Mitigating Distribution Shift in Stock Price Data via Return-Volatility Normalization for Accurate Prediction

TL;DR

This paper introduces ReVol, a novel normalization method that mitigates distribution shifts in stock price data, combining return-volatility normalization with neural networks and geometric Brownian motion to improve prediction accuracy.

Contribution

ReVol is the first method to explicitly address distributional discrepancies in stock data by normalizing features and integrating long-term and short-term modeling techniques.

Findings

ReVol improves prediction metrics significantly across multiple datasets.

It outperforms state-of-the-art models with an average IC increase of over 0.03.

ReVol effectively reduces the impact of market anomalies on predictions.

Abstract

How can we address distribution shifts in stock price data to improve stock price prediction accuracy? Stock price prediction has attracted attention from both academia and industry, driven by its potential to uncover complex market patterns and enhance decisionmaking. However, existing methods often fail to handle distribution shifts effectively, focusing on scaling or representation adaptation without fully addressing distributional discrepancies and shape misalignments between training and test data. We propose ReVol (Return-Volatility Normalization for Mitigating Distribution Shift in Stock Price Data), a robust method for stock price prediction that explicitly addresses the distribution shift problem. ReVol leverages three key strategies to mitigate these shifts: (1) normalizing price features to remove sample-specific characteristics, including return, volatility, and price scale, (2) employing an attention-based module to estimate these characteristics accurately, thereby reducing the influence of market anomalies, and (3) reintegrating the sample characteristics into the predictive process, restoring the traits lost during normalization. Additionally, ReVol combines geometric Brownian motion for long-term trend modeling with neural networks for short-term pattern recognition, unifying their complementary strengths. Extensive experiments on real-world datasets demonstrate that ReVol enhances the performance of the state-of-the-art backbone models in most cases, achieving an average improvement of more than 0.03 in IC and over 0.7 in SR across various settings.

Tables

Table 1. Table 1 . Symbols and descriptions.

Symbol	Definition
$S_t$	Stock price at time $t$
$B_t$	Value of Brownian motion at time $t$
$\mu$	Return of a stock
$\sigma$	Volatility of a stock
$S_t^c,S_t^o,S_t^h,S_t^l$	Closing, opening, the highest, and the lowest prices at time
	step $t$
${ϵ}_t^c,{ϵ}_t^o,{ϵ}_t^h,{ϵ}_t^l$	Error terms of closing, opening, the highest, and the lowest
	prices at time step $t$
$T$	The current time step
$r$	Time interval between opening and previous day’s
	closing prices
$u$	Time interval between the highest and previous day’s
	closing prices
$v$	Time interval between the lowest and previous day’s
	closing prices
${𝐱}_t$	Price feature vector at time step $t$
${\widetilde{𝐱}}_t$	Feature vector at time step $t$ from fully connected layer
${𝐡}_t$	Hidden state vector at time step $t$ from LSTM
${\alpha }_t$	Attention weight at time step $t$
$\theta$	Parameters of the neural network backbone
$f_{\theta }$	Neural network backbone
$\varphi$	Parameters of the return-volatility estimator module
RVEϕ	Return-volatility estimator module
RVN	Return-volatility normalization module
$w$	The window size
$\beta$	Hyperparameter to control the weight of guidance loss

Table 2. Table 2 . Summary of datasets.

Data	Stocks	Days	Dates
U.S.1	178	5035	2003-01-02 to 2022-12-30
China1	171	2430	2011-01-06 to 2020-12-31
U.K.1	21	2383	2014-01-06 to 2024-01-10
Korea1	220	2170	2014-01-02 to 2022-11-01

Table 3. Table 3 . An input feature vector 𝐱 t \mathbf{x}_{t} at day t t .

Feature	Calculation
$x_t^o$	$x_t^o=S_t^o/S_t^c-1$
$x_t^h$	$x_t^h=S_t^h/S_t^c-1$
$x_t^l$	$x_t^l=S_t^l/S_t^c-1$
$x_t^c$	$x_t^c=S_t^c/S_{t-1}^c-1$

Table 4. Table 4 . Stock prediction performance of ReVol and baselines measured by Information Coefficient (IC), Rank Information Coefficient (RIC), and Sharpe Ratio (SR). ReVol consistently improves the performance of backbone models in most cases, showing an average improvement of over 0.03 in IC and over 0.7 in SR.

Method	U.S.			China			U.K.			Korea
	IC	RIC	SR	IC	RIC	SR	IC	RIC	SR	IC	RIC	SR
LSTM (Nelson et al., 2017)	0.019	0.026	0.642	0.015	0.004	1.283	0.040	0.014	0.820	0.043	0.029	2.092
+ ReVol	0.022	0.021	1.001	0.042	0.030	2.628	0.108	0.056	1.035	0.044	0.045	2.338
GRU (Cho et al., 2014)	0.020	0.027	0.656	0.015	0.008	1.181	0.037	0.020	0.382	0.043	0.023	2.008
+ ReVol	0.022	0.022	0.901	0.032	0.024	2.090	0.092	0.058	1.289	0.044	0.046	2.348
ALSTM (Feng et al., 2019)	0.017	0.022	0.796	0.022	0.014	1.281	0.062	0.028	1.006	0.041	0.028	1.961
+ ReVol	0.022	0.021	0.943	0.039	0.026	2.436	0.112	0.059	1.358	0.047	0.049	2.728
Vanilla Transformer (Vaswani et al., 2017)	0.015	0.022	0.768	0.011	0.005	1.110	0.046	0.026	0.804	0.033	0.014	1.306
+ ReVol	0.020	0.019	0.865	0.032	0.017	2.608	0.127	0.062	1.594	0.039	0.046	1.315
DTML (Yoo et al., 2021)	0.016	0.017	0.834	0.013	0.010	1.109	-0.004	-0.004	0.115	0.006	-0.042	0.106
+ ReVol	0.018	0.016	0.886	0.048	0.043	2.454	0.095	0.046	0.879	0.035	0.044	1.674
MASTER (Li et al., 2024)	0.013	0.014	0.753	0.002	0.014	0.542	0.019	0.001	0.345	0.005	0.015	0.359
+ ReVol	0.015	0.015	0.794	0.035	0.019	2.248	0.102	0.050	1.213	0.038	0.042	1.688

Table 5. Table 5 . Performance comparison on Information Coefficient (IC), Rank Information Coefficient (RIC), and Sharpe Ratio (SR) with the state-of-the-art normalization methods. ReVol outperforms other methods, demonstrating better capability to reduce distributional discrepancy in stock price data.

