Self-Adaptive Scale Handling

• Self-Adaptive Scale-handling (AS) is a module that adaptively calibrates instance-level scale factors via Scale Calibrating (SC) and Scaling Selection (SS) to address scale heterogeneity.
• AS integrates into standard forecasting models like Informer, Transformer, and Performer, selectively balancing training stability with the preservation of scale semantics.
• Empirical results on fund sales datasets show that AS reduces WMAPE and RSE, outperforming vanilla scaling, global normalization, and other common preprocessing strategies.

Self-Adaptive Scale-handling (AS) is an add-on module for time series forecasting under scale heterogeneity, a regime in which different time series share similar temporal patterns but differ by orders of magnitude. In "Self-Adaptive Scale Handling for Forecasting Time Series with Scale Heterogeneity" (Zhang et al., 18 Jun 2026), AS is proposed as a self-adaptive scaling strategy that learns adaptive scale factors tailored to each input, with the stated goal of preserving semantic discriminability while reducing inverse-scaling errors. The module consists of Scale Calibrating (SC), which calibrates prior mean scaling factors through neural networks, and Scaling Selection (SS), which decides whether to apply calibration or retain the original factor. It is designed to integrate into standard forecasting backbones such as Informer, Transformer, Performer, and Autoformer.

1. Problem setting and the limits of conventional scaling

AS is formulated for multivariate time series forecasting with a shared backbone, especially in industrial scenarios such as fund sales forecasting, where different products can vary by up to three orders of magnitude while still exhibiting similar seasonality or trend patterns (Zhang et al., 18 Jun 2026). In this setting, joint modeling is desirable because the series share patterns and some series are data-sparse, but a large scale gap makes shared training difficult: models can overfit dominant scales and underrepresent small-scale signals.

The paper identifies two mainstream preprocessing strategies as insufficient. Global normalization or standardization assumes that the data distribution is roughly homogeneous. Under scale heterogeneity, low-scale series become over-compressed, which lowers signal-to-noise ratio and weakens gradients, so low-scale inputs become hard to learn. Window-based scaling, including DeepAR-style preprocessing by local mean, maps most windows to a similar numerical range, but the paper attributes two specific failures to this design: semantic discriminability is destroyed, because the model no longer observes that some series are inherently much larger than others; and inverse-scaling error is amplified, because small prediction errors in normalized space are multiplied back by large scale factors in original space.

The design objective is therefore not scale removal in the abstract, but a controlled compromise: scale enough to stabilize learning, but not so much that scale semantics vanish or inverse scaling becomes unstable. This framing is central to AS and distinguishes it from one-size-fits-all normalization rules.

2. Formal problem formulation and forecasting pipeline

The paper defines a historical window and future target as

$$\mathcal{X}^h=[x_1,x_2,\ldots,x_n]\in\mathbb{R}^{n\times c}, \qquad \mathcal{X}^f=[x_{n+1},x_{n+2},\ldots,x_{n+m}]\in\mathbb{R}^{m\times c}.$$

AS begins from a prior mean scaling factor $v_i$, following a DeepAR-style construction in which $\hat{t}$ is the number of non-zero time steps and the added $1$ ensures numerical stability (Zhang et al., 18 Jun 2026). Rather than directly replacing this prior with an unconstrained learned value, AS treats it as a baseline to be refined selectively.

The end-to-end pipeline is:

1. extract temporal features from the input window;
2. compute a calibrated scaling coefficient $\hat{\lambda}$ and a binary selection flag $d_i$;
3. apply either the calibrated scale factor $\hat{v}_i = v_i \times \hat{\lambda}$, or the original prior scale factor $v_i$, depending on the selection result;
4. forecast in scaled space and inverse-scale the output.

The final forecasting equation is

$$\hat{\mathcal{X}^f}= f(\mathcal{X}^h/\hat{v}_i)\times \hat{v}_i,$$

where $f(\cdot)$ is any forecasting backbone. The paper emphasizes that the same scalar $v_i$0 is used for all time steps in the instance. This choice suggests that AS targets instance-level scale adaptation rather than per-timestep amplitude modulation.

3. Internal structure: Scale Calibrating and Scaling Selection

AS is divided into two submodules with distinct roles (Zhang et al., 18 Jun 2026).

