Random Coefficients Research Articles

Normalizing Basis Functions: Approximate Stationary Models for Large Spatial Data

ABSTRACTIn geostatistics, traditional spatial models often rely on the Gaussian process (GP) to fit stationary covariances to data. It is well known that this approach becomes computationally infeasible when dealing with large data volumes, necessitating the use of approximate methods. A powerful class of methods approximate the GP as a sum of basis functions with random coefficients. Although this technique offers computational efficiency, it does not inherently guarantee a stationary covariance. To mitigate this issue, the basis functions can be ‘normalized’ to maintain a constant marginal variance, avoiding unwanted artefacts and edge effects. This allows for the fitting of nearly stationary models to large, potentially non‐stationary datasets, providing a rigorous base to extend to more complex problems. Unfortunately, the process of normalizing these basis functions is computationally demanding. To address this, we introduce two fast and accurate algorithms to the normalization step, allowing for efficient prediction on fine grids. The practical value of these algorithms is showcased in the context of a spatial analysis on a large dataset, where significant computational speedups are achieved. While implementation and testing are done specifically within the LatticeKrig framework, these algorithms can be adapted to other basis function methods operating on regular grids.

The relationship between maximum principle and dynamic programming principle for stochastic recursive control problem with random coefficients

The relationship between maximum principle and dynamic programming principle for stochastic recursive control problem with random coefficients

Bounds for Exponential Sums With Random Multiplicative Coefficient

Abstract For $f$ a Rademacher or Steinhaus random multiplicative function, we prove that $ \max _{\theta \in [0,1]} \frac{1}{\sqrt{N}} \bigl | \sum _{n \leq N} f(n) \mathrm{e} (n \theta ) \bigr | \gg \sqrt{\log N}$, asymptotically almost surely as $N \rightarrow \infty $. Furthermore, for $f$ a Steinhaus random multiplicative function, and any $\varepsilon> 0$, we prove the partial upper bound result $ \max _{\theta \in [0,1]} \frac{1}{\sqrt{N}} \bigl | \sum _{\substack{n \leq N \\ P(n) \geq N^{0.8}}} f(n) \mathrm{e} (n \theta ) \bigr | \ll{(\log N)}^{7/4 + \varepsilon }$, asymptotically almost surely as $N \rightarrow \infty $, where $P(n)$ denotes the largest prime factor of $n$.

Computationally efficient and error aware surrogate construction for numerical solutions of subsurface flow through porous media

Limiting the injection rate to restrict the pressure below a threshold at a critical location can be an important goal of simulations that model the subsurface pressure between injection and extraction wells. The pressure is approximated by the solution of Darcy’s partial differential equation for a given permeability field. The subsurface permeability is modeled as a random field since it is known only up to statistical properties. This induces uncertainty in the computed pressure. Solving the partial differential equation for an ensemble of random permeability simulations enables estimating a probability distribution for the pressure at the critical location. These simulations are computationally expensive, and practitioners often need rapid online guidance for real-time pressure management. An ensemble of numerical partial differential equation solutions is used to construct a Gaussian process regression model that can quickly predict the pressure at the critical location as a function of the extraction rate and permeability realization. The Gaussian process surrogate analyzes the ensemble of numerical pressure solutions at the critical location as noisy observations of the true pressure solution, enabling robust inference using the conditional Gaussian process distribution.Our first novel contribution is to identify a sampling methodology for the random environment and matching kernel technology for which fitting the Gaussian process regression model scales as O(nlogn) instead of the typical O(n3) rate in the number of samples n used to fit the surrogate. The surrogate model allows almost instantaneous predictions for the pressure at the critical location as a function of the extraction rate and permeability realization. Our second contribution is a novel algorithm to calibrate the uncertainty in the surrogate model to the discrepancy between the true pressure solution of Darcy’s equation and the numerical solution. Although our method is derived for building a surrogate for the solution of Darcy’s equation with a random permeability field, the framework broadly applies to solutions of other partial differential equations with random coefficients.

Estimating heterogeneous effects: Applications to labor economics

A growing number of applications involve settings where, in order to infer heterogeneous effects, a researcher compares various units. Examples of research designs include children moving between different neighborhoods, workers moving between firms, patients migrating from one city to another, and banks offering loans to different firms. We present a unified framework for these settings, based on a linear model with normal random coefficients and normal errors. Using the model, we discuss how to recover the mean and dispersion of effects, other features of their distribution, and to construct predictors of the effects. We provide moment conditions on the model’s parameters, and outline various estimation strategies. A main objective of the paper is to clarify some of the underlying assumptions by highlighting their economic content, and to discuss and inform some of the key practical choices.

