Crossing-Free Probabilistic K-Line Forecasts Without Retraining Runyao Yu∗
Yuchen Tao
Yujie Chen
London Business School London, United Kingdom [email protected]
RWTH Aachen University Aachen, Germany [email protected]
The Chinese University of Hong Kong Shenzhen, China [email protected]
Wentao Wang
Derek W. Bunn
University of Technology Sydney Sydney, Australia [email protected]
London Business School London, United Kingdom [email protected]
arXiv:2607.26792v1 [stat.ML] 29 Jul 2026
Abstract
forecasting, predicts four complementary price landmarks [9, 10]. The open price reflects the valuation at the beginning of a trading period, the high and low prices determine the realized trading range and extreme movements, and the close price summarizes the terminal valuation. Joint OHLC forecasts can consequently support entry and exit decisions, stop-loss placement, intraperiod risk assessment, and trading strategies that cannot be constructed from a single price index alone. However, existing K-line forecasting research remains predominantly pointwise. Structural VAR and VECM models forecast one OHLC vector for each period [10], while Transformer-based methods similarly produce point forecasts for the four prices [9]. Conversely, most probabilistic financial forecasting studies estimate the distribution of only one target [3, 18]. Probabilistic K-line forecasting combines these two settings by predicting multiple quantiles for all four prices, thereby describing the uncertainty of the opening price, closing price, and trading range simultaneously. Although financial econometric models commonly forecast returns as asset price levels are often nonstationary, we retain price levels to align with the existing K-line forecasting literature and preserve the direct interpretation of OHLC forecasts. Kronos represents a rare recent attempt to generate K-line paths [15]. Nevertheless, to our knowledge, the simultaneous reconciliation of probabilistic K-line forecasts has not been studied as a model-agnostic and training-free problem. Probabilistic K-line forecasts introduce two distinct consistency problems. As illustrated in Fig. 1, quantile crossing occurs when a higher quantile falls below a lower quantile; K-line crossing occurs when the predicted high is lower than the open or close, or the predicted low is higher than the open or close. These violations can occur simultaneously, as a probabilistic K-line forecast must preserve both the quantile order of every feature and the OHLC relations at every quantile level. K-line crossing has received comparatively limited attention. Existing solutions incorporate OHLC relations through model-specific designs and penalty terms in the training loss [9]. Quantile crossing has been studied more extensively through post-hoc reordering [5], penalized quantile-regression objectives [13], and hierarchical multi-quantile heads [19–21]. However, these methods are designed for one constraint family and do not independently guarantee both quantile and K-line consistency. Moreover, training-time approaches require model modification, retraining, or penalty selection, whereas reordering can alter the original quantile assignments and change forecasts more than necessary.
Probabilistic K-line forecasting describes uncertainty in four complementary prices, namely open–high–low–close (OHLC). However, it introduces two consistency problems: quantile crossing and K-line crossing. Quantile crossing occurs when a higher-quantile forecast falls below a lower-quantile forecast, while K-line crossing occurs when the forecast low exceeds the open or close, or the forecast high falls below the open or close. Existing solutions generally address only one problem through output reordering, specialized architectures, or penalized training objectives. We propose K-line–Quantile Sequential Projection (KQSP), a parameter-free and training-free reconciliation method applicable to forecasts produced by any model. Compared with other crossing solutions, KQSP preserves predictive accuracy while producing substantially smaller corrections to the original forecasts. To mitigate model bias, we evaluate KQSP using various models, including pretrained foundation models. KQSP reduces both quantile and K-line crossing rates to zero for all test data undertaken. These results show that probabilistic K-line consistency can be enforced independently of forecast generation and without retraining.
CCS Concepts • Computing methodologies → Machine learning; • Applied computing → Economics.
Keywords probabilistic k-line forecasting, forecast correction, time-series foundation models ACM Reference Format: Runyao Yu, Yuchen Tao, Yujie Chen, Wentao Wang, and Derek W. Bunn. 2026. Crossing-Free Probabilistic K-Line Forecasts Without Retraining. In Proceedings of (Preprint). ACM, New York, NY, USA, 8 pages. https://doi. org/10.1145/nnnnnnn.nnnnnnn
1
Introduction
Probabilistic forecasts are particularly valuable in uncertain financial markets as they describe possible outcomes and tail risks rather than providing only one expected value [7]. However, most probabilistic financial forecasting studies focus on a single price target [3, 4, 16], such as index level or close price [11, 12]. K-line forecasting, also known as candlestick or open–high–low–close (OHLC) ∗ Runyao Yu is also affiliated with Delft University of Technology and the AIT Austrian
Institute of Technology. 1
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Yu et al.
Crossing issues
Proposed KQSP ẐKQSP = PQ( PO(Z) )
K-Line Crossing --C --H --O --L
Price
Forecast correction
Stage 1 K-Line projection PO
Raw forecast matrix Z (may cause both crossing issues)
Horizon
q10
q50
q90
q10
q50
q90
Stage 2 Quantile projection PQ q10
q50
q90
KC Valid OHLC relations L ≤ min(O, C) ≤ max(O, C) ≤ H
O
95
100
105
O
95
100
105
O
95
100
105
H
99
99
105
H
99
100
105
H
99
100
105
Quantile crossing
L
95
98
99
L
95
98
99
L
95
98
99
99
C
99
100
99
C
99
99.5 99.5
C
Price
q90 q50 q10 Horizon
QC
Monotone quantiles qτ₁ ≤ qτ₂ ≤ ··· ≤ qτ K
99
101
Any forecaster
Each stage is an orthogonal projection ⇒ minimizes correction error subject to the respective constraint. Model-agnostic ⌘ Works with any forecaster
↻
No retraining Post-hoc projection only
No parameters ⇵ Not add additional model weights
ANN
TimesFM
TimesFM 2.5
Chronos
Chronos-2
Moirai
Time-MoE
TabPFN
Kronos
Raw matrix Z
No tuning ◎ Deterministic and reproducible
KQSP PQ(PO(·))
Repaired probabilistic K-Line forecasts (satisfies KC and QC)
Figure 1: Overview of the work. Left: illustration of the two crossing issues. Middle: a numerical example of KQSP, which sequentially applies the K-line projection and quantile projection to repair the raw forecast. Right: model-agnostic application of KQSP to probabilistic K-line forecasts generated by various models. (b)
(c)
BMW
Mercedes
Volkswagen
(d)
BP
PetroChina
Shell
Close price
Close price
Close price
Close price
(a)
Goldman
JPMorgan
LSEG
Meta
Microsoft
Figure 2: Min–max-scaled close prices over the common period from January 1 to July 23, 2026. The panels show three stocks from each category: (a) Automobile, (b) Energy, (c) Finance, and (d) Technology. • We evaluate KQSP across a fully trained ANN and various zero-shot foundation models. KQSP eliminates all remaining crossings and consistently improves AQL, MAE, and RMSE, demonstrating robustness across forecasting models with substantially different architectures.
