Phase-driven Generalizable Representation Learning for Nonstationary Time Series Classification
Authors: Payal Mohapatra
Lixu Wang
Qi Zhu
Venue: Accepted by TMLR ()

Abstract: Pattern recognition is a fundamental task in continuous sensing applications, but real-world scenarios often experience distribution shifts that necessitate learning generalizable representations for such tasks. This challenge is exacerbated with time-series data, which also exhibit inherent nonstationarity—variations in statistical and spectral properties over time. In this work, we offer a fresh perspective on learning generalizable representations for time-series classification by considering the phase information of a signal as an approximate proxy for nonstationarity and propose a phase-driven generalizable representation learning framework for time-series classification, PhASER. It consists of three key elements: 1) Hilbert transform-based augmentation, which diversifies nonstationarity while preserving task-specific discriminatory semantics, 2) separate magnitude-phase encoding, viewing time-varying magnitude and phase as independent modalities, and 3) phase-residual feature broadcasting, integrating 2D phase features with a residual connection to the 1D signal representation, providing inherent regularization to improve distribution-invariant learning. Extensive evaluations on five datasets from sleep-stage classification, human activity recognition, and gesture recognition against 13 state-of-the-art baseline methods demonstrate that PhASER consistently outperforms the best baselines by an average of 5% and up to 11% in some cases. Additionally, the principles of PhASER can be broadly applied to enhance the generalizability of existing time-series representation learning models.

Published in Transactions on Machine Learning Research (10/2025)
Phase-driven Generalizable Representation Learning for Nonstationary Time Series Classification
Payal Mohapatra'
Northwestern University, Evanston, Illoooi, USA
Lixu Wang*
liru hin..wwwnggnnorbbussteer....
Northwesterrn University, Evanston, Illonois USA
Qi Zhu
qi @i:hh@@nortbhwsseer...l
Northwesterr University, Evanston, Mlinois, USA
Reviewed on OpenReview: https: // openreview. net/ forum? id= cb3nwo
Abstract
Pattern recognition is fundamental task in continuous sensing applications, but real-world scenarios often experience distribution shifts that necessitate learning generalizable rep- resentations for such tasks. This challenge is exacerbated with time-series data, which also exhibit inherent nonstationarity variations statistical and spectral properties over time. In this work, we offer fresh perspective on learning generalizable representations for timo-scries classification by considering the phase information of signal as approximate proxy for nonstationarity and propose a phase-driven generalizable representation learning framework for time-series classification, PhASER. It consists of three key elements: 1) Hilbert transform-based augmentation, which diversifies nonstationarity while preserving task-specific discriminatory semantics, 2) seporate paggitudde-hasee encoding, viewing time-varying mag- nitude and phase as independent modalities, and 3) phase-residual feature broudcasting, integrating 2D phase features with residual connection the 1D signal representation, providing inherent regularization to improve inttributtinninvarinnn learning. Extensive eval- uations five datasets from sleep-stage classification, human activity recognition, and gesture recognition against 13 state-of-the-art bascline methods demonstrate that PhASER consistently outperforms the best baselines by an average of 5% and up to 11% in some cases. Additionally, the principles of PhASER can be broadly applied to enhance the generalizability of existing time-series representation learning models.
1 Introduction
Time-series data play a ubiquitous and crucial role in numerous real-world applications, such as continuous monitoring for human activity recognition (Li et al., 2020), gesture identification (Ozdemir et al., 2020), sleep tracking (Kemp et al., 2000), fatigue monitoring (Mohapatra et al., 2024a) and more. Continuous time series often exhibit non-stalionarily, i.e., the statistical and spectral properties of the data evolve over time. Another practical challenge is domain shift, where the lata-generating process systematically evolves over time due to factors such changes in sensor lype, sub-population shifts, or environmental variations, thereby degrading model performance previously unseen distributions. Thus, developing methods for more generalizable pattern recognition in nonstationary time scries classification is crucial.
Most existing methods (Ragab et al., 2023a;b; He el al., 2023) lackle distribution shifts in time-series applications via domain adaptation, assuming accessible target domain samples. Yet, obtaining data from
*These authors contributed equally.

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Published in Transactions on Machine Learning Research (10/2025)
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Figure 1: PhASER's components: I. Hilbert transform-based phase augmentation. II. Separate feature encoding of time-varying phase and magnitude derived from Short-Term Fourier Transform (STFT) using PMag and Fris- III. Key elements of the phase-residual broadcasting network, demonstrating design of depth-wise feature encoder (Fbep), temporal encoder (Frem), and incorporation of phase-projection head's output (gres) for broadcasting (annotated dimensions of intermediate feature maps). IV. Task-specific classification encoder (gens)-
unseen distributions in advance is not always feasible. To overcome this challenge, a few works (Gagnon- Audet et al., 2022; Xu et al., 2022) applied standard domain generalization (DG) algorithms (Volpi et al., 2018; Sagawa et al., 2019; Parascandolo et al., 2020) to temporally-varying time-scrics data, but reported a siguificant performance gap when compared with visual data. Recent research on DG tailored for time series explores aatent-domale Uchida, 2021; Li et al., 2021a), preservation of non-stationarity dictionary (Liu et al., 2022; Kim et al., 2021c), and utilization of spectral characteristies of time series (He et al., 2023; Yang & Hong, 2022; Kim et al., 2021a). While successful in some cases, these methods have their limitations. Tatent-domain characterization heavily relies on the hypotheses of latent domains, limiting its broader applicability. Augmentation strategies (shift, jittering, masking, etc.) for time series may not be universally applicable and can impair the task (Iwana & Uchida, 2021). For instance, in physiological signal analysis, morphological alterations from augmentations are harmful, and time-slicing is unsuitable for periodic signals. Advanced augmentation techniques like spectral perturbations
characterization (Lu al., 2023; Du et al., 2021), augmentation strategies (Iwana &
warping, decomposition techniques, etc.) are usually heavily parametric (Wen et al., 2021) and application-speeffe.. Other approaches specific to preserving non-stationarity are constrained by maintaining the same input-output space, making them unsuitable for multivariate time-series classification tasks. While some works (He et al., 2023; Yang & Hong, 2022) focus on frequency domain representations for robustness to feature shifts, they overlook cases with time- varying spectral responses. Another significant issue is that many of these studies rely on domain identity, which in practice expensive and intrusive to obtain, espocially in healtheare and finance (Yan et al., 2024; Bai et al., 2022). Thus, achieving generalizable time-series classification without access to unseen distributions and the labels of available distributions remains challenging yet crucial pursuit.
Our Approach and Contributions. We propose a novel Plase-Augmentted Separate Encoding and Residual (PHASER) framework to achieve generalizable representations for classification of nonstationary real-world time series. Figure 1 illustrates an overview of PhASER, which includes three key modules. First, we diversify the source domain data through an intra-instance phase shift by leveraging the generality and non- parametric nature of the Hilbert Transform (HT) (King, 2009) to handle nonstationary signals and introduce plase-shift-based augmentation. Next, we encode the time-varying magnitude and phase responses separately for enhanced integration of the time-frequency information. Finally, we design an effective broadcasting mechanism with non-linear residual connection between the phase-encoded embedding and the backbone representation to learn generalizable (He et al., 2020; Marion et al., 2023) task-specific features (He et al., 2016). We experiment with 13 baselines on 5 datasets lo quantitatively demonstrale PhASER's superiority in learning generalizable representations, even in challenging scenarios such as transferring from one domain multiple domains. Additionally, we provide detailed design insights through ablation analysis, explore PhASER's applicability other architectures, examine other augmentation schemes with PhASER, and present qualitative visualizations of its learned representations.
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2 Approach
2.1 Problem Formulation
Definition 2.1 (Nonstationary Time Scrics). Following the definition of mixed decomposition-basee
nonstationary signals in Dama Sinoquet (2021), we assume that a noustationary time-series sample
(ro, ... "t,... drawn from a domain Dx can be decomposed into components with mean f and variance or (both ir and or are not always zero) as:
Pix-D, (x) it + + 2, where VL. 2 1, 31, (eg + pete) V lo * att));
where 2 is a stationary stochastic component with a zero mean and a unit variance.
Definition 2.2 (Time-Serics Generalization to Unseen Domains). Suppose there is dataset S [(x, w)) with M nonstationary time-series samples drawn from set of Ns source domains S ] The joint distribution of S is Pr(As, i.e., X x~> ~ ~ Vss and x; E RR RVVT, where V is the number of time-series ura dimensions and T is the sequence length. E R 1x is the categorical label. Note that the joint distributions of different source domains are similar (with shared underlying patterns) but domain-specific distinctions:
+ Pr, 1
(2)
For any potential unseen target domain Dr, its joint distribution remains distinct like Eq. (2). In our problem, although the source dataset is assumed to contain multiple domains, the annotations that specify the domain identity are unavailable. Our goal to train a model consisting a fcature extractor F and classifier g using the given source dataset (F o g: Xs + Vs), such that
min (x.
[C(g(F(x)),
(3)
where ((-) is a certain cost that measures the errors between model predictions and the ground truth.
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Figure (Real-world time-series are nonstationary.) Example non-stationarity using a sample from a human activity recoguition dataset (HHAR) where (a) shows the temporal non-stationarity denoted by varying mean μ and variance a within a domain three regions color-coded and denoted as I, II, and III. (b) shows that the magnitude response (DDFT) of the Discrete Fourier Transform (DFT) for each region is distinct. There is a clear difference the dominant frequency for each region. (c) shows the phase responses (<(DFT)) for each region. The Z(DFT) of each region is also distinct.
Motivation. Consider human activity recognition (HAR) application, where non-stationarity is unavoidable due to changes in sensor characteristies (Bangaru et al., 2020). We illustrate an instance of hon-stationarity in Figure 2(a), which visualizes univariate accelerometer data sample from dataset called HHAR (Stisen et al., 2015) in the time domain. By segmenting this sample into sequential windows and conducting a Discrete Fourier Transform (DFT) to obtain its magnitude and phase responses, as shown in Figures 2(b) and (c), we observe shifts in the spectral domain. The Augmented Dockey-Fullee (ADF) statistic (Said & Dickey, 1984) also supports the signal's non-stationarity Thus, there is non-stationarity in terms of signal statistics and spectral propertics. Most real-world time-scrics datascts for activity recognition, sleep stage classification, and gesture recoguition applications are nonstationary, as indicated by their ADF statistics reported in Table 11 in the Appendix. Furthermore, more structured nonstationarities (in time domain statistics, spectral propertics etc.) arise from sensor and subpopulation shifts, which manifest as distribution shifts and highlight the need to learn generalizable representations for unseen distributions.
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Now we pose the question: What is the impact of non-stationarity of time series on a model's generalization ability to unseen distribulions? We conduct a simple empirical study on the HHAR dataset by varying the sequence length to synthesize increasing non-stationarity, measured by the ADF statistic (a higher ADF value indicales greater non-stationarity). More details of the ADF test are provided in Section B of the Appendix. We adopt DG model BCResNet from Kim et al. (2021a) for time- series classification to explore the relationship between the degree of non- stationarity and the model's generalization ability to unseen distributions. Figure 3 shows an evident drop in the accuracy of BeReeNet as non- stationarily increases, highlighting that non-stationarity has a detrimental impact on generalization performance, which is also observed in prior work model overfitting to the source domains' non-task-spoeific nonstationary attributes.
Out
Sequerice Langth oADr stati
Figure 3: (Nonstationarity im- pacts generalization) Compari- son between PhASER and BCResNet with increasing non-stationarity in
HHAR dataset.
(Lu et al., 2024), possibly due to
We are motivated to leverage the phase information of signal as a proxy for its non-stationarity, as the phase response captures underlying local time shifts and time-localized frequency variations characteristic of nonstationary signals (Klein et al., 2001; Oppenheim & Schafer, 1999). In this work, we propose PhASER, whose components are anchored in phase to achieve generalizable performance for nonstationary time series classification tasks, as demonstrated in Figure 3.
Overview of PhASER. Our proposed PhASER framework, shown in Figure 1, begins with an augmentation module that utilizes the Hilbert Transform to generate out-of-phase augmentations for time series, which diversify non-stationarity while preserving the category-discriminntoy semantics for classification tasks. We use the short-term Fourier Transform (STFT) to obtain temporal magnitude and phase responses. Two separate encoders process the magnitude and phase as distinct input modalities. Next, PhASER establishes a phase-feature broadcasting mechanism as residual connection to emphasize learning task-relevant information. Finally, the classifier extracts robust tasl-discriminntory reprosentations. In the following Scctions 2.2 to 2.4, we introduce the details of these three components of PhASER and discuss the theoretical insights that inspire our design.
2.2 Hilbert Transform based Phase Augmentation
Our motivating study in Figure 3, demon- strates the adverse effects of increasing non-task-specific non-stationarity on model's generalization ability, and Fig- ure 4 shows that the phase response of a signal encodes its non-stationarity. In- spired by these observations, we propose to employ a data augmentation technique that diversifies the non-stationarity in the training data while preserving the original data's discriminatory properties to ensure semantic differentiability.