Variability in Causal Effects and Noncompliance in a Multisite Trial: A Bivariate Hierarchical Generalized Random Coefficients Model for a Binary Outcome.

Within each of 170 physicians, patients were randomized to access e-assist, an online program that aimed to increase colorectal cancer screening (CRCS), or control. Compliance was partial: of the experimental patients accessed e-assist while no controls were provided the access. Of interest are the average causal effect of assignment to treatment and the complier average causal effect as well as the variation of these causal effects across physicians. Each physician generates probabilities of screening for experimental compliers (experimental patients who accessed e-assist), control compliers (controls who would have accessed e-assist had they been assigned to e-assist), and never takers (patients who would have avoided e-assist no matter what). Estimating physician-specific probabilities jointly over physicians poses novel challenges. We address these challenges by maximum likelihood, factoring a "complete-data likelihood" uniquely into the conditional distribution of screening and partially observed compliance given random effects and the distribution of random effects. We marginalize this likelihood using adaptive Gauss-Hermite quadrature. The approach is doubly iterative in that the conditional distribution defies analytic evaluation. Because the small sample size per physician constrains estimability of multiple random effects, we reduce their dimensionality using a shared random effects model having a factor analytic structure. We assess estimators and recommend sample sizes to produce reasonably accurate and precise estimates by simulation, and analyze data from a trial of a CRCS intervention.

Simulation of a temporally dynamic atmospheric wavefront using the modified von-Kármán spectrum and Zernike-based method

In the simulation of the atmospheric wavefront, the Kolmogorov spectrum facilitates the mathematical treatments, but this model gives inaccurate predictions in cases where the effects of outer- and inner-scale can not be neglected. Therefore, we derive the covariance formula of Zernike coefficients based on the modified von-Kármán turbulence spectrum to improve the Zernike-based method. Further, using the frozen flow hypothesis, the temporal power spectra of Zernike coefficients are derived, and the effects of wind velocity and direction, as well as the outer and inner scales, are also taken into account. Then, we introduce a method for generating the temporally dynamic atmospheric wavefront using the inverse Fourier transform to draw random time-evolving coefficients from the temporal spectra. We evaluate the performance on simulated atmospheric wavefront, and the results confirm that accurate spatial and temporal statistics are obtained.

Information Theoretic Evaluation of Raccoon's Side-Channel Leakage

Raccoon is a lattice-based scheme submitted to the NIST 2022 call for additional post-quantum signatures. One of its main selling points is that its design is intrinsically easy to mask against side-channel attacks. So far, Raccoon's physical security guarantees were only stated in the abstract probing model. In this paper, we discuss how these probing security results translate into guarantees in more realistic leakage models. We also highlight that this translation differs from what is usually observed (e.g., in symmetric cryptography), due to the algebraic structure of Raccoon's operations. For this purpose, we perform an in-depth information theoretic evaluation of Raccoon's most innovative part, namely the AddRepNoise function which allows generating its arithmetic shares on-the-fly. Our results are twofold. First, we show that the resulting shares do not enforce a statistical security order (i.e., the need for the side-channel adversary to estimate higher-order moments of the leakage distribution), as usually expected when masking. Second, we observe that the first-order leakage on the (large) random coefficients manipulated by Raccoon cannot be efficiently turned into leakage on the (smaller) coefficients of its long-term secret. Concretely, our information theoretic evaluations for relevant leakage functions also suggest that Raccoon's masked implementations can ensure high security with less shares than suggested by a conservative analysis in the probing model.