In this work, we propose K-line–Quantile Sequential Projection (KQSP), a training-free method that reconciles any existing probabilistic K-line forecast. KQSP first applies the minimum-distance Kline projection at each quantile level and then applies the minimumdistance quantile projection to each OHLC feature, which will be detailed in Section 4. KQSP introduces no additional parameter, hyperparameter, loss penalty, or retraining requirement. We evaluate KQSP on various stocks using forecasts from a fully trained Artificial Neural Network (ANN) and eight zero-shot foundation models, and compare it with multiple crossing solutions. KQSP eliminates both crossing types for every evaluated case without compromising predictive performance. Instead, these predictive metrics are frequently improved with statistical significance, while KQSP changes the original forecasts less than other crossing solutions. The main contributions are threefold:
2 Preliminary 2.1 Quantile Crossing Let T = {𝜏1 < · · · < 𝜏𝑄 } be 𝑄 ordered quantile levels, let F = {𝑂, 𝐻, 𝐿, 𝐶} denote open, high, low, and close, and let Z = [𝑧𝑞𝑓 ] ∈ R𝑄 ×4 be one raw probabilistic forecast, where 𝑧𝑞𝑓 is the forecast at level 𝜏𝑞 for feature 𝑓 ∈ F . Quantile consistency requires every feature-specific vector to belong to the monotone cone
• We propose K-line–Quantile Sequential Projection (KQSP), a model-agnostic method that removes quantile and K-line crossings, without introducing parameters, hyperparameters, loss penalties, or retraining. • We systematically compare KQSP with multiple crossing solutions. KQSP achieves comparable predictive accuracy to the other methods while producing significantly smaller corrections.
CQ = {v ∈ R𝑄 : 𝑣 1 ≤ · · · ≤ 𝑣𝑄 }.
(1)
A quantile crossing occurs when 𝑧𝑞𝑓 > 𝑧𝑞+1,𝑓 for at least one adjacent pair. Independent unconstrained outputs can violate this order under finite data and imperfect optimization. For example, at levels (0.1, 0.5, 0.9), open forecasts (95, 105, 100) cross because the 0.9-quantile forecast 100 is lower than the median forecast 105. 2
Crossing-Free Probabilistic K-Line Forecasts Without Retraining
Preprint, Under Review, -
Table I: Data set summary. Mean and standard deviation of daily OHLC prices in native units.
Open Stock
Category
Exchange Currency Ticker
High
Low
Close
Mean
Std.
Mean
Std.
Mean
Std.
Mean
Std.
81.90 55.05 155.69
12.57 11.32 51.26
80.16 53.85 151.67
12.53 11.21 49.23
81.04 54.45 153.62
12.57 11.27 50.27
85.80 441.86 1.85 7.15 513.93 2251.21
85.90 1.88 515.20
BMW Automobile Xetra Mercedes Automobile Xetra Volkswagen Automobile Xetra
EUR EUR EUR
BMW MBG VOW
81.05 54.46 153.75
12.56 11.27 50.35
BP Energy PetroChina Energy Shell Energy
LSE HKEX LSE
GBX HKD GBX
BP. 00857 SHEL
441.95 7.14 2251.40
85.97 446.89 1.87 7.22 514.40 2274.34
Goldman JPMorgan LSEG
Finance Finance Finance
NYSE NYSE LSE
USD USD GBX
GS JPM LSEG
367.03 206.59 371.23 209.41 362.96 203.84 367.20 206.76 153.79 68.84 155.29 69.55 152.34 68.19 153.85 68.90 7161.82 2405.01 7238.58 2429.66 7083.68 2377.85 7161.80 2403.31
Google Meta Microsoft
Technology Nasdaq Technology Nasdaq Technology Nasdaq
USD USD USD
GOOG META MSFT
117.48 308.56 249.07
2.2
K-Line Crossing
max(𝑂, 𝐶) ≤ 𝐻 }.
(2)
A K-line crossing occurs when 𝑧𝑞𝐻 < max{𝑧𝑞𝑂 , 𝑧𝑞𝐶 } or 𝑧𝑞𝐿 > min{𝑧𝑞𝑂 , 𝑧𝑞𝐶 }. Independent feature outputs need not preserve these semantic bounds. For example, (100, 99, 98, 101) in (𝑂, 𝐻, 𝐿, 𝐶) order crosses because high 99 is below close 101.
3
118.84 312.49 251.42
77.89 187.71 138.87
116.24 304.56 246.57
75.94 183.09 136.44
117.58 308.59 249.10
76.99 185.39 137.69
shared representation h = 𝑓𝜽 (x) is mapped by its existing output layer to Z, where x is the historical input and 𝜽 denotes the backbone parameters. Our primary goal is not to optimize the backbone, engineer features, or construct optimal factors for predictive accuracy. Instead, we isolate whether quantile and K-line crossings can be removed from existing density forecasts, regardless of whether their native predictive performance is strong or weak.
