KASPER: Kolmogorov Arnold Networks for Stock Prediction and Explainable Regimes
Authors: Vidhi Oad
Param Pathak
Nouhaila Innan
Shalini Devendrababu
Muhammad Shafique
Venue: Accepted by TMLR ()

Abstract: Forecasting in financial markets remains a significant challenge due to their nonlinear and regime-dependent dynamics. Traditional deep learning models, such as long short-term memory networks and multilayer perceptrons, often struggle to generalize across shifting market conditions, highlighting the need for a more adaptive and interpretable approach. To address this, we introduce Kolmogorov–Arnold networks for stock prediction and explainable regimes (KASPER), a novel framework that integrates regime detection, sparse spline-based function modeling, and symbolic rule extraction. The framework identifies hidden market conditions using a Gumbel-Softmax-based mechanism, enabling regime-specific forecasting. For each regime, it employs Kolmogorov–Arnold networks (KANs) with sparse spline activations to capture intricate price behaviors while maintaining robustness. Interpretability is achieved through symbolic learning based on Monte Carlo Shapley values, which extracts human-readable rules tailored to each regime. Applied to real-world financial time series from Yahoo Finance, the model achieves an $R^2$ value of 0.89, a Sharpe Ratio of 12.02, and a mean squared error as low as 0.0001, outperforming existing methods. This research establishes a new direction for regime-aware, transparent, and robust forecasting in financial markets.

Published in Transactions on Machine Learning Research (02/2026)
KASPER: Kolmogorov Arnold Networks for Stock Prediction and Explainable Regimes
Vidhi Oad"
vidlii.cc200mmaillcco com
Vishwakarma Government Engineering College, Ahmedabad, India
Param Pathak"
Sarem.pethhakOffaccche
QuantumAl Lab, Fractal Analytics, Mumbai, India
Nouhaila Innan
nouhailaa innan Dryu. edu
eBRAIN Lab, Division of Engineering, New York Universily Abu Dhabi (NYUAD), Abu Dhabi, UAE
Center for Quantam and Topological Systems (CQTS), NYUAD Research Institute, NYUAD, Abu Dhabi, UAE
Shalini D
shalini, heni.dddeeennbbue@ffcttt.l...
QuantumAl Lab, Fractal Analytics, Gurugram, India
mahhammad shafique @ryu edu eBRAIN Lab, Division of Engineering, New York University Abu Dhabi (NYUAD), Abu Dhabi, UAE Center for Quantum and Topological Systems (CQTS), NYUAD Research Institute, NYUAD, Abu Dhabi, UAE
Muhammad Shafique
Reviewed on OpenReview: https: openreviee. net/ forum? id- PD4JGJQQ1L
Abstract
Forccasting in financial markets remains significant challenge due to their nonlinear and regime-dependent dynamics. Traditional deep learning models, such as long short- term memory networks and multilayer perceptrons, often struggle to generalize across shifting market conditions, highlighting the need for a more adaptive and interpretable approach. To address this, we introduce Kolmogorov-Arrolo networks for stock prediction and explainable regimes (KASPER), novel framework that integrates regime detection, sparse spline- based function modeling, and symbolic rule extraction. The framework identifies hidden market conditions using Gumbel--Softmx---ene specific forecasting. For each regime, it employs Kolmogoror-Arnol networks (KANs) with sparse spline activations to capture intricate price behaviors while maintaining robustness. Interpretability achieved through symbolic learning based on Monte Carlo Shapley values, which extracts mnadabre rules tailored to each regime. Applied real-world financial time series from Yahoo Finance, the model achieves an R² score of 0.89, a Sharpe Ratio of 12.02, and mean squared error low 0.0001, outperforming existing methods. This research establishes new direction for regime-aware, transparent, and robust forecasting in financial markets.
mechanism, enabling regime-
1 Introduction
The stock market is a complex, dynamic system influenced by multiple factors, including economic factors, global events, and investor emotions. Predicting its behavior is a critical challenge with significant impact on financial decision-making, portfolio management, and risk assessment. However, the volatility, non-stationarity.
and abrupt regime shifts in market behavior make accurate forecasting challenging task (A & James 2023
The key problem we address is the accurate prediction of stock market behavior, particularly during regime shifts such as transitions between bullish, bearish, and stagnant phases. These shifts are important because
"These authors contributed equally.

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Published in Transactions on Machine Learning Research (02/2026)
they represent changes in market dynamics, and failing to adapt to them can lead to substantial financial losses (Kokare et al. 2022). Traditional models like Autoregressive Integrated Moving Average (ARIMA) (Ho & Xic, 1998), and Generalized Autoregressive Conditional Hcteroskcdasticity (GARCH) (Banwens et al. 2006), while effective for stationary data, struggle to capture the nonlinearities and temporal dependencies
inherent in financial time series (Alkhfajee & Al-Sultan 2024 Crawford & Fratantoni 2003).
State-of-the-art approaches to stock market prediction can be broadly categorized into statistical models, Machine Learning (ML) models, and hybrid methods, each with distinet strengths and limitations. Statistical models, such as Hidden Markov Models (HMMs) and regine-switching models (Yuan & Mitraa 2016), are interpretable and thcorctically grounded but rely on rigid paramctric assumptions, fixed transition probabilities, and Gaussian distributions, which fail to capture the nonlinearities and abrupt regime shifts of woal-world markets (Watorek et al., 2021), leading to poor performance during volatile periods. MT. models, including Long Short-Term Memory networks (LSTMs), transformers, and Large Language Models (LLMs), excel at capturing complex patterns and long-term dependencies, with LLMs even incorporating external textual data for enhanced predictions.
However, their black-box nature limits interpretability (Chen et al. 2023 and they often struggle to adapt to distinct market regimes, eit her overfitLing to historical patterns or introducing lookahead bias due to poor temporal aligument (Bhandari et al., 2022 Zhang et al. 2022). Hybrid methods aim to combine the strengths of both approaches, but they frequently fail to enforce sparsity, leading to overfitting (Islam et al. 2024). In this context, sparsity enforcement refers to applying L1 regularization (A]lW'))) to the model parameters W to drive many weights to zero, thereby reducing model complexity and mitigating overfitting, Most hybrid ensemble methods lack such explicit sparsity constraints their feature-importance weights ensemble coefficients. They also lack robust mechanisms for temporal aligument, which can result in data leakage and inflated performance metrics. More specifically, data leakage occurs when future information enters the training process, for example, by using future prices for normalization or allowing validation//ten windows to overlap with training periods. This improper temporal scparation inflates metries such as accuracy, Sharpe ratio, and R² by allowing the model to learn from information it would not have access to at prediction time, leading to overly optimistic assessment of real-world performance. Additionally, they often fail to provide clear, actionable insights into regime-specific drivers, limiting their practical utility (Haase & Neuenkirch, 2023). These limitations highlight the need for more advanced methods, particularly for modeling stock market behavior.
To address these challenges, we propose KASPER, a novel framework that uses KANs [Liu et al. 2024b and integrates adaptive regime detection with sparse, interpretable feature engineering. Specifically, we enforce ()) and dynamic feature masking,
sparsity through L1 regularization on regime-specific forecast weights wj ensuring that only the most predictive signals influence each regime. This sparse modeling is implemented within KANs, which are inspired by the Kolmogorov-Arnol representation theorem (Schmidt-Hieber (021), stating that any multivariate continuous function can be decomposed into finite sum of univariate functions. Accordingly, KANs implement this decomposition through learnable apline-based activation functions, enabling flexible function approxination. As illustrated in Fig. [ the KAN architecture maps raw inputs through learnable splinc-activated units (bottom squares) and then lincarly recombines the transformed signals across successive layers, producing regime-aware predictions (top node). Each activation Φ(x) is constructed from B-spline bases defined on a knot grid, ie., a set of breakpoints {ki, kg...... ... km that partition the input domain into intervals, and refined across progressively finer grid resolutions. The knot locations control the spline's local behavior by determining where basis functions are centered and how Φ(z) transitions between polynomial segments as x varies. This grid refinement improves expressive power while preserving interpretability, allowing KANs to represent high-dimensional mappings with fewer layers.
