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[W1] Additional quantitative metrics for evaluating PS loss performance

To provide a more comprehensive evaluation, we incorporated additional metrics: Dynamic Time Warping (DTW), Time Distortion Index (TDI), and Pearson Correlation Coefficient (PCC), to assess the performance of PS loss. A detailed explanation of these metrics is provided below:

On the iTransformer model, PS loss consistently improves all three shape-aware metrics, indicating better structural alignment. On the ETTh2 dataset, the iTransformer trained with MSE achieves lower DTW scores, reflecting smaller numerical differences after optimal alignment. However, the higher TDI in this case suggests that the alignment requires more extensive temporal warping, which indicates greater structural distortion compared to the forecasts generated using PS loss.

Due to space limitations, we report the average metric values across all forecasting lengths. Please refer to (Table 6) for full results.

[Q1] Drawbacks of fixed-length patching

Fixed-length patching requires grid search over a predefined set of patch lengths, which introduces computational overhead and lacks adaptability across datasets.

In our ETTh1 experiments, the best-performing fixed patch size was found to be $P = 12$, which matches the patch length estimated by our adaptive patching strategy using the dominant period $p$. This demonstrates that our method can automatically identify an appropriate patch length without manual tuning.

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[E1] Comparison between GDW and grid-search

To evaluate GDW against a traditional grid search for selecting loss weights, we conducted experiments on the ETTh1 dataset using iTransformer. Both methods use the same overall PS loss weight $\lambda$. For grid search, coefficients $\alpha$, $\beta$, and $\gamma$ were chosen from {0.3, 0.5, 0.7, 1.0}, totaling 64 runs per prediction length. We report the best and average performance from grid search for comparison:

GDW achieves performance comparable to the best grid-searched results, while avoiding exhaustive tuning. Moreover, it reflects the intuition that the weights of correlation, variance, and mean should evolve dynamically during training to maintain balanced attention across all three loss terms, which static coefficients cannot assure.

[E2] Visualization of loss weights generated by GDW

We visualize the evolution of weights generated by GDW in Figure 1 and made the following observations:

• Weight Range: The weights for correlation, variance, and mean have different ranges, reflecting the inherent variation in their gradient magnitudes, which highlights the need for adaptive balancing.
• Weight Evolution: Correlation weight tends to decrease, while variance and mean weights increase. This does not imply shifting focus, but rather ensures equilibrium among components, allowing structural alignment to be preserved during optimization.

This confirms that GDW adaptively balances multiple objectives throughout training, improving stability and convergence.

[W1] Hyperparameter settings

The PS loss weight $\lambda$ is selected from {0.1, 0.3, 0.5, 0.7, 1.0, 2.0, 3.0, 5.0, 10.0}. The patch length threshold $\delta$ is chosen from {24, 48}.

[Q1] PS loss as a standalone objective

Results show that PS loss alone yields comparable accuracy to MSE+PS, demonstrating its effectiveness as a standalone optimization objective (Table 8).

[Q2] PS loss complexity on large datasets

We report the empirical runtime cost by measuring the average seconds per epoch during training using iTransformer across three datasets: ETTh1 (small), Weather (medium), and ECL (large):

Despite the added cost, the runtime increase is modest and justified by performance gains.

[Q3] Zero-shot performance across forecasting lengths

We extended zero-shot experiments based on iTransformer to forecast lengths of 96, 336, and 720, in addition to 192 (reported in the paper). PS loss improved accuracy in 33 out of 36 settings, confirming its robustness across both short- and long-term horizons. See Table 9 for details.

[Q4] Combination of PS loss and FreDF loss

FreDF focuses on frequency-domain alignment to mitigate label autocorrelation, while PS loss emphasizes patch-wise structural alignment in the time domain. Their goals are complementary. We evaluated MSE+FreDF, MSE+PS, and MSE+PS+FreDF using iTransformer as backbone.

Results show that combining the two losses yields either improved or comparable performance, supporting their compatibility. Table 10 provides full results.

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[Q1] PS loss on ultra-long-term and short-term forecasting

We evaluated PS loss on ultra-long-term (T = {1080, 1440, 1800, 2160}) and short-term (T = {12, 24, 48}) forecasting tasks using iTransformer and DLinear. We report averaged MSE results. Please refer to Table 1 and Table 2 for full results.

• Ultra-long-term: MSE reduced by 7.38% (iTransformer) and 11.01% (DLinear).

• Short-term: MSE reduced by 3.43% (iTransformer) and 1.60% (DLinear).

These results demonstrate the effectiveness of PS loss across both short-term and ultra-long-term forecasting tasks.

[Q2] Comparison with Ada-MSHyper and FAN

The contributions of Ada-MSHyper and FAN differ from our PS loss, as they address distinct challenges:

• Ada-MSHyper introduces a hypergraph transformer with a graph constraint loss to enhance multi-scale interaction modeling through hypergraph learning.
• FAN proposes a frequency-based adaptive normalization method to address both trend and seasonal non-stationary patterns.
• PS (Ours) presents a novel loss function that enhances structural alignment between predictions and ground truth via patch-wise statistical metrics.

We also combined PS loss with both methods. Please refer to Table 3 and Table 4 for the full results.