For quantile index 𝑞, let z𝑞 = (𝑧𝑞𝑓 ) 𝑓 ∈ F denote its OHLC forecast. K-line consistency requires this vector to belong to CO = {(𝑂, 𝐻, 𝐿, 𝐶) ∈ R4 : 𝐿 ≤ min(𝑂, 𝐶),
76.88 185.62 137.73
85.88 437.12 1.91 7.07 515.77 2229.66
4.2
K-line–Quantile Sequential Projection (KQSP)
KQSP is a parameter-free sequential projection method that makes probabilistic K-line forecasts Z = [𝑧𝑞𝑓 ] ∈ R𝑄 ×4 crossing-free. Each row of Z contains the OHLC forecasts at one quantile level, while each column contains all quantile forecasts for one OHLC feature. KQSP first corrects each row to satisfy the K-line constraints and then corrects each column to satisfy the quantile order, using the minimum-distance correction at each stage. The projection is
Data
We use 30,174 daily OHLCV observations, where V denotes trading volume, from 12 stocks across four categories. The data span July 23, 2016 to July 23, 2026, with 2,427–2,537 observations per stock. Each eligible sample uses a historical OHLCV window to predict the next-day OHLC vector. Samples are ordered chronologically and split without shuffling into 70% training, 15% validation, and 15% testing subsets. All input and output scalers are fitted on training data only. Table I reports descriptive statistics of the OHLC prices in their native units. For visual comparability and to reveal the overall temporal trends, each category panel in Fig. 2 rescales every stock’s 2026 close prices to [0, 1] over the displayed period. Together, the summary table and figure demonstrate substantial diversity in price scale, temporal shape, and business domain, providing heterogeneous cases for evaluating forecast reconciliation.
b ZKQSP = PQ (PO (Z)) ,
(3)
where PO denotes the OHLC projection, PQ denotes the quantile projection, and b ZKQSP is the final forecast. The size of a correction is measured by 𝐸 corr (b Z, Z) =
𝑄 ∑︁ ∑︁
(b 𝑧𝑞𝑓 − 𝑧𝑞𝑓 ) 2 .
(4)
𝑞=1 𝑓 ∈ F
A smaller value means that the corrected forecast remains closer to the original forecast, which is critical when the original model is considered reliable and only the minimum correction required to remove crossings is desired. Stage 1: K-line projection. For each quantile level 𝑞, KQSP finds the valid OHLC vector closest to the original row:
4 Method 4.1 Backbone-Agnostic Forecast Interface The probabilistic K-line forecasts Z = [𝑧𝑞𝑓 ] ∈ R𝑄 ×4 may come from a fully trained ANN, a zero-shot foundation model (FM), or any other backbone with the same output interface. For an ANN, a
c𝑞★ = arg min ∥c − z𝑞 ∥ 22, c∈ CO
3
(5)
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Table II: Corrections for the K-line example (100, 99, 98, 101) and the quantile example (99, 100, 99). Bold values indicate corrected forecasts.
where CO contains all vectors satisfying 𝐻 ≥ 𝑂, 𝐻 ≥ 𝐶, 𝐿 ≤ 𝑂, and 𝐿 ≤ 𝐶. Because there are only four relations, KQSP checks every possible combination in which one or more relations become equalities and selects the valid result with the smallest correction. For example, consider (𝑂, 𝐻, 𝐿, 𝐶) = (100, 99, 98, 101). The high forecast is below both the open and close forecasts. One possible correction is to raise 𝐻 from 99 to 101, producing (100, 101, 98, 101) with squared correction (101 − 99) 2 = 4. KQSP instead makes a smaller joint correction: it raises 𝐻 from 99 to 100 and lowers 𝐶 from 101 to 100. It therefore returns c𝑞★ = (100, 100, 98, 100) with squared correction (100 − 99) 2 + (100 − 101) 2 = 2. Thus, KQSP satisfies all OHLC relations while changing the original forecast less than the one-sided correction. Stage 2: quantile projection. After correcting all rows, let c★:𝑓 contain the forecasts of feature 𝑓 across all quantile levels. KQSP finds the closest nondecreasing sequence: KQSP
b z:𝑓
= arg min ∥v − c★:𝑓 ∥ 22, v∈ CQ
Method
K-line forecast
Quantile forecast
Correction error
KQSP Cumulative Max. Reordering Joint Projection
(100, 100, 98, 100 ) (100, 101, 98, 101) ( 99, 101, 98, 100 ) (100, 100, 98, 100 )
(99, 99.5, 99.5 ) (99, 100, 100 ) (99, 99, 100 ) (99, 99.5, 99.5 )
2 + 0.5 = 2.5 4+1=5 6+2=8 2 + 0.5 = 2.5
5 Baselines 5.1 Crossing Solutions No Constraint. This baseline returns the raw forecast without correction: b ZNC = Z. (8)
(6)
Cumulative Maximum. This baseline first corrects each OHLC row while keeping its open and close forecasts unchanged: z𝑞CM = 𝑧𝑞𝑂 , max{𝑧𝑞𝐻 , 𝑧𝑞𝑂 , 𝑧𝑞𝐶 }, min{𝑧𝑞𝐿 , 𝑧𝑞𝑂 , 𝑧𝑞𝐶 }, 𝑧𝑞𝐶 . (9)
where CQ contains all sequences satisfying 𝑣 1 ≤ · · · ≤ 𝑣𝑄 . KQSP checks the quantiles from low to high. Whenever consecutive values decrease, it replaces the violating group with its mean and continues until the complete sequence is nondecreasing. For example, consider quantile levels (𝜏1, 𝜏2, 𝜏3 ) = (0.1, 0.5, 0.9) and the crossed sequence c★:𝑓 = (99, 100, 99). The forecasts at 𝜏2 and 𝜏3 are incorrectly ordered because 99 < 100. One possible correction is to raise the 𝜏3 forecast from 99 to 100, producing (99, 100, 100) with squared correction (99 − 100) 2 = 1. KQSP instead makes a smaller joint correction: it lowers the 𝜏2 forecast from 100 to 99.5 and raises the 𝜏3 forecast from 99 to 99.5. It therefore returns KQSP b z:𝑓 = (99, 99.5, 99.5) with squared correction (100 − 99.5) 2 + (99 − 99.5) 2 = 0.5. Thus, KQSP restores the quantile order while changing the original forecasts less than the one-sided correction. The exemplary correction errors are used in Section 5 to explain the intuition behind KQSP and compare it with alternative crossing solutions. Preservation of K-line consistency. Stage 2 does not reintroduce the K-line crossings removed in Stage 1. After Stage 1, the ★ ≤ 𝑐 ★ , 𝑐 ★ ≤ 𝑐 ★ , 𝑐 ★ ≤ 𝑐 ★ , and 𝑐 ★ ≤ 𝑐 ★ at forecasts satisfy 𝑐𝑞𝐿 𝑞𝑂 𝑞𝐿 𝑞𝐶 𝑞𝑂 𝑞𝐻 𝑞𝐶 𝑞𝐻 every quantile level 𝑞. Stage 2 applies the same minimum-distance projection to each feature column. This projection preserves the order between columns: if one column is no greater than another at every quantile level before projection, it remains no greater afterward. Therefore, n o n o KQSP KQSP KQSP KQSP KQSP KQSP b 𝑧𝑞𝐿 ≤ min b 𝑧𝑞𝑂 , b 𝑧𝑞𝐶 , b 𝑧𝑞𝐻 ≥ max b 𝑧𝑞𝑂 , b 𝑧𝑞𝐶 , ∀𝑞. (7) Hence, Stage 2 enforces quantile order while preserving the K-line consistency obtained in Stage 1. Method Characteristics. KQSP introduces no additional parameter, hyperparameter, penalty term, or loss modification and requires no retraining. It can be applied directly to probabilistic K-line forecasts produced by any forecasting model, regardless of its architecture or training procedure. Because KQSP operates only on the forecast matrix, the original model and its predictive process remain unchanged.
It then corrects each feature column by replacing every forecast with the largest value observed at that or any lower quantile level: CM b 𝑧𝑞𝑓 = max 𝑧 CM 𝑗𝑓 , 1≤ 𝑗 ≤𝑞
(10)
where 𝑗 indexes the quantile levels up to 𝑞. For the OHLC example (100, 99, 98, 101) and the quantile example (99, 100, 99) introduced in Section 4.2, Table II summarizes the corrected forecasts and corresponding correction errors. The cumulative maximum method increases the correction error by 100% compared with KQSP. Reordering. This baseline corrects an OHLC row by sorting its four values. Let 𝑎𝑞 (1) ≤ 𝑎𝑞 (2) ≤ 𝑎𝑞 (3) ≤ 𝑎𝑞 (4) denote the sorted values of z𝑞 . The minimum is assigned to low, the maximum to high, and the two middle values to open and close while retaining their original direction: ( 𝑎𝑞 (2) , 𝑎𝑞 (4) , 𝑎𝑞 (1) , 𝑎𝑞 (3) , 𝑧𝑞𝑂 ≤ 𝑧𝑞𝐶 , (11) z𝑞RE = 𝑎𝑞 (3) , 𝑎𝑞 (4) , 𝑎𝑞 (1) , 𝑎𝑞 (2) , 𝑧𝑞𝑂 > 𝑧𝑞𝐶 . It then sorts each feature column in ascending quantile order: RE RE (12) b zRE :𝑓 = sort↑ 𝑧 1𝑓 , . . . , 𝑧𝑄 𝑓 . Although Reordering removes both crossing types, it permutes the predicted values instead of minimizing their displacement. Consequently, a value originally produced for one OHLC feature or quantile level may be reassigned to another, breaking the association between the input and output. As shown in Table II, Reordering yields the highest total correction error for these examples. Hierarchical. The hierarchical head was proposed in [21]. Let U = [𝑢𝑞𝑓 ] ∈ R𝑄 ×4 denote the native output of the hierarchical model, where 𝑞 indexes the quantile level and 𝑓 ∈ F indexes the OHLC feature. The non-negative residual is 𝑟𝑞𝑓 = |𝑢𝑞𝑓 |. 4
(13)
Crossing-Free Probabilistic K-Line Forecasts Without Retraining
Preprint, Under Review, -
Let 𝑚 denote the median-quantile index satisfying 𝜏𝑚 = 0.5. Hierarchical1 uses the median output as its starting point and recursively adds or subtracts the residuals: 𝑢𝑚𝑓 , H1 H1 𝑧𝑞−1,𝑓 + 𝑟𝑞𝑓 , b 𝑧𝑞𝑓 = b H1 b 𝑧𝑞+1,𝑓 − 𝑟𝑞𝑓 ,
𝑞 = 𝑚, 𝑞 > 𝑚,
foundation models are compared later. For each stock, we sample 100 ANN configurations. The number of hidden layers is sampled from {1, 2, 3, 4, 5}; units per layer from the integers [2, 512]; learning rate log-uniformly from [10−5, 10−2 ]; dropout uniformly from [0, 0.9]. Each model uses the nine quantiles T = {0.1, 0.2, . . . , 0.9}, batch size of 256, and 500 epochs. The validation-best configuration is selected separately for each stock, and its untouched test split is used for Table III. TimesFM. Developed by Google Research, TimesFM is pretrained on large-scale real-world and synthetic time series from diverse domains [6]. We evaluate the original TimesFM and the newer 200-million-parameter TimesFM 2.5. Chronos. Developed by Amazon, the Chronos family provides generic probabilistic forecasting based on public and synthetic pretraining data [1, 2]. We evaluate both Chronos and Chronos-2, which support multivariate and covariate-informed forecasting. Moirai. Moirai is a universal time-series foundation model pretrained on the Large-scale Open Time Series Archive (LOTSA) [17]. LOTSA contains more than 27 billion observations collected from nine application domains. Time-MoE. Time-MoE is a sparse mixture-of-experts foundation model designed to investigate scaling and computational efficiency in time-series forecasting [14]. It is pretrained on Time-300B, which contains more than 300 billion time points from nine domains. TabPFN. TabPFN-TS adapts a tabular foundation model to zeroshot time-series forecasting [8]. Because the underlying model is pretrained entirely on synthetic data, its pretraining corpus does not contain observations from real-world forecasting benchmarks. Kronos. Kronos is a finance-specific foundation model pretrained on more than 12 billion OHLCV candlestick records from 45 global exchanges and seven temporal granularities [15]. The Kronos pretraining corpus ends in June 2024, whereas the shared test period used by Kronos and the fully trained ANN spans January 17, 2025 to July 23, 2026. Therefore, no test observations overlap the Kronos pretraining period. Input and Output Specification. The univariate models TimesFM, Chronos, and Time-MoE receive the historical sequence of each OHLC target separately. TimesFM 2.5 and TabPFN receive each target history together with the remaining OHLCV channels as lag covariates, whereas Chronos-2, Moirai, and Kronos receive multivariate historical sequences. TimesFM, TimesFM 2.5, Chronos-2, and TabPFN directly provide probabilistic outputs. Chronos, Moirai, and Kronos construct quantiles from sampled future paths, while Time-MoE converts its point forecast into quantiles using past one-step residuals.
