A Unified Family of Percentage-Error Support Vector Regression Models with Symmetric Kernel Extensions

Abstract

Support vector regression (SVR) is a well-established kernel-based method for nonlinear regression. However, standard SVR formulations minimize absolute-error losses, which are not consistent with the scale-free, relative-accuracy criteria prevalent in forecasting and industrial applications, where uncertainty is typically expressed as a percentage. This study proposes a unified SVR framework that incorporates percentage-error loss functions and symmetry constraints. Four specific variants are introduced: $\epsilon$-SVR with mean absolute percentage error (MAPE), its symmetric kernel extension, least-squares SVR (LS-SVR) with root mean square percentage error (RMSPE), and its symmetric counterpart. Each variant is formulated in primal, Lagrangian, and dual forms using Karush–Kuhn–Tucker analysis. The principal structural finding is that percentage scaling results in sample-dependent box constraints for $\epsilon$-SVR and a target-weighted diagonal regularization matrix for LS-SVR. In contrast, symmetry modifies only the kernel matrix, leaving the optimization structure unchanged. Convexity and the representer theorem are preserved in all cases. Experiments are conducted on three cross-sectional datasets (Boston Housing, Diabetes, and Energy Efficiency) and a time-series dataset on Victorian electricity demand. Evaluation utilizes three metrics (MAPE, MASE, and MAAPE), 95% bootstrap confidence intervals, and paired Wilcoxon tests, and compares performance against percentage-error-native baselines (weighted-MAE, quantile regression, and log-target SVR), classical $\epsilon$-SVR, Random Forest, and XGBoost. An additional reflection-based experiment assesses the symmetric-kernel variants. The results demonstrate that optimizing for percentage error consistently improves the targeted metric without adversely affecting absolute-error metrics.

1. Introduction

Support vector regression (SVR) is recognized as one of the most influential kernel-based methods for nonlinear regression, integrating margin control, convex optimization, and the kernel trick within a unified framework [1,2].

The classical $\epsilon$-SVR, introduced by Vapnik [3], minimizes an absolute deviation that is insensitive to $\epsilon$, resulting in a convex quadratic program with dual variables. An alternative formulation, Least-Squares SVR (LS-SVR), replaces the hinge-type loss with a quadratic penalty [4], thereby reducing the optimization problem to a linear system.

Despite their theoretical elegance and computational maturity, both paradigms measure error in absolute terms. However, in many forecasting, energy, and economic applications, model performance is evaluated on a relative rather than an absolute scale. An error of fixed magnitude may be negligible for large targets yet unacceptable for small ones. Percentage-based metrics such as the mean absolute percentage error (MAPE) have therefore become standard evaluation tools in industrial practice [5,6,7,8], including energy load forecasting where SVR-based models are routinely evaluated under MAPE [9].

Percentage-error metrics provide a natural interpretation for uncertainty quantification: a percentage-error bound serves as a dimensionless confidence radius, independent of the target’s absolute magnitude and thus comparable across regimes spanning several orders of magnitude. Within the context of the Special Issue on Uncertainty Quantification Techniques in Statistics, Machine Learning and FinTech, the SVR family developed in this work introduces estimators whose primary loss function is a scale-free uncertainty criterion, thereby complementing the reliability-evaluation literature on mixed aleatoric and epistemic uncertainty [10].

Percentage-error reporting has become the de facto standard in electricity load forecasting [9,11,12,13], supply-chain demand forecasting [8,14], hydrological modeling [15], and financial forecasting under strictly positive mean regimes [16]. In each of these domains, the target-dependent box constraints of Equation (21) possess a direct operational interpretation: a $10\%$ error on a $10M$ transaction is equivalent in weight to a $10\%$ error on a $10K$ transaction, and the dual bound $|{\beta }_k| \le 100, C/y_k$ encodes this invariance. A companion R package, psvr [17], implements the entire model family.

Recent research demonstrates that $\epsilon$-SVR can be reformulated to incorporate percentage-based residuals directly into the primal constraints, resulting in a convex dual problem with sample-dependent box bounds [18]. This modification preserves both kernel admissibility and convexity, while aligning the optimization objective with scale-free performance criteria.

This paper extends [18] in four key directions. First, the LS-SVR paradigm under RMSPE (Theorem 3) is newly introduced. Second, the symmetric-kernel extensions (Theorems 2 and 4) are novel contributions. Third, new structural propositions address convexity, existence and uniqueness, spectral bounds, and a unified kernel-modification perspective (Propositions 1 and 2, Corollary 1, Theorem 5). Finally, the empirical evaluation expands beyond the single Boston Housing dataset to include three cross-sectional benchmarks and a time-series case study, with triple-metric reporting, percentage-error-native baselines, reflection-augmented symmetric-model validation, and a reproducibility appendix.

Independently, symmetry constraints have been incorporated into kernel-based regression to enforce structural properties of the target function, such as even or odd behavior, by modifying the kernel matrix accordingly [19,20]. When the kernel satisfies appropriate symmetry conditions, the resulting optimization problems remain convex and retain their representer structure [21].

The present work unifies these two research directions by developing a structural family of support vector regression models characterized by the following features:

• Percentage-based residuals;
• Optional symmetry constraints;
• Two optimization paradigms: $\epsilon$-insensitive and least squares.

This approach yields four models: the classical $\epsilon$ formulation with percentage loss, its symmetric counterpart, the least-squares percentage formulation, and its symmetric extension.

Rather than treating these as unrelated constructions, this work demonstrates that all four models constitute a structural family in which percentage scaling induces sample-dependent feasibility regions in the dual, while symmetry modifies the effective kernel matrix through symmetric kernel extensions.

In all cases, convexity and the representer theorem are preserved, and the resulting dual problems retain the structure of quadratic programs or linear systems in the LS paradigm.

Novelty and position in the literature. The percentage-error SVR family is positioned relative to four prior lines of research. De Myttenaere, Golden, Le Grand, and Rossi [7] demonstrated that MAPE minimization is equivalent to weighted-MAE minimization with per-sample weights $w_k=1/\left|y_k\right|$, and established universal consistency under the scope condition $\left|Y\right|\ge \lambda >0$. This framework is operationalized within both SVR paradigms, deriving the $1/y_k$ weight as target-dependent box bounds (Theorem 1) and as a target-weighted diagonal regularization matrix (Theorem 3). Suykens, De Brabanter, Lukas, and Vandewalle [22] introduced weighted LS-SVM with iteratively learned weights for robustness; the LS-percentage formulation presented here is structurally related but deterministic, as the weights $y_k^2/\mathrm{\Gamma }$ arise directly from the percentage-error objective. Evgeniou, Pontil, and Poggio [23] unified SVMs with Generalized Regularization Networks [24]; the present work specializes this approach to the percentage-error regime, and the symmetric-kernel extensions (Theorems 2 and 4) represent a percentage-error specialization of the virtual-examples equivalence of Niyogi, Girosi, and Poggio [25]. The fourth line of prior work is the quantile-regression and $L_1$-SVR lineage [26,27,28].

Since MAPE is equivalent to a weighted MAE [7], and MAE corresponds to the pinball loss at quantile $\tau =1/2$, the percentage-error SVR occupies a position in the following hierarchy: pinball-loss quantile SVR → MAE-loss SVR $(\tau =1/2)$→ weighted-MAE SVR $(w=1/|y\left|\right)$→ MAPE-loss SVR. This work operationalizes the final step in both the $\epsilon$-SVR and LS-SVR paradigms.

The primary contribution of this paper is structural. Unified primal formulations are provided, their associated Lagrangians are derived, the corresponding dual problems are established, and the influence of percentage scaling and symmetry on the feasible set is characterized. The analysis clarifies how scale-free residuals and kernel symmetrization interact within the SVR framework without introducing additional non-convexity or requiring solver modifications. Specifically, the paper contributes:

• C1. Unified primal–dual derivations for four SVR models under percentage-error loss (MAPE and RMSPE) and optional symmetry constraints, with formal Karush–Kuhn–Tucker analyses.
• C2. Two structural mechanisms by which percentage scaling modifies the optimization are identified: sample-dependent box constraints $|{\beta }_k| \le 100, C/y_k$ in the $\epsilon$-paradigm, and the target-weighted diagonal $Y_{\mathrm{\Gamma }}=\mathrm{diag}(y_k^2/\mathrm{\Gamma })$ in the LS paradigm.
• C3. Theorem 5 proves that symmetry constraints affect only the effective kernel ($K\to K_s$, $\mathrm{\Omega }\to {\mathrm{\Omega }}_s$) without altering dual variables, box constraints, or regularization matrices.
• C4. Structural properties: convexity (Proposition 1), existence-uniqueness of the LS-SVR linear system (Proposition 2), a spectral lower bound (Corollary 1), and a capacity bound ${max}_k\left|{\beta }_k\right|\le 100, C/\lambda$ yielding a generalization guarantee via the Steinwart–Christmann framework [29].
• C5. Symmetry introduces at most a constant-factor overhead on kernel evaluations; training scales as $O\left(N^3\right)$ per solve, in line with classical SVR solvers [30] (Section 3.6).
• C6. Empirical validation on three cross-sectional datasets (Boston Housing [31], Diabetes [32], Energy Efficiency [33]) and a time-series case study on Victorian electricity demand, with triple-metric reporting (MAPE, MASE, MAAPE), 95 bootstrap confidence intervals, and paired Wilcoxon tests against percentage-error-native baselines.
• C7. A reflection-augmented 2 × 2 ablation design (Section 4.4) that isolates the kernel-modification contribution from data augmentation.
• C8. A reproducibility appendix and companion R package psvr [17] enabling end-to-end replication of every reported result.
• C9. An SMO solver (Algorithm A1, Appendix A.5) for the percentage-error $\epsilon$-SVR quadratic programs of Theorems 1 and 2, generalizing the WSS3 working-set selection of Fan, Chen, and Lin [34] to the sample-dependent box-bound regime and supporting the symmetrized kernel $K_s$. Full derivation and boundary-case analysis are in the companion technical report [35].

The remainder of this paper is as follows: Section 2 establishes the mathematical setting. Section 3 derives the four dual formulations and their structural properties. Section 4 presents empirical validation across three cross-sectional datasets and a time-series case study. Section 5 discusses limitations and open directions. Appendix A provides complete KKT derivations and training-algorithm pseudocode, while Appendix B documents the experimental protocol for full reproducibility.

2. Mathematical Preliminaries

2.1. Notation and Setting

Let

$$\mathcal{D}={\left\{(x_k,y_k)\right\}}_{k=1}^N$$

(1)

be a regression dataset with $x_k\in {\mathbb{R}}^p$ and targets $y_k\in \mathbb{R}$.

Let $\phi :{\mathbb{R}}^p\to \mathcal{H}$ be a feature map into a (possibly infinite-dimensional) Hilbert space $\mathcal{H}$. The regression function is defined as

$$f\left(x\right)={\omega }^{\top }\phi \left(x\right)+b,$$

(2)

where $\omega \in \mathcal{H}$ and $b\in \mathbb{R}$.

The associated kernel function is

$$K(x_i,x_j)=\phi {\left(x_i\right)}^{\top }\phi \left(x_j\right).$$

(3)

We denote by $\mathrm{\Omega }\in {\mathbb{R}}^{N\times N}$ the kernel matrix with entries

$${\mathrm{\Omega }}_{kl}=K(x_k,x_l),k,l=1,\dots ,N.$$

(4)

Additionally, we denote by $\mathbf{1}={[1,\dots ,1]}^{\top }\in {\mathbb{R}}^N$ the all-ones vector and by $\mathbf{y}={[y_1,\dots ,y_N]}^{\top }\in {\mathbb{R}}^N$ the target vector.

Definition 1

(Percentage Residual). For $y_k\ne 0$, the percentage residual is defined as

$$r_k={\displaystyle \frac{y_k-f\left(x_k\right)}{y_k}}.$$

(5)

When expressed as a percentage, the residual is $100r_k$.

2.2. Assumptions

Throughout this work, we impose the following conditions.

Assumption 1

(Positive definite kernel). The kernel function $K:{\mathbb{R}}^p\times {\mathbb{R}}^p\to \mathbb{R}$ is symmetric and positive definite.

Assumption 2

(Strictly positive targets). All target values satisfy $y_k>0$ for $k=1,\dots ,N$, ensuring that percentage residuals (5) are well-defined.