B-027 CLSI-Based Result Precision on the cobas ISE neo and cobas c 703 Analytical Units Across Two Sites in Europe

Abstract Background Novel cobas® pro integrated solutions analytical units, the cobas® ISE neo and cobas c 703 analytical units (all Roche Diagnostics International Ltd, Rotkreuz, Switzerland), were assessed at two sites in Europe (Aalst, Belgium and Heidelberg, Germany). During the system design validation, precision was evaluated under intended use conditions in a routine simulation setting. Methods Quality control (QC) materials at two analyte concentrations were measured per site for selected assays (ion selective electrodes [ISEs], general chemistries and specific proteins, and immunochemistry assays); analyte recovery per QC was closely monitored. Precision was evaluated at both study sites based on Clinical and Laboratory Standards Institute EP05-A3 guidelines over 21 days for 53 applications representing the entire assay menu: six ISE serum/plasma and urine applications; 40 clinical chemistry serum/plasma and urine applications (four were measured using cobas c 503 at one site only); and seven immunochemistry serum/plasma applications. Reproducibility under various stressed routine-like conditions, including provocations, was tested using routine simulation imprecision experiments. To test the interaction of new and existing analytical units under routine conditions, two test configurations were used: the first configuration comprised one cobas ISE neo, two cobas c 703, and one cobas e 801 analytical units; and the second configuration comprised one cobas ISE neo, one cobas c 703, one cobas c 503, and one cobas e 801 analytical units. The precision of batch type measurements was compared with the precision under routine simulated random access conditions. Random access coefficients of variance (CVs) that exceeded 1.5× reference batch CVs, or single measurements deviating by >10% from the batch mean, triggered in-depth analysis of the system components that contributed to the result. CVs for intermediate precision comprised measurements from ≤4 cobas c 703 reagent disks, dependent on the system set-up and application assignment. CVs for reproducibility were calculated across four ISE measuring units, ≤3 cobas c 703 reagent disks, or ≤4 cobas e 801 measuring cells. Results Using four analytical units, n=511 CVs for repeatability, intermediate precision, and reproducibility were calculated. For repeatability, the median CV for ISE (n=24) was 0.8%, for clinical chemistry (cobas c 703: n=146; cobas c 503: n=8) was 0.9%, and for immunochemistry (n=28) was 1.0%. For intermediate precision, the median CV for ISE (n=24) was 1.1%, for clinical chemistry (cobas c 703: n=146; cobas c 503: n=8) was 1.2%, and for immunochemistry (n=28) was 1.5%. For reproducibility, the median CV for ISE (n=12) was 0.9%, for clinical chemistry (cobas c 703: n=73) was 1.6%, and for immunochemistry (n=14) was 1.7%. Conclusions The novel cobas ISE neo and cobas c 703 analytical units demonstrated robust precision, comparable to the existing cobas pro integrated solutions analytical units, over a prolonged period.

B-028 Result Precision on the New cobas pro integrated solutions cobas ISE neo and cobas c 703 Analytical Units Under Routine-Like Conditions Across Two Sites in Europe

Abstract Background The precision of the new cobas® pro integrated solutions cobas® ISE neo and cobas c 703 analytical units (Roche Diagnostics International Ltd, Rotkreuz, Switzerland) was measured under intended use conditions in a routine simulation setting. Methods The interaction of hardware, software, assays, and samples was tested at two sites (Aalst, Belgium and Heidelberg, Germany) between October and December 2023. Reproducibility under various stressed routine-like conditions, including provocations, was tested using routine simulation imprecision experiments. To test the interaction of new and existing analytical units under routine conditions, test configurations including commercially available cobas c 503 and cobas e 801 analytical units were used. The first configuration comprised one cobas ISE neo analytical unit, two cobas c 703 analytical units, and one cobas e 801 analytical unit. The second configuration comprised one cobas ISE neo analytical unit, one cobas c 703 analytical unit, one cobas c 503 analytical unit, and one cobas e 801 analytical unit. The precision of batch type measurements was compared with the precision under routine simulated random access conditions. Random access coefficients of variance (CVs) that exceeded 1.5× reference batch CVs, or single measurements deviating by >10% from the batch mean, triggered in-depth analysis of the system components that contributed to the result. Results Overall, 56 applications were measured during the reference and random parts of seven routine simulation imprecision runs: six ion selective electrodes (ISE; three serum, three urine), 35 clinical chemistry applications on cobas c 703 (27 serum, eight urine), seven clinical chemistry applications on cobas c 503 (five serum, one urine, one whole blood) and eight immunochemistry (serum) applications. Of six ISE (serum and urine) applications, the mean reference and random CVs were 0.8% and 0.9%, respectively. Of 42 clinical chemistry (serum, urine, and whole blood) applications, the mean reference CV was 0.9% and the mean random CV was 1.0%. Of eight immunochemistry (serum only) applications, mean reference and random CVs were 1.1% and 1.4%, respectively. Conclusions The findings from this study demonstrate that the new cobas ISE neo and cobas c 703 analytical units show consistent results over a prolonged time period, with most CVs in this setting below 0.9%. These results also show the successful integration of the new analytical units into the cobas pro integrated solutions, with seamless interaction of existing and analytical units under stressed conditions.

