Nonlinear Prediction Research Articles

Integrating the A2DS2 Score with 24-Hour ASPECTS and red cell distribution width for enhanced prediction of stroke-associated pneumonia following intravenous thrombolysis: model development and internal validation

IntroductionStroke-associated pneumonia (SAP) is a major cause of mortality during the acute phase of stroke. The A2DS2 score is widely used to predict SAP risk but does not include 24-h non-contrast computed tomography-Alberta Stroke Program Early CT Score (NCCT-ASPECTS) or red cell distribution width (RDW). We aim to evaluate the added prognostic value of incorporating 24-h NCCT-ASPECTS and RDW into the A2DS2 score and to develop a novel prediction model for SAP following thrombolysis.MethodsThis retrospective cohort study included thrombolyzed AIS patients at Saraburi Hospital, Thailand. The combined A2DS2-MFP model incorporated 24-h NCCT-ASPECTS and RDW, along with non-linear continuous predictors, using multivariable fractional polynomial (MFP) regression. Predictive performance was evaluated using the area under the receiver operating characteristic curve (AuROC), calibration plots, and decision curve analysis (DCA), comparing it with the traditional A2DS2 model and a model with continuous predictors. The goodness of fit for logistic regression models in relation to the observed data was determined through the Hosmer–Lemeshow method, and the accuracy of the probability predictions was examined using a calibration curve. Internal validation was performed using a bootstrapping approach. The predicted probability equation obtained from the final model after optimism correction was developed into a web-based application for predicting the risk of SAP, using PHP and JavaScript.ResultsOf 345 AIS patients, 20.3% developed SAP. The combined A2DS2-MFP model demonstrated excellent discriminative performance (AuROC: 0.917) compared to the traditional A2DS2 model (AuROC: 0.868) and the model with continuous predictors (AuROC: 0.888). Both the calibration curve and the Hosmer–Lemeshow test indicated that the predicted probabilities and observed frequencies were in acceptable agreement. Incorporating 24-h NCCT-ASPECTS and RDW significantly improved risk prediction and clinical utility, as shown by improved reclassification indices and DCA. The model was internally validated with a C-statistic of 0.912, confirming its robustness.ConclusionsThe combined A2DS2-MFP calculation showed superior performance, enabling early SAP detection and improving survival outcomes. This novel model offers a practical tool for resource-limited settings, supporting better SAP risk stratification and clinical management.

Enhanced RTP method for nonlinear statistical prediction of ship structural response

Abstract In this study, the Enhanced RAO-based Translation Process (ERTP) method, which is an enhanced version of the RAO-based Translation Process (RTP) method proposed in the previous study, is newly proposed. The RTP method approximates the Extreme Value Distribution (EVD) in arbitrary irregular waves from the nonlinear response in regular waves based on the idea of the transformation process, in which the nonlinear process is regarded as a transformation of a linear process. However, it was found that the RTP method is inaccurate when the response spectrum has multiple peaks, which is often the case for stresses in ship structures in waves. To solve this problem, the author developed the ERTP method, which defines a nonlinear transformation based on the L p-norm of the response spectrum. Direct Load and Structure Analysis considering the nonlinearity of wave loads was conducted and confirmed that the proposed method can reasonably estimate the EVD even for structural elements with a multi-peaked response spectrum with respect to the wave direction.

Practical Nonlinear Prediction Model for Uneven Settlement of an Enlarged Levee Crest

Practical Nonlinear Prediction Model for Uneven Settlement of an Enlarged Levee Crest

Design of nonlinear rubber vibration isolators with low resonant frequency and high load bearing capacity

Designing a rubber vibration isolator to simultaneously meet the requirements on both low-frequency vibration isolation and high load carrying capacity has always been a major challenge. To tackle this problem, in this paper, a design strategy capable of effectively balancing these two effects to guide the design of such isolators is proposed. The rubber isolator static and dynamic behavior prediction models suitable for engineering applications are first established, where Yeoh model is adopted for rubber strong nonlinear feature prediction. Experiments are carried out to determine the material constants in Yeoh model and dynamic-to-static ratio of the rubber material. Combing with Multi-Objective Genetic Algorithm, a design strategy is proposed. The conducted experiments demonstrate that based on the proposed numerical model and design procedure, rubber isolators with enough load bearing capability and low-frequency vibration isolation performance can be successfully designed.

