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Manifold learning is a widely adopted nonlinear dimensionality reduction technique employed to discover low-dimensional representations from high-dimensional data and to explore the intrinsic data structure. It encompasses a range of nonlinear techniques, including Isomap, Local Linear Embedding (LLE), Local Tangent Space Alignment (LTSA), and Multidimensional Scaling (MDS), among others. Isomap, as a popular method in manifold learning, has exhibited promising results. However, it encounters several challenges, such as the Topological Stability Problem, Shortest Path Distance, and noise sensitivity, which compromise its accuracy and elevate its computational demands. This survey paper presents three novel existing approaches to address these challenges and enhance the performance of Isomap. Firstly, the FastIsomap method, an existing approach that combines state-of-the-art algorithms like the KD tree and NN-descent to increase graph accuracy while reducing computational cost. Secondly, the FastIsomapVis method, also an existing technique, employs a hierarchical divide-and-conquer strategy employing the KD tree and Dijkstra Buckets Double, which enhances accuracy and reduces computational time in Isomap. Lastly, the NR-Isomap method, another existing approach, used the Local Tangent Space Alignment algorithm to remove noise sensitivity and short-circuit edge problems. The existing methods improve the accuracy of Isomap but also streamline its computational requirements. Furthermore, this survey paper discusses the open issues and future research directions for Isomap and manifold learning methods. It can be a valuable resource for researchers seeking insights across diverse domains. [ABSTRACT FROM AUTHOR]

This paper proposes a nonlinear space dimension reduction method named Optimized Generative Topographic Mapping (OGTM). The Generative Topographic Mapping (GTM) method relies on the training sample set to capture the manifold of objective functions, and the generation of the training sample set causes an enormous computational burden. The choice of GTM hyperparameters has a significant influence on the design results. Traditional research has generally adopted the "cut-and-try" method to determine the corresponding hyperparameters and the best design, leading to wasted computational cost. The proposed OGTM overcomes this issue by minimizing the fitting error between the low-dimensional and high-dimensional samples, and the suitable hyperparameters are directly obtained by minimizing the fitting. In addition, the paper adopts a variable-fidelity sample filtration method to extract the promising regions with fewer sample points. To test and verify the effectiveness of the proposed method, it was then compared with the PCA and EGO methods in RAE2822 airfoil and ONERA M6 wing aerodynamic designs. The results demonstrate that the proposed method could capture the effective design space and generally take less computational cost to find the ideal results in all design optimizations. [ABSTRACT FROM AUTHOR]

Abstract Phenological shifts across plant species is a powerful indicator to quantify the effects of climate change. Today, mobile applications with automated species identification open new possibilities for phenological monitoring across space and time. Here, we introduce an innovative spatio‐temporal machine learning methodology that harnesses such crowd‐sourced data to quantify phenological dynamics across taxa, space and time. Our algorithm links individual phenological responses across thousands of species and geographical locations, using a similarity measure. The analysis draws on nearly ten million plant observations collected through the AI‐based plant identification app Flora Incognita in Germany from 2018 to 2021. Our method quantifies changes in synchronisation across the annual cycle. During the growing season, synchronised behaviour can be encoded by a few characteristic macrophenological patterns. Nonlinear spatio‐temporal changes of these patterns can be efficiently quantified using a data compressibility measure. Outside the growing season, the phenological synchronisation diminishes introducing noise into the patterns. Despite biases and uncertainties associated with crowd‐sourced data, for example due to human data collection behaviour, our study demonstrates the feasibility of deriving meaningful indicators for monitoring plant macrophenology from individual plant observations. As crowd‐sourced databases continue to expand, our approach holds promise to study climate‐induced phenological shifts and feedback loops.

