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Channel-Dependent State Space Model for Multivariate Time Series Forecasting
Yu-Cheng Wu, Fan-Keng Sun, Li-Chun Lu, Duane S. Boning
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Abstract
Abstract:Multivariate time series forecasting (MTSF) is critical across many real-world domains. Existing deep learning approaches fall into two paradigms with distinct limitations: channel-independent (CI) methods unconditionally ignore cross-variable dependencies and model only temporal dynamics, while channel-dependent (CD) methods consider both but typically rely on architectural compromises to mitigate overfitting and computational overhead. We therefore propose Chameleon, a specialized CD state space model (SSM) that enables data-dependent, fine-grained interactions across variables while scaling linearly with their number. By connecting selective SSMs with the Kalman filter, we leverage the missing measurement update in the former for cross-variable modeling while preserving the SSM backbone for robust temporal modeling. We further identify favorable inductive biases of GatedDeltaNet for time series, adapt it as our backbone, and improve generalization through additional techniques, including a previously unexplored stochastic perturbation of reversible instance normalization. On strongly dependent ODE and PEMS datasets, Chameleon achieves the best MSE and MAE across all settings, while its CI ablation and prior CD methods incur 61-178% higher MSE on average. Across 28 standard benchmark settings, Chameleon also achieves better MSE and MAE than each baseline in at least 27 and 22 cases, respectively. Training-time and peak-memory analyses on Traffic and ETT further demonstrate competitive efficiency and favorable memory scalability across different variable counts.
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- Source: https://arxiv.org/abs/2609.36453v1
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Channel-Dependent State Space Model for Multivariate Time Series Forecasting Yu-Cheng Wu1, Fan-Keng Sun1, Li-Chun Lu1,2, Duane S. Boning1 1Massachusetts Institute of Technology, 2National Taiwan University Abstract Multivariate time series forecasting (MTSF) is critical across many real-world domains. Existing deep learning approaches fall into two paradigms with distinct limitations: channel-independent (CI) methods unconditionally ignore cross-variable dependencies and model only temporal dynamics, while channel-dependent (CD) methods consider both but typically rely on architectural compromises to mitigate overfitting and computational overhead. We therefore propose Chameleon, a specialized CD state space model (SSM) that enables data-dependent, fine-grained interactions across variables while scaling linearly with their number. By connecting selective SSMs with the Kalman filter, we leverage the missing measurement update in the former for cross-variable modeling while preserving the SSM backbone for robust temporal modeling. We further identify favorable inductive biases of GatedDeltaNet for time series, adapt it as our backbone, and improve generalization through additional techniques, including a previously unexplored stochastic perturbation of reversible instance normalization. On strongly dependent ODE and PEMS datasets, Chameleon achieves the best MSE and MAE across all settings, while its CI ablation and prior CD methods incur 61–178% higher MSE on average. Across 28 standard benchmark settings, Chameleon also achieves better MSE and MAE than each baseline in at least 27 and 22 cases, respectively. Training-time and peak-memory analyses on Traffic and ETT further demonstrate competitive efficiency and favorable memory scalability across different variable counts. 1 Introduction Multivariate time series forecasting (MTSF) refers to predicting the future values of multiple continuous and potentially correlated variables based on their histories. Accurate forecasts are valuable across applications including energy demand prediction, traffic flow estimation, weather forecasting, financial asset prediction, and product quality estimation in manufacturing (Suganthi and Samuel, 2012; Hernandez et al., 2014; Jiang and Luo, 2022; Lana et al., 2018; Bi et al., 2023; Price et al., 2025; Elliott and Timmermann, 2008; Sezer et al., 2020; Ren et al., 2023). With the growth of data volume and complexity, deep neural networks (DNNs) have become widely adopted for MTSF. However, designing a generalizable architecture across diverse multivariate systems remains a long-standing challenge because cross-variable dependencies vary widely in strength and structure. Moreover, time series data are generally noisier, less regular, and less abundant than text or images, while both the temporal and variable dimensions can span hundreds of elements, making unconstrained joint modeling prone to overfitting, computationally expensive, and difficult to scale. One simplified paradigm, channel-independent (CI) modeling, has therefore recently emerged and proven unexpectedly effective on benchmark datasets. PatchTST (Nie et al., 2023) pioneered this line by forecasting each variable, also referred to as a channel or series, independently through a shared Transformer (Vaswani et al., 2017) that treats temporal patches as tokens while avoiding interactions across variables, achieving state-of-the-art (SOTA) performance at its release. Subsequent approaches, such as FITS (Xu et al., 2024) and SFNN (Sun et al., 2025), reinforced this trend by matching or even surpassing prior, more complex channel-dependent (CD) architectures through one or a few feedforward layers. Together, these results highlight the importance of effective intra-series modeling and suggest that many current benchmarks exhibit weak cross-variable dependencies. Nevertheless, information from other variables can be crucial in some settings, motivating renewed interest in CD models. Inspired by CI, a prominent family of CD methods embeds each variable independently over time before modeling dependencies across variables. In addition, to mitigate the overfitting risk, architectural compromises are commonly adopted, often at the cost of modeling granularity or capacity for temporal dynamics. iTransformer (Liu et al., 2024) popularized these strategies by representing each variable’s entire input window as a single token using only a linear layer, while reserving the more expressive Transformer for cross-variable interactions. This formulation is retained by approaches such as SOFTS (Han et al., 2024) and S-Mamba (Wang et al., 2025), which replace the Transformer with more scalable designs. Other variants, including Crossformer (Zhang and Yan, 2023), Leddam (Yu et al., 2024), and TimeMachine (Ahamed and Cheng, 2024), follow the same factorization but employ either self-attention or Mamba along both temporal and variable dimensions. However, this added capacity incurs higher computational cost without consistently translating into better performance, especially on benchmarks, where simple CI SFNN (Sun et al., 2025) and later, more constrained CD methods such as DUET (Qiu et al., 2025) and TimeFilter (Hu et al., 2025) remain empirically superior. To improve generalization, DUET and TimeFilter instead restrict interactions to selected variable or patch pairs, yet still inherit similar compromises or introduce greater scalability concerns by constructing pairwise dependencies across patches from all variables. These developments therefore leave an unresolved trade-off among modeling granularity, temporal capacity, generalization, efficiency, and scalability. These observations raise two natural questions: 1. Can we incorporate fine-grained cross-variable interactions into a strong temporal modeling backbone while remaining efficient and scalable? 