Stochastic Processes with Modified Lognormal Distribution Featuring Flexible Upper Tail

Anastassia Baxevani; Dionissios T. Hristopulos; Giorgio Kaniadakis

REVIEW 3 major objections 6 minor 86 references

Stochastic Processes with Modified Lognormal Distribution Featuring Flexible Upper Tail

T0 review · 3 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper introduces a three-parameter κ-lognormal distribution—the κ-exponential transform of a Gaussian—with a right tail lighter than the lognormal's, closed-form probability functions, and a warped Gaussian-process construction that…

desk verdict Solid extension of the authors' earlier kappa-lognormal work; the process-level construction is new and useful, but the noise specification in the real-data application is inconsistent with the warped-GP model and needs to be reconciled. read the letter →

arxiv 2505.14713 v1 pith:ORGCB4QC submitted 2025-05-17 stat.ME cs.LGmath.STphysics.data-anstat.MLstat.TH

Dionissios T. Hristopulos , Anastassia Baxevani , Giorgio Kaniadakis This is my paper · ORCID

classification stat.MEcs.LGmath.STphysics.data-anstat.MLstat.TH MSC 60E0560G1562F10

keywords kappa-lognormaldistributionkappa-exponentialfunctionkappa-logarithmtransformlighteruppertailhazardratewarpedGaussianprocessregressionstochasticbimodality

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

The paper tries to establish that a single deformation parameter, κ, can tune the lognormal's right tail from its usual heavy form to lighter alternatives while keeping the model tractable. It defines the κ-lognormal as $X=\exp_\kappa(Y)$ with $Y$ Gaussian, where $\exp_\kappa$ is the κ-exponential, and shows that the density, CDF, quantiles, moments, and hazard rate all have closed forms. Because the inverse transform $\ln_\kappa$ is explicit and monotone, the same construction extends to stochastic processes: a latent Gaussian process warped by $\exp_\kappa$ yields a joint density through the multivariate change-of-variables formula, and prediction can proceed in the latent space. If right, this gives practitioners a smooth alternative to truncated lognormals for skewed positive data with bounded or lighter tails, plus ready-made maximum-likelihood estimation and Gaussian-process prediction.

What carries the argument

The load-bearing object is the κ-exponential function $\exp_\kappa(y)=(\sqrt{1+\kappa^2 y^2}+\kappa y)^{1/\kappa}$ and its inverse, the κ-logarithm $\ln_\kappa(x)=(x^\kappa-x^{-\kappa})/(2\kappa)$. The transformation $X=\exp_\kappa(Y)$ turns a Gaussian latent variable into the κ-lognormal; the derivative of $\ln_\kappa$ supplies the Jacobian factor $x^{\kappa-1}+x^{-\kappa-1}$ in the density, and the same latent-Gaussian construction defines the multivariate joint density and the warped Gaussian process predictors. The characteristic polynomial $p_1(z)=z^6-az^5+bz^4-2az^3+cz^2-az-1$ with $a=2\mu\kappa$, $b=1-4\kappa\sigma^2(\kappa-1)$, and $c=4\kappa\sigma^2(\kappa+1)-1$ determines the number and location of modes.

What would settle it

Take a large sample from a process with a known physical upper bound, or from the paper's own model with $\kappa>0$, fit the κ-lognormal by maximum likelihood, and apply a normality test to the estimated values $\hat{y}_i=\ln_{\hat{\kappa}}(x_i)$. If the test rejects Gaussianity in data the model is claimed to describe, or if the fitted density assigns appreciable probability above the physical bound, the central lighter-tail and latent-Gaussian claim is not supported for that setting.