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Different from most existing time-series augmentation techniques, we introduce a phase shift to signal while preserving the magnitude response, thereby offering an augmented view. This intra-sample phase- augmentation technique is less studied in the context of time-series classification for domain generalization (although some re- cent works, like Demirel & Holz (2024), explore phase-mixup for contrastive learn- ing). We intuitively justify our design
Frequency [Hz]
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ADF statictics zigl: 2e14 ADF statistics for liTts 3e14
statist
ADF statisicss siga:
NDF stallst es for liftsig2): 3:50
KDF statisties for limssg331 4.74
Figure 4: (Phase a proxy for non-stationarity.) (a c) Time- domain signals: (a) Stationary sinusoid; (b, c) Non-stationary sinusoids with the same base frequency. (d-f) Magnitude spectra (DFT) of the corresponding signals, showing similar frequency content. (g i) Phase spectra, displaying distinct responses that reflect differences in underlying dynamics. ADF statisties below each column summarize the stationarity of the respective signals and show how changes with the proposed Hilbert Transform (HT) augmentation.
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choice by exploring question: Does shifting the phase response of time series change its non-stationarity? In Figure (a-c)), we illustrate stationary sinusoid and two non-stationary sinusoids all sharing the same frequency, as evidenced by their magnitude responses shown in Figure 4(d f). However, the distinct phase responses of each signal reveal changes in the underlying dynamics. These phase variations occur as time-local oscillometric fluctuations arise, motivating the use of phase information as a proxy for capturing the underlying non-stationarity (Wu et al., 2009).
We propose simple but effective data augmentation technique based the Hilbert Transform (HT) to diversify the non-stationarity and preserve discriminatory features. Specifically, for each time-series sample x in the source dataset S, we can assume it is real-valued signal x {ao. It.... E R that is characterized by a deterministic function I22 = x()). Then, HT(x(()) x(1) = _ x(-)=([)] dr. HT can be easily interpreted in the frequency domain via Fourier analysis:
[(-
(x(9) = =(x(0) = I
<<<<<000
1 Fx(E)e
100
x(t) = -()=)((((())
-00
-00 < <
where F,F-1 denote the Fourier transform and inverse, and & is the frequency variable. To interpret & in the frequency domain, the negative frequency spectrum of fx(E) is multiplied with the imaginary unit i, while the positive spectrum is multipled with -i. Then we have:
HT(x(t)) x(t) = =- (-((s==(((())
(4)
where sgn(-) is sign function. Applying HT signal results in a phase shift of -x/2, yiclding new out-of-phase signal. After obtaining the transformed & for across all feature dimensions, merge the augmented dataset S and the original S to form a new larger dataset S' = S.S For the rest of the design, there is no distinction among the samples in S', whether they belong to S or S.
Theoretical Motivation: Phase-Based Hilbert Transform Diversifies Non-Stationarity. Suppose there are MD samples (observations) available for a nonstationary time-series domain Dx, and each sample x; = (2i.0 Fi,ts }..} characterized by, (t) = Fi,t = x,),), i e [L, Mp]. If we apply Hilbert Transformation HT(x(()) = x(1) = - (())><_ to augment these time-series samples, the nonstationary statistics of augmented samples are different from the original ones, Pr Pr...,_ (x) + (i))D,
Implication. This result establishes that phase manipulation via the Hilbert transform can mcan- ingfully diversify the nou-stationarity of time series. Empirically (see Figure 4), we observe that phase- augmented signals exhibit distinct non-stationarity statistics, as measured by ADF tests. To demon- strate that the proposed augmentation produces meaningful diversification in real-world datasets, we conduct a domain discrepancy test (Saito et al., 2018), where we train classifier to distinguish between samples drawn from S or S. For all the datasets used in this work (more details on datasets are given in Section 3), we observe much higher-than-chance accuracy in Table 1, supporting that Š and S are indeed diverse.
Table 1: Domain discrep- ancy accuracy for S vs S.
Table 2: Generalization accuracy across domains (train + test).
Dataset
Accuracy
Dataset
S S
S S
WISDM UCIHAR
0.99
WISDM
0.88
0.88
0.99
UCIHAR
0.83 0.86
HHAR
1.00
HHAR
0.76
0.74
SSC
0.98
SSC
0.76
0.77
GR
0.97
GR
0.78
0.75
To verify that the proposed diversification does not compromise task-specific discriminative features, we conduct a controlled experiment: a model is trained on S and evaluated on held-out samples from both S and their augmented counterparts S. As shown in Table 2, the model exhibits minimal performance degradation across domains, indicating that the augmentation presorves task-relevant semantics. This aligns with our design motivation to preserve the magnitude response, which, as shown in the following section (Table 3, contains more prominent task-relevant information.
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2.3 Magnitude- Phase Separate Encoding
After augmenling the source domain with phase-shift using HT, next, we identify optimal ways to encode time series for generalization. While employing spectral transformation is a common approach, our perspective diverges from most existing methods which typically focus on separating time and frequency information. Rather, we unify the time and frequency context and instead consider the magnitude and phase information as distinct modalities of the original signals.
Intuition of treating phase and magnitude as separate modalities. Building on insights from prior studies (He et al., 2023; Kim et al., 2021a) highlighting the importance of spectral imput in generalizable learning, we conduct a small-scale empirical study on the WISDM HAR dataset (Kwapisz et al., 2011) to explore optimal time-frequency input methods. Specifically, we compare four approaches: magnitude- only, phase-only, concatenated magnitude and phase, and separate encoders for magnitude and phase. Results (see Table 3) demonstrate that using only phase input yields inferior performance compared to magnitudc-only input, suggesting the latter contains more discriminative information for classification tasks. Here the phase-only features achieve an accuracy of 0.62 in six-class classification task significantly higher than chance accuracy (0.17) supporting the presence of task-discriminating but time-varying attributes in the phase response; motivating us to use it as an approximate proxy for signal's vonstationarity in PhASER, Also, concatenating magnitude and phase does not improve performance, whereas separate encoding followed by late fusion proves superior in this case. This may be altributed to 1) the independent selection of high-level features from the magnitude and phase for the task of classification, and 2) the learning about non-stationarity from the phase information. Additionally, these magnitude and phase encoders for separate encoding can be deployed in parallel after training since they have no data dependencies, potentially optimizing computational efficiency during inference.
Table 3: Comparison of various time-
frequency input configurations.
Input Modality
Accuracy
Only Magnitude (Mag)
0.81 t 0.03
Only Phase (Pha)
0.62 0.03
Mag-Pha Concalenate
0.73 + 0.03
Mag-Pha Phaa Separate
0.85 1 0.01
We adopt STFT instead of DFT because the DFT is typically suited for stationary. periodic signals, whereas time-varying signals require a method that accounts for changes over time. The STFT addresses this by applying the DFT sequentially within moving window, capturing both the frequency and time information across the entire time series. Specifically, for each training sample x E S' with continuous time function x(t), sampling it at a fixed rate generates a discrete time series denoted as x[n] with a sequence length N, we have:
fx|n,k] k]
Σ win m]x[m]ee<5am
55)
1I-
The STFT of x(n), fx(n, k), is a function of both discrete time n and frequency bin indices k with lengths N and E, respectively. &a is a digital frequency variable given by & 2 and w[-) is a window function. = Without losing generality, we adopt the Hanning window with window length W, i.e., w|n] 0.5(1 - cos W-T where 0 < n & W- - 1. Note that the length and shape of the window determine the time-frequency resolution. A larger W provides better frequency resolution and a smaller W gives better temporal scale. We set W to be randomly sampled powers of for cach time-scries fcature, i.c., W. = 2D: < E,p, ~ U E Z E [1, V), where U denotes a uniform distribution for integers. After obtaining [x(n, k), we can compute its magnitude and phase as:
2nn
Mag(x)= = / Re k))² (f(Sf(n, Pha(x) = arctan 2 (Im()) Re(Sx k)));
(6)
where Im(-) and Re(-) indicate imaginary and real parts of complex number, and arctan 2(-) is the two- argument form of arctan. Then we take Mag(x), Pha(x) E VxEX as inputs of two separate encoders Fmag and Fri.. This approach is motivated by the viability of reconstructing a time-series signal using phase and magnitude responses (Hayes et al., 1980; Jacques & Feuillen, 2020), which is supported by our study below.
Before fusing the extracted embeddings of FMeg and Fpha: we incorporate sub-fcature normalization proposed by Chang et al. (2021). Specifically, the embeddings of FMaa and Frna are divided into B sub-feature spaces. We apply normalization in each sub-feature space for each time-series variate,
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(xtag()== = (x):: Fatag(X)) (x)6 Pstng(x); B
where (-) and o()) denote the computation the mean and variance of the given input. The same sub-feature normalization is also conducted on (x) Then, both Faag(* (x) and Fpha(e (x) are fused along the variale axis by multiplying with 2D convolution kernels denoted as fusing encoder Frus. The fused embeddings IFuu FFus (FMag (x), (x)) are then fed into the following modules.
af ((Fstou(X)y
b=1
2.4 Phase-Residus Feature Broadcasting
Lastly, outline our phase-based broadcasting approach to achieve domain generalizable representation learning. It starts with depthwise feature encoder, Depp, which transforms the fused embeddings, FFus, inlo 1D feature maps, FDep: along the temporal dimension, given as:
R C(rra))x D(eru ( IR (
where C(-), D(-), and T(-) represent the channel number, the feature dimensions, and the temporal dimensions of an embedding. Fep is implemented as several convolution layers followed by an average pooling operation to unify all features at each temporal index. Once the 1D feature map is obtained, we attach sequence-to- sequence (the dimension format of the feature map remains intact) temporal encoder, I'Tems to characterize its temporal dependency and semantics. The choice of backbone for Frem is not central to our design and a suitable toseuueeee encoder can be chosen. Here we leverage convolution layers to form Frem and we have also tested other architectures (please refer to Section B in the Appendix for details). We adopt this feature consolidation approach to enable specialized learning of spectral attribules by FDep and global temporal dependencies using Frem, resulting in more valuable overall semantic characterization.
We now introduce non-lincar projection of (x) broadcast with the out put dimensions of Frem; we use projection head, gRes for the transformation:
shortcut through FDep Frem- To suitably
R ( (TT
C(rou))) D( (x))aa((x))x((r xT(mm)))
We conduct a controlled experiment using different residual feature broad- casting no residual connection, using magnitude (FMag); using phase (FPha), and using the fused magnitude and phase (FFus)-and evaluate this on held-out samples from the source domain, denoted as in-domain ac- curacy, and on the target domain, denoted as out-of-domain accuracy. We present the results in Figure on a Gesture Recognition (GR) dataset. It is not surprising that the magnitude residual performs very well for in-domain evaluation; however, the drop in OOD accuracy can be indicative that the model has overfit to the in-domain non-task specific non-stationarity, Using phase residual especially since the phase component of the dataset is diversified thorugh the porosed augmentation helps implicitly regularize the model's learnt against non-task specific nonstationaritics.
Bede of
1_010
P.345
Figure 5: Comparison of gener- alization performance of different. residual broadcasting features.
A After the projection, we can broadcast. the output From to form the final representation r that is intended to learn discriminatory characteristics despite non-stationarity:
r= Frem (FDep) + 9nes (Frrn
A After these efforts to preserve and enhance the discriminatory characteristics amid input's 's non-stationarity, we now optimize for semantic distinction. This optimization is achieved with Cross-Entropy Loss applied to classification head JCls, which is attached to Frem as LCE 1 Σi y; goggc)r(r) where Na is the size of batch in the mini-batch training, and y; is the one-hot form of the label yi-
2.5 Theoretical Motivation
Here provide some theoretical insights to demonstrate that our method design is rigorously motivated. Detailed definitions and proofs are provided in Section A of the Appendix.
Code is available at https://github.c http::///tthhoo.co///y/aaaappppp
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Definition 2.3 (3-Divergence). Suppose two data domains D1, D2 are built on input variable and label variable y- Let > 0 be constant. The 3-Divergence between Dj and D2 is defined as:
D1 Di 8) 97 D2(x, y)
(,(Di [Bix))DDD
(8)
Per the definition in (Germain al., 2016), 3-Divergence can be linked to the Rényi Divergence (Van Erven & Harremos, 2014) R.2(.) as:
(D(D) = 2 - (P,
Lemma 2.4 (Bounding 3-Divergence in A Convex Hull). Let S a set of source domains, denoted as S No A convez hull As considered here consists of a mixture distributions As = ={ S(-)
Σe T.Si))), mi E ANs-th, where AN is the (Ns-1)-th dimensional simplex. Let (,(S: S for Vi.j E [Ns], and then we have the following relation for the B-Divergence between any pair of two domains D', D" Ag in the conver hull:
(D SE.
S.
(10)
Proposition 2.5 (Houristic Risk Bound for an Unseen Domain). Let H be a hypothesis space built from a set of source time-series domains, denoted as S = (Si) with the same value range (i.e., the supports of these source domains are the same). Suppose q > 0 is constant. For any unseen time-series domain Du from the conver hull As, we have ils closest element Du As, Le., DU = arg min (Pull n,S.).