2. If so, can we avoid overfitting and achieve SOTA performance in multivariate systems with both weak and strong dependencies across variables? These questions become even more compelling in light of recent advances in selective state space models (SSMs), pioneered by Mamba (Gu and Dao, 2024). Mamba retains the recency-emphasizing inductive bias of recurrent neural networks while overcoming their long-range memory limitations. Among its successors, GatedDeltaNet (Yang et al., 2025) further introduces delta-based memory updates (Schmidhuber, 1992), enabling both rapid forgetting and targeted memory rewrites, which are particularly desirable properties for noisy time series. Despite these advantages, GatedDeltaNet has not yet been explored for MTSF, while existing Mamba-based attempts largely inherit established architectures by replacing their original, often attention-based, backbones with Mamba (Ahamed and Cheng, 2024; Wang et al., 2025), leaving specialized cross-variable interaction mechanisms underexplored. This work therefore presents Chameleon, an MTSF model built on our proposed CD SSM blocks to address these gaps. Our contributions are: 1. We draw a parallel between selective SSMs and the Kalman filter (Kalman, 1960) for MTSF, exploiting this connection to design an explicit cross-variable measurement update while maintaining linear complexity in the number of variables. 2. We identify favorable inductive biases of GatedDeltaNet for time series, tailor it into an MTSF-specialized backbone, and further introduce a previously unexplored stochastic perturbation of reversible instance normalization (Kim et al., 2022) during training to improve generalization. 3. Chameleon achieves SOTA performance across standard long-horizon benchmarks and strongly dependent systems. 2 Preliminaries Problem Formulation. Given a historical window ∈ℝN×TX ^N× T of N variables observed at T regularly spaced time steps, MTSF aims to forecast the next F steps ∈ℝN×FY ^N× F. Exact timestamp information is not explicitly used, and stationarity is not assumed, as trends and seasonality are prevalent in real-world time series. Selective SSMs and GatedDeltaNet. Selective SSMs (Gu and Dao, 2024; Dao and Gu, 2024) have emerged as strong sequence learners through efficient long-context modeling and data-dependent state updates. GatedDeltaNet (Yang et al., 2025) further combines the global decay of Mamba2 (Dao and Gu, 2024) for rapid forgetting with the delta rule of DeltaNet (Yang et al., 2024) for targeted memory rewrites: t=αtt−1(−βttt⊤)+βttt⊤∈ℝDv×Dk,S_t= _tS_t-1(I- _tk_tk_t )+ _tv_tk_t ^D_v× D_k, (1) where αt,βt∈(0,1) _t, _t∈(0,1) control global decay and targeted updating, respectively, and t∈ℝDvv_t ^D_v and t∈ℝDkk_t ^D_k are input-dependent projections. This adaptive memory management provides a favorable inductive bias for filtering irrelevant information in noisy time series. The Kalman Filter. The Kalman filter (Kalman, 1960) estimates dynamic system states through a recursive two-step cycle. The “time update” produces the prior state estimate t−∈ℝDsS_t^- ^D_s and error covariance t−∈ℝDs×DsP_t^- ^D_s× D_s: t−=tt−1+tt,t−=tt−1t⊤+t,S_t^-=A_tS_t-1+B_tu_t, _t^-=A_tP_t-1A_t +Q_t, (2) where t∈ℝDs×DsA_t ^D_s× D_s is the state transition matrix, t∈ℝDuu_t ^D_u is the control input, t∈ℝDs×DuB_t ^D_s× D_u is the control matrix, and t∈ℝDs×DsQ_t ^D_s× D_s is the process noise covariance. The “measurement update” corrects this predicted prior state using new observations t∈ℝDzz_t ^D_z: t=t−t⊤(tt−t⊤+t)−1,t=t−+t(t−tt−),t=(−tt)t−,K_t=P_t^-H_t (H_tP_t^-H_t +R_t)^-1, _t=S_t^-+K_t(z_t-H_tS_t^-), _t=(I-K_tH_t)P_t^-, (3) where t∈ℝDz×DsH_t ^D_z× D_s maps state to measurement space, t∈ℝDz×DzR_t ^D_z× D_z is the measurement noise covariance, and t∈ℝDs×DzK_t ^D_s× D_z is the Kalman gain that weights the innovation by the relative confidence between prediction and measurement. The recurrent state update of the aforementioned SSMs naturally parallels the time update, with tv_t serving as the control input. However, explicit measurement updates for SSMs remain underexplored as domain-specific modeling mechanisms. This work leverages this connection to introduce a learnable measurement update for cross-variable modeling in MTSF. 3 Method The proposed framework processes the multivariate input through the following stages: optional reversible instance normalization (RevIN) (Kim et al., 2022) and decomposition, patching and embedding, an encoder built from our proposed Chameleon blocks, and an MLP-based decoder. Before entering the encoder, each variable is independently transformed into patch-level representations, which are then processed by the encoder and mapped to forecasts by the decoder. 3.1 Input Processing The input ∈ℝN×TX ^N× T optionally undergoes RevIN and decomposition to mitigate distribution shifts and preserve prominent trend structure, respectively. The criteria for their use and supporting ablations are provided in Appendix B. Reversible Instance Normalization. When enabled, the mean μi _i and standard deviation σi _i for each variable i are computed over a predefined normalization window and used to normalize the input as x~i,t=(xi,t−μi)/σi x_i,t=(x_i,t- _i)/ _i. When disabled, we instead set μi=0 _i=0 and σi=1 _i=1 for all instances and variables. These statistics are stored for denormalizing the final predictions. To further improve generalization under distribution shifts, we introduce a simple yet previously unexplored data augmentation that perturbs μi _i and σi _i during training: μ~i=μi+ϵμ,i,σ~i=σi⋅exp(ϵσ,i),ϵμ,i∼(0,δμ2),ϵσ,i∼(0,δσ2), μ_i= _i+ _μ,i, σ_i= _i· ( _σ,i), _μ,i (0, _μ^2), _σ,i (0, _σ^2), (4) where δμ _μ and δσ _σ control the perturbation magnitude. The multiplicative parameterization of σ~i σ_i ensures positivity without additional constraints. During training, μi _i and σi _i in both normalization and denormalization are replaced by μ~i μ_i and σ~i σ_i, respectively. Decomposition, Patching, and Embedding. Following Autoformer (Wu et al., 2021), when decomposition is applied, each series is separated into a trend component x~i,ttrend x_i,t^trend via a length-preserving moving average kernel and a seasonal component x~i,tseasonal=x~i,t−x~i,ttrend x_i,t^seasonal= x_i,t- x_i,t^trend. Each component is then divided into L=⌈T/Tp⌉L= T/T_p non-overlapping patches of length TpT_p following PatchTST (Nie et al., 2023), producing itrend,iseasonal∈ℝL×TpX_i^trend,X_i^seasonal ^L× T_p. Replication padding is used to preserve length