tasks with more exogenous variables. The results demonstrate that DeformTime improves accuracy against previous competitive methods across the vast majority of MTS forecasting tasks, reducing the mean absolute error by 7.2% on average. Notably, performance gains remain consistent across longer forecasting horizons.

Published in Transactions on Machine Learning Research (04/2025)
LeforrTimee capturing variable dependencies with deformable attention for time series forecasting
Yuxuan Shu
yuruan. www.sshh.....cc.cooo
Centre for Artificial Intelligence Deportment of Computer Science University College London
Vasilcios Lampos
v.lampos @ucl. ac..call.........oooooo uk
Centre for Artificial Intelligence Department of Computer Science
Universily College London
Reviewed on OpenReview: https:///penneeviaal net/ formm id= N62P74077
Abstract
In multivariable time series (MTS) forecasting, existing state-of-the-art deep learning approaches tend focus on autoregressive formulations and often overlook the potential of using exogenous variables in enhancing the prediction of the target endogenous variable. To address this limitation, present DEFORMTIME, neural network architecture that attempts to capture correlated temporal patterns from the input space, and hence, improve forecasting accuracy. It deploys two core operations performed by deformable attention blocks (DABs): learning dependencies across variables from different. time steps (variable DAB), and preserving temporal dependencies in data from previous time steps (temporal DAB). Input data transformation is explicitly designed enhance learning from the deformed series of information while passing through DAB. We conduct extensive experiments on MTS data sets, using previously established benchmarks well as challenging infectious disease modelling tasks with more exogenous variables. The results demonstrate that DEFORM improves accuracy against previous competitive methods across the vast majority of MTS forecasting tasks, reducing the mean absolute error by 7.2% on average. Notably, performance gains remain consistent across longer forecasting horizons.
1
Introduction
Time series forecasting models provide multifaceted solutions for many application domains, including health (Shaman & Karspeck 2012 loannidis et al. 2022), climate (Reichstein et al., 2019), energy (Deb)
et al. 2017), transport (Li et al. 2018), and finance Sezer et al. 2020 In their pursuit for greater accuracy, forecasting methods have always been trying to benefit from multimodal exogenous predictors (De Gooijer & Hyndman 2006 Athanasopoulos al. 2011). Over recent years, plethora of new digitised information sources has become available, accompanied by a rapid development of capable and efficient deep learning architectures (Rangapuram et al. 2018 Lai et al.) 2018). Current state-of-the-art (SOTA) time series forecasting models are drawing particular focus the deployment of many-to-many problem formulations (Nie) ct al. 2023 Wu et al.) 2023 Luo & Wang 2024b Wang et al. 2024a). However, this approach does not necessarily provide the right learning mechanism for multivariable time series (MTS) forecasting (Hidalgo & Goodman 2013), where we have definitive target (endogenous) variable and a considerable amount of external (exogenous) input variables. Furthermore, although various neural network (NN) architectures were
proposed address forecasting challenges with multiple inputs (Zhang & Yan 2023
Yi al. 2023a Huang
et al. 2023 Luo & Wang 2024b), many (Zeng al. 2023:
Nie al. 2023 Jia et al. 2023
Lee al. 2024