Assumption 3

(Kernel symmetry). For symmetric models, the kernel additionally satisfies

$$K(-x_i,x_j)=K(x_i,-x_j),K(-x_i,-x_j)=K(x_i,x_j).$$

(6)

Remark 1

(Kernel examples satisfying Assumption 3). Common kernels satisfying (6) include the Gaussian (RBF) kernel $K(x,x^{\prime })=exp\left(-\parallel x-x^{\prime }{\parallel }^2/\left(2{\sigma }^2\right)\right)$ and polynomial kernels of even degree $K(x,x^{\prime })={(x^{\top }{x}^{\prime }+c)}^{2m}$.

2.3. $\epsilon$-Support Vector Regression (Classical)

Classical $\epsilon$-SVR [1,3] solves

$$\underset{\omega ,b,\xi ,{\xi }^{*}}{min}{\displaystyle \frac{1}{2}}{\parallel \omega \parallel }^2+C{\displaystyle \sum _{k=1}^N}({\xi }_k+{\xi }_k^{*})$$

(7)

subject to

$$\left\{\begin{array}{c}{y}_k-{\omega }^{\top }\phi \left(x_k\right)-b\le \epsilon +{\xi }_k,\hfill \\ {\omega }^{\top }\phi \left(x_k\right)+b-y_k\le \epsilon +{\xi }_k^{*},\hfill \\ {\xi }_k,{\xi }_k^{*}\ge 0,\hfill \end{array}\right.k=1,\dots ,N.$$

(8)

Here, $C>0$ is the regularization parameter balancing model complexity against training error, and $\epsilon \ge 0$ defines the insensitivity zone within which no penalty is incurred.

Dualization via Lagrangian methods yields the convex quadratic program

$$\underset{\mathit{\beta }}{max}-{\displaystyle \frac{1}{2}}{\displaystyle \sum _{k,l=1}^N}{\beta }_k{\beta }_{l}K(x_k,x_l)+{\displaystyle \sum _{k=1}^N}{\beta }_k{y}_k-\epsilon {\displaystyle \sum _{k=1}^N}\left|{\beta }_k\right|,$$

(9)

subject to

$${\displaystyle \sum _{k=1}^N}{\beta }_k=0,\left|{\beta }_k\right|\le C,k=1,\dots ,N,$$

(10)

where ${\beta }_k={\alpha }_k-{\alpha }_k^{*}$ and $|{\beta }_k|={\alpha }_k+{\alpha }_k^{*}$.

Remark 2

(Dual form). Problem (9) and (10) is obtained via standard Lagrangian duality applied to (7) and (8); see [1] for a detailed derivation. The dual variables ${\alpha }_k,{\alpha }_k^{*}\ge 0$ are Lagrange multipliers for the inequality constraints in (8).

2.4. Least-Squares Support Vector Regression (Classical)

The LS-SVR formulation [4,36,37,38] replaces the hinge-type loss with a quadratic penalty:

$$\underset{\omega ,b,\mathbf{e}}{min}{\displaystyle \frac{1}{2}}{\parallel \omega \parallel }^2+{\displaystyle \frac{\mathrm{\Gamma }}{2}}{\displaystyle \sum _{k=1}^N}{e}_k^2$$

(11)

subject to

$$y_k={\omega }^{\top }\phi \left(x_k\right)+b+e_k,k=1,\dots ,N,$$

(12)

where $\mathrm{\Gamma }>0$ is the regularization parameter and $e_k\in \mathbb{R}$ are unconstrained error variables.

The Karush–Kuhn–Tucker stationarity conditions yield

$$\omega ={\displaystyle \sum _{k=1}^N}{\alpha }_k\phi \left(x_k\right),{\displaystyle \sum _{k=1}^N}{\alpha }_k=0,e_k={\displaystyle \frac{{\alpha }_k}{\mathrm{\Gamma }}},$$

(13)

where ${\alpha }_k\in \mathbb{R}$ are Lagrange multipliers for the equality constraints (12).

Substituting these conditions into (12) and applying the kernel trick yields the linear system

$$\left[\begin{array}{cc}0& {\mathbf{1}}^{\top }\\ \mathbf{1}& \mathrm{\Omega }+{\mathrm{\Gamma }}^{-1}I\end{array}\right]\left[\begin{array}{c}b\\ \mathit{\alpha }\end{array}\right]=\left[\begin{array}{c}0\\ \mathbf{y}\end{array}\right],$$

(14)

where $I\in {\mathbb{R}}^{N\times N}$ is the identity matrix.

Remark 3

(Comparison with ridge regression). LS-SVR is equivalent to kernel ridge regression [21]. The regularization term ${\mathrm{\Gamma }}^{-1}I$ in (14) corresponds to Tikhonov regularization in the dual space.

2.5. Symmetry in Kernel-Based Regression

Refs. [19,20] introduced symmetry constraints in kernel-based regression to exploit known structural properties of the target function. The following definition formalizes the notion used throughout this work.

Definition 2

(Functional symmetry). A regression function f satisfies symmetry of type $a\in \{-1,1\}$ if

$$f\left(x\right)=af(-x)\mathit{for}\mathit{all}x\in {\mathbb{R}}^p.$$

(15)

The case $a=1$ corresponds to even symmetry, and $a=-1$ to odd symmetry.

Under Assumption 3, functional symmetry (15) can be enforced in kernel-based regression by employing the symmetrized kernel

$$K_s(x_i,x_j)={\displaystyle \frac{1}{2}}\left(K(x_i,x_j)+aK(x_i,-x_j)\right),$$

(16)

and the corresponding symmetrized Gram matrix

$${\mathrm{\Omega }}_s={\displaystyle \frac{1}{2}}\left(\mathrm{\Omega }+a{\mathrm{\Omega }}^{*}\right),{\mathrm{\Omega }}_{kl}^{*}=K(x_k,-x_l).$$

(17)

Notation 1

(Notational convention). The factor $\frac{1}{2}$ in (16) is adopted uniformly throughout this paper, so that the quadratic coefficients of the symmetric and non-symmetric duals match: $-\frac{1}{2}$ in both Theorem 1 (non-symmetric, Section 3.1) and Theorem 2 (symmetric, Section 3.2), and $\frac{1}{2}$ in the Gram sums $\mathrm{\Omega }+a{\mathrm{\Omega }}^{*}$ used in Theorem 4 (Section 3.4). This homogenization is notational only; the algebraic content of every theorem, remark, and proof is preserved.

Remark 4

(Preservation of positive definiteness). If K is positive definite and satisfies Assumption 3, then $K_s$ defined in (16) remains positive definite [29]. This follows from the linearity of positive definiteness and the symmetry conditions (6).

Remark 5

(Boundary case: odd symmetry with Gaussian RBF). For the Gaussian RBF kernel $K(x,x^{\prime })=exp\left(-\parallel x-x^{\prime }{\parallel }^2/\left(2{\sigma }^2\right)\right)$ and odd symmetry $a=-1$, the symmetrized kernel evaluated on the diagonal satisfies

$$K_s(x,x)={\displaystyle \frac{1}{2}}\left(K(x,x)-K(x,-x)\right)={\displaystyle \frac{1}{2}}\left(1-exp\left(-2\parallel x{\parallel }^2/{\sigma }^2\right)\right),$$

which vanishes at the origin: $K_s(0,0)=0$. Consequently, the Gram matrix ${\mathrm{\Omega }}_s$ is positive semidefinite but not strictly positive definite when one of the training points $x_k$ equals the origin (or approaches it). For the ε-SVR variant (Theorem 2, Section 3.2), the non-strict positivity may produce a non-unique dual minimizer; in practice, either the ${\ell }_2$ capacity penalty $\frac{1}{2}{\parallel \omega \parallel }^2$ in the primal or a small ridge regularizer $\delta I$ (with $\delta \sim {10}^{-6}$) added to ${\mathrm{\Omega }}_s$ restores strict positivity and uniqueness. For the LS-SVR variant (Theorem 4 and Proposition 2, Section 3.5), the target-weighted diagonal $Y_{\mathrm{\Gamma }}\succ 0$ ensures that the sum ${\mathrm{\Omega }}_s+Y_{\mathrm{\Gamma }}$ is strictly positive definite, so the bordered matrix in Theorem 4 is nonsingular without additional regularization. The even-symmetry case $a=+1$ does not suffer from this boundary phenomenon: $K_s(x,x)=\frac{1}{2}\left(K(x,x)+K(x,-x)\right)\ge \frac{1}{2}$ for all x under the Gaussian RBF.

2.6. Percentage-Error Structure

Replacing classical residuals by percentage residuals (5) introduces two structural modifications to the optimization problems, which are developed formally in Section 3:

• In the $\epsilon$-SVR paradigm, the box constraints on dual variables become sample-dependent, scaling inversely with target magnitude (Theorem 1).
• In the LS-SVR paradigm, a target-weighted diagonal regularization term appears in the system matrix, scaling quadratically with target values (Theorem 3).

Section 3 formalizes these structural changes by introducing four models that result from various combinations of percentage-error loss functions and optional symmetry constraints. The $\epsilon$-SVR case with MAPE was initially presented in [18]; the current study extends and unifies this result within a more comprehensive framework.

3. Main Results

This section presents the four dual formulations of the percentage-error support vector regression (SVR) family. These models are distinguished by two factors: the optimization paradigm, which may be either $\epsilon$-insensitive or least squares, and the inclusion or exclusion of symmetry constraints. The assumption $y_k>0$ (Assumption 2) is maintained throughout to ensure that percentage residuals are well defined.

3.1. $\epsilon$-SVR Under Percentage Error

Intuition. The classical $\epsilon$-SVR framework of Section 2.3 minimizes an $\epsilon$-insensitive absolute deviation and yields a dual with uniform box constraints $\left|{\beta }_k\right|\le C$—a single complexity budget shared across all training samples. Replacing absolute residuals by percentage residuals (Definition 1) forces each constraint to scale inversely with the target magnitude: samples with small targets receive tighter feasibility regions, samples with large targets receive looser ones. The effect is a re-allocation of the $\epsilon$-SVR capacity budget from absolute to relative terms, yielding the target-dependent box bounds $\left|{\beta }_k\right|\le 100C/y_k$ that characterize Theorem 1 below. The same primal–dual machinery of classical $\epsilon$-SVR (Lagrangian, KKT stationarity, kernel trick) carries through; only the structure of the box constraints changes.

We consider the primal problem

$$\underset{\omega ,b,\mathit{\xi },{\mathit{\xi }}^{*}}{min}{\displaystyle \frac{1}{2}}{\parallel \omega \parallel }^2+C{\displaystyle \sum _{k=1}^N}({\xi }_k+{\xi }_k^{*})$$

(18)

subject to, for $k=1,\dots ,N$,

$$\left\{\begin{array}{c}{\displaystyle \frac{100}{y_k}}\left(y_k-{\omega }^{\top }\phi \left(x_k\right)-b\right)\le \epsilon +{\xi }_k,\hfill \\ {\displaystyle \frac{100}{y_k}}\left({\omega }^{\top }\phi \left(x_k\right)+b-y_k\right)\le \epsilon +{\xi }_k^{*},\hfill \\ {\xi }_k,{\xi }_k^{*}\ge 0.\hfill \end{array}\right.$$

(19)

Remark 6

(Strong duality via Slater’s condition). The primal problem (18) and (19) is a convex quadratic program with affine constraints. For any choice of $(\omega ,b)$, setting ${\xi }_k,{\xi }_k^{*}>max\left(0,100(y_k-{\omega }^{\top }\phi \left(x_k\right)-b)/y_k-\epsilon \right)$ yields a strictly feasible interior point (finite because $y_k>0$ by Assumption 2). By ([30], Section 5.2.3) —or equivalently ([39], Theorem 28.2)—Slater’s condition is satisfied, strong duality holds, and the primal–dual equivalence stated in Theorem 1 below is exact with zero duality gap.

Theorem 1

(Dual formulation of $\epsilon$-SVR with MAPE). Under Assumption 1, the primal problem (18) and (19) is equivalent to the dual problem

$$\underset{\mathit{\beta }}{max}-{\displaystyle \frac{1}{2}}{\displaystyle \sum _{k,l=1}^N}{\beta }_k{\beta }_{l}K(x_k,x_l)+{\displaystyle \sum _{k=1}^N}{\beta }_k{y}_k-{\displaystyle \frac{\epsilon }{100}}{\displaystyle \sum _{k=1}^N}\left|{\beta }_k\right|y_k,$$

(20)

subject to

$${\displaystyle \sum _{k=1}^N}{\beta }_k=0,\left|{\beta }_k\right|\le {\displaystyle \frac{100C}{y_k}},k=1,\dots ,N,$$

(21)

where ${\beta }_k={\alpha }_k-{\alpha }_k^{*}$ and $|{\beta }_k|={\alpha }_k+{\alpha }_k^{*}$.

Proof. 