Method	U.S.			China			U.K.			Korea
	IC	RIC	SR	IC	RIC	SR	IC	RIC	SR	IC	RIC	SR
RevIN (Kim et al., 2021)	0.010	0.011	0.749	0.030	0.024	2.096	0.013	0.015	0.283	0.023	0.021	1.511
Dish-TS (Fan et al., 2023)	0.012	0.020	0.518	0.020	0.032	1.477	0.015	0.019	0.250	0.015	0.002	1.296
ReVol (proposed)	0.018	0.016	0.886	0.048	0.043	2.454	0.095	0.046	0.879	0.035	0.044	1.674

Table 6. Table 6 . Empirical evidence that the attention mechanism down-weights noisy observations: (left) Pearson correlation coefficient between | S t c S t − 1 c | \left|\frac{S_{t}^{c}}{S_{t-1}^{c}}\right| and attention weight α t \alpha_{t} , (right) ratio 𝔼 [ α t | | S t c S t − 1 c | ≥ 0.1 ] 𝔼 [ α t | | S t c S t − 1 c | < 0.1 ] \frac{\mathbb{E}\left[\alpha_{t}\,\middle|\,\left\lvert\frac{S_{t}^{c}}{S_{t-1}^{c}}\right\rvert\geq 0.1\right]}{\mathbb{E}\left[\alpha_{t}\,\middle|\,\left\lvert\frac{S_{t}^{c}}{S_{t-1}^{c}}\right\rvert<0.1\right]} of attention weights for noisy inputs to normal inputs.

Data	Correlation	Attention Ratio
U.S.	–0.179	0.788
China	–0.498	0.940
U.K.	–0.056	0.966
Korea	–0.424	0.483

Table 7. Table 7 . An ablation study of ReVol on China dataset.

Method	IC	RIC	SR
ReVol-N	0.002	-0.011	0.158
ReVol-E	0.045	0.040	0.595
ReVol-D	0.035	0.020	0.168
ReVol	0.048	0.043	2.454

Equations

$$S_t=S_0{e}^{\left(\mu -\frac{1}{2}{\sigma }^2\right)t+\sigma B_t}$$

$$S_t=S_{t-1}{e}^{\left(\mu -\frac{1}{2}{\sigma }^2\right)+\sigma {ϵ}_t}$$

$$S_{t-1}=S_0{e}^{\left(\mu -\frac{1}{2}{\sigma }^2\right)(t-1)+\sigma B_{t-1}}.$$

$$\frac{S_t}{S_{t-1}}=e^{\left(\mu -\frac{1}{2}{\sigma }^2\right)+\sigma \left(B_t-B_{t-1}\right)}.$$

$$\widehat{\mu -\frac{1}{2}{\sigma }^2},\hat{\sigma }=RV{E}_{\varphi }\left(S_{T-w+1:T}^o,S_{T-w+1:T}^h,S_{T-w+1:T}^l,S_{T-w+1:T}^c\right)$$

$${ϵ}_t^c=\frac{1}{\hat{\sigma }}\left(\log \frac{S_t^c}{S_{t-1}^c}-\left(\widehat{\mu -\frac{1}{2}{\sigma }^2}\right)\right).$$

$$S_t^c=S_{t-1}^c{e}^{\left(\mu -\frac{1}{2}{\sigma }^2\right)+\sigma {ϵ}_t^c}.$$

$${ϵ}_t^c=\frac{1}{\sigma }\left(\log \frac{S_t^c}{S_{t-1}^c}-\left(\mu -\frac{1}{2}{\sigma }^2\right)\right).$$

$${ϵ}_t^o=\frac{1}{\hat{\sigma }}\left(\log \frac{S_t^o}{S_{t-1}^o}-\left(\widehat{\mu -\frac{1}{2}{\sigma }^2}\right)\right).$$

$$S_{t-1+r}=S_{t-1}{e}^{\left(\mu -\frac{1}{2}{\sigma }^2\right)r+\sigma {ϵ}_t^o}$$

$$\hat{r}=\frac{\mathbb{E}\left[\log \frac{S_t^c}{S_{t-1}^c}\log \frac{S_t^o}{S_{t-1}^c}\right]}{\mathbb{E}\left[{\left(\log \frac{S_t^c}{S_{t-1}^c}\right)}^2\right]}.$$

$$S_{t-1+r}=S_{t-1}{\left(\frac{S_t}{S_{t-1}}\right)}^r.$$

$$\log \frac{S_{t-1+r}}{S_{t-1}}=r\log \frac{S_t}{S_{t-1}}.$$

$$\mathbb{E}\left[{\left(r\log \frac{S_t}{S_{t-1}}-\log \frac{S_{t-1+r}}{S_{t-1}}\right)}^2\right].$$

$$0$$

$${ϵ}_t^o=\frac{1}{\hat{\sigma }}\left(\log \frac{S_t^o}{S_{t-1}^c}-\left(\widehat{\mu -\frac{1}{2}{\sigma }^2}\right)\hat{r}\right).$$

$$\frac{1}{\sqrt{\hat{r}}}{ϵ}_t^o\stackrel{\text{sample}}{\sim }\mathcal{N}(0,1).$$

$${ϵ}_t^o=\frac{1}{\sigma }\left(\log \frac{S_{t-1+r}}{S_{t-1}}-\left(\mu -\frac{1}{2}{\sigma }^2\right)r\right).$$

$${ϵ}_t^h=\frac{1}{\hat{\sigma }}\log \frac{S_t^h}{S_{t-1}^c}.$$

$$S_t^h=\underset{0<u\le 1}{\max }{S}_{t-1}^c{e}^{\left(\mu -\frac{1}{2}{\sigma }^2\right)u+\sigma B_u}.$$

$$S_t^h\approx \underset{0<u\le 1}{\max }{S}_{t-1}^c{e}^{\sigma B_u}\Rightarrow \frac{1}{\sigma }\log \frac{S_t^h}{S_{t-1}^c}\approx \underset{0<u\le 1}{\max }{B}_u.$$