Phenological shifts across plant species is a powerful indicator to quantify the effects of climate change. Today, mobile applications with automated species identification open new possibilities for phenological monitoring across space and time.Here, we introduce an innovative spatio‐temporal machine learning methodology that harnesses such crowd‐sourced data to quantify phenological dynamics across taxa, space and time. Our algorithm links individual phenological responses across thousands of species and geographical locations, using a similarity measure. The analysis draws on nearly ten million plant observations collected through the AI‐based plant identification app Flora Incognita in Germany from 2018 to 2021.Our method quantifies changes in synchronisation across the annual cycle. During the growing season, synchronised behaviour can be encoded by a few characteristic macrophenological patterns. Nonlinear spatio‐temporal changes of these patterns can be efficiently quantified using a data compressibility measure. Outside the growing season, the phenological synchronisation diminishes introducing noise into the patterns.Despite biases and uncertainties associated with crowd‐sourced data, for example due to human data collection behaviour, our study demonstrates the feasibility of deriving meaningful indicators for monitoring plant macrophenology from individual plant observations. As crowd‐sourced databases continue to expand, our approach holds promise to study climate‐induced phenological shifts and feedback loops. [ABSTRACT FROM AUTHOR]

The escalating incorporation of wind energy into power grids imposes constraints on the effective operation and control of power systems. Effective short-term wind power forecasting technology is essential for safe power supply and stable operation of the power grid. Thus, a UMAP-IVMD-ILSTM model for wind power forecasting is proposed based on the uniform manifold approximation and projection (UMAP) integrated improved variational mode decomposition (IVMD) and multi-level improved long and short-term memory (LSTM) networks. First, the UMAP approach was utilized to reduce the dimension of related meteorological indicators, and k-means clustering was applied to classify the weather categories after dimension reduction. Subsequently, through the improved variational mode decomposition, the wind power of the adjacent day is decomposed. Moreover, long and short-term memory networks were optimized using the sparrow search algorithm (SSA) as an improved LSTM (ILSTM), and a wind power forecasting model with multi-level LSTM was established for each mode, and the forecasting results of each mode were integrated. Through actual wind farm operation data, it was proven that the method proposed in this study is effective in processing high-dimensional numerical weather prediction (NWP) data. The model can better preserve the information of the original data and reduce the complexity of the original data sequence based on visualization. Compared with the comparative models, the proposed model reduces the RMSE and MAE by up to 18.2% and 21.6% respectively, and has the least running time, which can effectively improve the accuracy and efficiency of wind power prediction.

Nonlinear Dimensionality Reduction (NLDR) is a well-known approach of manifold learning to transform the data from high to low dimensional space. After studying various techniques proposed for the NLDR, we find that performance improvement is still required. Therefore, we adopt classical Isomap, which reduces Shortest Path Distance (SPD) and high computational time cost problems. These problems are occurring due to the Dijkstra algorithm. This paper presents the A*-FastIsomap method for SPD issues, which is based on the A* Search Algorithm with the Double Buckets algorithm. We compared the A*-FastIsomap with classical Isomap to verify its better efficiency and results for high dimensional datasets with much higher accuracy. The outcome of our current study demonstrates that as compared to classical Isomap, our proposed A*-FastIsomap is faster and more accurate. Furthermore, our proposed method can reduce the computation time for high and large-dimensional datasets. [ABSTRACT FROM AUTHOR]

Classical model reduction techniques project the governing equations onto linear subspaces of the high-dimensional state-space. For problems with slowly decaying Kolmogorov-nwidths such as certain transport-dominated problems, however, classical linear-subspace reducedorder models (ROMs) of low dimension might yield inaccurate results. Thus, the concept of classical linear-subspace ROMs has to be extended to more general concepts, like model order teduction (MOR) on manifolds. Moreover, as we are dealing with Hamiltonian systems, it is crucial that the underlying symplectic structure is preserved in the reduced model, as otherwise it could become unphysical in the sense that the energy is not conserved or stability properties are lost. To the best of our knowledge, existing literature addresses either MOR on manifolds or symplectic model reduction for Hamiltonian systems, but not their combination. In this work, we bridge the two aforementioned approaches by providing a novel projection technique called symplectic manifold Galerkin (SMG), which projects the Hamiltonian system onto a nonlinear symplectic trial manifold such that the reduced model is again a Hamiltonian system. We derive analytical results such as stability, energypreservation, and a rigorous a posteriori error bound. Moreover, we construct a weakly symplectic deep convolutional autoencoder as a computationally practical approach to approximate a nonlinear symplectic trial manifold. Finally, we numerically demonstrate the ability of the method to achieve higher accuracy than (non-)structure-preserving linear-subspace ROMs and non-structure-preserving MOR on manifold techniques. [ABSTRACT FROM AUTHOR]