(14)
𝑞 < 𝑚.
This construction enforces quantile order but does not enforce the OHLC relations. Inspired by Hierarchical1 , Hierarchical2 interprets 𝑢𝑞𝑂 and 𝑢𝑞𝐶 as the open and close forecasts and uses the remaining outputs as non-negative high and low residuals: b z𝑞H2 = 𝑢𝑞𝑂 , max{𝑢𝑞𝑂 , 𝑢𝑞𝐶 } + 𝑟𝑞𝐻 , min{𝑢𝑞𝑂 , 𝑢𝑞𝐶 } − 𝑟𝑞𝐿 , 𝑢𝑞𝐶 . (15) This construction enforces the OHLC relations but does not enforce quantile order. Hierarchical3 combines the recursive quantile construction of Hierarchical1 with the OHLC construction of Hierarchical2 to enforce both constraint families. The hierarchical decoders introduce no additional trainable variables or loss penalties, but they change the meanings of the native outputs from direct forecasts to residuals. They therefore require loading the corresponding model weights and retraining the model. Their correction errors cannot be determined from the two forecast examples alone because their outputs depend on the residuals learned during retraining. Errors may also accumulate along the recursive chain: an upper quantile depends on the preceding lower quantile, so an error in that lower quantile is propagated to all subsequent upper quantiles. The same propagation can occur from the median toward the lower quantiles. Joint Projection. To evaluate the stage-wise design of KQSP, we introduce Joint Projection as a comparison variant rather than an existing baseline. Like KQSP, JP corrects both crossing types, but it enforces both constraints simultaneously: b ZJP = arg min V∈ CJ
𝑄 ∑︁ ∑︁
(𝑣𝑞𝑓 − 𝑧𝑞𝑓 ) 2,
(16)
𝑞=1 𝑓 ∈ F
where 𝐶 J denotes the set satisfying both the quantile and K-line constraints. Because this is a strictly convex quadratic program with linear constraints, it has a unique global solution. Since KQSP is also feasible, 𝐸 corr b ZJP, Z ≤ 𝐸 corr b ZKQSP, Z . (17) In Table II, KQSP happens to attain the same correction error as Joint Projection. Joint Projection guarantees the minimum correction error but solves one coupled problem over all forecasts, even when only one crossing type is present. KQSP instead uses sequential projections, and an already-satisfied stage can be skipped. We include Joint Projection to test whether its theoretical reduction in correction error yields an empirical improvement over KQSP.
5.2
6
Evaluation Metrics
Let 𝑖 ∈ {1, . . . , 𝑁 } index the 𝑁 test samples, let 𝑦𝑖 𝑓 be the realized value of feature 𝑓 , and let 𝑧𝑖𝑞𝑓 be its raw quantile forecast. Quantile Crossing Rate (QCR) is the percentage of samples with at least one quantile crossing,
Forecasting Models
ANN. The artificial neural network (ANN) is trained from scratch for each stock and serves as the reference model. It represents a conventional task-specific approach against which the zero-shot
QCR = 5
𝑁 100 ∑︁ 1 ∃(𝑞, 𝑓 ) : 𝑧𝑖𝑞𝑓 > 𝑧𝑖,𝑞+1,𝑓 . 𝑁 𝑖=1
(18)
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Table III: Before–after test metrics for the validation-best ANN of each stock.
Stock
QCR (%) Before After 𝑃
BMW Mercedes Volkswagen
82.41 86.61 100.00
0.00 0.00 0.00
*** *** ***
10.76 35.70 94.75
0.00 0.00 0.00
*** *** ***
0.39 0.24 0.50
BP PetroChina Shell
73.95 44.66 84.21
0.00 0.00 0.00
*** *** ***
16.32 0.55 27.89
0.00 0.00 0.00
***
Goldman JPMorgan LSEG
100.00 100.00 100.00
0.00 0.00 0.00
*** *** ***
92.86 97.35 41.84
Google Meta Microsoft
100.00 100.00 80.42
0.00 0.00 0.00
*** *** ***
78.84 100.00 20.90
Improved or n.s.
KCR (%) Before After 𝑃
𝑃
QCE (%) Before After 𝑃
Before
MAE After
0.39 0.24 0.48
*** *** ***
3.49 3.43 3.98 3.32 10.87 11.15
0.96 0.60 1.12
***
2.22 2.21 0.04 0.04 10.93 10.88
*** *** ***
2.83 6.70 3.65
2.79 6.70 3.72
5.43 0.10 26.33
0.00 0.00 0.00
*** *** ***
3.83 3.79 1.21 1.18 52.34 51.55
*** *** ***
5.83 5.98 6.37
5.59 5.30 6.62
9.48 9.47 3.12 2.90 130.82 125.06
0.00 0.00 0.00
*** *** ***
1.45 4.88 1.94
*** *** ***
9.05 8.92 17.27 16.40 2.68 2.75
12/12
AQL Before After
1.42 4.64 1.93
12/12
12/12
**
3.71 11.95 4.73 12/12
𝑃
Before
RMSE After
0.95 0.59 1.13
*** ***
1.39 0.86 1.58
1.38 0.85 1.59
5.42 0.11 26.28
*
8.08 0.18 39.53
8.07 0.18 39.42
3.47 11.44 4.71
*** *** *** ***
13.62 13.62 4.17 4.01 217.15 211.68 5.28 16.54 7.09
4.97 16.06 7.08
12/12
𝑃 ** ***
*** *** *** *** 12/12
Table IV: Comparison of crossing baselines using the ANNs. Hierarchical1 enforces quantile order only, Hierarchical2 enforces OHLC validity only, and Hierarchical3 enforces both constraints.