Our approach uses dynamic opline-based activations (Vecci et al. 1998) to capture nonlincar price dynamics and adapt to regime shifts through robust percentile-based initialization of spline knots. Specifically, for each feature dimension, knot positions are initialized based on the effective range of the training distribution. Mathematically, given a feature vector x E R", we first determine the robust boundaries Zmin = quantile(x, Pmii) and Tmax = quantile(x, Pmax), where Pmin and Pmax are lower and upper percentiles (e.g., 0.01 and 0.99) chosen to exclude outliers. We then compute knot locations ] on uniform grid within these bounds: kg = Emin + (g- - Fmax where G is the total number of grid points and g indexes
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2,1
Spline Basis Function Approximation
4/11 2n Byx Bytx."")
KAN uses spline basis functions 9(k) defined over knot grid.
111.1
122,1
13311
1,,
115,11
enhance flexbilly: we extend the knot grid from coarse (3. to finer (Gr)
911.1
1121
113.1
$14.1
91.5.1
resolution.
fewer knots simpler etx) G: more knots - higher expmssivenees
11,
1,2
11,
X1,4
1,5
8,22.
80.24
K0.211
20,1,5
fo2.55
KASPER uses regime specific spline
1..111
1,.
0.13
functions with quanti--atttattzeedd
knots.
huu
hu2
Auz
Pazz
Auza
Grid extension enables grained
modeling of non- liner financial pantems
acress different market regimes,
10,1
0,2
KAN architecture with two input features and spline-activated units across layers. Inputs T0,1 and connect to spline units Φo.s, that produce transformed features 20, i.j. which are selectively routed hidden nodes æ1,j and aggregated to the output J2,1- Each Š1,i denotes learned univariate spline transformation applied to linearly projected input, supporting interpretable feature-to-node mappings and sparse connectivity. The right: inset illustrates spline basis- function approximation knot grid and the effect of grid extension (G1 G2), where increased knot resolution improves expressiveness and enables smoother approximations.
Figure 1:
the knot. This construction ensures that the spline bases the effective support of the distribution with consistent resolution and improved stability under extreme values. To ensure temporal alignment and prevent lookahead bias, we employ strict closed-left windows for rolling feature calculations and carefully shift historical data. To further improve generalization, sparsity is enforced through two complementary mechanisms. First, we apply L1 regularization to all model parameters, including spline basis coefficients and forecast weights. Second, we use dynamic masking that selectively zeros out maglltm spline coefficients ((() based on regime-specific thresholds. This dual approach retains only the most significant weights and basis functions, redueing overfitting. For interpretability, we use Monte Carlo Shapley values (Ghorbani & Zoo 2019) to quantify the contribution of each feature to regime-specific predictions, enabling the extraction of actionable rules, such identifying dominant features like High-Low (HL) in bearish regimes and Open Close (OC) in bullish regimes. In particular, HL is defined HL, In He ( where H;
and Le are the daily high and low prices, and OC is defined as OC, In (.): where O, and C2 are the opening and closing prices. This combination of adaptive modeling, sparsity en enforcement, and interpretability enables regine-specific analysis across diverse market conditions.
The key contributions of this study are as follows:
A novel Aegime-Adaptiee Forecasting Layer that makes use of sparsely activated splines to model distinct market dynamics across regimes. This mitigates the overfitting risk in financial time series by ensuring sparsity-constrained representations that generalize well.
An orthogonality constraint within the regime detection network enforce disentangled representa- tions for different market states. This prevents feature collapse, ensuring each regime retains distinct and interpretable characteristics.
For seamless discernibility of the regimes, Contrastive Regularization is used. helps maximize inter-regime dissimilarity while maintaining intra-regime coherence, thereby improving the stability of regime classification and reducing misclassification in volatile market conditions.
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Utilizing Monte Carlo Shapley method with temporal weighting to extract interpretable, regime- specific rules. This enhances the model's transparency by identifying dominant factors influencing each market regime.
The rest of the paper is organized as follows:
Sec. 2 reviews related work on market regime detection, KAN- based financial modeling, and interpretability in finance; Sec. 3 presents the proposed KASPER framework; Sec. 4 reports the experimental setup and results, including ablation studies and comparisons with bascline models; finally, Sec. concludes with a sunmary of the main findings and directions for future work.
2
Background and Related Work
Recent work in financial forecasting includes hybrid deep architectures for price prediction Liu et al. 2025 Elhoscny et al.) 2025). transformer-based models for capturing long-range temporal dependencics under volatility (Kabir et al., 2025 Hadizadeh et al. 2025), and weinforcement-learriing frameworks for portfolio decision-maie makinng and risk control non-stationary markets (Yao) 2025: Ye et al., 2020). Consistent with the scope of KASPER, this section focuses on market regime detection, KAN-based approaches for financial time series, and explaimablity//tterrrtttiiii methods in finance.
2.1 Market Regime Detection
Financial markets transition between distinct regimes, characterized by varying volatility, return distributions, and risk factors. Early work established regime switching via Markov models (Hamilton 1989):
Ti Ps. + ~~,000 2.
(1)
j(===11 22-1 ( Aij.
(2)
where Tt is the observed return at time t, is the latent regime indicator, and A is fixed transition matrix. While interpretable, this approach suffers from static regime centroids and rigid paramcters.
Ang and Bekaert (Ang & Bekaert 2002). showed that modeling asset returns as regime-dependent improves forecasting and risk management:
Tt 2 = k~ N (14,,
(3)
with portfolio weights dynamically adjusted according to the inferred regime.
Building on this, Guidolin and Timmermann (Guidolin & Timmermann 2007 exlended the framework using a multivariate Markov-switching model with four regimes (crash, slow growth, bull, and recovery). Regime transitions follow:
P(2 = 1 = Asa i,j = 1,
(4)
where Aja is the transition probability. In their portfolio choice setting, the investor's dynamic optimization problem at time to is formulated as:
W1-1 B
J J rb, 20, 0, to) max Es,
(5)
1-
where W, denotes the investor's wealth at time 4, l and 2/ are the vectors of historical asset returns and predictor variables, respectively, O, collects all parameters of the regine-switching process, ny is the vector of filtered regime probabilities at to. and tg a discrete decision epoch (rebalancing date) with b = 0,....,, B 1. The maximization performed over the vector of portfolio weights w allocated to risky assets, and the objective is the expected utility of terminal wealth WB with relative risk aversion 7, conditional information available 4. Bayesian updating of regime probabilities performed via:
(b+) (0.) n1 1; 0.) (1.) C 11 2.
(6)

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The Vector Smooth Transition Autoregressive (VLSTAR) model alternative that used a continuous transition function:
[uucii & Ciciretti 2021
was also an
y1 = po +
Po.JUr-s + Aoxr + Ge(slinic)
N1+ + Σ
P1cU1-s + Alos
+ t
(7)
5=1
with the logistic function defined by:
G.(s67.c)) = [1 cxp(->((<< - -c)
(8)
where 8t triggers regime shifts and y controls transition smoothness (low y yiclds gradual changes; high 7 produces abrupt shifts). This model dynamically captures volatility shifts in market conditions.
2.2 KANs for Finance
KANs are effective in financial modeling, particularly for option pricing and stock prediction. Existing Informed Neural Networks (FINNs) for option pricing suffer from poor learning efficiency, high computational costs, complex training processes, and limited interpretability due intricate loss functions requiring larger architectures and extensive training to maintain accuracy. To address these challenges, Liu et. al. Tiu et al. 2024a] proposed FINN KAFIN, that integrated financial equations into a neural framework. The Black Scholes formula (Barles & Soner 1998 which governed their pricing model can be represented as:
aC Ōt
+ 0² g² 02
CC 0s + +
ac
- - =0
0s
where t( is the price of a financial option, S is the price of the underlying asset, or is the volatility of the asset's returns, and r is the risk-free interest rate. The price function approximated using:
N C(S,t) = Σ (h)))))
(10)
where di and hz(z) are trainable functions. The model minimizes a loss function incorporating financial constraints:
I.(0) Aint Tinit + Aboundary /bouddayy Afnancial Liianccial-
(11)
On the other hand, to enhance stock price prediction, Yao Yao 2024 proposed an LSTM-KAN hybrid model, where LSTM captures temporal dependencies and KAN refines nonlinear patterns. KAN improves nonlinear mapping via:
2n++1 Σ
Σ hag(az)
f(x)
(12)
where 9i and hug trainable functions the model is trained using. Empirical results showed that KAFIN enhanced option pricing accuracy, while the LSTM-KAN model significantly reduced stock forecasting errors.