Coping with intensive training demands: a longitudinal investigation of the relationships between appraisal, emotion, coping effectiveness and engagement among elite fencers

ABSTRACT Training is, by essence, very challenging, both physically and mentally. The way athletes stay fully engaged in their intensive training programme and achieve their performance goals may depend on their capacity to deal effectively with such daily training demands. This study therefore aimed to further the understanding of the relationships between appraisal, emotion, coping effectiveness and engagement among elite fencers in relation to the demands of intensive training over the course of a competitive season. Twenty-three elite fencers (15 women, 8 men; age: 23.04 ± 3.54 years; fencing experience: 15.04 ± 3.74 years) completed an online questionnaire once a week, measuring perceived effort, perceived difficulty, pleasant and unpleasant emotions, athlete engagement, and coping effectiveness (i.e., 36 time-points). The results of the random coefficient regression models revealed the mediating role played by emotions in the dynamic relationships (a) between perceived difficulty and coping effectiveness (pleasant emotions: Sobel Test = −2.39, p < .05; unpleasant emotions: Sobel Test = −2.47, p < .05), and (b) between perceived effort and coping effectiveness (pleasant emotions: Sobel Test = 2.15, p < .05). Results also highlighted the partial mediating role played by coping effectiveness in the dynamic relationships (a) between perceived effort and engagement (Sobel Test = 2.27, p < .05), and (b) between perceived difficulty and engagement (Sobel Test = −2.16, p < .05). This study underlines the need to consider subjective appraisal during training over the course of a competitive season and its influence on psychological adaptation and engagement in intensive training settings.

School-level variation in children's moderate to vigorous intensity physical activity before and after COVID-19: a multilevel model analysis.

Schools play a crucial role in facilitating physical activity among children, but the COVID-19 pandemic has affected both children's physical activity and the school environment. It is essential to understand between-school differences in children's physical activity post lockdown, to determine if and how the role of schools has changed. Active-6 is a natural experiment comparing postlockdown accelerometer-estimated physical activity to a pre-COVID-19 comparator group. Accelerometer and individual data were collected on 1296 children aged 10-11 pre-COVID-19 (2017-8), with school characteristics collected from the 50 schools they attended. Post lockdown, we collected accelerometer, individual and school data from 393 children in 23 of the same schools and 436 children in 27 of the same schools in 2021 (Wave 1) and 2022 (Wave 2), respectively. Sources of variation (between-school, between-pupil and within-pupil) in child weekday moderate to vigorous physical activity at each wave were modelled using linear mixed-effects models with school-level wave random coefficients. We extended the model to estimate the proportion of between-school variation explained by school policy, curriculum and physical environment factors and school-aggregated pupil characteristics. We also explored the extent to which postlockdown differences in moderate to vigorous physical activity were mediated by individual or school factors. Between-school variation comprised 13% of the total variation pre-COVID-19, 7% in Wave 1 and 13% in Wave 2. School factors associated with moderate to vigorous physical activity were the following: whether physical education was compromised due to space (often: 9 minutes lower moderate to vigorous physical activity; sometimes: 5.4 minutes lower); high after-school club attendance (7 minutes higher moderate to vigorous physical activity for each additional club attended on average in the school); cycle training policy (4 minutes higher moderate to vigorous physical activity); and higher prevalence of active travel (1 minute higher moderate to vigorous physical activity for each 10% point increase in prevalence). These factors explained 22% of the between-school variation pre-COVID-19, and 72% at Wave 2. The relative importance changed, with cycle training policy and active travel being the most important pre-COVID-19 and cycle training policy, active after-school clubs and compromised physical education space most important in Wave 2. No factors were found to mediate the postlockdown differences in moderate to vigorous physical activity, except compromised physical education space, which had a suppressor effect in Wave 2. Only 27 of the initial 50 schools participated post lockdown, limiting our ability to make comparisons across waves. Sample sizes were additionally affected by missing data for some variables. While schools continue to play an important role in facilitating children's physical activity, the factors that contribute to this have changed post-COVID-19, with cycle training, active after-school clubs and ensuring physical education is prioritised even when space is limited now explaining nearly three-quarters of the between-school variation in children's moderate to vigorous physical activity. School-level interventions that focus on these areas, and policies that support them, may offer the potential to increase children's physical activity. This article presents independent research funded by the National Institute for Health and Care Research (NIHR) Public Health Research programme as award number NIHR131847.