Method
QCR (%)
KCR (%)
AQL
𝑃
Max AQL
𝑃
Mean corr.
𝑃
Max corr.
𝑃
87.69 0.00 0.00 0.00 94.85 0.00 0.00 0.00
51.48 0.00 0.00 21.95 0.00 0.00 0.00 0.00
6.66 6.57 6.56 6.96 7.26 8.53 6.56 6.56
***
67.14 66.36 66.97 57.96 64.80 64.59 66.06 66.06
**
– 1.07 1.60 9.26 11.55 14.90 0.91 0.91
– *** *** *** *** ***
– 43.29 33.59 81.78 99.98 125.41 22.94 22.94
– *** *** *** *** ***
No Constraint Cumulative Max. Reordering Hierarchical1 Hierarchical2 Hierarchical3 Joint Projection Proposed KQSP
* ** ***
K-Line Crossing Rate (KCR) is the percentage of samples with at least one K-line crossing,
We use standard RMSE and MAE for pointwise evaluation.
𝑁
KCR =
100 ∑︁ 1{∃𝑞 : 𝑧𝑖𝑞𝐻 < max(𝑧𝑖𝑞𝑂 , 𝑧𝑖𝑞𝐶 ) or 𝑁 𝑖=1
7 Case Study 7.1 Effectiveness in the Full-Training Setting
(19)
𝑧𝑖𝑞𝐿 > min(𝑧𝑖𝑞𝑂 , 𝑧𝑖𝑞𝐶 )}.
We first use the fully trained ANN to determine whether KQSP resolves both crossing issues without compromising predictive accuracy. The model chooses the optimal historical input length from {1, 3, 7, 30} trading days based on validation data. Within each stock, Table III compares paired test-day contributions before and after KQSP using a two-sided sign-flip test. Green stars denote significant improvement, red stars denote significant deterioration, and a blank denotes no significant difference. One, two, and three stars indicate 𝑝 < 0.05, 𝑝 < 0.01, and 𝑝 < 0.001, respectively. The last row counts improvement or no significant difference across stocks. First, the raw ANN forecasts exhibit severe crossings: the quantile crossing rate (QCR) ranges from 44.66% to 100.00%, while the K-line crossing rate (KCR) ranges from 0.55% to 100.00%. Second, KQSP reduces both crossing rates to zero for every stock. Third,
Here 1{·} is the indicator function. The pinball loss at quantile level 𝜏 for error 𝑒 is 𝜌𝜏 (𝑒) = 𝑒 (𝜏 − 1{𝑒 < 0}) . (20) Average Quantile Loss (AQL) averages quantile loss over samples, quantiles, and OHLC features, 𝑁
AQL =
𝑄
1 ∑︁ ∑︁ ∑︁ 𝜌𝜏𝑞 (𝑦𝑖 𝑓 − 𝑧𝑖𝑞𝑓 ). 4𝑄𝑁 𝑖=1 𝑞=1
(21)
𝑓 ∈F
Max AQL is the largest sample-level AQL. Quantile coverage error (QCE) averages the absolute empiricalcoverage deviation, QCE =
𝑄 𝑁 100 ∑︁ 1 ∑︁ ∑︁ 1{𝑦𝑖 𝑓 ≤ 𝑧𝑖𝑞𝑓 } − 𝜏𝑞 . 𝑄 𝑞=1 4𝑁 𝑖=1
(22)
𝑓 ∈F
6
Crossing-Free Probabilistic K-Line Forecasts Without Retraining
Preprint, Under Review, -
although KQSP is designed to enforce consistency rather than improve predictive accuracy, it significantly improves AQL for all 12 stocks (𝑝 < 0.001) and improves several MAE and RMSE results. A plausible explanation is that the violations partly reflect estimation noise: by projecting the forecasts onto economically valid OHLC and quantile constraints, KQSP removes infeasible variation while making the smallest possible correction. This shape-constrained correction can therefore move inconsistent forecasts closer to the realized values without changing already valid components unnecessarily.
7.2
difference between the fully trained ANN and Kronos, indicating strong zero-shot probabilistic performance from the finance-specific Kronos model. By contrast, the generic foundation models do not match the simple ANN on this dataset. One possible explanation is that these models are pretrained primarily on broad and generic time-series data and may not sufficiently capture stock-specific price dynamics. However, this experiment is not intended to establish a universal ranking of forecasting models. Its purpose is to determine whether KQSP eliminates both crossing issues without degrading predictive accuracy, regardless of whether the forecasts are produced by a strong or weak model.