2.3 Explainability and Interpretability in Finance
Explainable AI (XAI) techniques address the transparency challenge in black-box deep reinforcement learning (DRL. models for portsolio management, where deep NNs provide limited insight into their decision-making, et al. (de-la Rica- Escudero et al., 2025) noted that many DRL approaches provide explanations only during training and do not support monitoring the agent's 's bchavior at trading time; only a small number of prior studies explicitly considered explainability. They proposed a post hoc Explainable DRL framework that combines Proximal Policy Optimization with model-agnostic XAT methods (feature importance, SHAP, and LIME) by saving state action pairs during training and generating explanations at inference time. Evaluated on five U.S. technology stocks using daily OHLC data 2015-2018), the framework highlighted asset-specific variations in OHLC importance and produced instance-level explanations of portfolio
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allocations, with statistical validation of Shapley-value stability (variance below 10-?, < 0.01 across 50 repetitions), enabling real-time monitoring of agent decisions during trading.
Interpretability challenges also persist in financial Al systerss where complex architectures obscure the factors driving predictions. Ni et al. (Ni et al., 2025) argued that standard time-series visualization is often insufficient to reveal the temporal patterns underlying model decisions in financial risk assessment, which can reduce analyst trust and complicate regulatory compliance. They introduced contrastive visual analytics framework that combines entropy-based lemporal importance weighting with interactive dimensionality reduction (t-SNE, UMAP) for multivariate financial time series, enabling direct comparison of normal versus anomalous patterns and highlighting differences in feature attributions across risk scenarios. Empirical evaluation on credit risk assessment (5,248 corporate cases) and market volatility prediction reported reduced analyst decision time and improved agreement and anomaly detection, while maintaining sub-100 ms response times for interactive exploration.
Despite these advances, important gaps remain. Traditional regime detection methods rely fixed transition matrices that require manual specification, while many recent. Deep Learning (DL) models apply uniform processing without adapting to distinct market conditions. Hybrid architectures can achieve strong predictive accuracy, but they often lack explicit reginie-wwitching mechanisms and provide primarily global explanations rather than regime-specific interpretability. Our work addresses these gaps by jointly performing regime detection and forecasting via diffcrentiable Humbel-Softmas classification, introducing Monte Carlo Shapley values with temporal weighting for regime-specific interpretability, and enforcing regime-specifie sparsity through orthogonality constraints to mitigate feature collapse.
3 KASPER Framework
Dataset Preparation
Load Yahoo Dataset
KAN 1 Regime Detection Layer
KAN Regime Adaptive Forecasting Layer
Training & Optimization
Evaluation
Feature Engineering
Regime Analysis
One-Hot Encoding
Regime-Aware Splitting
Spline Activation Quantile-ormmalieed per-feature
Update Sparsity Mask Update masks & compute attention
Optimize parameters via gradient-based learning
Compute MSE, RMSE, MAE, & Sharpe Ratio
Training - 70%
Use Monte Carlo Shapley Values quantify individual feature contribution
Testing 15%
Feature Embedding Sttack output to create embedded features
Regime-specific forecast Combine activations & weight sum
Compute total loss using a regularized loss function
Validation 15%
Regime Classifier
Update model via backpropagation and Adam optimizer
Classily data via softmax- regimes
Embedded features go through: (L-BN-G-D)a G.())())
Aggregation & Connection Aggregate, refine
OrthoReg
Figure Workflow of the KASPER framework. The pipeline begins with Yahoo dataset preparation, feature engineering, one-hot encoding, and regime-aware splitting into training (70%), validation (15%), and testing (15%). KAN-1 performs regime detection using quantile-nornalized spline activations, learns embedded features, and assigns regimes via a Gumbel-Softmax classifier with orthogouality regularizer. KAN-2 produces regine-adaptive forecasts by updating a sparsity mask over spline components and computing attention-based aggregation before the final refinement. Parameters
learned with gradient-based optimization (AdamW) under regularized composite loss. Performance is evaluated using MSE, RMSE, MAE, and Sharpe Ratio, while interpretability is provided via Monte Carlo Shapley value estimation and regime-specific rule extraction.
The proposed KASPER framework is composed of two core stages: regime detection and regine-adaptive forecasting, illustrated in Fig. 2 and described in Algorithm [ The input consists of an n-day window of financial time series data, structured state matrix $ xxx where n denotes the number of historical

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days and f the number of features per day. Each row in P. captures log-transformed returns and volatility indicators to stabilize variance and enhance stationarity:
High s-++1 In
flow
Open t-i+1 Open Jti
L1,1 = In Pt-it1 Pi-i
,
.In
In
HLow
In
Vi-i+1
HH.
_i___sprcadd_s OC_ OC_spread, - volatility_raaiio...
(13)
Vi-i
stt.ectty.ipppcc.ccccccaaooa...r volume_change,.- volume
state volume_state,_j
where Pt-; denotes the closing price and LLig and fLow denote the high and low prices, respectively, on day i. The engineered features include HL (denoted as HL spread), OC (denoted OC spread), Average True Range (ATR), volatility ratio, price velocity, price acceleration, volume change, and volume state. These eight features are selected via SelectKBest with f regression scoring and form the final input to KASPER; further details are provided in Sec. 4.1
Algorithm 1: KASPER
Input: Raw financial dataset
Output: Trained model, regime-specifie rules, performance metrics
// Step 1: Preprocessing Feature Engineering
Load dataset D and forward-fill missing values
Extract features: lags, rolling statistics, volatility, volume, momentum, temporal dummies
Compute target as: Cyp-Ce
C2
Split data into Drrwin Deat, Drest
Apply feature selection via SelectXBest Standardize features using StandardScaler
Step Define KASPER Architecture
Define Spline Activation for nonlinear mapping
Create Regime Detection Layer to estimate soft regime probabilities using Gumbel-Softme Initialize Regime Adaptive Forecasting Layer with regime-specific basis functions and attention mechanism Combine both layers into KASPER model
10
// Step Train the Model
11 for epoch = to do for each batch (2i, Wk) E Dirain do
13
Compute predictions Út, regime probabilities embeddings
14
Compute loss: = Lpred + Aehomntraative + Ao AcLaparae + A Mongoua + Ao Coniaanee
Backpropagate and update parameters
16
Evaluate Deat and apply early stopping if needed
// Step 4: Extract degime-Specific Rules
17
for regime r= to R do Identify samples with highest from regime probabilities
18
Compute Shapley value o5 for each feature
Select top-3 contributing features as rules for regime r 20
// Step 5:
Evaluate Financial Metrics
Apply model to Drest to compute: Sharpe Ratio, Max Drawdown, Cumulative Returns, Win Rate, Direction Accuracy
22
3.1 KAN Layer 1: Regime Detection
The regime detection module is designed uncover latent market regimes using spline-activated KANs. This layer incorporates four components: spline activation functions, Cumbel-Softmax--asse regime classification, contrastive loss for representation separation, and orthogonality regularization to enforce disentangled regime-specific embeddings.

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3.1.1 Spline Activation Function
Each input feature is processed through a hybrid spline activation function that captures both linear and nonlinear trends:
f(x) L(x) + C(z),
(14)
where L(x) and C(x) are the linear and cubic components, respectively. These are defined as:
Nanear
[(x) =
Σ
lanh(wm) [ReLU([norm - km) ReLU kim+1)) 1)]
(15)
m=0
Neable
C(x)= =
(vm (lmm)=
(16)
where Wm and Um are trainable parameters, a(.) denotes the sigmoid function, and Znorm is the input normalized with respect to the knot sequence {km]. To improve robustness to outliers, we place the knots {km J on a uniform grid bounded by empirical percentiles of the imput feature x. Specifically, the knots are computed as km = 2min + (m - =- where minn quantile(x, Pmin) and Tmax = quantile(x, Pmax) are the lower and upper percentile bounds (e.g., Pmin 0.01 and Pmax 0.99), and G the grid size.