during decomposition and complete the final patch when needed. Rather than processing the two components through separate forecasting networks and combining their outputs only at the final stage, we embed them through separate linear projections and immediately sum the resulting representations, keeping the additional cost minimal: i,lembed=(s)i,lseasonal+(t)i,ltrend+l∈ℝDv,x^embed_i,l=W^(s)x^seasonal_i,l+W^(t)x^trend_i,l+e_l ^D_v, (5) where (s),(t)∈ℝDv×TpW^(s),W^(t) ^D_v× T_p are shared across patches and variables, and le_l is the fixed sinusoidal positional encoding (Vaswani et al., 2017). When decomposition is disabled, x~i,tseasonal=x~i,t x_i,t^seasonal= x_i,t, and the embedding reduces accordingly to i,lembed=(s)i,lseasonal+lx^embed_i,l=W^(s)x^seasonal_i,l+e_l. 3.2 Chameleon Blocks and Encoder The encoder is composed of a stack of repeated units. Each unit consists of root mean square layer normalization (RMSNorm) (Zhang and Sennrich, 2019) followed by our proposed Chameleon block and an optional feedforward network block. As illustrated in Figure 1, the Chameleon block unifies the sequential modeling of GatedDeltaNet (Yang et al., 2025) with a Kalman-filter-inspired measurement update for cross-variable state calibration. In the classical multivariate Kalman filter, both the time and measurement updates operate on the joint state of all variables at each time step. However, such joint modeling has been found suboptimal for Mamba in MTSF (Weng et al., 2025). Inspired by CI and recent factorized CD architectures, we accordingly assign temporal and variable modeling to the time and measurement updates, respectively. The “time update” first processes each patch token as a control input to drive intra-series evolution recurrently, after which the “measurement update” derives an observation signal from the same token to correct hidden states across variables. Figure 1: Overview of Chameleon, which preserves the recurrent GatedDeltaNet backbone for temporal modeling and introduces a Kalman-filter-inspired measurement update for cross-variable modeling through data-dependent observation signals and covariances (cov.). Time Update. The GatedDeltaNet block is leveraged as the temporal modeling backbone due to its advantageous inductive biases. For variable i, given the RMSNorm-processed l-th patch token ¯i,l∈ℝDv x_i,l ^D_v, the SSM computes a prior state via Equation 1: i,l−=αi,li,l−1(−βi,li,li,l⊤)+βi,li,li,l⊤∈ℝDv×Dk,S_i,l^-= _i,lS_i,l-1(I- _i,lk_i,lk_i,l )+ _i,lv_i,lk_i,l ^D_v× D_k, (6) where αi,l _i,l, βi,l _i,l, i,lk_i,l, and i,lv_i,l are all projected from ¯i,l x_i,l. Unlike the original GatedDeltaNet, the convolutional layers for deriving i,lk_i,l and i,lv_i,l are omitted for efficiency, as the patch-based embedding already captures local details. Structurally, αi,l(−βi,li,li,l⊤)∈ℝDk×Dk _i,l(I- _i,lk_i,lk_i,l ) ^D_k× D_k, βi,li,l⊤ _i,lk_i,l , and i,lv_i,l are respectively analogous to the state transition matrix, control matrix, and control input in the Kalman-filter time update of Equation 2, but the update here is performed independently within each variable’s state without computation across variables. Measurement Update. To capture dependencies across variables, we instantiate the measurement update as a learnable state calibration mechanism operating only along the variable dimension. The same patch tokens ¯i,l x_i,l are reused as sources of observation signals for this step. Under our factorized design, the two updates extract distinct representations for their respective modeling dimensions, avoiding suboptimal joint modeling and potential redundancies. The time update derives control inputs from the tokens to drive intra-series state evolution, whereas the measurement update performs cross-variable state correction based on the resulting observations. Unlike the original Kalman filter, where the two stages alternate recursively, we instead perform this correction once across all patch steps after completing the time update recurrence. This design is rationalized by feeding the outputs generated from all calibrated stepwise states into subsequent blocks, rather than relying only on the final one, as in PatchTST (Nie et al., 2023) and other CI methods with deep sequential backbones. These outputs can therefore still interact temporally in later blocks, while the separation preserves GatedDeltaNet’s chunk-parallel acceleration along the time dimension and maintains efficient cross-variable modeling. The remaining challenge is to construct a learnable and compatible observation signal lZ_l and Kalman gain l∈ℝN×NK_l ^N× N analogous to Equation 3, which involves specifying l,l−,l∈ℝN×NH_l,P_l^-,R_l ^N× N. Because the observations here are not actual physical measurements but conceptual signals derived from the input patch tokens, we produce them directly in the hidden-state space, naturally leading to l=H_l=I. Specifically, input projections i,l∈ℝDvo_i,l ^D_v and i,l∈ℝDkh_i,l ^D_k are introduced following the same design as i,lv_i,l and i,lk_i,l, and their outer products i,li,l⊤∈ℝDv×Dko_i,lh_i,l ^D_v× D_k are flattened and stacked across variables to form l∈ℝN×(DvDk)Z_l ^N×(D_vD_k). Since the time update operates independently for each variable and the corrected states are not fed back into the recurrence, we accordingly represent the prior error covariance l−P_l^- as a diagonal matrix diag(l)diag(p_l), where l∈ℝ+Np_l ^N_+. Meanwhile, the measurement noise covariance admits the covariance-correlation decomposition l=l1/2ll1/2R_l= _l^1/2C_l _l^1/2, where l=diag(l) _l=diag(d_l) with l∈ℝ+Nd_l ^N_+, and l∈ℝN×NC_l ^N× N is a correlation matrix. Both lp_l and ld_l are designed to be data-dependent and are elaborated later. To build lC_l, an additional input projection i,l∈ℝDη η_i,l ^D_η is normalized onto the unit sphere for each variable i. Stacking these projections yields l∈ℝN×Dη η_l ^N× D_η, whose Gram matrix ll⊤ η_l η_l contains pairwise cosine similarities that reflect dependencies among variables in the projected space. We further introduce a global learnable scale γ∈(0,1)γ∈(0,1) and define: l=(1−γ)+γll⊤∈ℝN×N,C_l=(1-γ)I+γ η_l η_l ^N× N, (7) which is symmetric positive semidefinite with unit diagonals, forming a valid correlation matrix. The parameter γ controls the strength of cross-variable interactions and can attenuate them when dependencies are weak. A small initial value of γ also limits unreliable interactions early in training before the learned dependencies become informative. Defining l=γl1/2lU_l= γ _l^1/2 η_l gives l=(1−γ)l+ll⊤R_l=(1-γ) _l+U_lU_l , and thus l−+l=diag(l+(1−γ)l)+ll⊤P_l^-+R_l=diag(p_l+(1-γ)d_l)+U_lU_l . The Kalman gain in Equation 3 therefore admits the Woodbury form: l=l−(l−+l)−1=diag(l)[l−ll(+l⊤ll)−1l⊤l]∈ℝN×N,K_l=P_l^-(P_l^-+R_l)^-1=diag(p_l) [ _l- _lU_l (I+U_l _lU_l )^-1U_l _l ] ^N× N, (8) where l=diag(l+(1−γ)l)−1 _l=diag(p_l+(1-γ)d_l)^-1. Although lC_l and lK_l are expressed as N×N× N matrices, neither is explicitly materialized for large N. Instead, the computation operates directly on the diagonal terms and the low-rank factor lU_l when applying the gain to the subsequent state correction. This avoids (N2)O(N^2) storage and reduces the