Nonlinear Prediction Problems Research Articles

Binary ant colony optimization algorithm in learning random satisfiability logic for discrete hopfield neural network

This study introduced a novel ant colony optimization algorithm that implements the population selection strategy of the Estimation of Distribution Algorithm and a new pheromone updating formula. It aimed to optimize the performance of G-type random high-order satisfiability logic structures embedded in Discrete Hopfield Neural Networks, thereby enhancing the efficiency of the Hopfield Neural Network learning algorithm. Through comparative analysis with other metaheuristic algorithms, our model demonstrated superior performance in terms of global convergence, time complexity, and algorithm complexity. Additionally, we evaluated the learning phase, retrieval phase, and similarity analysis using various ratios of literals and clauses. It was shown that our proposed model exhibits stronger search ability compared to other metaheuristic algorithms and Exhaustive Search. Our model enhanced the efficiency of the learning phase, resulting in the number of global solutions accounting for 100 %, and significantly improved the global solution diversity. These advancements contributed to the efficiency of the model in convergence, rendering it applicable to a wide range of nonlinear classification and prediction problems.

Dynamical system prediction from sparse observations using deep neural networks with Voronoi tessellation and physics constraint

Despite the success of various methods in addressing the issue of spatial reconstruction of dynamical systems with sparse observations, spatio-temporal prediction for sparse fields remains a challenge. Existing Kriging-based frameworks for spatio-temporal sparse field prediction fail to meet the accuracy and inference time required for nonlinear dynamic prediction problems. In this paper, we introduce the Dynamical System Prediction from Sparse Observations using Voronoi Tessellation (DSOVT) framework, an innovative methodology based on Voronoi tessellation which combines convolutional encoder–decoder (CED) and long short-term memory (LSTM) and utilizing Convolutional Long Short-Term Memory (ConvLSTM). By integrating Voronoi tessellations with spatio-temporal deep learning models, DSOVT is adept at predicting dynamical systems with unstructured, sparse, and time-varying observations. CED-LSTM maps Voronoi tessellations into a low-dimensional representation for time series prediction, while ConvLSTM directly uses these tessellations in an end-to-end predictive model. Furthermore, we incorporate physics constraints during the training process for dynamical systems with explicit formulas. Compared to purely data-driven models, our physics-based approach enables the model to learn physical laws within explicitly formulated dynamics, thereby enhancing the robustness and accuracy of rolling forecasts. Numerical experiments on real sea surface data and shallow water systems clearly demonstrate our framework’s accuracy and computational efficiency with sparse and time-varying observations.

Bidirectional Long Short-term Neural Network Based on the Attention Mechanism of the Residual Neural Network (ResNet-BiLSTM-Attention) Predicts Porosity through Well Logging Parameters.

Porosity is an integral part of reservoir evaluation, but in the field of reservoir prediction, due to the complex nonlinear relationship between logging parameters and porosity, linear models cannot accurately predict porosity. Therefore, this paper uses machine learning methods that can better handle the relationship between nonlinear logging parameters and porosity to predict porosity. In this paper, logging data from Tarim Oilfield are selected for model testing, and there is a nonlinear relationship between these parameters and porosity. First, the data features of the logging parameters are extracted by the residual network, which uses the "hop connections" method to transform the original data closer to the target variable. In addition, the residual blocks inside the residual network use jump connections, which alleviates the gradient vanishing problem caused by increasing depth in deep neural networks. The dynamic nature of data would necessitate LSTM in the first place. Then, a bidirectional long short-term network (BiLSTM) is used to predict the porosity of the extracted logging data features. Among them, the BiLSTM is composed of two independent reverse LSTMs, which can better solve the nonlinear prediction problem. In order to further improve the accuracy of the model, this paper introduces an attention mechanism to learn by weighting each of the inputs in proportion to their impact on the porosity. The experimental results also show that the data features extracted by the residual neural network can be better used as the input of the BiLSTM model.