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Jin et al. 2024 Lin et al. 2024 were not designed to incorporate inter-variable dependencies, key property
of conventional forecasting models (Pi & Peterson 1994 Hyndman & Ullah 2007 Bontemps et al. 2008)
There certainly exist neural forecasting models that attempt to leverage dependencies across variables and time. Solutions based on recurrent neural network (RNN) architectures (Lai et al. 2018 Lin et al. 2023 can capture information from previous time steps (up to an extent), but have a limited capacity in establishing
inter-variable dependencies. LightTS (Zhang et al., 2022) uses multi-layer perceptrons to model dependencies within the input data, first along the temporal dimension and then across variables. For transformer-baser models, Crossformer (Zhang & Yan, 2023) proposes to first partition the input into temporal patches, and then use a router module to capture information across variables from different time steps, improving performance compared to pre-established RNN and Graph NN architectures (Lai et al., 2018 Wu et al.) 2020). However, both LightTS and Crossformer have been outperformed by later proposed models that do not establish
dependencies between variables (Zeng et al., 2023 Nie et al.) 2023 Jia et al. 2023 Lee et al. 2024 Jin et al.
2024 Lin et al. 2024). The counterargument from these approaches that models without inter-variable dependency modules benefit from being able to use longer look-back windows without significantly increasing model complexity (Han et al. 2023). Nevertheless, common issue throughout the deep learning forecasting literature that casts doubt on some of these conclusions is the inappropriateness of various benchmark tasks (see Appendix A.4 In addition, recent work has argued that to improve performance while exploiting inter-variable dependencies, more effort. is required in temporally aligning the input time series (Zhao & Shen 2024 Attempts have also been made to capture dependencies with large receptive fields within segmented time scries patches (Luo & Wang 2024b). Motivated by this, we have introduced guided re-arrangements of the input to better capture inter- and intra-variall dynamics.
Deformable neural networks were initially proposed in computer vision Dai et al. 2017 to accommodate geometric transformations with Convolutional Neural Networks (CNNs). Combined with transformer-bsee NN architectures, deformation has achieved SOTA performance in various tasks (Zhu et al. 2021 Chen et al.,
2021 Xia ct al. 2022). Prior work (Wang ct al. 2024c) has also deployed deformable mechanisms within time series forecasting. However, their application ion was limited to establishing intra-variable dependencies, a structural design also shared in (Luo & Wang 2024a), making them less effective in capturing correlations between multiple variables over time. Importantly, this model was outperformed by later ones (Nie et al. Liu et al., 2024). Based on the aforementioned insights, this paper introduces DEFORMTIME, a novel MTS lorecasting model that deploys deformable module on top of transformer encoders to introduce some flexibility in the determination of receptive fields across different variables and time steps. The premise of DEFORM is the inclusion of learnable mechanisms that we refer to as deformable attention blocks (DABs). These enable transformations of the input information stream that enhance learning from key patterns within endo- but most importantly exogenous variables, an operation that ultimately improves forecasting accuracy (for the target endogenous variable). Additionally, input is transformed in alignment with the operational goal of the DAB modules, guiding the modelling of dependencies within time intervals and at various temporal resolutions. Our model also deploys both fixed and learnable positional embeddings to preserve the sequential order of the input time scries and capture relative positional information after deformation, respectively. These embeddings work in tandem within the proposed architecture.
2023
We summarise the main contributions of this paper as follows:
(a) We propose DEFORMTIME a novel MTS forecasting model, that bet captures inter- and intra-variable dependencies at different temporal granularities. It comprises two DABs which facilitate learning from adaptively transformed input across variables (V-DAB) and time (T-DAB).
(b) We assess the predictive accuracy of MTS forecasting models on 3 established benchmarks as well as 3 new infectious disease prevalence tasks, each across 4 increasing horizons. We argue that the disease prevalence tasks support a more thorough evaluation because data sets include a substantial amount of exogenous variables, and encompass multiple years, 4 of which are used as distinct annual test periods.
(c) In our experiments, DEFORMTIME reduces the mean absolute error (MAE) by 7.2% on average compared to the most competitive forecasting method (which may be a different one for cach evaluation). There is a 5% MAE reduction based on established benchmarks and 9.3% for the disease forecasting tasks. Overall, performance gains remain stable as the forecasting horizon increases.
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Published in Transactions on Machine Learning Research (04/2025)
Variable DAB (V-DAB)
yeR
Temporal DAB (T-DAB) E
Exg + Z,
Decoder
Transformation
W,
Zue é t
Projection
Eiger
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GRU
MLP
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2z Z,eR exglo
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Grouping
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Figure 1:
The architecture of DEFORMTIME. We use the notation introduced in sections and DEFORMTIME's core modules comprise two deformable attention blocks (DABs), variable DAB (V- DAB) and a temporal DAB (T-DAB) that respectively capture inter- and intra-variable dependencies. Both DABs reside in the Encoder module. We deploy a 2-layer GRU as the Decoder. Finally, we have visualised key data operations (Segment and Adapt blocks) that take place in the DABs.
(d) As an aside, we note that models that attempt to establish some form of inter-variable dependency performed better in evidently (based on the results) the more challenging task of discase forecasting. This also highlights the need for more appropriate methods of assessment.
2 MTS forecasting task definition
Across our experiments, we consider an MTS forecasting task where the focus is on a single target variable, i.e. there might exist multiple inputs, and optionally multiple outputs, but we are only concerned with the predictions of one output variable. All models follow a rolling window forecasting setting with a look-back window of L time steps. To be more precise, time step t, E holds the time series of C exogenous variables over the L time steps autoregressive historical values for the target variable for time steps that 8 e No introduces an optional delay of d time steps between the observed exogenous variables and the endogenous (target) variable. This becomes relevant in a real-time infectious disease forecasting task, where estimates for the rate an infectious disease are becoming available with a delay of 8 = or 14 days, when other indicators are not affected by this (sce Appendix A.2). Joining both input signals, i.c. the exogenous and endogenous (autoregressive) variables, we obtain the input matrix Z. = [Q,, E (CC(CC1) Commonly, the prediction target is denoted by Y++ where H indicates the mumber of time steps we are forecasting ahead. However, some (baseline) models follow multi-tass learning formulation whereby every input variable becomes a prediction target. On this occasion, the prediction targets are denoted by Y(+H= = [Q4HH,
1, In addition, yı-s E
Rd holds the corresponding L 1-8-1, Note
(C++1) Hence, the aim of the forecasting task is to learn a function f(-) such that f: : Z, + Yt+H or Y(+H- Irrespective of the number of outputs, forecasting accuracy is reported on the final estimate of Yt+H, # ER, which denotes the value of the target variable at time step t H. For notational simplicity, we have chosen to only imply temporal subscripts for the remainder of the manuscript (i.e. Z over Z.).
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3 Time series forecasting with DeformTinee
An overview of DEFORMTIME'S structure is presented in Figure Motivated by assumption that variables (endo- and exogenous) in an MTS task may not only be correlated with the historical values of the target variable, but also with other predictors within adjacent time steps, DEFORMTTMEE deploys a modified transformer encoder layer with two core modules that attempt to account for both dependencics: a Variable Deformable Attention Block (V-DAB) to capture inter-variable dependencies across time, and a Temporal Deformable Attention Block (T-DAB) to capture intra-variable dependencies across different time periods. In the following sections, we provide detailed description of all modules and operations of DEFORMTIMES
3.1 Multi-head attention
We first revisit the multi-head attention mechanism originally proposed by Vaswani et al. (2017p. Given an input Z Rexa Q K, and Ve Ruxa respectively denote the query, key, and value embeddings projected from Z with learnable linear projection matrices Wq. WK, and W (as Q = ZWq). Assuming M attention heads, these embeddings are each partitioned into M non-overlapping submatrices column-wise. For example, the m-th submatrix of Q, denoted by Qm. has dimensionality of (u/M). Following up this notational convention, the output of the m-th attention head, Am E (R/M))/M), is given by
A m softmax Qm m /T/MM//MM mm-
The outputs from each head are then concatenated along the second dimension and linearly projected into the hidden dimension u with a weight matrix W E Rux= to form the output A ER When we set M = 1, we refer to this method as single-head attention.
3.2 Neighbourhood-aware input embedding (NAE)
The receptive field of DEFORMTIME is determined by a CNN with sampling offset (see section 3.3 This introduces limitation in capturing relationships between neighbouring variables. We seek to minimise the impact of this by learning neighbourhood-wware input embedding (NAF) as follows. We first re-arrange the order of variables in the imput Z, ranking them based on their linear correlation with the target variable in a temporally aligned fashion (see also Appendix D.1]. We then embed the curelation-driven re-arranged version of the input that captures C' variables to a hidden dimensson d. However, we do not perform this embedding as a single holistic step, but embed G neighbouring groups of variables, cach one to shortened embedding of size d/G, using fully connected layer Our expectation is that the presence of stronger intra- intra-group correlations will be subsequently leveraged by our model. Both d and G are hyperparameters that we learn during training. The embeddings of cach input variable grouping are concatenated to form E 1/xd Finally, we also add a simusoidal position embedding (Vaswani et al. 2017 Zhou et al. 2021 )
P. E Lxd to E, and normalise within layer as widely adopted in prior work (Dosovitskiy et al. 2021
Zhang
& Yan 2023 Hence, the embedding Z. E gxx of the input Z given by
Z. = LN(E + Pa)
(2)
where LN(.) denotes layer normalisation Ba et al. 2016
3.3 Variable deformable attention block (V-DAB)
V-DAB aims to capture inter-variable dependencies across time. This is achieved by performing cross- aross-attention over emboddings of both endo- and exogenous variables within proximal time steps. The embedded input sequence, Ze, is first segmented into palches along the temporal dimension that encompasses total of L time steps. This reduces model complexity By deploying a segmentation length l and a stride s < f (to ensure all data points are included), we obtain
only consider dependencies within adjacent time steps.
The source code of DRFORMITIME available githuttu.bb..c ccm//laaddaahuu/efoottt
2The input pero-padded to proper length, if the number variables divisible by G.
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[/s0/s] patches, denoted by Zp Ex xd We augment the vanilla Transformer attention (Vaswani et al. 2017) with deformable mechanism which allows the network to adaptively attend across time and variables. Each data point in patch Zp can be represented by a pair of (row, column) indices, = (i.j). We sample from a deformed position p + Ap over Zp to obtain the key and value embeddings, Ka, VaeRlxd respectively[
To determine the position offset A we conduct the following series of operations. We first obtain a query embedding, Qp ER over patch of the input as described in section 3.1 i.c. Qp Zp Q. We then pass Qp from 2D CNN, denoted by fotz, comprising convolutional layer with k k kernel that captures neighbouring information, and 1 x convolutional layer that projects the embodded feature. We use the tanh activation function and a learnable amplitude a E (that controls the upper bound of the sampling range) to obtain Áp. Hence, Ap is given by
Ap a tanh (fors (Qp))
3
We use Ap and bilinear interpolation (v) over 2x2 grid determined by p + Ap DDii et al. 2017 to obtain the deformed patch Za:
Za = 3 (Zp: + Ap)
5
(4)
We then obtain Ka Va from Za with
Kg Wa and Va ZaWv Pv,
55)
where ExeRed is relative positional bias (hha et al. 2018) that is added to assist in maintaining some pairwise positional information after the deformation.
A single head self-attention layer is applied to attend Qp to K4, Va given by
A: = (soffmaa tmax QQ,K] /vd V.) W.,
(6)
where W, e the weight matrix, to form the attention embedding E for the i-th segmented patch. The output from all patches is concatenated across the temporal dimension, forming A (( We conduct. cross-attention over the original latent variable captured in Qp and the deformed latent variables Ka, Va. Specifically, clements at position p can attend to elements at position p + Ap. This enables the model to learn from interactions between neighbouring variables across time.
The final output of the V-DAB module, Z, Lxd is given by
Z, = W. A,
(7)
where W, ( denotes the weight matrix of fully connected layer.
3.4 Temporal deformable attention block (T-DAB)
We use the T-DAB module in parallel with V-DAB to capture intra-variable dependencies across different. time periods. This is achieved by applying cross-attention embeddings that hold information from the same input embedding over neighbouring time steps. The T-DAB module receives the same input the V-DAB module (Z.). Based on insights from prior work (Wu et al., 2023 Jia et al.) 2023 we first adapt Z, to support learning from different temporal resolutions. Specifically, every column Z, ERL of Z. is transformed to x ER*x* where r denotes the time window (amount of time steps) that T-DAB is using for capturing temporal relationships, and λ = L/r This converts a vector representing L time steps, {t1, (f1,t2... ttth, to an r X matrix where the starting time for cach row is f2,,t2,.... 12,,,,,,,,, b, and the time stamps for the elements of the j-th row are 14,,4tm 1.+2r...; (ot.:..::-(xxxxx))) (++((x1111) 1) Therefore, Z. becomes Z grxxxd which comprises r patches; a patch for T-DAB denoted by ERodd
3z. is zero-padded into proper length, if L not divisible by
4Soc Appendix B for detailed example of how deformation works V-DAB.
fff sampling position out-oo-bbundh we interpolate using zero values.
6 is padded proper length with the last available value, L. divisible by r.
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The factor (time steps) can be set to different values across encoder layers to capture dependencies at various temporal granularities (see Appendix D). If we set r r- Ze remains unchanged. Similarly to V-DAB, we obtain the query embedding Q E over the transformed input using Q. Z, UQ, where Uq e Rddd is a weight matrix. We then implement an augmented Transformer attention that contains deformed information across the temporal dimension. We work our way column-wise, and for every column 22 E ZZe we sample (lincarly as opposed to the bilincar approach in V-DAB) from an index position p E.
The position offset (Ap) is shared among groups of correlated embedded sequences they may have similar temporal dependencies. Given that the original imput variables were already grouped based their correlation with the target signal (section 3.2) we expect that correlations are already captured in feature neighbourhoods of Zy. Therefore, to obtain Ap, we first divide Qr into the same G subgroups (column-wise) as in the
NAE operation. Each subgroup, Qg E (*x((/)) with = {1....,G}, G}. shares the same temporal deformation controlled by Aplo). Similarly to Eq. Apli) defined as
Apis) = a (
(8)
with Mote denoting ID CNN (with a kx1 and 1x1 convolution layers), and a being the same offset amplitude
as in V-DAB. We then obtain the matrix of temporally deformed sequences Za = [z!) ..... z!
E ERaxd
each submatrix, Z!!) E gxx((d/G), given by
Z 20 2r (2 Ap(a))
where ((-) denotes a lincar interpolation function over 2 adjacent points of 2, E ZZ determined by the deformed index p= + Ap(9). If sampling position is outside of the matrix boundaries, we interpolate with O.
Similarly to V-DAB, K,, V. eR are obtained from Zg, using K₂ = K. and V, Uy + Pr. where R is relative positional bias. Kg and V. thus contain elements sampled from the neighbouring time steps (zi.jo= zi.jto) We then apply multi-head attention to have elements at the position p allend to elements at the deformed position + Ap(s), This is achieved by first splitting K, and V, into G groups (submatrices) denoted by K, and V, E [=x((/()) and then by using A, = sof tmax ///TT// / Vg- The outputs from different heads are concatenated column-wiss wise and linearly projected with weight matrix W, E Rdddd to obtain the attention embedding of the i-th patch A, E Rxxd To form the output of T-DAB, denoted by Z. E the outputs of all patches are concatenated along the first dimension and re-arranged back to the original temporal structure (matching the input Z. as well as V-DAB's output Z.).
3.5 Encoder
The encoder of DEFORMTIME may consist of more than one layer. In our experiments, we deploy 2 encoder layers, similarly to related work (Liu et al. 2024 Wu et al. 2023). Each encoder layer contains two DAB branches (on the left and right side, respectively see also Figure D. comprising stacked transformer blocks (Dosovitskiy et al. 2021) with layer normalisation placed within the residual connection (Xiong el al.) 2020 Xia et al. p022). p. Instead of using vanilla attention, we are introducing a V-DAB (Ieft side) and a T-DAB (right side) module to capture inter- and intra-variable dependencics, respectively. The operations of branch can be summarised by
Z, Drop(DAB (Z.)) + Ze and
(10)
Z. Drop(MLP(LN (Z.)) + Ze.
(11)
where DAB(-) denotes the V-DAB or T-DAB operation, Z, Z. RRLxd are an intermediate output and the oulput of the DAB encoder layer, respectively, Drop(-) denotes stochastic depth (Huang et al. 2016) where layers within the network are randomly dropped during training with a learnable probability, and MLP(-) is a
2-layer perceptron with ReLU activation and hidden size of d. The outputs from the two encoder branches are concatenated over the second dimension. We feed the concatenated output into a fully connected layer form the output Zy E Lxd of the j-th encoder layer.
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3.6 Encoder-decoder structure
In T-DAB (section 3.4 we set the time window r to different values across encoder layers, enabling the model to attend to information at multiple temporal gramularities. To effectively learn from this encoding, we construct hicrarchical encoder structure motivated by prior related work (Zhou et al. 2021 Liu et al.,
2022a: Zhang & Yan 2023). Specifically, the output from encoder layer j-1, 1 25-1 E RR2 becomes the input. to encoder layer j. This is formulated by
Z Enc (Ze) Enc (Zej-))
j=
(12)
otherwise
where Enc(-) denotes the operations of an encoder layer with deformable attention (scction 3.5 of the final encoder layer is fed to a 2-layer Gated Recurrent Unit (GRU) NN with a hidden dimension of d that acts as the decoder. The output of the GRU decoder, Zeat, maintains the same dimensionality as the input, i.. dxd. Finally, we use 2-layer perceptron with hidden dimension of d and LeakyReLU activation to project zet along the temporal dimension, forming an intermediate output Zaut E ER xd where H denotes the number of steps we are forecasting ahead as defined in section [ We then feed Zou to a fully connected layer to lincarly project it into the target forccasting output ý R
The output
4 Results
We present forecasting accuracy results across 3 benchmark and disease rate prediction tasks, comparing DEFORNTIMME to 9 competitive baseline models. In addition, we provide an ablation study and computational complexity / efficiency analysis.
4.1 Experimental settings
Forecasting tasks and baseline methods. Experiments are conducted on real-world data sets. These include 3 established benchmarks from previously published papers (Wu et al. 2021 Zeng et al. 2023 Yi ct al.] 2023b Wu et al., 2023; Nie et al.] 2023 Luo & Wangl 2024b), and specifically weather-related and 2 electricity transformer temperature (ETThl and ETTh2) forecasting tasks with respective temporal resolutions of 10 minutes and hour. In addition, we have formed 3 discase prevalence modelling tasks, focusing on the prediction of influenza-like illness (ILI) rates in England (ILI-ENG) and US Health & Human Services (HHS) Regions 2 and 9 (ILI-US2 and ILI-US9). For the ILI rate forecasting tasks, we also introduce the frequency time series of web searches as exogenous predictors. In the ETT and weather tasks, we consider oil temperature and carbon dioxide level, respectively, as the target variables; we refer to the rest indicators as exogenous variables. We note that the ILI ing lasks have considerably more exogenous variables compared to the ETTh1, ETTT2, and weather tasks. Based on the evaluation settings in prior work (Nie) ct al. 2023 Lin et al. 2024), for the ETT and weather tasks, we set the forccasting horizon H to 96, 192, 336, and 720 time steps ahead, and use single test fold of consecutive unseen instances. For the ILI forecasting task, we conduct experiments on 4 consocutive influenza seasons (2015/16 to 2018/19) as separate test sets (4 test folds); the forecasting horizons we consider are Hl [7, 14, 21, 28} days ahead. Further details, including the evaluation setup are available in Appendix A While we are aware of other datasets used in MTS forecasting, not all of them are suitable because they may not have an explicitly defined target. variable or pursuc an ill-defined prediction task (see Appondix A.4 for further justification).
We compare DEFORMTIME to 9 competitive time series forecasting models that to the best of our knowledge, form the current SOTA methods. These are LightTS (Zhang et al. 2022), DLinear (Zeng) ct al. 2023), Crossformer (Zhang & Yan. 2023). PStchTST (Nic et al.] 2023). iTransformer (Liu et al. 2024), TimeMixer (Wang et al.] 2024a), Modern (Luo & Wang] 2024b), CycleNet (Lin et al. (0024), and TimeXer (Wang et al. 2024b). We also include a naive persistence model baseline. Further details and justification for the selection of basclines can be found in Appendix [ For the ILI forccasting tasks, we conduct hyperparameter tuning for all models. In the other benchmark tasks, with the exception of Crossformer and Lig we adopt the settings from their official repositories to reproduce results.
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Table 1: Forecasting accuracy results across all tasks and methods. H denotes the forecasting horizon time steps. For the ILI forecasting tasks, the table enumerates the average error across the 4 test scasons. Complete % denotes sMAPE. The best results are in bold font and the
results per season are shown in Appendix E
socond best are underlined.
Models DEFORNTONE Modera/TNN
CycleNet
Tinexer
PatchTST
TTrnnsoruer
TimeMiixrr
Crossformer
LighTS
DLinew
Pensistence
11
MAE
<%
MAE
<%
MAE < %
MAE
<%
MAE
c%
MAE
<%
MAE
<
%
MAE
<%
MAE
<%
MAE
<%
MAE
c%
96
0.1941
14.96
0.2047
15.06
0.1977
15.05
0.2135
16.03
0.2017
15.41
0.2052
15.46
0.2112
16.32 0.2126 16.52 0.2215
17.24
0.2399
20.82
0.2371
18.47
192
0.2116
16.08
0.24177
18.32
0.2411
18:25
0.2322
16.96
0:2409
18.29
0.2429
18.13
0:2382
17.91 0,2820
21.63
0.2636
20.55
0.3749
31.78
0:2803
21.46
336
0.2158
16.27
0,2415
18.52
0.2425
18.35
0.2414
17.34
0.2559
19.29
0,2583
19.11
0.2625
19.72
0.2947
22.65
0.2807
22.15
0.6328
08.94
0.3028
22.90
720
0.2862
21.81
0.2785
20.44
0.3018
22.96
0.2617
18.58
0.3087
23.59
0,2886
22.05
0.3055
23.25
0.3350
24.54
0.5331
44.57
0.7363
09.52
0.3222
25.29
ETT32
96
0.3121
40.07
0.3150
40.68
0.3282
40.74
0.3346
41.04
0.3145
39.25
0.3420
42.41
0.3454
41,27 0.3486
40.71
0.3507
41.80
0.3349
41.68
0.3522
43.85
192
0.3281
37.90
0.3867
47.08
0. 0.3687
41.78
0.4154
47.07
0.3830
45:45
0.4253
47.44
0.4183
47.49 O. 0.4035
63.16
0.4022
48.01
0.4084
50.67
0.4416
50.24
336
0.3450
87.00
0.3904
50.54
0,3811
0,4041
42.26
46.77
0.4032
45.95
0,4380
46.79
0.4487
49.44
0.4425
51.35
0.4710
05.03
0.4836
03.70
720
0.56-10
34.99 0.5722
63.04
0. 1828
89.99
0.51885
56.17
0.4960
65:27
0.4565
45.40
0.4729
46.37
0.5832
61.45
0.6252
70.50
0.7981
94.67
0.5199
58. 75
ILI-US9 ILI-US2 ILL-ENG Weather
0.0244
0.0279
42.49
0.0253
38.67
0,0301
44.04
0.0258
39.37
0.0277
42.39
0.0322
45.90
0.0271
44.92
0.0293
48.48
0.0251
39.
39.03
0.0529
51.63
192
0.0260
39.33 0.0314
20.05
0.0270
11.72
0.0326
06.85
0.0279
42.02
0.0277
42.77
0.0311
68.62 O. 0.0308
54.14
0.0319
51.45
0.0270
12.68
0.0361
$4.92
336
0.0291
44.26
50.12
0,0801
45,31
0,0348
49.12
45.31
46.01
62.53
0.0817
60.83
47.
0.0361
65.14
720
0.0363
53.72
0.0114
57.66
0.0383
56.17
0.0155
58.70
0.0389
56.04
0.0396
57.01
0.0457
59.82
0.0396
65.
0.0386
26.96
0.0552
51.51
0.0394
56.01
7
1.6417
28.61
1.9489
28,
2,5054
29.50
2,8094
33.06
2.3115
27.0
2,0084
26.38
1.1748
25.08
25.71
2. 2399
52.25
2.82114
43.02 2. 2.1710
24.96
2.2308
33.98
2,7050
36.01
3.3011
9.49197
0.2547
37.1
36.67
35.00
2.6544
30.97
2.6879
38.29
3.7022
65.28
3.0625
33.77
21
2.6500
32.70
3.0400
20.02
4.4519
52.83
4.3337
51.56
4.3192
51.11
4.2347
48.93
3.5501
49.36
3.0014
40.57
3.3616
51.78
4.4739
61.25
3.8617
42.03
2.7228 28
40.44
3,3611
47.87
5.0259
39.93
1.9011
61:60
1.9961
59.60
1.8125
55.35
4,0000
51.27
3.1963
16.14
3.1152
55.59
5.0347
67.75
4.5857
49.19
7
0.4122
16.01
10.55
0.8961
24.67
0.6088
23.38
D. 0.7097
21.52
0.0507
23.24
0.5281
20.07
0.0000
10.16
0. 16366222
16.74
0.7355
27 91
0.6474
22. 18
0.4752
17.73
20.22
29.15
0,7725
29.07
30.11
28.17
24.61
0.5862
91.96
0.5827
23.11
0.8155
28.24
21
0.5425
23.13
0.5781
23.85
1.0469
37.98
0.8243
.4
1,02866
36.70
0.8042
30.03
0.6794
27.68
0.6245
22
0.6683
20.27
0.9124
34.93
0.0633
33.51
0.5538
22.25
0.3710
23.66
1,1388
0.9074
34.72
42.61
0.9619
36.75
0.6512
23.91
0.7175
27.78
0.9805
38.54
7
0.2622
12:26 0,2899
14.17
0,4480
21.05
0.3813
18.20
0.4116
19.34
0.4057
18.57 0,3239
15.21 0.3149
14.44
0.3185
15.65
0.4675
23.47
0.4057
18.49
0.3084
13.80 0,3117
15:29
0.5072
24.02 0.4605
22.11
24.09
0,1702
22.44 8.4060
0.3571
17:23
0.3791
19.04
27.
0.5008
23.07
21
0.31
79
14:24 0.3710
15.45
0.58226
28.
28.47
0.5715
27.42
0.5035
29.
0.5106
24.11
0.4376
21.40
0.3418
15:90
0.4764
23.74
0.6001
29.
0.3006
27.41
0.3532
15.74
0.3940
17.19
0.7081
34.50
0.6535
31.32
33.35
0.6196
31.04
0.5121
21.11
0.3747
16.41
0.4769
23.22
32.16
31.07
Note that due to different task formulations, DEFORMMTIME and TimeXer focus on forecasting the endogenous variables only, whereas the other baseline models also predict future values of the variables that we consider as exogenous. Notably, we only use the last value of the predicted target variable y1+H (see section 2 for model evaluation, regardless of the number of outputs each model makes (sce also Appendix
DeformTime's sctup. We use G=4 variable neighbourhoods in all our experiments (sce scction 3.2 This setting (G'=4) 4) also determines the number of group partitions for offset generation and multi-head attention in T-DAB (section 3.4p. We provide a more detailed analysis of the effect of this specific parameter in Appendix E.4 The learnable amplitude a E (3, 5, 7.9} (see sections 3.3 and 3.4 is tuned collectively for both V-DAB and T-DAB. We keep the kernel size & of the interpolation functions ((-) and v-(() identical to a throughout the experiments. In all experiments, DEFORM
has encoder layers (see section 3.5p. For the ETT and weather tasks, we set r=1 in the first encoder layer and seloct from r6,6, 12, 24) time steps in the second encoder layer (see section 3.4p. In addition, we select the segmentation length from LE (6, 24] time steps (see section (.33 collectively for both encoder layers. Given the more direct weekly temporal resolution in the reported ILI rates and the fact that web scarch activity trends do have weekly patterns, in the ILI forecasting task, we set 8-7 time steps (days), and also set r=1 and r=7 7 time steps in the first and second encoder layer, respectively. The segmentation stride 8 is kept the same as e throughout the experiments.
Hyperparameters specific to forecasting tasks and optimisation settings. The look-back window L is set to 336 time steps for the ETTh1, ETTh2, and weather forecasting tasks, based on NNe et al. (003)) For influenza. forecasting, we set L to {28, 28, 56, 56} days for forecasting horizons H = {7, 14, 21, 28} days ahead, respectively. Only for influenza forecasting, we also set the batch size and d to 64. For the other tasks, these become Icarnable parameters. The number of training epochs is set to a maximum of 100 for ETTh1, ETTh2, and weather lasks and 50 for the influenza forecasting lask. Neural networks are optimised with Adam using mean squared error (MSE) loss on all outputs. As an exception, we use MAE loss in all models for the weather forecasting task as this gives better performance on the tuned basclines. Note that the output can be a time series (sequence) of one variable (the target) or more variables (target and exogenous) depending on the forecasting method.
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Forecasto
28 days ahead ILI rate forecasting, 2018/19, England (ILHENG)
Ground Inuth
DeromeeTiMe
ModemTCN
Crossfomer
LighTTS
22-
rate (per took people)
Timeliiee
Timeker
iransformer
CycleNer
PwichTST
Inwar
Persisencs
101500
2018-11
21801
20161
271866
2015-07
2018-0
Days
201-11
amaio
2019-0
201443
2013-0
mitel
Days
Days
Figure 2: 28 days ahead forecasting results for infiuenza season 2018/19 in England (ILL-ENG) for all models. The black line denotes the ground truth, ie. the reported (actual) II.I rates.
For the ETT and weather forecasting tasks, the batch size and the hidden dimension d are both selected from {16, 32, 64]. For all tasks, Adam's initial learning rate is selected from {2, 1, .5. 22. .1, .053 x 10-33 For the ILI forecasting task, the learning rate is decayed to 0 with a linear decay scheduler applied after every epoch, whereas for the ETT and weather forccasting tasks, it is docayed by half every epoch following prior work (Zhou et al. 2021 Nie et al. 2023 Luo & Wang 2024b). Depending on the model, the dropout (most baselines) or the layer drop rate (DEFORMTIMEE is selected from {0, 21. Deciding the number of exogenous predictors in the ILI forecasting task (Lampos et al., 2017 Morris et al. 2023) introduces another group of hyperparameters that require validation (see section D.2 Hyperparameter values are determined using grid search. Training stops early when the validation error of the target variable over time (I+1,... -1+ ,... [++H) does not decrease further (compared its lowest value) for 5 consecutive epochs. We then use the model with the lowest validation loss. Please refer to section D.3 for the random seed initialisation settings.
4.2 Forecasting accuracy
Forecasting accuracy for all tasks, models, and horizons is enumerated in Table I Mean Absolute Error (MAF) and symmetric Mean Absolute Percentage of Error (sMAPF or e %) are used our evaluation metrics. Note that for the ILL rate prediction tasks, the error metrics represent the average MAE and sMAPE across 4 consccutive test scasons (per scason results are available in Tables S1 and 53 where we also include the linear correlation with the larget variable as an additional performance metric), whereas for the remaining benchmark tasks performance metrics are obtained from one train / test split for each forecasting horizon, following previously reported evaluations.
DEFORMTIME displays an overall superior accuracy. It outperforms the most competitive method (that might be different one per task and horizon), reducing MAE by 7.2% and sMAPE by 4.5% average across all tasks and forecasting horizons. In more absolute terms, DEFORMTIME is the best performing model in all but two occasions based on MAE, and 20 out of 24 times based on sMAPE. If we exclude the weather task that uses a 1-year data set from a single location and hence is only expected to offer limited insights while attempting to prodict 720 time steps (7.5 days) ahcad (sce also Appendix A.4], our mothod offers on average 9.3% of MAE reduction in the most challenging forecasting tasks (maximum forecasting horizon H). Specifically, MAE reduced by 5.5% in the ETT tasks (for =2720 hours), and by 14.9% (England) or 4.4% (US Regions) in the ILI tasks (for = days). Therefore, performance gains compared to other forecasting methods do not decrease as the forecasting horizon increases.
When comparing DEFORMTIME to a specific forecasting method (as opposed to the best performing one per task and horizon), we notice that MAE reduction is >11% on average across all tasks and forecasting horizons, ranging from a 11.9% reduction vs. ModernTCN to 34.8% vs. DLinear These significant performance gains
7MAPE (Appendix F.1 can provide more balanced insights when metrics are averaged across different tasks.
AA brief note further explain Dinnar''s performance is provided in Appendix R.7
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highlight our model's capacity to consistently perform well under different tasks and horizons compared to other SOTA models. Interestingly, DLinear fails to surpass the average accuracy of naive persistence model (DEFORM'TIME reduces persistence's MAE by 30.1%).
Turning our focus to the more interpretable task of predicting ILI rates, we first notice that forecasting models that do not model inter-variable dependencies (DLinear, PatchTST, and CycleNet) or do so after embedding the input along the sequence dimension (iTrransformer, TimeXer) showcase an overall inferior performance. This highlights the potential benefit of incorporating variable dependencies over time steps slices that preserve the underlying temporal patterns. Such structure is common design choice amongst the better-performing models (DEFORMTIME, ModernTCN, Crossformer, LightTS, and TimeMixer). Compared to DEFORMT IIM'S average sMAPE (across all locations) ranging from 18.96% to 26.14% for H - 7 to 28 days ahead forecasting respectively, the two consistent competitors are Crossformer (18.87% to 28.83%) and Modern TCN (19.67% to 29.57%). The rest of the methods do not perform well as the task becomes more challenging, reaching average sMAPES ranging from 35.51% (LightTS) to 50.39% (TimeXer) for = days. Pending a more detailed evaluation (oul-of-scope for this work), there is at least partial evidence to support. that DEFORMTIME is SOTA forecaster for ILI (Reich et al.) 2019 Morris et al., 2021 Osthus & Moran 2021 Morris ct al. 2023). It not only demonstrates great regression fit,, but also captures the overall trend while forecasting 28 days ahead across 3 different geographical regions in 2 countries (average correlation is
> >90,, see Tables $1 $2 and S3
We depict the ILI rate forecasts of all models in Appendix G snapshot is presented in Figure 2 showcasing 28 days ahead predictions for influenza season 2018/19 in England. We observe that many models resemble the naïve persistence model that predicts the last seen value of the target variable in the autoregressive input, providing smooth but uninformative and increasingly inaccurate (as H increases) forecasts. DLinear provides the overall worst fit. With the exception of TimeXer and iTrranffrmer (see also Appendix C competitive baseline models that capture inter-variale dependencies provide more informative predictions. However, their estimates- -at least in this example- appear to be more noisy, and are either not capable of capturing the onset of the influenza season (TimeMixer) or its intensity (ModernTCN, Crossformer and less so Li)h(TS). Contrary to that, DEFORMTIME provides smoother and more accurate forecasts that corroborate our choice to account for both inter- and intra-variable dependencies.
Further experiments, presented in Appendix E.6 provide evidence that DEFORMTIME is also robust to random seed initialisation (Table S5 In Appendix E.8 and Table S6 we also provide results where the forecast error is averaged over the entire output sequence as opposed to only considering predictions at the target forecasting horizon time step, with additional discussion offering some insights as to why this may not be the most appropriate evaluation approach. Nevertheless, DEFORMTIME remains the best-performing forecasting model.
4.3 Ablation analysis
We perform an ablation study to understand the contribution of various operations in DEFORM'TIME. namely the V-DAB and T-DAB modules, the position embeddings used in DABs (we conventionally denote both by Pv.:), and the NAF module with and without position embedding P.. Experiments are conducted on the ILI rate forecasting task for 2 locations, England (ILIEENG) and US Region 9 (ILI-US9), across all forecasting horizons (H = (7, 14, 21, 28} days ahead) the 4 test seasons. Hyperparameters are re-tuned using the same validation approach separately for each ablation variant. Further details are provided in Appendix E.2
Table 2 cnumerates the ablation outcomes, showing the average MAE across all test periods. Evidently, each component or operation contributes to the reduction of MAE. On average, the V-DAB module provides stronger performance improvements (7.7%) compared to the T-DAB module (3.9%). However, for the longest forecasting horizon (H (H=28), the difference between the level of contribution from these modules is lower (respectively, 9.8% and 8.0%). Hence, establishing inter-variable dependencies is always useful, but intra-variable dependencies appear to become more important in longer forecasting horizons.
The inclusion of the NAE component is equally important as it improves MAE by 6.4% on average. Interestingly, the level of relative improvement increases as the forecasting horizon extends (from 6.7% to 7.1%). This polentially highlights that longer forecasting horizons require more abstraction over the
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Table 2: Ablation study for DEFORMTIME. We report the average MAE over the test periods for 1L1 tasks across all forecasting horizons, H = (7, 21, 28} days ahead. '-' denotes the absence of a module or an operation. P., denotes both position embeddings used in DABs (V-DAB, T-DAB). Pa denotes the position embedding used in NAE.
H
DEFORMTIME
V-DAB
T-DAB
P
NAF
P
7
1.6417
18836
1,8043
1.9420
1.8238
1.8849
ILI-ENG
14
2.2308
2,4978
2,3570
2.2646
2,4280
2.8982
2.6500
2.9173
2,6840
2.8838
2.9523
2.9971
28
2.7228
3.3685
3,2848
3.1082
3.1016
2.8446
Avg.
2.3113
2.6593
2.5325
2.5497
2.5764
2.6562
7
0.4122
0.4600
0.4165
0.4507
0.4414
0.4612
ILI-US2
14
0.4752
0.4796
0.4758
0.5099
0.5033
0.4973
0.5425
0.5790
0.5563
0.5696
0.5474
0.5545
28
0.5538
0.5924
0.5794
0.5902
0.5960
0.5881
Avg.
0.4959
0.5278
0.5070
0.5301
0.5220
0.5253
7
0.2622
0.2709
0,2649
0.2753
0.2665
0.2737
ILI-US9
14
0.3084
0.3138
0.3103
0.3174
0.3161
0.3270
21
0.3179
0.3493
0.3236
0.3452
0.3273
0.3355
28
0.3532
0.3664
0.3623
0.3689
0.3638
0.3693
Avg.
0.3104
0.3251
0.3153
0.3267
0.3184
0.3264
feature space, favouring learning from correlated groups of variables. Finally, the use of position embeddings, albeit not entailing very sophisticated operation, significantly enhances the impact of the V-DAB SEDAB/T--AAL T-DAB and NAE modules by 6.4% and 7.9%, respectively. We argue that position embeddings, obtained before (Pa) and after (Pv,) deformation, work in tandem to maintain key information that evidently further improves the overall forecasting accuracy.
Interestingly, comparing the results in Tables 2 and 1 DEFORMTIME without the V-DAB module, which results in a simpler model that captures inter-variable dependencies via the NAF module and a GRU decoder, yields similar accuracy as ModerTTCN, We consider that the added value of V-DAB stems from its ability to flexibly capture variable dependencies that may not be temporally aligned. These dependencies, if they exist, have been shown to potentially contribute to better forecasting performance when aligned (Zhao & Shen 2024). ModernTNN, however, does not specifically model such unaligned temporal dependencies. Our analysis suggests that this omission is an important factor for the observed performance degradation.
4.4
Assessing the impact of an increasing number of exogenous predictors
We also assess the accuracy of models as the number of exogenous predictors increases. For this we use the ILI rate forecasting task given its relatively large amount exogenous predictors. We set the number of exogenous input variables, i.c. the frequency time scries for different scarch querics, to C to (32, 64, 128, 256, 512) Queries selected by obtaining the top-C most correlated ones with the target variable (ILI rate) in the 5 last flu seasons in the training data (a detailed data set description is provided in Appendix A.2 We hypothesise that models that consistently give better performance as C' increases are more robust to handling varti-variable input and capture inter-variable dependencies better. To this end, we focus our analysis on models that. have components that aim to capture information across variables, namely iTransformer, TimeXer, TimeMixer, LightTS, Crossformer, ModernTIN, and DEFORMTIME Experiments are conducted the ILI forecasting task for England across all test scasons and we report the avcrage MAE and sMAPE scores. All hyperparameters, except the number of input variables (see Appendix D.2), are re-tuned for each model (for all train wtest splits).
9The total number of input variables C+1 if also factor in the antoregressive part the input.
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Published in Transactions on Machine Learning Research (04/2025)
Table 3: Forecasting accuracy for models that capture vnter-variable dependencies on the ILI-ENG task under increasing amount of input (exogenous) variables. The best results are in bold font and the second best are underlined.
Models
iTransformer
TimeXer
TimeMixer
LightTS
Crossformer
Modern
DEFORM
C
MAE
e %
MAE
< %
MAE
< %
MAE
< %
MAE
%
MAE
%
MAE
e e%
32
2.5141
28.67
2.7315
32.62
2.1333
24.68
1.9263
33.12
1.6659
27.51
1.9468
37.37
1.6790
23.56
64
2.3658
27.79
2.9042
33.52
2.1768
25.03
2.2359
41.94
1.9090
30.72
1.7984
41.54
1.4904
23.23
128
2.3346
27.15
2.8816
33.29
2.4315
27.96
2.1155
46.60
1.6660
24.80
1.8501
31.70
1.4468
22.68
256
2.3325
27.06
2.7711
33.60
2.48422
28.34
2.5989
35.87
180188
22.17
1.9449
33.33
1.5797
25.03
512
2,4466
27.98
2.9864
35.19
2.8672
32.94
2.7775
51:29
1.7930
22.53
2.1583
46.45
1.6314
23.60
32
3.4327
39.69
3.4781
40.86
3.1351
35.30
2.8046
38.21
2.5710
34.28
2.6098
41.93
2.1651
30.25
64
3.2482
37.76
3.5437
41.27
3.2472
37.42
2.9960
48.12
2.5851
35.37
2.4487
34.03
2.0055
29.36
14
128
3./9923
39.94
3.5398
41.56
3.3225
37.71
2.9178
40.72
2.5510
32.87
2.4755
37.70
2.0362
29.55
256
3.3671 1
38.98
3.4928
41.95
3.3600
39.06
2. 2.6980
47.30
3.1166
38. 46
2.7400
38.99
2.1626
31.91
512
3.3427
38 53
3.6936
44.14
3.7326
41.78
2.7095
39.44
3.1111
39.70
2.9224
43.48
2.2848
33.15
32
4.6290
55.13
4.4526
54.44
4.0293
56.15
3.2743
45.51
2.8971
33.86
2.7548
41.25
2.7363
37.05
64
4.1034
48.80
4.3821
51..9
3.9984
54.02
3.45119
52.31
3.0058
36.95
2.7769
51.61
2.5821
37.15
21
128
4.2907
50.79
4.3504
51.15
3.7035
53.27
4.5949
60.69
2,8090
36.49
2.9763
40.86
2.6646
37.65
256
4.4976
53.14
4.5287
56.65
3.7519
52.49
3.5740
32.22
3.0846
38 88
2.8275
41.56
2.5746
34.64
512
4.2563
4.6257
56.27
3.5404
45.61
3.3258
34.37
3.6267
44.76
2.9788
49.13
2.6973
35.55
32
4.8528
58.22
4.9145
61.07
4.8393
60.72
3.7597
46.94
3.5441
44.01
3.1705
41.17
2.8000
34.75
64
4.6811
56.08
4.8851
61.35
4.5416
58.54
3.6672
37.08
3.6433
46.64
3.3598
49.85
2.7637
34.17
28
128
4.8265
56.90
4.8329
60.65
4.7736
57.94
3.7655
56.16
3.4786
44.29
3.0837
47.69
2.7945
36.90
256
5.0621
60.88
4.9689
02.48 48
4.4152
57.56
3.6055
53.02
3.6987
45.14
3.1944
46.01
2.9963
41.02
512
5.0417
59.91
5.3472
08.10
4.0968
54.52
3.5609
46.86
4.0750
48.02
3,2858
47.23
3.1113
38.94
Forccasting accuracy results are enumerated in Table 3 First, considoring the entirety of reported outcomes, we note that. DEFORMTIME provides superior performance, redueing the average MAE and sMAPE by 11.8% and 10.6% respectively compared to the best- pest-performing baseline(s). When consider outcomes for a consistent setting of the number of input variables, DEFORMTIME yields the lowest MAE in all but one forecasting task (vs. Crossformer for C==2 and = days ahead). This re-affirms our previously reported results and also highlights that DEFORMTIME tends to outperform competitive baselines irrespectively of the number of exogenous predictors, maintaining its superiority as the forecasting horizon increases.
Consistent with the results presented in section 4.2 the most competitive baselines are ModernTCN and Crossformer, where DEFORMTIME reduces their MAF by 13.9% and 17.1% average, respectively. iTransformer and TimeXer are the two least competitive models, with DEFORMTIME reducing the average MAE by 37.9% and 41.9% respectively. This provides some evidence for the importance of structural design when it comes to time series modelling as highlighted in section 4 Specifically, disrupting the input's 's temporal patterns prior to modelling is likely to cause model collapse, resulting in lcarning a simplistic forecasting choice resembling the persistence model, as also reflected in Figure - It also underscores the need for conducting evaluation on diverse datasets and metrics to comprehensively assess model performance (TimeXer is the best-performing baseline for ETThl but less competitive for other tasks).
4.5 Computational complexity and efficiency of DeformTime
The computational complexity of DEFORMTIME is O(L2d + Ld)), where L and d respectively denote the size of the look-back window and the number of hidden layers throughout the method. A detailed derivation of this is provided in Appendix E.5) We note that the number of operations for an encoder layer reduces quadratically as we increase r (the size of T-DAB's time window) to values greater than 1. Likewise, segmenting the L input time steps to patches of a smaller length h (in V-DAB) results in quadratic computational benefits.
Figure 3 depicts the GPU VRAM memory consumption of DEFORMTIME during training compared to other transformer- based models wrr.t. the length of the look-back window (L; part A) and the number of input variables (C + 1; part B). In these experiments, we set both the batch size and the hidden dimension d to 64.
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3.5
A
DEFORMTIME
25
CrossFormer
GPU Memory (GB)
3.0
PatchTST
20
ITranformer
2.5
TimeXer
15
2.0
10
1.5
1.0
0.5
25
75
100
125
100
200
300
Look back window
Number of input variables
Figure 3: GPU memory (VRAM) consumption based on (A) the length (time steps) of the look-back window (L), and (B) the number of input variables (C+1).
While assessing the impact of L, we set C+1 32, and while assessing the impact of the amount of input variables, set L to 56. We then present the average memory consumption across 50 training epochs.
Overall, DEFORMTIME has relatively small (GPU VRAM) memory footprint. As the length of the look-back window increases (from 28 to 140 time steps), i'Transfrmmer, TimeXer, and DEFORMTIME are showing a stable memory consumption that does not exceed GB. We note that the low memory consumption of iTransformer TimeXer expected as they encode all input / exogenous input over the sequence dimension, which guarantees consistent GPU memory usage regardless of the value L. In DEFORMTIME, the input partitioning and adaptive transformations within the V-DAB and T-DAB modules reduce compulational complexity, leading to reduced memory consumption, especially when L is large. At the same time, the memory footprints of PatchTST and Crossformer lincarly increase with L.
As we now increase the number of input variables from (32 to 512), other methods display a linear increase in memory consumption (delayed and considerably lower for iTransformer and TimeXer compared to PatchTST and Crossformer), whereas for DEFORMTIME memory consumption is almost unaffected and stable (e.g. GB for DEFORMTIME, but 24 GB for PatchTST). Hence, we argue that our method can handle a greater amount of exogenous predictors more efficiently.
Compared to other transformer-based MTS forecasting methods (Crossformer, PatchTST, and iTransformer), our method exhibits consistently low memory consumption as the number of input variables increases (vs. a linear increase for the other methods). Furthermore, increasing the look-back window does not overly affect memory consumption either (vs. a more accelerated linear increase for Crossformer and PatchTST). Hence, we deduce that DEFORMTIME: is adept to MTS forecasting tasks with a considerable amount of exogenous predictors and longer look-back windows.
5 Conclusions
We propose DEFORMTIME, a novel deep learning architecture for multi-variable time series forecasting that uses deformable attention blocks to effectively learn from exogenous predictors, deploying specific operations that aim to capture inter- and intra-variable dependencics. We assess forecasting accuracy using established benchmark tasks well as ILI rate prediction tasks with web search frequency time series exogenous variables, covering 3 locations (in countries) and spanning time period longer than 12 years. DEFORMTIME yields strong forecasting accuracy across the board, reducing mean absolute error average by 7.2% compared to the best basclinc model (different for cach task and horizon) and by at least 11.9% (for Modern (Luo & Wang 2024b)) when compared to a specifie model. Importantly, performance gains remain stable for longer forecasting horizons. Our experiments, including the ablation study, highlight that modelling variable dependencics an important attribute. Specifically the discase forecasting task, where more exogenous predictors are present, the most competitive baselines capture variable dependencies some extent, whereas models that do not many occasions cannot surpass the performance of a naïve persistence
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model (see also Appendix E.9 In contrast to other methods, DEFORMTIME'S GPU memory footprint is not significantly affected by the amount of exogenous variables.
Acknowledgements
The authors would like to thank the RCGP for providing 1LI rates for England. V. Lampos would like to acknowledge all levels of support from the EPSRC grant EP/X031276/1.
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Supplementary Material / Appendix
A Data sets for time series forecasting
This section offers a detailed description of the data sets used in our experiments assess the performance of scries of methods for MTS forccasting. We provide links to data sets that are publicly available. For the ILI data sets formed by us, while we hope that will be able to share the exact data set with the broader community in the future, cannot do so due to copyright restrictions at the time of publication. As an alternative, we provide instructions as to how rescarch teams can obtain access directly from the data provider.
A.1 Established time series forecasting benchmarks
ETTh1 and ETTh2. ETThl and ETTh2 are electricity transformer temperature data sets that were obtained from 2 different counties in China (Zhou et al.) 2021 T0 Each data set contains exogenous variables capturing power load attributes, and the target variable which is the temperature of oil. The data set covers a period from July 1, 2016 to June 26, 2018 (although not all data points are used in our experiments match the setup in related methods that we compare to). The temporal resolution of these data sets is hourly.
Adopting the sctup in prior work (wu et al. 2023 Nie et al. 2023 Liu ct al. 2024), we use total amount of 14,400 time steps (starting from July 1, 2016) where the first 8,640 are used for training, the next 2,880 for validation, and the last 2,880 for testing.
Weather. The weather data set (Wu et al. 2021 contains meteorological measurements collected from weather station at the Max Planck Biogeochemistry Institute. It covers a year from Jamuary 1, 2020 to January 1, 2021 The temporal resolution is 10 minutes and 20 meteorological indicators (exogenous variables) are being reported; the target variable is carbon dioxide concentration level. This results a total of 52,696 time steps (samples). Based on prior work (Zhang & Yan 2023), we use 70%, 10%, and 20% of the time steps for training, validation, and testing, respectively.
A.2 Forecasting influenza- like illness rates using web search activity
Important components of our data set are shared in the paper's online repository. At the time of publication, these allow replication of our work for teams who have obtained the same data-sharing agreements (mainly access to the Google Health Trends API).
Influenza-like illness (ILI) rates. ILI is defined the presence of common influenza symptoms (e.g. sore throat, cough, headache, runny nose) in conjunction with high fever. We obtain ILI rates for England (in the United Kingdom) and two US HHS Regions, specifically Region 2 (states of New Jersey and New York) and Region 9 (states of Arizona and California). For England, we obtain ILI rates from the Royal College of General Practitioners (RCGP) that monitors ILI prevalence via an established sentinel network of GP practices throughout the country. An RCGP ILI rate represents the amount of ILI infections in every 100,000 people the population of England. For the US Regions, data obtained from the Centers for Disease Control and Prevention (CDC) An ILI rate from CDC represents the proportion of ILI-related doctor consultations over the total amount of consultations (for any health issue). Hence, the units of the ILI rate in these two monitoring systems (RCGP and CDC) are different. The time span for the obtained III rates is the same as the time span for the web search data (see next paragraphs).
10 and ETTh2, github, www.gp..c....oouaaaa.. hhtpubb..cn/hhhhaaoyyyy/aaaaatt 1 Weather data set, withub.cco///e hatbb.cco//mmmm//tttooreeee 12US II.I rates www..ddc.oo/greeee..eeeee httpad/dcc./////aa/a//aattttttaa
edc,
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Influenza season definition. An annual influenza season for the HHS US Regions is assumed to start August of year X and end on July 31 of year x+1. For England, this is shifted by a month, i.e. an annual influenza season starts on September 1 of year x and ends on August 31 of year x+1.
Linear interpolation of weekly ILI rates. IIII rates are reported on a weekly basis. To generate daily data (that increases the amount of training samples by a factor of 7), we use linear interpolation. It should be noted that for RCGP, Monday determines the start of a week, while for CDC this is determined by Sunday (different ISO specification). We assume that the weekly reported ILI rate is representative of the middle day of week (Thursday for England and Wednesday for the US Regions) and the ILI rates are then lincarly interpolated accordingly.
Real-time delay in ILI rate availability. We also note that in practice, the reported ILI rate is delayed, i.e. assuming that it is published by RCGP or CDC at time t, it actually refers to an ILI rate representing a previous time t 5. Specifically, we consider that there is delay of 5 days for RCGP//nnlann
and a delay of 8= 14 days for CDC/US Regions (Wagner et al. 2018 Reich et al. 2019). This delay, 6, has an impact on the autoregressive time series (of the target variable, in this case, the ILI rate) in a forecasting task (see also section 2). As the delay increases, the forecaster is expected to rely more on the exogenous prodictors (that are not delayed) rather than the autoregrossive time scries of the target variable.
Web search frequency time series. Web search activity trends, if modelled appropriately (Ginsberg ct al.) 2009 Lazer et al., 2014 Lampos et al. 2015 Kandula & Shaman 2019). are a good indicator of influenza rates in a population and can become a strong exogenous predictor for influenza forecasting (Duaas ct al., 2013 Morris et al.] (023). We obtain web search activity data from the Google Health Trends API; this is not a publicly available API, but access can be provided via an application process 13 For a day and a certain location, the frequency of search query is determined as the ratio of searches conducted for a particular term or set of terms divided by the overall scarch volume. We use pool of 22,071 unique health-related search queries and obtain their daily frequency from August 1 (US regions) or September 1 (England), 2006 to July 31 (US regions) or August 31 (England), 2019. However, we note that not all search queries are used as exogenous variables in our forecasting models. feature selection process is described in Appendix A.3
Given that for the US, we can only obtain data at the state level (as opposed to regional), we use a weighted average of the state-level search query frequencies. The weights are based the population of each state. In particular, for Region 2, the states of New Jersey and New York have weights of 0.32 and 0.68, whereas for Region 9, the states of Arizona and California have weights of 0.16 and 0.84, respectively. Note that smaller locations (or distant ones) that may be part of an HHS Region are excluded from the web search data sets; given their relatively small population, these locations do not have a significant impact on the reported ILI rates and the data obtained from the Google Health Trends API are quite sparse.
Training, validation, and test sets. We assess the accuracy of forecasting models across the last 4 influenza seasons (4 separate test sets) in England and the US regions. Each test period is a complete influenza season (as previously defined). For each test period, we train models based on the 9 influenza seasons that precede it. A part of each training set is used for validating model decisions, including hyperparameters. We construct validation sets using the following strategy. Each validation set has 180 days in total, using 60-day periods to capture the onset, peak, and the period after the peak (or the outset) of an influenza scason. Each 60-day period comes from a different influenza season in the training set; we use the last 3 influenza seasons in the training data to make sure our validation process has a recency effect. The last, penultimate, and third from last influenza seasons are used to define the outset, peak, and onset validation periods, respectively. We use CDC's definition to determine the onset of an influenza season, i.e. a time point is deemed to be the onset when the subsequent ILI rates exceed threshold for 2 consecutive weeks The peak point is simply the highest ILI rate within an influenza season. The outset has the inverse definition from the onset, i.c. the last time point where the ILI rate excoods the onset threshold for two consccutive weeks. Once these
Gongle Health Trends API, support. www.orttggogge..o www.gge..c////eeeaeeeeeeeeeeeeee hoogg//.cmm/reen//ntttt///rreete
14 CCCD's influenza season onset delinition, rab.gov//lluviee//ovvriiee
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IIJ rate [per 100K people)
Train
40
20
2011 2012
2013
2019
2020
Days
Figure S1: An example of how the training, validation, and test sets are constructed when the test influenza season is 2018/19 (England). The lines blue, red, and orange colour denote the training, validation, and test periods, respectively. To form the validation set from our training data, we select the period after the peak (outset) from the third to last influenza season, the period around the peak from the penultimate season, and a period around influenza onset from the last season.
time points are determined, they become the 30th point in a 60-day validation period. Figure ST shows an example of the validation periods that were determined when training an ILI rate model for England using (training) data from September 2009 to August 2018 (the test period being September 2018 to August. 2019).
A.3 Feature selection for web search activity time series
Originally, we consider 22,071 search queries. We then perform two feature selection steps to maintain more relevant queries the ILI forccasting task. First, we apply semantic filter to remove queries that are not related to the topic of influenza, similarly to the approach presented by Lampos al. (2017). We obtain embedding representation for each search query using pre-trained sentence BERT model 15 We also obtain the embeddings of teattenzz--rllteed (base) terms and expressions, such as "flu", "flu symptoms" and so on T6 We compute the cosine similarity between each base term and search query and maintain the 10p-1000 (England) or top-500 (US regions) search queries (per base term). This leaves 4,398 and 2,479 queries respectively for England and the US regions. The second selection step dynamic, i.e. might have a different oulcome as the training data change. this step, we compute the linear correlation between the remaining search queries and the target variable (III rate) in the 5 most recent influenza seasons in the training data (to impose recency). We maintain search queries with a correlation that exceeds correlation threshold т, which is a learnable hyperparameter (see also Appendix D.2
A.4 Criticism of data sets used benchmark forecasting methods
Although present results using the ETTh1, ETTh2, and weather data sets to compare with other forecasting methods in the literature, consider these data sets to have several shortcomings. Clearly, weather data set that covers only 1-year period and is based on the sensors from just one geographical location cannot be expected to provide good enough insights into meteorological forecasting, It captures a negligible portion meteorological data and, as a result, derived forecasting models cannot have any practical impact (compared actual models that are used metcorology). Hence, using this data set for training and evaluating forocasting model may, unfortunately, result mislcading or at least inconclusive outcomes. Similar issues, but perhaps to smaller extent, are present in the ETTh1 and ETTh2 data sets. On the one hand, these data sets seem to have more solid practical application. However, experiments their ETTml and ETTm2 variants, where the temporal resolution changes from 1 hour to 15 minutes (samc data, shorter temporal resolution), showcase part of the problem. When we compared the forecasting performance of SOTA baselines between the two different temporal resolutions, we surprisingly found that the prediction based on the hourly sampled data set has better accuracy compared to the hourlyr-hourly sampled data set even longer forecasting horizons. For example, models that conduct 192 or even 336 time steps (cquivalent
18 18gentence BERT, batgggingaae..cobbeeee ca/bert-base-uccaee
16The complete list base terms provided at gitggt. Stthhb..cm//taaiiisss///eemmmmma
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to 8 or 14 days) ahead forecasting with ETTh1 and ETTh2 achieve a lower MAE compared to 720 time steps ahead forecast with ETTml and m2 (equivalent to 7.5 days ahead forecast), respectively (Nie) et al. 2023 Wu et al. 2023 Luo & Wang 2024b).
We also noticed that some papers run experiments on US national weekly ILI rates obtained from the CDC (Zhou et al. 2022b Wang et al. 2023: Nie et al., 2023). The first concerning observation was that these experiments, the input variables were based on variates (columns) from the CDC extract (spreadsheet) that only provide redundant insights compared to the ILI rate (e.g. the raw numerator and denominator of an ILI rate the population unweighted ILI rate). In addition, some forecasting horizons that were explored in these benchmarks were practically unrcasonable (even from a biological perspective, there are limits in predicting the future, especially when it comes to viruses). SOTA forecasting models for influenza generally provide good accuracy at approximately 2 weeks ahead forecasting horizon (Osthus & Moran 2021 Morris et al. (023). Anything beyond that with satisfactory performance should be considered as a very important development within the discase modelling community. We do think DEFORMTIME might fall in that category as delivers good results for 3 or weeks ahead forecasting horizons. Contrary to that, in the aforementioned ILI rate benchmark, there were forecasting horizons of 24 or 60 weeks ahead (i.e. even more than a year!). Of course, the accuracy of the forccasting models under these forccasting horizons was reportedly detrimentally poor (influenza does not have strong periodicity), and consequently could not be used for any meaningful comparative conclusions. If all forecasting models are performing poorly, even the least poorly performing one still is an inadequate forecaster (for the underlying task).
Beyond this, we have also excluded data sets, such as traffic (Wu et al. 2021 and electricity consumption (Zhou et al. 2021), ) that do not have explicitly defined target variable, but instead focus on multivariate predictions (multi-task learning). Consequently, these data sets are not entirely compatible with our forecasting task or method (DEFORM makes predictions about one target variable).
To compensate for that we have developed a set of 3 influenza forecasting tasks. Models from these tasks can find direct real-world applications, i.e. become part of influenza monitoring systems in England or the US. The evaluation process is rigorous, i.e. across 4 consecutive influenza seasons as opposed to using fixed test. period that may again lead to biased insights.
B Detailed explanation in deformation using V-DAB
In this section, we provide detailed description of how positional deformation is conducted in DEFORMTIME. taking V-DAB as an example. Suppose we have an element Zp(m,,) n) that was originally on position (m, n) of the patch matrix Zp. In this example, let's 's assume that m = 5 and = 10 and that the learned positional offset for that position is An), = (-0.1, 1.5). Then, the value for Za(m, n) is the value sampled from (m + Am, n + An) or, in this example, (5 0.1, 10 + 1.5) with bilincar interpolation. Specifically, the deformed point p = (4.9, 11.5) lies within a x index matrix of actual discrete positions, i.e. (4, 11), (4, 12), (5, 11), and (5, 12). We first linearly interpolate the values across the rows of the index matrix. We note that the deformed position in this example is equal to 11.5 and that mcans that there is equal distance between the indices 11 and 12. We therefore perform:
Z, 11.5) = (4, x 0.5 + (2, x (1 0.5) and
Zz 11.5) (5, 11) x 0.5 (5, 12) (1 0.5)
We then conduct lincar interpolation along the column to obtain the final value of the bilincar interpolation. Here we note that the deformed position is equal to 4.9 and hence it does not have the same distance between indices 4 and 5. We therefore perform:
Zz 11.5) 2p(4, x 0.1 (, 11.5) x (1 -01))
This operation is conducted on every p E Zp to obtain Za-
Za then forms the key (K) and value (V) embeddings with learnable weight matrices. Given a scalar element 2(.5) E Zp. K and V contain elements sampled from the neighbouring latent variables (Zi-a: Zita) in V-DAB neighbouring time steps (Z())-a); 2((j+a)) in T-DAB, where is the offset amplitude.
21