Comparison of Existing Crossing Solutions 8
No Constraint, Cumulative Maximum, Reordering, Joint Projection, and KQSP are applied directly to the same saved forecasts without further training. The three hierarchical variants load the same base ANN weights and are retrained using the same hyperparameters. Table IV compares each baseline with KQSP using a two-sided paired Wilcoxon test across the stocks. Mean and maximum correction measure absolute changes in native price units and are particularly important when the original forecasts are trusted and only minimal corrections are desired. First, Cumulative Maximum, Reordering, Hierarchical3 , Joint Projection, and KQSP eliminate both crossing types. Especially, KQSP achieves a significantly lower AQL than No Constraint (𝑝 < 0.001), showing that consistency can be imposed while even improving the predictive accuracy with statistical significance (𝑝 < 0.001). Second, Cumulative Maximum and Reordering have no significant AQL difference from KQSP. Nevertheless, both produce significantly larger mean and maximum correction errors (𝑝 < 0.001). This distinction is critical when the forecasting model is trusted, and the objective is to resolve crossings with the smallest possible modification. Third, the hierarchical AQL values follow Hierarchical3 > Hierarchical2 > Hierarchical1 , with Hierarchical3 performing worst. The deterioration caused by combining both recursive constructions is consistent with error propagation across the quantile and OHLC hierarchies. Lastly, Joint Projection does not differ significantly from the proposed KQSP across any metric (all 𝑝 > 0.05), as indicated by the absence of significance stars in the table. This shows that KQSP achieves empirically equivalent correction performance while offering greater stage-wise flexibility.
7.3
Conclusion
This study introduced KQSP, a parameter-free method for reconciling probabilistic K-line forecasts without retraining the forecasting model. Each stage of KQSP minimizes the squared correction, allowing KQSP to remove both crossing types while preserving the original forecasts as closely as possible. Three case studies establish the effectiveness of KQSP. First, for fully trained ANNs across diverse stocks, KQSP reduces both QCR and KCR to zero and significantly improves AQL for every stock. Second, KQSP achieves predictive accuracy comparable to other strong baselines while producing significantly smaller mean and maximum corrections. Third, experiments involving zero-shot foundation models confirm that KQSP eliminates crossings and can even improve partial forecasting metrics across different models. A limitation of KQSP is that it applies the minimum correction required for consistency, regardless of whether the original forecasts are strong or weak. Future work should investigate methods that not only eliminate crossings but also improve predictive accuracy when the original forecasts are inaccurate.
References [1] Abdul Fatir Ansari, Oleksandr Shchur, Jaris Küken, Andreas Auer, Boran Han, Pedro Mercado, Syama Sundar Rangapuram, Huibin Shen, Lorenzo Stella, Xiyuan Zhang, Mononito Goswami, Shubham Kapoor, Danielle C. Maddix, Pablo Guerron, Tony Hu, Junming Yin, Nick Erickson, Prateek Mutalik Desai, Hao Wang, Huzefa Rangwala, George Karypis, Yuyang Wang, and Michael Bohlke-Schneider. 2025. Chronos-2: From Univariate to Universal Forecasting. arXiv:2510.15821 [cs.LG] https://arxiv.org/abs/2510.15821 [2] Abdul Fatir Ansari, Lorenzo Stella, Caner Turkmen, Xiyuan Zhang, Pedro Mercado, Huibin Shen, Oleksandr Shchur, Syama Sundar Rangapuram, Sebastian Pineda Arango, Shubham Kapoor, Jasper Zschiegner, Danielle C. Maddix, Hao Wang, Michael W. Mahoney, Kari Torkkola, Andrew Gordon Wilson, Michael Bohlke-Schneider, and Yuyang Wang. 2024. Chronos: Learning the Language of Time Series. arXiv:2403.07815 [cs.LG] https://arxiv.org/abs/2403.07815 [3] Jozef Baruník, Martin Hronec, and Ondřej Tobek. 2026. Forecasting stock return distributions around the globe with quantile neural networks. International Journal of Forecasting (2026). doi:10.1016/j.ijforecast.2026.04.004 [4] Derek Bunn, Arne Andresen, Dipeng Chen, and Sjur Westgaard. 2016. Analysis and forecasting of electricity price risks with quantile factor models. The Energy Journal 37, 1 (2016), 101–122. [5] Victor Chernozhukov, Iván Fernández-Val, and Alfred Galichon. 2010. Quantile and probability curves without crossing. Econometrica 78, 3 (2010), 1093–1125. [6] Abhimanyu Das, Weihao Kong, Rajat Sen, and Yichen Zhou. 2024. A decoder-only foundation model for time-series forecasting. In Proceedings of the 41st International Conference on Machine Learning (Vienna, Austria) (ICML’24). JMLR.org, Article 404, 20 pages. [7] Tilmann Gneiting and Matthias Katzfuss. 2014. Probabilistic Forecasting. Annual Review of Statistics and Its Application 1 (2014), 125–151. doi:10.1146/annurevstatistics-062713-085831 [8] Shi Bin Hoo, Samuel Müller, David Salinas, and Frank Hutter. 2026. From Tables to Time: Extending TabPFN-v2 to Time Series Forecasting. arXiv:2501.02945 [cs.LG] https://arxiv.org/abs/2501.02945
Sensitivity to Different Models
To mitigate model-specific bias and investigate whether KQSP can consistently improve AQL across different forecasting models, we compare the fully trained ANN with zero-shot foundation models using identical test observations and quantile levels. Each model’s raw forecast is evaluated before and after the KQSP operator. Table V reports paired before–after results aggregated across stocks. Before is the raw forecast and After is the same forecast after KQSP. Each 𝑃 cell is a two-sided paired Wilcoxon test of all stocks Before and After metrics. Table VI compares post-KQSP AQL between models. First, KQSP removes all quantile and K-line crossings for every model. Surprisingly, the results confirm that KQSP can consistently improve predictive metrics, including AQL, MAE, and RMSE, with statistical significance. Second, Table VI shows no significant AQL 7
Preprint, Under Review, -
Yu et al.
Table V: Before–after results aggregated across stocks.