We introduce the structure optimized proximity scaling (STOPS) framework for hyperparameter selection in parametrized multidimensional scaling and extensions (proximity scaling; PS). The selection process for hyperparameters is based on the idea that we want the configuration to show a certain structural quality (c-structuredness). A number of structures and how to measure them are discussed. We combine the structural quality by means of c-structuredness indices with the PS badness-of-fit measure in a multi-objective scalarization approach, yielding the Stoploss objective. Computationally we suggest a profile-type algorithm that first solves the PS problem and then uses Stoploss in an outer step to optimize over the hyperparameters. Bayesian optimization with treed Gaussian processes as a an apt and efficient strategy for carrying out the outer optimization is recommended. This way, hyperparameter tuning for many instances of PS is covered in a single conceptual framework. We illustrate the use of the STOPS framework with three data examples. [ABSTRACT FROM AUTHOR]

We present a data-driven point of view for rare events, which represent conformational transitions in biochemical reactions modeled by overdamped Langevin dynamics on manifolds in high dimensions. We first reinterpret the transition state theory and the transition path theory from the optimal control viewpoint. Given a point cloud probing the manifold, we construct a discrete Markov chain with a Q-matrix computed from an approximated Voronoi tesselation via the point cloud. We use this Q-matrix to compute a discrete committor function whose level set automatically orders the point cloud. Then based on the committor function, an optimally controlled random walk on point clouds is constructed and utilized to efficiently sample transition paths, which become an almost sure event in O(1) time instead of a rare event in the original reaction dynamics. To compute the mean transition path efficiently, a local averaging algorithm based on the optimally controlled random walk is developed, which adapts the finite temperature string method to the controlled Monte Carlo samples. Numerical examples on sphere/torus including a conformational transition for the alanine dipeptide in vacuum are conducted to illustrate the data-driven solver for the transition path theory on point clouds. The mean transition path obtained via the controlled Monte Carlo simulations highly coincides with the computed dominant transition path in the transition path theory. [ABSTRACT FROM AUTHOR]

Here we present a mathematical model of movement in an abstract space representing states of cellular differentiation. We motivate this work with recent examples that demonstrate a continuum of cellular differentiation using single cell RNA sequencing data to characterize cellular states in a high-dimensional space, which is then mapped into ℝ 2 or ℝ 2 with dimension reduction techniques. We represent trajectories in the differentiation space as a graph, and model directed and random movement on the graph with partial differential equations. We hypothesize that flow in this space can be used to model normal and abnormal differentiation processes. We present a mathematical model of hematopoeisis parameterized with publicly available single cell RNA-Seq data and use it to simulate the pathogenesis of acute myeloid leukemia (AML). The model predicts the emergence of cells in novel intermediate states of differentiation consistent with immunophenotypic characterizations of a mouse model of AML.

We introduce a method for the nonlinear dimension reduction of a high-dimensional function |$u:{\mathbb{R}}^d\rightarrow{\mathbb{R}}$|⁠ , |$d\gg 1$|⁠. Our objective is to identify a nonlinear feature map |$g:{\mathbb{R}}^d\rightarrow{\mathbb{R}}^m$|⁠ , with a prescribed intermediate dimension |$m\ll d$|⁠ , so that |$u$| can be well approximated by |$f\circ g$| for some profile function |$f:{\mathbb{R}}^m\rightarrow{\mathbb{R}}$|⁠. We propose to build the feature map by aligning the Jacobian |$\nabla g$| with the gradient |$\nabla u$|⁠ , and we theoretically analyze the properties of the resulting |$g$|⁠. Once |$g$| is built, we construct |$f$| by solving a gradient-enhanced least squares problem. Our practical algorithm uses a sample |$\{{\textbf{x}}^{(i)},u({\textbf{x}}^{(i)}),\nabla u({\textbf{x}}^{(i)})\}_{i=1}^N$| and builds both |$g$| and |$f$| on adaptive downward-closed polynomial spaces, using cross validation to avoid overfitting. We numerically evaluate the performance of our algorithm across different benchmarks, and explore the impact of the intermediate dimension |$m$|⁠. We show that building a nonlinear feature map |$g$| can permit more accurate approximation of |$u$| than a linear |$g$|⁠ , for the same input data set. [ABSTRACT FROM AUTHOR]

