GeoConformal prediction: a model-agnostic framework of measuring the uncertainty of spatial prediction

Liqiu Meng; Peng Luo; Xiayin Lou

REVIEW 4 major objections 4 minor 7 references

GeoConformal prediction: a model-agnostic framework of measuring the uncertainty of spatial prediction

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that replacing the equal weights of conformal prediction with distance-decayed geographic weights yields prediction intervals with local, location-specific coverage guarantees that hold for any spatial prediction model.

desk verdict A useful empirical package for spatially varying conformal intervals, but the advertised coverage guarantee is assumed rather than proven, so the paper needs major revision before it deserves publication. read the letter →

arxiv 2412.08661 v3 pith:UY4KUVVS submitted 2024-12-05 stat.ML cs.LGphysics.data-anstat.AP

Xiayin Lou , Peng Luo , Liqiu Meng This is my paper · ORCID

classification stat.MLcs.LGphysics.data-anstat.AP MSC 62M3062G1562G05

keywords GeoConformalPredictionspatialuncertaintycovariateshiftKrigingAIinterval

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

GeoConformal Prediction (GeoCP) claims to fix a blind spot in spatial prediction: most models report accuracy but not trustworthy uncertainty, and standard conformal prediction gives one global interval that ignores location. The paper's proposal is to weight each calibration point's nonconformity score by its geographic proximity to the test point, producing a prediction interval that varies across space while retaining a coverage guarantee. In a housing-price regression, GeoCP reached 93.67% coverage at the 90% level, whereas bootstrapping topped out at 81.00% even after 2000 resamples, with far less computation. The authors also show the uncertainty aligns with Kriging variance in interpolation and that adding explicit local features to a GeoAI model shrinks uncertainty most where local spatial dependence is strong.

What carries the argument

The carrying object is the geographic weight function w(X_i) = K(|l_i - l_test| / b), a kernel that decays with distance, applied to nonconformity scores in the weighted conformal prediction quantile. The paper uses a Gaussian decay kernel throughout. The geographically weighted quantile (GeoQuantile) computes the (1 - epsilon) quantile of the weighted empirical distribution of calibration scores, and this threshold becomes the local prediction interval radius; Appendix C states a localized weighted exchangeability assumption under which this yields the local coverage theorem.

What would settle it

Run GeoCP on a synthetic field with a sharp spatial discontinuity, such as two distinct regimes separated by a boundary, using the paper's Gaussian weight with a moderate bandwidth; if coverage at test points near the boundary falls substantially below 1 - epsilon, then the localized weighted exchangeability assumption, rather than the underlying predictor, is the failing component.

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Extended reading notes

Core claim

The central claim is equation (4): conditional on the test point's location (u_test, v_test), the probability that the true value lies in the GeoCP interval is at least 1 - epsilon. This is achieved by replacing the empirical quantile of calibration nonconformity scores with a geographically weighted quantile, GeoQuantile, where each calibration score is weighted by a kernel of distance to the test point. The authors argue this converts conformal prediction from a marginal, dataset-level guarantee into a local, place-specific one, without modifying the underlying predictor.

Load-bearing premise

The coverage guarantee rests on the premise that a location-based kernel weight captures the actual distribution shift from calibration to test; the paper assumes this localized weighted exchangeability and does not estimate or report the bandwidth that would make it true.