$${ϵ}_t^l=\frac{1}{\hat{\sigma }}\log \frac{S_t^l}{S_{t-1}^c}.$$

$$\widehat{\mu -\frac{1}{2}{\sigma }^2}$$

$$\hat{\sigma }$$

$$\widehat{\mu -\frac{1}{2}{\sigma }^2}$$

$$\hat{\sigma }$$

$${\alpha }_t=\frac{e^{{\mathbf{h}}_t^{\top }{\mathbf{h}}_T}}{{\sum }_{i=T-w+1}^T{e}^{{\mathbf{h}}_i^{\top }{\mathbf{h}}_T}}.$$

$${\widehat{ϵ}}_{T+1}^c=f_{\theta }({ϵ}_{T-w+1:T}^o,{ϵ}_{T-w+1:T}^h,{ϵ}_{T-w+1:T}^l,{ϵ}_{T-w+1:T}^c)$$

$${\hat{S}}_{T+1}^c=S_T^c{e}^{\left(\widehat{\mu -\frac{1}{2}{\sigma }^2}\right)+\hat{\sigma }{\widehat{ϵ}}_{T+1}^c}.$$

$$\mathbb{E}\left[{\left(\frac{{\hat{S}}_{T+1}^c}{S_T^c}-\frac{S_{T+1}^c}{S_T^c}\right)}^2+\beta {\left(\frac{1}{w}\sum _{t=T-w+1}^T\log \frac{S_t^c}{S_{t-1}^c}-\widehat{\mu -\frac{1}{2}{\sigma }^2}\right)}^2\right].$$

Full text

We propose ReVol (Return-Volatility Normalization for Mitigating Distribution Shift in Stock Price Data), a robust method for stock price prediction that explicitly addresses the distribution shift problem. ReVol leverages three key strategies to mitigate these shifts: (1) normalizing price features to remove sample-specific characteristics, including return, volatility, and price scale, (2) employing an attention-based module to estimate these characteristics accurately, thereby reducing the influence of market anomalies, and (3) reintegrating the sample characteristics into the predictive process, restoring the traits lost during normalization. Additionally, ReVol combines geometric Brownian motion for long-term trend modeling with neural networks for short-term pattern recognition, unifying their complementary strengths. Extensive experiments on real-world datasets demonstrate that ReVol enhances the performance of the state-of-the-art backbone models in most cases, achieving an average improvement of more than 0.03 in IC and over 0.7 in SR across various settings.

††copyright: acmlicensed††journalyear: 2025††copyright: rightsretained††conference: Proceedings of the 34th ACM International Conference on Information and Knowledge Management; November 10–14, 2025; Seoul, Republic of Korea††booktitle: Proceedings of the 34th ACM International Conference on Information and Knowledge Management (CIKM ’25), November 10–14, 2025, Seoul, Republic of Korea††doi: 10.1145/3746252.3761067††isbn: 979-8-4007-2040-6/2025/11

1. Introduction

How can we address distribution shifts in stock price data to improve predictive accuracy? Distribution shifts — distribution discrepancies across samples — pose a fundamental challenge in data mining and machine learning. This issue is especially critical in stock price prediction, where market conditions evolve continually due to economic factors, geopolitical events, and investor sentiment. These shifts occur not only over time as market conditions change but also across different stocks, each exhibiting unique statistical properties such as return and volatility. Successfully overcoming these shifts is crucial for constructing robust and trustworthy predictive models, which in turn support more reliable investment strategies and enhance financial outcomes.

Despite extensive work on stock price prediction, existing methods (Feng et al., 2019; Li et al., 2021; Yoo et al., 2021) often fail to adequately deal with distribution shifts. They do not explicitly mitigate distributional differences across samples, limiting their generalization in non-stationary financial time series. Methods for addressing distribution shifts (Kim et al., 2021; Fan et al., 2023) unify distribution parameters across samples, while failing to align distribution shapes. Figure 1 illustrates how raw price data and normalized data distributions differ between training and test periods. Others (Lin et al., 2021; Zhan et al., 2024; Zhao et al., 2023) rely on adaptation to the representation space, increasing the complexity of the model without directly resolving the distribution shift. These limitations hinder the robustness and accuracy of the prediction under changing market conditions.

We propose ReVol (Return-Volatility Normalization for Mitigating Distribution Shift in Stock Price Data), a method for robust stock price prediction that explicitly addresses distribution shifts by normalizing individual stock characteristics. The main ideas of ReVol are as follows. First, ReVol mitigates distribution shifts by removing sample-specific attributes, such as return, volatility, and price scale, from the input data. Second, it accurately estimates these characteristics through an attention-based weighted averaging technique that alleviates the impact of transient market anomalies. Third, ReVol reintegrates these characteristics into a prediction pipeline to recover the information removed during normalization.

Our contributions are summarized as follows: {itemize*}

Addressing distribution shifts in stock price prediction. We propose ReVol, a novel approach to address distribution shifts by normalizing individual sample characteristics, including return, volatility, and price scale. Figure 1 depicts that the distribution gap between training and test data is mitigated by ReVol.

Integration of short-term patterns and long-term trends. ReVol unifies geometric Brownian motion (GBM) for modeling long-term price behavior with a neural network for capturing nonlinear short-term patterns, thereby leveraging the complementary strengths of both methods.

Experiments. We demonstrate that ReVol improves the performance of the state-of-the-art backbone models in most scenarios, delivering an average gain of more than 0.03 in IC and over 0.7 in SR. Moreover, ReVol effectively narrows the distribution gap between training and test data via the proposed GBM-based normalization.

The definitions of the symbols in this paper are listed in Table 1. The code and datasets are available at https://github.com/AnonymousCoder2357/ReVol.

2. Related Works

2.1. Geometric Brownian Motion

Geometric Brownian motion (GBM) (Bjork, 2019) is one of the most popular models for stock price dynamics. GBM is widely used in various financial tasks, such as option pricing and portfolio optimization, due to its suitability and simplicity. The GBM model assumes that the logarithm of stock prices follows a Brownian motion with drift, allowing for continuous price movements. In GBM, the stock price $S_{t}$ at time $t$ is modeled as:

[TABLE]

where $\mu$, $\sigma$, and $B_{t}$ are the return, volatility of the stock, and the value of Brownian motion at time $t$, respectively. Brownian motion (BM) $B$ is a stochastic process with the following properties:

1. (1)

$B_{0}=0$ 2. (2)

if $0\leq s_{1}\leq t_{1}\leq s_{2}\leq t_{2}$, then $B_{t_{1}}-B_{s_{1}}$ and $B_{t_{2}}-B_{s_{2}}$ are independent 3. (3)

$B_{t}-B_{s}\sim\mathcal{N}(0,t-s)\quad\text{(for }0\leq s\leq t\text{)}$ 4. (4)

$t\to B_{t}$ is almost surely continuous

where $B_{t}$ is the value of BM at time $t$.