Watch

Extended reading notes

Core claim

The central claim is that the κ-lognormal family, $\kappa\mathrm{LN}(\mu,\sigma,\kappa)$, defined by $X=\exp_\kappa(Y)$ with $Y\sim N(\mu,\sigma^2)$, is a continuous deformation of the lognormal: the limit $\kappa\to 0$ recovers $\ln_\kappa\to\ln$ and the lognormal density, while for $\kappa>0$ the right tail is lighter than the lognormal's and controlled by κ. The paper derives the marginal PDF $f_X(x)=\frac{1}{2\sqrt{2\pi}\sigma}e^{-(\ln_\kappa(x)-\mu)^2/2\sigma^2}(x^{\kappa-1}+x^{-\kappa-1})$, the CDF $\Phi((\ln_\kappa(x)-\mu)/\sigma)$, the quantile function $Q_X(p)=\exp_\kappa(\mu+\sqrt{2}\sigma\,\mathrm{erf}^{-1}(2p-1))$, and asymptotic results: for $\kappa=0.5$ the hazard rate tends to a constant, for $\kappa>0.5$ it increases at infinity, and for $\kappa<0.5$ it declines. It further claims that certain parameter triples give bimodal densities, that moments of all integer orders follow from the first-order moment by scaling, and that a κ-lognormal stochastic process can be defined by applying $\exp_\kappa$ to a latent Gaussian process, with joint density from the multivariate change-of-variables theorem and prediction from warped Gaussian process regression.

Load-bearing premise

The load-bearing premise is that, for real data, the transformed variable $\ln_\kappa(X)$ is exactly Gaussian and that noise enters only as a diagonal variance term added to the latent Gaussian covariance; if that fails, the predictive density, quantile intervals, and likelihood are misspecified.

Editorial extensions

If this is right

Skewed positive data with lighter-than-lognormal tails can be modeled without truncation; the κ parameter interpolates continuously to the lognormal at $\kappa=0$.
Closed-form quantiles and CDF make simulation, quantile fitting, and prediction intervals straightforward, including quantile-invariant intervals in the observation space.
The hazard-rate result at $\kappa=0.5$ gives a simple diagnostic: data whose tail hazard increases support $\kappa>0.5$, where the model is suitable for failure-time analysis, unlike the lognormal.
Warped Gaussian process regression with the κ-logarithm gives median and mode predictors for time series and spatial fields, with likelihood-based estimation of all parameters.

Reading between the lines

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

If the latent Gaussianity assumption holds only approximately, the same framework could be paired with a goodness-of-fit check on $\ln_\kappa(X)$; a rejected normality test would indicate a different warping or a heavier-tailed latent model.
Because $\exp_\kappa$ is defined for negative arguments, the κ-logarithm is a Box-Cox-type transform whose inverse never breaks; this may make it attractive for zero-inflated or censored positive data, though the paper does not develop that case.
The lighter right tail changes extreme-value behavior: for matched mean and variance, κ-lognormal typical extremes are smaller than lognormal ones, so the model may reduce overestimation of upper quantiles in environmental and engineering data.
The bimodal regime suggests a possible use for two-state or switching phenomena, but the paper notes that a single three-parameter family may not capture all peak shapes; a testable extension would fit the model to such datasets and compare peak locations against empirical modes.

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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

Short version: this is a real paper, not a repackaging. The kappa-lognormal marginal itself is already in the same group's 2022 Entropy paper, so the family is not new. But the process construction (joint density via Jacobi, hazard-rate transition at kappa=0.5, moment bounds, MLE machinery) is a genuine step beyond that work. The math checks out by my reading: the PDF/CDF/quantile derivations are clean, the scaling relation (28) is proven, the asymptotic hazard analysis is correct, and the mode analysis via Descartes is plausible. The synthetic estimation experiments are honest and the parameter recovery is good.