Editorial extensions

If this is right

Any regression or classification spatial model can be wrapped with GeoCP without retraining or changing its internal structure.
Uncertainty becomes mappable: prediction intervals shrink or grow across space instead of being a single fixed width.
GeoCP enables direct comparison of uncertainty across different spatial models fitted to the same data.
Adding explicit spatial features to a GeoAI model reduces prediction uncertainty, with the largest reductions where local spatial dependence is strongest.
GeoCP uncertainty, as the range of the error distribution at each location, can serve as a basis for measuring geographic bias.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The coverage guarantee depends on the kernel bandwidth b matching the true covariate shift; because the paper does not estimate or report b, a misspecified bandwidth is the main practical risk to equation (4).
Distance-only weighting may mislead when the distribution shift is driven by features correlated with location but not smoothly distance-decayed, such as a sharp urban-rural boundary.
A natural extension is to calibrate the bandwidth adaptively per test point, for example by cross-validating coverage on held-out spatial data.
The simulation result that recovery degrades as the spatial autocorrelation of true uncertainty approaches zero suggests a floor beyond which no location-based method can reconstruct locally varying uncertainty.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things about this one. First, the empirical core is genuinely useful: GeoCP is weighted split conformal prediction with a geographic kernel, and the authors validate it with a simulation study, a Seattle housing-price regression, and a temperature interpolation task. They also ship code and data. That is real work, and the qualitative finding that local features reduce prediction uncertainty is plausible and well illustrated. Second, the central theoretical claim does not hold as stated. Equation (4) promises conditional coverage given the test location, but the proof in Appendix C rests on Assumption 1, which simply assumes localized weighted exchangeability. That is the exact condition needed to get the result; it is not derived from Tobler's law or from any spatial generative process. Worse, as printed, Assumption 1 is not a normalized probability law: for A equal to the full sample space, the right-hand side is E[prod w_i / sum w_i], which is generically less than 1. Theorem 2 is also truncated mid-sentence, ending at “≥ 1 −”. So the finite-sample coverage guarantee advertised in the abstract is unsupported. This is a load-bearing flaw, not a cosmetic one. The empirical coverage rates do not repair it, since they come from a single data split and cannot validate a conditional guarantee. The bandwidth b is never reported, nor is there sensitivity analysis, so the kernel weighting is effectively a free parameter. The bootstrap comparison is also weakened by the usual confidence-interval versus prediction-interval mismatch, though the authors at least acknowledge the distinction in an appendix. What the paper does well: it frames spatial heterogeneity as covariate shift, connects to Tibshirani et al.’s weighted conformal prediction, and demonstrates a computationally cheap way to get spatially varying intervals. The simulation framework with known uncertainty is a nice contribution on its own. The citation pattern is fine; they engage the relevant conformal and spatial-statistics literature. My take: this is a promising applied paper with a broken theoretical core. It deserves a serious referee, not a desk reject, because the method is a special case of existing weighted conformal prediction and the fix is clear: either correct the assumption, prove the guarantee under explicit and verifiable conditions, or honestly reposition GeoCP as a heuristic and support it with more thorough empirical validation, including bandwidth sensitivity. As written, I would not accept it. But I would send it to review and ask the authors to address the theory head-on.

Referee Report

4 major / 4 minor

Summary. The paper proposes GeoConformal Prediction (GeoCP), a split-conformal method that replaces the equally weighted empirical distribution of calibration nonconformity scores with a geographically weighted distribution, using a distance-decay kernel (e.g., Gaussian) between the test location and calibration locations. The authors claim a conditional coverage guarantee in Eq. (4), supported by an appendix theorem under a 'localized weighted exchangeability' assumption. The empirical sections apply GeoCP to a Seattle housing-price regression with XGBoost and to Los Angeles temperature interpolation with Ordinary Kriging and a deep spatial interpolation model, reporting coverage rates, correlations with bootstrap errors, and analyses of how spatial features affect uncertainty. The paper also releases code and data on GitHub.

Significance. If the coverage guarantee were valid, GeoCP would be a useful, model-agnostic tool for spatially varying uncertainty quantification, filling a genuine gap in GeoAI. The simulation framework that constructs biased models with known uncertainty is a thoughtful way to benchmark uncertainty estimates, and the empirical explorations of spatial feature effects on uncertainty are interesting. The paper's strengths include the clear demonstration that original conformal prediction gives constant intervals, the reported negative results for kriging variance as a spatial uncertainty measure, and the public release of code and data. However, the central advertised property, the finite-sample coverage guarantee of Eq. (4), is not established by the manuscript: the stated assumption is not a valid probability law, the theorem is truncated, and the empirical coverage claims rest on an unfair comparison with a confidence interval rather than a prediction interval. These are load-bearing issues for the paper's main contribution.