3.1.2 Differentiable Regime Classification via Gumbel-Softman
The classification of regime probabilities is achieved through the Gumbel--Softmax function, which provides a differentiable approximation to categorical sampling. We denote regimes by the index r E (1...., R), where R the total number of regimes (set to 3 in our experiments):
P(
exp (fr(2, + gr)/T)
(17)
Dt exp (fa 9 + 9T)
where 2, represents the input feature matrix time t, fr denotes the polnnetransffrmed activation corresponding to regime r, Gr is the Gumbel noise sampled from Gumbel(0, 1), the temperature parameter, and R is the total number of regimes.
3.1.3 Contrastive Loss for Regime Separation
To ensure that the latent embeddings corresponding the same regime remain well clustered, a contrastive loss is introduced:
Ccontrastive E ||| 2yl1 Yayl
(18)
where 2; and žz denote the latent embeddings of samples i and j, and Vıy E {0, 1} is binary indicator defined as Vij = samples and j belong to the same regime and yı = 0 otherwise. Specifically, 2; = $ detect (x;) is the output of the spline-based regime detection layer for input x; E R", before the softiax classification step. The expectation E is taken over the empirical distribution of the training mini-batch B, with inputs x treated as random samples from the market data distribution D. This term enforces intra-regime compactness by minimizing the squared distance between embeddings assigned to the same regime.
3.1.4 Orthogonality Regularization
To enforce distinctiveness across regime-specific transformations, orthogonality is imposed on the regime weight vectors:
Larth |WWT - IR
(19)
where W E Ie RxF is the global weight matrix formed by vertically stacking the regime-specific row vectors W, for all R regimes, and IR is the R x Rt identity matrix. Specifically, the r-th row of W is W, = (r) ("))) wh))) ....,wF") w.f]]] where w is the learnable scalar weight for the j-th feature regime r, and F the total number of features.
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3.2
KAN Layer 2: Regime-Adaptive Forecasting
Following regime identification, forecasting is performed using second KAN layer with regime-specific parameterization. This layer employs sparse spline basis functions and is optimized using composite objective function incorporating robustness, sparsity, and structural regularization.
3.2.1
Forecasting Model
The predicted return 8!")
for regime is defined
0f Σ (
(20)
where Φ, is the feature matrix, of are spline basis functions and wnth are trainable coefficients.
3.2.2 Spline Basis Functions
Each basis function of is constructed using regime-specific B-splines:
8 Σ 2 Ba (b:;))))
(21)
where B& denotes the k-th B-spline basis function with knots g(t), and 8/2 spline coefficients learned from data.
3.2.3 Sparsity Enforcement
To promole interprelability and reduce overfitting, an lr-regularization term is applied:
ReLU([w)")) g(r)),
(22)
where ((r) denotes a regime-specific sparsity threshold. validation-based tuning of A.
The sparsity level is adaptively controlled through
3.2.4 Composite Loss Function
The full objective function for training integrates multiple loss components:
L = LHuher + As Σ |pl + Acfrontrastive + NoLorth + AoLralances
(23)
with O representing the set of all trainable model parameters, |pl denotes the absolute value (L1 norm) of each individual parameter p. This composite formulation ensures: (1) robustness to outliers through Huber
loss, (2) regime disentanglement via contrastive loss, (3) model sparsity through L11 regularization, (4) distinct regime representations via orthogonality constraints, and (5) balanced regime distribution across training samples.
3.3 Interpretability through Shapley- Based Rule Extraction
To enhance transparency, KASPER employs a Shapley value-based approach to interpret regime-specific forecasts.

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3.3.1 Shapley Value Estimation
For given feature j in regime r, its contribution quantified as:
(r)
E
|S - |S| - x
- (
(24)
FF|
SCAVO
where F denotes the full feature set, and (() is the regime-r model output with subset S.
3.3.2 Monte Carlo Approximation
approximate Shapley values efficiently, Monte Carlo sampling used:
M E U() (f
(25)
where Sm denotes randomly selected feature subsets (coalitions) and M is the number of Monte Carlo samples.
3.3.3 Temporal Weighting Scheme
To emphasize recent market behavior, temporal weighting is applied to the sequence of past Shapley values:
T
35 (r).t wro
Σ
Wt
(26)
E T-
where 7 E (0,1) is the decay factor controlling the emphasis on recent time steps.
3.3.4 Regime-Specifit Rule Extraction
For each regime r, the top three most influential features are selected:
R,
arg max JEF
((r)
j 1..3. Vr R),
(27)
resulting in interpretable rules of the form:
Regime r: X + X, + Njs Y
(28)
where X2y Xjs are the dominant features and Y, denotes the forecasted market response under regime r.
4 Results and Discussion
4.1 Experimental Setup
We evaluate KASPER on the Yahoo Finance dataset (ARORA 2023 for regime-aware stock return forecasting. The full experimental configuration summarized in Table To ensure leakage- setting, missing values are forward-filled and all rolling features are computed using strict closed-left windows, that only information available up to time t is used to construct the inputs at t.
Under this setup, KASPER is instantiated as a two-layer KAN model with two main components: a RegimelDetecttonLayer which learns spline-based representations and infers market states via Gumbel Softmax classification, and a Orthogonality regularization promates distinct regime representations, while the contrastive objective increases inter-regime separability, improving the stability of regime assignments.
www.aappiiioorrsstnnnggggeeeeee which produces regime-conditional forecasts.
Given the emphasis temporal consistency, engineer the input features using only historical observations to capture price dynamics, volatility structure, volume behavior, and regime context, with all rolling
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Table 1:
Experimental configuration of KASPER, including dataset details, preprocessing, model architecture, training setup, and evaluation metrics.
Category
Parameter
Value Salue/DDssin Desccripionn
Source
Yahoo Finance Datasel
Dataset
Time Range
Splitting Strategy
70% 1TK-15%--552 temporal train-val-test split
Data
Feature Engineering
15 engineered features (lags, rolling statistics, volatility, price dynamics)
Preprocessing
Feature Selection
8 features via Select with f_ [_ggression
Scaling Method
StandardScaler for both features and target
Number of Regimes
Model
Hidden Dimension
64
Architecture
Spline Configuration
Hybrid linear (3) and cubic (2) splines with quantile-based initialization AplineActivation and GELU
Activation Functions
Optimizer
AdamW (Ir=0.001, weight decay=le-5)
Batch Size
32
Loss Function
Huber loss with composite regularization
Training
Regularization Weights
Contrastive: 0.01, Sparsity: 0.001, Orthogonality; 0.01, Regime Bal- ance: 0.05
Parameters
Epochs
100 with early stopping (patience=15)
Learning Rate Scheduler
ReduceLROnPlateau (faclor=0.7, patience=7)
Gradient Clipping
0.5
Evaluation Metrics
Statistical
R, MAE, R²
Financial
Sharpe Ratio, Direction Accuracy, Max Drawdown, Win Rate, Profit Factor
computations respecting the closed-left window constraint. The feature set includes HL and OC spreads, k- day log-returns (R))) In for & E {1,7}, and a 21-day rolling volatility computed on past returns only: F E.- r and V ΣE (1 F))² with 21. We further compute the volatility ratio VR, and the Average True Range ATR, = - E
|H,-; Cr-i-ll,
to capture gap-aware price variability. Price momentum is summarized via velocity U. = C. - Ct-1 and acceleration A, U. Ut-1- Volume dynamics are represented using the normalized volume change AVol, (Vol, (ol.)/ + +), where Vol,
E Vol,-s and V is the standard deviation of [Vol,-)) We also compute volume-state ratio to flag unusual trading activity. Finally, regime-context. indicators computed from rolling summaries of volatility and momentum to reflect broader market conditions.
The experiments are conducted using Torch library for DL model implementation. Key supporting libraries include NumPy and Pandas for data manipulation, scikit-learn for preprocessing and feature selection; Matplotlib and Seaborn are used for visualization. The implementation and testing are conducted on PC with AMD Ryzen processor, 8 GB RAM, 128 GB SSD storage, and AMD Radeon graphics card.