Lossless Image Compression Using Context-Dependent Linear Prediction Based on Mean Absolute Error Minimization.

This paper presents a method for lossless compression of images with fast decoding time and the option to select encoder parameters for individual image characteristics to increase compression efficiency. The data modeling stage was based on linear and nonlinear prediction, which was complemented by a simple block for removing the context-dependent constant component. The prediction was based on the Iterative Reweighted Least Squares (IRLS) method which allowed the minimization of mean absolute error. Two-stage compression was used to encode prediction errors: an adaptive Golomb and a binary arithmetic coding. High compression efficiency was achieved by using an author's context-switching algorithm, which allows several prediction models tailored to the individual characteristics of each image area. In addition, an analysis of the impact of individual encoder parameters on efficiency and encoding time was conducted, and the efficiency of the proposed solution was shown against competing solutions, showing a 9.1% improvement in the bit average of files for the entire test base compared to JPEG-LS.

Deep interval type-2 generalized fuzzy hyperbolic tangent system for nonlinear regression prediction

Recently, due to the rapid rise of artificial intelligence (AI), considerable progress has been made in the field of nonlinear regression prediction. However, many existing methods suffer from the issues of rule and parameter explosion and poor accuracy, particularly for high-dimensional data with uncertainty. To address these limitations, this paper proposes a deep interval type-2 generalized fuzzy hyperbolic tangent system (DIT2GFHS). First, a novel neural network-based implementation of the interval type-2 fuzzy generalized fuzzy hyperbolic tangent system (IT2GFHS) is introduced to improve the efficiency of system parameter updates and optimization. Then, using a hierarchical and block-based framework, multiple IT2GFHSs are stacked layer by layer from bottom to top to construct the DIT2GFHS, with each layer’s fuzzy subsystems being independent of the others. Additionally, DIT2GFHS incorporates optimization algorithms and the Adam optimizer for training, thereby avoiding the tedious manual parameter tuning process. The detailed analysis of the construction manner and internal mechanisms for DIT2GFHS indicates that it features a reduced number of parameters, a transparent and clear structure, strong capability in handling uncertainty, and favorable accuracy. Notably, the small number of parameters and the explicit structure reduce computational and hardware burdens while maintaining interpretability. Finally, extensive experimental studies on both relatively low-dimensional and high-dimensional datasets are conducted. The results demonstrate that DIT2GFHS achieves excellent performance with fewer parameters compared to many deep-structured models, including deep fuzzy systems and deep learning models. This highlights its potential impact in addressing practical nonlinear regression problems.

Multi-Objective Optimization of Friction Stir Processing Tool with Composite Material Parameters

Compared to base aluminum alloys, the surface composites of aluminum alloys are more widely used in the automotive, aerospace, and other industries. The ability to yield enhanced physical properties and a smoother microstructure has made friction stir processing (FSP) the method of choice for developing aluminum-based surface composites in recent times. In this work, the Goal Programming (GP) approach is adopted for the Multi-Objective Optimization of FSP processes with three Artificial Intelligence (AI)-based metaheuristics, viz., Artificial Bee Colony (ABC), Particle Swarm Optimization (PSO), and Teaching–Learning-Based Optimization (TLBO). Three parameters, copper percentage (Cu%), graphite percentage (Gr%), and number of passes, are considered, and multi-factor non-linear regression prediction models are developed for the three responses, Tool Vibrations, Power Consumption, and Cutting Force. The TLBO algorithm outperformed the ABC and PSO algorithms in terms of solution quality and robustness, yielding significant improvements in tool life. The results with TLBO were improved by 20% and 14% compared to the PSO and ABC algorithms, respectively. This proves that the TLBO algorithm performed better compared with the ABC and PSO algorithms. However, the computation time required for the TLBO algorithm is higher compared to the ABC and PSO algorithms. This work has opened new avenues towards applying the GP approach for the Multi-Objective Optimization of FSP tools with composite parameters. This is a significant step towards toll life improvement for the FSP of composite alloys, contributing to sustainable manufacturing.