For discrete time $t$, $S_{t}$ is expressed in terms of $S_{t-1}$ by representing the BM part as $\epsilon_{t}\sim\mathcal{N}(0,1)$ with the following Lemma.

Lemma 2.1.

When time is discretized, the stock price $S_{t}$ at time $t\in\mathbb{N}$ is expressed as follows:

[TABLE]

where the error term $\epsilon_{t}\sim\mathcal{N}(0,1)$. For any $s,t\in\mathbb{N}$ with $s<t$, the error terms $\epsilon_{s}$ and $\epsilon_{t}$ corresponding to timepoints $s$ and $t$, respectively, are independent.

Proof.

By Equation (1), the stock price $S_{t-1}$ at time $t-1$ is given as follows:

[TABLE]

By dividing Equation (1) by Equation (3), we obtain the following:

[TABLE]

Let $\epsilon_{t}=B_{t}-B_{t-1}$. Then, $\epsilon_{t}$ and $\epsilon_{s}$ are independent for $s<t$ and $s\in\mathbb{N}$, and $\epsilon_{t}\sim\mathcal{N}(0,1)$ due to properties 2 and 3 of BM, respectively. By replacing $B_{t}-B_{t-1}$ with $\epsilon_{t}$ in Equation (4), we obtain the result. ∎

The assumption that $\epsilon_{t}$ and $\epsilon_{s}$ are independent for $s<t$ and $t,s\in\mathbb{N}$ means that future stock prices cannot be predicted from past prices. In contrast, we relax this assumption to allow learning from historical dependencies.

2.2. Deep Learning-Based Stock Price Prediction

With the advancement of deep neural networks, attempts to predict stock prices using the time series processing capabilities of recurrent neural networks have emerged (Nelson et al., 2017; Sunny et al., 2020). The attention mechanism has been used to enhance the performance of RNN-based models (Qin et al., 2017; Li et al., 2018; Feng et al., 2019; Yoo and Kang, 2021; Soun et al., 2022; Kim et al., 2024). GCN or Transformer architectures have been used to leverage the correlations between stocks (Li et al., 2021; Yoo et al., 2021; Li et al., 2024). However, these methods struggle to address the distribution shift problem since they fail to alleviate distributional discrepancy between training and test data. Instead, we focus explicitly on mitigating distribution shift rather than solely learning representations.

2.3. Methods for Addressing Distribution Shift

As distribution shift has emerged as a critical issue in time series analysis, various methods have been proposed to address it in the time series and financial domains. RevIN (Kim et al., 2021) and Dish-TS (Fan et al., 2023) equalize the distribution parameters, mean and standard deviation, for each sample through instance normalization. However, financial time series data do not generally follow a normal distribution. As a result, even if the mean and standard deviation are equalized, the distribution shapes still differ across samples due to higher-order characteristics such as skewness and kurtosis. There have been also methods to detect distribution shifts that may occur during the evolution of stock time series, and adapt to various market conditions (Lin et al., 2021; Zhao et al., 2023; Zhan et al., 2024). However, these methods rely on the representation space to adapt to the distribution on its own, which not only increases the burden on the model but also fails to explicitly address distribution shift. In this work, we aim to resolve the distribution shift problem by making the distribution of each sample identical to improve the accuracy of stock price prediction.

3. Proposed Method

We propose ReVol (Return-Volatility Normalization for Mitigating Distribution Shift in Stock Price Data), an accurate method for stock price prediction by mitigating the distribution shift problem.

3.1. Overview

We give a formal definition of the stock price prediction problem as follows:

Problem 1 (Stock Price Prediction.).

Given the historical prices $S_{T-w+1:T}^{o}$, $S_{T-w+1:T}^{h}$, $S_{T-w+1:T}^{l}$, and $S_{T-w+1:T}^{c}$ from day $T-w+1$ up to $T$, where $S_{t}^{o}$, $S_{t}^{h}$, $S_{t}^{l}$, and $S_{t}^{c}$ are opening, the highest, the lowest,and closing prices on day $t$, respectively, find the closing price $S_{T+1}^{c}$ for the next day.

To achieve accurate prediction mitigating distribution shift, ReVol is designed to address the following challenges.

{enumerate*}

Distribution shift in stock price data. The distribution of stock price data varies across stocks and over time. This distribution shift problem prevents accurate stock price prediction. However, existing methods focus only on matching the mean and standard deviation across samples, leaving the distribution shape unaligned. How can we reduce the discrepancies between distributions?

Estimating sample characteristics. We must accurately estimate and remove individual sample characteristics, from stock price data to reduce distribution discrepancies. However, due to external shocks, stock prices sometimes deviate temporarily from their true distribution. Such anomalies hinder the estimation of the true distribution when distribution parameters are naively estimated using arithmetic means. How can we estimate the true distribution accurately?

Reintegrating sample characteristics. Removing sample characteristics from the input data interrupts accurate prediction since the model cannot properly reconstruct the data distribution. How can we reintegrate the individual sample characteristics into the prediction?

Our proposed ReVol addresses the challenges with the following ideas. Figure 2 depicts the overall process of ReVol.

{enumerate*}

Return-volatility normalization (Section 3.2). Inspired by GBM, we normalize the input stock price data with sample characteristics (return, volatility, and price scale), to remove them from the data. This effectively reduces distributional discrepancies across samples. As a result, we obtain historical error terms, which serve as inputs to the neural network backbone.

Return-volatility estimator (Section 3.3). We calculate return and volatility using a weighted average with attention weights. This approach mitigates the impact of anomalies that disrupt characteristic estimation. Consequently, we acquire precisely estimated sample characteristics.

Backbone and Return-volatility denormalization (Section 3.4). We predict the future error term with the neural network backbone and denormalize it with sample characteristics to obtain the final prediction. By doing so, we restore the sample characteristics lost during the normalization process.

Each of the above idea corresponds to each main module of ReVol. First, $RVN$ (Return-Volatility Normalization) module normalizes historical stock prices with sample characteristics to generate historical error terms, which act as inputs for the neural network backbone. The backbone model does not suffer from distribution shift since the error terms follow an identical distribution across samples. Second, $RVE$ (Return-Volatility Estimator) module accurately estimates sample characteristics through an attention mechanism, which are subsequently used in $RVN$ and $RVD$ (Return-Volatility Denormalization). Lastly, the backbone predicts the future error term, and $RVD$ module denormalizes it to reintegrate information removed by $RVN$. The denormalized output provides the final future price prediction.