inversion complexity from (N3)O(N^3) to (NDη2+Dη3)O(ND_η^2+D_η^3), which is linear in N for fixed DηD_η and substantially improves scalability. Since lp_l and ld_l reflect the uncertainty of the prior state i,l−S_i,l^- and observation signal, respectively, we make both dependent on the input tokens, with lp_l additionally conditioned on the prior state. We therefore introduce another input projection to produce the log observation variance logdi,l d_i,l for each variable i. Because multiplying lp_l and ld_l by the same positive scalar leaves the Kalman gain unchanged, their common scale is unidentifiable. We remove this ambiguity by centering logdi,l d_i,l across variables, fixing the geometric mean of ld_l to one. With this common scale fixed, rather than estimating pi,lp_i,l directly, we model its ratio to di,ld_i,l through τi,l=log(pi,l/di,l) _i,l= (p_i,l/d_i,l). Following the state readout design of GatedDeltaNet, an additional normalized query vector extracts a compact representation from i,l−S_i,l^-, which is combined with the input token to produce τi,l _i,l, and the prior variance is recovered as pi,l=di,lexp(τi,l)p_i,l=d_i,l ( _i,l). Both the centered logdi,l d_i,l and τi,l _i,l are bounded by predefined values for numerical stability, while exponentiation guarantees di,l,pi,l>0d_i,l,p_i,l>0. The prior state l−∈ℝN×Dv×DkS_l^- ^N× D_v× D_k is then flattened into ~l−∈ℝN×(DvDk) S_l^- ^N×(D_vD_k) and calibrated via the data-dependent Kalman gain lK_l and observation signal lZ_l: ~l=(−l)~l−+ll∈ℝN×(DvDk). S_l=(I-K_l) S_l^-+K_lZ_l ^N×(D_vD_k). (9) After restoring the Dv×DkD_v× D_k state shape, the posterior i,lS_i,l and prior i,l−S_i,l^- are fused and read out independently for each variable through another global learnable scalar gate ω∈(0,1)ω∈(0,1): i,lenc=[ωi,l+(1−ω)i,l−]i,l∈ℝDv,y^enc_i,l=[ _i,l+(1-ω)S_i,l^-]\,q_i,l ^D_v, (10) where i,l∈ℝDkq_i,l ^D_k is the GatedDeltaNet output query vector projected from ¯i,l x_i,l. The prior branch provides a gated skip connection around the measurement update step to improve stability and flexibility. Similar to γ, ω can reduce cross-variable influence when dependencies are weak, while a small initialization keeps the Chameleon block close to the CI backbone early in training. 3.3 Decoder The decoder maps the encoded representations to the forecast horizon F in a CI manner. For each variable i, the patch representations ienc∈ℝL×DvY^enc_i ^L× D_v are flattened into a vector of dimension LDvLD_v, followed by a SiLU activation and a linear projection to produce ^′∈ℝN×F Y ^N× F. Optional decoder blocks consisting of RMSNorm, SiLU, and a width-preserving linear layer can be appended but are typically not used. When RevIN is enabled, the final predictions ^∈ℝN×F Y ^N× F are obtained by reversing the normalization with the stored statistics; otherwise, ^=^′ Y= Y . 4 Experiments Chameleon is trained with smooth L1 (Huber) loss using Adam (Kingma and Ba, 2014) under a StepLR schedule, with parameters additionally smoothed via exponential moving average. All experiments are conducted on a single NVIDIA L40S (46GB) or H100 (80GB) GPU. Full hyperparameters and code are available at https://github.com/anonymous/Chameleon-TSF. 4.1 Datasets Based on Granger causality analysis and empirical comparisons of prior CI and CD methods, Abdelmalak et al. (2025) argue that current long-horizon MTSF benchmarks exhibit weak dependencies, and accordingly suggest several simulation datasets with strong coupling governed by physical ordinary differential equations (ODEs) (Gilpin, 2021). To examine the proposed measurement update, we therefore select Double Pendulum and Lorenz Coupled, two of the strongest datasets identified in these analyses. As a complementary real-world setting with a different dependency structure and higher dimensionality, PEMS (Song et al., 2020) contains traffic flow measurements from hundreds of spatially distributed sensors on California highways. Traffic propagation can induce approximately linear relationships between sensors at temporal offsets, making maximum pairwise lagged correlation particularly informative. Although Granger causality does not indicate stronger dependencies than in the benchmarks, our analysis shows that all PEMS datasets exhibit an average maximum pairwise lagged correlation above 0.8, compared with around 0.45 for the Traffic benchmark. PEMS03 and PEMS08 are selected because they show the strongest dependencies under both lagged-correlation and Granger-causality analyses. While PEMS provides additional spatial information, it is not used by Chameleon or any of the compared baselines. Still, we test Chameleon on seven long-horizon benchmarks to evaluate whether it can achieve SOTA performance in these common settings, where more competitive baselines are available. These experiments further evaluate the effectiveness of the GatedDeltaNet backbone, the proposed RevIN perturbation, and other temporal modeling designs, and examine whether the measurement update can exploit relatively stronger dependencies without degrading performance when they are weak. The evaluated benchmarks include ETTm1, ETTm2, ETTh1, ETTh2 (Zhou et al., 2021), Electricity, Solar Energy, and Traffic (Lai et al., 2018). Following SFNN (Sun et al., 2025), we exclude Weather, Illness, and Exchange Rate due to known issues, such as low predictability or insufficient samples. We follow the standard practice of splitting all datasets chronologically into training, validation, and test partitions. Appendix A provides the split ratios and brief descriptions of all datasets. 4.2 Setup Protocol. We use the common forecast horizons F∈96,192,336,720F∈\96,192,336,720\ for all benchmarks, while the ODE and PEMS datasets are evaluated at two horizons each. We exclude longer horizons for the chaotic ODE systems, where forecast errors grow rapidly, and use F∈48,96F∈\48,96\ for PEMS due to better baseline availability. Because the conventional look-back size T=96T=96 can omit important periodic context and disadvantage methods with greater long-range temporal modeling capacity, as analyzed in Appendix C, we follow recent works (Abdelmalak et al., 2025; Sun et al., 2025; Liu et al., 2024; Qiu et al., 2025; Hu et al., 2025; Xu et al., 2024) in evaluating multiple look-back sizes and adopt SFNN’s grids (Sun et al., 2025) for the benchmarks. Full dataset-specific T and F settings are provided in Appendix A. For each dataset-horizon-model combination, we evaluate every T over five independent runs, average the mean squared error (MSE) and mean absolute error (MAE) across runs, and report the results from the best T. Baselines. We compare against six competitive methods spanning the major modeling paradigms. S-Mamba (Wang et al., 2025) with adaptive channel normalization (ACN) (Lee et al., 2025) is, to our knowledge, currently the strongest CD SSM-based method, while TimeFilter (Hu et al., 2025) and DUET (Qiu et al., 2025) are SOTA CD approaches leading under both conventional and longer look-back sizes. Crossformer (Zhang and Yan, 2023), the strongest model evaluated by Abdelmalak et al. (2025) on Double Pendulum and Lorenz Coupled at shorter horizons, is included for the strongly dependent datasets. For the benchmarks, the primarily CI SFNN (Sun et al., 2025) and CI FITS (Xu et al., 2024) provide additional strong references. When no public configuration is available, mostly for the ODE datasets, we align shared hyperparameters with Chameleon and otherwise retain each baseline’s defaults. 