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Published in Transactions on Machine Learning Research (04/2025)
C
Summary of other forecasting models
Here provide list of the models that are comparing DEFORMTTIME to:
2022 uses MLP layers the building blocks to extract both inter- vattill and intra-variable dependencics upon time scries with different temporal resolutions.
LightTS (Zhang et al.
DLinear ZZeng et al. 2023) uses fully connected layer along the temporal dimension with seasonal-trend decomposition to regress the historical values for future predictions.
Crossformer [Zhang & Yan 2023 is a transformer-bent based model that divides the input into patches and
proposes a two-stage attention layer to capture inter- and intra-variable dependencics.
PatchTST (Nie et al. 2023 is a transformer-based model that takes segmented patches as input tokens for the model. The prediction of each variable is designed to be independent of one another, i.e. with no inter-variable dependencies established.
iTransformer (Liu et al., 2024) is transformer-based model that embeds the sequence dimension rather than the input variable dimension. We note that although iTransformer captures inter-variable dependencies, it achieves that by embedding the input along the sequence dimension. This disrupts the temporal structure within each variable, leading to a sub-optimal performance in our empirical evaluation.
TimeMixer (Wang et al. 2024a is an MLP-based model that predicts the seasonal and trend components at different sampling scales and mixes forecasts to form the final prediction. Note that the authors consider variable mixing optional for their model. We opted to deploy the version that conducts variable mixing for the ILI rate forecasting tasks.
TimeXer (Wagg et al., 2024b) is a transformer-basee model that captures exogenous information with cross-attention. It currently claims to offer SOTA accuracy on longer-term forecasting tasks Similarly to iTransformer, Ter embeds the exogenous variables along the sequence dimension which disrupts the temporal structure.
ModernTCN (Luo & Wang 2024b) is a CNN-based model that convolutes over both temporal and variable dimensions respectively with large receptive fields to capture inter- and intra-variable dependencies.
CycleNet (Lin et al. 2024 is an MLP-based model that discntangles periodic patterns to capture temporal dependencies. Similarly to PSTchTST, CycleNet predicts each variable independently. Note that the authors propose two versions of CycleNet. We report results for the version that has linear layer, which has higher accuracy for overlapping tasks (ETT and weather).
The persistence model is naive bascline that uses the last seen value of the target variable in the input as the forecast.
D Supplementary experiment settings
This section is supplementary to section 4.1 in the main paper. All experiments were conducted using a Linux server with 2 NVIDIA A40 GPUs (and AMD EPYC 7443 CPUs), except for the ablation experiments which were conducted using a Linux server with NVIDIA L40S GPUs (and 2 AMD EPYC 9354 CPUs).
D.1 Input variable re-arrangement based on correlation with the target variable
In the ETT and weather forecasting tasks, we rearrange variables (see NAE in section 3.2 based on their linear correlation with the target variable across all time steps preceding the validation set In the ILI rate forecasting tasks where training sets cover a period years prior to the test period, we only use the last influenza seasons in the training set to obtain correlations. This because both user search behaviour and search engine characteristics (e.g, automatic search suggestions) evolve, and this approach to feature selection reinforces weak recency effect that is ultimately beneficial to prediction accuracy (Yang et al. 2015 Morris
et al. 2023)
17 "ong-term forecast tasks ranking, wwt.e/aaa//ttttneeeeec htt..///ttuuu//tte--rrre---r--a-
22