Where it gets soft: the paper oversells novelty in the abstract by presenting the marginal family without flagging ref [61] as the source. The real-data support is mixed—Berea's AIC/BIC gains are modest, and on Jura the lognormal still wins for four of seven metals. The most substantive problem is the noise model. Section IV-B builds the warped-GP prediction on the assumption that noise lives in latent Gaussian space, so ln_kappa(x) is Gaussian. But Section VI-D models the Berea permeability as P(s)=X(s)+epsilon(s) with Gaussian noise added in observation space. Under Definition 3, ln_kappa(P) is then not Gaussian, so the predictive density (48), the intervals (49), and the likelihood (54) are all misspecified for that application. The paper never checks whether the kappa-log transform of the noisy real data is approximately Gaussian, nor does it simulate this noise regime. That is not a fatal flaw for the distributional part, but it does undermine the practical predictive claims.

Minor but real: citation markers for warped GPs point to the wrong references, the Hessian is incompletely typeset, and no code or data are shipped. Remark 10 documents a genuine failure of free-kappa optimization—the global optimizer returns essentially the lognormal with worse likelihood—and this deserves a more prominent treatment than a remark.

Bottom line: reliability and geostatistics practitioners will get value from the tail flexibility and the process extension. If the authors fix the noise-model inconsistency, or clearly restrict the prediction claims to noiseless or latent-space-noise settings, this is a citable contribution. As it stands it deserves peer review, but I would send it back for major revision.

Referee Report

3 major / 6 minor

Summary. The paper introduces the kappa-lognormal distribution, defined by X = exp_kappa(Y) with Y Gaussian, and develops its marginal properties: closed-form PDF, CDF, quantile function, hazard rate, moment bounds, power-series moment expansions, mode analysis via Descartes' rule, and explicit MLE gradient and Hessian. It then defines kappa-lognormal stochastic processes via Jacobi's theorem applied to a latent Gaussian process and proposes warped Gaussian process (w-GP) prediction, with applications to synthetic time series, Berea sandstone permeability, and Jura heavy-metal data. The central claim is that the family provides continuous deformations of the lognormal with lighter right tails controlled by kappa, plus tractable estimation and prediction machinery.

Significance. If the claims hold, the paper contributes a parametric family with several closed-form statistical functions and a warped GP framework for skewed positive data; the hazard-rate asymptotics (increasing for kappa>0.5) and the explicit MLE derivatives are practically useful. The construction is definitional, so the lighter-tail and hazard properties are analytic consequences rather than fitted conclusions; the scaling relation (28) is proven by substitution, and the mode analysis via Descartes' rule is a plausible contribution. The synthetic experiments correctly test estimation/prediction within the assumed model. However, the practical predictive claims for real spatial data rest on a noise specification that is internally inconsistent between the latent-space formulation and the observation-space data model used in the application, and the moment series expansion lacks convergence justification.

major comments (3)

[IV-B and VI-D] The predictive machinery in Section IV-B assumes noise is added in latent space: equations (46)-(47) add sigma_epsilon^2 I to the latent covariance C_Y, so y_S = ln_kappa(x_S) is treated as a Gaussian latent vector and the predictive density (48) and intervals (49) follow. In contrast, Section VI-D defines the permeability field as P(s) = X(s) + epsilon(s) with epsilon Gaussian in observation space. Under Definition 3, ln_kappa(P) is then not Gaussian, so the conditional density of P(s*) is not the kappa-lognormal density (48), and the intervals (49) are not the correct predictive intervals. The time-series experiments in Section VI-C are noiseless, so they do not exercise this assumption. The authors should either change the spatial data model to latent-space noise or derive and validate the correct predictive distribution and interval coverage under observation-space noise; the paper currently validates neither on the real data.
[Theorem 4, Eq. (31a)] The power-series expansion of the moments is obtained by expanding exp_{kappa/l}(l y) in a Taylor series around mu and integrating term-by-term against the Gaussian density. The kappa-exponential Taylor series, given in the Supplement Eq. (90), has finite radius of convergence (kappa y)^2 < 1, and no justification is provided for interchanging the sum and the integral over the full real line. As stated, the series may be only asymptotic; the paper should either prove convergence of the resulting moment series or explicitly characterize it as an asymptotic expansion. This is load-bearing because Section III-D uses the truncated series to approximate moments (Figure 8) and presents the expansion as a closed-form contribution.
[Theorem 1 and Appendix A] Theorem 1 states that the stationary points satisfy R in {1,3} and that the PDF has at most three modes, but item 4 of the theorem says that R=5 positive roots (which would imply five stationary points and potentially three modes) is not excluded by Descartes' rule, still leaving a logical inconsistency in the mode classification. The text should reconcile this by either excluding R=5 rigorously or stating that the theorem's classification is conditional on the empirical observation that five positive roots were not found.