Spatiotemporal deep learning rainfall-runoff forecasting combined with remote sensing precipitation products in large scale basins

Rainfall-runoff modeling is a complex nonlinear spatiotemporal prediction problem. However, few studies have considered the spatial characteristics of rainfall-runoff relationship in runoff forecasts based on machine learning. With the emergence of high-resolution Satellite-based Precipitation Products (SPPs) and the continuous improvement of rainfall estimation accuracy, the shortcoming of sparse spatial information for in-situ rainfall monitoring has been made up. Therefore, this study developed a large scale spatiotemporal deep learning rainfall-runoff (SDLRR) forecasting model for hydrological stations in the upper Yangtze River, and evaluated the positive impact of utilizing spatial information of three SPPs on reducing errors of runoff forecasts. The adopted remote sensing precipitation products are bias-corrected Climate Hazards Group InfraRed Precipitation with Station data (CHIRPS), Integrated Multi-satellite Retrievals for Global Precipitation Measurement data (IMERG) and Tropical Rainfall Measuring Mission Multi-satellite Precipitation Analysis data (TMPA). For runoff forecasting at the Luoduxi (LDX) hydrological station, compared to regular Long Short Term Memory Network (LSTM) model, the proposed SDLRR model that utilizing IMERG data as precipitation input (IMERG_SDLRR) improved 15% in terms of Coefficient of Determination (R2) and improved 25% in terms of Root Mean Squared Error (RMSE). Compared to the best performance model among models using area-averaged precipitation as input, IMERG_SDLRR improved 5% in terms of R2 and 11% in terms of RMSE. Good performance was also acquired in the other hydrological stations. For extreme flood forecasts, IMERG_SDLRR decreased Mean Relative Error (MRE) by 0.29 and increased Qualified Rate (QR) by 53% compared to LSTM, and decreased MRE by 0.08 and increased QR by 6% compared to the best performance model using area-averaged precipitation as input. The utilization of IMERG or TMPA spatial information improved the accuracy of runoff forecasting. The accuracy evaluation of SPPs based on the results of spatiotemporal rainfall-runoff forecasts method was also demonstrated. The research is of great significance for developing runoff forecasting methods and optimizing water resources management.

Heart Disease Prediction Using Genetic Algorithm with Machine Learning Classifiers

Based on the world health organization (WHO), heart disease is the major reason for the loss of life everywhere on earth. Heart attacks are the leading reason of loss of life among heart diseases. This disease is called a silent disease. The person does not feel any pain until the last level of sickness and may arrive at death if not saved at the right time. The datasets for this disease are becoming available, so it is a particularly good branch of study. Predicting a heart attack for a medical practitioner is difficult since it requires increased expertise. However, over the last few decades, resolving complicated, extremely non-linear classification and prediction problems have been using machine learning algorithms (ML). Hence, it is feasible to establish a prediction model that would see the existence or nonexistence of heart disease based on many heart-related symptoms (features). The essential contribution of this research is to introduce various prediction models for heart disease using a genetic algorithm (GA) to find optimal features combined with classical machine learning techniques. The optimized prediction model uses a genetic algorithm that performs better than classical models. The execution of the algorithms is tested using Cleveland and Framingham datasets. The prediction models' performance is standardized using three evaluation criteria: accuracy, precision, and recall. The proposed system showed superior performance compared with other related systems. It reached an accuracy of 100% for the Cleveland dataset and 91.8% for the Framingham dataset.

Robust Least Squares Twin Support Vector Regression With Adaptive FOA and PSO for Short-Term Traffic Flow Prediction

Accurate short-term traffic flow prediction plays an important role in the field of modern Intelligent Transportation Systems. Since various uncontrollable factors ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e.g.</i> weather, traffic jams or accidents), collected traffic data inevitably contain outliers. This makes it challenge to achieve satisfactory results for traffic flow prediction. Least Squares Twin Support Vector Regression (LSTSVR) has been shown to provide a powerful potential in nonlinear prediction problems. This is especially true when using appropriate heuristic algorithms to determine the parameters of nonlinear LSTSVR. In view of this, a novel LSTSVR model based on the robust <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{L}_{2,\mathrm {p}}$ </tex-math></inline-formula> -norm ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0&lt; p\le 2$ </tex-math></inline-formula> ) distance is proposed to alleviate the negative effect of traffic data with outliers, called PLSTSVR. An iterative algorithm is designed to solve the optimization problem of PLSTSVR, which has great potential for solving other relevant optimization problems. To search the parameters of constructed PLSTSVR, this paper constructs two traffic flow prediction models based on PLSTSVR and heuristic algorithms (Fruit Fly Optimization Algorithm and Particle Swarm Optimization), called PLSTSVR-FOA and PLSTSVR-PSO. Extensive experiments demonstrate that the constructed models are more effective and robust than other competing models in various experimental settings.