Then the risk of Du any in H is:
Rou [e] s _doc(o) + [epo (p))
(11)
where dp(p) and ep(p) are expected disagreement and an expected joint error of domain D, respectively. The is value larger than the marimum 3- 3-Diveryence in As;
max (JE(N)))
2 2-1 RD, (S:((((()))))00)
(12)
Vl (1-9))2,+ o, "0
9(, 11,e)
(13)
where (S)(((())))))) 2(1 - 2,1-q)0i, + 207,
Insights. Theorem 2.5 indicates potential efforts to reduce the generalization risk of an unseen target domain. According to Eq. (11), the risk bounded by two terms. The first term (p) is the expected disagreement of Du and we are unable to conduct any approximation without accessing the data from Du- Regarding the second term, the coefficient can be viewed as the maximum 3-Divergence of source domains, and according Eq. (13), the nonstationary statistics of time series are arguments of the 3-Divergence. We regard the 3-Divergence as proxy for non-stationarity. Our architectural design leverages implicit regularization effect, achieved by reintroducing the phase dictionary through phase-residual broadcasting al deeper network layers. In this mechanism, the Fria features, which approximately encapsulate diverse (due to augmentation) non-stationarity (shown earlier in Figure 4), are used to modulate the task-specific information learned so far using Fremp- This forces the network to repeatedly disentangle and emphasize task-relevant representations deeper in the layers as well (Noh et al., 2017), resulting implicit regularization that contributes to a tighter generalization risk bound according to Theorem 2.5. Empirically, this translates to enhanced robustness and improved transferability to unseen target domains, shown in Figure 5.
3 Experiments
We extensively evaluate our proposed PhASER framework against 13 saate-of-the-art approaches (including a large foundation timo-scrics model), on datasots across three time-scrics applications. Our evaluation metric is per-segment accuracy. More implementation-speee details are provided in Section D of the Appendix. Our source codes are provided in the Supplementary Materials.
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Table Classification accuracy of Target 1~4 scenarios for pross-person generalization Human Activity Recognition WISDM, HHAR, and UCIHAR (Best in bold, second-best underlined).
Dataset
WISDM
HHAR
UCIHAR
HAR
Target
1
2
3
4
Avg.
1
2
3
Avg.
2
3
Avg.
Avg.
ERM
0.57
0.50
0.51
0.55
0.53
0.49
0.46
0.45
0.47
0.47
0.72
0.64
0.70
0.72
0.70
0.57
GroupDRO
0.71
0.67
0.60
0.67
0.66
0.60
0.53
0.59
0.64
0.59
0.91
0.84
0.89
0.85
0.87
0.71
DANN
0.71
0.65
0.65
0.70
0.68
0.66
0.71
0.67
0.69
0.68
0.84
0.79
0.81
0.86
0.83
0.73
RSC
0.69
0.71
0.64
0.61
0.66
0,32
0.49
0.44
0.47
0.48
0.82
0.73
0.74
0.81
0.78
0.64
ANDMask
0.74
0.73
0.69
0.69
0.71
0.63
0.64
0.66
0.69
0.66
0.86
0.80
0.76
0.78
0.80
0.72
InceptionTime
0.83
0.82
0.80
0.77
0.81
0.77
0.80
0.82
0.83
0.80
0.91
0.82
0.88
0.91
0.88
0.82
BCReNNet
0.83
0.79
0.75
0.78
0.79
0.66
0.70
0.75
0.68
0.70
0.81
0.77
0.78
0.80
0.76
NSTrans
0.43
0.40
0.37
0.37
0.40
0.21
0.22
0.27
0.28
0.24
0.35
0.35
0.51
0.47
0.42
0.35
Koopa
0.63
0.61
0.72
0.57
0.63
0.72
0.63
0.72
0.69
0.69
0.81
0.72
0.81
0.77
0.78
0.70
Ours+RReII
0.86
0.85
0.84
0.84
0.85
0.82
0.82
0.92
0.85
0.85
0.96
0.90
0.93
0.97
0.94
0.88
MAPU
0.75
0.69
0.79
0.79
0.75
0.73
0.72
0.81
0.78
0.76
0.85
0.80
0.85
0.82
0.83
0.78
Diversify
0.82
0.82
0.84
0.81
0.82
0.82
0.76
0.82
0.68
0.77
0.89
0.84
0.93
0.90
0.89
0.83
Chronos
0.71
0.66
0.65
0.62
0.66
0.66
0.73
0.75
0.66
0.72
0.56
0.57
0.50
0.82
0.61
0.67
Ours
0.86
0.85
0.85
0.82
0.85
0.83
0.83
0.94
0.88
0.87
0.96
0.91
0.95
0.97
0.95
0.89
We conduct experiments three common time-series applications Human Activity Recognition (HAR), Sleep-Stage Classification (SSC), and Gesture Recognition (GR). For HAR, we use 3 benchmark datasets: WISDM (Kwapisz el al., 2011) collected from 36 different users with 3 univariate dimensions, UCIHAR (Bulbul et al., 2018) collected from 30 people with 9 variates, and HHAR (Stisen et al., 2015) collected from 9 users with 3 feature dimensions, comprising distinct activitics with a sequence length of 128. For SSC, the dataset (Goldberger et al., 2000) consists of single-channel EEG data from 20 healthy individuals with a sequence length of 3000. For GR, the dataset (Lobov et al., 2018) is 8-channel EMG data for 6 different gestures, with sequence length of 200, prepared similarly as in (Lu et al., 2022b). We follow the sctup of ADATime (Ragab et al., 2023a) for HAR and SSC. More data-specific details are provided in Table 11 of the Appendix. Specifically, the class distributions of the considered datasets in Figure D.1, as well as the trends of two performance metries, segment-wise Area-under-thecuure (AUC) and accuracy, for the WISDM dataset in Figure D.1, are provided in the Appendix to justify the choice of performance metrics in accordance with previous works (Ragab et al., 2023a; Lu et al., 2022b).
Datasets.
Experimental Sctup. Each dataset is divided into four distinct non-overlapping cross-domain scenarios, following the approach in (Lu et al., 2023). Details are provided in Section D.1 of the Appendix. 20% of the training data is reserved for validation. Mean results from three trials are reported in the main text, with full statistics in Section E of the Appendix.
Comparison Baselines. We conduct comparison with a range of state-of-the-art approaches including domain generalization algorithms - ERM, DANN (Ganin et al., 2016), GroupDRO (Sagawa et al., 2019), RSC (Huang et al., 2020) and ANDMask (Parascandolo al., 2020) implemented based on the DomainBed benchmarking suite (Gulrajani & Lopez-Paz, 2020); an audio domain generalization method BCResNet (Kim et al., 2021b); a time-series representation ion learning method MAPU (Ragab et al., 2023b); song deep-learning time-series classification model (top ranked by Middlehurst et al. (2024)), Inception Time (Ismail Fawaz et al., 2020), a time-series domain generalizable learning method Diversify (Lu et al., 2022b); and a large time-series foundation model Chronos (Ansari et al., 2024). We also adapt the time-series forecasting models Nonstationary Transformer (NSTrans) (Tiu et al., 2022) and Koopa (Liu et al., 2024), and integrate a network-agnostic statistical technique RevIN (Kim et al., 2021c) with our method (denoted as Rurs+RevIN). We follow the default setups of these works and only conduct necessary modifications for our problem setting. Details are in Sections D.2 and D.6 of the Appendix.
3.1 Effectiveness of PhASER across Applications
Human Activity Recognition. We assess the generalization ability of PhASER framework in two settings:
1) cross-person generalization, where the model is trained on Ns (Ns > 1) source domains and evaluated on unseen target domains, and 2) one-person-to-anote and evaluated on another person. In the pross-person setting, as shown in Table 4, we find that existing
where the model is trained on one person (Ns = 1)
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Table 5:
Classification accuracy with Source 0-8 person for cee-peroon-t-anottiie generalization on the HHAR dataset (Best in bold, second-best underlined).
Table 6: generalization (Target 1~4) Sleep-Stage Classification (EEG) and Gesture Recognition (EMG) (Best in bold, second-best underlined).
Classification accuracy for cross-person
Source
2
3
6
8
Avg.
Application
Sloep- Stage Classification
Gesture Rocognittio
ERM
0.27
0.40
0.41
0.44
0.42
0.44
0.45
0.44
0.48
0.42
Target
Avg.
Avg.
GroupDRO
0.33
0.33
0.38
0.48
0.47
0.51
0.17
0.48
0.19
0.16
DANN
RSC
ANDMask
0.32
0.27 0.45
0.34
0.50
0.41
0.38
0.37
0.42
0.43
0.45
0.45
0.40
0.46
0.12
0.47
0:48
0.50
0.46
0.49
0.44
0.45
0.52
0.53
0.51
0.43
0.11
GroupDRO
ERM
DANN
0.50 0.46
0.37 0.56
0.64 0.63
0.55
0.69
0.19
0.39
0.45
0.63
0.37
0.65
0.47
0.60
0.45
0.53
0.66
0.58
0.59
0.65
0.57
0.45
0.54
0.64
0.64
0.18
0.551
DnceptionTime
0.52
0.62
0.41
0.69
0.60
0.66
0.61
RSC
0.50 0.48 0.52 0.46
0.19
0.50
0.60
0.61
0.56
0.59
BCResNet
NSTrans
0.20
0.28
0.48 0.32
0.22 0.17 0.20
0.47
0.42
0.21
0.52
0.22
0.44
0.26
0.45
0.17
0.49
0:20
0.43
0.21
Inception Time
ANDMask
0.74 0.78 0.72 0.80
0.55 0.50 0.54 0.57
0.76
0.54
0.68
0.41
0.70
0.51
0.45
0.72
0.39
0.69
0.70
0.45
Koopa
0.32
0.42
0.37
0.40
0.42
0.45
0.35
0.43
0.48
0.40
BCRosNet
0.79 0.82
0.81
0.62
0.65
0.61
0.64
Ours+ RevEN
0.48
0.66
0.57 0.65
0.61
0.64
0.65
0.64
0.63
0.62
NSTrans
0.43 0.37 0.42 0.35
0.39
0.31
0.34
0.34
0.32
0.33
Divensify
MAPU
Chronce
0.39
0.42
0.32 0.23
0.62
0.57 0.35
0.26
0.32
0.62
0.52
0.25
0.49
0.56
0.27
0.61
0.23
0.49
0.53
0.21
0.52
0.24
0.61
0.52
0.25
0.53
0.25
Ours+ DevIN
Koopa
MAPU
0.69 0.68 0.65 0.69
0.58 0.62 0.53 0.49
0.82 0.79 0.78 0.81
0.56
0.68
0.80
0.47
0.64
0.68
0.54
0.81
0.69
0.60
0.77
0.71
0.70
0.76
0.68
0.68
0.58
0.76
Diversify
0.73 0.76 0.68 0.77
0.73
0.68
0.75
0.76
0.75
Ous
0.53 0.70 0.63
0.66
0.64
0.67
0.65
0.67
0.62
0.66
Chronos
0.33 0.47 0.47 0.57
0.49
0.54
0.51
0.48
0.51
Ous
0.85 0.80 0.83
0.70 0.77
0.75 0.76
statc-of-tho-art domain generalization methods, popular in vision-based domains, do not perform as well in time-scries classification (such obscrvation is consistent with previous works (Gagnon-Audet et al., 2022; Lu et al., 2022b)). PhASER achieves superior out-of-domain generalization performance across all cases, notably outperforming the best baseline on WISDM, HHAR, and UCIHAR by 3%, 9%, and 6%, respectively. In the more challenging one-person- anc-person-to-another setting, as shown in Table 5, we select the HHAR dataset due to its high non-stalionnarty, and the results show that PhASER excels in this setting well, outperforming Diversify by almost 20% and Inception'iiee Time by almost 8%.
Sleep-Stage Classification. Next, we evaluate PhASER for cross-person generalization in five types of sleep-stage classification using EEG. Past methods (Ragab et al., 2023a; He et al., 2023) generally report the lowest performance in their respective settings for SSC tasks indicating its inherent complexity. The results in Table (left) show that PhASER provides the best performance in all cases, outperforming the best baseline (BCResNet) by 2% and the time-series domain generalization baseline (Diversify) by almost 11%
Gesture Recognition. In GR, the used bio-electronic signals are heavily influenced by user behavior and sensor time-varying properties, which correspond to natural non-stationarity. We follow the approach in (Lu et al., 2023) to use 6 common classes when conducting evaluations in cross-person setting. The results in Table 6 (right) show that PhASER again offers the best overall performance.
3.2 Further Analysis
Ablation Study. We examine the impact of our proposed design components in two cases: WISDM and GR (Table 7). The first row represents the performance of the complete PhASER framework, with subsequent rows showing performance with specific components detached or modified (details in Section D of the Appendix). When phase augmentation is omitted (row 2), performance notably decreases (by 11.6% on WISDM and 5.8% on GR). This finding along with results from Tables 2 and 1, is consistent with recent vision domain rescarch showing that Hilbert Transform-based augmentations indeed preserve task-relevant features while generating diverse representations (Wang et al., 2025). Comparing the results of row 6 with that of row 5 confirms the importance of separate mlase-magnitude encoding, aligning with earlier findings in Table 3. Under identical conditions (comparing row 5 with row 1), phase-residual broadcasting boosts the performance of PhASER by 4%, aligning well with our design motivation that phase can be considered proxy for non-stationarily. Reintroducing this phase-dictionary deeper in the layers enables the model to learn spec-fpe representations that are more robust to non-stationarity, making it better equipped to handle unseen non-stationarity in the target domains. Removing the phaso-based residual and scparate encoding structure (rows 3-7 in Table 7) results in average performance drops of 10.6% and 13.7%, respectively. This demonstrates the value of all the components in PhASER.