minor comments (6)

[Theorem 3 and Figure 7] Theorem 3 assumes mu > 0, but Figure 7 presents results for mu = -2 and claims the lower bound is accurate; the assumption should be relaxed or the figure should be framed as an extrapolation outside the theorem's stated conditions.
[Theorem 5 heading] The heading reads 'Jabobi's multivariate theorem' and should read 'Jacobi's multivariate theorem'.
[Appendix A] The phrase 'umimodal' appears in the proof of Theorem 1 and should read 'unimodal'.
[Section III-G] The heading 'Failure of Simple Scaling Invariance' contains the typo 'lognornal' in the first sentence; it should be 'lognormal'.
[Table VII and Section VI-D] The text says 'the relative RMSE is 16% for N_tr=500 and 14% for N_te=1100', but when N_tr=1100 the test set size is N_te=500; the labels in the table and surrounding text should be checked for consistency.
[Eq. (3)] The asymptotic notation exp_kappa(y) ~ (2 kappa y)^{±1/kappa} is ambiguous; specifying the positive and negative branches separately would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the kappa-lognormal construction is definitional, and the analytic results and simulation studies do not reduce to their inputs by construction.

full rationale

The paper defines the kappa-lognormal variable by X=exp_kappa(Y) with Y Gaussian (Definition 2 and Definition 3, Eq. 21), so the PDF (22), CDF (24), quantile function (25), hazard rate (40), and moment expressions (27)-(31) are derived by standard change-of-variables and calculus from that definition; they are analytic consequences rather than fitted predictions. The synthetic experiments in Section VI-A and VI-C generate data from the same model and then estimate parameters, which tests the estimators but is not presented as external validation; that is standard simulation practice and does not make the derivation circular. The real-data Berea and Jura applications assume the latent Gaussianity of ln_kappa(X) and a particular noise model; even if those assumptions are misspecified (e.g., observation-space noise in Section VI-D versus latent-space noise in Section IV-B), that is a correctness or robustness concern, not a circularity. Self-citations (e.g., LDHO kernel [72], earlier warping [61], [68]) are used as modeling choices or background, not as load-bearing evidence that forces the central result. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled in via citation. Hence no circular step can be identified.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new particles, forces, dimensions, or conserved quantities. The kappa-deformed functions are borrowed from Kaniadakis' earlier framework. The only invented mathematical objects are the kappa-lognormal distribution itself and its process-level construction, which are defined by the transform X=exp_kappa(Y) with Gaussian Y. The free parameters are the usual distribution and kernel parameters fitted to data.

free parameters (3)

kappa (tail shape) = 0.556 (Berea), 0.43 (Cd), 1.04 (Co), 0.70 (Cr), 0.82 (Ni), 2.89 (synthetic A), 0.52 (synthetic B)
Controls the right-tail lightness and whether the density is unimodal or bimodal. Estimated by MLE or quantile fitting on each dataset; no independent physical measurement fixes it.
latent Gaussian mean mu and variance sigma^2 = mu approx 8.26 and sigma approx 1.37 for the Berea marginal fit; varies by dataset
These set the location and scale of the latent Gaussian process that is warped by the kappa-exponential. They are estimated from data and are not derived from theory.
covariance kernel and noise hyperparameters = tau_c approx 23.63 and omega_d approx 0.13 in one forecast realization; xi approx 5.42, rho approx 16.53, phi approx…
The latent Gaussian process covariance parameters and observation noise variance are fitted to data for forecasting and spatial interpolation. They are not implied by the kappa-lognormal distribution theory.