Principal asymmetric least squares (PALS) is introduced as a novel method for sufficient dimension reduction with heteroscedastic error. Classical methods such as MAVE [Xia et al. (2002), 'An Adaptive Estimation of Dimension Reduction Space' (with discussion), Journal of the Royal Statistical Society Series B, 64, 363-410] and PSVM [Li et al. (2011), 'Principal Support Vector Machines for Linear and Nonlinear Sufficient Dimension Reduction', The Annals of Statistics, 39, 3182-3210]maynot perform well in the presence of heteroscedasticity, while the new proposal addresses this limitation by synthesising different expectile levels. Through extensive numerical studies, we demonstrate the superior performance of PALS in terms of estimation accuracy over classical methods including MAVE and PSVM. For the asymptotic analysis of PALS, we develop new tools to compute the derivative of an expectation of a non-Lipschitz function. [ABSTRACT FROM AUTHOR]

This paper investigates the theoretical foundations of the t-distributed stochastic neighbor embedding (t-SNE) algorithm, a popular nonlinear dimension reduction and data visualization method. A novel theoretical framework for the analysis of t-SNE based on the gradient descent approach is presented. For the early exaggeration stage of t-SNE, we show its asymptotic equivalence to power iterations based on the underlying graph Laplacian, characterize its limiting behavior, and uncover its deep connection to Laplacian spectral clustering, and fundamental principles including early stopping as implicit regularization. The results explain the intrinsic mechanism and the empirical benefits of such a computational strategy. For the embedding stage of t-SNE, we characterize the kinematics of the low-dimensional map throughout the iterations, and identify an amplification phase, featuring the intercluster repulsion and the expansive behavior of the low-dimensional map, and a stabilization phase. The general theory explains the fast convergence rate and the exceptional empirical performance of t-SNE for visualizing clustered data, brings forth the interpretations of the t-SNE visualizations, and provides theoretical guidance for applying t-SNE and selecting its tuning parameters in various applications. [ABSTRACT FROM AUTHOR]

Topological Data Analysis (TDA) provides novel approaches that allow us to analyze the geometrical shapes and topological structures of a dataset. As one important application, TDA can be used for data visualization and dimension reduction. We follow the framework of circular coordinate representation, which allows us to perform dimension reduction and visualization for high-dimensional datasets on a torus using persistent cohomology. In this paper, we propose a method to adapt the circular coordinate framework to take into account the roughness of circular coordinates in change-point and high-dimensional applications. To do that, we use a generalized penalty function instead of an L2 penalty in the traditional circular coordinate algorithm. We provide simulation experiments and real data analyses to support our claim that circular coordinates with generalized penalty will detect the change in high-dimensional datasets under di_erent sampling schemes while preserving the topological structures. [ABSTRACT FROM AUTHOR]

Proximity scaling methods such as multidimensional scaling represent objects in a low-dimensional configuration so that fitted object distances optimally approximate object proximities. Besides finding the optimal configuration, an additional goal may be to make statements about the cluster arrangement of objects. This fails if the configuration lacks appreciable clusteredness. We present cluster optimized proximity scaling (COPS), which attempts to find a configuration that exhibits clusteredness. In COPS, a flexible parameterized scaling loss function that may emphasize differentiation information in the proximities is augmented with an index (OPTICS Cordillera) that penalizes lack of clusteredness of the configuration. We present two variants of this, one for finding a configuration directly and one for hyperparameter selection for parametric stresses. We apply both to a functional magnetic resonance imaging dataset on neural representations of mental states in a social cognition task and show that COPS improves clusteredness of the configuration, enabling visual identification of clusters of mental states. Online are available including an R package and a document with additional details. [ABSTRACT FROM AUTHOR]