QC-ODKLA: Quantized and Communication-Censored Online Decentralized Kernel Learning via Linearized ADMM.

This article focuses on online kernel learning over a decentralized network. Each agent in the network receives online streaming data and collaboratively learns a globally optimal nonlinear prediction function in the reproducing kernel Hilbert space (RKHS). To overcome the curse of dimensionality issue in traditional online kernel learning, we utilize random feature (RF) mapping to convert the nonparametric kernel learning problem into a fixed-length parametric one in the RF space. We then propose a novel learning framework, named online decentralized kernel learning via linearized ADMM (ODKLA), to efficiently solve the online decentralized kernel learning problem. To enhance communication efficiency, we introduce quantization and censoring strategies in the communication stage, resulting in the quantized and communication-censored ODKLA (QC-ODKLA) algorithm. We theoretically prove that both ODKLA and QC-ODKLA can achieve the optimal sublinear regret O(√T) over T time slots. Through numerical experiments, we evaluate the learning effectiveness, communication efficiency, and computation efficiency of the proposed methods.

ETTrack: enhanced temporal motion predictor for multi-object tracking

Many Multi-Object Tracking (MOT) approaches exploit motion information to associate all the detected objects across frames. However, traditional tracking-by-detection (TBD) methods, relying on the Kalman Filter, often work well in linear motion scenarios but struggle to accurately predict the locations of objects undergoing complex and non-linear movements. To overcome these limitations, we propose ETTrack, a novel motion prediction method with an enhanced temporal motion predictor. Specifically, the motion predictor integrates a transformer model and a Temporal Convolutional Network (TCN) to capture both long-term and short-term motion patterns, and it predicts the future motion of individual objects based on the historical motion information. Additionally, we propose a novel Momentum Correction Loss function that provides additional information regarding the motion direction of objects during training. This allows the motion predictor to rapidly adapt to sudden motion variations and more accurately predict future motion. Our experimental results demonstrate that ETTrack achieves a competitive performance compared with state-of-the-art trackers on DanceTrack and SportsMOT, scoring 56.4%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\%$$\\end{document} and 74.4%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\%$$\\end{document} in HOTA metrics, respectively. Our work provides a robust solution for MOT in complex dynamic environments, which enhances the non-linear motion prediction capabilities of tracking algorithms.

Serum total calcium levels as a non-linear predictor of in-hospital mortality in heart failure patients: insights from a retrospective cohort study

BackgroundCalcium is pivotal in the regulation of bodily homeostasis, with numerous studies highlighting its link to cardiovascular disease in the adult population. However, the relationship between serum calcium levels and the prognosis of heart failure (HF) patients is not clear. This study explored the association between serum total calcium (STC) and in-hospital mortality in patients with HF.MethodsClinical data of 1,176 patients with HF were obtained from the Multiparametric Intelligent Monitoring in Intensive Care III (MIMIC-III) database. The patients were categorized into STC quartiles, and baseline characteristics were comprehensively analyzed. Univariate and multivariate analyses were employed to identify factors associated with in-hospital mortality. To explore the non-linear relationship between STC and mortality, a two-piecewise linear regression model was applied. Subgroup analyses were conducted to identify potential confounding variables.ResultsIn this cohort, 159 (13.53%) patients experienced in-hospital mortality. Significant differences in various parameters were observed among STC quartiles. Univariate analysis identified numerous factors associated with mortality. Multivariate analysis confirmed STC as an independent predictor of in-hospital mortality, with a negative association persisting even after adjusting for confounding factors (odds ratio [OR]: 0.49, 95%CI: 0.32–0.76; P = 0.0016). Non-linear analysis revealed an inflection point at 8.41 mg/dL, below which the risk of in-hospital death significantly increased (OR: 0.26, 95%CI: 0.12–0.55; P = 0.0005). Subgroup analyses indicated a pronounced inverse association in patients without atrial fibrillation or chronic obstructive pulmonary disease, as well as those with a left ventricular ejection fraction ≤ 50%.ConclusionThis study identified STC as an independent predictor of in-hospital mortality in HF patients, with a non-linear relationship.