major comments (4)

[Appendix C, Assumption 1] Assumption 1 as written is not a valid probability law. Taking A to be the full sample space, the right-hand side equals E[∏_{i=1}^{n+1} w(X_{π(i)}) / ∑_{i=1}^{n+1} w(X_{π(i)})], which is generically less than 1 and can be undefined whenever a weight is zero; it does not reduce to P((Z_1,...,Z_{n+1}) ∈ Ω) = 1. Thus the weighted exchangeability condition is not correctly formulated, and Theorem 2 cannot be verified from the stated assumption. Moreover, the sentence 'This holds under covariate shift or when sampling is locally randomized around the test point' is asserted without proof; the assumption is not derived from Tobler's law or from any explicit spatial data-generating process.
[Appendix C, Theorem 2; Eq. (4)] Theorem 2 is incomplete: the conclusion reads 'P(Y_{n+1} ∈ C(X_{n+1})) ≥ 1 −' with the miscoverage level missing, and no proof is provided. In addition, the theorem as stated would give at best a marginal coverage guarantee, whereas Eq. (4) claims the stronger conditional guarantee P[y_test ∈ C_geo(X_test) | (u_test, v_test)] ≥ 1 − ε. Conditional coverage at each test location is a much stronger property and is not a consequence of the (even corrected) weighted exchangeability result. The central coverage claim of the paper is therefore unsupported.
[Section on 'Evaluation of coverage ratio' (GeoCP in spatial regression)] The headline comparison between GeoCP and bootstrapping is not apples-to-apples. The main text describes computing bootstrap intervals by taking the 5th and 95th percentiles of 2000 fitted-model predictions at each location, which is a confidence interval for the mean prediction. Appendix A itself notes that prediction intervals are wider than confidence intervals. Although the text says a prediction interval was computed per Appendix B, the reported procedure omits the residual resampling step described there. As a result, the maximum bootstrap coverage of 81.00% is expected to be below the 90% target even if the bootstrap method were working correctly, and the claimed 93.67% versus 81.00% advantage is not a valid demonstration of GeoCP's superiority.
[Algorithm 1 and Algorithm 2 (GeoQuantile)] The geographic weights depend on a kernel K and a bandwidth b, but the bandwidth is never estimated, reported, or subjected to sensitivity analysis. The algorithms list a 'decay function β' as input without specifying b. Since the coverage guarantee, if it existed, would require the weights to match the true covariate likelihood ratio, and since the empirical intervals are sensitive to b, the absence of bandwidth reporting makes the experimental results non-reproducible and the claimed guarantee vacuous in practice.

minor comments (4)

[Section 'GeoCP in spatial interpolation', Figure 12] The text refers to 'boxplot in Figure 8b' when discussing GeoCP in the spatial interpolation section; this appears to be a reference to Figure 12b, and the figure numbering should be corrected.
[Eq. (2) and Eq. (6)] The notation Quantile_{1−ε}((1/m)∑δ_{α_i}) is nonstandard; it should be stated explicitly that this is the (1−ε)-quantile of the empirical distribution of calibration scores, and similarly for the weighted empirical distribution in Eq. (6).
[Introduction/abstract terminology] The abstract uses 'GeoConformal Prediction' while the rest of the paper uses 'GeoCP'; the terminology should be unified throughout.
[Section 'GeoCP in spatial interpolation'] When describing 'DGSI with local features', the model uses the average of k-nearest neighbors' target values as an input; the authors should clarify how this is handled to avoid train/test leakage in the interpolation evaluation.