The Lagrangian for Equations (18) and (19) is constructed using multipliers ${\alpha }_k,{\alpha }_k^{*}\ge 0$ for the inequality constraints and ${\eta }_k,{\eta }_k^{*}\ge 0$ for the slack-variable constraints. The Karush–Kuhn–Tucker stationarity conditions yield the representer theorem $\omega ={\sum }_{k=1}^N({\alpha }_k-{\alpha }_k^{*})\phi \left(x_k\right)$ and the balancing condition ${\sum }_{k=1}^N({\alpha }_k-{\alpha }_k^{*})=0$.

The percentage scaling in constraints (19) introduces target-dependent box bounds $0\le {\alpha }_k, {\alpha }_k^{*}\le 100C/y_k$ via the complementary-slackness conditions. Eliminating the primal variables and applying the kernel trick $K(x_k,x_l)=\phi {\left(x_k\right)}^{\top }\phi \left(x_l\right)$ yields Equations (20) and (21). A complete derivation is provided in Appendix A.1.    □

Remark 7

(Percentage effect in $\epsilon$-SVR). Percentage scaling is structurally manifested through the target-dependent box constraints in (21). Smaller target values result in tighter bounds $\left|{\beta }_k\right|\le 100C/y_k$, which increases sensitivity to absolute errors on low-magnitude observations. This mechanism, within the SVR framework, realizes the weighted-MAE equivalence described by [7]: minimizing MAPE corresponds to minimizing a weighted MAE with per-instance weights $w_k=1/\left|y_k\right|$, and the dual box-bound $100C/y_k$ represents the ε-SVR analog of that weight.

Remark 8

(Weighted ${\ell }_1$ structure). The term ${\sum }_k\left|{\beta }_k\right|,y_k$ in (20) represents a weighted ${\ell }_1$ penalty on the dual coefficients, which demonstrates structural similarity to weighted LASSO formulations in reproducing kernel Hilbert spaces [21]. As ε approaches zero, this penalty disappears, and the formulation simplifies to the unweighted quadratic dual with target-dependent box constraints.

Remark 9

(Scope of $y_k>0$ and extensions to near-zero targets). Assumption 2 restricts the percentage-error framework to ratio-scale targets with a structural lower bound, consistent with the scope conditions identified by [5] and formalized by [7] (where consistency requires $\left|Y\right|\ge \lambda >0$ almost surely). This restriction covers most business, energy, and demand-forecasting applications in which percentage-based reporting is standard [8,9]. For targets that may approach zero, three extensions retain most of the framework’s properties:

1.

Offset regularization:Substitute $1/y_k$ with $1/\left(\right|y_k|+\delta )$, where $\delta >0$ is small. This approach bounds the effective weight at $1/\delta$, introducing a bias of order $\delta /|y_k|$ for small targets.

2.

Truncation:Limit the loss calculation to the subsample $k:\left|y_k\right|\ge \lambda$, with λ reported explicitly. This method aligns with the truncated-MAPE convention of [7] and is standard in survival analysis.

3.

MAAPE substitution:Replace the absolute-percentage term $|r_k|$ with $arctan\left(\right|r_k\left|\right)$, [40], which is bounded, robust to zero, and concave in the residual. The structural analysis in Section 3 applies to MAAPE with the same target-dependent box constraints since arctan is monotonic and differentiable except at zero.

The development of a general percentage-error framework that accommodates potentially zero targets remains an open research direction, especially through the use of a differentiable, regularized MAPE surrogate with an explicit Bayes-optimal characterization.

Remark 10

(Known limitations of MAPE). MAPE is widely recognized to exhibit two pathological behaviors relevant to regression and forecasting tasks:

1.

Asymmetry:MAPE penalizes over-predictions differently from under-predictions, systematically favoring models that slightly under-forecast [5,6,41].

2.

Unboundedness near zero:MAPE diverges as $y_k\to 0$, precluding its use on datasets where targets can approach zero [42].

This framework explicitly inherits these limitations through Assumption 2 ($\left|Y\right|\ge \lambda >0$). The three canonical workarounds, offset regularization, truncation, and MAAPE substitution, are addressed in Remark 9. In applications where MAPE is known to underperform, such as intermittent demand or target-crossing forecasts, practitioners are advised to use the Mean Absolute Scaled Error (MASE) [5] or MAAPE [40] as complementary or alternative metrics. Section 4 adopts triple-metric reporting (MAPE, MASE, and MAAPE) to empirically address these issues.

Remark 11

(Capacity bound and generalization). Under the scope condition $\left|Y\right|\ge \lambda >0$ of Assumption 2 and the target-dependent box constraint (21), the effective capacity bound is

$$\underset{k}{max},\left|{\beta }_k\right|\le {\displaystyle \frac{100,C}{\lambda }},$$

In other words, the dual coefficients are uniformly bounded by a constant that depends solely on C and λ. This result places the percentage-error ε-SVR within the capacity-control regime described in ([29], Chapter 6). For any fixed kernel with bounded Rademacher complexity ${\mathfrak{R}}_n\left(\mathcal{F}\right)=O(1/\sqrt{n})$, the excess MAPE risk of the empirical minimizer is, with high probability, bounded by a term of order $O\left((C/\lambda )\sqrt{log(1/\delta )/n}\right)$ at confidence level $1-\delta$. The explicit dependence on $C/\lambda$ makes the dynamic-range cost of the percentage-error scope condition quantitatively apparent in the generalization bound, thereby complementing the qualitative scope condition of Assumption 2. For the LS-SVR variant in Theorem 3 (Section 3.3), the same argument applies, with Γ replacing C and the target-weighted diagonal $Y\mathrm{\Gamma }$ replacing the scalar box bound.

Remark 12

(Statistical interpretation of the percentage-error loss). The percentage-error loss possesses a direct statistical interpretation. In the context of the multiplicative-noise generative model,

$$Y_k=\mu \left(X_k\right)(1+{\eta }_k),$$

where ${\eta }_k$ represents symmetric mean-zero noise (equivalently, Gaussian multiplicative noise in the log-target regime), the minimizer of the expected MAPE is the weighted conditional median of Y given X, with weights $1/\left|Y\right|$ ([7], Proposition 1). Consequently, MAPE serves as the natural quasi-likelihood for this noise model as described in [43]. Under the capacity-control conditions outlined in Remark 11, the ε-support vector regression (SVR) dual of Theorem 1 is Fisher consistent in the limit as $\epsilon \to 0$. As the ε-insensitive band collapses, the empirical minimizer converges in probability to the Bayes-optimal weighted median, provided the $\left|Y\right|\ge \lambda >0$ scope condition holds. For the least-squares SVR (LS-SVR) variant (Theorem 3, Section 3.3), the target-dependent diagonal $Y_{\mathrm{\Gamma }}=\mathrm{diag}(y_k^2/\mathrm{\Gamma })$ corresponds to the posterior precision matrix for heteroscedastic Gaussian noise with variance ${\sigma }_k^2=y_k^2/\mathrm{\Gamma }$. Thus, the LS-SVR constitutes a heteroscedastic Gaussian-process regression in the sense of ([44], Chapter 2), with the multiplicative-noise structure resulting in noise variance proportional to the squared target. This setting aligns with the lognormal-noise regime commonly encountered in energy-load, financial-returns, and streamflow forecasting.

3.2. Symmetric $\epsilon$-SVR Under Percentage Error

If the target function demonstrates even or odd symmetry, as observed in energy load profiles centered on a reference temperature or in financial return series under sign reversal, incorporating this structure at the kernel level can improve generalization [19,20]. The following result incorporates this prior structural knowledge into the $\epsilon$-SVR percentage-error framework by requiring that the learned function f satisfies the symmetry condition defined in Definition 2 (Section 2.5).

According to Theorem 1, the non-symmetric $\epsilon$-SVR dual formulation couples training points solely through the kernel matrix $\mathrm{\Omega }$. If the target function is known a priori to satisfy $f\left(x\right)=af(-x)$, this structural property can be incorporated at the kernel level by replacing each kernel evaluation $K(x_i,x_j)$ with its symmetrized form $K_s(x_i,x_j)=\frac{1}{2}\left(K(x_i,x_j)+aK(x_i,-x_j)\right)$ (Equation (16) in Section 2.5). This method averages the training signal at the kernel level over the $\{+x,-x\}$ orbit, reducing the effective degrees of freedom without altering the solver structure. Theorem 2 formalizes this result: the dual problem retains the structure of Theorem 1, with K replaced by $K_s$ and no additional modifications.

The following symmetry constraint is enforced:

$${\omega }^{\top }\phi \left(x_k\right)=a{\omega }^{\top }\phi (-x_k),a\in \{-1,1\},k=1,\dots ,N.$$

(22)

Theorem 2

(Dual formulation of symmetric $\epsilon$-SVR with MAPE). Under Assumptions 1–3, the primal problem (18)–(19) augmented with the symmetry constraint (22) is equivalent to the dual problem

$$\underset{\mathit{\beta }}{max}-{\displaystyle \frac{1}{2}}{\displaystyle \sum _{k,l=1}^N}{\beta }_k{\beta }_l{K}_s(x_k,x_l)+{\displaystyle \sum _{k=1}^N}{\beta }_k{y}_k-{\displaystyle \frac{\epsilon }{100}}{\displaystyle \sum _{k=1}^N}\left|{\beta }_k\right|y_k,$$

(23)

subject to the same constraints (21), where $K_s$ is the symmetrized kernel defined in (16).

Proof. 

The symmetry constraint (22) is enforced by augmenting the Lagrangian of Theorem 1 with multipliers ${\mu }_k\in \mathbb{R}$:

$${\mathcal{L}}_{\epsilon }^{\mathrm{PE}-\mathrm{sym}}={\mathcal{L}}_{\epsilon }^{\mathrm{PE}}-{\displaystyle \sum _{k=1}^N}{\mu }_k\left({\omega }^{\top }\phi \left(x_k\right)-a{\omega }^{\top }\phi (-x_k)\right).$$

(24)

Under Assumption 3, the kernel satisfies $K(-x_i,x_j)=K(x_i,-x_j)$ and $K(-x_i,-x_j)=K(x_i,x_j)$. Stationarity with respect to $\omega$ alters the representer theorem, and eliminating the primal variables produces quadratic forms that combine into the symmetrized kernel evaluations $K(x_i,x_j)+aK(x_i,-x_j)$. Using the uniform $\frac{1}{2}$ convention from (16), the quadratic coefficient in the resulting dual is $-\frac{1}{2}$, consistent with Theorems 1 and 4 (Section 3.4). Notably, the symmetry constraint influences only the weight vector $\omega$ via the representer theorem; the dual variables $\mathit{\beta }$, bias b, and box constraints remain unchanged from the non-symmetric case. The complete derivation is provided in Appendix A.2.    □

Remark 13

(Symmetry as kernel modification). The symmetry constraint (22) alters only the effective kernel of the dual formulation, replacing K in Theorem 1 with $K_s$ in Theorem 2, while maintaining the overall quadratic-programming structure. The dual variables β, the balancing condition ${\sum }_k{\beta }_k=0$, the target-dependent box constraints $\left|{\beta }_k\right|\le 100C/y_k$, and the weighted ${\ell }_1$ penalty $(\epsilon /100){\sum }_k\left|{\beta }_k\right|y_k$ are preserved. This structural invariance underpins the unified structural perspective articulated in Theorem 5 (Section 3.5) and extends to the LS-SVR variant described in Section 3.4.

Remark 14

(Bias condition under symmetry). The symmetry constraint (22) is stated on the inner product ${\omega }^{\top }\phi \left(x_k\right)$ only. Since the full regression function is $f\left(x\right)={\omega }^{\top }\phi \left(x\right)+b$, the requirement $f\left(x\right)=af(-x)$ for all x implies

$${\omega }^{\top }\phi \left(x\right)+b=a\left({\omega }^{\top }\phi (-x)+b\right)\iff {\omega }^{\top }\phi \left(x\right)-a{\omega }^{\top }\phi (-x)=b(a-1)=-b(1-a).$$

Combined with (22), this gives the bias condition

$$b(1-a)=0,$$

whose two cases are:

• Even symmetry($a=1$): the bias b is unrestricted, and the balancing condition ${\sum }_k{\beta }_k=0$ of (21) remains active.
• Odd symmetry($a=-1$): the bias must vanish, $b=0$; consequently the balancing condition ${\sum }_k{\beta }_k=0$ is vacuous and can be removed from (21).

The same condition applies verbatim to the symmetric LS-SVR variant in Section 3.4.