• Ada-MSHyper + PS. PS loss improves the average performance by 8.28% (MSE) and 4.68% (MAE).

• FAN + PS. When using DLinear as the backbone, PS loss further improves the average performance of FAN by 2.07% (MSE) and 2.31% (MAE).

These results demonstrate that while FAN and Ada-MSHyper focus on different aspects, PS loss can still further improve their performance by enhancing the structural alignment of the forecasted series.

[Q3] PS loss on LLM-based models for zero-shot forecasting

We conducted zero-shot forecasting experiments with LLM-based models: OFA, AutoTimes, and Time-LLM. PS loss improved forecasting accuracy with average MSE reductions of 2.07% (OFA), 6.33% (AutoTimes), and 7.29% (Time-LLM). Please refer to Table 5 for the full results.

[Q4] Time complexity analysis of PS loss

We analyze the time complexity of PS loss with respect to the forecast length $T$, number of channels $C$, and hidden dimension $d$. The overall complexity arises from three main components:

• Fourier-based Adaptive Patching (FAP): The complexity of this component is dominated by the Fast Fourier Transform (FFT), which is $O(T\log T)$ per channel. Since FFT is applied to each of the $C$ channels, the total time complexity is $O(C\cdot T \log T)$.
• Patch-wise Structural Loss (PS): The series is split into $N \approx \frac{2T}{P}$ patches, where $P$ is the patch length. Calculating correlation, variance, and mean over each patch requires $O(P)$ operations. Given $C \cdot N$ patches, the total complexity becomes $O(C \cdot N \cdot P) = O(C\cdot T)$.
• Gradient-based Dynamic Weighting (GDW): The gradient computation for each loss component w.r.t. the model output has shape $d \cdot T$, leading to a complexity of $O(d \cdot T)$.

Therefore, the overall time complexity of PS loss is $O(C \cdot T \log T + C \cdot T + d \cdot T)$.

[Q5] Performance gains in terms of percentage improvements

We now report percentage improvements throughout the paper and summarize them in updated Table.

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[M1] PS loss performance on non-periodic targets

When there is no clear periodic pattern in the target series, the dominant frequency—i.e., the one with the highest amplitude in the FFT spectrum—does not necessarily correspond to a true periodic component in the data. Instead, it typically falls into one of two categories:

• Short period ($p<\delta$): This often results from high-frequency components such as local fluctuations or noise. In this case, Equation (3) yields a short patch length, which still allows the model to focus on finer-grained local structure.
• Long period ($p>\delta$): This typically corresponds to low-frequency background components or weak global trends. In this case, the patch length will be capped at $\delta$ to prevent excessively large patches that could hinder fine-grained comparisons.

This design allows PS loss to adapt to both periodic and non-periodic series, using frequency content to guide patch granularity, while $\delta$ prevents overly large patches. On the Exchange dataset, which lacks a clear periodic pattern, PS loss still improved MSE by 6.43% on DLinear, demonstrating its effectiveness.

[M2] Purpose of mean loss refinement

The purpose behind this design is to first focus on aligning the shape of the series, and then gradually increase attention to the value offset. As correlation and variance alignment improve during training (indicated by increasing $c$, $v$), the model allocates more weight to $L_{mean}$ to refine value-level offsets. Ablation on iTransformer and TimeMixer (ETTh1) confirms its effectiveness.

[M3] Additional metrics for structural evaluation

Beyond MSE/MAE, we include additional shape-aware metrics: DTW, TDI, and PCC, using iTransformer for evaluation (Please see Reviewer Pd9N [W1] for metric details). PS loss consistently improves all three metrics, indicating better structural alignment (Table 6).

[E4] Time complexity analysis of PS loss

1. Theoretical time complexity analysis

We analyze the time complexity of PS loss with respect to the forecast length $T$, number of channels $C$, and hidden dimension $d$. The overall complexity arises from three main components:

• Fourier-based Adaptive Patching (FAP): The complexity of this component is dominated by the Fast Fourier Transform (FFT), which is $O(T\log T)$ per channel. Since FFT is applied to each of the $C$ channels, the total time complexity is $O(C\cdot T\log T)$.
• Patch-wise Structural Loss (PS): The series is split into $N \approx \frac{2T}{P}$ patches, where $P$ is the patch length. Calculating correlation, variance, and mean over each patch requires $O(P)$ operations. Given $C\cdot N$ patches, the total complexity becomes $O(C\cdot N\cdot P) = O(C\cdot T)$.
• Gradient-based Dynamic Weighting (GDW): The gradient computation for each loss component w.r.t. the model output has shape $d\cdot T$, leading to a complexity of $O(d\cdot T)$.

Therefore, the overall time complexity of PS loss is $O(C\cdot T\log T+C\cdot T+d\cdot T)$.

2. Actual run time overhead

We report the empirical runtime cost by measuring seconds per epoch using iTransformer across three datasets: ETTh1 (small), Weather (medium), and ECL (large):

Despite the added cost, the runtime increase is modest and justified by performance gains.

[E5] PS loss performance without MSE

Results show that PS loss alone yields comparable accuracy to MSE+PS, demonstrating its effectiveness as a standalone optimization objective (Table 8).