Circularity Check

1 steps flagged · score 8.0 of 10

The Eq. (4) local coverage guarantee is forced by Assumption 1 in Appendix C, which is assumed rather than derived; the geographic kernel is never shown to equal the covariate likelihood ratio, so the central theoretical claim reduces to its own input by construction.

self definitional [Appendix C, Assumption 1 and Theorem 2]
"Assumption 1 (Localized weighted exchangeability). Let Zi=(Xi,yi) for i=1,…,n+1, and fix a measurable, nonnegative weight function ... w(Xi)=K(|li-ln+1|/b) ... We assume P((Z1,…,Zn+1)∈A)=E[∏ w(Xπ(i))/∑ w(Xπ(i)) 1{(Z1,…,Zn+1)∈A}] ... This holds under covariate shift or when sampling is locally randomized around the test point. ... This result is identical in form to the global weighted case, despite weights here being local."

Theorem 2 is the standard weighted-exchangeability coverage result restated with local weights. The only spatial ingredient, the kernel K(|li-ln+1|/b), is placed inside the assumption rather than derived from Tobler's law, from any spatial data-generating process, or from the data. The paper promises 'a mathematical proof demonstrating that the distance-weighted split conformal prediction satisfies the local weighted exchangeability condition' in Appendix C, but the appendix contains only the assumption and the assertion that it 'holds under covariate shift.' The bandwidth b is never estimated or reported. Thus the advertised Eq.

full rationale

GeoCP is, at the algorithmic level, weighted conformal prediction with the covariate likelihood ratio replaced by a geographic kernel. The empirical simulations and case studies are non-circular and self-contained, and I do not count them against the paper. The circularity is concentrated in the theoretical coverage claim. Section 2 says a proof that distance-weighted split conformal prediction satisfies localized weighted exchangeability is in Appendix C; Appendix C instead assumes exactly that condition in Assumption 1 and then states Theorem 2, whose own text says it 'is identical in form to the global weighted case.' So the local coverage guarantee in Eq. (4) is not derived from spatial dependence; it is the assumption repackaged as a theorem. The added claim that the condition 'holds under covariate shift or when sampling is locally randomized around the test point' is not proved, and the kernel bandwidth is neither estimated nor subjected to sensitivity analysis. Because the central theoretical claim reduces to the assumption by construction, while the empirical content remains independent, the score is 8 rather than 10.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The central claim rests on Assumption 1, an ad hoc weighted exchangeability condition, and on Tobler's law as a justification for geographic weights. The bandwidth b is a free parameter with no reported value. No new entities are introduced.

free parameters (1)

Gaussian decay bandwidth b = not reported
Algorithm 1 accepts a decay function beta as input; the paper says Gaussian kernels are used but gives no bandwidth value, estimator, or selection rule, so the method is under-specified.

assumptions (3)

domain assumption Tobler's first law of geography: nearby observations have similar values; therefore a kernel of geographic distance can stand in for the covariate likelihood ratio under spatial covariate shift.
Used in Section 2 to justify replacing likelihood ratio weights in weighted conformal prediction with distance-based weights; not derived for any specific dataset.
ad hoc to paper Localized weighted exchangeability: P[(Z_1,...,Z_{n+1}) in A] = E[ product_i w(X_{pi(i)}) / sum_i w(X_{pi(i)}) 1{(Z_1,...,Z_{n+1}) in A} ] for all permutations pi.
This is the exact condition needed for the coverage theorem in Appendix C; it is assumed rather than shown to follow from Tobler's law or from a data-generating process.
domain assumption Calibration set is drawn from the same distribution as the training set, and P_{Y|X} is unchanged between training and test under covariate shift.
Stated in Section 2 framework; inherited from weighted conformal prediction without additional justification for the geographic setting.