Wind Energy Resource Prediction and Optimal Storage Sizing to Guarantee Dispatchability: A Case Study in the Kenyan Power Grid

Kenya is experiencing a fast increase in grid-connected intermittent renewable energy sources (RESs) to meet its increased power demand, and at the same time be able to fulfill its Paris Agreement obligations of abating greenhouse gas emissions. For instance, Kenya has 102 MW of grid-tied solar power and 410 MW of grid-tied wind power. However, these sources are very intermittent with low predictability. Thus, after their installation and integration into the grid, they impose a new challenge for the secure, reliable, and economic operation of the system. To mitigate these and to ensure proper planning of the system operations, accurate and faster prediction of the generation output of the wind energy resources and optimal design and sizing of storage for the large-scale wind energy integration into the grid are of paramount importance. Artificial intelligence (AI) and metaheuristic techniques have proven to be efficient and robust in offering solutions to complex nonlinear prediction and optimization problems. Therefore, this study aims to utilize backpropagation neural network (BPNN) algorithm to conduct hourly prediction of the generation output of Lake Turkana Wind Power Plant (LTWPP), a 310 MW plant connected to the Kenyan power grid, and optimally size its battery energy storage system (BESS) using genetic algorithm (GA) to guarantee its dispatchability. The historical weather data, namely wind speed, ambient temperature, relative humidity, wind direction, and generation output from LTWPP, are employed in the training, testing, and validation of the neural network. LTWPP and BESS are modelled in MATLAB R2016a software. Thereafter, the developed BPNN and GA algorithms are applied to the modelled systems to predict the wind output and optimize the storage system, respectively. BESS optimization with neural prediction reduces the BESS capacity and investment costs by 59.82%, while the overall dispatchability of LTWPP is increased from 73.36% to 90.14%, hence enabling the farm to meet its allowable loss of power supply probability (LPSP) index of 0.1 while guaranteeing its dispatchability.

Optimization of Interval Type-2 Intuitionistic Fuzzy Logic System for Prediction Problems

Derivative-based algorithms have been adopted in the literature for the optimization of membership and non-membership function parameters of interval type-2 (T2) intuitionistic fuzzy logic systems (FLSs). In this study, a non-derivative-based algorithm called sliding mode control learning algorithm is proposed to tune the parameters of interval T2 intuitionistic FLS for the first time. The proposed rule-based learning system employs the Takagi–Sugeno–Kang inference with artificial neural network to pilot the learning process. The new learning system is evaluated using some nonlinear prediction problems. Analyses of results reveal that the proposed learning apparatus outperforms its type-1 version and many existing solutions in the literature and competes favorably with others on the investigated problem instances with low cost in terms of running time.

A new grey prediction model and its application in landslide displacement prediction

Developing a grey prediction model with high nonlinear prediction accuracy is an important issue in grey system theory. A new grey prediction model was developed that was the first to combine the idea of twin support vector regression with Hausdorff derivative operator. The new model is a non-linear data-driven model. An improved salp swarm algorithm is used to determine parameters of the model. Two numerical examples show that the error of the new model is smaller than the existing grey prediction models and least square support vector machine model. Moreover, with the displacement, precipitation, reservoir level elevation, variation velocity of reservoir level elevation, and displacement velocity of the previous month as the input variables, the new model was successfully used to predict the displacement of a landslide in the real-world. The new model is a powerful tool for solving nonlinear prediction problems.

Mechanical Fault Diagnosis of an On-Load Tap Changer by Applying Cuckoo Search Algorithm-Based Fuzzy Weighted Least Squares Support Vector Machine