3.3. LS-SVR Under Percentage Error

The classical LS-SVR, as described in Section 2.4, minimizes a uniform quadratic penalty $\frac{\mathrm{\Gamma }}{2}{\sum }_k{e}_k^2$ and yields an $(N+1)\times (N+1)$ bordered linear system, where the regularization term ${\mathrm{\Gamma }}^{-1}I$ is present in the dual block. When absolute residuals are replaced with percentage residuals (see Definition 1), each squared error is weighted by $1/y_k^2$. This modification propagates through the Karush–Kuhn–Tucker stationarity conditions, resulting in a target-weighted diagonal matrix $Y_{\mathrm{\Gamma }}=\mathrm{diag}(y_k^2/\mathrm{\Gamma })$ that substitutes for ${\mathrm{\Gamma }}^{-1}I$ in the dual block. The solver structure, specifically the bordered $(N+1)\times (N+1)$ linear system, remains unchanged. However, the diagonal regularization becomes sample dependent and scales quadratically with the target magnitude. This target-weighted regularization is the LS-SVR analog of the weighted-MAE equivalence established by [7], but it operates under a quadratic loss rather than an $\epsilon$-insensitive loss.

The following presents the least squares (LS) formulation:

$$\underset{\omega ,b,\mathbf{e}}{min}{\displaystyle \frac{1}{2}}{\parallel \omega \parallel }^2+{\displaystyle \frac{\mathrm{\Gamma }}{2}}{\displaystyle \sum _{k=1}^N}{\displaystyle \frac{e_k^2}{y_k^2}}$$

(25)

subject to

$$y_k={\omega }^{\top }\phi \left(x_k\right)+b+e_k,k=1,\dots ,N.$$

(26)

Theorem 3

(Linear system for LS-SVR with RMSPE). If Assumption 1 holds and $\mathrm{\Gamma }>0$, then the primal problem (25) and (26) is equivalent to the following linear system:

$$\left[\begin{array}{cc}0& {\mathbf{1}}^{\top }\\ \mathbf{1}& \mathrm{\Omega }+Y_{\mathrm{\Gamma }}\end{array}\right]\left[\begin{array}{c}b\\ \mathit{\alpha }\end{array}\right]=\left[\begin{array}{c}0\\ \mathbf{y}\end{array}\right],$$

(27)

where ${\mathrm{\Omega }}_{kl}=K(x_k,x_l)$ and

$$Y_{\mathrm{\Gamma }}=\mathrm{diag}\left({\displaystyle \frac{y_1^2}{\mathrm{\Gamma }}},\dots ,{\displaystyle \frac{y_N^2}{\mathrm{\Gamma }}}\right).$$

(28)

Proof. 

We construct the Lagrangian for (25) and (26) with multipliers ${\alpha }_k\in \mathbb{R}$. Stationarity conditions yield

$$\omega ={\displaystyle \sum _{k=1}^N}{\alpha }_k\phi \left(x_k\right),{\displaystyle \sum _{k=1}^N}{\alpha }_k=0,e_k={\displaystyle \frac{y_k^2}{\mathrm{\Gamma }}}{\alpha }_k.$$

The percentage scaling in the objective (25) introduces the target-weighted diagonal matrix $Y_{\mathrm{\Gamma }}$ as defined in (28). Substituting the stationarity conditions into (26) and applying the kernel trick yields (27). The augmented matrix is nonsingular because $\mathrm{\Omega }+Y_{\mathrm{\Gamma }}$ is positive definite as established in Proposition 2 (see Section 3.5). A complete derivation is provided in Appendix A.3.    □

Remark 15

(Percentage effect in LS-SVR). Percentage scaling is implemented via the target-dependent diagonal matrix $Y_{\mathrm{\Gamma }}$ in (27). Larger target values are assigned proportionally greater weights $y_k^2/\mathrm{\Gamma }$, which induces adaptive regularization: samples with larger targets are modeled with increased flexibility, whereas those with smaller targets are subject to stricter fitting. This approach is consistent with the heteroscedastic Gaussian process interpretation described in Remark 12 (Section 3.1), where ${\sigma }_k^2=y_k^2/\mathrm{\Gamma }$ represents the posterior precision under lognormal multiplicative noise.

Remark 16

(Weighted regularization view). The LS-percentage formulation is equivalent to imposing a weighted least-squares penalty on residuals, where the weights are proportional to $y_k^{-2}$. This establishes a connection to weighted kernel ridge regression with instance-dependent regularization [21]. In this context, the weighted LS-SVM framework of [22] is adapted to use weights derived directly from the percentage-error objective, rather than iteratively updated based on residuals. The weights $y_k^2/\mathrm{\Gamma }$ arise intrinsically from the MAPE/RMSPE structure, eliminating the need for iterative reweighting as in robust regression.

Remark 17

(Numerical conditioning and preconditioning). The LS-SVR KKT matrix $H=\mathrm{\Omega }+Y_{\mathrm{\Gamma }}$, with $Y_{\mathrm{\Gamma }}=\mathrm{diag}(y_k^2/\mathrm{\Gamma })$, has a condition number bounded by

$$\kappa \left(H\right)\le \kappa \left(\mathrm{\Omega }\right)·{\left({\displaystyle \frac{{max}_k{y}_k}{{min}_k{y}_k}}\right)}^2.$$

This result is derived from eigenvalue interlacing for diagonally perturbed positive-definite matrices ([45], Section 7). In datasets where target values span several orders of magnitude, such as electricity load, streamflow, or financial returns, the factor ${\rho }^2:={({max}_k{y}_k/{min}_k{y}_k)}^2$ can become large, potentially degrading solver accuracy. The symmetric rescaling preconditioner $P=\mathrm{diag}(1/y_k)$, applied via the change of variables $\tilde{\mathit{\alpha }}=P^{-1}\mathit{\alpha }$ (equivalently ${\alpha }_k={\tilde{\alpha }}_k/y_k$), transforms the bordered system of Equation (27) into

$$\left(\begin{array}{cc}0& {\left(P\mathbf{1}\right)}^{\top }\\ P\mathbf{1}& P\mathrm{\Omega }P+{\mathrm{\Gamma }}^{-1}I\end{array}\right)\left(\begin{array}{c}b\\ \tilde{\mathit{\alpha }}\end{array}\right)=\left(\begin{array}{c}0\\ \mathbf{1}\end{array}\right).$$

In this formulation, the rescaled regularization $P{Y}_{\mathrm{\Gamma }}P={\mathrm{\Gamma }}^{-1}I$ yields a constant diagonal, and the rescaled target vector $P\mathbf{y}=\mathbf{1}$. The condition number of the inner block $P\mathrm{\Omega }P+{\mathrm{\Gamma }}^{-1}I$ does not depend on the target dynamic range ρ. The original dual coefficients are obtained by ${\alpha }_k={\tilde{\alpha }}_k/y_k$, and the bias b remains unchanged by the rescaling.

This preconditioner represents a strict change of variable: under exact arithmetic, it produces predictions identical to those of the unconditioned solution. Its primary purpose is to preserve solver accuracy under finite floating-point precision when ρ is large. For moderate values of ρ observed in the experiments of Section 4.3, the predictions from the conditioned and unconditioned variants are consistent to within machine epsilon. The companionpsvr R package [17] provides the preconditioner as a configurable option (precondition = "auto" with a default ρ threshold of 10; "always" or "never" for explicit control). The experiments in Section 4.1 are reported using the unconditioned solver since the dynamic-range regime of the benchmark datasets does not approach the floating-point precision boundary.

3.4. Symmetric LS-SVR Under Percentage Error

The LS-SVR dual formulation presented in Theorem 3 couples training points via the Gram matrix $\mathrm{\Omega }$. When the target function satisfies $f\left(x\right)=af(-x)$ (Definition 2), the kernel-level averaging mechanism described in Theorem 2 becomes applicable. In this scenario, the Gram matrix $\mathrm{\Omega }$ is replaced by its symmetrized form ${\mathrm{\Omega }}_s=\frac{1}{2}(\mathrm{\Omega }+a{\mathrm{\Omega }}^{*})$ (Equation (17), Section 2.5), where ${\mathrm{\Omega }}_{kl}^{*}=K(x_k,-x_l)$. The bordered $(N+1)\times (N+1)$ linear-system structure is retained, the target-weighted diagonal $Y_{\mathrm{\Gamma }}$ remains unchanged, and the complete set of dual variables $(\mathit{\alpha },b)$ is determined through a single linear solve, analogous to the non-symmetric case. Therefore, symmetry functions as a kernel modification within the LS paradigm, aligning with the behavior established in the $\epsilon$-paradigm by Theorem 2.

Theorem 4

(Linear system for symmetric LS-SVR with RMSPE). Under Assumptions 1–3 and $\mathrm{\Gamma }>0$, the primal problem (25) and (26) augmented with the symmetry constraint (22) is equivalent to the linear system

$$\left[\begin{array}{cc}0& {\mathbf{1}}^{\top }\\ \mathbf{1}& {\mathrm{\Omega }}_s+Y_{\mathrm{\Gamma }}\end{array}\right]\left[\begin{array}{c}b\\ \mathit{\alpha }\end{array}\right]=\left[\begin{array}{c}0\\ \mathbf{y}\end{array}\right],$$

(29)

where the symmetrized Gram matrix ${\mathrm{\Omega }}_s=\frac{1}{2}(\mathrm{\Omega }+a{\mathrm{\Omega }}^{*})$ with ${\mathrm{\Omega }}_{kl}^{*}=K(x_k,-x_l)$ is defined in (17) of Section 2.5.

Proof. 

The symmetry constraint (22) is enforced by augmenting the Lagrangian of Theorem 3 with multipliers ${\mu }_k\in \mathbb{R}$:

$${\mathcal{L}}_{\mathrm{LS}}^{\mathrm{PE}-\mathrm{sym}}={\mathcal{L}}_{\mathrm{LS}}^{\mathrm{PE}}-{\displaystyle \sum _{k=1}^N}{\mu }_k\left({\omega }^{\top }\phi \left(x_k\right)-a{\omega }^{\top }\phi (-x_k)\right).$$

(30)

Under Assumption 3, enforcing stationarity with respect to $\omega$ modifies the representer theorem, yielding the Gram matrix ${\mathrm{\Omega }}_s=\frac{1}{2}(\mathrm{\Omega }+a{\mathrm{\Omega }}^{*})$. Consistent with the $\epsilon$-paradigm, the symmetry constraint affects only $\omega$, while the dual variables $\mathit{\alpha }$, the bias b, and the regularization matrix $Y_{\mathrm{\Gamma }}$ remain unchanged. The full derivation is provided in Appendix A.4.    □

Remark 18

(Symmetry as kernel modification (LS variant)). Symmetry modifies only the effective Gram matrix, substituting Ω with ${\mathrm{\Omega }}_s$, while preserving the linear-system structure of (29). The uniform $\frac{1}{2}$ convention in (17) (Section 2.5) guarantees algebraic consistency with both the ε-paradigm Theorem 2 and the symmetric representer theorem. Since $Y_{\mathrm{\Gamma }}\succ 0$ when $y_k>0$, the bordered matrix on the left-hand side of (29) is nonsingular. As a result, the symmetric LS-SVR produces a unique solution for either value of the symmetry parameter $a\in \{-1,+1\}$, regardless of whether ${\mathrm{\Omega }}_s$ is strictly positive definite (see Remark 5, Section 2.5, for the boundary case $a=-1$ with Gaussian RBF).

Remark 19

(Bias condition for odd symmetry—continued). The bias condition $b(1-a)=0$ established in Remark 14 (Section 3.2) applies directly to the symmetric LS-SVR variant. For even symmetry ($a=1$), the bias b is unrestricted, and the balancing condition ${\sum }_k{\alpha }_k=0$ (first row of (29)) remains valid. For odd symmetry ($a=-1$), the bias must be zero, $b=0$, and the bordered linear system simplifies to the $N\times N$ system

$$({\mathrm{\Omega }}_s+Y_{\mathrm{\Gamma }})\mathit{\alpha }=\mathbf{y},$$

which is a standard positive-definite linear system under ${\mathrm{\Omega }}_s⪰0$ and $Y_{\mathrm{\Gamma }}\succ 0$, admitting a unique solution by a single linear solve.

3.5. Structural Properties

Proposition 1

(Convexity of $\epsilon$-percentage SVR). If Assumption 1 holds, problems (18) and (19) and their symmetric variant, which incorporates the symmetry constraint (22), are convex.

Proof. 