Purpose: Multimodal Cardiac Image (MCI) registration is one of the evolving fields in the diagnostic methods of Cardiovascular Diseases (CVDs). Since the heart has nonlinear and dynamic behavior, Temporal Registration (TR) is the fundamental step for the spatial registration and fusion of MCIs to integrate the heart's anatomical and functional information into a single and more informative display. Therefore, in this study, a TR framework is proposed to align MCIs in the same cardiac phase. Materials and Methods: A manifold learning-based method is proposed for the TR of MCIs. The Euclidean distance among consecutive samples lying on the Locally Linear Embedding (LLE) of MCIs is computed. By considering cardiac volume pattern concepts from distance plots of LLEs, six cardiac phases (end-diastole, rapid-ejection, end-systole, rapid-filling, reduced-filling, and atrial-contraction) are temporally registered. Results: The validation of the proposed method proceeds by collecting the data of Computed Tomography Coronary Angiography (CTCA) and Transthoracic Echocardiography (TTE) from ten patients in four acquisition views. The Correlation Coefficient (CC) between the frame number resulted from the proposed method and manually selected by an expert is analyzed. Results show that the average CC between two resulted frame numbers is about 0.82±0.08 for six cardiac phases. Moreover, the maximum Mean Absolute Error (MAE) value of two slice extraction methods is about 0.17 for four acquisition views. Conclusion: By extracting the intrinsic parameters of MCIs, and finding the relationship among them in a lower-dimensional space, a fast, fully automatic, and user-independent framework for TR of MCIs is presented. The proposed method is more accurate compared to Electrocardiogram (ECG) signal labeling or time-series processing methods which can be helpful in different MCI fusion methods.

While tomographic neuroimaging data is information rich, objective, and with high sensitivity in the study of brain diseases such as Alzheimer's disease (AD), its direct use in clinical practice and in regulated clinical trial (CT) still has many challenges. Taking CT as an example, unless the relevant policy and the perception of the primary outcome measures change, the need to construct univariate indices (out of the 3-D imaging data) to serve as CT's primary outcome measures will remain the focus of active research. More relevant to this current study, an overall global index that summarizes multiple complicated features from neuroimages should be developed in order to provide high diagnostic accuracy and sensitivity in tracking AD progression over time in clinical setting. Such index should also be practically intuitive and logically explainable to patients and their families. In this research, we propose a new visualization tool, derived from the manifold-based nonlinear dimension reduction of brain MRI features, to track AD progression over time. In specific, we investigate the locally linear embedding (LLE) method using a dataset from Alzheimer's Disease Neuroimaging Initiative (ADNI), which includes the longitudinal MRIs from 562 subjects. About 20% of them progressed to the next stage of dementia. Using only the baseline data of cognitively unimpaired (CU) and AD subjects, LLE reduces the feature dimension to two and a subject's AD progression path can be plotted in this low dimensional LLE feature space. In addition, the likelihood of being categorized to AD is indicated by color. This LLE map is a new data visualization tool that can assist in tracking AD progression over time.

Gaussian process (GP)-based automatic relevance determination (ARD) is known to be an efficient technique for identifying determinants of gene-by-gene interactions important to trait variation. However, the estimation of GP models is feasible only for low-dimensional datasets (~200 variables), which severely limits application of the GP-based ARD method for high-throughput sequencing data. In this paper, we provide a nonparametric prescreening method that preserves virtually all the major benefits of the GP-based ARD method and extends its scalability to the typical high-dimensional datasets used in practice. In several simulated test scenarios, the proposed method compared favorably with existing nonparametric dimension reduction/prescreening methods suitable for higher-order interaction searches. As a real-data example, the proposed method was applied to a high-throughput dataset downloaded from the cancer genome atlas (TCGA) with measured expression levels of 16,976 genes (after preprocessing) from patients diagnosed with acute myeloid leukemia. [ABSTRACT FROM AUTHOR]