To improve adaptability, feature resolution, and identification accuracy when diagnosing mechanical faults in an on-load tap changer (OLTC) of a transformer, in the present research, wavelet packet energy entropy is used to describe the information comprising vibration signal in the switch process of an OLTC, and a fuzzy weighted least squares support vector machine (CSA-fuzzy weighted LSSVM) model based on the cuckoo search algorithm is proposed to identify mechanical fault types. Specifically, according to the different importance of the sample data in different periods, the idea of fuzzy weighting of training samples is proposed. The cuckoo search algorithm is used to optimise regularisation parameters, kernel function width, and weight control factor of CSA-fuzzy weighted LSSVM. Finally, the real experimental platform for typical mechanical faults of an OLTC is established, and the vibration signals of several typical mechanical faults under different degrees of fatigue are obtained. The results show that the new method achieves a higher accuracy rate of fault identification compared with other common methods. It can better deal with small sample and nonlinear prediction problems and shows higher fitting accuracy than CSA-LSSVM, single LSSVM, and radial basis neural network methods and is thus better suited for mechanical fault diagnosis in OLTCs. This paper presents a new intelligent diagnosis scheme for mechanical faults of on-load tap changers, which can achieve noninterruption and nonintrusive detection. The proposed diagnosis method would change the traditional diagnosis method of the on-load tap changer and improves the power supply quality and the detection efficiency under the premise of ensuring the safety of the staff.

Study on Intelligent Identification Method of Coal Pillar Stability in Fully Mechanized Caving Face of Thick Coal Seam

The combination of coal precise mining and information technology in the new century is one of the important directions for the future development of coal mining. Taking the fully mechanized top coal caving condition of a thick coal seam in the 90,101 working face of Baoshan Yujing Coal Mine in Shanyin City, Shanxi Province as an example, the intelligent identification method of section coal pillar stability was studied. The load transfer law of overlying strata in the upper part of coal pillar was analyzed, and the coal pillar values of each index were obtained by using an empirical formula, mean impact value-genetic algorithm-BP neural network (MIV-GA-BP) simulation experiment, and finite difference algorithm. The Delphi index evaluation system was used to calculate the optimal value of the coal pillar. The results showed that the non-contact cantilevered triangle on the two wings of the coal pillar in the goaf reduced the stress on the coal pillar; according to the width of the coal pillar at 10 m, 14 m, 16 m, and 20 m, combined with the relationship between the plastic zone and the core zone of coal pillar, and the relationship between the stress field and the ultimate strength of coal pillar, the numerical simulation value of the coal pillar was determined. The MIV (mean impact value) characteristics screened out the influencing factors of coal pillar width in the section near the horizontal fully mechanized top coal caving face order of importance; the relative error between the predicted value and the expected value of the MIV-GA-BP simulation experiment was less than 5%, which has good stability for the multi-factor nonlinear coupling prediction problem; and the optimal value of the coal pillar was 16.03 m by the intelligent identification method of the coal pillar. When the 16 m pillar was used, the surrounding rock deformation of the roadway was small, and the control effect was good. The research results provide a theoretical basis and reference for the parameter determination of similar projects.

Trajectory prediction of cylinders freely dropped into water in two dimensions based on state space model

Experiments show that the behavior of dropped cylindrical objects may be stochastic if small disturbances exist at the initial moment. An envelope of various trajectories can be obtained from experimental tests. In addition, numerical results also indicate that small changes of initial states, e.g., drop angle, rolling frequency, and the selection of drag coefficient, affect the trajectory greatly. If the initial state follows the Gaussian distribution,it can be considered as a nonlinear prediction problem. Then the Unscented Transformation (UT) method, as a deterministic sampling method, can be used to predict the trajectory. In this paper, we firstly present a systematic formulation of the cylinder's three-degree-of-freedom (3DOF) of motions as a state space model. Secondly, the UT method and the interval estimation method are employed to solve this prediction problem. Finally, the predicted trajectories will be compared with the experimental data. Meanwhile, two performance parameters are first proposed to evaluate the accuracy of predicted trajectory. Both parameters can be directly applied to the salvage and recovery of dropped objects in the offshore operations. In addition, the numerical results also show that the UT method is an efficient way to predict the trajectory of cylinders freely dropped into water. • The state space model of a dropped cylinder has been successfully constructed in this paper. • The trajectory of a dropped cylinder is considered as the stochastic process due to the uncertainty at the initial moment. • The Unscented Transformation (UT) method is employed to solve the nonlinear trajectory prediction. • Two performance measures R A and R L have been first proposed to evaluate the accuracy of numerical trajectory prediction, in order to meet the requirement from the offshore engineering industry.