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Published in Transactions on Machine Learning Research (04/2025)
Table S1: ILI rate forecasting results in England (ILI-ENG) using web search frequency time series as exogenous variables. P and e % denote linear correlation and sMAPE, respectively, The best results are in bold font and the second best are underlined.
2015/16 MAE
2016/17
2017/18
2018/19 MAE
Average MAE
11 Model
%
MAE
%%
P
MAE
5%
%%
%
Persistencee
0.9072
1.7077
22.
(1.1129
1.2287
22.67
0.8539
4.0444
31.08
0.8941
1.7037
23.67
0,8921
2.1710
24.96
DLincar
0.8710
2.6985
38.86
018766
1.6879
33.77
0.8231
4.2734
50.14
0.5906
2.6258
49.30
0.8654
2.8214
43.02
PatciTTT
0.8632
1.9823
26.13
0.871L
1.1307
25.39
0,8175
3.7925
31.23
0,8526
2,0103
27.70
0.8611
2.3115
27.61
CycleNet
0.8730
2.0579
27.08
0.8508
1.5396
27.54
0.7891
4.5348
35.72
0.8473
2 2.0922
28.02
0.8428
2.5554
29.59
irranfoommrr
0.8773
2.0615
27.26
0.8911
1.3731
21.06
0,8150
1.107T
31.26
0,8911
L.688
22.91
0.8691
2.3084
7.
TineXer
0.8927
1.9800
26.29
(18773
1.5327
29.94
0.6974
5.1695
43.18
0.7564
2.5507
35.22
0.8060
2.80184
11.66
TimeMMier
0.9056
1.7179
22.27
0.8616
1.4676
21.97
0.8694
3.1628
28. 09
0,8388
2,0310
27.39
0.8688
2.1718
25.
LightTS
0.8385
2.2625
38.87
0.94:5
2,1880
89.24
0.9441
2.6512
52,01
0,9426
1.5570
28.87
0.9171
2.2397
52.25
Croesformer
0.8879
2,3314
36.08
0.9591
0.8606
17.82
0.9234
2.9133
27.04
0.94136
1.3737
21.89
0.9285
1.8998
25.71
ModeraiINN
0.9190
1.6013
17.82
(1.9000
1.1988
22. 75
0.9679
3.233
40.35
0.9316
1.7581
91.98
0.9521
1.9488
DEPORNTIME
0.9728
1.3786
19.57
0.9665
1,383
29. 49
0,9585
2.6313
411.40
0.9433
1.2485
23.96
0.9603
1.6417
28. 61
Persistence
0.8292
2.3161
20.16
0.8414
L6867
29.
0.7269
6.8180
42.30
0.5031
2.4291
33.08
0.8006
3.0625
23.12
DLinesr
0.7810
3.6351
52.
0.7857
2.1872
47
0.6834
5.5368
0.8351
3,8092
65.17
0.7713
3.7922
55.28
Patc
0.8078
2.6267
35.59
0.7130
1.9662
33.55
0.6722
5.6075
43.64
0.7218
2.8284
38.25
0.7437
3.2547
37. 76
CycleNet
0.7970
2.6843
35.97
0.7402
2.1182
36.50 59
0.6127
6.0212
47.53
0.7326
2.7
37.61
0.7206
3.3911
39. 12
Transformer
0.7731
2.7620
36.46
0.7776
L90229
34.17
0.6376
5.7796
44.34
0.7850
2.3862
31.71
0.7483
3.2301
36.67
14 Timexer
0.7891
2,6580
35.27
0.7910
19994
36.99
0.5503
6.2678
53.64
0.6646
3 3.0626
41.61
0.6988
3.4937
41. 88
TineMixeer
0.7706
2.6783
34.25
0.8084
1.5621
29.06
0.7164
5.0940
33.50
0.7503
2.7491
38.66
0.7764
3.0200
35.30
LightTS
0.8109
2.5494
32.28
0.9233
LG292
36 46
0,7821
4.4367
30.46
0,8016
2.1365
44.97
0.8619
2.6879
38. 29
Crossforme
0.7940
2.9124
41.75
0.8843
1.4711
23.39
0.8470
3.8295
30.88
0.9007
2.4044
27.86
0.8565
2.6543
30.97
MaderTTON
0.8614
2.1727
31
0.8964
1.40553
28.61
0.8906
4.6735
48.74
0.8441
2.4700
34.76
0.8756
2.7050
36.01
DEFORATIME
0.9259
2.0556
25.36
0.9400
1,3180
33,86
0.8964
3.7631
47.33
0.9154
1.7863
29.36
0.9194
2,2308
33:98
Persistetce
0.7357
2.8614
36.98
0.7465
2,1552
36.45
0.60022
7.9119
52.40 46
0.6052
3.1193
42.23
0.8944
3,8617
42.03
DLincur
0.4466
4.1958
56.02
0.6441
2,8204
52.56
0.4920
6.9514
71.66
0,6608
3.9192
64.75
0.5609
4.4739
61.25
PateTTT
0.6126
3.6607
48.57
0.6214
2,6377
44.77
0.4427
7,1628
60.54
0.5861
3.5956
50.55
0.5657
4.3192
51.11
CycleNer
0.5596
3.9238
52.84
0.5857
2.7003
46.01
0,4114
7.7331
65.57
0.5088
3.3905
46.92
0.5389
4.4519
52.83
Trnnsformer
0.5310
4,1390
55.62
0.3987
2.6718
43.66
0,1403
7,0816
55.17
0,6556
3.0161
41:29
0.3564
4.2347
48.
21
Tinneerr
0.5958
3.7854
49.46
0.6981
2.4045
41.17
0.4377
7.1688
62.71
0.5301
3.7962
52.91
0.5634
4.33137
51.
TimeMixer
0.5398
3.8805
56.71
0,7799
1.79.0
30,45
0,7497
5.5338
0,7539
2.9919
51.58
0,7058
3.5501
19.36
LightTS
0.7082
3.3630
45.16
(188806
1.9335
58.65
0,8132
5.6422
51.74
0,8504
2.5078
51.59
0.8126
3.3616
51.78
Croesformer
0.8113
2,6710
36.12
0.8014
1. 1.835
31.29
0.8501
1.5615
19.08
0.7331
2 9.9999
15.48
0.7991
3.00111
10.57
ModeralINN
0.7869
2.5283
27.42
0.8500
1.6790
30.86
0,90185
5.7489
61.11
0.8357
2.4040
10.67
0.8417
3,0400
10.02
DEFORMTTMEE
0.8819
2.1700
27.01
0.8859
18890
32.65 65
0,9110
1 1981
11.78
0.9092
1.6335
26.36
0.8970
2.6500
32.70
Persistence
0.6408
3,3786
43.17
0.6344
2.6018
42.68
0.4813
8.5676
00.33
0.5733
3. 3,8047
51.79
0.5825
4.5857
19. 49
DLinear
0.3917
1.5039
60.30
0.5077
3,3928
60.91
0.3911
T.7738
77.58
0,5771
1.1683
72.19
0.1670
5.0317
67.5 75
TST
0.4271
4.5240
60:22
0.44177
3.2658
53.01
0.3203
8.0693
65.91
0.4486
4.12
59.27
0.4109
4.9964
59. 60
CycleNet.
0.4681
1.3576
0,1710
3.1149
0.3109
8.7191
73.07
0.5037
3.8816
56.73
iTrnnformmr
0.3492
4.9542
66.45
0.6761
2.2745
38.38
0.2947
8.3463
03.38
0.5286
3.6750
48.18
0.4622
4.8125
55.35
28 TimeXer
0.1978
1.2217
56.56
0.3523
2.8581
18.91
0.3413
8.14993
71.44
0.3337
1.3758
09.50
0,/3113
4.9013
61. 60
TimeMixer
0.5395
3.8587
57.17
0.6800
2.4063
45.76
0.4862
6.7783
62:93
0.7502
2.9567
51.21
0.6190
4.0000
54.27
LighTTS
0.5812
3.4254
43.91
0.8311
1.9020
36.26
0.7151
6.0675
83.54
0,8900
2.2678
58.66
0.7543
3.4132
55.59
Creesformr
0.7299
155
41.56
0.8082
18197
33.57
0.7841
5.4882
55.97
0.9349
2.3500
53.48
0,8143
3.1953
46.14
ModernTNN
0.8017
2. 2.6488
36.17
0.7628
2.6508
51.36
0.7544
4.8228
49.19
0.8152
3.3222
54.78
0.7835
3.3611
47. 47.87
DEPORATTIME
0.7919
2.6540
49.29
0.9229
1.8100
41.87
0.8493
4.8324
75.
0.9499
1.5947
33.06
0.8785
2.7228
40.44
D.2 Hyperparameters specific to the ILI forecasting task
In the ILI rate forecasting task, the number of exogenous variables we use is a learnable parameter that depends the linear correlation threshold 7 (see (A..). For US regions, is selected from {.3, .5.).. .5). Given these thresholds, a model can select from 13 to 51 search queries for US regions and from 80 to 752 for England (the higher the correlation threshold, the fewer queries are being selected). Note that for ModernTCN, PatchTST and Crossformer, we restricted TE {.3, .4 .5) for all locations as the models required an excessive amount of GPU memory for larger sets of exogenous variables (see also scction 4.5 for PST and Crossformer).
For England, T is selected from 1,0,, 33
D.3 Random seed initialisation
For the ILI rate forecasting tasks, we use a fixed seed equal to '42' during training. For the ETTh1, ETTh2, and weather tasks, we set this to '2021' based on previous work (Nic et al. 2023 Zeng et al. 2023 Wang et al. 2024a). with the exception of iTransformr ('2023'), and CycleNet, where results are averaged across 5 seeds ('2024' $2025', "2026', $2027', "2028") in accordance with their official configuration.
23