4.3 Results Table 1: Results on strongly dependent datasets. Chameleon (CI) removes the proposed cross-variable measurement update. Best and second-best results are highlighted. Underlining denotes statistical significance over all other baselines (p<0.05p<0.05). The Avg./Max. increase rows report each baseline’s average/maximum relative error increase over Chameleon across all settings, with larger values indicating worse performance. Model Chameleon Chameleon (CI) S-Mamba+ACN TimeFilter DUET Crossformer Dataset (F) MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE Double Pendulum 96 0.0678 0.1176 0.1258 0.1880 0.0927 0.1642 0.0870 0.1616 0.2510 0.3271 0.1208 0.1975 192 0.3053 0.3216 0.3845 0.3836 0.3325 0.3701 0.3522 0.3787 0.5044 0.5052 0.4774 0.4717 Lorenz Coupled 48 0.0179 0.0389 0.0493 0.0761 0.0708 0.1247 0.0424 0.0934 0.1570 0.2137 0.0581 0.1152 96 0.1569 0.1842 0.3368 0.3213 0.3666 0.3837 0.2935 0.3230 0.5026 0.4758 0.3019 0.3299 PEMS03 48 0.0905 0.1920 0.1045 0.1991 0.0927 0.1968 0.1048 0.2099 0.1028 0.2050 0.0968 0.1989 96 0.1156 0.2157 0.1359 0.2229 0.2309 0.3330 0.1407 0.2389 0.1447 0.2407 0.1211 0.2208 PEMS08 48 0.0975 0.1783 0.1187 0.1924 0.1266 0.1985 0.1617 0.2047 0.1244 0.1929 0.1525 0.2111 96 0.1307 0.1910 0.1703 0.2115 0.1869 0.2064 0.2960 0.3193 0.1663 0.2087 0.2201 0.2335 Avg. Increase – – 61% 34% 81% 57% 62% 47% 178% 110% 74% 55% Max. Increase – – 175% 96% 296% 221% 137% 140% 777% 449% 225% 196% Table 2: Results on long-horizon benchmarks, averaged across four forecasting horizons. Highlighting and Avg./Max. increase conventions follow Table 1. Model Chameleon S-Mamba+ACN TimeFilter DUET SFNN FITS Dataset MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE Traffic 0.3636 0.2409 0.4012 0.2744 0.3738 0.2580 0.3868 0.2548 0.3836 0.2656 0.4074 0.2783 Solar 0.1783 0.2204 0.1936 0.2531 0.2020 0.2438 0.1884 0.2150 0.1831 0.2434 0.2086 0.2495 Electricity 0.1522 0.2418 0.1590 0.2576 0.1573 0.2541 0.1575 0.2469 0.1554 0.2452 0.1609 0.2552 ETTh1 0.3870 0.4168 0.4484 0.4538 0.4477 0.4431 0.4078 0.4214 0.3964 0.4208 0.4087 0.4207 ETTh2 0.3310 0.3773 0.3850 0.4074 0.3928 0.4155 0.3440 0.3864 0.3481 0.3832 0.3331 0.3820 ETTm1 0.3298 0.3673 0.3666 0.3952 0.3509 0.3812 0.3365 0.3694 0.3329 0.3687 0.3501 0.3759 ETTm2 0.2356 0.2995 0.2654 0.3252 0.2673 0.3213 0.2512 0.3108 0.2438 0.3034 0.2416 0.3098 Avg. Increase – – 11.3% 9.8% 10.5% 7.2% 4.8% 1.9% 3.2% 3.8% 7.1% 6.0% Max. Increase – – 16.3% 14.8% 18.7% 10.6% 6.6% 5.8% 5.5% 10.4% 17.0% 15.5% Across all ODE and PEMS datasets and horizons, Chameleon achieves the best MSE and MAE over all baselines, with statistical significance over all baselines in seven of the eight settings, as shown in Table 1. Specifically, removing the measurement update raises MSE and MAE by 61% and 34% on average, reaching maximum increases of 175% and 96%, respectively, on Lorenz Coupled (F=48F=48), supporting the effectiveness of the proposed update in capturing cross-variable dependencies. The gaps to prior CD methods are even larger, with MSE and MAE increases of 62–178% and 47–110%, respectively, relative to Chameleon. Table 2 presents the benchmark results averaged across horizons for each dataset, with individual results, cross-run standard deviations, and statistical significance reported in Appendix D. Under weaker dependencies, Chameleon still ranks first in MSE on all seven benchmarks and in MAE on six, while the baselines incur 3.2–11.3% higher MSE and 1.9–9.8% higher MAE. Notably, prior CD methods frequently trail Chameleon’s CI variant on the strongly dependent datasets or CI methods such as FITS (Xu et al., 2024) and the primarily CI SFNN (Sun et al., 2025) on the benchmarks, whereas Chameleon remains strongest across datasets with varying dependency strengths, highlighting the importance of preserving strong temporal modeling capacity. Table 3: Ablation studies across five datasets. Chameleon reports the average MSE across horizons, while the other columns report the relative MSE increase over Chameleon. Chameleon (CI) deactivates the cross-variable measurement update, w/o RevIN Pert disables the stochastic RevIN perturbation, and w/o Decomp removes decomposition. GatedDeltaNet denotes the vanilla backbone without these three designs. w/ Mamba2 replaces the GatedDeltaNet backbone with Mamba2. Dataset Chameleon GatedDeltaNet Chameleon (CI) w/o RevIN Pert w/o Decomp w/ Mamba2 PEMS08 0.1141 +26.8% +28.5% +2.16% +1.09% +0.38% Traffic 0.3636 +2.47% +2.31% +2.67% +2.53% +1.82% Solar 0.1783 +7.20% +3.89% +1.07% +3.05% +3.76% ETTh2 0.3310 +0.77% +0.55% +0.77% +0.80% +0.85% ETTm2 0.2356 +1.00% +0.04% +0.77% +0.27% +0.83% Figure 2: Learned effective Kalman gains ωl _l on ETTh2, Traffic, and Double Pendulum. Larger off-diagonal magnitudes indicate stronger interactions across variables, while ω∈(0,1)ω∈(0,1) is a learned gate controlling the reliance on the raw Kalman gain lK_l. The same Double Pendulum patch yields different gains when serving as the first and last patch in two temporally offset input windows. We further conduct ablation studies across multiple datasets and evaluate changes in test MSE averaged across horizons, as shown in Table 3. We ablate the measurement update, RevIN perturbation, and decomposition, compare against vanilla GatedDeltaNet (Yang et al., 2025) without these designs but with its original convolutional layers, and replace the GatedDeltaNet backbone with Mamba2 (Dao and Gu, 2024). Together with Table 1, these results suggest that the benefit of the measurement update broadly aligns with dependency strength. Removing it increases MSE by 28.5% on PEMS08 and by even larger margins on the ODE datasets. Among the benchmarks, Traffic and Solar have higher average Granger F-scores at lag 24 (6.3 and 5.5) than ETTh2 and ETTm2 (2.2 and 1.4), respectively, and correspondingly show larger MSE increases of around 2–4% under the ablation. Even on the weakly dependent ETT datasets, the measurement update provides incremental improvements without degrading performance, with only a 4.3% increase in parameter count on average. Figure 2 provides complementary evidence through the learned effective Kalman gains ωl _l. For ETTh2 and Traffic, we visualize the gains with the largest mean off-diagonal magnitudes during testing, whereas the Double Pendulum example is randomly sampled. Even under this selection, Double Pendulum exhibits substantially stronger cross-variable interactions and greater reliance on lK_l, with ω=0.901ω=0.901, compared with 0.5800.580 for Traffic and 0.1080.108 for ETTh2. Moreover, the same Double Pendulum patch yields distinct gains in temporally offset input windows because the corresponding prior states differ. Such conditioning cannot be achieved by simply applying attention or an SSM along the variable dimension to patchwise representations, as in many prior