Influence of Modal Decomposition Algorithms on Nonlinear Time Series Machine Learning Prediction Models in Engineering: A Case Study of Subway Tunnel Settlement

The settlement values of subway tunnels during the construction period exhibit significant nonlinear and spatial‒temporal variation characteristics. To overcome the problems of historical data interference and spatiotemporal characteristics in tunnel settlement prediction models, this paper proposes a tunnel settlement prediction method based on data decomposition, reconstruction, and optimization. First, the original data are optimized via the SSA, which has global optimization capability, high noise immunity, and high adaptivity. The original signal is subsequently decomposed into multiple subsignal sequences via a variational modal decomposition (VMD) algorithm combined with a rolling decomposition technique. Finally, the decomposed signals are fed into the machine learning model to construct a high-precision settlement prediction model based on rolling decomposition. The prediction accuracy of different models was analyzed via the measured settlement data during the construction period of the Beijing subway as an example. The results show that the prediction model with the integrated decomposition algorithm reduces the RMSE and MAE by 33% and 37%, respectively, which significantly improves the prediction accuracy and generalization ability of the neural network to meet the demand of practical engineering prediction and simultaneously enhances the risk warning ability of the model.

A self-driven ESN-DSS approach for effective COVID-19 time series prediction and modelling.

Since the outbreak of the COVID-19 epidemic, it has posed a great crisis to the health and economy of the world. The objective is to provide a simple deep-learning approach for predicting, modelling, and evaluating the time evolutions of the COVID-19 epidemic. The Dove Swarm Search (DSS) algorithm is integrated with the echo state network (ESN) to optimize the weight. The ESN-DSS model is constructed to predict the evolution of the COVID-19 time series. Specifically, the self-driven ESN-DSS is created to form a closed feedback loop by replacing the input with the output. The prediction results, which involve COVID-19 temporal evolutions of multiple countries worldwide, indicate the excellent prediction performances of our model compared with several artificial intelligence prediction methods from the literature (e.g., recurrent neural network, long short-term memory, gated recurrent units, variational auto encoder) at the same time scale. Moreover, the model parameters of the self-driven ESN-DSS are determined which acts as a significant impact on the prediction performance. As a result, the network parameters are adjusted to improve the prediction accuracy. The prediction results can be used as proposals to help governments and medical institutions formulate pertinent precautionary measures to prevent further spread. In addition, this study is not only limited to COVID-19 time series forecasting but also applicable to other nonlinear time series prediction problems.

Sampling error mitigation through spectrum smoothing: First experiments with ensemble transform Kalman filters and Lorenz models

In data assimilation, an ensemble provides a way to propagate the probability density of a system described by a nonlinear prediction model. Although a large ensemble size is required for statistical accuracy, the ensemble size is typically limited to a small number due to the computational cost of running the prediction model, which leads to a sampling error. Several methods, such as localization and inflation, exist to mitigate the sampling error, often requiring problem-dependent fine-tuning and design. This work introduces a nonintrusive sampling error mitigation method that modifies the ensemble to ensure a smooth turbulent spectrum. It turns out that the ensemble modification to satisfy the smooth spectrum leads to inhomogeneous localization and inflation, which apply spatially varying localization and inflation levels at different locations. The efficacy of the new idea is validated through a suite of stringent test regimes of the Lorenz 96 turbulent model.

Pore-throat structure, fractal characteristics and permeability prediction of tight sandstone: the Yanchang Formation, Southeast Ordos Basin