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1.0
Phase
Separate
Pinxx
Accuracy
Augueutation
Encoders
Residual
WISDM
GR
0.80
0.70 0.00
Accuracy
Accuracy
0.81
0.61
0.6
0.6
XSaa Res.)
0.8222.0
0.00
Fii Ros.)
0.84ga01
2000
0.4
0.4
0.82g
0.05
KiMag Ouly)
x
g.73
0.59
0.2
NSTrans
0.2
NSsrrans
X(Mag Only)
0.83
0.66
Durs Iistaans
Ours NSTrans
X(Mag- Pha Concat.)
x
0.73ga.s
0.61
0.0
0.0
(a) Target (WISDM dataset)
(b) larget (HHAR dataset)
Figure 6: Improvement in average cross-person general-
Table 7: Ablation of PhASER WISDM and GR. The inclusion of a component is denoted as /f and exclusion (modification).
ization performance NSTrans in (a) WISDM from 0.40 to 0.83 and (b) HHAR from 0.25 to 0.78, with our phase- driven approach.
General Applicability of PhASER. We demonstrate the general applicability and flexibilityy of PhASER by incorporating three proposed design elements into the NSTrans model for classification: phase-based augmentation for nonstationarity diversification, separate magnitudo-phase feature encoding, and phase incorporation with residual connection. Significant performance improvements on WISDM and HHAR (Figure 6) highlight the effectiveness of these designs and the flexibility of PhASER with different backbone models. Further details are provided Section D.4 of the Appendix. Additionally, our experiments integrating statistical modules for nonstationary time series, such as RevIN, into PhASER yield performance that is on par with slightly below that of PhASER alone. This corroborates recent findings (Chen et al., 2025) that incorporating RevIN offers clear advantage for classification tasks. Overall, the nonstationary basclincs-such as NSTrans, Koopa, and RevIN originalyy designed for forecasting tasks-do not perform competitively on classification tasks (Tables 4, 5, 6), which is unsurprising (Chen et. al., 2025; Wang et al., 2024); however, we include them for completeness of nonstationary time-series baselines.
Visualization. We provide t-SNE visualizations of our mothod (PhASER), Diversify, and BCResNet on the HHAR dataset for left-out domains in scenario 1 (Figure 7). The plots depict out-of-domain data, with colors representing the six activity classes, showcasing PhASER's superior separability without domain labels target domain data. Further details are in Section D.8 of the Appendix.
(a)
(b)
(c)
50
20
25
-20
-25
-40
-40
-60
-40
-20
0
20
40
-75
-50
-25
50
-40
-20
20
40
60
Figure 7: t-sne visualization for (a) PhASER, (b) Diversify, and (c) BCResNet for HHAR scenario 1.
PhASER with Other Augmentation Strategies. Here, we explore a random phase augmentatiion-varien using Hilbert Transform under certain signal periodicity assumptions (more details in Section D.7.2 the Appendix). Additionally, we adopt traditional augmentations like rotation, permutation, and circular time-shift proposed by past works (Qin et al.,
0.95
Mean Accuracy
0.90
0.80
0.75
2023; Um et al., 2017); the HHAR dataset with the PhASER framework. The results are illustrated in Figure 8 and imple- mentation details are provided in Section D.7 of the Appendix. The rotation and permutation augmentations perform 5% worse than the no augmentation scenario in this case possibly due to semantic corruption (Mintun al., 2021). Time-shift may be viewed linear phase shift for pure sinusoid (for example,
No Augmantation
Botation Perm
Random Linear Phase Shift
Random Phase Shift HT
Fied Phose Shift HT
Augmentation Techniques with PLASER
Figure 8: Brief comparison between different. augmentation strategies with PhASER.
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for an imput x(t) sin(wt), time-shifted version by T time units is given by x(t x(t- T) sin(w(t - T)) which incurs a phase shift Φ = w however, most real-world signals are not stationary or pure tone. In such a case, a time shift introduces varied phase shifts for each frequency, and past works like Umapathy et al. (2010) expose the difficulty in the correct choice of a time-shift amount for retaining the signal's spectral properties of interest. This highlights the overall motivation of Hilbert Transform lo provide an accurate phase shift of all frequency components by- -#/2 without any explicit signal characterization. Our further exploration to induce random phase shift using HT does not show any particular advantage, hence we stick to the choice of using the fixed phase-shift augmentation followed by other phase-anchored components for domain generalization in nonstationary time-series classification tasks in the proposed PhASER framework.
4 Related Works
Nonstationary Time-Series Analysis. In real-world scenarios, nonstationary time-series data pose challenges for forecasting and classification (Esling & Agon, 2012; Ismail Fawaz et al., 2019; Dama & Sinoquet, 2021). While various solutions cxist, including Bayesian models, normalization techniques, recurrent neural networks, and transformers, systematic works addressing non-stationarity's impact on time-series classificatio are limited (Liang, 2005; Chen & Sun, 2021; Tuu et al., 2023c; Chang et al., 2021; Passalis et al., 2019; Tang et al., 2021; Du et al., 2021; Liu al., 2022; Wang et al., 2022a). Our study builds upon prior empirical findings and, to the best of our knowledge, is the first to investigate the impact of non-stationarity generalizable time-series classification. (Zhao et al., 2020; Tonekaboni et al., 2020; Eldele et al., 2023).
Domain Generalizable Learning. While domain generalizable learning is well-established in visual data (Wang et al., 2022b), applying it to time-series data poses unique challenges. Traditional approaches like data augmentation (Wang et al., 2021) and domain discrepancy minimization (Zhang & Chen, 2023; Li et al., 2018) face limitations in time series due to less flexible augmentation and broader domain concepts (Wen et al., 2021; Wilson et al., 2020). Some studies explore domain-invariane representation learning (Lu et al., 2023; Wang et al., 2023) and learnable data transformation (Qin et al., 2023). We highlight the non-stationarity of time scries and its role in domain discrepancy, drawing on evidence from the visual domain regarding the importance of phase (Kim et al., 2023; Xu el al., 2021). A handful of works hint at phase's role in domain-invariant learning time-series applications (Tu et al., 2022a), and there is evidence in traditional signal processing that phase-only information sufficient to reconstruct signal (Masuyama et al., 2023; Jacques & Feuillen, 2020; 2021). Inspired by these insights, we propose a novel phaso-driven framework with an augmentation module and a phase-anchorel representation learning to address non-slationarity and minimize domain discrepancy.
Spectral Features for Time-Series Analysis. Spectral representation of time series data is generally used for feature extraction (Zhang et al., 2022a; Woo et al., 2022; Ma et al., 2024; Yi et al., 2025; Mohapatra et al., 2023) and compression (Zhou et al., 2022; Rippel et al., 2015), which effectively captures the periodicity and global dependencies in the data. Commonly, Discrete Fourier Transform is used to obtain these spectral features (Yang & Hong, 2022; Zhang et al., 2022b; Wu et al., 2023). However, for nonstationary signals where the spectral content is also time-dependent, a time-frequency representation is more suitable, and Short-term Fourier Transform (Li et al., 2021b; Yao et al., 2019), Discrete Wavelet Transform (Wang et al., 2018; Khan & Yener, 2018), and Empirical Mode Decomposition (Cai et al., 2025; Van Jaarsveldt et al., 2023) are used in such cases. More recently, time-series foundation models have also explored spectral representation as a tokenization scheme (Masscrano al., 2025). While most prior work leverages the magnitude response, some works especially in audio denoising (Paliwal et al., 2011) and beamforming applications have demonstrated the benefit of incorporating phase information. Other works on time series have shown promising basse- augmented views for contrastive learning settings (Qian et al., 2022; Liu et al., 2023b; Demirel & Holz, 2023). Our work contributes to this line of investigation by demonstrating a design paradigm anchored in phase to learn generalizable time series representations.
5 Limitations and Future Work
PhASER achieves domain generalization without explicit domain characterization or accessing target domain samples, by diversifying non-stationarity and anchoring design to signal's 's phase information. Our evaluation
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is currently limited to categorical tasks due to scarcity of publicly available datasets with distinct domain definitions for continuous tasks like regression. Our future work aims to develop universal representation for generalization actoss various tasks in dynamic conditions (Mohapatra et al., 2025; 2024b).
6 Conclusion
We address the generalization problem for nonstationary time-series classification using phase-driven approach without accessing domain labels of source domains samples from unseen distributions. Our approach conducts thase-based augmentation, treats time-varying magnitude and phase as separate modalities, and incorporates a phase-dcrived residual connection in the network. We support our design choices with rigorous theoretical and empirical evidence. Our method demonstrales significant improvement over baselines across 13 benchmarks on 5 real-world datasets.
Acknowledgements
We gratefully acknowledge support in part from the National Science Foundation under Grants 2038853, 2324936, and 2328973.
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Appendix
This Appendix includes additional details for the paper "Phase-driven Domain Generalizable Learning for Nonstationary Time Series", including the reproducibility statement, theoretical proofs (Section A), additional details of PhASER (Section B), detailed dataset introduction (Section C), implementation details (Section D), and detailed results (Section E) of main experiments.
A Theoretical Proofs
Lemma 2.4. Let a set S of source domains S = N A convex hull As is considered here that consists of misture distributions As = 18 S(.) = -i (,), E ANs-1), where AN is the (Ns-1)-M dimensional simplez. Let By S for Vi,j [Ns], we have the following relation for the 3-Divergence between any pair of two domains D', D As in the convex hull,
(D Se.
(14)
Proof. Suppose two unseen domains D' and D, on the convex hull As of Ns. source domains with support Ω. More specifically, let these two domains be D' NN (+))(+) and D" Ei=i (+)(()), then the 3-Divergence between 'D' and D is
Ns
8 DD = RD,
(15)
Let us consider the part of Rényi Divergence as follows,
RD,
1
In [ (z)
D (x))"" dx
1
No
1-9
1 In R
(x))
Σ miS((1))
dr
q
Lk=1
No
1-9
In
S
ΣΣ TAT)S(()))
dx
(16)
N3 N₂
Th / [Sx(z)] [S(())) dx
In
k= 1=1
No No
T max kJE[[Ns] / (Se( (S(x))) dr
In
In max
[S(()))[S((<)))' dr.
According the given assumption that (5,1)))) S for Vi,j E [Ns), we have,
RD,(D'IDD) < In max KJE[NN 2
4 [Se(t)] (S()))) do max RD,(S)((S) < q-1 AJE[NNs)
logg c. (17)
Thus (D(D"DD")) <c.
Theorem 2.5. Let H be a hypothesis space built from a set of source time-series domains S (S.),N with the same value range (i.e., the supports of these source domains are the same). Suppose q > is a constant, for any unseen time-series domain Du from the conver hull As, we have its closest element Dj in As, i.e., Dū = arg min (2 EN,, S,S.). Then the risk of Du on any p in H is,
Row [p] < do (p) + [eD, (p)
(18)
2
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where dp(p) and ep(p) are an expected disagreement and an expected joint ervor of domain D, respectively, and they are defined as follows,
do (p) = Rx Bn (b)(x) # 11 (x)),
(19)
eD (okkg)>DBBBnpppl up [1(x)) + y] (x) + u),
(20)
where I[-] is on indicalor function with [[True] = 1 and [[alse]] = 0. The in Eq. (11) is a volue larger than the marimam B- B-Dinergence in Ag,
2
max (FE
2 2¹ RD, (5,0(,,(0)))
(21)
where
(1 q]0; t0.
In
(D)((,())))))))) = aupra - 11,1)2
o,e "0"
(22)
+
2(1 -00,, + 207
1 ④
Proof. According to Theorem 3 of Germain et al. (2016), if H is a hypothesis space, and S, T respectively are the source and target domains. For all p in H.
- do) (T]IS) [es(p)]' 1Tss RT - 2
(23)
where IT denotes the distribution of (x. ~ T conditional to (x, E SUPP(S). But because it is hardly conceivable to estimate the joint error (()) without making extra assumptions, Germain et al. (2016) defines the worst risk for this unknown area,
TTI PT(x.)~ [(x,)) V) & SUPP(S)) sup [r). hEw
(24)
In Theorem 2.5, all domains from the convex hull As have the same value range, in other words, their supports are continuous and fully overlapped. In this case, Pr(x.)>T [(x, y) & SUPP(S)) = 0, i.e, ITS = O.
With Eq. (23), if the target domain T is assumed as an unseen domain Du from the convex hull As, and we select its closest element D = arg min ( Ns T&S,)) and regard it as the source domain, we can aNo
derive Eq. (23) into
RD [p] S 2 do (p) (P(DullP)) (ep, (p)] 1- +0.
(25)
Then according to Lemma 2.4, as both DU and Du are from the convex hull As, (,(Dul(Du) S. As for acquiring Eq. (13), we only need to substitute the time series domains in the form of random variable distributions into the Rényi Divergence.
Theorem A.1 (Non-stationarriy Change of Hilbert Transform). Suppose there are Mp samples (observations) available for a nonstationary time-series domain Dx, and each sample x, f...o...., is characterized by its deterministic function, i.e., (t) F₁,t = (t),, E [p]. If we apply Hilbert Transformation HT(x(t)) (t) I_ x(1)= dr to augment these time-series samples, the nonstationary stotistics of augmented samples are different from the original ones, Pr Pr (x) + (x)
Proof. Proof. According to Definition 2.1, the statistics of the non-stationary lime-series domain consist of non-stationary mean and variance. To prove Theorem A.1, we only need to prove that the mean of the time-sories domain changes after applying Hilbert Transformation (HT). HT can only be conducted on deterministic signals, thus we use the empirical statistics of Mp samples to approximate the real statistics,
Mr
(x)(1) Σ E E
(1()) Dee (x) = Σ (1) = 4c.