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Published in Transactions on Machine Learning Research (04/2025)
Table S2: ILI rate forecasting results in US Region 2 (ILI-US2) using web search frequency time series as exogenous variables. P and e % denote linear correlation and sMAPE, respectively. The best results are in bold font and the second best are underlined.
2015/16 MAE
2016/17 MAE
2017/18
2018/19 MAE
Average MAE
11 Model
%
P
%%
P
MAE
%
c%
%
Persistetcee
0.7758
0.4114
22.26
(182711
0.6803
24.57
0,7389
1.11281
25.48
0.8951
0.4699
17.59
0.8143
0.6474
22. 08
DLincar
0.7456
0.4952
29.23
0.7958
0.5210
31.85
0.7133
1.0910
30.82
0.5955
0.5318
19.84
0.7881
0.7355
27.
PatciTTT
0.7666
0,1290
23.78
0.7973
0,7516
26.97
0.6937
1.1018
26.52
0,8621
0.5533
20.79
0.7800
0.7097
21.52
CycleNet
0.7577
0.4461
24.63
0.8134
0.7300
26.99
0.7345
1.0530
27.47
0.5782
0.5133
19.58
0.7935
0.6951
24.67
irranfoommrr
0.7986
0.3865
21.30
0.8171
0.7367
27.07
0,7781
0,9000
23.22
0,8545
0,5717
21.38
0.8122
0.6507
23.21
7.
TineXer
0.8024
0.4214
24.76
(1.8-80
0,6882
27.16
0.8494
0.8233
22.91
0,8859
0.5001
18.70
0.8464
0.6018
23. 23.38
TimeMiier
0.8237
0,38
21.76
0.8710
0.61119
21.16
0.9121
0.6235
17.87
0.8877
0. 0.4588
16.19
0.8737
0.5284
20.07
LightTS
0.7930
0.4045
23.25
0.90156
0,5806
18.01
0,9539
0.5688
15.39
0.9451
0,2988
10.33
0,8949
0.4652
16.74
Croesformer
0.8796
0.3382
21 21.39
0.9107
0.1976
32
0
0.6309
15.61
0.9623
0. 0.2966
10.52
0.9109
0,4400
16. 16.46
ModeraiINN
0.8745
0.3874
21.99
(1.9359
0,5216
17.45
0.9569
0.4782
14.12
0.9635
0. 0.3722
12.64
0,9302
0.4398
16.55
DEPORNTIME
0.8887
0.3428
21.86
0.9463
0.4796
18.66 66
0,9008
0.5369
12.88
0.9622
0.2894
10.64
0:9245
0.4122
16.01
Persistence
0.6872
0,5000
26.67
0.7617
0.8546
31:23
0.6331
1.3022
32.42
0.8436
0.5006
22.65
0.7314
0.8135
28. 28.24
DLinear
0.6546
0.5761
33 63.62
0.7335
0.9406
36.90 90
0.6307
1.2213
0.8261
0.6300
23.74
0.8135
32:22
PStekTST
0.6628
0.5103
27.
0.7350
0.9267
34.44
0.5983
1.3088
31.82
0.7922
0.7085
26.50
0.6973
0.8633
30.11
CycleNet
0.7019
0.4754
25.40
0.7208
0.9457
35.28
0.6201
1.2199
30.54
0 0.8265
0. 1.6168
25.39
0.7173
29. 0.8219
Transformer
0.6864
0.4894
26.35
0.7548
0.8034
33.26
0.6025
1.1633
30.28
0.8294
0.6115
22.78
0.7183
0.7896
28.17
14 Timexer
0.7178
0.4823
27. 24
0.7446
0.9211
36.47
0.7118
1.0505
09
0.8164
0. 136322
24.47
0.7476
0.7725
29.07 07.07
TineMixeer
0.7741
0.4438
25.01
0.8403
0.7314
27.32
0.8043
0.8911
25.13
0.8622
0.5560
20.59
0.8201
0.6556
24.61
LightTS
0.7203
0.5086
28.85
0,8601
0,7079
22.96
0,9191
0.6608
23.47
0.9409
0.4445
17.40
0.8001
0.5827
23:11
Crossformer
0.7672
0.4357
24.61
0.8640
0.6502
22.95
0.8057
0.8159
20.80
0.9241
0.4380
15.54
0.8453
0.5852
30.98
MaderiTON
0.8093
0.4647
25.78
0.9000
0.6482
28.18
0,9360
0.6020
17.05
0,9672
0.3008
14.87
0:9006
0.3279
20.22
DEFORATIME
D. 0.4050
0.4404
23.82
0.9271
0.5029
18.53
0,9126
D.6351
16.34
0.9652
0.3226
12.21
0.9025
0.4752
17.73
Persistetace
0.5918
0,3880
30.04
0.7017
1.0055
37.20
0.5197
1.5443
38.94
0.7792
0.7361
27.88
0.6466
0,9635
33.51
DLincur
0.6016
0.6023
35.11
0.6897
0.9671
38.06
0.4966
1.3272
38.76
0.7936
0.7530
27.80
0.6454
0.9124
34.93
PateTTS
0.5775
0.5592
29. 11
0.6288
1,1661
43.41 41
0.5189
1395
38, 13
0.6478
0.9495
35,86
0.5932
1.0286
36.
CycleNer
0.6097
0.5216
27.54
0.5883
1,1875
44.72
0.4663
1.4768
41.27
0.5079
L.0088
38.41
0.5656
1.049
37. 97.98
Trnasformer
0:6118
0,4790
24.54
0.7515
0,8784
31.22
0,8230
1.08845
31,85
0,7321
0.7748
29.52
0.7371
0.8112
30.03
21
Tinneerr
0.4856
0.5815
32.17
0.7739
0.8636
33.97
0.6529
1.1238
32.12
0.7482
0.7225
27.56
0.6651
0.8243
31.46
TimeMixer
0.7368
0.4655
27.79
0,8288
0,7853
31.00
0,8509
0.9275
31.07
0,8803
0,5391
21.86
0,8240
0.6794
27
LightTS
0.8484
0.5606
32.47
0.8936
0.6749
34.80
0.9177
0.8065
28.47
0.8951
0.63320
21.34
0.8887
0.6883
29.27
Croesformer
0.7729
0.1631
26.54
018999
0.6088
25.62
0.7621
0.8682
23. 17
0
0.4777
13.82
0,8125
0.6215
22,
ModeralINN
0.8104
0.4656
26.17
0.8746
0.6790
301.30
0.9105
7,58
20.75
0.9487
0.4121
18.19
0.8860
0.5781
23.
DEFORM(TIME
0.7114
0.1568
25.91
0.9028
0.6189
26 11
0.9313
0.6981
21.01
0.9:108
0.3963
15.09
0.8791
0.5-125
22.13
Persistence
D.4800
0.6393
33.84
0.6374
1.1444
42.55
0.40-66
1.7516
44.79
0.7071
0. 0.8674
32.98
0.5688
1. 1.1007
38.
DLinear
0.4636
0.6512
$7.96
0.6112
1.0328
41.19
0,1125
1.4015
10.27
0.6810
0. 0.8337
31.08
0.5179
0.9805
37.
TST
0.4013
0.6678
36.55
0.5274
1.2615
48 85
0.3903
1.6234
44.67
0.5679
1.0676
40.39
0.4892
1.1525
42.61
CycleNet
0.4197
0.6150
33.71
0.5823
1.2092
0.3997
1.5371
43.37
0,1523
1.1910
15.34
0.1588
1.1388
12.31
iTrnnformmr
0.5258
0.5949
32.54
0.6567
1.0632
41.38
0.5726
1.3053
38.82
0.6738
0.8842
34.26
0.6072
0.9619
36.75
28 TimeXer
0.3979
0.5856
32. 82
0.7017
0.97/18
39.33
0.6137
1.2462
35.29
0.7037
0. 0.8229
31.45
0.6012
0.9074
31.72
TimeMixer
0.5756
015835
36.06
0.7251
1.0106
30.21
0.7694
1.2470
43.49
0.7467
0.7000
27.37
0.7042
0.8853
36.53
LighTTS
0.8262
0.6018
33.56
0.8332
0,8180
31.54
0.8556
0.8913
27.03
0.9011
0.5589
18.78
0.85-00
0.7175
27.73
Creesformrr
0.7102
0.46997
27.38
0.9064
0.7624
28.93
0.8432
0.8253
20.08
0,8782
0.5474
18.64
0.8345
0.6512
23:91
ModernTNN
0.7635
0.4702
27.09
0.8667
0,6742
27.31
0,9223
0.6618
19.83
0.9604
0.4776
20.41
0.8782
0.5710
23.66
DEPORATTIME
0.8369
0.4716
27.21
0.9562
0.6017
25.66
0.9253
0.5954
17.62
0.9294
0.5463
18.52
0.9119
0.5538
22.25
D.4 Data normalisation
Tn all forecasting tasks, we standardise all variables (zero mean, unit standard deviation), each time based on the training data. The prediction output is de-normalised back to its original scale prior to being compared with the (de-normalised) ground truth. In addition to that, for the ETT and weather data sets, we also standardise each variable within the input's look-back window, in accordaacce with previous work [Liu et al.
2022b Zhou et al. 2022a Nic et al. 2023
E
Supplementary results
E.1 Symmetric Mean Absolute Percentage Error
To compare results across different tasks that have different units, we use the sMAPE metric. We note the general (and correct) perception that occasionally sMAPE may mislead (by over- or under-estimating error), and hence it can only be used for partial insights in conjunction with more robust error metric. Hence, our
24