3.2. Return-Volatility Normalization

We propose $RVN$ (Return-Volatility Normalization) module to reduce the distributional discrepancy, inspired by GBM (Bjork, 2019). $RVN$ normalizes historical stock prices $S_{T-w+1:T}^{o}$, $S_{T-w+1:T}^{h}$, $S_{T-w+1:T}^{l}$, and $S_{T-w+1:T}^{c}$ for $w$ days up to day $T$ into normalized inputs $\epsilon_{T-w+1:T}^{o}$, $\epsilon_{T-w+1:T}^{h}$, $\epsilon_{T-w+1:T}^{l}$, and $\epsilon_{T-w+1:T}^{c}$, which serve as inputs to the backbone model. Note that $\epsilon_{T-w+1:T}^{o}$, $\epsilon_{T-w+1:T}^{h}$, $\epsilon_{T-w+1:T}^{l}$, and $\epsilon_{T-w+1:T}^{c}$ are not random variables as in Equation (2), but rather samples in this section. The backbone model is unaffected by distribution shift since the error terms share an identical distribution across samples.

For example, consider a stock observed over two different time periods. In the first period, the prices generally range between 100 and 120, and the daily returns have a mean of 0.02 and a standard deviation of 0.05. In the second period, the prices decrease and range between 60 and 80, and the daily returns have a mean of -0.01 and a standard deviation of 0.12. The raw daily returns in the first period, measured on a higher price scale (around 110), are tightly concentrated around 0.02 with low variability, while in the second period, the returns observed on a lower price scale (around 70) are more dispersed around -0.01 with higher variability. If we train the model on the first period and test it on the second period without any normalization, the model encounters a significant distribution shift in the level and dispersion of daily returns, as well as in the its price scale. After applying return-volatility normalization, the daily returns in both periods are transformed to have an identical distribution. This process enables the backbone network to learn predictive patterns that remain stable even when the distribution of the same stock changes over time.

To normalize the stock prices with the sample characteristics, we must first define the sample characteristics. In the widely used stock modeling method GBM, stock characteristics are defined as return, volatility, and price scale. We assume that stock prices follow a GBM, except for the independence assumption (refer to property 2 of GBM in Section 2.1). We relax this assumption since temporal dependencies exist due to market microstructure effects and autocorrelation in prices. Observing Equation (2), the equation consists of the variables $\mu$ (return), $\sigma$ (volatility), $S_{t-1}$ (price scale), and $\epsilon_{t}$ (error term). Note that 1) $\mu$, $\sigma$, and $S_{t-1}$ represent sample characteristics, and 2) $\epsilon_{t}$ is sampled from a standard normal distribution. Therefore, we eliminate distributional discrepancy by constructing an input with error terms.

We first briefly explain the process of estimating sample characteristics in Section 3.2.1 and then describe normalization methods for closing, opening, and the highest/lowest prices in Sections 3.2.2, 3.2.3, and 3.2.4, respectively.

3.2.1. Return and Volatility Estimation.

We first estimate the sample characteristics—return and volatility—based on GBM, using proposed $RVE$ (Return-Volatility Estimator), which is an attention-based module designed to accurately estimate return and volatility (see Section 3.3 for details). The estimation is expressed as follows.

[TABLE]

where $\mathbb{\phi}$ denotes the parameters of the $RVE$ module.

3.2.2. Normalizing Closing Price.

We obtain the error term $\epsilon_{t}^{c}$ by normalizing the logarithmic daily return using the estimated return and volatility, as shown in the following Lemma.

Lemma 3.1.

An error term $\epsilon_{t}^{c}$ which corresponds to the closing price at day $t$, is obtained as follows:

[TABLE]

Furthermore, $\epsilon_{t}^{c}$ is a sample drawn from a standard normal distribution: $\epsilon_{t}^{c}\overset{\text{sample}}{\sim}\mathcal{N}(0,1)$.

Proof.

By Equation (2), the closing price $S_{t}$ on day $t$ is given as follows:

[TABLE]

Note that $\epsilon_{t}^{c}\overset{\text{sample}}{\sim}\mathcal{N}(0,1)$ by Lemma 2.1. We rearrange Equation (6) as follows:

[TABLE]

We get the result by replacing $\mu-\frac{1}{2}\sigma^{2}$ and $\sigma$ with their estimations. ∎

The error term $\epsilon_{t}^{c}$ represents the error of the closing price on day $t$ relative to the closing price on day $t-1$. Thus, we exploit $\epsilon_{t}^{c}$ as a feature for training in place of $S_{t}^{c}$, the closing price on day $t$.

3.2.3. Normalizing Opening Price.

Capturing the relationship between the previous day’s closing price and today’s opening price is essential since overnight information—e.g., disclosures, news, and foreign market movements after the close—is immediately reflected in today’s opening price. A simple comparison of the opening–opening relationship fails to account for this overnight drift. Therefore, modeling the closing–opening relationship first is critical to improve daily prediction performance.

A simple way to normalize the opening price is to apply Equation (5) in the same way, as follows:

[TABLE]

However, in this case, the error term $\epsilon_{t}^{o}$ represents the change from the opening price on day $t-1$ to the opening price on day $t$. Then, the relationship between the opening and closing prices cannot be captured. Therefore, we propose to map the opening and closing time onto a continuous time axis $[0,\infty)$, and associate each time with the corresponding price. Then, we calculate error terms that capture the relationship between them.

Let $S_{t}$ represent the stock price at time $t$ on a continuous time axis $[0,\infty)$. In this axis, $t\in\mathbb{N}$ represents the closing time of day $t$ where we regard $t$ as an integer. That is, $S_{t}=S_{t}^{c}$ for $t\in\mathbb{N}$. We associate the opening, highest, and lowest prices on day $t$ with their respective time points on this time axis. Let $0\leq r,u,v\leq 1$ represent the time intervals between the closing price on day $t-1$ and the opening, highest, and lowest prices on day $t$, respectively, as shown in Figure 3. Then, on the continuous time axis, $S_{t-1+r}=S_{t}^{o}$, $S_{t-1+u}=S_{t}^{h}$, and $S_{t-1+v}=S_{t}^{l}$.