(26)
1=1
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According to the standard definition of HT (King, 2009) and the linear property of integral operation, we
have
Mp
MD
Mp
x-
(x)(t) = Σ ==1 1=1 -c T(--T)
(t(
Σ (T(
di
Xi(T)
T(t - T)
-
i=1
(27)
1
H7 dr.
π
To interpret Eq. (27), we can assume there is new signal = "M...... with the deterministic function 1t = u(t), and we next apply proof by contradiction for the following proof. Suppose the non-stationary statistics the original and #T-trranformedd samples are identical, i.e., E x (x)(t) = Ex (xx we can derive the following formula,
00 u(T)
t dr = u(t),
(28)
which indicates that the HTT-transformed is identical to the original HT has property called Orthogo- nality (King, 2009): if xt) is real-valued energy signal, then x(t) and its HT-transformed signal x(t) are orthogonal, i.e.,
DC x(t)&(t)dt = 0.
(29)
To prove the properly of Orthogonality, we need to Plancherel's Formula,
Theorem A.2 (Plancherel's Formula (Lang & Lang, 1985)). Suppose that u, E (R) (2R), then
f u(t)o(t)dt 2T
Fulw)
(30)
where (-), denote the LP spaces with 1, p 2 respectively, R represents the real-valued space, and F denotes the Plancherel transformation.
With Plancherel's Formula, we can prove the property of Orthogonality as follows,
/ x(t)x(tt)t
(ww))=issnw))F((() dw
2T
((w( (w))dw
2n
(31)
((w)
2
0,
where sgn(-) is sign function. After proving the Orthogonality, can use it with the condition of Eq. (28),
i.e.,
u(t)a(t)dt= = | u²(t)dt = 0.
(32)
Eq. (32) holds true only if E [0, +oo), u(l)= = 0, which contradict to our initial assumption that 14, = u(l) is not always zero in Definition 2.1. As result, the assumption of ğ. = pr is false.
Insights. This theorem illustrates that HT does change the nonstationary statistics of time series, proving that our phase augmentation can diversify the non-stationarity of time series.
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B
Additional Details on PhASER
Augmented Dickey Fuller (ADF) Test. This is a statistical tool to assess the non-stationarity of a given time-series signal. This test operates under a null hypothesis Hg where the signal has a unit-root. The existence of unit-root is a guarantee that the signal is non-stationary (Said Dickey, 1984). To reject Hip the statistic value of the ADF test. should be less than the critical values associated with a significance level of 0.05 (denoted by p, the probability of observing such test statistic under the null hypothesis). Throughout the paper, for multivariate time series, the average ADF statistics across all variates are reported. Besides, since this is a statistical tool to evaluate non von-stationarity for each instance of time-series data, we provide an average of this number across a dataset to give the reader a view of the degree of non-stationarity.
Phase Augmentation. In this work, we are particularly interested in learning representations robust to temporal distribution shifts. Incorporating a phase shift in signal is less-studied augmentation technique. One of the main challenges is that real-world signals are not composed of single frequency component and accurately estimating and controlling the shifting of the phase while retaining the magnitude spectrum of signal is difficult. To solve this, we leverage the analylic transformation of a signal using the Hilbert Transform. The key advantages of this technique are maintaining global temporal dependencies and magnitude spectrum, no exploration of design paramcters and being extendible to non-stationary and periodic time series.
Lets walk through a simple example for a signal, x(1) = 2cos(wpl) which can be written in the polar coordinates as x(t) euto + -inot Applying the HT conditions from Equation 4, HT(x(tt) = 2stn) Essentially, HT shifts the signal by x/2 radians. We conduct this instance-level augmentation for each variate of the time series input. The aim is to diversify the phase representation. We use the scipy (Virtanen et al., 2020) library to implement this augmentation.
STFT Specifications. Non-stationary signals contain time-varying spectral properties. We use STFT to capture these magnitude and phase responses in both time and frequency domains. There are three main arguments to compute STFT length of each segment (characterized by the window size and the ratio for overlap), the number of frequency bins, and the sampling rate. We use the scipy library implement this operation and use k < as multiplier to the length of the window W to give the segment length k x W with no overlap between segments. The complete list of STFT specifications is given in Table 8. We also demonstrate a sensitivity analysis concerning the number of frequency bins and the segment length in Figure 9.
(a) 0.88
(b)
0.875
WISDM
HHAR
0.850
Accuracy
Accuracy
0.86
0.825
0.84
0.800
WISDM
0.775
HHAR
0.82
128
256
512
1024
4 8
16
32
Number Frequency bins
Length of each segment STFT
Figure 9: Illsstration of the sensitivity of performance to the design choices of STFT by varying a) the number of frequency bins with fixed segment length of 4 and b) by varying the segment lengths with a 1024 frequency bins.
Note: It is tempting to empirical mode transformation and then apply a Hilbert-Huan transformation to obtain an instantancous phase and amplitude response in the case of non-stationary signals. It absolves us from a finite time-frequency resolution for the STFT spectra. However, our initial results indicate a high dependence on the choice of the number of intrinsic mode functions (Huang, 2014) for signal decomposition. Hence, for a generalizable approach, we choose STFT as the tool for the line-frequency spectrum.
Additional analyses with wavelet transform. We present comparison of Discrete Wavelet Transform (DWT) based analyses of the WISDM dataset with STFT within the PhASER architecture in Table B.
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Table 8: Arguments for SIFT computation
Dataset
Sampling Rate
Sequence Length STFT segment length
Number of frequency bins
WISDM
20 Hz
128
4
1024
HHAR
100 Hz
128
4
1024
UCIHAR
50 Hz
128
4
1024
SSC
100 Hz
3000
16
1024
GR
200 Hz
200
4
1024
Table 9: Accuracy and standard deviation for different methods.
Method Accuracy + Std Dev
STFT
0.87 + 0.01
DWT
0.82 + 0.01
We also present the spectrograms obtained using STFT, DWT, and EMD for sample from the WISDM dataset in Figure 10.
Backbones for Temporal Encoder. The choice of temporal encoder, Frem, not central to our design. Table 10 demonstrates the performance of PhASER under the identical settings for four cross- person settings using WISDM datasets using different backbones for From- For the convolution-base self-attention (second row in Table 10) we use three encoders to compute query (W₂), key (W&), and value(V) matricos for rDep following the guidelines from Vaswani et al. (2017). Then compute self-attention as,
QK?
A = smft
V, where da is the temporal dimension of rDpp Subsequently, use DDDe = IDDep + A,
Vd.
as the input to Frem For more details on the convolution and transformer backbones refer to Section D.3.
Table 10: Rosults for different cross-person settings for WISDM dataset.
Backbones for Frem
2
2D Convolution based
0.86
0.85
0.86
0.84
2D Convolution based with self-attention
0.88
0.83
0.84
0.81
Transformer
0.87
0.84
0.87
0.84
C
Dataset Details
Past works (Gagnon-Audet et al., 2022; Ragab al., 2023a) have shown that the datasets used in our work suffer from distribution shift across users and also within the same user temporally. This makes them suitable for evaluating the efficacy of our framework. In this section, provide more details on the datasets. Table 11 summarizes the average ADF statistics of the datasets along with their variates and their number of classes and domains.
WISDM (Kwapisz et al., 2011): It originally consists of subjects performing 18 activities but we follow the ADATime (Ragab al., 2023a) suite to utilize subjects comprising of activity classes given as walking, climbing upstairs, climbing downstairs, sitting, standing, and lying down. The dataset consists of 3-axis accelerometer measurements sampled at 20 Hz to predict the activity of each participant for segment of 128-time steps. According to Ragab et al. (2023a), this is the most challenging dataset suffering from the highest degree of class imbalance.
HHAR (Stisen et al., 2015): To remain consistent with the existing AdaTime benchmark we leverage the Samsung Galaxy recordings of this datasct from 9 participants from a 3-axis accelerometer sampled 100 Hz. The activity classes, in this case, are biking, sitting, standing, walking, climbing up the stairs, and climbing down the stairs.
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Original Time bes
1.5
1.0
Tone
STFT Magritude
STFT Phose
0.30
0.35
DWT Magnituee
DIN Phase
20.
204
15.9
L25
12.
102
255
2.5
EMD Nagnitude Spectrum
END Average Instantanezus Phace
10
14
4.0
3.2
1.0
A2
Tie
Tima
Figure 10: Comparison of STFT, DWT, and EMD-based frequency domain transformations of time series sample from WISDM.
Table 11: Summary of the dataset attributes. Higher value of ADF stat indicates greater non-stationarity within signal.
Representative AtF-Statistic (mean across all variates)
Category
Dataset
Variates
Domains
Classes
Human Activity recognition
UCIHAR
-2.58 (0.014)
9
31
6
Human Activity recognition
HHAR
-1.74 (0.062)
3
Human Activity recognition
WISDM
-0.78 (0.051)
3
36
6
Gesture Recoguition
EMG
-33.14 14 (0.011)
8
36
6
Sleep Stage Classification
EEG
-3.7 (0.047)
1
UCIHAR (Bulbul al., 2018): This dataset is collected from 30 participants using 9-axis incrtial motion unit using waist-mounted cellular device sampled at 50 Hz. The six activity classes the same as WISDM
dataset.
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HHAR
UCI HAR
WISDM
SSC
GR
2000
Frequency
Frequency
Frequency
Frequency
Frequency
40000
2000
2000
10000
1000
1000
1000
2000
0
bike
sit
stand
walk stairs stairs down
walk
downstairg
sit
stand
life
walk
jog
stand downstale
W
N1
N2
N3
REM
Figure 11: Class Distributions of the datasets used for evaluation.
SSC (Goldberger et al., 2000): This is single channel EEG dataset collected from 20 subjects classify five sleep stages wake, non-rapid eye movement slages N1, N2, N3, and rapiddeye--ovveeen.
GR (Lobov et al., 2018): For surface-EMG based gesture recognition we follow Lu et al. (2023)'s preprocessing and use an 8-channel data recorded from 36 participants for six types gestures sampled at 200 Hz. Note, that this the least, stalionary dalaset (see Table 11, yet PhASER performs as well as or better than the stat-of-the-art techniques as shown in Table in the main paper.
Implementation Details
All experiments are performed on an Ubuntu OS server equipped with NVIDIA TITAN RTX GPU cards using PyTorch framework. Every experiment is carried out with 3 different seeds (2711, 2712, 2713). During model training, we use Adam optimizer (Kingma et al., 2020) with learning rate from 1e-5 to 1e-3 and maximum number of epochs is set 150 based on the suitability of each setting. We tune these optimization-related hyperparameters for each setting and save the best model checkpoint based on early exit based the minimum value of the loss function achieved on the validation set.
D.1 Dataset Configuration
There is standard benchmarking for domain generalization for time-series where the domain labels and target samples inaccessible. We leverage past works of Ragab et al. (2023a); Lu et al. (2023) for preprocessing steps. For each datasct, we use cross-person setting in four scenarios. The details of the target domains chosen in each scenario are given in Table 12, the rest are used as source domains. Note for GR we use the same splits as Lu et al. (2023). Our method not influenced by domain labels as we do not require them for our optimization.
Table 12: Target domain splits for 4 scenarios of each dataset.
Target
Scenario 1
Scenario 2
Scenario 3
Scenario 4
Domains
WISDM
(--
10-17
18-27
28-35
HHAR
0,1
2,3
4,5
6-8
UCIHAR
0-7
8-15
16-23
24-29
GR
0-8
9-17
18-26
27-35
SSC
0-5
5-9
10-14
15-20
Figure D.1 illustrates the class distribution for each dataset. Only the WISDM and Sleep Stage Classification (SSC) datasets exhibit notable imbalances among certain classes. To validate the consistency of our conclusions, we compare the Arca Under the Curve (AUC) with the adoptod accuracy metric in Figure D.1. Generally, past works (Lu al., 2023; Gageon-Audet et al., 2022), utilizing these datasets have adopted accuracy the primary performance metric, and we follow the same approach.
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1.0
0.8
Scores
0.6
0.4
0.2
0.0
ours
BCResnet
Diversify
ANDMask
MAPU
GroupDRO
AUC
ACC
Figure 12: Illusraaioon of additional performance metric, Area Under the ROC Curve (AUC), along with Accuracy- for Scenario of the WISDM dataset, for the top-performing baselines. These metrics demonstrate consistency and justify our choice of accuracy the primary evaluation metric.
D.2 Baseline Methods
General Domain Generalization Methods. For all the standard domain generalization baselines we use conv2D layers for feature transformation of multivariate time series. It worth mentioning that DANN is actually domain adaptation study, which requires access certain unlabeled target domain data. For cross-person generalization, the source domain consists of data from multiple people, in which we divide the source domain data into two parts with equal size and view one of them as the target domain to leverage DANN for domsin-invarian training. As for sue-person-to-anotee cases, we randomly sample small nunber of unlabeled instances from each target person and merge them into the target set that is needed for running DANN.