Manifold learning (ML) is a research topic of great interest in the field of machine learning that aims to determine the appropriate low-dimensional embeddings of data. The embeddings should preserve the intrinsic structure of the data manifold. Many ML techniques have been proposed to learn the underlying manifold of data. It is crucial to effectively evaluate the quality of the corresponding embedding results when selecting an appropriate ML technique in practice. However, there is a lack of effective embedding quality assessment (EQA) criteria to evaluate the embedding quality. In this paper, a new local included angles preservation (LUNA) criterion is proposed to evaluate the embedding quality. Unlike previous EQA methods that mainly focus on local neighborhood preservation performance or the preservation of the global geometric structure, the proposed LUNA criterion incorporates both an assessment of the neighborhood preserving capacity and local included angle holding performance. By introducing an effective evaluation of the performance of local included angle preservation, we show that the LUNA criterion could provide a more reasonable quality assessment than conventional criteria. The implementation of the LUNA criterion is direct and simple. To the best of our knowledge, this is the first EQA method that explicitly takes into account the preservation of local included angles. The effectiveness of the LUNA criterion is experimentally supported by its outstanding performance on a series of benchmark datasets. [ABSTRACT FROM AUTHOR]

Here we present a mathematical model of movement in an abstract space representing states of cellular differentiation. We motivate this work with recent examples that demonstrate a continuum of cellular differentiation using single-cell RNA-sequencing data to characterize cellular states in a high-dimensional space, which is then mapped into $ \mathbb {R}^2 $ or $ \mathbb {R}^3 $ with dimension reduction techniques. We represent trajectories in the differentiation space as a graph, and model directed and random movement on the graph with partial differential equations. We hypothesize that flow in this space can be used to model normal and abnormal differentiation processes. We present a mathematical model of haematopoiesis parameterized with publicly available single-cell RNA-Seq data and use it to simulate the pathogenesis of acute myeloid leukaemia (AML). The model predicts the emergence of cells in novel intermediate states of differentiation consistent with immunophenotypic characterizations of a mouse model of AML.

BRIC is an acronym coined by Jim O'Neill from Goldman Sachs in 2001 to abbreviate four emerging economies, Brazil, Russia, India and China, based on economic data at the time. Later, as new data became available, Goldman Sachs updated this list to include Mexico, Indonesia, Nigeria and Turkey, which was referred to as MINT. This list, as well as some other similar lists of emerging economies, is based on descriptive statistics of the economic data combined with economists' insights. The purpose of this study is twofold: to see if these insights into the global economic trends can be learned with statistical learning tools, and, if so, to identify the next emerging countries. We apply both unsupervised and supervised learning methods, which include linear and nonlinear principle component analysis, and nonlinear sufficient dimension reduction, to 13 years worth of economic data. Our results show that these statistical learning techniques, and in particular the kernel sliced inverse regression algorithm, can serve as a useful tool for economists and policy-makers for analyzing global economic trends, by its ability to incorporate large amount of economic data and previous experts' judgments, which otherwise may take years of experiences to acquire. [ABSTRACT FROM PUBLISHER]

Simultaneous recordings of electroencephalogram (EEG) and functional magnetic resonance imaging (fMRI) are at the forefront of technologies of interest to physicians and scientists because they combine the benefits of both modalities—better time resolution (hdEEG) and space resolution (fMRI). However, EEG measurements in the scanner contain an electromagnetic field that is induced in leads as a result of gradient switching slight head movements and vibrations, and it is corrupted by changes in the measured potential because of the Hall phenomenon. The aim of this study is to design and test a methodology for inspecting hidden EEG structures with respect to artifacts. We propose a top-down strategy to obtain additional information that is not visible in a single recording. The time-domain independent component analysis algorithm was employed to obtain independent components and spatial weights. A nonlinear dimension reduction technique t-distributed stochastic neighbor embedding was used to create low-dimensional space, which was then partitioned using the density-based spatial clustering of applications with noise (DBSCAN). The relationships between the found data structure and the used criteria were investigated. As a result, we were able to extract information from the data structure regarding electrooculographic, electrocardiographic, electromyographic and gradient artifacts. This new methodology could facilitate the identification of artifacts and their residues from simultaneous EEG in fMRI.