By Equation (1), the opening price $S_{t-1+r}$ is expressed in terms of the previous day’s closing price $S_{t-1}$:

[TABLE]

where $\epsilon_{t}^{o}\overset{\text{sample}}{\sim}\mathcal{N}(0,r)$ is the error term sampled from $B_{t-1+r}-B_{t-1}$. $\epsilon_{t}^{o}$ is determined if $r$ is known. However, determining the time interval $r$ between the opening and previous day’s closing prices is not straightforward. We might consider setting $r$ to 0 since the opening price is the immediate price following the previous day’s closing price. However, the opening price differs from the previous day’s closing price due to price fluctuations during after-hours trading, meaning that $r$ cannot be set to 0. Therefore, we need to estimate $r$ before normalizing the opening price. We estimate $r$ based on the assumption that stock prices follow a geometric modeling with respect to time.

Lemma 3.2.

By minimizing a mean squared error, $r$ is estimated as follows:

[TABLE]

Proof.

According to the geometric modeling of stock prices, we express the opening price on day $t$ as follows:

[TABLE]

This expression is based on a simplifying assumption that the stock price follows a deterministic path: $S_{t}=S_{0}e^{\kappa t}$ for some $\kappa\in\mathbb{R}$. While this assumption does not fully align with the GBM and is slightly inconsistent with Equations (2) and (7), it offers a tractable approximation that allows for a closed-form estimation of $r$, which is useful in practice.

By taking the log of Equation (9), we rewrite it as follows:

[TABLE]

To find the best $r$ that satisfies the equation above, we minimize the following mean squared error:

[TABLE]

Equation (10) attains its minimum at the point where the derivative with respect to $r$ is equal to zero since it is convex with respect to $r$:

[TABLE]

Solving Equation (11), we obtain the result. ∎

As a result, we obtain the error term $\epsilon_{t}^{o}$ with the following Lemma.

Lemma 3.3.

Assume that Equation (8) accurately estimates $r$, i.e., $\hat{r}=r$. Then, the error term $\epsilon_{t}^{o}$ corresponding to the opening price on day $t$, is obtained as follows:

[TABLE]

Furthermore, $\frac{1}{\sqrt{\hat{r}}}\epsilon_{t}^{o}$ is a sample drawn from a standard normal distribution:

[TABLE]

Proof.

Rewriting Equation (7), we obtain the following:

[TABLE]

Since $\epsilon_{t}^{o}\overset{\text{sample}}{\sim}\mathcal{N}(0,r)$, $\frac{1}{\sqrt{\hat{r}}}\epsilon_{t}^{o}\overset{\text{sample}}{\sim}\mathcal{N}(0,1)$. We obtain the result by replacing $\mu-\frac{1}{2}\sigma^{2}$, $\sigma$, and $r$ with their estimations. ∎

3.2.4. Normalizing the Highest and Lowest Prices.

The highest price also cannot be normalized to the error term using the previous day’s highest price, since it fails to capture the relationship with the closing price. Furthermore, it is also impossible to normalize it in the same way as the opening price, as shown in Equation (12), since the time interval $u$ between the highest and previous day’s closing prices is unknown.

To address this, we define $\epsilon_{t}^{h}$ and normalize the highest price as follows:

[TABLE]

This definition is justified as follows. Since $S_{t}^{h}$ is the highest price on day $t$,

[TABLE]

We simplify Equation (13) by ignoring the drift term $\left(\mu-\frac{1}{2}\sigma^{2}\right)u$, as its contribution is negligible compared to the volatility term $\sigma B_{u}$ on a daily scale (Andersen and Bollerslev, 1998; Andersen et al., 2001). These small values indicate that the drift’s effect is minor in intraday settings. To empirically justify this simplification, we approximate the distribution of the ratio $\left|\frac{\left(\mu-\frac{1}{2}\sigma^{2}\right)u}{\sigma B_{u}}\right|$. The results show that this ratio is less than $\frac{1}{5}$ in 74.1% of the cases in the U.S., 78.7% in China, 75.3% in the U.K., and 73.0% in Korea. Thus, we get

[TABLE]

Since $\max_{0<u\leq 1}B_{u}$ follows a distribution independent of specific samples, the resulting $\epsilon_{t}^{h}$ also follows a consistent distribution across different samples.

We also normalize the lowest price in the same way:

[TABLE]

Note that $\epsilon_{t}^{l}$ also follows a consistent distribution across different samples.

3.3. Return-Volatility Estimator

$RVE$ (Return-Volatility Estimator) is designed to accurately estimate sample characteristics from historical stock prices. Specifically, $RVE$ takes historical prices as input and estimates the distribution parameters: return and volatility. Stock prices sometimes deviate temporarily from their true distribution due to external shocks. For example, stock prices experience a temporary surge due to expectations around a new product launch, or a sharp decline due to institutional investors liquidating their positions for short-term gains. These temporary fluctuations create noise in the data, making it difficult to accurately estimate the underlying distribution of stock prices. We propose to estimate the distribution parameters exploiting a weighted average based on attention weights to handle such anomalies flexibly.

We observe that $\log\frac{S_{t}^{c}}{S_{t-1}^{c}}=\mu-\frac{1}{2}\sigma^{2}+\sigma\epsilon_{t}^{c}\sim\mathcal{N}(\mu-\frac{1}{2}\sigma^{2},\sigma^{2})$ by applying Equation (2) to closing prices. According to this observation, a simple approach to estimate $\mu-\frac{1}{2}\sigma^{2}$ and $\sigma$ is to compute the arithmetic mean as follows:

[TABLE]

However, this approach is significantly affected by anomalies that deviate from the true distribution, making it difficult to accurately estimate the distribution parameters. Our main idea is to estimate $\mu-\frac{1}{2}\sigma^{2}$ and $\sigma$ using a weighted average based on attention weights as follows:

[TABLE]

where $\alpha_{t}$ is the attention weight at time step $t$, calculated by Equation (16). Note that $\sum_{t=T-w+1}^{T}\alpha_{t}=1$. This approach flexibly handles anomalies by dynamically adjusting the average weights.

Attention module structure. We describe the structure of the attention module to compute the attention weight $\alpha_{t}$. We exploit attention LSTM (Feng et al., 2019) to generate query and key vectors that effectively capture sequential dependencies in time-series data.