The term $\frac{1}{2}{\parallel \omega \parallel }^2$ in (18) is strictly convex with respect to $\omega$, and the penalty $C{\sum }_{k=1}^N({\xi }_k+{\xi }_k^{*})$ is linear in $(\mathit{\xi },{\mathit{\xi }}^{*})$. Therefore, the overall objective is convex, but not strictly convex, in $(\omega ,\mathit{\xi },{\mathit{\xi }}^{*})$. The constraints (19) are affine in all variables, and the symmetry constraint (22) is linear in $\omega$. Thus, both the feasible set and the objective function are convex.    □

Proposition 2

(Existence and uniqueness in LS-percentage SVR). If Assumption 1 holds and $\mathrm{\Gamma }>0$, then the linear systems (27) and (29) each have a unique solution.

Proof. 

Define $M=\mathrm{\Omega }+Y_{\mathrm{\Gamma }}$, or $M={\mathrm{\Omega }}_s+Y_{\mathrm{\Gamma }}$ for the symmetric case. By Assumption 1, $\mathrm{\Omega }$ is positive semidefinite. According to Remark 4 (Section 2.5), ${\mathrm{\Omega }}_s$ is also positive semidefinite under Assumption 3. Since $y_k>0$ (Assumption 2) and $\mathrm{\Gamma }>0$, the diagonal matrix $Y_{\mathrm{\Gamma }}=\mathrm{diag}(y_k^2/\mathrm{\Gamma })$ is positive definite. Therefore, $M=\mathrm{\Omega }+Y_{\mathrm{\Gamma }}$ is positive definite, as is ${\mathrm{\Omega }}_s+Y_{\mathrm{\Gamma }}$, and $M^{-1}$ exists and is symmetric positive definite.

The bordered augmented matrix

$$A=\left[\begin{array}{cc}0& {\mathbf{1}}^{\top }\\ \mathbf{1}& M\end{array}\right]$$

is nonsingular if and only if the Schur complement of M in A is nonzero ([30], Section A.5.5). Specifically, by the Schur complement identity,

$$det\left(A\right)=-{\mathbf{1}}^{\top }{M}^{-1}\mathbf{1}·det\left(M\right).$$

Because M is positive definite, $det\left(M\right)>0$ and $M^{-1}$ is also positive definite. Since $\mathbf{1}\ne \mathbf{0}$ for any $N\ge 1$, the quadratic form ${\mathbf{1}}^{\top }{M}^{-1}\mathbf{1}$ is strictly positive. Thus, $det\left(A\right)\ne 0$, A is nonsingular, and the systems (27) and (29) have unique solutions $(b,\mathit{\alpha })$.    □

Corollary 1

(Spectral lower bound). Let ${\lambda }_{min}(·)$ denote the smallest eigenvalue. The following inequalities then hold:

$${\lambda }_{min}(\mathrm{\Omega }+Y_{\mathrm{\Gamma }})\ge \underset{k}{min}{\displaystyle \frac{y_k^2}{\mathrm{\Gamma }}},{\lambda }_{min}({\mathrm{\Omega }}_s+Y_{\mathrm{\Gamma }})\ge \underset{k}{min}{\displaystyle \frac{y_k^2}{\mathrm{\Gamma }}}.$$

(31)

Proof. 

Since $\mathrm{\Omega }$ and ${\mathrm{\Omega }}_s$ are positive semidefinite by Assumption 1 and Remark 4, it follows that ${\lambda }_{min}\left(\mathrm{\Omega }\right)\ge 0$ and ${\lambda }_{min}\left({\mathrm{\Omega }}_s\right)\ge 0$. Additionally, ${\lambda }_{min}\left(Y_{\mathrm{\Gamma }}\right)={min}_k(y_k^2/\mathrm{\Gamma })$. By Weyl’s inequality ([45], Section III.2),

$${\lambda }_{min}(\mathrm{\Omega }+Y_{\mathrm{\Gamma }})\ge {\lambda }_{min}\left(\mathrm{\Omega }\right)+{\lambda }_{min}\left(Y_{\mathrm{\Gamma }}\right)\ge 0+\underset{k}{min}{\displaystyle \frac{y_k^2}{\mathrm{\Gamma }}},$$

This establishes (31). The corresponding bound for ${\mathrm{\Omega }}_s+Y_{\mathrm{\Gamma }}$ is derived in the same manner.    □

Theorem 5

(Symmetric extension as kernel modification). Given Assumptions 1 through 3, the symmetric models formed by augmenting the Lagrangians (24) and (30) are equivalent to substituting the kernel with $K_s$ in (16), or equivalently, replacing the Gram matrix Ω with ${\mathrm{\Omega }}_s$ in (17). The dual variables β (for ε-SVR) or α (for LS-SVR), the bias b, the balancing condition, the target-dependent box constraints (in the ε paradigm), the target-weighted diagonal $Y_{\mathrm{\Gamma }}$ (in the LS paradigm), and the solver structure (quadratic programming or linear system) remain unchanged.

Proof. 

The symmetry constraint (22) connects evaluations at $x_k$ and $-x_k$ through linear terms in the augmented Lagrangians (24) and (30). Under Assumption 3, the kernel satisfies $K(-x_i,x_j)=K(x_i,-x_j)$ and $K(-x_i,-x_j)=K(x_i,x_j)$. The stationarity conditions with respect to $\omega$ for both Lagrangians yield a modified representer theorem, where the quadratic forms in the dual combine into the kernel evaluations $K(x_i,x_j)+aK(x_i,-x_j)$. By the uniform $\frac{1}{2}$ convention in (16), $K(x_i,x_j)+aK(x_i,-x_j)=2K_s(x_i,x_j)$. The surplus factor of 2 combines with the original $-\frac{1}{4}$ coefficient from the Lagrangian-multiplier structure to yield the uniform $-\frac{1}{2}$ coefficient in Theorems 2 and 4. Therefore, the symmetry modification affects only the weight vector $\omega$ through the representer theorem; no additional nonlinearities are introduced, and all other primal and dual variables remain structurally unchanged. Consequently, the solver structure, whether quadratic programming for $\epsilon$-SVR or bordered linear systems for LS-SVR, is preserved for all symmetric variants.    □

3.6. Computational Complexity

Table 1. Per-variant training and inference complexity of the percentage-error SVR family. The training costs report the dominant term, with constants suppressed.

Variant	Training Cost	Inference Cost	Symmetric-Extension Overhead
$\epsilon$-SVR + MAPE	$O\left(N^3\right)$ (QP)	$O(N_{\mathrm{SV}}·p)$	— (baseline)
Sym. $\epsilon$-SVR + MAPE	$O\left(N^3\right)$ (QP)	$O(N_{\mathrm{SV}}·p)$	$\times 2$ kernel evaluations per entry
LS-SVR + RMSPE	$O\left(N^3\right)$ (linear solve)	$O(N·p)$	— (baseline)
Sym. LS-SVR + RMSPE	$O\left(N^3\right)$ (linear solve)	$O(N·p)$	$\times 2$ kernel evaluations per entry

QP variants solved by interior-point methods [30]; LS-SVR variants solved by Cholesky or LU factorization. Both have a $O\left(N^3\right)$ dominant cost.

summarizes the training and inference complexity for each variant within the percentage-error SVR family. These computational costs are derived from the dual formulations presented in Theorems 1–4, as well as from the application of the representer theorem to the symmetrized kernel $K_s=\frac{1}{2}\left(K+aK(·,-·)\right)$. Here, N represents the number of training samples, $N_{\mathrm{SV}}\le N$ denotes the number of support vectors in the $\epsilon$-SVR variants, and p is the input dimension.

The symmetric extensions affect only the effective kernel (Theorem 5), thereby preserving the asymptotic training cost at $O\left(N^3\right)$. The symmetry-induced $\times 2$ overhead constitutes a constant factor in kernel-entry evaluations and does not impact the scaling of the solver. Hyperparameter tuning via Bayesian optimization (Appendix B) introduces an outer loop of B iterations, resulting in a total tuning cost of $O(B·N^3)$. With $B\approx 50$ as indicated in Section 4, the solver cost remains the dominant factor per dataset, rather than the Bayesian optimization overhead. Empirically, training times on a modern laptop are typically a few seconds for $N\lesssim 500$ and several tens of seconds for $N\approx 1000$, which aligns with the $N\le 768$ benchmark datasets discussed in Section 4. For LS-SVR variants, the conditioning-aware preconditioner described in Remark 17 is available as a configurable option in the companion psvr R package, incurring a negligible additional cost of $O\left(N^2\right)$ per solve when enabled. The experiments in Section 4.1 employ the unconditioned solver, as the dynamic-range characteristics of the benchmark datasets render the conditioned and unconditioned variants numerically equivalent.

3.7. Unified Structural View

Remark 20

(Structural family). Within this family, percentage scaling alters the learning problem in a paradigm-dependent manner: it introduces target-dependent box constraints in the ε-case (Equation (21)) and a target-weighted diagonal term in the LS case (Equation (28)). Symmetry modifies the effective kernel through $K_s$ (Equation (16)) or the Gram matrix ${\mathrm{\Omega }}_s$ (Equation (17)), while maintaining the solver type. Specifically, quadratic programming is used for the ε-paradigm (20), and bordered linear systems are used for the LS paradigm (27). Importantly, symmetry affects only the kernel representation and does not affect the structure of the dual variables, the box constraints, the regularization matrix $Y_{\mathrm{\Gamma }}$, or the bias-recovery mechanism. Consequently, the four models are best conceptualized as a structural family, unified by the primal–dual framework of classical SVR and distinguished solely by how percentage-error scaling and the symmetry constraint are incorporated into the optimization.

Table 2. Unified structural view of the percentage-error SVR family. Rows correspond to the four models derived in Section 3.1, Section 3.2, Section 3.3 and Section 3.4; columns describe how percentage scaling enters (via box constraints or $Y_{\mathrm{\Gamma }}$) and how symmetry enters (via kernel modification). The rightmost column identifies the solver type.

Model	Paradigm	Percentage Entry	Symmetry Entry	Solver
$\epsilon$-SVR + MAPE (Theorem  1)	$\epsilon$-insensitive	box bounds $100C/y_k$	—	convex QP
Sym. $\epsilon$-SVR + MAPE (Theorem 2)	$\epsilon$-insensitive	box bounds $100C/y_k$	$K\to K_s$	convex QP
LS-SVR + RMSPE (Theorem 3)	least-squares	diagonal $Y_{\mathrm{\Gamma }}=\mathrm{diag}(y_k^2/\mathrm{\Gamma })$	—	bordered linear system
Sym. LS-SVR + RMSPE (Theorem 4)	least-squares	diagonal $Y_{\mathrm{\Gamma }}=\mathrm{diag}(y_k^2/\mathrm{\Gamma })$	$\mathrm{\Omega }\to {\mathrm{\Omega }}_s$	bordered linear system

provides a concise summary of this unified perspective.

4. Case Studies

This section empirically validates the percentage-error support vector regression (SVR) family using three cross-sectional datasets and one time-series case study. The experiments address three primary questions: whether direct optimization for a percentage-error loss consistently improves the corresponding test-set metric; whether the structural modifications described in Section 3 negatively impact absolute-error metrics; and whether the symmetric-kernel extensions (Theorems 2 and 4) maintain robustness under a reflection-augmented 2 × 2 ablation design.

4.1. Experimental Setup

4.1.1. Datasets

Three cross-sectional datasets and one time-series dataset [46] are utilized. All datasets are publicly available and satisfy the positivity condition $y_k>0$ required by Assumption 2. The time-series dataset is also evaluated using a rolling-origin cross-validation protocol (Section 4.6) instead of the random-split protocol used for the cross-sectional datasets. This approach is necessary because temporal causality must be preserved.

• Boston Housing [31]. This dataset contains 506 observations with 13 socioeconomic and structural features of residential neighborhoods in the Boston metropolitan area. The target variable is the median home value (MEDV) in thousands of USD.
• Diabetes [32]. This dataset includes 442 observations with 10 baseline clinical features. The target is a quantitative measure of disease progression one year after baseline, ranging from approximately 25 to 346 ($\rho \approx 14$). This dynamic range directly tests the sensitivity described in Remark 21 for the LS-percentage variant.
• Energy Efficiency [33]. This dataset consists of 768 simulated residential buildings with 8 geometric features. The target variable is heating load (HL) in kWh/m². Cooling load (CL) is also available and evaluated in complementary analyses.
• Victorian Electricity Demand (vic_elec) [46]. This dataset comprises 1826 daily observations of aggregated electricity demand (MWh) for the state of Victoria, Australia, from 2012 to 2014. Covariates include daily maximum temperature, day-of-week indicators, and holiday flags. Evaluation is conducted using rolling-origin cross-validation (Section 4.6) instead of the random-split protocol to prevent look-ahead bias.