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Published in Transactions on Machine Learning Research (04/2025)
Table S3: ILI rate forecasting results in US Region 9 (ILI-US9) using web search frequency time series as exogenous variables. P and e % denote linear correlation and sMAPE, respectively. The best results are in bold font and the second best are underlined.
2015/16 MAE
2016/17
2017/18
2018/19 MAE
Average MAE
11 Model
%%
P
MAE
%%
P
MAE
%
c%
%
Persistencee
0.8384
0.3945
18.58
(1.8191
0.2791
16.25
0.731H1
0.5949
21.35
0.8898
0.3544
17.57
0.8270)
0.4157
18. 19.49
DLincar
0.8071
0.4769
25.07
01880192
0.3727
22.02
0.6759
0.5704
22.96
0.5925
0.4490
23.82
0.7062
0.4875
23.47
PatciTTT
0.8248
0.3993
19.30
0.8318
0,28012
16.91
0.7068
0.6078
23.25
0,5796
0.3592
17.91
0.8107
0.1116
19.94
CycleNet
0.8101
0.4156
20.13
0.7754
0.3265
19.50
0.6885
0.6897
26.34
0.8843
0.3603
18:22
0.7846
0,4450
21.05
irranfoommrr
0.8380
0.3736
17.55
0.8116
0,2721
16.22 22
0.6861
0.6856
26,00
0,9230
0.2915
14.52
0,8180
0.1057
18.57
7.
TineXer
0.8257
0.4071
20.04
(18847
0,2669
15.50
0.7638
0.5235
21.92
0,9021
0.3277
16,36
0.8344
0.3813
18.20
TimeMiier
0.9079
0,2892
13.61
0.8521
0.2733
16.29
0.7923
0.1896
18.47
0,9525
0.2133
12.48
0.8762
0.3239
15.21
LightTS
0.8596
0,2790
13.66
0.8858
0,2931
16.95
0,8277
0.4171
16.82
0,9453
0.2544
15.20
0,8866
0.3185
15.65
Croesformer
0.9305
0,2556
12.21
0.9521
0,2874
14.89
0,8213
0-1624
17.08
0.94135
0.2611
13.58
0.9119
0.31
14.44
ModeralLNN
0.9263
0.2615
13.08
0.8888
0,2452
14.86
0,8700
0.4745
21.37
0.9770
0.1754
8.37
0.9156
0,2859
11.17
DEPORNTIME
0:9161
0.2437
12:46
0.9356
0.2364
11.69
0.8744
0.3664
13.61
0,9676
0:2023
11:29
0.9234
0, 0.2622
12.26
Persistence
0. 0.7672
0.4876
23.00
0.7800
0.3386
19.84
0.6439
0.7283
26.96
0.8230
0.4489
22.49
0.7530
0.5008
23.
DLinear
0.7323
0.5767
29.99
0.7371
0,1381
25.
0.6105
0.6635
0.8216
0.5085
26.27
0.7254
0.5-167
PSte
0.7626
0.4810
23.33
0.7411
0.3524
21.48
0.6246
0.7164
28.11
0.8:02
0.4683
23.44
0.7346
0.5020
24.00
CycleNet
A.7309
0,5006
24.59
0.7279
0.3086
22.02
0.6372
0.7082
26. 12
0,8256
0.4513
23.34
0.7319
0.5072
24.02
Transformer
0.7837
0.4310
20.63
0.7620
0.3448
20.49
0.6138
0.6853
27.21
0,8495
0.4196
21.41
0.7525
0.4702
22.44
14 Timexer
0.7423
0.4796
23. 23.26
0.7872
0.3078
18.07
0.6443
0.6440
25.49
0.8261
0.4345
21.76
0.7500
0. 0.4665
22. 14
TineMixeer
0.5319
0.3600
17.41
0.8221
0.3153
18.22 22
0.6668
0.6111
23.48
0.9139
0.3377
17.21
0,8087
0.4060
19.08
LightTS
0.8697
0.4018
20.65
0.8137
0.34188
20.50 50
0,7802
0.4509
20.31
0.9396
0.2730
14.70
0.8625
0.3791
19.04
Crossformer
0.8787
0.3335
16.05
0.8965
0,2080
16.74
0.8339
0.4325
16.91
0.9169
0.3647
19.21
0.8815
0.3571
17:23
MaderiTON
0,9250
0,3582
15:44
0.8543
0.3176
17:00
0.8696
0.4244
15.32
0.9320
0.2006
13.40
0.8052
0.3417
15.29
DEFORATIME
0.9031
0.2962
14.39
0.9374
0.2893
13.15
0.8490
0.3897
14.62
0.9378
0.2584
13.05
0.9069
0.3084
13.80
Persistetaee
0.6682
0.5831
27.54
0.7150
0,4013
23.26
0.5734
0.8466
31.93
0.7520
0.53333
26.92
0.6772
0.3916
27.41
DLincur
0.6687
0.6032
30.66
0.6920
0,4060
29.49
0.5830
0.7252
30.15
0,7778
0.5760
29.33
0.6804
0.0001
29.66
PateTTST
0. 0.6899
0.5705
28. 11
(163199
0.4255
26.14
0.5214
0.7966
32.881
0.7137
0.5815
30.27
0.6412
0.5933
29. 40
CycleNer
0.7092
0.5656
28.42
0.6895
0,4043
24.79
0.5731
0.8140
30.35
0.0097
0.5864
30.31
0.6629
0.5926
28.47
Trnnsformer
0.8659
0.3645
17.75
0.7078
0,3712
21:73
0,1561
0,8642
33.91
0,8298
0.4
23.01
0.7149
0.5106
21.11
21
Tinneerr
0.6201
0.5890
29:37
0.6373
0.4192
24.19
0.5158
0.7825
31.10
0,7460
0.5006
25.04
0.8298
0.5715
27.42
TimeMixer
0.8280
0.1112
19.07
0.8180
0,3058
17.67
0.6555
0.6812
0,5631
0,4322
23.
0.7912
0.4376
21.10
LightTS
0.8354
0.4252
20.19
0.7748
0,0021
25.59
0.6920
0.6201
25.57
0.8699
0.4542
23.80
0.7933
0.4754
23.74
Croesformer
0.9565
0.3267
15.19
0.8809
0.3168
15.95
0.8777
0.3991
15.11
0.9135
0.3273
17.07
0.9036
0.31118
15.90
ModeralINN
1.8695
0.3118
12.39
(18777
0.3895
16.88
0,7372
0.50ST
18.36
0.8987
0. 2968
14.08
0,8510
0.3710
15.43
DEFORM/TIME
0.8331
0.35
15.62
0.8966
0.3039
14.99
0.91411
0.3536
13.06
0.9241
0.2598
13.27
0,8922
0.3179
14:24
Persistence
0.3763
0.6781
32.15
0.6261
0.4733
27.09
0.5059
0.9541
36,29
0.6705
0. 0.6144
31.14
0.5047
0.6790
31. 67
DLinear
0:6083
0.6196
92.81
0.6016
0.5329
30.118
0.5506
0.7893
32.75
0,7138
0. 0.6533
$2.89
0.6261
0.6564
32.16
TST
0. 3.6067
0.6467
32.75
0.5192
0.4903
30.18
0.4424
0.8509
36.70
0.6556
0. 0.6492
33.78
0.5660
0. 0.6965
33:35
CycleNet
0.6075
0.6136
$1.91
0.5300
0,1860
0.1990
0.8940
0,1816
0.7887
0.5321
0.7031
31.50
iTrnnformmr
0.6984
0.5531
27.80
0.5779
0.4444
25.70
0.3211
0.9942
40.08
0.6797
0.6077
30.60
0.5003
0.6498
00
28 Timexer
0.5345
0.6372
32.39
0.6320
0.1302
23.80
0.3998
0.9296
37.46
0.6127
0. 1.62288
31.61
0.5448
0.6355
31.32
TimeMixer
0.8130
0.4056
21.18
0.8184
0.3244
19.25
0.6444
0.7787
31.60
0.7816
0.5409
24.41
0.7644
0.5124
24.11
LighTTS
0.8462
0.1765
24.19
0.6962
0.1831
25.81
0.7176
0.5616
22.25
0.9033
0.3862
20.33
0.7907
0.4769
23. 22
Creesformrr
0.8969
0.3398
16.47
0.8346
0.3647
16.99
0.8514
0.4365
15.42
0.8023
0.3578
16.87
0.8688
0.3747
16.44
ModernTNN
0.8552
0.3611
17.38
0.9076
0.3748
17.59
0.8068
0.5156
18.98
0.8645
0.3245
14.79
0.8685
0.3940
17.19
DEPORATTIME
0.8588
0.3718
15.55
0.9046
0,3153
14.57
0.9037
0.4185
16. 18
0.9260
0.3069
16.68
0.9058
0.3532
15.74
main error metric is MAE. For the ILI rate forecasting lasks, we additionally show linear correlation as an established metric in related literature (Ginsberg et al. 2009 Lampos et al., 2015 Yang et al.] 2015).
We use the following definition of sMAPE. For series of estimates E and a corresponding series of true values y E R", sMAPE is given by
sMAPE(S.y) = 100 n2
10; Mjl
(S1)
0.5 ( 05 + 105l)
j=1
In Tables throughout the manuscript, we denote sMAPE using e %.
E.2 Ablation study for DeformTime - Additional information
Here we provide additional details to supplement the ablation study presented in section 4.3
V-DAB: Denotes that we train a DEFORMTIME model without the V-DAB encoder branch (see section 3.3
Within each encoder layer, we only use the T-DAB branch for sequence encoding. Instead of projecting the
25