This is a competitive benchmark for several audio-scene recognition challenges and demonstrates many useful techniques for domain generalization. BCResNet originally required www.ffuuuunccccpsssteee coefficients but it is not suitable for time-series, hence, use standard STFT of the multivariate-time series input in this case.
BCResNet.
Kon-Stationary Transformer and Koopa. These are forecasting baselines that particularly address non-stationarity in short-term time sequences, Non-stationary transformer (NSTrans)) (Liu et al., 2022) and Koopa (Liu et al., 2024). To adapt it to our setting we use the encoder part NSTrans followed by a classification head composed of fully connected layers. We simply average the encoder's output from all time steps and feed it this classifier head.
RevIN. Further, demonstrate that statistical techniques like Reversible Instance Normalization (RevIN) (Kim et al., 2021c) may be used of using RevIN is that the input and output dimensions of this module must have the same dimensions to de-normalize the instance in the feature space. This may limit the usability of the module, however, we find that applying this module around the fusion encoder specifying the same number of input and output channels in the 2D convolution layer suitable. We do not observe any significant benefit of incorporating this module from the experiments, however, if an application can specifically benefit from such RevIN, PhASER framework can support it.
alug-and-play module with our framework. One limitation
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Diversify. The goal of this design is to characterize the latent domains and use proxy- voxx-ttaining schema to assign pseudo-domain labels to the samples to learn generalizable representations. It is an end-to-end version of the adaptive RNN (Du et al., 2021) method which also proposes to identify sub-domains within a domain for generalization. is intcresting to note that for time-scries generalizable representation viewing the non-slationarityy or intra-domain shifts is crucial. Both diversily and PhASER address this problem from completely different approaches and demonstrate improvement over other standard methods or even domain adaptation methods that have the advantage of accessing samples from unseen distributions. While diversify aims to characterize latent distributions and uses a parametric setting, PhASER forces the model to learn domain-invariant features by anchoring the design to the phase which is intricately tied to non-stationarity. It also highlights that time-series domain generalization is a unique problem (compared to the more popular visual domain) and dedicated frameworks need to be designed in this case.
MAPU. MAPU the statc-of-the-art source-free domain adaptation study for time series, thus, fact, it does not apply to the timo-serics domain generalizable learning problem. However, we still view it effective approach that can address distribution shifts and achieve domain-invariant learning. In our implementation, in addition to the source domain data, still provide MAPU with the unlabeled target domain data for both cross-person generalization and one-person- to-anothhr cases. The training procedure is identical to the default MAPU design, which is to pre-train the model on labeled source domain data and then conduct the training on unlabeled target domain data.
Chronos. Large foundation models are a sought-after approach in many domains and Chronos is one such most recent candidate for time-series. It trained on 42 datasets and presents impressive zero-shot and few-shot abilities. Although it is largely targeted a forecasting tool, the authors indicate its universal representation ability for variety of tasks. Four variants of Chronos model checkpoints from 20M to 70M parameters and embedding sizes from 256 to 1024. Based on pilot testing with scenario on WISDM dataset (accuracies with a 1M parameter downstream model for the three variants: tiny-0.65, basc-0.41, largc-0.36), we find that the smallest version of the model, Chronos-tiny best suits our conservative dataset sizes for downsstrean fine-tuning. We use a few layers of 2D convolution layers with max-pooling reduce the feature size which is dependent of the length of the sequence and then flatten and input fully-connected layers
available ranging
downstream model.
Nole: A few works (Jin et al., 2024; Liu et al., 2023a) use large language models directly to analyze raw time- series despite the obvious modality gap and can report comparable performance. However, our preliminary testing with ChatGPT (Radford et al., 2019) with in-context-learning by prompting similar to Jin et. al (Jin et al., 2024) using the HHAR dataset does not provide satisfactory results and we do not pursue that direction. Instead, we use a domain-specifie large foundation model like Chronos as a fair baseline.
Table 13: Complete set of results from three trials on each baseline for WISDM cross-person generalization setting.
Baselines
Scenario 1
Scenario 2
Scenario
Scenario 4
Mean
Std
Mean
Std
Mean
Std
Mean
Std
ERM
0.57
0.02
0.50
0.02
0.51
0.02
0.55
0.02
GroupDRO
0.71
0.06
0.67
0.06
0.60
0.07
0.67
0.04
DANN
0.71
0.02
0.65
0.01
0.65
0.06
0.70
0.03
RSC
0.69
0.05
0.71
0.07
0.64
0.10
0.61
0.11
ANDMask
0.74
0.01
0.73
0.03
0.69
0.06
0.69
0.03
InceptionTime
0.83
0.01
0.82
0.02
0.80
0.04
0.77
0.01
BCResNet.
0.83
0.00
0.79
0.04
0.75
0.04
0.78
0.04
NSTrans
0.43
0.02
0.40
0.01
0.37
0.02
0.37
0.03
Koopa
0.63
0.02
0.61
0.04
0.72
0.03
0.57
0.01
MAPU
0.75
0.02
0.69
0.04
0.79
0.06
0.79
0.03
Diversify
0.82
0.01
0.82
0.01
0.84
0.01
0.81
0.01
Chronos
0.71
0.01
0.67
0.01
0.65
0.01
0.62
0.01
Ours + *
0.86
0.01
0.85
0.01
0.84
0
0.84
0.03
Ours
0.86
0.01
0.85
0.01
0.85
0.01
0.82
0.02
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Table 14: Complete set of results from three trials on each baseline for HHAR cross-person generalization
setting.
Basclines
Sccnario 1
Sccnario 2
Sccnario 3
Scenario 4
Mean
Std
Mean
Std
Mean
Std
Mean
Std
ERM
0.49
0.05
0.46
0.01
0.45
0.02
0.47
0.03
Gr
0.60
0.01
0.53
0.02
0.59
0.02
0.64
0.03
DANN
0.66
0.01
0.71
0.01
0.67
0.09
0.69
0.03
RSC
0.52
0.05
0.49
0.04
0.44
0.03
0.47
0.03
ANDMask
0.63
0.02
0.64
0.06
0.66
0.11
0.69
0.05
InceptionTime
0.77
0.04
0.80
0.01
0.82
0.03
0.83
0.01
BCResNet
0.66
0.05
0.70
0.06
0.75
0.04
0.68
0.04
NSTrans
0.21
0.02
0.22
0.03
0.27
0.04
0.28
0.02
Koopa
0.72
0.04
0.63
0.03
0.72
0.05
0.69
0.02
MAPU
0.73
0.02
0.72
0.03
0.81
0.01
0.78
0.03
Diversify
0.82
0.01
0.76
0.01
0.82
0.01
0.68
0.01
Chronos
0.73
0.04
0.75
0.03
0.73
0.01
0.66
0.12
Ours RevIN*
0.82
0.05
0.82
0.02
0.92
0.04
0.85
0.03
Ours
0.83
0.02
0.83
0.02
0.94
0.03
0.88
0.02
Table 15: Complete set of results from three trials on each baseline for UCIHAR cross-person generalization
setting.
Baselines
Scenario Mean
Scenario
Scenario
Scenario
Std
Mean
Std
Mean
Std
Mean
Std
ERM
0.72
0.09
0.64
0.05
0.70
0.01
0.72
0.03
GroupDRO
0.91
0.02
0.84
0.01
0.89
0.04
0.85
0.07
DANN
0.84
0.02
0.79
0.01
0.81
0.02
0.86
0.03
RSC
0.82
0.13
0.73
0.07
0.74
0.03
0.81
0.06
ANDMask
0.86
0.08
0.80
0.06
0.76
0.13
0.78
0.09
Inception'Time
0.91
0.03
0.82
0.07
0.88
0.02
0.91
0.04
BCResNet
0.81
0.02
0.77
0.02
0.78
0.02
0.83
0.02
NSTrans
0.35
0.02
0.35
0.01
0.51
0.02
0.47
0.01
Koopa
0.81
0.02
0.72
0.05
0.81
0.06
0.77
0.03
MAPU
0.85
0.03
0.80
0.01
0.85
0.02
0.82
0.03
Diversify
0.89
0.03
0.84
0.04
0.93
0.02
0.90
0.02
Chronos
0.56
0.05
0.57
0.01
0.50
0.02
0.82
0.13
Ours + *
0.96
0.01
0.90
0.01
0.93
0.03
0.97
0.01
Ours
0.96
0.01
0.91
0.01
0.95
0
0.97
0.01
D.3 Implementation Details of PhASER
The magnitude and phase encoders, Fiag and Fria are implemented using 2D convolution layers with the number of input channels equal to the variates, V, and the out channels as 2c with (5 x 5) kernels. is a hyperparameter used to conveniently control the size of the overall network. For all HAR and GR models we adopt e as and for SSC 4. For more specific details please refer to our code. The sub-spectral feature normalization uses a group number of 3 and follows Equation 2.3 for operation. This is inspired by Chang et. al (Chang al., 2021) subspectral normalization for audio applications with frequency spectrum input. The key idea is to conduct sub-band normalization (across fixed set of frequency bins along time and examples for each channel). We find merit in using this technique for domain generalizable applications, as it can help overcome the low-frequency drifts arising due to device differences (for eg. DC drifts in various sensors). One implementationnsseeee that if the number of sub-bands not divisible by the total number of features then we choose apply the remainder bands with batch-nore nocec-oomaiizaiinn. The output from the respective encoders then fused along the channel/ variate axis by multiplying with 2D convolution kernels to provide feature map which is the
modification we carried out to ensure a generalizable framework is
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input to our phase-driven residual network. The Frus similarly is implemented using 2D convolution layers with the number of input channels as 4c and output channels to be 2c.
Subsequently for the depth-wise encoder, DDep we use 2D convolution layers with batch normalization and SiLU (Elfwing et al., 2018) activation function. This style of architecture closely adapted from the basic building blocks BCResNet. (Kim al., 2021a). After average pooling the Frem can assume any backbone as per the requirements of the application. As demonstrated previously in Section B, the choice of backbone is not central to our design here. We find that some applications (like WISDM and GR) benefit from attention-basee temporal encoding more than others. For the attention- based version of Frem used multi-hcaded attention based on a transformer encoder (Vaswani et al., 2017). Regarding positional encoding, we used a simple sinusoid-basel encoding and added it to the sequence representation rDpp- However, arriving at the best positional encoding for numerical time-series data is an active area of research (Kazemi et al., 2019; Tang et al., 2023; Mohapatra et al., 2023) given its uniqueness compared to typical natural language inputs and further optimizations can be carried out. For the the convolution-basse basod Frem we simply use a x 3) in 2D convolution layer to conduct temporal convolutions.
kernel of size
For the classification head, ICls, apply 2D convolution layers to have the number of output channels equal to the number of classes in an application, followed by softmax operation. Interestingly, if the choice of From remains convolutional the entire network can be implemented in a purely convolutional form allowing applicability to real-time problems. The model sizes across the different datasets range from 40k- 100k trainable parameters (based on the number of variates, temporal encoding etc.) which modest and can be further tuned for resource- ewource-conssrainee applications by adjusting the e parameter.
Table 16: Complete set of results from three trials each bascline for SSC cross-person gencralization setting.
Baselines
Scenario 1
Scenario 2
Scenario 3
Scenario 4
Mean
Std
Mcan
Std
Mean
Std
Mean
Std
ERM
0.50
0.05
0.46
0.04
0.49
0.02
0.45
0.03
GroupDRO
0.57
0.07
0.56
0.03
0.55
0.05
0.59
0.06
DANN
0.64
0.02
0.63
0.02
0.69
0.03
0.63
0.04
RSC
0.50
0.09
0.48
0.02
0.52
0.07
0.46
0.01
ANDMask
0.55
0.10
0.50
0.09
0.54
0.07
0.57
0.08
Inception Time
0.74
0.04
0.78
0.03
0.72
0.05
0.80
0.02
BCResNet
0.79
0
0.82
0.01
0.79
0.01
0.81
0
NSTrans
0.43
0.02
0.37
0.04
0.42
0.06
0.35
0.03
Koopa
0.58
0.02
0.62
0.01
0.53
0.04
0.49
0.06
MAPU
0.69
0.01
0.68
0.01
0.65
0.03
0.69
0.02
Diversify
0.73
0.03
0.76
0.02
0.68
0.05
0.77
0.02
Chronos
0.53
0.04
0.47
0.04
0.47
0.01
0.57
0.03
Ours R
0.82
0.01
0.79
0.02
0.78
0.01
0.81
0.01
Ours
0.85
0.01
0.80
0.01
0.79
0.01
0.83
0.01
D.4 Ablation Details of PhASER
For row 1 in Table 7, the modification to PhASER is straightforward by simply omitted the Hilbert transfor- mation during data preprocessing. When the separate encoders are not used (rows 6 and 7 in Table 7), we only use FMgg and connect the output of the sub-feature normalization block directly to the FDep When the residual removed entirely (rows and 6 in Table 7), we cannot broadcast the 1D input to 2D anymore so we take the mean across all the temporal indices of From (FDep) and flatten it to input to fully connected layers. Based on the dataset choose few fully connected layers truncating to the number of classes finally.
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Table 17: Complete sot of results from three trials on cach bascline for GR cross-person generalization sctting.