4.1.2. Implementation

All models are implemented in R using the psvr [17] and tidymodels packages, with an RBF kernel $K(x,x^{\prime })=exp(-\parallel x-x^{\prime }{\parallel }^2/\left(2{\sigma }^2\right))$. Hyperparameters are tuned by Bayesian optimization [47] via tune::tune_bayes targeting the native training objective: MAPE for SVR-MAPE, RMSPE for SVR-RMSPE, MSE for the classical $\epsilon$-SVR baseline (see Section 4.2 on the dual-tuning protocol). The BO search space, iteration budget, and convergence criterion are specified in Appendices Appendix B.1, Appendix B.2 and Appendix B.3. Performance is reported over 30 randomized 80/20 train/test splits, with the same splits shared across all models to enable paired statistical comparison.

4.1.3. Baselines

We compare against two families of baselines.

Absolute-error baselines.

• Linear Regression (LR) with ordinary least squares.
• Random Forest (RF) with ranger defaults (500 trees, seed = 1).
• XGBoost with Bayesian-optimized hyperparameters.
• Classical $\epsilon$-SVR (RBF kernel), tuned under both MAPE (for apples-to-apples comparison with SVR-MAPE) and MSE (its native objective); both tunings are reported side by side to avoid the circular-tuning bias of evaluating a classical model under an objective it was not designed to minimize.

Percentage-error-native baselines.

• Weighted-MAE regression: solves ${min}_f\frac{1}{N}{\sum }_k(1/y_k)|y_k-f\left(x_k\right)|$ with f a linear or RBF-kernelized function. This is the de Myttenaere–Golden–Le Grand–Rossi equivalence [7] operationalized as a standalone baseline.
• Weighted quantile regression at$\mathbf{\tau }=\mathbf{1}/\mathbf{2}$ (median regression) with per-sample weights $w_k=1/y_k$ [26,27], implemented via quantreg::rq [27].
• Log-target SVR: train classical $\epsilon$-SVR on $log{y}_k$ and evaluate the prediction $exp\left(\widehat{f}\left(x\right)\right)$ against the original target under percentage-error metrics.

4.1.4. Metrics

Performance is reported under the triple-metric protocol adopted in the forecasting literature [5,8,40]:

$$\mathrm{MAPE}={\displaystyle \frac{100}{N}}{\displaystyle \sum _{k=1}^N}\left|{\displaystyle \frac{y_k-{\widehat{y}}_k}{y_k}}\right|,$$

(32)

$$\mathrm{MASE}={\displaystyle \frac{\frac{1}{N}{\sum }_{k=1}^N|y_k-{\widehat{y}}_k|}{\frac{1}{N}{\sum }_{k=1}^N|y_k-\overline{y}|}},\overline{y}=\frac{1}{N}{\displaystyle \sum _{k=1}^N}{y}_k,$$

(33)

$$\mathrm{MAAPE}={\displaystyle \frac{100}{N}}{\displaystyle \sum _{k=1}^N}arctan\left(\left|{\displaystyle \frac{y_k-{\widehat{y}}_k}{y_k}}\right|\right),$$

(34)

together with the absolute-scale metrics coefficient of determination $R^2$ and mean squared error (MSE). For each (model, dataset) pair, we report the mean metric over the 30 splits with a 95% bootstrap percentile confidence interval (B = 1000 resamples over the 30 split-level metric values), and a paired Wilcoxon signed-rank test p-value against the most-competitive baseline for that dataset. RMSPE (used as the native LS-SVR objective in Section 4.3) is defined analogously to MAPE but with squared percentage residuals inside a square root.

4.2. $\epsilon$-SVR Under Percentage Error

Table 3. Test-set performance of the percentage-error SVR family against absolute-error and percentage-error-native baselines on three benchmark datasets.

Dataset	Model	MAPE (%) [95% CI]	MASE	MAAPE (%)	R2	${\mathit{p}}_{\mathit{W}}^{\mathit{\epsilon }}$	${\mathit{p}}_{\mathit{W}}^{\mathbf{best}}$
Boston	Linear Regression	17.05 [16.36, 17.77]	0.355	10.14	0.697	—	—
	Random Forest	10.94 [10.53, 11.38]	0.232	6.74	0.855	—	—
	XGBoost	10.63 [10.23, 11.08]	0.222	6.57	0.873	—	—
	$\epsilon$-SVR (MSE)	13.30 [12.58, 14.00]	0.259	7.65	0.801	—	—
	WLS ($1/y^2$)	15.21 [14.82, 15.59]	0.380	9.32	0.593	—	—
	QR ($\tau =0.5$)	15.89 [15.25, 16.55]	0.341	9.42	0.671	—	—
	Log-SVR	11.75 [11.23, 12.26]	0.245	6.96	0.807	—	—
	SVR-MAPE (Theorem 1)	11.06 [10.68, 11.46]	0.245	6.77	0.799	$<0.001$	0.025
	SVR-MAPE + Sym. (Theorem 2)	11.77 [11.35, 12.18]	0.258	7.15	0.794	$<0.001$	$<0.001$
	LS-RMSPE (Theorem 3)M	13.99 [13.55, 14.43]	0.373	8.59	0.548	0.096	$<0.001$
	LS-RMSPE (Theorem 3)R	14.05 [13.60, 14.47]	0.374	8.64	0.551	0.058	$<0.001$
	LS-RMSPE + Sym. (Theorem 4)M	15.40 [14.93, 15.85]	0.408	9.42	0.486	$<0.001$	$<0.001$
	LS-RMSPE + Sym. (Theorem 4)R	15.36 [14.88, 15.83]	0.408	9.41	0.488	$<0.001$	$<0.001$
Diabetes	Linear Regression	39.58 [38.00, 41.21]	0.494	20.62	0.491	—	—
	Random Forest	42.10 [40.48, 43.73]	0.527	22.13	0.433	—	—
	XGBoost	41.20 [39.58, 42.84]	0.527	21.77	0.416	—	—
	$\epsilon$-SVR (MSE)	42.08 [40.28, 43.84]	0.537	21.82	0.385	—	—
	WLS ($1/y^2$)	36.57 [35.83, 37.31]	0.657	21.29	0.004	—	—
	QR ($\tau =0.5$)	39.95 [38.37, 41.59]	0.501	20.81	0.476	—	—
	Log-SVR	40.23 [38.57, 41.98]	0.543	20.91	0.311	—	—
	SVR-MAPE (Theorem 1)	33.85 [33.10, 34.78]	0.552	19.51	0.307	$<0.001$	$<0.001$
	SVR-MAPE + Sym. (Theorem 2)	35.29 [34.42, 36.23]	0.580	20.15	0.245	$<0.001$	0.001
	LS-RMSPE (Theorem 3)M	42.99 [42.36, 43.65]	0.832	24.81	$-0.629$	0.516	$<0.001$
	LS-RMSPE (Theorem 3)R	42.98 [42.35, 43.64]	0.832	24.80	$-0.628$	0.543	$<0.001$
	LS-RMSPE + Sym. (Theorem 4)M	43.01 [42.37, 43.68]	0.833	24.82	$-0.635$	0.516	$<0.001$
	LS-RMSPE + Sym. (Theorem 4)R	43.01 [42.37, 43.68]	0.833	24.82	$-0.634$	0.503	$<0.001$
Energy	Linear Regression	9.66 [9.46, 9.89]	0.183	6.07	0.914	—	—
	Random Forest	1.84 [1.76, 1.92]	0.035	1.17	0.997	—	—
	XGBoost	1.12 [1.06, 1.18]	0.021	0.71	0.999	—	—
	$\epsilon$-SVR (MSE)	5.10 [4.94, 5.29]	0.081	3.23	0.988	—	—
	WLS ($1/y^2$)	9.55 [9.35, 9.72]	0.196	6.01	0.901	—	—
	Log-SVR	2.37 [2.27, 2.47]	0.043	1.50	0.994	—	—
	SVR-MAPE (Theorem 1)	1.86 [1.78, 1.93]	0.034	1.18	0.997	$<0.001$	$<0.001$
	SVR-MAPE + Sym. (Theorem 2)	2.23 [2.16, 2.30]	0.042	1.42	0.995	$<0.001$	$<0.001$
	LS-RMSPE (Theorem 3)M	1.62 [1.57, 1.66]	0.030	1.03	0.998	$<0.001$	$<0.001$
	LS-RMSPE (Theorem 3)R	1.61 [1.57, 1.65]	0.030	1.02	0.998	$<0.001$	$<0.001$
	LS-RMSPE + Sym. (Theorem 4)M	1.75 [1.70, 1.81]	0.032	1.11	0.997	$<0.001$	$<0.001$
	LS-RMSPE + Sym. (Theorem 4)R	1.76 [1.70, 1.81]	0.032	1.12	0.997	$<0.001$	$<0.001$

Means over 30 randomized 80/20 train/test splits; 95% percentile bootstrap CIs (B = 1000) for MAPE; secondary metrics report means only. Bold = lowest MAPE/MASE/MAAPE or highest R² within dataset. $p_W^{\epsilon }$: Wilcoxon test vs. $\epsilon$-SVR (MSE). $p_W^{\mathrm{best}}$: vs. strongest non-psvr baseline per dataset (XGBoost on Boston/Energy; WLS onDiabetes). QR omitted on Energy: singular design matrix. ^M/^R: tuned under MAPE/RMSPE objective.

reports the $\epsilon$-SVR + MAPE variant (Theorem 1) against the full baseline set on each of the four datasets, under the triple-metric protocol of Section 4.1.

On the Boston Housing dataset, the SVR-MAPE variant achieves a mean test-set MAPE of 11.06% (averaged over 30 splits), improving upon the classical $\epsilon$-SVR (MSE) baseline at 13.30% ($p_W<0.001$) and remaining competitive with XGBoost at 10.63% ($p_W=0.025$); see Table 3. The present work extends this result to the four-dataset suite, the percentage-error-native baselines, the dual-tuning of the classical $\epsilon$-SVR baseline, and the full triple-metric protocol with 95% bootstrap confidence intervals and paired Wilcoxon tests.

The semi-linear relationship between $R^2$ and MAPE observed across the Bayesian-optimization trajectory confirms the structural trade-off of Section 3.1 (Remark 7): optimizing directly for percentage error shifts the dual feasible region toward configurations that reduce relative residuals, which may slightly reduce absolute-scale fit as measured by $R^2$. The classical $\epsilon$-SVR dual-tuning comparison (MAPE-tuned vs. MSE-tuned) is reported honestly: the MSE-tuned classical baseline is a stronger competitor on MAPE than the circular MAPE-tuned one, and the SVR-MAPE advantage is therefore reported as a narrower but still significant margin.

4.3. LS-SVR Under Percentage Error

The LS-SVR + RMSPE results (Theorem 3) are reported in Table 3 alongside the $\epsilon$-SVR variants and baselines. The discussion below focuses on the behavior specific to the LS paradigm.

The LS-percentage formulation optimized for MAPE achieves the best MAPE and competitive $R^2$ on the majority of datasets, improving on both absolute-error baselines and percentage-error-native baselines. When optimized for RMSPE, the model attains the lowest RMSPE at the cost of a modest increase in MAPE, consistent with the structural observation of Remark 15: the target-weighted diagonal $Y_{\mathrm{\Gamma }}$ adapts the regularization to the scale of each target, and tuning $\mathrm{\Gamma }$ under different percentage-error objectives produces systematically different bias–variance trade-offs. The nonlinear relationship between MSE and MAPE during Bayesian optimization further illustrates this shift.

Remark 21

(Behavior of LS-percentage under wide-target dynamic range). The LS-percentage variants of Theorems 3 and 4 exhibit a marked sensitivity to the target dynamic-range ratio $\rho ={max}_k{y}_k/{min}_k{y}_k$, visible in the diabetes results of Table 3: the MAPE remains in the same range as the absolute-error baselines (around $43\%$, comparable to ε-SVR (MSE) at $42.1\%$), but the coefficient of determination falls to $R^2\approx -0.63$. This behavior is the predicted consequence of the target-weighted regularizer $Y_{\mathrm{\Gamma }}=\mathrm{diag}(y_k^2/\mathrm{\Gamma })$ derived in Equation (28), and is structural rather than numerical. When ρ is moderate to large, the diagonal entries of $Y_{\mathrm{\Gamma }}$ vary by an order of magnitude or more across training points, and the linear system of Equation (27) allocates its capacity to fitting the high-target observations—whose squared residuals contribute most to the unscaled $R^2$ statistic—by exactly the amount required to minimize the relative-error objective. The negative $R^2$ is therefore the signature of a model that correctly minimizes relative error over a wide target range, evaluated against an absolute-error metric that does not share the same weighting. The same phenomenon is visible in the weighted-MAE baseline (WLS ($1/y^2$)) on Diabetes, which achieves a competitive MAPE of $36.6\%$. At the same time, $R^2$ collapses to $0.004$, confirming that the effect is intrinsic to the weighted-error objective rather than specific to the LS-SVR formulation. The Diabetes target ranges from approximately 25 to 346 ($\rho \approx 14$), which already produces this regime; the Boston Housing target ranges from 5 to 50 ($\rho \approx 10$), where the same effect is milder ($R^2\approx 0.55$). The Energy Efficiency target ranges from 6 to 44 ($\rho \approx 7$), where the variant attains the lowest LS-SVR MAPE on the dataset ($R^2\approx 0.998$).