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Published in Transactions on Machine Learning Research (04/2025)
Table S4: Ablation study on different groups for DEFORM'TMEE using the ILI-ENG and ILI-US9 data sets. Results are averaged across all scasons.
G=2
G=4
G=8
G = 1G
==
Q=4
G=3
G
MAR
MAR
%
MAR
%
MAE
%
MAE
MAE
%
MAP:
%
MAE
1,7733
34.79
1.6417
28.61
1.7325
23.52
1,7085
24.73
0
7
0.2965
12.80
0.2622
12.26
0.2010
11.85
012675
12. 12.84
32:92
2,2308
2.1954
11.52
29.50
14.82
0,3084
0.2926
13.39
0,2995
13.71
21
2.7963
35.23
2.6500
32.70
2.7750
43.23
2.8962
34.70
21
0.3475
16.27
0.3179
14.24
0.3273
15.35
0.3534
16.12
3.1379
44.84
2.7228
2.9545
3,0996
41.41
28
0.3532
0.3550
0.3679
16.73
concatenated output of two encoder branches, we take Ze directly from T-DAB the output Zj of the j-th encoder layer.
- T-DAB: Denotes that we train a REFORMTIME model without the T-DAB encoder branch (see section 3.4 We take Ze directly from V-DAB as the output Z of the j-th encoder layer.
Pv.t: Denotes that we do not use the relative positional biases (both P, and P.) when obtaining the value embedding for both the V-DAB and T-DAB modules.
NAE: Denotes that train DEFORMTIME model without the NAE module. For imput Z ( simply use fully connected layer to embed the C+1 variables into hidden dimension of size d and obtain E
PR: Denotes that we do not use the fixed position embedding Pa from the embedding procodure, i.c. instead of Equation we use Ze = LN(E).
E.3 Detailed ILI forecasting results
Complete results (all test periods, models, and forecasting horizons) in the ILI rate forecasting tasks for England (ILI-ENG), US Region 2 (TLI-US2), and US Region (ILI-US9) presented in Tables S1] S2 and S3
respectively. In addition to MAE' and sMAPE, also show the linear correlation (denoted by p) between estimates and the target variable throughout each test period.
E.4 The effect of variable grouping in the NAE module
We further explored the effect of the grouping parameter G in DEFORMTIME, which determines both the number of groups NAE and the number of heads the multi-head attention of T-DAB. Experiments are conducted using the ILJ forecasting task for England (TLI-ENG) and US Region 9 (TLI-US9)). Hyperparameters are re-tuned for each value of Ge (2, 4, 16).
Results are enumerated in Table S4 It can be argued that more groups tend to benefit performance for shorter forecasting horizons, whereas a more moderate level of grouping is required for the optimal performance as the forecasting horizon increases. Notably, extensive (G = 16) or overly small (G = 2) groupings result in performance degradation. This observation could be further interpreted in conjunction with the results presented in Table 3 where we see that DEFORMTIME performs better shorter-term forecasting tasks when it uses more variables (C). Presumably, the model benefits from an adaptive G, where G proportional the number of input variables C. Nevertheless, to reduce the complexity of hyperparameter optimisation in our experiments have chosen to use constant value, setting =
We note that the observed influence of grouping may pose a potential limitation of DEFORMTIME. as it affects how dependencies are captured within the latent variables. Specifically, smaller grouping numbers allow the dependencies to be captured among variables holding less linear correlations and vice versa. However, there likely exists a trade-off between allowing more variables to be able to be captured by V-DAB and the model being overfitted as we have empirically shown in Tables and S4 where both removing the NAE and using overly small number of groups lead to inferior performance.
26

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Published in Transactions on Machine Learning Research (04/2025)
Table S5: Seed control (5 seeds) for DEFORMTIME using the ILI-ENG data set across all forecasting horizons (H). j and denote the mean and standard deviation of the obtained MAEs per test period. The results in the main paper were obtained for seed '422.
2015/16
2016/17
2017/18
2018/19
Average
2015/16
2016/17
2017/18
2018/19
Average
02
1,379
1,3088
2,031
1.215
1.612
02
2.170
1,898
1.598
1,633
2,650
1.416
1.063
2.266
1,360
1.526
2,079
1.507
4.871
1.974
2.683
111
1.476
1,386
0.586
1.401
2,315
1,086
1,583
1,690
1,600
1.509
21
112
2.250
2.354
1.825
1.913
4.158
1.088
1,941
1.625
2.191
2,007
1.443
1.109
2:107
1.226
1.404
2.147
1.505
4.406
1.842
2.408
(o)
1.42 (0.04)
1,15 (0)18)
2:28 (0.21)
1.41 (0.18)
1.57 (0.06)
[o]
2.20 (0.09)
1,75 (0.17)
4.00 (0.28)
1,78 (0.13)
2.58 (0.09)
4.532
1,696
1.514
3.439
1.712
2,000
2.486
1,692
5.194
1,907
2,755
1,881
1.297
3,556
1.812
2.139
112
2:655
1.825
4.587
1.907
2.819
1111
1,621
1.195
3.327
1.396
1.900
111:
2:251
1.709
4.529
2.082
1234
1,837
1.137
3.870
1.391
2.061
1234
2,002
1,808
4.000)
1.613
2,763
(0.15)
1,35 (0.14) 3.39 (0.20) (0.19) 2.10 (0.10)
[e)
(0.17)
1.79 (0.05) (0.21)
(0.18)
(0.06)
E.5
Computational complexity and efficiency of DeformTime Additional information
The total number of operations (based multiplications) for DEFORMTIME is given by
(c+1)dL+d+
2d
NAE
Decoder
n [Nt + 1)6d+336 438 anndl) +r + ()(d+1)n++3))) and 3ndd +2d+6d
(S2)
V-DAB
T-DAB
Encoder
The main components that affect the order of operations are the following:
V-DAB: V= n = 3ed 3 nldz] operations
(S3)
Le = ( L dl?))
T-DAB: T T (k+ Rd 3rd 3n operations
(84)
L (PP).)
Encoder: 8= T + 2d r.) operations
(S5)
= o dL?)
Decoder: G 2d2 operations
(S6)
= odd
Hence, conclude that the order of operations for DEFORMTIME is Old dL?). where L denotes the length (time steps) of the look-back window, and d is the size of the hidden layers used throughout our method.
E.6
Seed robustness for Deform Time
We have examined the robustncss of DEFORMTIME across different seeds with hyperparameter tuning by re-running the experiments on the ILI-ENG data set across all forecasting horizons and lest. sets. The results are enumerated in Table S5 The seed used for the results presented the main paper ('42") provided a performance that did not deviate significantly compared to other seeds. Hence, we conclude that DEFORMTIME robust to random seed initialisation. We note that compared the average performance across the explored seeds, seed '42" provides rather conservative estimate.
27

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Table S6: Forecasting accuracy results (MAE) using the entire output sequence (as opposed to using the last prediction only at the target forecasting horizon) across all tasks and methods. H denotes the forecasting horizon. For the ILI tasks, errors are averaged across all 4 test seasons for each region.
Thsk
ModemnTNN
QyeleNet
Timexor
PaschTST
(Trorsformer
TimeMixer
Croseformce
T4ghe:S
DMdnow
96
0.1810
0.1771
0.1746
0.1807
0.1757
0.1825
0.1831
0.1989
0.2089
0.2045
192
0.1988
0:2013
0.1990
0.2043
0.1999
0.2051
0.2040
0:2149
0.2267
0.2534
336
0.2024
0.2123
0.2189
0.2236
0.2248
0.2522
0,2310
0.2577
0.2187
0.3877
720
0,2590
0.2288
0,2367
0,2292
0,2500
0,2449
0.2163
0.3000
0.1150
0.5007
ETT12
96
0.2924
0,2790
0.2849
0.2797
0.2765
0.2991
0.2963
0.3254
0.3125
0.2840
192
0.3010
0.3153
0.3223
0.3331
0.3174
0.3163
0.3:28
0.3531
0.3196
0.3313
336
0.3172
0.3350
0.3536
0.3780
0.3411
0.3857
0.3758
0.3891
0.3989
0.4056
720
0.3577
0:4259
0.3976
0,3069
0,3887
0.3945
0.39
0:4218
0,1339
0.5391
CS. USL- ILIGENNG Wother
0.0226
0.0235
0.0228
0.0270
0.0204
0.0218
0.0272
0.0217
0.0263
0.0206
192
0.0237
0.0265
0.0257
0.0294
0.0239
0.0253
0.0309
0.0271
0.0292
0.0239
336
0.0258
0,0296
0.0279
0,0312
0.0259
0.0269
0,0325
0.0311
0.0294
0.0261
720
0.0508
0.0365
0.0333
040355
0.0316
0.0512
0.0369
0,0363
0.0333
0.03407
1.2802
1.7351
2,0997
L.7086
1.9402
1.7740
L.7745
1.3243
1,9713
2.1892
1.6521
2.2656
2,6170
1.9703
2.4857
2,3080
2 2.2577
1.9826
2,3676
2.7452
2.3246
2.3351
3,2327
2.8852
2,8158
3,0904
2:9285
2,8733
2.7742
3.2963
28
2,7829
2.58-88
3. 3,6091
2.7894
3.2161
3,4784
3. 3,2/25
3,0740
3,2539
3.6805
0.41442
0.4523
0.0974
0.4931
0.6400
0.5455
0.5156
0.4464
0.4323
0.7214
11
0.1544
0.5501
0.7601
0.5
0.7302
0.6357
0.5:99
0.4695
0.1723
0.7899
0.5374
0.3576
0,8130
0.6190
0.7489
0.6554
0. 5.566
0.5720
0.6432
0.8611
28
0.5554
0.5892
0.8729
0,0806
0.8150
0.7370
0,6888
0.58
0.6886
0.8920
0,2614
0.1126
0.3135
0,3882
0.3725
0.3322
0.5222
0,3078
0,1311
0.3040
0.33MS
0.4384
0.3715
0.4334
0.4049
0.3500
0.3254
0.4322
0-4780
0.3187
0.3558
0.5224
0,1130
0. 10.6681
0.1501
0.5711
0.5285
0.1385
0.5315
28
0.3913
0.3786
0.5727
0-4774
0.5002
0.5533
0:4236
0.4018
0.5663
0.5022
E.7
A brief note about the performance and evaluation of DLinear
Although DLinear achieved competitive results in prior work (Zeg et al. 2023 the performance drop in our paper originates from the fact that we evaluate models based on their estimate for yt+H Lee. the target variable's 'ss value at time step t+ H (sec a detailed description of the forccasting task in section 22) In fact, for most tasks this is the larget forecast (represented by the corresponding forecasting horizon H) and that is what forecasting accuracy should be measured on. Contrary to common sense, DLinear (as well as other works, c.g. Wang et al.] (2024b) or Liu et al.] (2024)) was evaluated on the entire time scries (entire sequence of predictions), i.e. from yc+1 to Y++H. Oddly, this evaluation was conducted in a uniform way, i.e. the error for each time step incurred the same penalty. That may be relevant for some tasks, but it is not relevant to tasks where forecasting H time steps ahead really necessitates obtaining an accurate forecast H time steps ahead (and that should be the case for most, if not all, forecasting tasks). The reason behind this is that forecasting errors tend to increase as the forecasting horizon extends and the forecasting task becomes harder (Hyndman & Kochler 2006). Models may perform differently at short versus long horizons. Averaging the error across all the outputs may favour forecasters that are very accurate early on, but very inaccurate closer to and at the target forecasting horizon. DLinear was obviously more accurate in early time steps (lower degree of difficulty), but very inaccurate in later ones (greater degree of difficulty). Hence, in our experiments, DLinear displayed the worst forecasting performance. The following section E.8 offers additional insights.
E.8 Evaluating forecasts across the entire output time series
For more comprchensive assessment of model performance, we provide forccasting accuracy results where we consider the entire output sequence (the error is averaged across the entire output sequence). Results are enumerated in Table SG DEFORMTIME outperforms baselines on 17 out of 24 tasks, with averaged MAE reduction by 1.7%. When compared to MCdernTNN, the best-performing bascline, DEFORMTING reduces the MAE by 7.2% on average across all tasks.
For the ETT and weather benchmark datasets, the best-performing baseline is PatchTST; DEFORMTIME reduces PatchTST's MAE by 1.5% on average. If we compare that to obtaining MAE based on the forecasts at the target forecasting horizon only (Table []. we can see a major discrepancy: DEFORMTIM TIME reduces PSTchTST MAF by 9.8% (on the ETT and weather forecast tasks; reduction is greater
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for the ILI forecasting task) when forecasting accuracy is evaluated based on the actual target forecast. Going back to using the entire output time series, for the ILI tasks, DEFORMTIMR reduces the MAF by 3.4% on average. For these tasks, Crossformer achieves the best MAE performance amongst baselines and DEFORMTIME reduces that by 8.2% on average.
Notably, while PStchTST outperforms other baselines when evaluated over the entire sequence in the ETT and weather data sets, CycleNet was the best-performing baseline in our main assessment (Table D.. Hence, by this observation alone, it is evident that while CycleNet is producing more accurate forecasts at the target horizon compared to PSt it yields an inferior average MAE when considering the entire series of forecast outputs from time step 1+1 to time step H+H. A similar ranking difference of bascline performance is also observed in the ILI tasks, where ModernTCN was the most competitive baseline when evaluating the results over the target horizon (DEFORMTIME reduces ModernTCN MAE by 11.2% on the ILI tasks as enumerated in Table D). The overall changes in the ranking of basclines consolidate our argument section E.7 that convoluting the error at the target forecasting horizon with errors in time steps prior to that can produce a distorted picture of the actual forecasting capacity of model.
E.9 Should forecasting models mix variables in the embedding space?
The paper that presented the PatchTST method (Nie al. 2023) provided some empirical evidence for the potential benefits of not modelling inter-variable dependencies. Contrary to that, our results demonstrate that this may not always hold for better-performing models well different time scries forccasting tasks. Based our ILI rate forecasting experiments, we deduce that models that mix variables in the embedding space while preserving the temporal structure (Modern'TCM, Crossformer, LightTS, TimeMixer, and DEFORMTTMME) outperform the rest. Hence, and perhaps as expected, modelling inter- variall relationships should still be considered a viable (and arguably essential) approach in time series forecasting.
F Societal impact
As with any scientific development, our work may have positive but also negative societal impact. From a positive perspective, forecasting models can be used to better predict outcomes in various domains and improve quality of life. Contrary to that, forecasting models could also be exploited by malicious actors in various forms of malpractice. We do think this should be common knowledge. We do also note that there is thin line between safeguarding Al malpractice and significantly restricting scientific progress (and hence, quality of life improvements). This is beyond the scope of our work and is obviously broader topic of discussion.
We would like to note that outcomes of this work (c.g. an improved variant of DEFORMTIME) may be used in infectious disease monitoring systems by interested stakeholders (e.g. public health organisations such as CDC, ECDC, UKHSA etc.). These, of course, do not constitute malicious actors. However, potential risk may arise when (if) forecasting model provides inaccurate insights. Nevertheless, this is shared risk among various other disease models (including established mechanistic models). To mitigate this risk, public health agencies use multiple endpoints to determine their course of action (e.g. sentinel and syndromic surveillance, laboratory tests, rapid tests, and vetted epidemiological models). Hence, any decision is based on a collection of different predictors. Nonetheless, prior to the adoption of any forccasting model within public policy, a more thorough and focused (lo the said application domain) evaluation of the forecasting method is required.
G Supplementary ILI rate forecasting figures
Here we present a series of figures for the ILI rate forecasting task. Each figure shows the estimates from all models for certain forccasting horizon and location.
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Ground tuch
days forecasting fing (LLENG)
Persioance
54
DUnear
Cyslehir.
Tineker
LighTS
50
Crossfomer
ModernTCH
56-
Drremv/ha
2015
2014-0
201744
201707
20k
201847
Figure S2: days ahead forecasts for all influenza seasons and models for England (ILI-ENG).
Foreanta uss IuP
days England
Persioencce
50
Dlinwar
Cysteher:
031000100 1990
Thneker
TimeMber
LighrT
50
Crossfomer
ModernToN
Doron/Ton
201647
201-43
203748
2014:1
Figure S3: days ahead forecasts for all influenza seasons and models for England (ILI-ENG).
30