works. Meanwhile, removing the RevIN perturbation or decomposition, or replacing GatedDeltaNet with Mamba2, increases MSE by around 1–4% on Traffic and Solar. Although the effects of the perturbation and decomposition on mean MSE are sometimes incremental, especially on ETTh2 and ETTm2, they reduce cross-run standard deviation by 64% and 72% on average, respectively, providing substantial stability gains with little additional overhead. Specifically, the perturbation introduces negligible computational cost, while decomposition and using GatedDeltaNet instead of Mamba2 add only 3.6% and 0.03% more parameters on average, respectively. The gap between vanilla GatedDeltaNet and full Chameleon also largely matches or exceeds the largest individual ablation gap on each dataset, suggesting little interference among the designs and confirming that the overall gains extend beyond backbone selection alone. (a) Test MSE versus training time, with bubble areas proportional to training peak memory. (b) Training peak memory versus number of variables. Figure 3: Efficiency and scalability analyses on Traffic (F=96F=96) with a batch size of one. Finally, Figure 3 evaluates training efficiency and scalability on Traffic, the dataset with the most variables (862). Chameleon achieves the best test MSE in Figure 3(a) while keeping training time and peak memory within the same order of magnitude as TimeFilter (Hu et al., 2025) and S-Mamba (Wang et al., 2025), and substantially below DUET (Qiu et al., 2025). More importantly, Figure 3(b) varies the number of variables by retaining subsets or duplicating all variables and shows that Chameleon’s peak memory scales comparably to S-Mamba, consistent with their theoretical (N)O(N) complexity, whereas TimeFilter and DUET grow considerably faster with their quadratic scaling in N. Parameter counts and corresponding efficiency and scalability analyses on the ETT datasets with fewer variables are provided in Appendix D. 5 Conclusion We presented Chameleon, a CD SSM that integrates a Kalman-filter-inspired cross-variable update mechanism into a strong GatedDeltaNet-based temporal backbone, enabling data-dependent, fine-grained interactions with linear complexity in the number of variables. On strongly dependent ODE and PEMS datasets, Chameleon achieves the best MSE and MAE in every case, while removing the update raises MSE by 61% on average. Across 28 standard benchmark settings, it also ranks first in 25 for MSE and 20 for MAE. Traffic and ETT analyses further show competitive training efficiency and favorable memory scalability across different variable counts. Together, these results suggest that Chameleon exploits strong cross-variable dependencies without sacrificing temporal modeling capacity or scalability across diverse MTSF settings. AI use statement In this work, we used generative AI tools for partial code revision. We have not used generative AI tools for developing theoretical models or conceptual frameworks, proposing or refining hypotheses, designing or providing feedback on research methodology or experiments, implementing the full method, supporting qualitative or thematic data analysis, or interpreting the results, and assistance in formulating or proving mathematical claims and cleaning or reformatting datasets are not applicable to this work. Additionally, we used generative AI tools for visualizing the analyses and polishing the writing. We have reviewed all AI-assisted work and take responsibility for the final content of this work, including text, claims, or artifacts produced with the aid of generative AI. References Abdelmalak et al. (2025) I. Abdelmalak, K. Madhusudhanan, J. Choi, C. Kloetergens, V. K. Yalavarit, M. Stubbemann, and L. 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The last four are standard long-horizon benchmarks. Dataset Description Double Pendulum Simulated double pendulum dynamics with angles and angular velocities. (Gilpin, 2021) Lorenz Coupled Two coupled simulated Lorenz systems. (Gilpin, 2021) PEMS Traffic flow collected from hundreds of sensors in the Caltrans PeMS. (Song et al., 2020) Traffic Road occupancy rates from 862 sensors on Bay Area freeways. (Lai et al., 2018) Solar Solar power production from 137 photovoltaic plants in Alabama. (Lai et al., 2018) Electricity Electricity consumption of 321 clients recorded in kWh from 2012 to 2014. (Lai et al., 2018) ETT Oil temperature and six load indicators of electricity transformers. (Zhou et al., 2021) Table 5: Experimental setup. Split: train/validation/test ratio by time; Var.: number of variables; Freq.: sampling frequency. Dataset Total Length Split Freq. Var. Look-back Window Size (T) Horizon (F) Double Pendulum 60,000 7:1:2 N/A 4 96, 192, 336 96, 192 Lorenz Coupled 60,000 7:1:2 N/A 6 96, 192, 336 48, 96 PEMS03 26,208 6:2:2 5min 358 96, 288 48, 96 PEMS08 17,856 6:2:2 5min 170 96, 288 48, 96 Traffic 17,544 7:1:2 1hour 862 168, 336, 672, 1344 96, 192, 336, 720 Solar 52,560 7:1:2 10min 137 144, 288, 576, 1008 96, 192, 336, 720 Electricity 26,304 7:1:2 1hour 321 168, 336, 672, 1344 96, 192, 336, 720 ETTh 14,400 6:2:2 1hour 7 168, 336, 672, 1344 96, 192, 336, 720 ETTm 57,600 6:2:2 15min 7 96, 192, 384, 672, 1344 96, 192, 336, 720 Descriptions and detailed settings for all datasets are provided in Table 4 and Table 5, respectively. The T grids for the benchmarks and ODE datasets follow Sun et al. (2025) and Abdelmalak et al. (2025), respectively. However, we omit 512,720\512,720\ for the latter because neither yields the best performance across all settings tested by Abdelmalak et al. (2025). Appendix B Preprocessing Guidance for Reversible Instance Normalization and Decomposition Table 6: Trend and distribution-shift diagnostics computed from the training and validation splits. All values report the 95th percentile across variables. Values exceeding the selection thresholds (Trend/Mean Shift/KS >0.2>0.2; Std. Shift >1.2>1.2) are highlighted. Dataset Trend Mean Shift Std. Shift KS Double Pendulum 0.058 0.093 1.105 0.074 Lorenz Coupled 0.089 0.076 1.010 0.043 PEMS03 0.471 0.213 1.173 0.174 PEMS08 0.815 0.467 1.283 0.270 Traffic 0.782 0.507 1.604 0.303 Solar 0.222 0.056 1.058 0.072 Electricity 1.271 0.633 1.823 0.381 ETTh1 1.427 0.498 1.754 0.182 ETTh2 2.317 1.465 5.486 0.685 ETTm1 1.427 0.478 1.746 0.183 ETTm2 2.317 1.465 5.512 0.689 Table 7: RevIN ablation results at F=96F=96. Better results in each column are highlighted. Dataset ETTh2 ETTm2 Solar Double Pendulum RevIN MSE MAE MSE MAE MSE MAE MSE MAE Yes 0.2628 0.3271 0.1528 0.2409 0.1534 0.1981 0.1158 0.1753 No 0.2968 0.3623 0.1695 0.2671 0.1725 0.2110 0.0678 0.1176 This section provides practical guidance for deciding whether to apply RevIN and decomposition to a given dataset. RevIN improves robustness to temporal distribution shifts in level and scale, while decomposition can preserve prominent trend structure. We therefore examine four diagnostics using only the training and validation splits. For each variable i, the Trend diagnostic is computed by fitting an ordinary least squares line over the combined training and validation segment: Trendi=|b^i|(Ttv−1)σi,Trend_i= | b_i | (T_tv-1 ) _i, (11) where b^i b_i is the fitted slope, TtvT_tv is the number of time steps in the combined segment, and σi _i is its standard deviation. This measures the magnitude of the fitted net change in units of the variable’s own scale. To capture scale changes between splits symmetrically, we define: Std.Shifti=max(σi,valσi,train,σi,trainσi,val),Std.