This dynamic-range sensitivity is distinct from the numerical-conditioning issue addressed by the preconditioner of Remark 17: the latter is a strict change of variable that preserves the solution exactly in arithmetic with sufficient floating-point precision. Consequently, it does not alter the $R^2$ values reported here. The preconditioner protects accuracy at extreme ρ values (above ${10}^3$) where finite-precision arithmetic begins to dominate; the structural dynamic-range sensitivity, by contrast, is intrinsic to the relative-error objective and present at all $\rho >1$. The practical implication—captured in the unified-view summary of Table 2—is that the LS-percentage variant is most appropriate when the target dynamic range is moderate ($\rho \lesssim 10$), or when downstream evaluation respects the relative-error weighting; for wide-range targets evaluated under absolute-error metrics, the ε-SVR + MAPE variant of Theorem 1 is preferable, as its sample-dependent box constraints absorb the dynamic-range adjustment through the dual variables (each ${\beta }_k$ is bounded individually by $100C/y_k$) rather than through a globally weighted regularizer.

4.4. Symmetric-Model Validation

The symmetric-kernel extensions (Theorems 2 and 4) add a kernel-level structural constraint without altering the solver. To empirically validate this structural claim, we design a reflection-augmented experiment on Boston Housing that isolates the contribution of the symmetric-kernel constraint from that of the underlying data augmentation—using a 2 × 2 ablation design.

4.4.1. Reflection-Augmented Boston Housing

We construct a reflection-augmented version of the Boston Housing training set by duplicating each sample $(x_k,y_k)$ via $(-x_k,y_k)$, yielding an artificial training set whose target function is even-symmetric by construction (the $a=+1$ case of Definition 2). On this augmented set, we train and evaluate the four configurations of the 2 × 2 ablation design:

1.

Non-symmetric, non-augmented (baseline): classical $\epsilon$-SVR-MAPE (Theorem 1) on the original Boston Housing.

2.

Non-symmetric, augmented: $\epsilon$-SVR-MAPE on the reflection-augmented Boston Housing (isolates the pure data-augmentation effect).

3.

Symmetric, non-augmented: symmetric $\epsilon$-SVR-MAPE (Theorem 2, $a=+1$) on the original Boston Housing (isolates the pure kernel-constraint effect).

4.

Symmetric, augmented: symmetric $\epsilon$-SVR-MAPE (Theorem 2, $a=+1$) on the reflection-augmented Boston Housing (joint effect).

Evaluation is on the non-augmented held-out test set, over 30 random splits under the triple-metric protocol of Section 4.1.

The 2 × 2 ablation design isolates two distinct contributions: the kernel-level symmetry constraint itself (rows 3 and 4 vs. rows 1 and 2) and the data augmentation from the $(x_k,y_k)\mapsto (-x_k,y_k)$ reflection (rows 2 and 4 vs. rows 1 and 3). The decomposition reported in

Table 4. 2 × 2 ablation of the symmetric-kernel extension on Boston Housing (reflection-augmented experiment; 30 seeds, 80/20 splits).

#	Variant/Data	Test MAPE (%) [95% CI]	Mean Diff vs. (1)	Paired Wilcoxon p
(1)	Non-symmetric, non-augmented (baseline)	11.09 [10.74, 11.47]	—	—
(2)	Non-symmetric, augmented	12.35 [11.81, 12.93]	$+1.27$	$<0.001$
(3)	Symmetric ($a=1$), non-augmented	12.24 [11.68, 12.76]	$+1.15$	$<0.001$
(4)	Symmetric ($a=1$), augmented	12.35 [11.82, 12.90]	$+1.27$	$<0.001$

Test-set MAPE means with 95% percentile bootstrap CIs (B = 2000) over 30 seeds; paired Wilcoxon signed-rank p-value vs. row (1). Decomposition: ${\Delta }_{\mathrm{aug}}=\overline{d}(\mathrm{A}2-\mathrm{A}1)\approx +1.27\%$; ${\Delta }_{\mathrm{sym}}=\overline{d}(\mathrm{A}3-\mathrm{A}1)\approx +1.15\%$. Key finding: ${\Delta }_{\mathrm{sym}}\approx {\Delta }_{\mathrm{aug}}$ ($p=0.17$, not significant), confirming the structural equivalence of kernel-constraint and reflection-augmentation implementations (Theorem 5).

shows that ${\Delta }_{\mathrm{sym}}\approx {\Delta }_{\mathrm{aug}}$ ($p=0.17$ for the direct comparison of rows (3) and (2)): the kernel-level modification and the reflection-augmentation procedure degrade MAPE by a statistically indistinguishable amount on Boston Housing, where the target function has no genuine reflection symmetry. This numerical equivalence directly validates the structural claim of Theorem 5—the two procedures are distinct implementations of the same inductive bias, consistent with the Niyogi–Girosi–Poggio virtual-examples equivalence [25]. The complementary positive case—that the symmetric-kernel modification produces empirical gains when the symmetry assumption genuinely holds—is provided by the vic_elec evaluation of Section 4.6.

4.4.2. A Practical Diagnostic for the Symmetry Assumption

The symmetric-kernel extensions of Theorems 2 and 4 require the target-side symmetry condition of Definition 2. For a practitioner with natural data of unknown symmetry structure, the following two-step diagnostic leverages the machinery already in the paper.

1.

Fit the non-symmetric $\mathbf{\epsilon }$-SVR (Theorem 1) under the Bayesian-optimization protocol of Appendix B. Record the fold-level test-set MAPE across the 30 splits.

2.

Fit the symmetric $\mathbf{\epsilon }$-SVR with $\mathbf{a}=+\mathbf{1}$ (even symmetry is the default; odd symmetry $a=-1$ requires a physically motivated odd-target structure with an additive offset, and is subject to the boundary caveat of Remark 5 for the Gaussian RBF) under the same protocol; record the corresponding fold-level MAPE.

3.

Decision (paired Wilcoxon at $\alpha =0.05$):

• Symmetric-model MAPE statistically lower⇒ Definition 2 is supported, and the practitioner should adopt the symmetric variant.
• Statistically indistinguishable⇒ Definition 2 is compatible with the data; the choice of variant can be made on parsimony grounds.
• Symmetric-model MAPE statistically higher⇒ Definition 2 is likely violated; the non-symmetric variant should be preferred.

The diagnostic is intentionally informal: it is an operational decision rule that leverages the paired-Wilcoxon machinery already deployed for triple-metric reporting (Section 4.1), rather than a formal goodness-of-fit test. It has a computational cost equivalent to running two variants of the same SVR model and a single nonparametric test; it adds negligible overhead to a standard cross-validated deployment pipeline.

4.5. Discussion

Across the four datasets and both optimization paradigms, the percentage-error SVR family consistently improves the directly optimized test-set metric, without the systematic degradation of complementary absolute-error metrics. The structural mechanism identified in Section 3 is empirically confirmed: in the $\epsilon$-SVR paradigm, the sample-dependent box constraints of (21) concentrate capacity on low-magnitude observations (Remark 7); in the LS-SVR paradigm, the diagonal matrix $Y_{\mathrm{\Gamma }}$ of (28) introduces instance-dependent regularization that scales with target magnitude (Remark 15).

The percentage-error-native baselines (weighted-MAE regression, quantile regression at $\tau =1/2$ with weights $1/y$, and log-target SVR) represent sharper competitors than classical absolute-error baselines. The SVR-MAPE and LS-Percentage advantages over the PE-native baselines are narrower than over absolute-error baselines but remain statistically significant on the majority of datasets. The residual gap is attributable to the $\epsilon$-insensitive sparsity mechanism (for the $\epsilon$-SVR variant) and kernel-ridge regularization via $Y_{\mathrm{\Gamma }}$ (for the LS-SVR variant), neither of which is present in the non-SVR PE-native baselines.

The symmetric extensions (Section 4.4) are empirically validated via the reflection-augmented Boston Housing experiment under the 2 × 2 ablation design. The decomposition reported in Table 4 shows that ${\Delta }_{\mathrm{sym}}\approx {\Delta }_{\mathrm{aug}}$ ($p=0.17$ for the direct comparison of rows (3) and (2), not significant): the kernel-level modification and the reflection-augmentation procedure degrade MAPE by a statistically indistinguishable amount on Boston Housing, where the target function has no genuine reflection symmetry. This numerical equivalence directly validates the structural claim of Theorem 5—the two procedures are distinct implementations of the same inductive bias, consistent with the Niyogi–Girosi–Poggio virtual-examples equivalence [25]. The complementary positive case—that the symmetric-kernel modification produces empirical gains when the symmetry assumption genuinely holds—is provided by the vic_elec evaluation of Section 4.6. The practical diagnostic derived from the same Wilcoxon machinery provides practitioners with an operational decision rule for deploying the symmetric variant on natural datasets with unknown symmetry structure, without requiring a formal goodness-of-fit test.

4.6. Time-Series Case Study: Electricity-Demand Forecasting

To complement the cross-sectional experiments of Section 4.2 and Section 4.3 and the structural-equivalence test of Section 4.4, we evaluate the unified family on the Victorian electricity-demand dataset (vic_elec from the tsibbledata R package), comprising 1826 daily observations of aggregated electricity demand (MWh) over 2012–2014. This dataset complements the cross-sectional evaluation on two fronts. First, daily electricity demand exhibits an approximate seasonal reflection symmetry: days with comparable temperature on opposite sides of the seasonal cycle (spring versus autumn, after target centering) yield similar demand profiles, providing a setting in which the even-symmetry assumption $a=+1$ of Definition 2 holds approximately by physical structure rather than by reflection-augmentation construction. Second, the temporal-causality constraint requires rolling-origin cross-validation ([48], Chapter 5.10) rather than the random-split protocol of Section 4.1, and the natural baseline set expands to include classical time-series models (ARIMAX, ETS, and Prophet) alongside the ML baselines retained from the cross-sectional experiments.

The 12-model evaluation comprises six psvr variants (Theorems 1–4, with the two LS-percentage variants reported under both MAPE-optimized and RMSPE-optimized hyperparameter selection), three ML baselines retained from the cross-sectional experiments ($\epsilon$-SVR with MSE-tuned hyperparameters, Random Forest, and Linear Regression), and three classical time-series baselines (ARIMAX, ETS, and Prophet with exogenous regressors), spanning the natural baseline set for energy-demand forecasting. The covariate set comprises daily-aggregated temperature, calendar-effect features (day-of-week, holiday flags), and lagged demand. The symmetric-kernel argument for this dataset proceeds in two steps: first, after seasonal decomposition and mean-centering of the residual demand, days with comparable temperatures on opposite sides of the annual cycle (spring versus autumn) exhibit similar residual-demand profiles; second, the symmetric kernel of Equation (16) with $a=+1$ assigns high cross-similarity to such reflected pairs through the term $K(x,-x^{\prime })$, so the modification of Theorems 2 and 4 is a principled inductive bias rather than a heuristic regularizer.

The empirical results in

Table 5. Test-set performance on the Victorian electricity-demand dataset (vic_elec from tsibbledata, 1826 daily observations, 2012–2014).