As a typical tight reservoir and an important site for unconventional hydrocarbon accumulation, the Chang 6 member of the Yanchang Formation is characterized by complex pore structures and strong heterogeneity. Analysing and characterizing the pore-throat structure is highly important for optimizing oil recovery processes. To clarify the nonhomogeneity and structural characteristics of the pore throats in the southeastern Ordos Basin, tight sandstone from the Chang 6 member was selected for analysis. Casting thin section (CTS), scanning electron microscopy (SEM), cathodoluminescence (CL), X-ray diffraction (XRD), and mercury intrusion capillary pressure (MICP) analyses were conducted. The results, revealed that intergranular pores, feldspar-dissolved pores, intergranular-dissolved pores, and microfractures were the predominant pore types found within the samples. By combining the results of MICP analysis with those of fractal theory, the pore–throat structure of each sample can be categorized into two types: large-scale and small-scale. Fractal theory was employed to characterize the complex and irregular pore–throat structure of the reservoir quantitatively. The average fractal dimension of large pores (D1) was 2.8094, whereas for small pores (D2), it was slightly lower than that of large pores, averaging 2.5325. These findings underscore that large-scale pore-throat structures are more complex and exhibit greater heterogeneity. Compared with those of large pores, the pore–throat structure parameters of small pores exhibit a more significant correlation with reservoir properties and fractal dimensions. Therefore, small pores are the primary contributors to the reservoir storage space and are key factors influencing the pore-throat structure of the Chang 6 tight sandstone. Based on the pore-throat radius and considering the influence of fractal characteristics on the pore structure, a nonlinear permeability prediction model was created using multiple regression analysis. Among these equations, the pore–throat radius corresponding to a mercury saturation of 40% (r40) emerged as the most effective predictor of permeability for tight sandstone. The investigation of micropore–throat structures in tight oil reservoirs holds practical significance for comprehending the distribution and accumulation of tight oil and guiding oil field development.

Bifurcation Analysis and Multiobjective Nonlinear Model Predictive Control of Sustainable Ecosystems

<i>Objective: </i>All optimal control work involving ecological models involves single objective optimization. In this work, we perform multiobjective nonlinear model predictive control (MNLMPC) in conjunction with bifurcation analysis on an ecosystem model. <i>Methods: </i>Bifurcation analysis was performed using the MATLAB software MATCONT while the multi-objective nonlinear model predictive control was performed by using the optimization language PYOMO. <i>Results: </i>Rigorous proof showing the existence of bifurcation (branch) points is presented along with computational validation. It is also demonstrated (both numerically and analytically) that the presence of the branch points was instrumental in obtaining the Utopia solution when the multiobjective nonlinear model prediction calculations were performed. <i>Conclusions: </i>The main conclusions of this work are that one can attain the utopia point in MNLMPC calculations because of the branch points that occur in the ecosystem model and the presence of the branch point can be proved analytically. The use of rigorous mathematics to enhance sustainability will be a significant step in encouraging sustainable development. The main practical implication of this work is that the strategies developed here can be used by all researchers involved in maximizing sustainability The future work will involve using these mathematical strategies to other ecosystem models and food chain models which will be a huge step in developing strategies to address problems involving nutrition.

Prediction of performance and turbulence in ITER burning plasmas via nonlinear gyrokinetic profile prediction

Abstract Burning plasma performance, transport, and the effect of hydrogen isotope (H, D, D-T fuel mix) on confinement has been predicted for ITER baseline scenario (IBS) conditions using nonlinear gyrokinetic profile predictions. Accelerated by surrogate modeling (Rodriguez-Fernandez et al 2022 Nucl. Fusion 62 076036), high fidelity, nonlinear gyrokinetic simulations performed with the CGYRO code (Candy et al 2016 J. Comput. Phys. 324 73), were used to predict profiles of Ti , Te , and ne while including the effects of alpha heating, auxiliary power (NBI + ECH), collisional energy exchange, and radiation losses inside of r / a = 0.9. Predicted profiles and resulting energy confinement are found to produce fusion power and gain that are approximately consistent with mission goals ( P fusion = 500 MW at Q = 10) for the baseline scenario and exhibit energy confinement that is within 1σ of the H-mode energy confinement scaling. The power of the surrogate modeling technique is demonstrated through the prediction of alternative ITER scenarios with reduced computational cost. These scenarios include conditions with maximized fusion gain and an investigation of potential resonant magnetic perturbation (RMP) effects on performance with a minimal number of gyrokinetic profile iterations required (3–6). These predictions highlight the stiff ITG nature of the core turbulence predicted in the ITER baseline and demonstrate that Q > 17 conditions may be accessible by reducing auxiliary input power while operating in IBS conditions. Prediction of full kinetic profiles allowed for the projection of hydrogen isotope effects around ITER baseline conditions. The gyrokinetic fuel ion species was varied from H, D, and 50/50 D-T and kinetic profiles were predicted. Results indicate that a weak or negligible isotope effect will be observed to arise from core turbulence in IBS conditions. The resulting energy confinement, turbulence, and density peaking, and the implications for ITER operations will be discussed.