Model
QCR (%) Before After 𝑃
ANN TimesFM TimesFM 2.5 Chronos Chronos-2 Moirai Time-MoE TabPFN Kronos
87.69 1.07 3.14 0.00 0.04 0.00 0.00 0.00 0.00
0.00 *** 0.00 ** 0.00 *** 0.00 0.00 0.00 0.00 0.00 0.00
KCR (%) Before After 𝑃 51.48 52.66 62.46 83.09 46.27 69.00 76.12 63.18 2.52
AQL Before After 𝑃
0.00 *** 0.00 *** 0.00 *** 0.00 *** 0.00 *** 0.00 *** 0.00 *** 0.00 *** 0.00 ***
6.66 8.37 8.01 7.79 7.71 8.15 8.02 6.90 6.55
6.56 *** 8.33 *** 7.83 *** 7.68 *** 7.68 *** 8.12 *** 7.92 *** 6.86 *** 6.55 ***
QCE (%) MAE Before After 𝑃 Before After 𝑃
RMSE Before After 𝑃
16.53 15.96 ** 20.84 20.73 *** 19.88 19.36 *** 19.16 18.95 *** 19.17 19.09 *** 20.05 20.01 *** 19.76 19.54 *** 16.87 16.78 *** 15.73 15.73 *
26.29 25.74 ** 31.19 31.05 *** 29.89 29.08 *** 29.51 29.24 *** 28.98 28.90 *** 30.10 30.05 *** 29.70 29.44 *** 26.56 26.41 *** 24.92 24.91 *
6.56 4.16 2.33 2.28 1.90 3.49 3.86 1.37 7.64
6.39 4.12 2.27 * 2.18 * 1.93 3.46 3.70 ** 1.36 7.63
Table VI: Upper-triangular pairwise comparison of post-KQSP AQL across stocks. Diagonal entries are mean AQL. Each upper cell is a two-sided paired Wilcoxon test between its row and column models: green stars favor the row model, red stars favor the column model, and a blank denotes no significant difference.
Model
ANN
TimesFM
TimesFM 2.5
Chronos
Chronos-2
Moirai
Time-MoE
TabPFN
ANN TimesFM TimesFM 2.5 Chronos Chronos-2 Moirai Time-MoE TabPFN Kronos
6.56
*** 8.33
*** *** 7.83
** *** ** 7.68
*** *** ***
*** *** * *** *** 8.12
*** *** *** *** ***
** *** *** *** *** *** *** 6.86
7.68
7.92
[9] Wenyang Huang, Tianxiao Gao, Yun Hao, and Xiuqing Wang. 2023. TransformerBased Forecasting for Intraday Trading in the Shanghai Crude Oil Market: Analyzing Open-High-Low-Close Prices. Energy Economics 127 (2023), 107106. doi:10.1016/j.eneco.2023.107106 [10] Wenyang Huang, Huiwen Wang, and Shanshan Wang. 2024. A Structural VAR and VECM Modeling Method for Open-High-Low-Close Data Contained in Candlestick Chart. Financial Innovation 10, 1 (2024), 97. doi:10.1186/s40854-02400622-6 [11] Junkyu Jang, Taehwan Kim, and Sung-Hyuk Park. 2024. Stock Index Forecasting Using an Explainable TAFT Model with Online Data-Driven Social Sentiment Index. In Proceedings of the 5th ACM International Conference on AI in Finance (Brooklyn, NY, USA) (ICAIF ’24). Association for Computing Machinery, New York, NY, USA, 787–794. doi:10.1145/3677052.3698618 [12] Jinwoong Kim and Sangjin Park. 2025. IKNet: Interpretable Stock Price Prediction via Keyword-Guided Integration of News and Technical Indicators. In Proceedings of the 6th ACM International Conference on AI in Finance (ICAIF ’25). Association for Computing Machinery, New York, NY, USA, 709–717. doi:10.1145/3768292. 3770343 [13] Guohao Shen, Yuling Jiao, Yuanyuan Lin, Joel L. Horowitz, and Jian Huang. 2024. Nonparametric Estimation of Non-Crossing Quantile Regression Process with Deep ReQU Neural Networks. Journal of Machine Learning Research 25, 88 (2024), 1–75. https://jmlr.org/papers/v25/22-0488.html [14] Xiaoming Shi, Shiyu Wang, Yuqi Nie, Dianqi Li, Zhou Ye, Qingsong Wen, and Ming Jin. 2025. Time-MoE: Billion-Scale Time Series Foundation Models with Mixture of Experts. arXiv:2409.16040 [cs.LG] https://arxiv.org/abs/2409.16040 [15] Yu Shi, Zongliang Fu, Shuo Chen, Bohan Zhao, Wei Xu, Changshui Zhang, and Jian Li. 2026. Kronos: A foundation model for the language of financial markets. In Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 40. 25366–25373. [16] James W Taylor and Derek W Bunn. 1999. A quantile regression approach to generating prediction intervals. Management Science 45, 2 (1999), 225–237. [17] Gerald Woo, Chenghao Liu, Akshat Kumar, Caiming Xiong, Silvio Savarese, and Doyen Sahoo. 2024. Unified training of universal time series forecasting transformers. In Proceedings of the 41st International Conference on Machine
Kronos *** *** *** *** *** *** *** 6.55
Learning (Vienna, Austria) (ICML’24). JMLR.org, Article 2178, 25 pages. [18] Xuanling Yang, Zhoufan Zhu, Dong Li, and Ke Zhu. 2024. Asset pricing via the conditional quantile variational autoencoder. Journal of Business & Economic Statistics 42, 2 (2024), 681–694. [19] Runyao Yu, Chenhui Gu, Jochen Stiasny, Qingsong Wen, Wasim Sarwar Dilov, Lianlian Qi, and Jochen L. Cremer. 2026. PriceFM: Foundation Model for Probabilistic Electricity Price Forecasting. arXiv:2508.04875 [cs.CE] [20] Runyao Yu, Julia Lin, Derek W. Bunn, Jochen Stiasny, Wentao Wang, Yujie Chen, Tara Esterl, Peter Palensky, and Jochen L. Cremer. 2026. A market-rule-informed neural network for efficient imbalance electricity price forecasting. Advanced Engineering Informatics 76 (2026), 105083. doi:10.1016/j.aei.2026.105083 [21] Runyao Yu, Yuchen Tao, Fabian Leimgruber, Tara Esterl, Jochen Stiasny, Derek W. Bunn, Qingsong Wen, Hongye Guo, and Jochen L. Cremer. 2026. OrderFusion: Encoding Orderbook for End-to-End Probabilistic Intraday Electricity Price Forecasting. arXiv:2502.06830 [q-fin.CP]
8