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Cite this review

Pith. "Pith review of GeoConformal prediction: a model-agnostic framework of measuring the uncertainty of spatial prediction." pith.science (2026). https://pith.science/paper/UY4KUVVS

@misc{pith2026241208661,
  author       = {Pith},
  title        = {Pith review of: GeoConformal prediction: a model-agnostic framework of measuring the uncertainty of spatial prediction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UY4KUVVS}},
  note         = {Machine review of arXiv:2412.08661}
}

read the original abstract

Spatial prediction is a fundamental task in geography. In recent years, with advances in geospatial artificial intelligence (GeoAI), numerous models have been developed to improve the accuracy of geographic variable predictions. Beyond achieving higher accuracy, it is equally important to obtain predictions with uncertainty measures to enhance model credibility and support responsible spatial prediction. Although geostatistic methods like Kriging offer some level of uncertainty assessment, such as Kriging variance, these measurements are not always accurate and lack general applicability to other spatial models. To address this issue, we propose a model-agnostic uncertainty assessment method called GeoConformal Prediction, which incorporates geographical weighting into conformal prediction. We applied it to two classic spatial prediction cases, spatial regression and spatial interpolation, to evaluate its reliability. First, in the spatial regression case, we used XGBoost to predict housing prices, followed by GeoConformal to calculate uncertainty. Our results show that GeoConformal achieved a coverage rate of 93.67%, while Bootstrap methods only reached a maximum coverage of 81.00% after 2000 runs. Next, we applied GeoConformal to spatial interpolation models. We found that the uncertainty obtained from GeoConformal aligned closely with the variance in Kriging. Finally, using GeoConformal, we analyzed the sources of uncertainty in spatial prediction. We found that explicitly including local features in AI models can significantly reduce prediction uncertainty, especially in areas with strong local dependence. Our findings suggest that GeoConformal holds potential not only for geographic knowledge discovery but also for guiding the design of future GeoAI models, paving the way for more reliable and interpretable spatial prediction frameworks.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

7 extracted references · 6 canonical work pages

[3]

Fit spatial prediction model 𝑓𝑓(𝑥𝑥) on each training set and predict the target values on corresponding test set

work page
[4]

For each point, we can get a 90% confidence interval by taking 5% and 95% percentile {�𝑞𝑞0.5,𝑖𝑖, 𝑞𝑞0.95,𝑖𝑖�}𝑖𝑖∈𝐼𝐼 of its K predicted values

In this way, all N points can get K predicted values {𝑦𝑦�𝑖𝑖,𝑗𝑗}𝑖𝑖∈𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡,𝑗𝑗∈1…𝐾𝐾. For each point, we can get a 90% confidence interval by taking 5% and 95% percentile {�𝑞𝑞0.5,𝑖𝑖, 𝑞𝑞0.95,𝑖𝑖�}𝑖𝑖∈𝐼𝐼 of its K predicted values. If we want to get the prediction interval from Bootstrapping method, we need to add random residuals ϵ to predicted values 𝑦𝑦�. Th...

work page
[6]

Randomly sample with replacement from the dataset to generate a training set 𝐼𝐼𝑡𝑡𝑡𝑡𝑐𝑐𝑖𝑖𝑛𝑛 of size M and a test set 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 of size N - M

work page
[7]

Repeat the resampling process for K times to create K bootstrap datasets {𝐼𝐼𝑡𝑡𝑡𝑡𝑐𝑐𝑖𝑖𝑛𝑛,𝑖𝑖, 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡,𝑖𝑖}𝑖𝑖=1 𝐾𝐾

work page
[8]

Also, the residuals {ϵ𝑖𝑖}𝑖𝑖∈𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑖𝑖𝑡𝑡 on one of this training sets will be obtained

Fit spatial prediction model 𝑓𝑓(𝑥𝑥) on one training set and predict the target values on corresponding test set. Also, the residuals {ϵ𝑖𝑖}𝑖𝑖∈𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑖𝑖𝑡𝑡 on one of this training sets will be obtained. Then, the final predicted values on the test set will be the predicted values plus a random error sampled from the residuals computed above 𝑦𝑦𝑖𝑖 + = 𝑦𝑦�𝑖𝑖 +...

work page
[9]

C．Localized weighted exchangeability and coverage validity in the geospatial context Assumption 1 (Localized weighted exchangeability)

For each point, we can get a 90% confidence interval by taking 5% and 95% percentile {�𝑞𝑞0.5,𝑖𝑖, 𝑞𝑞0.95,𝑖𝑖�}𝑖𝑖∈𝐼𝐼 of its K predicted values. C．Localized weighted exchangeability and coverage validity in the geospatial context Assumption 1 (Localized weighted exchangeability). Let Zi=(Xi,yi) for i=1,…,n+1, and fix a measurable, nonnegative weight function ...