4.2 Performance Across Regimes
In Fig. 3 breakdown of regime distribution patterns in the KASPER model is presented. The bar chart shows the percentage of samples classified into different market states, with each state defined by three characteristics, i.e., market direction (bearish, bullish, or neutral), confidence level (high or low), and regime number (0, 1, or 2). The most striking pattern is the dominance of neutral high-confidence states across all regimes. Neutral high regime 0 accounts for the largest portion at 18.7% of samples, followed by neutral high regime 2 at 16.6% and neutral high regime at 13.9%. This demonstrates that the model identifies stable market conditions where neither bullish nor bearish signals predominate. Among the bearish classifications, bearish high regime 2 appears most frequently at 9.2%, significantly more common than other bearish states. This suggests that regime 2 captures bearish patterns that the model can identify with high confidence. The
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beerthh low megine
9.7%
①: Low confidence classfitcatons rare, indicating the model's strong dedisiveness in assigning regimes.
bocishh low regits
0.7%
bdithh low rigine- 0-
0.7%
2 Bullish states evenly distributed reflecting the model's condistent detection of upward trends.
regimes.
ngiite
① Bearish states show a distinct concentration in Regime 2, suggesting
nginet
restral low nigine
1:22
model captures unique bearish dynamics.
becch law rigitse
a High-confidence neutral states dominate all regimes, highllighting the madel's preference for identifying stable market conditions.
nectral miiire
beltsh hight moline
bellsh bagk regiar
beerah lagh reginn
bellab high regine
metine
3
bearah aish reiine
neatt aist mainne
10.0%
nutral biyh relite
berrall high regite
Purceniage of Samples
Figure 3: Regime distribution across market states predicted by KASPER. The bar chart reports the percenlage of samples assigned to each state, defined by market direction (bullish, bearish, neutral), confidence level (high/losg /low), and regime index (0, 1, 2). High-confidence neutral states dominate, indicating the model frequently identifies stable market conditions.
consistent distribution of bullish states across regimes, ranging from 6.3% to 7.3% for high confidence, indicates a similar ability to identify rising prices in all types of markets. Notably, low-confidence classifications rarely occur across all market directions and regimes, with most falling below 1.5% of samples. This demonstrates the model's decisiveness in regime assignments, preferring to make high-confidence classifications.
As shown in Fig. 4 detailed breakdown of feature contributions is done within each identified regime. This visualization provides crucial insights into what drives market behavior in different states. In Regime 0, OC spread dominates at 88.9% contribution, with price velocity at 3.9%, momentum state at 3.1%, and other features contributing minimally. This regime represents a relatively stable market setting where intraday price movements between open and close are the primary predictive factor. Regime shows a somewhat different pattern. While DC_spraad remains dominant at 83.6%, momentum tattnnttuttttttteetteeeeee shows increasod importance at 7.0%, followed by HL spread al 4.9% and price velocity al 3.2%. This suggests an increased momentum in the market, because of shifts in sentiment triggered by news events or short-term trader behavior. Regime 2 exhibits the highest dominance of OC spread at 89.4%, with price_ velocity the second most important feature at 5.4%. This shows the market is moving strongly in one direction, with prices changing faster each day. The reduced importance of range-based indicators (HT. spread at just 1.0%) suggests that in this regime, direction matters more than volatility range. These feature contribution patterus, align well with established market behaviors, where trending periods show directional signals and transitional periods display increased momentum indicators. The clear differentiation between feature importance across regimes validates our model's ability to identify meaningful, and distinct market states.
price_vvlocity
Regime
Regime
Regime
0C spredd
88.0%
OC spreud
8..8%
OC, speind
89.0%
prose
CC spread overwhelmingy dominent, indicating that ovamight open-close differences primary driver in this stable regime.
jarend-
1.9%
saroedreddd remains influeniial, with contribution from HL spread and price velocity suggesting some intraday volitility
price welocity
OC spread dominates onca more, but an incressed price. velocity hints more dynamic daily price movements
momentum state
prico velecity
3.35
momentum ptate
13.4%
momentum, state
1.3%
proce acosteration
1.7%
velaciity rotlo 11.7
peico acceloration 1.0%
speead
1.0%
20
40
60
80
100
20
40
60
80
100
20
40
60
80
100
Contrabutionn
Contribution
15 Costribution
Figure 4: Relative contribution of features across identified market regimes. Each regime exhibits distinct feature im- portance patterns: Regime 0 shows balanced influence of directional and volatility indicators, Regime demonstrates increased momentum siguificance while maintaining @C_spreed dominance, and Regime 2 displays the strongest OC_ spread contribution, with price_velocity as a secondary factor, representing different phases of market behavior, from consolidation to trending states.
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4.3 Financial Metrics and Risk Analysis
Our ovaluation reveals that KASPER delivers excoptional performanco across all financial metrics, out- performing existing approaches in both statistical accuracy and practical trading applications. The model achieves precision with an MSF of 0.0001, an RMSE of 0.0046, and an MAE of 0.0033. The R2 value of 0.8953 + 0.0030 explains over 89% of the variance in market returns.
A key highlight of KASPER's performance is its Sharpe Ratio of 12.02. This exceptional risk-adjusted return is due to three core innovations in our approach; orthogonal regularization which minimizes volatility by decorrelating regime-specific features, preventing risk factors from affecting each other across regimes. Cubic spline activations, which boost prediction signals (particularly OC spread, with Soc = 0.25) during stable periods by using nonlinear patterns that simple models can't detect. And, contrastive loss, which keeps different market regimes separated, allowing risk and return to be managed separately (piy 0.15). Unlike transformer models that mix features and cause high variability, KASPER controls volatility and avoids bad investments during market changes.
0.002
Drewndow
Std. band widens durlngg volatile regime transitions.
@: Maximum drawdown point during market turbulence.
-0.03
@: Drawdown stabilizes near zero after the shock.
-0015
Averoge Traadow
21 Std
60
75
100 Time Stops
126
150
175
Figure 5: Drawdown analysis of KASPER model across the testing period. The system maintains exceptional capital preservation characteristics, with maximum drawdown limited to -0.09%, demonstrating robust risk management capabilities. Notable features include widened standard deviation bands during regime transitions (timesteps 40-60) and rapid recovery periods following temporary losses, reflecting the model's adaptive response to changing market conditions.
The drawdown analysis in Fig. 5 highlights KASPER's effectiveness in managing risk. Throughout the testing period, the maximum drawdown remains remarkably contained at just -0.09%, nearly two orders of magnitude lower than traditional approaches. Even during periods of heightened market volatility, as seen the point of maximum loss, the model remains stable. Following this challenging period, we observe the loss reducing by approximately 0.4%, indicating the model's ability adapt. The wider standard deviation band between regimes shows the natural uncertainty during these shifts. This drawdown profile indicates that KASPER rarely makes consecutive prediction errors in the same direction, limiting potential losses during tough market movements.
Fig. e compares actual versus predicted returns through various market conditions. During stable periods, the prediction accuracy is exceptional, with errors below 0.5%. Midway through the test period, we observe volatility surge where actual uncertainty exceeds predictions with a standard deviation of 0.015. This corresponds to market regime transition where historical patterns temporarily lose reliability. The sudden down-spike represents bearish signal triggered by unexpected negative market news, an outlier event that no model could reasonably predict. Based on this single held-out test evaluation, the cumulative returns of 2.76% over the test period, combined with a win rate of 83.17% and a profit factor 1.53, show that the forecasting system works very well. For every dollar risked, the strategy generates $1.53 in profits, creating a favorable risk-reward profile attractive even to conservative investment approaches. KASPER's regime-aware approach enables it to navigate different market conditions with appropriate strategies, maintaining profitability while limiting downside exposure. KASPER bridges the gap between black-box DL and interpretable financial modeling by adapting to changing market dynamics.
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Actual Baturta
Predicted Herumo Actual zt St2
0.02
Predicted al
RuCures
0.00
-0.02
Volability Surger Actual uncertainty esceeds market prediction (a 0.015), regime transition period, during which historical patterns become temporartly less reliable.
-0.00
The sudden nown--pple shows bearish signal triagered by unixppected negative market news that the model couldn't anticipate.
-0.04
-0.05
75
100
125
150
1.76
Time Stepo
Figure 6: Comparison actual versus predicted returns across testing period. The model demonstrates varying forecasting precision under different market conditions: exceptional accuracy during stable periods (timesteps 75-100), increased standard deviation during regime transitions (timesteps 110-130); and limited ability to anticipate extreme market events evidenced by the outlier at timestep 27.