Short-Term Traffic Flow Prediction Based on Least Square Support Vector Machine with Hybrid Optimization Algorithm

Accurate short-term traffic flow prediction plays an indispensable role for solving traffic congestion. However, the structure of traffic data is nonlinear and complicated. It is a challenge to get high precision. The least square support vector machine (LSSVM) has powerful capabilities for time series and nonlinear regression prediction problems if it can select appropriate parameters. To search the optimal parameters of LSSVM, this paper proposes a hybrid optimization algorithm which combines particle swarm optimization (PSO) with genetic algorithm. The main contributions are twofold: (1) A hybrid optimization method is proposed, which can skip the local optimal pitfall with less learning time by introducing a selection strategy, crossover and mutation operators into PSO; (2) the crossover and mutation operators are controlled by adaptive probability functions. The crossover and mutation probabilities increase when the population fitness is concentrated, and decrease when the fitness is dispersed. It can effectively improve the precision and speed of convergence. The proposed model is verified based on the measured data. The experimental results show that our new model yields better prediction ability and relatively high computational efficiency compared with other related models.

Global sensitivity analysis using support vector regression

• An orthogonal polynomial kernel function is proposed for SVR model. • An iteration scheme is introduced to optimize the proposed kernel function. • Sobol’ indices are obtained by post-processing the coefficients of the SVR model. • The proposed method is efficient for complex problems. Global sensitivity analysis (GSA) plays an important role in exploring the respective effects of input variables on response variables. In this paper, a new kernel function derived from orthogonal polynomials is proposed for support vector regression (SVR). Based on this new kernel function, the Sobol’ global sensitivity indices can be computed analytically by the coefficients of the surrogate model built by SVR. In order to improve the performance of the SVR model, a kernel function iteration scheme is introduced further. Due to the excellent generalization performance and structural risk minimization principle, the SVR possesses the advantages of solving non-linear prediction problems with small samples. Thus, the proposed method is capable of computing the Sobol’ indices with a relatively limited number of model evaluations. The proposed method is examined by several examples, and the sensitivity analysis results are compared with the sparse polynomial chaos expansion (PCE), high dimensional model representation (HDMR) and Gaussian radial basis (RBF) SVR model. The examined examples show that the proposed method is an efficient approach for GSA of complex models.

ADAPTIVE NEURO-FUZZY INFERENCE SYSTEM FOR PREDICTING ALPHA BAND POWER OF EEG DURING MUSLIM PRAYER (SALAT)

The features of electroencephalographic (EEG) signals include important information about the function of the brain. One of the most common EEG signal features is alpha wave, which is indicative of relaxation or mental inactivity. Until now, the analysis and the feature extraction procedures of these signals have not been well developed. This study presents a new approach based on an adaptive neuro-fuzzy inference system (ANFIS) for extracting and predicting the alpha power band of EEG signals during Muslim prayer (Salat). Proposed models can acquire information related to the alpha power variations during Salat from other physiological parameters such as heart rate variability (HRV) components, heart rate (HR), and respiration rate (RSP). The models were developed by systematically optimizing the initial ANFIS model parameters. Receiver operating characteristic (ROC) curves were performed to evaluate the performance of the optimized ANFIS models. Overall prediction accuracy of the proposed models were achieved of 94.39%, 92.89%, 93.62%, and 94.31% for the alpha power of electrodes positions at O1, O2, P3, and P4, respectively. These models demonstrated many advantages, including efficiency, accuracy, and simplicity. Thus, ANFIS could be considered as a suitable tool for dealing with complex and nonlinear prediction problems.

A dual method for solving the nonlinear structured prediction problem

In this paper, we present a perceptron-based algorithm and have developed a dual formulation to solve the nonlinear structured prediction problem, which we called Dual Structured Incremental Margin Algorithm (DSIMA). The proposed formulation allows the introduction of kernel functions enabling the efficient solution of nonlinear problems. In order to verify the correctness and applicability of the algorithm, we consider an inverse approach to the path planning problem. The problem mapped on a grid environment can be solved by a search process that essentially depends on the definition of the transition costs between states. In this context, we develop and apply a learning algorithm that is able to perform the reverse path, i.e., the prediction of these costs in a direct space for the linear form. However, considering the nonlinear form, the problem is solved in a space of high dimension and where it is possible to learn a path instead of the transition costs. This learning problem is usually formulated as a convex optimization problem of maximum margin. Several tests to solve the costs prediction problem were carried out and the results compared to other structured prediction techniques. The proposed algorithm demonstrated greater efficiency in terms of computational effort and quality of prediction.