We begin by transforming price feature vector $\mathbf{x}_{t}\in\mathbb{R}^{D}$ at each time step using a fully connected layer with a hyperbolic tangent activation function, $\tilde{\mathbf{x}}_{t}=tanh(\mathbf{W}\mathbf{x}_{t}+\mathbf{b})$, where the parameters $\mathbf{W}\in\mathbb{R}^{E\times D}$ and $\mathbf{b}\in\mathbb{R}^{E}$ are shared across all time steps $t$. This transformation produces a new feature representation before the data are passed into the LSTM, improving the network’s ability to capture temporal patterns (Feng et al., 2019).

The feature vectors $\tilde{\mathbf{x}}_{T-w+1},\ldots,\tilde{\mathbf{x}}_{T}$ are then passed into the LSTM and transformed into hidden state vectors $\mathbf{h}_{T-w+1},\ldots,\mathbf{h}_{T}$. The hidden state vector $\mathbf{h}_{T}$ from the last time step represents past information and serves as the query vector. Finally, the attention weight $\alpha_{t}$ at time step $t$ is calculated as follows:

[TABLE]

3.4. Backbone and Return-Volatility Denormalization

The neural network backbone $f_{\mathbf{\theta}}$ receives the normalized error terms of historical stock prices as input, specifically $\epsilon_{T-w+1:T}^{o}$, $\epsilon_{T-w+1:T}^{h}$, $\epsilon_{T-w+1:T}^{l}$, and $\epsilon_{T-w+1:T}^{c}$. These error terms are generated by the $RVN$ module and share an identical distribution across samples, effectively mitigating distribution shift. The backbone model outputs the predicted future error term $\hat{\epsilon}_{T+1}^{c}$, which does not contain any sample-specific characteristics such as return or volatility.

[TABLE]

The backbone can be instantiated with any time series prediction model, such as LSTM, GRU, or Transformer variants, allowing ReVol to leverage their strengths while controlling for distribution shift.

With the predicted future error term $\hat{\epsilon}_{T+1}^{c}$, the $RVD$ (Return-Volatility Denormalization) module denormalizes it to produce the final prediction $\hat{S}_{T+1}^{c}$. It reintegrates the information of sample characteristics lost in the normalization process into the prediction of the backbone model. $RVD$ works by replacing the BM part of the GBM with the predicted future error term.

[TABLE]

Due to the independence assumption that future prices are independent of past prices, GBM cannot predict future prices. However, the model gains forecasting ability by eliminating the independence assumption and replacing the BM part with the predicted future error term. ReVol considers both learned patterns of the historical error terms and inherent price dynamics reflected in the sample characteristics.

3.5. Loss Function

The final loss function is designed to optimize the parameters of the backbone model $\mathbf{\theta}$ and the parameters of the $RVE$ module $\mathbf{\phi}$. The loss is given by

[TABLE]

The loss is composed of two parts. The first part is the MSE loss for daily return prediction. The second part is a guidance loss to give robustness to $RVE$. We observe that ReVol with returns calculated through arithmetic averages (Equation (14)) also leads to the performance improvement. Therefore, we propose to guide $\widehat{\mu-\frac{1}{2}\sigma^{2}}$ to resemble the arithmetic mean of the logarithmic daily return through minimizing guidance loss. These joint objectives enable end-to-end training of the backbone and the $RVE$ module. $\beta$ is a hyperparameter to control the effect of the guidance loss. Algorithm 1 illustrates the whole training process.

4. Experiments

We present experimental results to answer the following research questions about ReVol:

{itemize*}

Performance (Section 4.2). Does ReVol outperform previous methods in stock price prediction?

Comparison with other normalization methods (Section 4.3). How well does ReVol address distribution shifts in stock price data compared to other normalization methods?

Hyperparameter sensitivity (Section 4.4). How sensitive is ReVol to hyperparameters?

Noise-aware behavior of attention weights (Section 4.5). Does the attention mechanism in the RVE module effectively suppress noisy inputs during return-volatility estimation?

Ablation study (Section 4.6). Does each module of ReVol contribute to improving the performance for stock price prediction?

4.1. Experimental Setup

Datasets. We use stock market datasets collected from the stock markets of the United States, China, United Kingdom, and Korea. The summary of the datasets is presented in Table 2. The goal is to predict the daily return $\frac{S_{T+1}^{c}}{S_{T}^{c}}$ given the price features up to day $T$. We split the datasets in chronological order into 70% for training, 10% for validation, and 20% for testing.

Data normalization. We generate an input vector $\mathbf{x_{t}}$ at day $t$ for baselines using widely used price ratio-based normalization method (Feng et al., 2019; Yoo et al., 2021). In contrast, ReVol exploits the normalization technique described in Section 3.2. The normalization method for competitors is shown in Table 3. $x_{t}^{o}$, $x_{t}^{h}$, and $x_{t}^{l}$ indicate the relative values of the opening, highest, and lowest prices in comparison to closing price at day $t$, respectively. $x_{t}^{c}$ indicates the relative value of the closing price at day $t$ in comparison to closing price at day $t-1$.

Baselines. ReVol is a model-agnostic method applied to neural network backbones for stock price prediction. We apply ReVol to the following baselines to validate its performance.

{itemize*}

LSTM (Nelson et al., 2017) is a recurrent neural network designed to capture long-term dependencies in time series data.

GRU (Cho et al., 2014) is a simplified variant of LSTM that uses gating mechanisms to effectively capture sequential dependencies in time series data with fewer parameters.

ALSTM (Feng et al., 2019) enhances the LSTM model by incorporating an attention mechanism, allowing it to focus on the most relevant time steps for improved stock price prediction.

Vanilla Transformer (Vaswani et al., 2017) is a self-attention-based architecture originally proposed for machine translation, which can model long-range dependencies in sequences without relying on recurrence.

DTML (Yoo et al., 2021) leverages a transformer structure to capture temporal and inter-stock correlations with global market context for accurate prediction.

MASTER (Li et al., 2024) is a transformer-based approach for stock price forecasting that effectively captures momentary and cross-time stock correlations through two-step self-attentions in temporal and stock dimensions, respectively.

Evaluation metrics. We conduct 10 experiments with different random seeds ranging from 0 to 9, and report the average across all cases. We evaluate the results of stock price prediction using Information Coefficient (IC) and Rank Information Coefficient (RIC). IC and RIC are the dominant metrics in finance, which measure the linear relationships between the predicted values and the actual values with Pearson correlation coefficient and Spearman correlation coefficient, respectively.