In order to reduce the loss of dimension reduction in high dimensional image dataset and improve the effect of data visualization, this paper proposed a new image dataset visualization method based on the combination of deep features and nonlinear dimension reduction. This method firstly designed and trained a convolutional neural network model,and the single model on the MNIST dataset achieved the highest recognition accuracy. Secondly,it used the high precision model to extract the deep middle layer features as an effective representation of the image dataset. Finally,it used nonlinear dimension reduction method to reduce the features into two-dimensional and realized data visualization. Experimental results show that this method can effectively avoid the loss of traditional dimension reduction in image dataset visualization, and the visualization result is very obvious. [ABSTRACT FROM AUTHOR]

In this study, a hybrid neural network predictor is proposed to predict spatiotemporal dynamics of the nonlinear distributed parameter systems (DPSs) with unwanted disturbance or slow set point changes. First, a nonlinear principal component analysis (NL-PCA) network is designed to transform the high-dimensional spatiotemporal data into a low-dimensional time domain, which can better represent the nonlinearity of the system compared to the linear time/space separation method. Then the hybrid NN models are built to identify the low-dimensional temporal data. To capture the spatiotemporal dynamics of DPS, the four-step recursive algorithm is used to obtain the time-varying weights of the model, while the parameters of NN model does not need to online update. The simulations demonstrated show that the proposed approach can achieve a good performance on prediction with system slow time-varying dynamics. [ABSTRACT FROM AUTHOR]

Abstract: Directional regression is an effective sufficient dimension reduction method which implicitly synthesizes the first two conditional moments. In this paper, we extend directional regression to a general family of estimators via the notion of general empirical directions. Data-driven method is used to identify the optimal estimator within this family. Based on the proposed general directional regression estimators, we develop a new methodology for nonlinear dimension reduction. Improvement of general directional regression over classical directional regression is demonstrated via simulation studies and an empirical study with the wine recognition data. [Copyright &y& Elsevier]

In this paper, a new nonlinear fault detection technique based on locally linear embedding (LLE) is developed. LLE can efficiently compute the low-dimensional embedding of the data with the local neighborhood structure information preserved. In this method, a data-dependent kernel matrix which can reflect the nonlinear data structure is defined. Based on the kernel matrix, the Nyström formula makes the mapping extended to the testing data possible. With the kernel view of the LLE, two monitoring statistics are constructed. Together with the out of sample extensions, LLE is used for nonlinear fault detection. Simulation cases were studied to demonstrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

Using metagenomics to detect the global structure of microbial community remains a significant challenge. The structure of a microbial community and its functions are complicated because of not only the complex interactions among microbes but also their interactions with confounding environmental factors. Recently dimension reduction methods have been employed extensively to investigate the complex structure embedded in metagenomic profiles which summarize the abundance of functional or taxonomic categorizations in metagenomic studies. However, metagenomic profiles are not necessary to meet the “Assumption of Linearity” behind these methods. Therefore it is worth to investigate whether nonlinear methods are appropriate methods which can be utilized in metagenomic analysis. In this paper, we compare the applications of several methods, including two linear methods (Principle component analysis and nonnegative matrix factorization) and a nonlinear manifold learning method—Isomap on visualizing and analyzing metagenomic profiles. These methods are applied and compared on a taxonomic profile from 33 human gut metagenomes and a large-scale Pfam profile which are derived from 45 metagenomes in Global Ocean Sampling expedition. We find that all three methods can discover interesting structures of the taxonomic profile from human gut. Furthermore, Isomap identified a novel nonlinear structure of protein families. The relationships among the identified nonlinear components and environmental factors of global ocean are explored. The results indicate that nonlinear methods could be a complementary technique to current linear methods in analyzing metagenomic dataset. [ABSTRACT FROM PUBLISHER]