assumptions (4)

domain assumption The transformed variable Y = ln_kappa(X) is exactly Gaussian.
Definition 3 and Section IV-B(c) assume the latent process is Gaussian; this is the engine of the warped Gaussian process predictors and is not validated for real datasets.
standard math Kaniadakis kappa-exponential and kappa-logarithm properties from refs [55]-[59] are taken as background.
Used for monotonicity, asymptotic behavior, and composition laws. These are published results not re-derived in this paper.
standard math Jacobi's multivariate theorem applies to the componentwise kappa-logarithm transform.
Section IV-A uses a diagonal Jacobian; this requires a one-to-one differentiable transform, which exp_kappa and ln_kappa satisfy.
domain assumption Noise in observation space can be modeled as a diagonal noise term in latent space.
Section IV-B(c) adds sigma_epsilon^2 to the latent covariance while the observation process includes additive Gaussian noise. The equivalence is an approximation and is not derived.

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Cite this review

Pith. "Pith review of Stochastic Processes with Modified Lognormal Distribution Featuring Flexible Upper Tail." pith.science (2026). https://pith.science/paper/ORGCB4QC

@misc{pith2026250514713,
  author       = {Pith},
  title        = {Pith review of: Stochastic Processes with Modified Lognormal Distribution Featuring Flexible Upper Tail},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ORGCB4QC}},
  note         = {Machine review of arXiv:2505.14713}
}

read the original abstract

Asymmetric, non-Gaussian probability distributions are often observed in the analysis of natural and engineering datasets. The lognormal distribution is a standard model for data with skewed frequency histograms and fat tails. However, the lognormal law severely restricts the asymptotic dependence of the probability density and the hazard function for high values. Herein we present a family of three-parameter non-Gaussian probability density functions that are based on generalized kappa-exponential and kappa-logarithm functions and investigate its mathematical properties. These kappa-lognormal densities represent continuous deformations of the lognormal with lighter right tails, controlled by the parameter kappa. In addition, bimodal distributions are obtained for certain parameter combinations. We derive closed-form analytic expressions for the main statistical functions of the kappa-lognormal distribution. For the moments, we derive bounds that are based on hypergeometric functions as well as series expansions. Explicit expressions for the gradient and Hessian of the negative log-likelihood are obtained to facilitate numerical maximum-likelihood estimates of the kappa-lognormal parameters from data. We also formulate a joint probability density function for kappa-lognormal stochastic processes by applying Jacobi's multivariate theorem to a latent Gaussian process. Estimation of the kappa-lognormal distribution based on synthetic and real data is explored. Furthermore, we investigate applications of kappa-lognormal processes with different covariance kernels in time series forecasting and spatial interpolation using warped Gaussian process regression. Our results are of practical interest for modeling skewed distributions in various scientific and engineering fields.

Figures

Figures reproduced from arXiv: 2505.14713 by the authors.

**Figure 1.** Plots of the difference expκ(x) − exp(x) for κ = 0.1, 0.5, 0.9 based on the κ-exponential definition (2) for x ≥ 0, which confirm that the natural exponential is the upper bound of the κ-exponential, namely exp(x) ≥ expκ(x), for x ≥ 0 (cf. Proposition 1). For x < 0 (not shown) the sign of the difference is reversed, that is, expκ(x) ≥ exp(x), and the natural exponential is a lower bound of the κ-exponential [PITH_F… view at source ↗

**Figure 2.** Plots of the difference expκ(y) − exp(y) for κ = 0.1, 0.5, 0.9 based on the κ-exponential definition (2) for y ≥ 0, which confirm that the natural exponential is the upper bound of the κ-exponential, namely exp(y) ≥ expκ(y), for y ≥ 0 (cf. Proposition 1). For y < 0 (not shown) the sign of the difference is reversed, that is, expκ(y) ≥ exp(y), and the natural exponential is a lower bound of the κ-exponential. Proposi… view at source ↗