We evaluate the portfolio’s performance using annualized Sharpe Ratio (SR). The portfolio is rebalanced at the close of each day to allocate a weight of 0.2 to the top 5 stocks with the highest predicted returns.

**Hyperparameters. ** We optimize the network parameters using the Adam (Kingma, 2014) optimizer and apply early stopping based on the IC of the validation set. We conduct a hyperparameter search on learning rate, weight decay, the number of attention heads (for transformer-based models), window size, hidden layer size, and guidance loss weight in $\{0.00001,0.0001,0.001\}$, $\{0,0.0001,0.001,0.01\}$, $\{1,2,4,8\}$, $\{8,16,24,32,40\}$, $\{128,256,512\}$, and $\{0,0.25,0.5,1\}$, respectively.

4.2. Performance (Q1)

Table 4 presents the performance comparison of ReVol and the baseline models for stock price prediction across four markets (U.S., China, U.K., and Korea). ReVol consistently improves performance across all backbone models and metrics, demonstrating its effectiveness in addressing distribution shifts. On average, ReVol improves IC by over 0.03 and SR by more than 0.7 compared to the baselines, confirming its ability to enhance predictive accuracy and profitability.

Notably, ReVol improves model stability and generalization across diverse market conditions, achieving consistent gains across multiple backbones. This robustness to distributional variations makes ReVol a practical and scalable solution for real-world financial forecasting tasks.

4.3. Comparison with Other Normalization Methods (Q2)

Table 5 compares the performance of ReVol with two state-of-the-art normalization methods, RevIN (Kim et al., 2021) and Dish-TS (Fan et al., 2023). RevIN, Dish-TS and ReVol utilize DTML as the backbone model.

ReVol shows consistent improvement in IC and SR across all markets, outperforming the other normalization methods. As illustrated in Figure 1, ReVol accurately aligns the distribution shapes between the training and test sets while the other method matches only distribution parameters. This precise alignment reduces distributional discrepancies that degrade predictive accuracy. Additionally, unlike other methods which normalize features independently and lose inter-feature correlations, ReVol preserves these relationships. This enables more accurate and stable predictions.

4.4. Hyperparameter Sensitivity (Q3)

Figure 4 compares the hyperparameter sensitivity of ReVol with DTML as the backbone to standalone DTML across four key hyperparameters: (a) hidden layer size, (b) window size, (c) number of attention heads, and (d) weight decay. DTML shows significant sensitivity to these hyperparameters, resulting in unstable performance and fluctuations in IC, especially for window size and weight decay settings.

In contrast, ReVol consistently maintains high IC values across all hyperparameter settings. This stability is attributed to ReVol ’s ability to mitigate distribution shifts, reducing overfitting and improving generalization. By aligning distributions across samples, ReVol decreases the burden on DTML, making the model more robust and less dependent on precise hyperparameter tuning. Consequently, ReVol achieves more reliable and consistent performance under various training conditions.

4.5. Noise-Aware Behavior of Attention Weights (Q4)

Table 6 shows that ReVol consistently assigns lower attention weights to time steps with high return magnitudes, indicating its overall tendency to suppress noisy inputs. Here, noisy inputs refer to observations in which stock prices deviate temporarily from their true distribution due to external shocks, such as unexpected news events or large-scale liquidations. These results suggest that our attention mechanism effectively identifies and down-weights noisy observations that degrade the estimation of return and volatility.

We quantify this behavior using two complementary analyses. First, the Pearson correlation coefficient between the absolute return and attention weight is negative across all markets, showing that higher-volatility inputs receive lower attention weights. Second, the attention ratio between noisy and less noisy time steps is consistently below 1, confirming that noisy observations are down-weighted.

This behavior is attributed to the optimization dynamics of ReVol. During training, the attention weights are adjusted to minimize the final loss. If noisy time steps receive high attention weights, the estimated return and volatility become less accurate, leading to larger prediction errors. Through backpropagation, the model receives gradients that penalize high attention weights assigned to noisy inputs since these inputs increase prediction error. Hence, the attention mechanism implicitly filters out noise by learning to trust stable time steps more.

4.6. Ablation Study (Q5)

We analyze the performance of ReVol with DTML as the backbone, and its variants by removing each of the three primary modules, as shown in Table 7:

{itemize*}

ReVol-N: ReVol without the return-volatility normalization

ReVol-E: ReVol without the return-volatility estimator

ReVol-D: ReVol without the return-volatility denormalization

Each module contributes to improved prediction performance, with ReVol incorporating the three modules achieving the highest results. $RVN$ module contributes the most to the overall performance as it directly addresses the fundamental issue of distribution shift in stock price data. $RVN$ effectively reduces distribution discrepancies across samples by normalizing the input data with sample characteristics. This enables the model to focus on learning normalized patterns from historical error terms. This process stabilizes the input distribution, facilitating more robust and efficient feature learning.

5. Conclusion

We propose ReVol, an accurate prediction method for alleviating distribution shift in stock price data. The main ideas of ReVol are:

1. normalizing price features to address distribution shifts by eliminating individual sample characteristics such as return, volatility, and price scale,
2. precisely estimating these characteristics through an attention-based module to minimize the influence of market anomalies, and
3. reintegrating the sample characteristics into the prediction process to restore lost information. Additionally, ReVol leverages geometric Brownian motion to capture long-term trends and neural networks for short-term patterns, effectively integrating their strengths for stock price modeling. ReVol boosts the performance of state-of-the-art backbone models in most scenarios, demonstrating notable improvements in predictive accuracy and stability through the proposed normalization approach.

GenAI Usage Disclosure

We have used a generative AI tool (ChatGPT by OpenAI) solely to check for grammatical errors and enhance the clarity of the manuscript during the writing stage. No part of the research design, experiments, data analysis, or results interpretation was conducted using generative AI. All technical contributions are the authors’ own work.

Acknowledgements

This work was supported by the National Research Foundation of Korea(NRF) funded by MSIT(2022R1A2C3007921). This work was also supported by Institute of Information & communications Technology Planning & Evaluation(IITP) grant funded by the Korea government(MSIT) [No.RS-2021-II211343, Artificial Intelligence Graduate School Program(Seoul National University)], and [NO.RS-2021-II212068, Artificial Intelligence Innovation Hub (Artificial Intelligence Institute, Seoul National University)]. The Institute of Engineering Research at Seoul National University provided research facilities for this work. The ICT at Seoul National University provides research facilities for this study. U Kang is the corresponding author.