Sliced inverse regression (SIR) was developed to find effective linear dimension-reduction directions for exploring the intrinsic structure of the high-dimensional data. In this study, we present isometric SIR for nonlinear dimension reduction, which is a hybrid of the SIR method using the geodesic distance approximation. First, the proposed method computes the isometric distance between data points; the resulting distance matrix is then sliced according to K-means clustering results, and the classical SIR algorithm is applied. We show that the isometric SIR (ISOSIR) can reveal the geometric structure of a nonlinear manifold dataset (e.g., the Swiss roll). We report and discuss this novel method in comparison to several existing dimension-reduction techniques for data visualization and classification problems. The results show that ISOSIR is a promising nonlinear feature extractor for classification applications. [ABSTRACT FROM AUTHOR]

Abstract: During the last decades, many studies have been conducted on performing reliable prediction for high-dimensional data that are usually non-linearly correlated with complex patterns. In this paper, we propose a novel Bayesian regression method via non-linear dimensionality reduction. The method incorporates prior information on the underlying structure of original input features to preserve input–output patterns on reduced features, and to provide distributions of predicted values. To verify the effectiveness of the proposed method, we conducted simulations on benchmark and real-world data. Results showed that the method not only better predicts a distribution of forecast estimates compared with other methods, but also more robust and consistent performance on prediction. [Copyright &y& Elsevier]

A major challenge in the current computer-aided detection (CAD) of polyps in CT colonography (CTC) is to reduce the number of false-positive (FP) detections while maintaining a high sensitivity level. A pattern-recognition technique based on the use of an artificial neural network (ANN) as a filter, which is called a massive-training ANN (MTANN), has been developed recently for this purpose. The MTANN is trained with a massive number of subvolumes extracted from input volumes together with the teaching volumes containing the distribution for the “likelihood of being a polyp;” hence the term “massive training.” Because of the large number of subvolumes and the high dimensionality of voxels in each input subvolume, the training of an MTANN is time-consuming. In order to solve this time issue and make an MTANN work more efficiently, we propose here a dimension reduction method for an MTANN by using Laplacian eigenfunctions (LAPs), denoted as LAP-MTANN. Instead of input voxels, the LAP-MTANN uses the dependence structures of input voxels to compute the selected LAPs of the input voxels from each input subvolume and thus reduces the dimensions of the input vector to the MTANN. Our database consisted of 246 CTC datasets obtained from 123 patients, each of whom was scanned in both supine and prone positions. Seventeen patients had 29 polyps, 15 of which were 5–9 mm and 14 were 10–25 mm in size. We divided our database into a training set and a test set. The training set included 10 polyps in 10 patients and 20 negative patients. The test set had 93 patients including 19 polyps in seven patients and 86 negative patients. To investigate the basic properties of a LAP-MTANN, we trained the LAP-MTANN with actual polyps and a single source of FPs, which were rectal tubes. We applied the trained LAP-MTANN to simulated polyps and rectal tubes. The results showed that the performance of LAP-MTANNs with 20 LAPs was advantageous over that of the original MTANN with 171 inputs. To test the feasibility of the LAP-MTANN, we compared the LAP-MTANN with the original MTANN in the distinction between actual polyps and various types of FPs. The original MTANN yielded a 95% (18/19) by-polyp sensitivity at an FP rate of 3.6 (338/93) per patient, whereas the LAP-MTANN achieved a comparable performance, i.e., an FP rate of 3.9 (367/93) per patient at the same sensitivity level. With the use of the dimension reduction architecture, the time required for training was reduced from 38 h to 4 h. The classification performance in terms of the area under the receiver-operating-characteristic curve of the LAP-MTANN (0.84) was slightly higher than that of the original MTANN (0.82) with no statistically significant difference (p-value=0.48). [ABSTRACT FROM PUBLISHER]