4.4 Interpretability via Shapley Values
The Monte Carlo Shapley approach reveals different feature attribution patterns. In Regime 0, OC_spread (0.016++0001) and HI, spread (0.011++0.06)) exhibit balanced contributions, indicating market state where both directional movements and volatility range influence price determination. This suggests a complex environment requiring multi-factor analysis for accurate forecasting. Regime shows shift towards directional dominance, with OC OC_spread increasing to 0.039:0.017 while HL_ HL_spread becomes negligible. The emergence of volatility ratio (0.002++0.001) as a secondary factor shows transitional market entering directional trends. Regime displays the most distinctive pattern, with OC ___sppeed reaching 0.083 while ATR (0.003++0.002) and price acceleration (0.001++.001) emerge secondary factors. These attributions align with established financial theory while providing quantitative precision for practical application. As the market moves from calm periods to breakouts and then to strong trends, the model shifts from using balancod features to focusing more on direction. For traders, this means using different strategies for each phase: a balanced approach in Regime 0, a focus on direction in Regime 1, and followoo with size adjustments in Regime 2. Changes which features matter most can also act as early warnings for market shifts, helping manage risk in advance.
4.5 Robustness and Generalization
To analyze KASPER's generalization capabilities, we conduct multiple walk-forward analyses across different market conditions. As shown Table 2 the model demonstrates remarkable stability across evaluation runs, with consistently low standard deviations in all performance metrics. We report the following evaluation metrics:
Sharpe Ratio is a standard measure of risk-adjusted return. It defined as:
Sharpe Ratio- Ps ro
(29)
where Ma is the mean of strategy returns, ro is the risk-free rate, and as is the standard deviation of the returns. In our implementation, this is anmualized by multiplying by vT, where T is the number of trading days per year (252).
Win Rate denotes the percentage of profitable trades, defined as:
N+ x 100,
Win Rate
(30)
N
where N4 is the number of trades with positive return, and N is the total number of trades.
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Directional Accuracy evaluates how often the predicted return direction matches the actual market
movement:
DA H (sign(0r) = sign(1)) x 100,
(31)
where is is the predicted return at time t, Vo is the actual return, and [[-] is the indicator function. Cumulative Return measures the compounded growth of the strategy over time:
Cumulative Return
I1) 1,
(32)
where Tt is the return at time t.
Maximum Drawdown (MDD) quantifies the largest observed loss from a peak to a trough in the
cumulative return curve:
Vi
-
MDD = min mmxx,s V,
(33)
where V2 is the cumulative portfolio value time
Table 2: Walk-forward aggregated financial performance of KASPER, reporting directional accuracy, Sharpe ratio, maximum drawdown, cumulative and average returns, and trade-level statistics (mean + standard deviation when applicable).
Metric
Value
Standard Deviation
Direction Accuracy (%)
80.94%
7.27
Sharpe Ratio
11.24
2.78
Max Drawdown (%)
-0.10%
Cumulative Returns (%)
2.66%
Win Rate (%)
80.94%
+ 7.27
Total Trades
700
Profitable Trades
566
Average Return (%)
0.0069%
Average Win (%)
0.0102%
Average Loss (%)
-0.0070%
Profit Factor
1.45
These metrics are calculated per walk-forward test window, then averaged across multiple runs (with each analysis averaged over 5 runs), with their respective standard deviations also recorded to assess statistical consistency. As shown in Fig. 7 our walk-forward validation confirms that KASPER maintains its predictive power when facod with regime transitions, preserving both directional accuracy and win rate across the entire lesting period. The model's returns stay consistent over time, with steady average performance per trade and minimal losses.
Examining the specific performance patterns, the Sharpe Ratio exhibits remarkable peak period 2.0, reaching approximately 15.0, indicating exceptional risk-adjusted returns where the model generates 15 units of excess return for every unit of risk taken. This peak coincides with optimal risk control, as evidenced by the maximum drawdown approaching ncar-zero lovels during the same period, demonstrating nearly perfect capital preservation. The win rate simultaneously achieves its highest performance at periods 2.0-2.5, reaching approximately 87%, where nearly out of 10 trades are profitable, indicating that the Cumbel-Softmax-Bl regime classification is operating at peak efficiency. While the Sharpe Ratio gradually moderates around 6.5 by period 4.0, the cumulative returns exhibit compelling trajectory, initially stabilizing around 1.5% before demonstrating sharp recovery and substantial growth to approximately 5.0% by the final period. This
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0.00
Purtill growth sarges te 3.0 i irtnl-catture mechoriere setlvate
Sharpe Ratio
14-
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Max Drawdoww
-0.05
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Win Paate
10-
Kil-efficianey
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-0.10
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6-
drrecionaal beads.
markrr dyrveree trsond
Sore 79% thoohhldd all mackel condiitons.
-0.20
Period
Period
Period
Period
Figure 7: Per-period performance of KASPER across key financial metrics over forwad validation windows. Each subplot displays the mean value (solid line) and one standard deviation (shaded area) for Sharpe Ratio, Cumulative Returns, Win Rate, and Maximum Drawdown.
divergent pattern, characterized by declining Sharpe ratio alongside rising cumulative returns, highlights KASPER's adaptive nature: shifting from conservative, precisreciionn approach during periods of market uncortainty to a more aggressive, trend-following strategy clearer directional regimes emerged, all while consistently maintaining win rates above 79% and maximum drawdowns below -0.10% across all periods.
The model's robustness arises from two key architectural innovations: our quantile morttile-nommlizeed spline ini- tialization strategy, which ensures appropriate boundary handling across diverse feature distributions and orthogonality regularization (enforcing WWWW?- - 7 minimization), which prevents feature collapse during training and maintains distinct regime characteristics as market conditions evolve. This enables KASPER to capture both linear and nonlinear relationships across market conditions without overfitting to past data. The consistent performance across varying market regimes, combined with the previously discussed financial metrics, confirms that KASPER delivers reliable performance under diverse conditions, making it suitable for practical deployment in real-world financial applications.
Robustness checks under realistic trading and temporal validation. To contextualize the unusually high Sharpe ratio and near-zero drawdown observed in the main experiment, we conduct additional robustness checks that account for market frictions and stricter temporal validation. Specifically, we run two additional evaluations. First, we evaluate KASPER under three cost settings reflecting different execution environments: optimistic setting with 0.05% transaction costs, 0.02% slippage, and same-day execution; more typical retail setting with 0. 1.10% transaction costs, 0.05% slippage, and a one-day execution delay; and conservative setting with 0.20% transaction costs, 0.10% slippage, and a one-day delay. Across these scenarios, KASPER maintains strong risk-adjusted performance (Sharpe = 13.25, 12.71, and 11.81, respectively), with stable win rates (88.42% (...%%%....% 90.53%) indicating that the performance is not an artifact of frictionless backtesting.
Second, to mitigate lookahead bias arising from overlapping bolling-windwwd computations, we perform purged walk-forward validation with 7-day purge gap between training and test segments, using the same trading-cost assumptions above. Standard folk-forward evaluation yields Sharpe 9.89 with 80.19% directional accuracy and 219.56% cumulative returns, while the purged protocol achieves Sharpe = 10.30 with 80.13% directional accuracy and 147.17% cumulative returns. The Sharpe change between the two protocols is 4.1%, which is below the magnitude typically associated with leakage-driven inflation, supporting that the reported gains are not driven by temporal information contamination. The consistency of Sharpe ratios across cost assumptions (11.81-13.25) and the limited sensitivity to purging provide evidence that the strong performance reflects stable regime-dependent structure rather than overfitting evaluation artifacts.
Cross-asset generalization. To assess generalization beyond the primary Yahoo Finance evaluation, we test KASPER on thrce markets with distinct microstructures: an equity index (S&P 500), a single large-cap technology stock (AAPL-MAAN)), and a commodity series (daily gold prices). Table summarizes the results. On the S&P 500, KASPER achieves strong performance (R² 0.80, Sharpe= 9.50); the reduction relative the main setting (R²² 0.89, Sharpe= 12.02) is consistent with the higher constituent heterogeneity and the effects of market-cap weighting. On AAPL, performance decreases to R² 0.56 and Sharpe= 7.02, reflecting increased idiosyncratic volatility and firm-specific shocks that are not fully captured by technical
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inputs alone. On gold, performance further degrades (R² = 0.33, Sharpe= 1.56), highlighting limitations when transferring an equity-opimmized regime formulation to macro-driven and more mean-reverting dynamics. Nevertheless, Sharpe ratios remain positive across all datascts, suggesting that the regimo-aware forccasting and risk-control components relain practical value across asset classes. These findings molivate asset-specific feature augmentation, such as macro indicators for commodities and fundamental/ewe signals for single stocks, and, where appropriate, regime definitions tailored to mean-reverting versus trending behavior.