Uncertainty quantification analysis of bifurcations of the Allen–Cahn equation with random coefficients

In this work we consider the Allen–Cahn equation, a prototypical model problem in nonlinear dynamics that exhibits bifurcations corresponding to variations of a deterministic bifurcation parameter. Going beyond the state-of-the-art, we introduce a random coefficient in the linear reaction part of the equation, thereby accounting for random, spatially-heterogeneous effects. Importantly, we assume a spatially constant, deterministic mean value of the random coefficient. We show that this mean value is in fact a bifurcation parameter in the Allen–Cahn equation with random coefficients. Moreover, we show that the bifurcation points and bifurcation curves become random objects. We consider two distinct modelling situations: (i) for a spatially homogeneous coefficient we derive analytical expressions for the distribution of the bifurcation points and show that the bifurcation curves are random shifts of a fixed reference curve; (ii) for a spatially heterogeneous coefficient we employ a generalized polynomial chaos expansion to approximate the statistical properties of the random bifurcation points and bifurcation curves. We present numerical examples in 1D physical space, where we combine the popular software package Continuation Core and Toolboxes (COCO) for numerical continuation and the Sparse Grids Matlab Kit for the polynomial chaos expansion. Our exposition addresses both, dynamical systems and uncertainty quantification, highlighting how analytical and numerical tools from both areas can be combined efficiently for the challenging uncertainty quantification analysis of bifurcations in random differential equations.

Sum rate optimization in STAR-RIS assisted multiuser massive MIMO-OFDM VLC systems

Visible Light Communication (VLC) offers a promising solution for future networks, leveraging existing lighting infrastructure in indoor environments. However, VLC requires a direct line of sight to function which can be limiting. Reconfigurable Intelligent Surface (RIS) is a new technology that can bend light and radio waves, addressing this limitation in VLC. RIS come in three types: passive which reflects signals, active that boosts and reflects signals, and Simultaneous Transmission and Reflection (STAR)-RIS which can both reflect and transmit signals simultaneously. STAR-RIS offers the most control over the signal. In this paper, we propose a new multi-user VLC system with a massive Multiple-Input, Multiple-Output Orthogonal Frequency-Division Multiplexing (MIMO-OFDM) architecture, leveraging STAR-RIS to optimize data rates and improve coverage, particularly in non-line-of-sight scenarios. We formulate a system model and solve a convex optimization problem to determine the optimal transmission and reflection coefficients for STAR-RIS elements, aiming to maximize the total sum rate for all users. By employing maximum ratio transmission precoding, we minimize interference among users and demonstrate significant performance gains. Simulation results show that our proposed energy splitting-based STAR-RIS configuration outperforms traditional mode selection and time switching approaches with fixed or random coefficients, yielding substantial improvements in data rates and user experience. This study offers the first detailed exploration of STAR-RIS in VLC systems, highlighting its potential for future high-speed, multi-user communication networks. Our findings set the stage for further research into optimizing VLC systems using STAR-RIS, particularly in complex environments with non-line-of-sight users and massive MIMO-OFDM configurations.

The local solubility for homogeneous polynomials with random coefficients over thin sets

AbstractLet and be natural numbers greater or equal to 2. Let be a homogeneous polynomial in variables of degree with integer coefficients , where denotes the inner product, and denotes the Veronese embedding with . Consider a variety in , defined by . In this paper, we examine a set of integer vectors , defined by where is a nonsingular form in variables of degree with for some constant depending at most on and . Suppose has a nontrivial integer solution. We confirm that the proportion of integer vectors in , whose associated equation is everywhere locally soluble, converges to a constant as . Moreover, for each place of , if there exists a nonzero such that and the variety in admits a smooth ‐point, the constant is positive.