work page
[265]

Guo, X., Q

PMLR. Guo, X., Q. Zhang, M. Peng, M. Zhua, and others. 2024. Explainable traffic flow prediction with large language models. arXiv preprint arXiv:2404.02937. Hagenauer, J., and M. Helbich. 2022. A geographically weighted artificial neural network. International Journal of Geographical Information Science 36 (2):215–235. Han, X., Z. Tang, J. Ghosh, and Q. ...

arXiv 2024

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [3]

Fit spatial prediction model 𝑓𝑓(𝑥𝑥) on each training set and predict the target values on corresponding test set

work page

[2] [4]

For each point, we can get a 90% confidence interval by taking 5% and 95% percentile {�𝑞𝑞0.5,𝑖𝑖, 𝑞𝑞0.95,𝑖𝑖�}𝑖𝑖∈𝐼𝐼 of its K predicted values

In this way, all N points can get K predicted values {𝑦𝑦�𝑖𝑖,𝑗𝑗}𝑖𝑖∈𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡,𝑗𝑗∈1…𝐾𝐾. For each point, we can get a 90% confidence interval by taking 5% and 95% percentile {�𝑞𝑞0.5,𝑖𝑖, 𝑞𝑞0.95,𝑖𝑖�}𝑖𝑖∈𝐼𝐼 of its K predicted values. If we want to get the prediction interval from Bootstrapping method, we need to add random residuals ϵ to predicted values 𝑦𝑦�. Th...

work page

[3] [6]

Randomly sample with replacement from the dataset to generate a training set 𝐼𝐼𝑡𝑡𝑡𝑡𝑐𝑐𝑖𝑖𝑛𝑛 of size M and a test set 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 of size N - M

work page

[4] [7]

Repeat the resampling process for K times to create K bootstrap datasets {𝐼𝐼𝑡𝑡𝑡𝑡𝑐𝑐𝑖𝑖𝑛𝑛,𝑖𝑖, 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡,𝑖𝑖}𝑖𝑖=1 𝐾𝐾

work page

[5] [8]

Also, the residuals {ϵ𝑖𝑖}𝑖𝑖∈𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑖𝑖𝑡𝑡 on one of this training sets will be obtained

Fit spatial prediction model 𝑓𝑓(𝑥𝑥) on one training set and predict the target values on corresponding test set. Also, the residuals {ϵ𝑖𝑖}𝑖𝑖∈𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑖𝑖𝑡𝑡 on one of this training sets will be obtained. Then, the final predicted values on the test set will be the predicted values plus a random error sampled from the residuals computed above 𝑦𝑦𝑖𝑖 + = 𝑦𝑦�𝑖𝑖 +...

work page

[6] [9]

C．Localized weighted exchangeability and coverage validity in the geospatial context Assumption 1 (Localized weighted exchangeability)

For each point, we can get a 90% confidence interval by taking 5% and 95% percentile {�𝑞𝑞0.5,𝑖𝑖, 𝑞𝑞0.95,𝑖𝑖�}𝑖𝑖∈𝐼𝐼 of its K predicted values. C．Localized weighted exchangeability and coverage validity in the geospatial context Assumption 1 (Localized weighted exchangeability). Let Zi=(Xi,yi) for i=1,…,n+1, and fix a measurable, nonnegative weight function ...

work page

[7] [265]

Guo, X., Q

PMLR. Guo, X., Q. Zhang, M. Peng, M. Zhua, and others. 2024. Explainable traffic flow prediction with large language models. arXiv preprint arXiv:2404.02937. Hagenauer, J., and M. Helbich. 2022. A geographically weighted artificial neural network. International Journal of Geographical Information Science 36 (2):215–235. Han, X., Z. Tang, J. Ghosh, and Q. ...

arXiv 2024