Table 3: Cross-asset performance of KASPER equity index (S&P 500), large-cap stock (AAPL/MAANG), and commodity series (daily gold prices). Reported metrics include R², MAE, Sharpe ratio, maximum drawdown (Max DD), MSE, and RMSE.
Dataset
R
MAE
Sharpe
Max DD
MSE
RMSE
S&P 500 (Kaggle 2025
0.80
0.0045
9.50
-0.25%
0.00015
0.0055
AAPL-MAANG Upaddyay 2022
0.56
0.0121
7.02
-3.52%
0.0003
0.0174
Daily Gold Prices
[Chodvadiya
2025
0.33
0.0082
1.56
-8.54%
0.0001
0.0112
4.6 Ablation Analysis
To assess the role of the two regularizers in regime identification, we ablate orthogonality regularization and the contrastive loss, disabling each component while keeping the rest of KASPER unchanged. Removing orthogonality regularization leads to regime collapse: the regime-specific weight matrices become nearly identical during training and the detector assigns the majority of samples to a single regime.
In contrast, disabling the contrastive loss produces a different failure mode, referred as regime entanglement. As shown in Fig. the regime assignments become weakly separated and are dominated by cow-confidence predictions, indicating substantial overlap of regime embeddings in the latent space. Without the contrastive objective, the model does not form a stable separation between market states, which degrades regime discrimination and leads to unreliable regime assignments.
Alentical commatise reporriation, the mudel bilk distiguust apposing sucket diections and dctaila so sebin clextifiation
23.7%.
bellat reglise
22.0%
②
Iecurra--sttte dominance (53.2%) adieates that the medd delauls antiguous dlssifiation when directionl pattemee carnel be ediialy inseeifial
bearssh
22,74
10
33
Regime distribution under contrastive-loss ablation, illustrating regime entanglement. The near-identical bullish and bearish proportions and the dominance of low-confidence assignments indicate weak regime separation.
Figure 8:
4.7 Comparative Analysis
We compare KASPER against cstablished and recent forocasting basclines to evaluate predictive accuracy, risk- adjusted performance, and interpretability under consistent experimental conditions. To ensure methodological consistency and completeness, all baselines are implemented and evaluated on the same Yahoo Finance dataset using identical preprocessing, train test splits, and normalization, while preserving each method's algorithmic components. Moreover, VLSTAR and AGNES are formulated for binary regime classification (calm versus volatile) rather than continuous return prediction; since they output regime probabilities instead
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of numerical forecasts, regression-bant based metrics such as R², MAE, MSE, and RMSE are not represented for these methods and are therefore omitted from the comparative summary.
Table 4: Comparative evaluation of KASPER against established and recent baselines on the Yahoo Finance dataset using Rt, MAE, Sharpe ratio, Max DD, MSE, and RMSE. Baseline results are reproduced from the cited works.
Model/Framewerr Framework
R
MAE
Sharpe
Max DD
MSE
RMSE
RF + Monte Carlo
Zhao 2025
0.78
0.095
0.93
-28.1 28.11%
0.015
0.122
Single Layer LSTM Bhandari et al.
2022
0.79
0.0072
0.85
-1.446%
0.00085
0.0292
LSTM KAN Yao
2024
0.48
0.0057
1.97
112.00%
0.00014
0.0082
AE-LSTM + DRL (Sagiraju u & Mogal la
2022
0.50
0.010
1.85
-33.00%
0.00017
0.013
VLSTAR (uucci & Ciciretti 2021
0.93
AGNES Bueci & Ciciretti
2021
0.82
PatchTST [ie et al. 2023
0.10
0.0082
2.24
-6.68%
0.0001
0.0108
DLinear (Zeng et al. 2023
0.03
0.0071
3.40
-9.98%
0.0001
0.0103
DQS [Li & Ming
2023
0.78
0.0149
3.65
-7.44%
0.000377
0.0194
KASPER (Ours)
0.89
0.0033
12.02
-0.09%
0.0001
0.0046
As shown in table 4 the results indicate that KASPER benefits from coupling regime identification with sparse and interpretable forecasting: aethogonality-constrannee regime scparation paired with Monte Carlo Shapley-based attribution, enabling regime-specific explanations while maintaining strong predictive accuracy. We further include two recent forecasting baselines, PatchTST and DLinear. PatchTST achieves R² = 0.10 with Sharpe 2.24, which is consistent with transformer-based models being sensitive to data availability and with the absence of an explicit regime-adaptation mechanism in the baseline formulation. DLinear yields R² = 0.03 with Sharpe = 3.40, suggesting limited explanatory power when a single linear decomposition is applied across heterogeneous market states. By contrast, KASPER explicitly detects the prevailing regime and applics regime-specific spline transformations, capturing 89% of the return variance (R² 0.89), while its interpretability module provides actionable regime-conditional insights into feature relevance. These rosults support that integrating regimo-aware modeling with sparse forccasting and attribution yiclds more accurate and informative framework for stock return prediction under changing market conditions.
4.8 Discussion
The comprehensive evaluation of KASPER across multiple dimensions, including regime behavior, financial performance, interpretability, and robustness, highlights the strength its architecture in addressing the challenges of financial forecasting. Unlike static models or black-box NNs, KASPER employs modular and transparent design that adapts effectively to evolving market dynamics. The regime segmentation results demonstrate that the model reliably distinguishes between bullish, bearish, and neutral market states, with a strong lendency toward confident regime assignments. This decisiveness, combined with feature altribution analysis, confirms that KASPER identifies meaningful market signals, such as a dominance of directional indicators in trending phasos and a greater emphasis on volatility during periods of transition. Such patterns align with financial theory and support more targeted forecasting and risk management decisions.
The model's financial resilience, reflected in its low drawdown and consistent performance across forward analysis, is enabled by its architectural innovations. The use of orthogonality constraints helps preserve regime-specifie representations, minimizing the risk of overlapping or diluted features across distinct market conditions. In parallel, spline-based activations enhance the model's ability to capture fine-grained nonlinear effects that conventional models often miss. Contrastive learning further reinforces regime separation, improving generalization over time.
Crucially, KASPER addresses a key need financial modeling: aligning strong predictive capabilities with practical deployability. Its interpretability, enabled through Shapley-based feature attribution, completes the transparency requirements of real-world applications, including those subject to regulatory oversight. Compared to traditional statistical models, which may fail under regime shifts, DL models, which often
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lack interpretability, KASPER offers a well-rounded solution that balances accuracy, adaptability, and explainability.
5 Conclusion
The proposed KASPER framework addresses key challenges in financial market prediction by combining adaptive regime modeling, sparse spline-based KANs, and Monte Carlo Shapley-based interpretability. Unlike conventional models, KASPER adapts dynamically to market conditions, improving predictive accuracy while avoiding overfitting, Evaluation on Yahoo Finance data reveals strong performance, with an R of 0.8953, Sharpe Ratio of 12.02, and MSE of 0.0001. Low drawdowns and consistent profitability confirm its practical relevance. Orthogonal regularization and rastive loss ensure clear regime separalion, while Shapley-based attribution offers transparent insight into key market drivers. KASPER achieves a strong balance between accuracy and interpretability, making it well-suited for real-world docision-making.