Submodule	Role	Principal output
SC	Learns a refined multiplicative correction to the prior mean scaling factor	$v_i$1
SS	Decides whether calibration should be applied at all for a given window	$v_i$2 or $v_i$3

Scale Calibrating (SC) receives the historical window $v_i$4. Feature extraction uses 3 convolution blocks, each consisting of padding, convolution, batch normalization, and ReLU, summarized by

$v_i$5

The resulting feature representation $v_i$6 is passed to a fully connected network $v_i$7, followed by a sigmoid: $v_i$8 The paper specifies that the first three FC layers form the MLP, the fourth layer discards bias, and the fifth layer outputs the sigmoid-gated coefficient. To stabilize training and testing, $v_i$9 is averaged over the batch to obtain $\hat{t}$0, and the calibrated factor is then

$\hat{t}$1

Scaling Selection (SS) exists because calibration is not uniformly beneficial. Some original prior mean scale factors $\hat{t}$2 may already be adequate, and forced calibration can harm performance. SS therefore uses a separate FC network $\hat{t}$3, with the same architecture as $\hat{t}$4 but independent parameters, to produce Bernoulli probabilities via a differentiable Gumbel-Softmax relaxation: $\hat{t}$5 The final scale factor is

$\hat{t}$6

Conceptually, SS implements a differentiable if-then gate over scaling strategy: calibration when useful, retention of the prior when not.

4. Learning dynamics and the rationale of adaptive scaling

The paper does not define a separate SC-only loss. Instead, SC is trained end-to-end through the forecasting loss together with the backbone, and the reported default training loss is WMAPE, not MSE, because WMAPE is scale-invariant and more suitable for heterogeneous scale data (Zhang et al., 18 Jun 2026). The ablation reported in the paper states that WMAPE is better, whereas MSE is more sensitive to residual scale differences and can be dominated by high-scale samples.

The conceptual claim for AS has two parts. First, it aims at preserving semantic discriminability. Unlike aggressive window scaling that forces every input near $\hat{t}$7, AS learns adaptive scale factors, so two series with very different original magnitudes need not be collapsed into the same normalized range. Second, it aims at reducing inverse-scaling error. If prediction errors in normalized space are multiplied by $\hat{t}$8, then smaller or better-calibrated scale factors reduce the final absolute error in original space.

SS is crucial in this argument. The paper states that, without SS, SC can over-calibrate some already good scale factors. In that sense, SC provides the main adaptive scaling benefit, while SS prevents harmful calibration. A plausible implication is that AS should be read less as a pure normalization scheme than as a learned decision layer over competing scale priors.

5. Backbone integration, datasets, and empirical findings

AS is presented as a plug-in scale-handling layer that can be inserted into several forecasting backbones without redesigning the backbone itself. The paper evaluates AS attached to Informer, Transformer, Performer, and Autoformer (Zhang et al., 18 Jun 2026). Variants compared include AS (full module), VS (vanilla scaling using $\hat{t}$9), nop (no preprocessing), and, in some comparisons, std (global standardization) and nor (global normalization).

The experiments use real-world fund sales data from Ant Fortune and Alipay:

• Fund$1$0: 66 fund sales datasets
• Fund$1$1: 106 fund sales datasets

Each dataset includes applying amount and redeeming amount. Series are jointly constructed using sliding windows with train/validation/test split 7:1:2. The forecasting setting uses input length 30 and forecast horizons 5 and 10. Reported metrics are WMAPE, RSE, and PMAE, with lower values being better.

The main empirical pattern is categorical rather than isolated to one architecture: AS performs best across all backbones and both datasets; VS is better than nop; and global normalization/standardization perform poorly, sometimes worse than no preprocessing. The paper further states that representative results from the tables show consistent gains: on Autoformer, AS improves over VS and nop on both Fund$1$2 and Fund$1$3; on Informer, AS yields clear reductions in WMAPE and RSE relative to VS and nop; and similar trends hold for Performer and Transformer.

The paper also notes that joint modeling of variables with similar patterns outperforms a fully multivariate setting, though detailed numbers are omitted. This observation is consistent with the broader premise of AS: scale heterogeneity is problematic precisely because shared structure remains worth exploiting.

6. Ablation evidence and technical scope

The ablation study isolates the contributions of SC and SS (Zhang et al., 18 Jun 2026). The notation $1$4 denotes SC only, using $1$5, whereas $1$6 denotes full AS, using $1$7. The reported findings are structurally important. In some cases, SC alone already improves over vanilla scaling. In other cases, SC alone over-calibrates and hurts performance. Adding SS restores robustness and yields the best overall results. This directly supports the internal division of labor claimed by the method: SC supplies calibration capacity; SS constrains that capacity.

Technically, AS occupies a specific position in the design space of forecasting systems. It is not a new backbone, not a replacement for sequence modeling, and not a global preprocessing rule applied uniformly across a dataset. It is an instance-level adaptive scaling module that mediates between a handcrafted prior scale factor and a learned correction. Because it leaves the forecasting function $1$8 generic, it is architecture-agnostic at the backbone level. Because it uses a single scalar $1$9 per instance, it does not attempt fine-grained temporal reweighting within a window.

Within that scope, the contribution of AS is to replace fixed scaling heuristics with a learned, selective mechanism that better balances training stability, preservation of scale semantics, and inverse-scaling accuracy. In the formulation of the paper, this is the core reason AS consistently improves forecasting performance on scale-heterogeneous time series.