Baselines
Scenario
Scenario 2
Scenario 3
Scenario
Mean
Std
Mean
Std
Mean
Std
Mean
Std
ERM
0.45
0.02
0.58
0.03
0.57
0.03
0.54
0.04
GroupDRO
0.53
0.08
0.36
0.11
0.59
0.05
0.45
0.13
DANN
0.60
0.01
0.66
0.04
0.65
0.02
0.64
0.03
RSC
0.50
0.10
0.66
0.05
0.64
0.03
0.56
0.03
ANDMask
0.41
0.13
0.54
0.20
0.45
0.15
0.39
0.12
InceptionTime
0.68
0.07
0.70
0.09
0.72
0.03
0.69
0.02
BCResNet
0.62
0.06
0.67
0.09
0.65
0.05
0.61
0.07
NSTrans
0.31
0.01
0.34
0.01
0.34
0.01
0.32
0.02
Koopa
0.47
0.03
0.54
0.02
0.60
0.05
0.70
0.06
MAPU
0.64
0.02
0.69
0.03
0.71
0.01
0.68
0.04
Diversify
0.69
0.01
0.80
0.01
0.76
0.02
0.76
0.01
Chronos
0.49
0.01
0.54
0.03
0.51
0.05
0.48
0.02
Ours RevIN*
0.68
0.03
0.81
0.04
0.77
0.03
0.76
0.02
Ours
0.70
0.02
0.82
0.02
0.77
0.04
0.75
0.01
Table 18: Complete set of results from three trials on each baseline for HHAR pnoe-prson--t-anotteo setting.
Baselines
1
4
5
Mean Std
Mean
Std
Mean
Std
Mean
Std
Mean
Std
Mean
Std
Mean
Std
Mean
Std
Mean
Std
ERM
0.27 0.01 0.40
0.05
0.41
0.05
0.44
0.05
0.42
0.08
0.44
0.01
0.45
0.04
0.44
0.04
0.48
0.02
GroupDRO
0.33 0.02 0.53
0.02
0.38
0.05
0.48
0.04
0.47
0.04
0.51
0.08
0.47
0.03
0.48
0.02
0.49
0.05
DANN
0.32 0.03
0.44
0.05
0.42
0.03
0.45
0.06
0.42
0.03
0.48
0.04
0.49
0.02
0.05 0.45
0.51
0.01
RSC
0.27 0.03 0.45
0.06
0.38
0.05
0.45
0.09
0.40
0.08
0.47
0.02
0.50
0.06
0.44
0.08
0.53
0.01
ANDMask
0.34 0,06 0.50
0.03
0.37
0.04
0.43
0.05
0.46
0.04
0.51
0.07
0.03 0.46
0.02 0.47
0.52
0.03
Inception Time
0.52 0.05 0.62
0.02
0.44
0.03
0.04
0.60
0.09
0.57
0.05
0.03 0.66
0.01
BCResNet
0.28 0.03 0.48 0.08
0.32
0.04
0.47
0.03
0.42
0.06
0.52
0.05
0.44
0.02
0.45
0.02
0.49
0.06
NSTrans
0.20 0.01 0.22
0.02
0.17
0.02
0.20
0.01
0.21
0.01
0.22
0.01
0.26
0.07
0.17
0.05
0.20
0.01
Koopa
0.32 0.02 0.42
0.04
0.37
0.01
0.40
0.01
0.42
0.02
0.45
0.05
0.35
0.02
0.43
0.03
0.48
0.02
MAPU
0.39 0.05 0.57
0.05
0.35
0.06
0.52
0.03
0.49
0.04
0.54
0.02
0.49
0.01
0.50
0.06
0.52
0.04
Diversify
0.42 0.04 0.62
0.04
0.32
0.09
0.62
0.56
0.03
0.61
0.01
0.53
0.52
0.10
0.05
Chronos
0.32 0.03 0.23
0.05
0.26
0.04
0.25
0.03
0.27
0.09
0.23
0.08
0.24
0.06
0.21
0.08
0.21
0.05
Ours *
0.48
0.02
0.66
0.08
0.57
0.05
0.04
0.64
0.05
0.65
0.63
0.03
Ours
0.53 0.04
0.70
0.03
0.63
0.01
0.66
0.03
0.64
0.06
0.67
0.01
0.65
0.03
0.67
0.04
0.62
0.02
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D.5 Phase-driven NSTrans
Non-stationary transformer, NSTrans (Liu et al., 2022), applies destationarizing attention around the transformer block. Since it is typically used for forecasting tasks, it comprises of encoder and decoder module. For adapting this model to classification around the encoder block. We use this modified version of NSTrans as the Frem module in PhASER and observe significant improvement in performance as shown in Figure 6.
update the design to conduct normalization and denormalization
Note: The poor performance of the Nonstationary transformer can be attributed to two main reasons:
(1) Originally, the Nonstationary transformer was designed for forccasting time-sories tasks and employs encoder-decoder style architecture. To successfully apply the core module of the Nonstationary trans- former (Liu et al., 2022), araatoonrraaiioodddttttttnnaaiii
the input-output space needs to remain consistent. This consistency is naturally ensured in encoder-docoder design. However, in our classification applications, we only utilize the encoder module. Although we maintain the input-output dimensions, the semantics of the latent space and input space are not the same. Hence, destationarization is not very successful.
(2) Nonstationary transformer inputs consist of raw time-scrics data with positional encoding. Given the fine- nie-grained nature of current tasks, such an approach can be more data-hungy as they try establish a relation (attention) among every time step. Therefore, it may not perform well short-range classification tasks that focus domain generalization. This indicates limitation in its direct usage for optimizing categorical objective function using only the encoder part with a classification head.
D.6 Computational Analyses
To assess the resource utilization of PhASER against other baselines, we offer two metries 1) Number of Multiply and Accumulate operations per sample (MACs) for approximate computational complexity at mun-time and 2) Number of trainable parameters to determine the memory footprint. We compute these for the HHAR dataset in Table 19 (these metries are dependent on input dimensions, hence different choices of dataset, sequence length, and modalities can yield different numbers).
Table 19: Model comparison based on MACs and number of trainable parameters. Trainable Parameters (x10²)
Model
MACs (x109)
ERM
19.5
98.1
GroupDRO
19.5
98.1
DANN
21.7
102.9
RSC
19.5
98.1
ANDMask
19.5
98.1
BCResNet
55.3
154.7
NSTrans
35.3
75.6
Koopa
32.7
118.7
MAPU
46.9
128.3
Diversify
35.7
922.9
Chronos
345.5
1049.8
Ours
48.6
81.4
Our computation cost is comparable to the other methods, achieving much better performance. We also determine the asymptotic time complexity the PhASER modules in Table 20. For multi-layer neural network modules, the representative time complexity for layer is provided (rows 3-7).
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Table 20: Complexity per module and imput notation for each module.
Module
Complexity
Hilbert augmentation (using Fast-Fourier transform)
O(V N log N) O(V N W log W) O N d Cin Cous)
2 Short-Term Fourier Transform
3 Magnitude Encoder (Fmag), Phase Encoder (Fraa); Phase Projection
Head (gRes) - 2D Convolution Layers Depthwise Feature Encoder (FDep) 2D Convolution Layers with aver-
O N d C.n Cout) O(d)
age pooling along feature axis
5 Temporal Encoder (Frem) (worst case backbone) Transformer Encoder
O(N d) Odd h)
Classffiaaton Encoder (gcls) fully connected layers
D.7 Additional Analyses
D.7.1 Traditional Augmentation
For time series, brute augmentations like scaling, reverting, cropping, and jittering may not be always suitable as they may alter the morphological properties that are important for the task. Even more advanced techniques like frequency-
warping and additive noise, need deliberate characterization of the signal's frequency response to meaningfully provide an augmented view while retaining the task-relevant semantics. This is one of the key motivating factors for us to explore gencral-purpose augmentation strategy that diversifies the non-stationarity in a signal without altering its task-specific semantics (magnitude and frequency responses).
To demonstrate the use of traditional augmentations with PhASER for human-activity recognition, we incorporate the following augmentations proposed by past works (Qin el al., 2023; Um al., 2017) on the HHAR dataset.
Rotation incorporating arbitrary rotation matrices to simulate different sensor locations.
Permutation random temporal perturbation for fixed window within each sample (Um al., 2017).
Circular Time-shift shifting the signal by a random time interval, constrained by a predefined maximum time-shift parameter (20% of the sample length in this case) for each sample. The shifted time points from the trailing edge are wrapped around and padded to the leading edge of the signal
We incorporate these augmentations in place of the Hilbert augmentation and apply the PhASER. We also experiment with identical settings with augmentations and illustrate in Figure 8. These results are indicative that arbitrary augmentations in the time domain do not necessarily diversify the non-stationarity of a signal. Hence, PhASER principles like residual connections to re-introduce nonstationary dictionary as chase-projection and broadcasting (using gnt)) do not bode well here, and even the performance of a no-augmentation scenario sometimes better than the traditional temporal augmentations for domain- generalization tasks in this case. However, the future, we may encounter applications where established augmentation strategics, in combination with Hilbert augmentation, might be the best choice. In this work, aim to propose more generic framework that can benefit most time-series classification tasks to achieve better generalizability.
D.7.2 Random Phase Augmentation using Hilbert Transform
We aimed to explore a random phase augmentation while ensuring minimal distortion to the signal's magnitude response to preserve important task-relevant properties. To achieve this, we leverage an adaptation of the Hilbert Transform. We illustrate our approach using a simple example: let the input signal be x(t) = sin(wt), and its Hilbert Transform be HT(x(tt) x(t) cos(wt). For an arbitrary phase shift o, the following trigonometric identity holds:
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sin(wt Φ) sin(wt) cos(o) + cos(wi) sin(Ø).
(33)
This gives the desired randomly phase-shifted version of x(t), expressed y() ax(() bR((), where cos(Ø) and b = sin(o). The following constraint is imposed on the scalars and b:
a + b² = 1,
(34)
which defines a valid phase shift Φ as:
Φ arctan
(35)
We solve for a and b, and apply them as shown in Figure 13 to obtain an approximately identical random phase shift. across all frequency components of nonstationary signal. The desired randomly sampled from the range [- //2..
a
x(t)
Hilbert Transform
X
+
ytt)
X
b
Figure 13: Schema illustrating the process for obtaining random phase augmentation by leveraging the Hilbert Transform of the original input x()..
As shown Figure 8, we observe no significant benefit from this randomization on the generalization performance of the current classification tasks. However, we are interested in exploring this direction future by imposing additional constraints inspired by underlying processes for other time-series tasks.
D.8 Visualization
We present some visualizations using the I-distributed stochastic neighbor embedding (t-sne) analyses on our PhASER, Diversify, and BCResNet for the HHAR dataset for the left-out domains scenario in Figure 7. We illustrate the t-sne plots for in-domain and out-of-domain data and the different colors indicate the six activity classes of this dataset. In all the cases, we only make necessary modifications extract the embeddings from the last layer of the network before categorical score assignment and tune the perplexity parameters during the t-sne plotting for optimal 2-dimensional projection. Figure 14. (a,d) shows that the clustering for cach class is distinct and clearly scparable for both in-domain and out-of-domain data using PhASER. The accuracy disparity for unseen domains also very low, 0.97 for in-domain PhASER accuracy and 0.94 for oft-of-domain, which justifies the overall strong generalization ability PhASER. without access to any target domain samples. We would also like to point out that t-sne plots are susceptible to hyperparameters, hence, even though the accuracy of Diversify is better than BCResnet for out-of-domain data, visually Figure 14. (f) may convey better separation between classes than Figure 14. (e).
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60
40
20
20
20
-20
-20
-40
-40
(8)
-30
(b)
(c)
-40
-20
20
40
60
-40
-20
20
-30
-20
-10
0
10
20
30
60
40
50
40
20
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-20
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(d)
(0)
(f)
-40
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40
-75
-50
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25
50
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-20
0
20
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60
Figure 14: t-sme plots for visualizations using embeddings from HHAR scenario 1 for in-domain samples in (a) PhASER with an accuuray of 0.97, (b) Diversify with in-domain accuracy 0.82 and (c) BCResNet with in-domaín accuracy of 0.78; and out-of-domain samples in (c) PhASER with accuracy of 0.94, (d) Diversify with accuracy of 0.77 and (e) BCResNet with accuracy of 0.74.
E Supplementary of Main Results
We conduct all experiments with three random seeds (2711, 2712, 2713), and present the error range in this section. Tables 13, 14 and 15 represent the mean and standard deviation corresponding to the main paper's Table for the WISDM, HHAR and UCTHAR datasets respectively. Tables 16 and 17 are the complete representations all the runs corresponding to Table in the main paper for sleep stage classification and gesture recognition respectively. Table 18 corresponds the Table in the main paper for the complete performance statistics for one person to another generalization using HHAR dalaset.
F Experimental Setup for Figure 4
We analyze three synthetic signals sampled at 100 Hz with 500 points: stationary sinusoid xx (t) sin (2m33),,, and two nonstationary signals (t) 5000 sin00inn22m111 - 101 and =t( 5t sin sin(2 + 10t. Each is normalized by its maximum amplitude. The Hilbert transform is applicd to extract imaginary components representing instantaneous phase information. Using FFT, we compule normalized magnitude and unwrapped phase spectra for original and Hilbert- imaieerrry signals.