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Reviews:
[{"review_id": "pM74w6KxPY", "reviewer": "Reviewer_PVwp", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes an algorithm for predicting future stock prices given past stock prices (from a window of $n$ days). The proposed algorithm begins by extracting _features_ from raw stock price data (various functions of the $n$ past stock prices) and then goes through a series of processing steps to produce a prediction for the stock value on the next day. The main technical contribution in this work is to combine the classical idea of keeping track of a latent _regime_ variable which approximately denotes the prevailing trend in the stock market (going up, crashing, stable, etc.), with feature processing using neural network with spline-based activations. The paper empirically demonstrates that the resultant algorithm performs favorably compared to existing baselines. However, the core details in the description of the algorithm is inadequate often containing ill defined mathematical quantities and their combinations, making it difficult for a reader to understand the core details of the work.", "claims_and_evidence_yn": "No", "claims_explanation": "The present paper writing significantly lacks clarity and, as such, it is very hard for a reader to understand the technical details of the main algorithm: \n\nIntroduction and Contributions:\n\n1. P2: \"but frequently fail to enforce sparsity\": What is the mathematical quantity that is subject to _sparsity_ here?\n\n2. P2: \"alignment, resulting in data leakage and inflated performance metrics.\": What is _data leakage_ and Why are test metrics inflated when previous models are non-robust.\n\n3. Fig 1: What is the _knot grid_? I presume the paper is talking about the arguments to the spline function in some form. However, this is not clarified and unclear.\n\n4. P2: \"percentile-based initialization of spline knots\": What is _percentile-based_ initialization?\n\n5. P2: \"Sparsity is enforced through ...\": What is the quantity for which sparsity is enforced?\n\n6. P2: \"dominant features like HL (High-Low) in bearish regimes and OC (Open-Close) in bullish regimes\": Mathematical definitions of features are not provided yet. \n\nBackground:\n\n7. Eq 3: Is $r_t$ the same as $y_t$? Consistent notation should be used. \n\n8. \"Regime transitions follow Eq 4. $P(z_t = i | z_{t−1} = j) = p_{ji}$\": $p$ was earlier denoted by $A$ in Eq 2. Notation needs to be made consistent.\n\n9. \"wealth, $r_b$ returns, and $\\pi_b$ the regime probability distribution\". Eq 5: What is $t_b$? What are we optimizing over?\n\n10. \"particularly for option pricing and stock prediction\", Eq 9, \"asset-related parameters\": What is the problem statement for options pricing? What is the quantity $C$? What is an \"asset-related parameter\"?\n\nMain Algorithm:\n\n11. Eq 13: Is is very important to be mathematically explicit in what the features are, especially because the performance of many stock prediction models heavily depends on the input feature engineering. Specifically the \". . .\" in Eq 13 needs to be replaced with the actual expressions for the features in terms of the input $x$.\n\n12. Eq 15: There is some problem with the brackets -- currently the equation evaluates to $k_{i + 1} - k_i$ completely ignoring the input features $x$. \n\n13. Eq 16: What is the knot sequence $k_i$, precisely, in terms of $x$?\n\n14. Eq 17: $i$ referred to the number of activations earlier in Eq 15. Now, $i$ corresponds to the index of the regime. Notation should be made consistent. \n\n15. Eq 18: \"where $z_i$, $z_j$ represent the embeddings of samples $i$ and $j$\": How are $z_i$, $z_j$ defined in terms of the input. What is the expectation being taken over? What are the random variables here?\n\n16. Eq 19: How is the matrix $W_r$ defined? Only the scalars $w_i$ have been defined earlier as weights. \n\n17. Eq 24: So far the letter $S$ was being used for the feature matrices. Now it seems to be used for sets. Notation should be made consistent.", "audience_yn": "Yes", "audience_explanation": "Stock prediction is a difficult time series prediction problem, and the community would be interested in works that move the field forward by proposing novel architectures that perform well.", "requested_changes": "Overall, the paper needs a significant rewrite to fill in all the gaps in the description of the algorithm, and the description of related work. \n\n1. Notation needs to be made consistent, and mathematical quantities need to be defined clearly see the \"claims\" concerns above. \n\n2. Once notation is consistent, and quantities are clearly specified, the algorithm needs to be expressed end to end in a coherent fashion, moving high level quantities to Fig 2, and pointing to their explanations to an appendix (e.g., details of feature engineering), or the main text.\n\n3. Once the main quantities and the flow is defined in a clear fashion, related work should be compared to in greater detail, comparing each component of the proposed algorithm (regime selection, spline based activations) to their corresponding related papers. (Perhaps Table 3 already suffices, but this is unclear currently without a clear description of the main Algorithm)", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "0uzsACW9pb", "reviewer": "Reviewer_2Pvr", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "## Summary\nForecasting in financial markets remains a significant challenge. This paper introduces Kolmogorov–Arnold networks for stock prediction and explainable regimes (KASPER), a novel framework that integrates regime detection, sparse spline-based function modeling, and symbolic rule extraction. Applied to real-world financial time series from Yahoo Finance, the model achieves an R2 score of 0.89, a Sharpe Ratio of 12.02, and a mean squared error as low as 0.0001, outperforming existing methods.\n\n## Strengths\n1. The paper combines regime-aware forecasting with spline-based KANs, enforcing sparsity and orthogonality constraints to improve generalization. This is a meaningful improvement over prior deep learning approaches, which often ignore regime dynamics.\n2. By employing Monte Carlo Shapley values with temporal weighting, the model extracts regime-specific rules. This provides transparent insights for practitioners, addressing a common weakness of black-box models.\n\n## Weaknesses\n1. The reported Sharpe ratio (12.02) and near-zero drawdown are unusually high compared to real-world benchmarks. This raises concerns about overfitting to the Yahoo Finance dataset, data leakage, or overly optimistic evaluation settings.\n2. The evaluation relies solely on a Yahoo Finance dataset (2018–2023) with US equities. Without testing on multiple markets or longer time horizons, it is unclear whether the framework generalizes beyond this narrow context. For example, this paper uses six different datasets in their experiments: Accurate Multivariate Stock Movement Prediction via Data-Axis Transformer with Multi-Level Contexts (Yoo et al., KDD 2021).\n3. In Table 3, which does the main comparison, I believe the performances of all competitors should be fully reported without missing values for the completeness of experiments. Currently, more than the half of all cells are missing.\n\n## Minor comments\n1. Citations are in a wrong format.\n2. Eq. (15) is written in two lines unnecessarily.\n3. Algorithm 1 is very far from where it is mentioned.", "claims_and_evidence_yn": "Yes", "claims_explanation": "I believe most of the claims are well supported, but the experimental evaluation needs to be strengthened to fully support the surprisingly strong performance of the proposed method.", "audience_yn": "Yes", "audience_explanation": "Financial AI is being widely studied in the research community, and this work represents a meaningful application of XAI within that domain.", "requested_changes": "Please refer to the weaknesses above.", "broader_impact_concerns": "I have con concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "0alNnwwEfM", "reviewer": "Reviewer_iu4b", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "# Contributions \nThis paper presents a framework named KASPER for stock prediction and explainable market regimes. The approach first identifies latent market regimes using spline-activated KANs, and subsequently performs regime-specific forecasting with an additional KAN layer. To improve interpretability, the authors design a Monte Carlo Shapley method to extract regime-dependent rules. Experimental results on real-world financial time series demonstrate both the generalization capability and the interpretability of the proposed framework.\n\n# Strengths \n- The motivation is practical. Non-stationarity and regime shifts are central challenges in financial time series.\n- The proposed method strikes a balance between predictive accuracy and interpretability.\n# Weaknesses\n- The Related Work section does not provide a comprehensive overview of the research in this field, with insufficient discussion of other studies. Furthermore, it fails to highlight the innovations of this work compared to existing approaches.\n- The selection of baseline methods is limited. Stronger approaches, such as PatchTST and DLinear, should be included for a more comprehensive comparison. Additionally, the results of the Comparative Analysis are incomplete; for instance, some methods report only the $R^2$ metric, while others report only the MAE. The authors should either conduct experiments to provide a complete set of results or offer a reasonable explanation for the missing metrics.\n- Ablation studies on key modules, such as no-orthogonality and no-contrastive settings, are missing. Including such experiments would provide deeper insights into the contributions of these components.\n- The Method section requires more detailed explanations. Many modules are merely presented without adequately discussing the rationale behind their design or the advantages they offer.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The proposed method in the paper is reasonable, and the results are promising. However, the work suffers from the absence of strong baselines and ablation studies.", "audience_yn": "Yes", "audience_explanation": "This financial time series forecasting is of significant interest to industry and academic audiences.", "requested_changes": "Please address the Weaknesses above.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)