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Published in Transactions on Machine Learning Research (04/2025)
Oround tuch
days forecasting England
Persioence
54
DUnear
Cyslehir.
Tineker
LighTS
50
Crossfomer
ModernTCH
Drremd/ha
201507
2016-0
201746
2017477
Figure S4: 21 days ahead forecasts for influenza seasons and models for England (ILI-ENG).
Foreanda Grs Iuph
days England
Persioencce
50
Dlinwar
Cystehir:
1990
Tineker
TimeMber
LighTS
50
Croesfomer
ModernTCN
Doron/Ton
201647
201-43
Nas-07
203748
2014:1
201e
Figure S5: days ahead forecasts for all influenza seasons and models for England (ILI-ENG).
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Published in Transactions on Machine Learning Research (04/2025)
Foreoeds Ground tuch
days forecasting Ragion (114452)
30-
Persioance
DUnear
Cyslehir.
16
Tineker
30
LighTS
Crossfomer
ModernTCH
Drrem//ha
2019-97
2016a
20174)
201967
2019-7
Daps
Figure S6: 7 days ahead forecasts for all influenza seasons and models for US Region 2 (ILI-US2).
Foreanda Grs Iup
days forecasting Region
30
Persioeence
Dlinwar
Cystehir:
10
Tineker
ThmeMbee
LighrT
Crossfomer
ModernTCN
Dorom/Ton
201547
20164t
207443
zoledt
201447
Josd:1
Days
Figure S7: 14 days ahead forecasts for all in infuenza seasons and models for US Region 2 (ILI-USS)..
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Published in Transactions on Machine Learning Research (04/2025)
Foreoeds Ground tuch
days forecasting Region (LAUS2]
Persioance
DUnear
10
Cyslehir.
10
Tineker
LighTS
Crossfomer
ModernTCH
Drrem//ha
2019-97
2016a
201967
2019
Figure S8: 21 days ahead forecasts for all influenza seasons and models for US Region 2 (ILI-US2).
Foreanda uss Iup
days Region
30
Persioencce
Dlinwar
15
Cystehir:
10
Thneker
ThreMbee
Lighre
Crossfomer
ModernToN
201547
201641
zoledt
201447
201-:1
Days
Figure S9: 28 days ahead forecasts for all in infueezaa seasons and models for US Region 2 (ILI-USS)..
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Published in Transactions on Machine Learning Research (04/2025)
Foreoeds Oround tuch
daye forecaeting Region (ILAUSS)
Persioance
50
DUnear
2.3
Cyslehir.
5-
25
Tineker
LighTS
50
Crossfomer
ModernTCH
255
5.
Drrem//ha
2015e1
201600
2016-1
201333
201707
201443
201077
Figure S10: 7 days ahead forecasts for all influenza seasons and models for US Region 9 (ILI-US9).
Foreanda uss tup
Region
Persioeence
5.3
Dlinwar
25
Cystehar:
50
Tineker
ThreMbber
LighrT
Crossfomer
5.
ModernToN
2.5
5.
201547
201643
201-41
201741
2017et
2014-7
301501
Days
Figure S11: 14 days ahead forecasts for all influenza seasons and models for US Region (ILI-US9).
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Published in Transactions on Machine Learning Research (04/2025)
Toreoeds Oround tuch
forecasting Region (L1-U5))
Persioance
235
5.
DUnear
5.0
2.5
1.5
Cyslehir.
25
Tineker
LighTS
Crossfomer
ModernTCH
Drrem//ha
2015e7
2016
2016-3
201333
201707
201443
Figure S12: 21 days ahead forecasts for all influenza seasons and models for US Region (ILI-US9).
Foreanda uss tup
Region
Persioennce
Dlinwar
Cystehar:
5-3
2.5
TineXer
TireMbbe
LighrT
2.5
Croesfomer
ModernToN
2.7
5.
Doron/Ton
201547
2016-3
201-41
201741
201het
20/447
301501
Days
Figure S13: 28 days ahead forecasts for all influenza seasons and models for US Region (ILI-US9).
35


Reviews:
[{"review_id": "zdcaZxAuw1", "reviewer": "Reviewer_EKtq", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength:\n- The experimental results are convincing. The authors utilize various datasets and ablation studies.\n- The proposed method is intuitive and the paper is easy to follow.\n\nWeakness:\n- The experimental settings are somewhat less practical (using disjoint exogenous predictors to forecast \"single\" target variables).\n- If the authors can provide the extension of the proposed works that can expand the experimental settings (including multiple target variable predictions as well as target variables are also included in the predictors), that makes this paper more impactful.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "- The authors propose a novel time-series prediction framework that can utilize the exogenous predictors efficiently so that it can capture inter- and intra-variable dependencies.\n- The authors tackled a specific problem setting where the model predicts a single target variable using disjoint exogenous predictors.\n- The experimental results show that the proposed method show better results than the alternatives (with lower computational complexity).", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Extension to multiple output prediction\n- As the authors mentioned in Section 2, the authors mainly focus on the setting where the inputs are multivariate but the model only predicts \"one\" output variable. \n- Could we expand the proposed method to multivariate output predictions? If yes, please provide some procedure to expand and provide some \"empirical results\".\n- If we can easily do that, it would significantly improve the impact of this paper because in practice, there are many applications where we need multiple variable predictions simultaneously.\n\n2. Inputs and targets variables\n- I agreed that there are some settings where target variable and input variables are different (i.e., Q and y are different).\n- But in many cases, Q and y are overlapping. Could we easily expand this method to the setting where Q and y are overlapping? If yes, please provide some procedure to expand and provide some \"empirical results\".\n\n3. Methods\n- As far as I know, there are various other methods which can also capture both temporal relationship as well as inter-variable relationship. (A straightforward example is RNN). In that point of view, what is the main contribution of this work?", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "dscxXZSf8W", "reviewer": "Reviewer_Z84T", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n\n-  S1. Tackling time series forecasting with exogenous variables is relevant, especially since most recent literature has focused on purely autoregressive approaches. The application of deformable attention blocks to time series forecasting appears to be both novel and well-motivated. \n\n-  S2. The paper is clearly written and easy to follow, except for the section on the variable deformation attention block. The numerous ablation studies and the detailed appendix effectively highlight the contribution of each architectural component. \n\n-  S3. The experimental results are strong, even though the evaluation setting differs slightly from the traditional forecasting setup. Instead of assessing performance over an entire prediction window, the model is evaluated on a pointwise forecast at horizon H.\n\n\n**Weaknesses**\n\n-  W1. The T-DAB mechanism is difficult to follow, particularly the interpretation of Δp in Section 3.3. Is the deformable patch a transformation of the currently considered patch, or is it relative to a neighboring patch? In other words, can Δp represent indices from other patches? Clarifying this section with additional explanations and possibly some figures would be helpful.\n\n-  W2. The paper considers only multivariate datasets of the form C×T, where C represents different channels (e.g., temperature, humidity for weather) and T is the number of timestamps. While this setting is interesting, the authors should discuss its specificity compared to other common setups, such as datasets structured as N×T (where N represents different samples, like households in the Electricity dataset or roads in the Traffic dataset). Have you evaluated your method against baselines like PatchTST on these alternative dataset structures (Electricity or Traffic) ?\n\n-  W3. In the discussion on DLinear and the forecasting setup, the paper states:\n“That may be relevant for some tasks, but it is not relevant to tasks where forecasting H time steps ahead really necessitates obtaining an accurate forecast H time steps ahead (and that should be the case for most, if not all, forecasting tasks).”\nI have two concerns regarding this statement:\n    - I do not fully understand the reasoning behind it. Could you clarify?\n    - Could you provide forecasting results using the traditional evaluation setting (evaluating over the entire horizon) for the ETT and weather datasets?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper addresses the multivariate time series forecasting problem by leveraging information from exogenous variables. The authors introduce two core components: the Variable Deformation Attention Block and the Temporal Deformable Attention Block. The proposed architecture is evaluated against numerous baselines across six datasets and demonstrates strong performance.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please see above.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "O2JsCxBpe4", "reviewer": "Reviewer_rgws", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The strength of the paper is the completeness of a method paper in it's empirical components: from basic baseline comparisons to memory and time analysis, as well as ablation study. \n\nThe biggest weakness is lack of motivation and clarity for methods, the \"why\" question. The current method section is a list of statements without explaining why the authors choose to do the things that way. Hence, it is hard to follow. See more in the Requested Changes Section.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a new multi-variable time series forecasting method called DeformTime. It uses Deform Attention Blocks on both variable and temporal dimensions. On both existing and proposed new benchmarks, DeformTime demonstrates lower errors than other baselines, as well as better memory usage. Ablation study also highlights the important of the proposed Deform Attention Blocks.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Section 3.2, are the grouping of variables just sequentially taking the d/G variables as one group after the reordering? What motivates such grouping? Why does it make sense to use them?\n\nI didn’t get the Section 3.3. For a patch $Z_p$ of size (l, d), why adding a shift $\\Delta P$ to a position p = (i, j) helps to learn the correlation between different variables? What does $\\Delta P$ mean? What does $Z_d$ mean? In this section, I think it lacks an overall introduction on how the authors are going to leverage multi-variable information for forecasting and why they believe that’s the right way. The current writing is simply a list of steps without explaining the motivation and the why, which makes it hard to follow. Section 3.4 has the same problem. \n\nWhy just compute errors on the last time step in the prediction window? Is there any particular reason? Unless there are strong reasons, I suggest the authors use a more standard way to compute error, that is over all the time steps in the prediction window. Then we can compare numbers to see if they match the baselines in their original papers so that we know the baselines are implemented and used properly.\n\nMinor: Why does the delay need to be explicitly modeled? Can it be part of the preprocessing step?\n\nMinor: How do the authors find so many exogenous variables in Section 4.4? By different search queries?", "broader_impact_concerns": "N/A.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper proposes DeformTime, a transformer-based architecture for multivariable time series forecasting. Specifically, given a number of exogenous variables and a single endogenous variable, DeformTime predicts future values of the endogenous variable. The paper’s main contribution is the design of two types of deformable attention blocks (DABs): temporal DABs for intra-variable dependencies and variable DABs for inter-variable dependencies. Ablation studies and baseline comparisons validate this contribution on established and new benchmark tasks.

The paper initially received mixed reviews. While the reviewers acknowledged the paper’s novel perspective on deformable attention blocks, comprehensive experiments, and promising results, they also raised important concerns. The presentation of the deformable attention blocks (T-DAB, V-DAB) and DeformTime's evaluation at a single forecasting horizon, in particular, spurred discussions. The authors responded well to this criticism and provided an insightful DAB example, an evaluation of DeformTime over entire forecasting windows, and motivation/intuition for key architectural components. The scope of DeformTime, with its single target variable but multiple exogenous predictors, also caused confusion, both presentation-wise and regarding the applicability of the proposed framework overall. While multivariate predictions remain out of scope, the authors reworked their presentation and emphasized DeformTime’s focus on a single target. Some concerns could not be resolved during the discussion phase, most importantly the consequences of grouping variables based on linear correlation. The authors agreed that this might indeed be a limitation, although at least from an empirical standpoint the negative effects can likely be contained.

Ultimately, the majority of reviewers recommended acceptance and I agree with this sentiment. However, it is paramount that the important clarifications made during the discussion phase find their way into the paper; they are a prerequisite for acceptance. While some of these updates have already been incorporated in the latest revision, others have not. 

The paper can be accepted under the following conditions:

- Improvements to the positioning of the paper, especially concerning the paper’s focus on a single target variable.
- Improvements to the separation of deformable attention blocks as a prior concept and the specific innovations of this paper.
- Improvements to the technical descriptions of V-DAB (3.3) and T-DAB (3.4), including motivation and intuition. The latest revision is a good step, but I highly encourage the authors to take a third look.
- Inclusion of all experiments related to DeformTime’s evaluation over entire prediction windows (ETT, weather, and ILI).
- Inclusion of the discussion points (positive and negative) related to grouping variables based on linear correlation.
- Inclusion of ILI-US2 in Table 2 (ablation study), with a structure identical to ILI-ENG and ILI-US9.
Decision: Accept (poster)