Online prediction of glucose concentration in type 1 diabetes using extreme learning machines.

We propose an online machine-learning solution to the problem of nonlinear glucose time series prediction in type 1 diabetes. Recently, extreme learning machine (ELM) has been proposed for training single hidden layer feed-forward neural networks. The high accuracy and fast learning speed of ELM drive us to investigate its applicability to the glucose prediction problem. Given that diabetes self-monitoring data are received sequentially, we focus on online sequential ELM (OS-ELM) and online sequential ELM kernels (KOS-ELM). A multivariate feature set is utilized concerning subcutaneous glucose, insulin therapy, carbohydrates intake and physical activity. The dataset comes from the continuous multi-day recordings of 15 type 1 patients in free-living conditions. Assuming stationarity and evaluating the performance of the proposed method by 10-fold cross- validation, KOS-ELM were found to perform better than OS-ELM in terms of prediction error, temporal gain and regularity of predictions for a 30-min prediction horizon.

Semi-widely linear estimation of [formula omitted]-proper quaternion random signal vectors under Gaussian and stationary conditions

Recursive algorithms for estimating quaternion signal vectors are provided. Assuming stationarity, Gaussianity and Cη-properness, two estimation problems are treated: the prediction and fixed-point smoothing problems. The problems are formulated in a very general way and solved following a semi-widely linear processing. The suggested solutions are derived taking additionally into account the information supplied by a square version of the quaternion observations. This extra information makes it possible to give improved estimators which outperform the corresponding full-widely linear estimators that ignore such information. A numerical example pertaining to the nonlinear prediction and fixed-point smoothing problems illustrates the application of the algorithms.

An improved particle filtering for time-varying nonlinear prediction of biomass and grain protein content

This paper addresses the problem of time-varying nonlinear prediction of biomass and grain protein content. The objectives of this paper are threefold. The first objective is to use an improved particle filter with better proposal distribution for nonlinear prediction. The second objective is to extend the state and parameter estimation techniques (i.e., particle filter (PF) and improved particle filter (IPF)) to better handle nonlinear and non-Gaussian processes without a priori state information, by utilizing a time-varying assumption of statistical parameters. The third objective is to apply the state estimation techniques PF and IPF for predicting and modeling biomass and grain protein content. In a first step, we present an application of PF and IPF to a simple dynamic crop model with the aim to predict a single state variable, namely winter wheat biomass. In a second step, we apply PF and IPF for updating predictions of complex nonlinear crop models in order to predict protein grain content.The comparative analysis is conducted to study the effects of two practical challenges (measurement noise, and the number of states and parameters to be estimated) on the estimation performances of PF, and IPF. To study the effect of measurement noise on the estimation performances, several measurement noise contributions are considered. Then, the estimation performances of PF and IPF are compared for different noise levels. Similarly, to investigate the effect of the number of states and parameters to be estimated on the estimation performances of PF and IPF, the estimation performance is analyzed for different numbers of estimated states and parameters.The simulation results of both comparative studies show that the IPF provides a significant improvement over the PF. This is because, unlike the PF which depends on the choice of sampling distribution used to estimate the posterior distribution, the IPF yields an optimum choice of this distribution, which also utilizes the observed data. The results of the comparative studies show also that, for all the techniques, estimating more model parameters affects the estimation accuracy as well as the convergence of the estimated states and parameters. The performance of the proposed method is evaluated on a synthetic example in terms of estimation accuracy, root mean square error and execution times.

Nonparametric Uncertainty Quantification for Stochastic Gradient Flows

This paper presents a nonparametric statistical modeling method for quantifying uncertainty in stochastic gradient systems with isotropic diffusion. The central idea is to apply the diffusion maps algorithm to a training data set to produce a stochastic matrix whose generator is a discrete approximation to the backward Kolmogorov operator of the underlying dynamics. The eigenvectors of this stochastic matrix, which we will refer to as the diffusion coordinates, are discrete approximations to the eigenfunctions of the Kolmogorov operator and form an orthonormal basis for functions defined on the data set. Using this basis, we consider the projection of three uncertainty quantification (UQ) problems (prediction, filtering, and response) into the diffusion coordinates. In these coordinates, the nonlinear prediction and response problems reduce to solving systems of infinite-dimensional linear ordinary differential equations. Similarly, the continuous-time nonlinear filtering problem reduces to solving a syst...