Natural gradient hybrid variational inference with application to deep mixed models

Stochastic models with global parameters and latent variables are common, and for which variational inference (VI) is popular. However, existing methods are often either slow or inaccurate in high dimensions. We suggest a fast and accurate VI method for this case that employs a well-defined natural gradient variational optimization that targets the joint posterior of the global parameters and latent variables. It is a hybrid method, where at each step the global parameters are updated using the natural gradient and the latent variables are generated from their conditional posterior. A fast to compute expression for the Tikhonov damped Fisher information matrix is used, along with the re-parameterization trick, to provide a stable natural gradient. We apply the approach to deep mixed models, which are an emerging class of Bayesian neural networks with random output layer coefficients to allow for heterogeneity. A range of simulations show that using the natural gradient is substantially more efficient than using the ordinary gradient, and that the approach is faster and more accurate than two cutting-edge natural gradient VI methods. In a financial application we show that accounting for industry level heterogeneity using the deep mixed model improves the accuracy of asset pricing models. MATLAB code to implement the method and replicate the results can be found at https://github.com/WeibenZhang07/NG-HVI

Psychiatric predictors of quality of life in Parkinson's disease: A three-year longitudinal study

IntroductionParkinson's disease (PD) is associated with worsened quality of life (QOL) over time. Few longitudinal studies exist investigating the relationship of psychiatric comorbidities with QOL in people with PD (PwP). We sought to determine specific psychiatric symptoms associated with decreasing QOL in PwP over time. MethodsWe recruited PwP without dementia from a movement disorders clinic at an academic medical center. Participants were evaluated annually with motor and neuropsychological assessments at each visit. QOL was measured using the Parkinson's Disease Questionnaire-39 (PDQ-39). We assessed psychiatric symptoms, including depression (Beck Depression Inventory II, BDI-II), anxiety (Beck Anxiety Index, BAI), and apathy (Apathy Scale). Psychosis and impulse control disorders (ICDs) were recorded as present or absent. Using random coefficient regression, we analyzed psychiatric features associated with worsened QOL in PwP over three years. ResultsFrom the 105 participants enrolled at baseline, 67 completed three years of follow up. Mean PDQ-39 scores increased from 16.0 at baseline to 19.8 at year three. In multivariate analysis, higher BDI-II scores, BAI scores, and apathy scores were uniquely associated with worsened QOL over time (p < 0.001 for all measures), while presence of ICDs (p = 0.18) or psychosis (p = 0.10) were not. Changes in the BAI score and the BDI-II score exerted similar effects on the overall PDQ-39 score. ConclusionDepression, anxiety, and apathy are all associated with worsening quality of life over time in PwP, while presence of ICDs and psychosis are not. Treatment of these symptoms may lead to improved QOL in PwP.

Identification of a Triangular Random Coefficient Model using a Correction Function

Abstract Previously, identification of triangular random coefficient models required a restriction on the dimension of the first stage heterogeneity or independence assumptions across the different sources of the heterogeneity. This note proposes a new identification strategy that does not rely on either of these restrictions but rather assumes conditional means have a conditional linear projection representation in order to construct “correction functions” to address endogeneity and gain identification of the average partial effect. This identification strategy allows for both continuous and discrete instruments. Finally, the proposed identification method is illustrated in estimating the returns to education.

Limit Theorem for a Rough Differential Equation with a Negative Long-Range Random Coefficient

Limit Theorem for a Rough Differential Equation with a Negative Long-Range Random Coefficient

Power-law localization in one-dimensional systems with nonlinear disorder under fixed input conditions

We conduct a numerical investigation into wave propagation and localization in one-dimensional lattices subject to nonlinear disorder, focusing on cases with fixed input conditions. Utilizing a discrete nonlinear Schrödinger equation with Kerr-type nonlinearity and a random coefficient, we compute the averages and variances of the transmittance, T, and its logarithm, as functions of the system size L, while maintaining constant intensity for the incident wave. In cases of purely nonlinear disorder, we observe power-law localization characterized by 〈T〉∝L−γa and 〈lnT〉≈−γglnL for sufficiently large L. At low input intensities, a transition from exponential to power-law decay in 〈T〉 occurs as L increases. The exponents γa and γg are nearly identical, converging to approximately 0.5 as the strength of the nonlinear disorder, β, increases. Additionally, the variance of T decays according to a power law with an exponent close to 1, and the variance of lnT approaches a small constant as L increases. These findings are consistent with an underlying log-normal distribution of T and suggest that wave propagation behavior becomes nearly deterministic as the system size increases. When both linear and nonlinear disorders are present, we observe a transition from power-law to exponential decay in transmittance with increasing L when the strength of linear disorder, V, is less than β. As V increases, the region exhibiting power-law localization diminishes and eventually disappears when V exceeds β, leading to standard Anderson localization.