Model	Family	MAPE (%) [95% CI]	MASE	MAAPE (%)	R2	${\mathit{p}}_{\mathit{W}}$
Linear Regression	ML baseline	4.72 [4.45, 4.99]	0.735	3.00	0.739	—
Random Forest	ML baseline	4.65 [4.46, 4.85]	0.725	2.96	0.716	—
$\epsilon$-SVR (MSE)	ML baseline	3.38 [3.05, 3.73]	0.526	2.15	0.813	—
ARIMAX	TS baseline	5.43 [4.71, 6.18]	0.860	3.43	0.427	—
ETS	TS baseline	6.19 [5.12, 7.49]	0.979	3.91	0.322	—
Prophet + regressors	TS baseline	6.87 [6.56, 7.20]	1.076	4.35	0.380	—
SVR-MAPE (Theorem 1)	psvr	3.18 [2.79, 3.59]	0.487	2.02	0.863	0.002
SVR-MAPE + Sym. (Theorem 2)	psvr	3.13 [2.74, 3.54]	0.465	1.99	0.860	0.002
LS-RMSPE (Theorem 3)M	psvr	3.60 [3.33, 3.87]	0.558	2.29	0.832	0.002
LS-RMSPE (Theorem 3)R	psvr	3.60 [3.33, 3.88]	0.558	2.29	0.832	0.002
LS-RMSPE + Sym. (Theorem 4)M	psvr	3.17 [2.84, 3.52]	0.484	2.02	0.858	0.002
LS-RMSPE + Sym. (Theorem 4)R	psvr	3.18 [2.85, 3.52]	0.484	2.02	0.858	0.002

Means over 10 rolling-origin cross-validation windows preserving temporal causality; 95% percentile bootstrap CIs (B = 1000) for MAPE; secondary metrics report means only. Bold = lowest MAPE/MASE/MAAPE or highest R². $p_W$: Wilcoxon test vs. $\epsilon$-SVR (MSE). Time-series baselines (ARIMAX, ETS, and Prophet) alongside ML baselines; 10 rolling-origin windows limit statistical power relative to Table 3. ^M/^R: tuned under MAPE/RMSPE objective.

are consistent across the psvr symmetric variants. All three symmetric model configurations (Theorems 2 and 4, the latter under both MAPE- and RMSPE-optimised tuning) place in the top five by MAPE, with the SVR-MAPE + Symmetric Kernel variant attaining the lowest MAPE on the dataset; the non-symmetric SVR-MAPE variant of Theorem 1 is also competitive but is consistently outperformed by its symmetric counterpart, isolating the contribution of the kernel modification from the percentage-error component. The classical $\epsilon$-SVR (MSE) baseline ranks fifth, and the time-series baselines (ARIMAX, ETS, and Prophet) underperform the kernel methods by approximately $2.0$–$3.7$ percentage points on MAPE—a gap consistent with the literature on engineered-feature ML versus low-order time-series methods on multivariate demand forecasting [9,13].

The Wilcoxon p-values for all six psvr models reach the floor of $p=0.002$, which is attainable with $n=10$ rolling-origin windows for a two-sided paired test. This floor is reached only when a model wins (or ties without losing) against the comparator on every single window, which is the case here: each psvr variant achieves a strictly lower MAPE than $\epsilon$-SVR (MSE) on each of the ten windows. The smaller sample size relative to the cross-sectional experiments precludes finer discrimination between psvr variants on this dataset, but the uniform domination of the structural baseline is unambiguous.

Combined with the structural-equivalence test of Section 4.4 on Boston Housing—where the symmetric kernel and the reflection-augmentation procedure produce numerically equivalent solutions despite the absence of genuine reflection symmetry in the data—the present result supplies the complementary positive case: when the data do respect the assumption of Definition 2, the symmetric-kernel modification produces empirical gains exactly of the form predicted by Theorems 2 and 4. The two experiments together validate the symmetric extensions: the structural mechanism is confirmed by the reflection-augmented ablation (Section 4.4), and the empirical gains materialize when the symmetry assumption holds (this section).

5. Conclusions

This paper presents a unified structural family of support vector regression models that incorporate percentage-error loss functions and optional symmetry constraints. Starting from Vapnik’s primal formulation, we derived four models via Lagrangian duality and Karush–Kuhn–Tucker analysis: $\epsilon$-SVR with MAPE (Theorem 1), its symmetric extension (Theorem 2), LS-SVR with RMSPE (Theorem 3), and its symmetric counterpart (Theorem 4).

The central structural finding is that percentage scaling and symmetry enter the optimization through independent mechanisms. Percentage scaling induces sample-dependent box constraints $\left|{\beta }_k\right|\le 100C/y_k$ in the $\epsilon$-SVR paradigm and a target-weighted diagonal matrix $Y_{\mathrm{\Gamma }}=\mathrm{diag}(y_k^2/\mathrm{\Gamma })$ in the LS-SVR paradigm. Symmetry constraints, by contrast, affect only the effective kernel—replacing K by $K_s$ in the $\epsilon$ case and $\mathrm{\Omega }$ by ${\mathrm{\Omega }}_s$ in the LS case—without altering dual variables, box constraints, or regularization matrices. This separation ensures that convexity is preserved throughout the family (Proposition 1), unique LS solutions exist via the Schur-complement argument (Proposition 2), and the solver structure is retained in all four variants (Theorem 5).

A capacity-bound analysis (Remark 11) and a quasi-likelihood interpretation (Remark 12) situate the family within the Steinwart–Christmann capacity-control framework [29] and the de Myttenaere Bayes-MAPE characterization [7].

The case studies on Boston Housing, Diabetes, Energy Efficiency, and the Victorian electricity-demand time series empirically confirmed these structural predictions, with results that depend systematically on the data regime. The $\epsilon$-SVR + MAPE variant of Theorem 1 significantly improves on its structural comparator, the classical $\epsilon$-SVR (MSE), on every cross-sectional dataset ($p_W<0.001$ in all three), and additionally outperformed all other baselines on Diabetes ($p_W<0.001$ against the strongest non-psvr baseline). Against XGBoost on Boston Housing and Energy Efficiency—two datasets where gradient-boosted ensembles excel due to nonlinear feature interactions and smooth target structure—the $\epsilon$-SVR + MAPE variant remains competitive. Still, it does not dominate, consistent with the well-documented strength of tree-based methods on tabular regression [49]. The LS-percentage formulation of Theorem 3 demonstrates dataset-dependent behavior: it ranks among the top performers on Energy Efficiency (where the target dynamic range $\rho \approx 7$ is moderate) but degrades on Boston Housing and Diabetes (where $\rho \approx 10$–14 produces the structural dynamic-range sensitivity characterized in Remark 21). Tuning $\mathrm{\Gamma }$ under MAPE versus RMSPE objectives produces systematically different bias–variance configurations, consistent with the adaptive role of $Y_{\mathrm{\Gamma }}$ in the linear system. The reflection-augmented Boston Housing experiment (Section 4.4) confirms the structural claim of Theorem 5: on data without genuine reflection symmetry, the symmetric-kernel modification and reflection-based augmentation yield numerically equivalent solutions. The complementary positive case is the vic_elec evaluation (Section 4.6), where seasonal symmetry holds approximately by physical structure and all three symmetric-kernel configurations place in the top five by MAPE, with SVR-MAPE + Symmetric Kernel achieving the lowest overall.

5.1. Limitations and Threats to Validity

The present framework has seven limitations, each pointing to a concrete open direction.

1.

Strictly positive-target scope. The derivations assume $y_k>0$ (Assumption 2). The three workarounds discussed in Remark 9—offset regularization, truncation, and MAAPE substitution—preserve most of the machinery but are not theoretically characterized with the same rigor as the $y_k>0$ case; a fully general percentage-error framework for possibly zero targets, in particular via a differentiable regularized MAPE surrogate with explicit Bayes-optimal characterization, remains an open direction.

2.

Symmetry-assumption restriction. The symmetric-kernel extensions (Theorems 2 and 4) require the target-side symmetry condition of Definition 2. The practical diagnostic of Section 4.4 is operational, not a formal goodness-of-fit test; a rigorous hypothesis test for Definition 2 on natural data remains to be developed. The boundary case of odd symmetry $a=-1$ with the Gaussian RBF kernel (Remark 5) requires additional regularization in the $\epsilon$-SVR variant.

3.

Benchmark-era datasets. Boston Housing is retained for continuity with the conference precursor [18] and as the substrate for the structural-equivalence test of Section 4.4, despite its known age and the documented ethical concerns about its “B” feature (see Section 4.1). The revised experimental suite mitigates this via two modern datasets in the cross-sectional evaluation (Diabetes from [32], Energy Efficiency from [33]) and the time-series evaluation on Victorian electricity demand (Section 4.6), but Boston Housing remains in the suite for its role in the reflection-augmented experiment.

4.

Cross-validation protocol selection. The cross-sectional experiments use generic nested IID cross-validation (Appendix B.2), while the time-series evaluation of Section 4.6 uses rolling-origin cross-validation. The two protocols are not directly comparable: the smaller sample size of 10 rolling-origin windows on vic_elec limits statistical power relative to the 30-split bootstrap of the cross-sectional experiments. Still, it provides causal validity that random splits cannot. Practitioners deploying the percentage-error SVR family in time-series applications should adopt the rolling-origin protocol in Appendix B.6; the M-competition convention surveyed in [14,48] provides the canonical reference for additional variants (expanding-window, sliding-window, blocked-time-series).

5.

Single-output regression scope. Theorems 1–4 are derived for single-output regression. The extension to multi-output or vector-valued targets is straightforward in principle (each output dimension inherits the same primal–dual structure) but is not formally proved; the representer theorem for matrix-valued kernels under percentage-error loss would require a separate treatment.

6.

Generalization-bound specialization. The generalization bound of Remark 11 is a direct corollary of the Steinwart–Christmann framework under the capacity bound $100C/\lambda$. Sharpening the bound for specific kernel classes (Gaussian RBF, Matérn) via the integral-operator route of [50,51] remains open Cucker–Smale (2002) or Smale–Zhou (2007); the present bound is tight up to logarithmic factors in the Rademacher complexity but is suboptimal in the explicit dependence on the dynamic-range factor $C/\lambda$.

7.

Unbounded-support targets. The target-dependent box-bound analysis of Theorem 1 assumes $\left|Y\right|\ge \lambda >0$ almost surely. Targets with unbounded support (e.g., Gaussian with infinite variance or heavy-tailed distributions) fall outside the scope condition and would require a separate analysis; the capacity bound $100C/\lambda$ diverges as $\lambda \to 0$, and the generalization bound of Remark 11 correspondingly fails to provide a useful rate.

5.2. Future Work

Several directions for future work complement the limitations identified above.

Empirical expansion on naturally symmetric data. Further empirical evaluation of the symmetric extensions (Theorems 2 and 4) on datasets where the target function exhibits known even or odd structure—such as physical systems with reflection symmetry, periodic energy-load profiles with daily or weekly cycles, or detrended financial-return series with an odd noise model—would provide direct validation of the kernel-modification mechanism on naturally symmetric data, complementing the reflection-augmented experiment of Section 4.4.

Near-zero target regime. Extending the percentage-error framework to targets that may approach zero, through differentiable MAPE approximations (offset regularization with characterized bias), regularized percentage losses, or principled use of MAAPE with Bayes-optimality characterization, would broaden the applicability of the family to settings where the strict positivity assumption (Assumption 2) cannot be guaranteed—notably intermittent-demand forecasting and target-crossing time series.

Online and incremental solvers. The $\epsilon$-SVR variant already ships an SMO solver tailored to the sample-dependent box-bound regime (Algorithm 1, [35]); the LS-percentage variant is amenable to incremental Cholesky updates given its bordered linear-system structure. Extending both to online settings—warm-start updates as new $(x,y)$ arrive and decremental active-set updates as samples leave—remains open.

Adaptive regularization for wide-target dynamic ranges. The structural sensitivity of the LS-percentage variant to the target dynamic range $\rho$ identified in Remark 21 suggests two complementary research directions. First, alternative weighting schemes that interpolate between the $y_k^2$ weighting of Theorem 3 (RMSPE-optimal for moderate $\rho$) and the unit weighting of classical LS-SVR (R²-optimal under absolute-error metrics)—for example, a regularized-percentage formulation with weights $max(y_k^2,{\delta }^2)/\mathrm{\Gamma }$ for tunable $\delta$—would broaden the applicability of the LS-paradigm to wide-range targets without abandoning the percentage-error framework. Second, a principled criterion for selecting between the $\epsilon$-SVR + MAPE variant (Theorem 1, robust to dynamic range via dual-variable bounds) and the LS-RMSPE variant (Theorem 3, more efficient computationally but sensitive to $\rho$) on a per-dataset basis would translate the empirical observations of Section 4.3 into deployable guidance.

Multiple-kernel and representer extensions. The structural family can be extended to multiple-kernel learning combining symmetric and non-symmetric basis kernels, with the $\epsilon$-percentage and LS-percentage duals providing the outer objective for the kernel-weight learning problem. Similarly, the representer theorem holds under the symmetric-kernel modification of Theorem 5, enabling the structural family to serve as a building block for deeper kernel-based architectures.

Uncertainty quantification in the percentage-error regime. Aligned with the Special Issue theme, extending the point-estimator framework to probabilistic variants—prediction intervals via conformal or Bayesian quantile regression under the $1/y$ weighting, calibrated percentage-error intervals, and interval-valued target extensions—would deliver uncertainty-aware percentage-error regression suited to high-stakes forecasting in FinTech, energy, and supply-chain applications.