Optimised Deep Learning for Time-Critical Load Forecasting Using LSTM and Modified Particle Swarm Optimisation

Short-term electric load forecasting is critical for power system planning and operations due to demand fluctuations driven by variable energy resources. While deep learning-based forecasting models have shown strong performance, time-sensitive applications require improvements in both accuracy and convergence speed. To address this, we propose a hybrid model that combines long short-term memory (LSTM) with a modified particle swarm optimisation (mPSO) algorithm. Although LSTM is effective for nonlinear time-series predictions, its computational complexity increases with parameter variations. To overcome this, mPSO is used for parameter tuning, ensuring accurate forecasting while avoiding local optima. Additionally, XGBoost and decision tree filtering algorithms are incorporated to reduce dimensionality and prevent overfitting. Unlike existing models that focus mainly on accuracy, our framework optimises accuracy, stability, and convergence rate simultaneously. The model was tested on real hourly load data from New South Wales and Victoria, significantly outperforming benchmark models such as ENN, LSTM, GA-LSTM, and PSO-LSTM. For NSW, the proposed model reduced MSE by 91.91%, RMSE by 94.89%, and MAPE by 74.29%. In VIC, MSE decreased by 91.33%, RMSE by 95.73%, and MAPE by 72.06%, showcasing superior performance across all metrics.

In situ stress prediction model in complex geology: A hybrid GA-ANN with nonlinear boundary condition

In regions characterized with great mining depths, complex topography, and intense geological activities, solely relying on lateral pressure coefficients or linear boundary conditions for predicting the in situ stress field of rock bodies can induce substantial deviations and limitations. This study focuses on a typical karst area in Southwest Guizhou, China as its research background. It employs a hybrid approach integrating machine learning, numerical simulations, and field experiments to develop an optimization algorithm for nonlinear prediction of the complex three-dimensional (3D) in situ stress fields. Through collecting and fitting analysis of in situ stress measurement data from the karst region, the distributions of in situ stresses with depth were identified with nonlinear boundary conditions. A prediction model for in situ stress was then established based on artificial neural network (ANN) and genetic algorithm (GA) approach, validated in the typical karst landscape mine, Jinfeng Gold Mine. The results demonstrate that the model's predictions align well with actual measurements, showcasing consistency and regularity. Specifically, the error between the predicted and actual values of the maximum horizontal principal stress was the smallest, with an absolute error 0.01-3 MPa and a relative error of 0.04-15.31%. This model accurately and effectively predicts in situ stresses in complex geological areas.

Longitudinal and Lateral Cooperative Control of Preview Intelligent Vehicles with Stabilization

Aiming at the highly nonlinear and coupled longitudinal and transverse motion control problems of intelligent vehicle path tracking control, a new method of preview MPC path tracking control is proposed by combining visual preview model and model predictive control theory. The method first locally linearizes the nonlinear prediction model and transforms the MPC optimization solution problem into a quadratic programming problem. The longitudinal vehicle speed and the preview distance are then treated as known in each control time domain, an exponential model is applied to characterize the longitudinal vehicle speed, and a transverse angular velocity-center of mass lateral deflection (r-β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r-\\beta $$\\end{document}) phase plane plot is applied to characterize the vehicle stability. Finally, proportional integral differential stability controller is designed to improve vehicle stability. Simulation and test results show that: Compared with the MPC tracking control method, the preview MPC control method proposed in this paper can optimize the vehicle tracking performance on different adhesion coefficient road surfaces and improve the tracking accuracy, and this enhancement effect is more obvious at high speeds.

Optimal asset allocation and nonlinear return predictability from the dividend-price ratio

AbstractWe study non-linear predictability of stock returns arising from the dividend-price ratio and its implications for asset allocation decisions. Using data from five countries — U.S., U.K., France, Germany and Japan — we find empirical evidence supporting non-linear and time-varying models for the equity risk premium. Building on this, we examine several model specifications that can account for non-linear return predictability, including Markov switching models, regression trees, random forests and neural networks. Although in-sample return regressions and portfolio allocation results support the use of non-linear predictability models, the out-of-sample evidence is notably weaker, highlighting the difficulty in exploiting non-linear predictability in real time.