**Figure 3.** Left: Unimodal probability density functions of κ-lognormal distributions defined by (22) with parameters µ = 1 and σ = 0.5. The lognormal probability density is recaptured at the limit κ = 0. Right: Approximately Gaussian κ-lognormal PDFs with parameters µ = 20 and σ = 2 for 0.7 ≤ κ ≤ 0.9 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (25 more)

**Figure 4.** Figure 4: Left: Bimodal probability density functions of κ-lognormal distributions defined by (22) with parameters µ = 1 and σ = 1 for 1.5 ≤ κ ≤ 3. Right: Bimodal κ-lognormal PDFs with parameters µ = 1 and σ = 1.5 for 1.5 ≤ κ ≤ 3. A comparison of the κ-lognormal with the lognorm…

**Figure 5.** Figure 5: Moments of order ℓ ∈ {1, 2, . . . , 10} for the κ-lognormal distribution with different κ values. All curves are obtained by numerical evaluation of the integral (27) for µ = 0 and σ = 1. The horizontal axis shows the moment order ℓ. The vertical axis uses a logarithmi…

**Figure 6.** Figure 6: Left: Mean of the lognormal distribution (continuous line, blue online) with µ = 5 and σ = 2 calculated by means of numerical integration (cf. (27)) and lower bound of the first-order moment based on (29b) (circle markers, red online). Right: Root of order ℓ of the exp…

**Figure 7.** Figure 7: Left: Mean of the κ-lognormal distribution (continuous line) with µ = −2 and σ = 2 versus κ (continuous line, blue online). The expectation is calculated by means of numerical integration (cf. (27)). The lower bound of the mean as a function of κ, based on (29b), is sh…

**Figure 8.** Figure 8: Left: Mean of the κ-lognormal distribution with µ = 5 and σ = 2 (continuous line) and σ = 0.9 (dashed line) calculated by means of numerical integration (cf. (27)) for κ ∈ [0.05, 1.95]. The Taylor-series moment approximation (31) is also shown (circles for σ = 2 and sq…

**Figure 9.** Figure 9: Left: Ratio q∗(κ) ≜ xmax/µ0.5 for the κ-lognormal distribution with different κ including the lognormal limit κ = 0 based on (34) ; xmax = Q(1 − 1/N) where N = 2L and µ0.5 = Q(0.5). The quantile function (25) is used to calculate xmax. Different L values correspond to …

**Figure 10.** Figure 10: Hazard function hκ(x) for the κ-lognormal distribution based on (39) for five different values of κ. All curves are obtained with µ = σ = 1. The transition from asymptotically declining to increasing hκ(x) at κ = 0.5 is evident. G. Failure of Simple Scaling Invariance…

**Figure 11.** Figure 11: Histograms and ML-estimated PDFs obtained from samples comprising 1000 random numbers from the [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]

**Figure 12.** Figure 12: Profile NLL curve (NLL per site) as a function of [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]

**Figure 13.** Figure 13: NLL Streamlines for the Berea permeability data on the [PITH_FULL_IMAGE:figures/full_fig_p021_13.png]

**Figure 14.** Figure 14: Left: Histogram of Berea permeability data and MLE fits of the lognormal PDF (blue broken line) and the κ-lognormal PDF (continuous line). The horizontal axis measures permeability and the units used are millidarcy (mD). Right: Q-Q plot of the lognormal quantiles (hor…

**Figure 15.** Figure 15: Each plot shows the distribution of the ratio [PITH_FULL_IMAGE:figures/full_fig_p022_15.png]

**Figure 15.** Figure 15: Violin plots displaying the distribution of the estimated model parameter estimates based on 500 realizations of the [PITH_FULL_IMAGE:figures/full_fig_p023_15.png]

**Figure 16.** Figure 16: Violin plots displaying the distribution of cross-validation measures based on 500 realizations of the [PITH_FULL_IMAGE:figures/full_fig_p023_16.png]