G Broader Impacts
PhASER, with its advanced approach time-series domain-gemeralizable learning, offers significant societal benefits various fields and domains, such healthcare, environment monitoring, and manufacturing domains, by enabling negative social impacts, its application within these critical areas necessitates thoughtful examination of ethical concerns. In healthcare, the application of PhASER could usher in a new era of patient monitoring and Ireatment, leading to improved experiences and outcomes for individuals across diverse demographics. Its robust generalization capabilities, even with limited access to source domains (see Table 5), offer the potential to bridge gaps and foster inclusivity, particularly in minority communitics, while enabling insights from rare occurrences. Moreover, for applications in environmental monitoring- from continuous sensing of ambient living conditions to remote and sporadic sensing of inaccessible geological sites PhASER's
precise and dependable data analysis. While PhASER itself does not directly cause
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principles hold promise for samplc-cfficiet, generalizable analysis. Similarly, in manufacturing applications, PhASER can be deployed for both qualitative and quantitative analyses of physical components, as well as for enhancing workers' safety through continuous sensing instrumentation. However, the implementation of PhASER in such vital arcas brings to the forefront ethical considerations like data privacy, bias prevention, and the careful management automation reliance. Addressing these issues important to leverage PhASER's benefits across these domains while ensuring ethical integrity and maintaining public trust in these arcas.
37


Reviews:
[{"review_id": "7fyfG09Pyz", "reviewer": "Reviewer_MkRX", "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\n1.\tThe use of the Hilbert Transform to generate phase-shifted augmentations represents a novel contribution beyond conventional time-series augmentation techniques.\n\n2.\tThe explicit incorporation of phase information for improving time-series classification performance is a relatively unexplored direction and introduces a fresh perspective.\n\n3.\tThe empirical evaluation is comprehensive, covering multiple benchmarks across diverse application domains. The proposed method consistently outperforms competitive baselines, demonstrating promising generalization capabilities.\n\nWeaknesses:\n\n1.\tWhy can phase information be considered as an approximate proxy for non-stationarity? The current explanation feels insufficiently grounded. Could the authors provide a more intuitive or theoretical justification for this assumption?\n\n2.\tWhy would employing data augmentation techniques to diversify non-task-specific non-stationarity help address the generalization challenge?\n\n3.\tThe proposal to treat magnitude and phase as separate modalities is effective but not entirely novel. Several recent works already explore time-frequency decomposition in time-series tasks. The manuscript should more explicitly describe its contributions beyond this.\n\n4.\tSome of the theoretical content feels overly redundant and hinder readability.\n\n5.\tWhile the empirical gains of the Phase-Residual Broadcasting module are reported, it remains unclear why this specific design is critical. How does it compare to simpler alternatives such as concatenation or skip connections?\n\n6.\tIn Section 2.2, the authors assume that each time-series sample is characterized by a deterministic function. This assumption seems counterintuitive, especially given the stochastic nature of many real-world signals.\n\n7.\tIn Section 4, the paper claims to be the first to address out-of-distribution time-series classification. However, the experimental setup appears to focus on domain generalization within known classes, rather than evaluating on novel classes at test time.", "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 PhASER, a novel framework designed to tackle non-stationarity and domain shift challenges, thereby enabling more generalizable pattern recognition for time series classification. PhASER comprises three key components: (1) a Hilbert transform-based augmentation module that addresses non-stationary signals by introducing phase-shifted variants; (2) a separated magnitude-phase encoding mechanism that facilitates richer integration of time-frequency information; and (3) a phase-residual feature broadcasting strategy that promotes the learning of domain-invariant and task-specific features. Comprehensive experiments on five benchmark datasets against 13 baselines demonstrate the effectiveness and generalization ability of PhASER.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1.\tThe authors claim that phase can serve as an approximate proxy for non-stationarity, which is central to the proposed methodology. However, this claim is not rigorously supported by theory or strong empirical justification. A clearer explanation, potentially supported by signal processing theory or experiments directly analyzing phase shifts under varying non-stationarity levels, is necessary.\n\n2.\tThe authors argue that Hilbert Transform–based augmentation diversifies non-stationarity, but no statistical analysis (e.g., stationarity tests before/after augmentation, divergence metrics) is provided to support this claim. Without such evidence, it is unclear whether the augmentation meaningfully contributes to robustness.\n\n3.\tThe overall writing quality makes the paper difficult to follow in several places. Key terms such as \"out-of-distribution classification\" are used imprecisely, and some explanation lack clarity and logical flow. Improving the exposition, tightening the structure, and clearly defining technical concepts would greatly enhance readability and help readers better understand and assess the contributions.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "JYoVJh1lmm", "reviewer": "Reviewer_A8Mk", "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\n1. The proposal to treat phase as a proxy for nonstationarity is well-motivated. The integration of Hilbert transforms for augmentation and residual broadcasting offers a principled approach to handle time series domain shifts.\n\n2. Ablation studies (Table 5) conclusively validate each component’s impact, while experiments integrating PhASER with NSTrans (Figure 4) show its flexibility to enhance existing architectures.\n\nWeaknesses\n\n1. The paper lacks sufficient discussion of related work that leverages frequency-domain features for time series modeling (e.g., [R1], [R2]), especially in the Introduction and Related Work sections.\n\n2. While the Hilbert transform is used to extract amplitude and phase, the paper does not adequately compare it against other established transforms such as FFT or wavelets, leaving its unique advantages for classification tasks unclear.\n\n3. The choice of datasets is not well justified. Commonly used benchmarks for time series classification [R3], including UCR, UEA, and other eight datasets like SleepEEG, Epilepsy, and ECG, are not considered or discussed.\n\n[R1] Temporal-frequency co-training for time series semi-supervised learning. AAAI, 2023.\n\n[R2] A survey on deep learning based time series analysis with frequency transformation. arXiv, 2023.\n\n[R3] A Survey on Time-Series Pre-Trained Models. TKDE, 2024.", "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": "This paper proposes PhASER, a novel phase-aware framework for domain-generalizable classification of nonstationary time series. PhASER combines Hilbert-based augmentation, separate magnitude-phase encoding, and residual phase broadcasting. Extensive experiments across multiple domains show strong performance gains over 13 baselines. Theoretical insights and ablations further support its effectiveness.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. [R1] shows that combining amplitude and phase improves classification performance over using either alone, consistent with Table 1. However, [R1] uses a single frequency encoder, whereas this paper employs two separate encoders. Does this increase training cost significantly? Additionally, can the authors attempt to use the method in [R1] to input amplitude and phase into an encoder for classification training?\n\n2. Prior work [R4] indicates that RevIN is not beneficial for classification, as it is designed for forecasting. Yet, Table 2 shows results using RevIN. Can the author further explain whether the addition of RevIN mentioned above is beneficial for time series classification? If it is invalid, why should it be placed in Table 2?\n\n3. Can the authors provide a comparative analysis of different frequency-domain methods (e.g., FFT, wavelets) for extracting amplitude and phase in classification tasks? Why was the Hilbert transform specifically chosen?\n\n4. Given [R1]'s recommendations on benchmarks, can the authors report results on part standard datasets such as UCR, UEA, SleepEEG, or Epilepsy? If not, please justify the dataset choices and discuss their generalizability.\n\n5. See weaknesses.\n\n[R4] FIC-TSC: Learning Time Series Classification with Fisher Information Constraint. ICML, 2025.\n\n\nThe above issues should be addressed through corresponding revisions in the main manuscript.", "broader_impact_concerns": "None.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "bQ3sAubnzu", "reviewer": "Reviewer_pFjB", "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* The paper is well written, context is properly conveyed and the related works section is adequate.\n* A lot of ablations were performed to empirically justify their architecture choices, and further justify their techniques through theoretical insights.\n* Overall results is strong, well constructed and convincing.\n\n# Weakness\n**Weak phase-residual layer justification**\n\nThe justification for the necessity of phase-residual feature broadcasting a bit weak.\n* Theoretical insights from Theorem 2.5 is not connected to any of the experimental results, thus hard to validate interpretation\n* There is empirical evidence in the ablation presented in Table 5, but this doesn't provide much intuition\n\nFrom this quote in Section 3.2:\n\n> Reintroducing this phase-dictionary deeper in the layers enables the model to learn task-specific representations that are more robust to non-stationarity, making it better equipped to handle unseen non-stationarity in the target domains.\n\nIt would've been interesting to run the Table 1 experiment in the OoD setting to see whether the generalization gap is smaller for the Phase input than the magnitude input. This might have justified the choice a bit more.\n\n**Unclear effect of phase data augmentation on discriminative features**\n\n The following statement\n\n> We propose a simple but effective data augmentation technique based on the Hilbert Transform (HT) to diversify the non-stationarity and preserve discriminatory features\n\nis a bit opaque due to the uninterpretable nature of time series. It is hard to conceptualize if this phase shift augmentation truly preserves discriminative features. Clearly from Table 5 this augmentation seems to be useful, but I think there was a missed opportunity there to have conducted a more controlled experiment. For example (using Section 2.2 notation), can models trained on $\\mathbf S$ transfer to $\\mathbf{\\hat{S}}$?", "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 presents an approach to tackle Out-of-Domain (OoD) generalization in time series. They propose three key contributions: First an Hilbert transform-based data augmentation technique to enrich the training set with phase shifts. Second they make use of separate magnitude and phase (obtained through STFT) encoding, which they subsequently fuse together. Lastly they use a residual style layer to propagate the phase encoding (exclusively) to the end of the network.\n\nUsing this combination of technique, they obtain strong performance on a wide set of (classification) benchmarks and outperform common baselines.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "**Critical**\n* This claim in Section 2.1 \"Consequently, the classifier learns domain-agnostic task-discriminatory representations\" is very strong, it essentially means that the OoD task in time series is solved. Please consider reducing the claim a bit or provide sufficient proof of this. Possible proof could be to show a negligible gap in performance between an oracle model (that have been trained on a dataset that includes the test domain) and a model where the same domain is held out.\n\n**Strengthen**\n* Run an experiment to confirm that the Hilbert Transform based Phase Augmentation preserves discriminative features. An example of such experiment is presented in the Weakness section.\n* Further insights from Theorem 2.6.\n* Run the experiment to further justify the phase-residual layer, see \"Weak phase-residual layer justification\" Weakness section.", "broader_impact_concerns": "No broader impact concerns", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "56ZuUByXfz", "reviewer": "Reviewer_VAfw", "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\n1. The problem addressed in the paper is of importance to the time series community.\n2. The method is simple and easy to follow.\n\n**Weaknesses**\n\n- The main motivation of this paper is not clear.\n  - Relationship of non-stationarity between generalization ability.\n    (1) Non-stationarity is the most important property in time series forecasting, as it reveals the changes in the statistical characteristics of observations over time. However, for classification problems, it does not represent a generalization across the time dimension, so the impact of non-stationarity on classification performance is counterintuitive. Although experiments in this paper have been conducted to reveal this phenomenon, some intuitive or theoretical explanations are still needed.\n- The motivations of some designs about phase are not clear.\n  -  Phase Augmentation: \n    (1) Why can the phase shift preserve discriminatory properties? Can changing the amplitude diversify non-stationarity？\n    (2) Why is it better to use only -π/2 as a phase shift? Wouldn't it be better to use different phase shifts like π/4 to diversify the sample set? In the section of PhASER with Other Augmentation Strategies, the authors mention that random phase shifts with HT do not show any particular advantage, which seems counterintuitive. Diversified random phase shifts should provide more valuable samples. \n    (3) In Figure 1 I, why is the horizontal axis labeled as amplitude and the vertical axis labeled as sequence length, with the range of  -1.5 ~ 1.5? Shouldn't the sequence length be larger than 0?\n  - Phase-Residual Feature Broadcasting\n    (1) Theoretical Insights mention that Phase-Residual Feature Broadcasting approximates the β-Divergence, but the method does not seem to directly reduce the β-Divergence. The overall training objective is to minimize the classification loss $\\mathcal L_{\\text{CE}}$. How is the β-Divergence being reduced?\n    (2) Is the shape of $\\mathbf{r}$ in Equation 7 same with $g_{\\text{Res}}(F_{\\text{Pha}}(\\text{Pha}(\\mathbf{x)})) $ ?\n    (3) Section 2.4 should emphasize the advantages of FPha Res compared to FMag Res or FFus Res to better articulate the motivation behind this design.\n\n- The Experiments:\n\n  - The results reported in the paper of Diversify (75) are worse than the original result of Diversify [1] with the same setting.\n\n  - To better compare different methods, could the authors provide more experiments on the datasets used in the existing works, e.g. DSADS, USC-HAD, and PAMAP datasets used in Diversify [1].\n\n    [1] Lu, Wang, et al. Out-of-distribution Representation Learning for Time Series Classification. In *ICLR*,2023", "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": "The paper proposes PhASER,  a framework aiming at learning generalizable representations under the setting of domain generalization for time series classification. PhASER includes three key modules: (1) Phase Augmentation to diversify non-stationarity. (2) Magnitude-Phase Separate Encoding to extract amplitude and phase features separately based on Short-Term Fourier Transform. (3) Phase-Residual Feature Broadcasting. Extensive experiments demonstrate PhASER’s superiority.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- Main concerns:\n  - Intuition about how non-stationarity boosts generalization ability in classification.\n  - Better clarification of the Phase Augmentation's motivation (questions in weaknesses).\n  - Better clarification of the Phase-Residual Feature Broadcasting's motivation (questions in weaknesses).\n  - Fair comparisons with baselines.\n  - More experiments on the datasets used in the existing works.", "broader_impact_concerns": "No", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)