\ Shift_i= ( _i,val _i,train, _i,train _i,val ), (12) where σi,train _i,train and σi,val _i,val are the corresponding standard deviations, and a value of one indicates no scale change. Mean Shift measures the change in level as MeanShifti=|μi,val−μi,train|σi,train,Mean\ Shift_i= | _i,val- _i,train | _i,train, (13) where μi,train _i,train and μi,val _i,val are the corresponding means. Finally, KSiKS_i denotes the Kolmogorov–Smirnov statistic between the training and validation marginal distributions as a broader measure of distribution shift. Each diagnostic is computed per variable, and the 95th percentile across variables is reported for each dataset. As practical selection criteria, we treat Trend, Mean Shift, and KS values above 0.20.2, and Std. Shift values above 1.21.2, as noticeable. RevIN is used when any diagnostic exceeds its threshold, while decomposition is applied when Trend exceeds 0.20.2. These diagnostics provide fast and interpretable guidance rather than definitive rules, while directly comparing validation performance with and without each preprocessing step can be more reliable, especially for borderline cases. As shown in Table 6, Double Pendulum and Lorenz Coupled are the only datasets that fall below all four thresholds. We therefore remove both RevIN and decomposition for these two ODE datasets and retain them for all remaining datasets. Table 7 further supports the RevIN choice at F=96F=96, where RevIN improves both MSE and MAE on ETTh2, ETTm2, and Solar, while disabling it substantially improves both metrics on Double Pendulum. The corresponding decomposition ablation is reported in Table 3. Appendix C Periodicity Analyses Figure 4: Traffic periodicity analysis, identifying 168 time steps (one week) as the strongest period. Figure 5: PEMS08 periodicity analysis, identifying 288 time steps (one day) as the strongest period. Figure 6: Double Pendulum periodicity analysis. No clear dominant period is identified. Figure 7: Lorenz Coupled periodicity analysis. No clear dominant period is identified, with even weaker periodicity than Double Pendulum. We conduct periodicity analyses for the Traffic, PEMS08, Double Pendulum, and Lorenz Coupled datasets (Lai et al., 2018; Song et al., 2020; Gilpin, 2021). Each analysis includes a Fourier transform, autocorrelation, and lag-copy test. Traffic and PEMS08 exhibit dominant periods of 168 and 288 time steps, corresponding to one week and one day, respectively. Both exceed the conventional look-back size of 96, supporting evaluation with larger T that preserve important periodic context. In contrast, while all seven benchmarks are periodic, the two chaotic ODE datasets have no clear dominant period. Without such regular temporal structure, their forecast errors grow much faster with horizon, and we therefore retain shorter horizons for evaluation. Together with their strong and distinct dependencies across variables, this contrasting temporal behavior makes the ODE datasets especially informative complements to the benchmarks. Appendix D Full Experimental Results and Additional Analyses Table 8: Detailed experimental results (mean ± std) on long-horizon benchmarks. Highlighting and underlining follow Table 1. Model Chameleon S-Mamba+ACN TimeFilter DUET SFNN FITS Dataset (F) MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE Traffic 96 ± 0.3309 ± 0.2247 ± 0.3653 ± 0.2563 ± 0.3428 ± 0.2412 ± 0.3493 ± 0.2367 ± 0.3484 ± 0.2478 ± 0.3820 ± 0.2674 ± 0.0018 ± 0.0021 ± 0.0009 ± 0.0007 ± 0.0003 ± 0.0003 ± 0.0008 ± 0.0003 ± 0.0002 ± 0.0003 ± 0.0008 ± 0.0011 192 ± 0.3485 ± 0.2328 ± 0.3755 ± 0.2664 ± 0.3609 ± 0.2505 ± 0.3691 ± 0.2456 ± 0.3697 ± 0.2582 ± 0.3944 ± 0.2713 ± 0.0026 ± 0.0014 ± 0.0029 ± 0.0019 ± 0.0003 ± 0.0002 ± 0.0010 ± 0.0002 ± 0.0031 ± 0.0013 ± 0.0003 ± 0.0003 336 ± 0.3651 ± 0.2418 ± 0.4055 ± 0.2746 ± 0.3775 ± 0.2600 ± 0.3922 ± 0.2592 ± 0.3853 ± 0.2661 ± 0.4077 ± 0.2777 ± 0.0039 ± 0.0009 ± 0.0014 ± 0.0003 ± 0.0003 ± 0.0002 ± 0.0003 ± 0.0002 ± 0.0014 ± 0.0006 ± 0.0003 ± 0.0004 720 ± 0.4105 ± 0.2647 ± 0.4587 ± 0.3002 ± 0.4141 ± 0.2804 ± 0.4366 ± 0.2778 ± 0.4310 ± 0.2903 ± 0.4454 ± 0.2969 ± 0.0047 ± 0.0026 ± 0.0045 ± 0.0016 ± 0.0004 ± 0.0003 ± 0.0038 ± 0.0002 ± 0.0010 ± 0.0007 ± 0.0004 ± 0.0006 Solar 96 ± 0.1533 ± 0.1979 ± 0.1743 ± 0.2218 ± 0.1827 ± 0.2282 ± 0.1647 ± 0.1875 ± 0.1553 ± 0.2001 ± 0.1880 ± 0.2400 ± 0.0017 ± 0.0020 ± 0.0027 ± 0.0019 ± 0.0100 ± 0.0058 ± 0.0013 ± 0.0007 ± 0.0027 ± 0.0010 ± 0.0004 ± 0.0008 192 ± 0.1750 ± 0.2198 ± 0.1867 ± 0.2519 ± 0.1974 ± 0.2396 ± 0.1864 ± 0.2190 ± 0.1855 ± 0.2521 ± 0.2059 ± 0.2483 ± 0.0029 ± 0.0019 ± 0.0035 ± 0.0046 ± 0.0140 ± 0.0097 ± 0.0044 ± 0.0002 ± 0.0019 ± 0.0045 ± 0.0002 ± 0.0006 336 ± 0.1899 ± 0.2291 ± 0.1993 ± 0.2617 ± 0.2056 ± 0.2466 ± 0.1950 ± 0.2345 ± 0.1935 ± 0.2638 ± 0.2194 ± 0.2546 ± 0.0024 ± 0.0022 ± 0.0026 ± 0.0019 ± 0.0059 ± 0.0056 ± 0.0017 ± 0.0023 ± 0.0024 ± 0.0031 ± 0.0003 ± 0.0003 720 ± 0.1951 ± 0.2346 ± 0.2141 ± 0.2771 ± 0.2225 ± 0.2606 ± 0.2077 ± 0.2189 ± 0.1980 ± 0.2574 ± 0.2211 ± 0.2552 ± 0.0019 ± 0.0018 ± 0.0021 ± 0.0032 ± 0.0151 ± 0.0096 ± 0.0015 ± 0.0006 ± 0.0011 ± 0.0015 ± 0.0001 ± 0.0003 Electricity 96 ± 0.1246 ± 0.2146 ± 0.1339 ± 0.2311 ± 0.1282 ± 0.2220 ± 0.1275 ± 0.2179 ± 0.1261 ± 0.2166 ± 0.1323 ± 0.2291 ± 0.0001 ± 0.0001 ± 0.0026 ± 0.0017 ± 0.0002 ± 0.0001 ± 0.0002 ± 0.0002 ± 0.0003 ± 0.0003 ± 0.0001 ± 0.0001 192 ± 0.1432 ± 0.2324 ± 0.1534 ± 0.2486 ± 0.1484 ± 0.2413 ± 0.1467 ± 0.2355 ± 0.1448 ± 0.2337 ± 0.1472 ± 0.2424 ± 0.0001 ± 0.0001 ± 0.0012 ± 0.0010 ± 0.0006 ± 0.0001 ± 0.0002 ± 0.0002 ± 0.0002 ± 0.0002 ± 0.0000 ± 0.0001 336 ± 0.1566 ± 0.2475 ± 0.1637 ± 0.2645 ± 0.1554 ± 0.2598 ± 0.1623 ± 0.2518 ± 0.1601 ± 0.2503 ± 0.1631 ± 0.2586 ± 0.0001 ± 0.0000 ± 0.0023 ± 0.0027 ± 0.0006 ± 0.0001 ± 0.0005 ± 0.0006 ± 0.0003 ± 0.0001 ± 0.0000 ± 0.0001 720 ± 0.1842 ± 0.2727 ± 0.1849 ± 0.2861 ± 0.1972 ± 0.2932 ± 0.1936 ± 0.2825 ± 0.1908 ± 0.2802 ± 0.2012 ± 0.2906 ± 0.0029 ± 0.0026 ± 0.0026 ± 0.0024 ± 0.0005 ± 0.0006 ± 0.0010 ± 0.0009 ± 0.0002 ± 0.0004 ± 0.0000 ± 0.0001 ETTh1 96 ± 0.3428 ± 0.3844 ± 0.3889 ± 0.4082 ± 0.3801 ± 0.3992 ± 0.3579 ± 0.3830 ± 0.3503 ± 0.3846 ± 0.3739 ± 0.3952 ± 0.0003 ± 0.0003 ± 0.0015 ± 0.0005 ± 0.0040 ± 0.0022 ± 0.0010 ± 0.0009 ± 0.0004 ± 0.0004 ± 0.0001 ± 0.0001 192 ± 0.3808 ± 0.4095 ± 0.4242 ± 0.4470 ± 0.4365 ± 0.4333 ± 0.3986 ± 0.4098 ± 0.3880 ± 0.4087 ± 0.4065 ± 0.4138 ± 0.0006 ± 0.0004 ± 0.0011 ± 0.0007 ± 0.0078 ± 0.0039 ± 0.0010 ± 0.0010 ± 0.0005 ± 0.0003 ± 0.0001 ± 0.0001 336 ± 0.4083 ± 0.4267 ± 0.4754 ± 0.4586 ± 0.4859 ± 0.4612 ± 0.4245 ± 0.4274 ± 0.4119 ± 0.4241 ± 0.4294 ± 0.4276 ± 0.0006 ± 0.0005 ± 0.0028 ± 0.0014 ± 0.0112 ± 0.0052 ± 0.0016 ± 0.0018 ± 0.0025 ± 0.0012 ± 0.0001 ± 0.0002 720 ± 0.4162 ± 0.4468 ± 0.5051 ± 0.5016 ± 0.4884 ± 0.4788 ± 0.4503 ± 0.4656 ± 0.4355 ± 0.4660 ± 0.4249 ± 0.4462 ± 0.0010 ± 0.0004 ± 0.0147 ± 0.0083 ± 0.0274 ± 0.0135 ± 0.0159 ± 0.0104 ± 0.0010 ± 0.0016 ± 0.0002 ± 0.0003 ETTh2 96 ± 0.2628 ± 0.3271 ± 0.3016 ± 0.3510 ± 0.3110 ± 0.3551 ± 0.2846 ± 0.3356 ± 0.2734 ± 0.3315 ± 0.2707 ± 0.3359 ± 0.0004 ± 0.0005 ± 0.0082 ± 0.0045 ± 0.0173 ± 0.0148 ± 0.0022 ± 0.0016 ± 0.0032 ± 0.0019 ± 0.0002 ± 0.0001 192 ± 0.3250 ± 0.3688 ± 0.3887 ± 0.4028 ± 0.3824 ± 0.4018 ± 0.3333 ± 0.3758 ± 0.3392 ± 0.3743 ± 0.3305 ± 0.3742 ± 0.0007 ± 0.0003 ± 0.0061 ± 0.0028 ± 0.0096 ± 0.0062 ± 0.0012 ± 0.0007 ± 0.0052 ± 0.0032 ± 0.0001 ± 0.0001 336 ± 0.3523 ± 0.3915 ± 0.4089 ± 0.4254 ± 0.4584 ± 0.4639 ± 0.3662 ± 0.4038 ± 0.3817 ± 0.4036 ± 0.3541 ± 0.3955 ± 0.0005 ± 0.0004 ± 0.0080 ± 0.0037 ± 0.0149 ± 0.0107 ± 0.0014 ± 0.0011 ± 0.0014 ± 0.0010 ± 0.0001 ± 0.0000 720 ± 0.3837 ± 0.4219 ± 0.4410 ± 0.4505 ± 0.4195 ± 0.4411 ± 0.3917 ± 0.4306 ± 0.3981 ± 0.4235 ± 0.3772 ± 0.4222 ± 0.0023 ± 0.0010 ± 0.0077 ± 0.0043 ± 0.0047 ± 0.0030 ± 0.0049 ± 0.0029 ± 0.0023 ± 0.0015 ± 0.0002 ± 0.0001 ETTm1 96 ± 0.2755 ± 0.3309 ± 0.3067 ± 0.3578 ± 0.2942 ± 0.3432 ± 0.2905 ± 0.3332 ± 0.2790 ± 0.3306 ± 0.3039 ± 0.3455 ± 0.0007 ± 0.0004 ± 0.0046 ± 0.0027 ± 0.0009 ± 0.0006 ± 0.0003 ± 0.0003 ± 0.0003 ± 0.0002 ± 0.0002 ± 0.0002 192 ± 0.3133 ± 0.3570 ± 0.3451 ± 0.3807 ± 0.3289 ± 0.3680 ± 0.3188 ± 0.3592 ± 0.3197 ± 0.3585 ± 0.3372 ± 0.3649 ± 0.0014 ± 0.0005 ± 0.0010 ± 0.0009 ± 0.0017 ± 0.0015 ± 0.0001 ± 0.0006 ± 0.0007 ± 0.0003 ± 0.0001 ± 0.0001 336 ± 0.3413 ± 0.3768 ± 0.3769 ± 0.4026 ± 0.3597 ± 0.3905 ± 0.3463 ± 0.3776 ± 0.3514 ± 0.3807 ± 0.3611 ± 0.3860 ± 0.0008 ± 0.0005 ± 0.0006 ± 0.0002 ± 0.0012 ± 0.0007 ± 0.0013 ± 0.0002 ± 0.0002 ± 0.0003 ± 0.0001 ± 0.0001 720 ± 0.3891 ± 0.4044 ± 0.4378 ± 0.4398 ± 0.4208 ± 0.4231 ± 0.3902 ± 0.4076 ± 0.3814 ± 0.4050 ± 0.3981 ± 0.4071 ± 0.0003 ± 0.0004 ± 0.0024 ± 0.0013 ± 0.0014 ± 0.0009 ± 0.0001 ± 0.0003 ± 0.0003 ± 0.0008 ± 0.0000 ± 0.0000 ETTm2 96 ± 0.1528 ± 0.2409 ± 0.1759 ± 0.2661 ± 0.1706 ± 0.2545 ± 0.1677 ± 0.2513 ± 0.1568 ± 0.2423 ± 0.1617 ± 0.2553 ± 0.0004 ± 0.0003 ± 0.0013 ± 0.0024 ± 0.0013 ± 0.0019 ± 0.0004 ± 0.0006 ± 0.0007 ± 0.0005 ± 0.0003 ± 0.0002 192 ± 0.2075 ± 0.2805 ± 0.2360 ± 0.3066 ± 0.2306 ± 0.2968 ± 0.2264 ± 0.2891 ± 0.2118 ± 0.2819 ± 0.2170 ± 0.2905 ± 0.0002 ± 0.0004 ± 0.0041 ± 0.0029 ± 0.0021 ± 0.0022 ± 0.0000 ± 0.0001 ± 0.0018 ± 0.0020 ± 0.0001 ± 0.0001 336 ± 0.2554 ± 0.3141 ± 0.2800 ± 0.3365 ± 0.2860 ± 0.3349 ± 0.2704 ± 0.3258 ± 0.2642 ± 0.3185 ± 0.2593 ± 0.3242 ± 0.0006 ± 0.0004 ± 0.0007 ± 0.0005 ± 0.0028 ± 0.0028 ± 0.0001 ± 0.0002 ± 0.0012 ± 0.0011 ± 0.0002 ± 0.0001 720 ± 0.3268 ± 0.3623 ± 0.3698 ± 0.3915 ± 0.3820 ± 0.3991 ± 0.3401 ± 0.3771 ± 0.3422 ± 0.3711 ± 0.3283 ± 0.3691 ± 0.0004 ± 0.0002 ± 0.0023 ± 0.0010 ± 0.0075 ± 0.0089 ± 0.0004 ± 0.0007 ± 0.0050 ± 0.0023 ± 0.0004 ± 0.0001 1st1^st Count ± 25 ± 20 ± 0 ± 0 ± 1 ± 0 ± 0 ± 4 ± 1 ± 3 ± 1 ± 1 w/ p<0.05p<0.05 ± 23 ± 18 ± 0 ± 0 ± 1 ± 0 ± 0 ± 3 ± 1 ± 2 ± 1 ± 1 Table 9: Detailed experimental results (mean ± std) on strongly dependent datasets. Highlighting and underlining follow Table 1. Model Chameleon Chameleon (CI) S-Mamba+ACN TimeFilter DUET Crossformer Dataset (F) MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE Double Pendulum 96 ± 0.0678 ± 0.1176 ± 0.1258 ± 0.1880 ± 0.0927 ± 0.1642 ± 0.0870 ± 0.1616 ± 0.2510 ± 0.3271 ± 0.1208 ± 0.1975 ± 0.0029 ± 0.0026 ± 0.0043 ± 0.0040 ± 0.0019 ± 0.0013 ± 0.0097 ± 0.0101 ± 0.0053 ± 0.0046 ± 0.0074 ± 0.0039 192 ± 0.3053 ± 0.3216 ± 0.3845 ± 0.3836 ± 0.3325 ± 0.3701 ± 0.3522 ± 0.3787 ± 0.5044 ± 0.5052 ± 0.4774 ± 0.4717 ± 0.0051 ± 0.0033 ± 0.0035 ± 0.0027 ± 0.0032 ± 0.0024 ± 0.0144 ± 0.0123 ± 0.0050 ± 0.0036 ± 0.0209 ± 0.0217 Lorenz Coupled 48 ± 0.0179 ± 0.0389 ± 0.0493 ± 0.0761 ± 0.0708 ± 0.1247 ± 0.0424 ± 0.0934 ± 0.1570 ± 0.2137 ± 0.0581 ± 0.1152 ± 0.0016 ± 0.0014 ± 0.0015 ± 0.0017 ± 0.0004 ± 0.0005 ± 0.0025 ± 0.0059 ± 0.0036 ± 0.0037 ± 0.0027 ± 0.0014 96 ± 0.1569 ± 0.1842 ± 0.3368 ± 0.3213 ± 0.3666 ± 0.3837 ± 0.2935 ± 0.3230 ± 0.5026 ± 0.4758 ± 0.3019 ± 0.3299 ± 0.0075 ± 0.0040 ± 0.0052 ± 0.0042 ± 0.0013 ± 0.0014 ± 0.0098 ± 0.0075 ± 0.0043 ± 0.0025 ± 0.0085 ± 0.0045 PEMS03 48 ± 0.0905 ± 0.1920 ± 0.1045 ± 0.1991 ± 0.0927 ± 0.1968 ± 0.1048 ± 0.2099 ± 0.1028 ± 0.2050 ± 0.0968 ± 0.1989 ± 0.0007 ± 0.0007 ± 0.0007 ± 0.0003 ± 0.0012 ± 0.0013 ± 0.0004 ± 0.0004 ± 0.0054 ± 0.0062 ± 0.0028 ± 0.0020 96 ± 0.1156 ± 0.2157 ± 0.1359 ± 0.2229 ± 0.2309 ± 0.3330 ± 0.1407 ± 0.2389 ± 0.1447 ± 0.2407 ± 0.1211 ± 0.2208 ± 0.0015 ± 0.0014 ± 0.0005 ± 0.0005 ± 0.0385 ± 0.0337 ± 0.0011 ± 0.0008 ± 0.0045 ± 0.0044 ± 0.0074 ± 0.0050 PEMS08 48 ± 0.0975 ± 0.1783 ± 0.1187 ± 0.1924 ± 0.1266 ± 0.1985 ± 0.1617 ± 0.2047 ± 0.1244 ± 0.1929 ± 0.1525 ± 0.2111 ± 0.0019 ± 0.0008 ± 0.0039 ± 0.0008 ± 0.0067 ± 0.0097 ± 0.0094 ± 0.0063 ± 0.0076 ± 0.0017 ± 0.0049 ± 0.0043 96 ± 0.1307 ± 0.1910 ± 0.1703 ± 0.2115 ± 0.1869 ± 0.2064 ± 0.2960 ± 0.3193 ± 0.1663 ± 0.2087 ± 0.2201 ± 0.2335 ± 0.0024 ± 0.0003 ± 0.0034 ± 0.0006 ± 0.0041 ± 0.0026 ± 0.0258 ± 0.0167 ± 0.0068 ± 0.0011 ± 0.0104 ± 0.0078 1st1^st Count ± 8 ± 8 ± 0 ± 0 ± 0 ± 0 ± 0 ± 0 ± 0 ± 0 ± 0 ± 0 w/ p<0.05p<0.05 ± 7 ± 7 ± 0 ± 0 ± 0 ± 0 ± 0 ± 0 ± 0 ± 0 ± 0 ± 0 (a) Test MSE versus training time, with bubble areas proportional to training peak memory. Results are averaged across the four ETT datasets. (b) Training peak memory versus number of variables on ETTh2. Figure 8: Efficiency and scalability analyses on ETT at F=96F=96 with a batch size of one. (a) Test MSE versus training time, with bubble areas proportional to parameter counts. Results are averaged across the four ETT datasets. (b) Test MSE versus training time on Traffic, with bubble areas proportional to parameter counts. Figure 9: Additional parameter-count analyses at F=96F=96 with a batch size of one. Tables 8 and 9 report the complete means and standard deviations across runs, together with statistical significance for each dataset-horizon combination, corresponding to Tables 2 and 1, respectively. Besides achieving the best mean MSE and MAE in most cases, Chameleon generally exhibits lower or comparable cross-run standard deviations than the other baselines, except for the lightweight and robust FITS (Xu et al., 2024). We further conduct the same training efficiency and scalability analyses on the ETT datasets, which contain only seven variables, complementing those on Traffic in Figure 3. Figure 8(a) reports results averaged across the four ETT datasets at F=96F=96 and shows that Chameleon achieves the best MSE while requiring the least training time and peak memory among all evaluated CD methods, with only the CI reference SFNN (Sun et al., 2025) requiring less of both. Figure 8(b) follows the same procedure as the Traffic analysis by duplicating all seven variables of ETTh2 to progressively increase their number. Chameleon’s peak memory again scales comparably to S-Mamba (Wang et al., 2025), whereas TimeFilter (Hu et al., 2025) and DUET (Qiu et al., 2025) grow considerably faster. Figure 9 further replaces peak memory with parameter counts for both ETT and Traffic, providing a complementary view of model efficiency. Together, these additional analyses suggest that Chameleon maintains competitive efficiency and strong scalability across datasets with different numbers of variables and model configurations, rather than reflecting a single favorable setting.