**Figure 17.** Figure 17: Time series generated from the κ-LN process with joint PDF (45) and LDHO covariance kernel (57) with parameter set ζ = (µ, σ, κ, τc, ωd) = (1, 1, 3, 30, 0.13). TABLE VI CROSS-VALIDATION MEASURES OVER THE 51-STEP FORECASTING HORIZON BASED ON A SINGLE RANDOM REALIZATION…

**Figure 18.** Figure 18: Left: Sample histogram of κ-lognormal process (bars), κ-lognormal PDF corresponding to the parameter set ζ = (µ, σ, κ, τc, ωd) = (1, 1, 3, 30, 0.13) (dash-dot line), κ-lognormal PDF obtained for the MLE ζˆ = (ˆµ, σ, ˆ κ, ˆ τˆc, ωˆd) ≈ (0.82, 0.79, 2.82, 23.63, 0.13) (…

**Figure 19.** Figure 19: Multi-step forecast for the time series shown in Fig. 17. [PITH_FULL_IMAGE:figures/full_fig_p025_19.png]

**Figure 20.** Figure 20: Spatial configuration of the gridded Berea permeability dataset. The colorbar measures permeability in mD. [PITH_FULL_IMAGE:figures/full_fig_p026_20.png]

**Figure 21.** Figure 21: Ratios xmax/µ0.5 representing normalized typical extreme values for N = 2L independent random variables versus L for the lognormal (continuous lines) and the normal (broken line) distributions— xmax = Q(1 − 1 N ) where N = 2L and µ0.5 = Q(0.5). Different L values corr…

**Figure 22.** Figure 22: Histograms of Jura heavy metal datasets (bins) with the optimal [PITH_FULL_IMAGE:figures/full_fig_p040_22.png]

**Figure 23.** Figure 23: Quantile-quantile plots of the lognormal distribution (horizontal axis) versus the [PITH_FULL_IMAGE:figures/full_fig_p041_23.png]

**Figure 24.** Figure 24: Left: Berea permeability data. Right: Reconstructed permeability field using Matern covariance kernel and a training set comprising 500 randomly ´ selected values. 20 40 60 80 100 True 30 40 50 60 70 80 90 GPR Median Predictor 5 10 15 20 25 30 35 40 5 10 15 20 25 30 3…

**Figure 25.** Figure 25: Left: Scatter plot of test set permeability values versus the median-based warped-GPR predictions using the Matern covariance kernel. ´ Right: Spatial distribution of absolute values of prediction errors. Empty sites correspond to training set points. The training set…

**Figure 26.** Figure 26: Left: Berea permeability data. Right: Reconstructed permeability field using Matern covariance kernel and a training set comprising 1100 randomly ´ selected values. 20 40 60 80 100 True 30 40 50 60 70 80 90 GPR Median Predictor 5 10 15 20 25 30 35 40 5 10 15 20 25 30 …

**Figure 27.** Figure 27: Left: Scatter plot of test set permeability values versus the median-based warped-GPR predictions using the Matern covariance kernel. ´ Right: Spatial distribution of absolute values of prediction errors. Empty sites correspond to training set points. The training set…

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write newline

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[3] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.state := if if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTIO...

[4] [3]

write newline

" write newline "" initialize.prev.this.status FUNCTION begin.bib " write newline preamble empty 'skip preamble write newline if " thebibliography " longest.label * " " * write newline " [1] #1 " write newline " url@samestyle " write newline " " write newline " [2] #2 " write newline " =0pt " write newline " " ALTinterwordstretchfactor * " " * write newli...

[5] [4]

write newline

" write newline "" before.all 'output.state := FUNCTION blank.sep after.quote 'output.state := FUNCTION fin.entry output.state after.quoted.block = 'skip 'add.period if write newline FUNCTION new.block output.state before.all = 'skip output.state after.quote = after.quoted.block 'output.state := after.block 'output.state := if if FUNCTION new.sentence out...

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