Dimensional Vector Research Articles

Pseudo-Fubini entire functions on the plane in the sense of Riemann

We prove the existence of a maximal dimensional vector space of real functions on the real plane all of whose nonzero members are bounded, entire, non-Lebesgue-integrable, and satisfy the equality of the two iterated integrals given in the conclusion of the Fubini theorem, with the additional property that these iterated integrals exist in the Riemann sense, but not in the Lebesgue one. Moreover, this vector space is dense in the space of smooth functions on the plane under its natural topology.

A proxy learning curve for the Bayes classifier

In this paper, a theoretical learning curve is derived for the multi-class Bayes classifier. This curve fits general multivariate parametric models of the class-conditional probability density. The derivation uses a proxy approach based on analyzing the convergence of a statistic which is proportional to the posterior probability of the true class. By doing so, the curve depends only on the training set size and on the dimension of the feature vector; it does not depend on the model parameters. Essentially, the learning curve provides an estimate of the reduction in the excess of the probability of error that can be obtained by increasing the training set size. This makes it attractive in order to deal with the practical problems of defining appropriate training set sizes.

Nonparametric Two-Sample Tests of High Dimensional Mean Vectors via Random Integration

Testing the equality of the means in two samples is a fundamental statistical inferential problem. Most of the existing methods are based on the sum-of-squares or supremum statistics. They are possibly powerful in some situations, but not in others, and they do not work in a unified way. Using random integration of the difference, we develop a framework that includes and extends many existing methods, especially in high-dimensional settings, without restricting the same covariance matrices or sparsity. Under a general multivariate model, we can derive the asymptotic properties of the proposed test statistic without specifying a relationship between the data dimension and sample size explicitly. Specifically, the new framework allows us to better understand the test’s properties and select a powerful procedure accordingly. For example, we prove that our proposed test can achieve the power of 1 when nonzero signals in the true mean differences are weakly dense with nearly the same sign. In addition, we delineate the conditions under which the asymptotic relative Pitman efficiency of our proposed test to its competitor is greater than or equal to 1. Extensive numerical studies and a real data example demonstrate the potential of our proposed test. Supplementary materials for this article are available online.

Two-Stage Feature Selection Method Created for 20 Neurons Artificial Neural Networks for Automatic Breast Cancer Detection

Breast cancer is a common deadly diseases in women. Initial recognition of breast cancer using mammogram images is a challenging task. Hence, this paper proposed a unique automatic diagnosis model for breast cancer. Initially, the mammogram images are preprocessed with a median filter and contrast limited adaptive histogram equalization (CLAHE). The pre-processed image is automatically segmented using the multilevel threshold method. Subsequently, statistical, texture, shape, and geometric features are extracted from the segmented image. So, the length of the feature vector is high, and it is important to identify optimum features. In this paper, the dimension of the feature vector is reduced by 2-stage feature selection methods. Initially, the feature vector is applied to the best first search method information gain (IG) with rank feature method, and then secondly, apply the Pearson correlation method (PCM). Artificial neural networks (ANNs) are used to increase the classification accuracy of a breast cancer diagnosis. In this model, the selection of appropriate neurons in a single hidden layer is used to avoid overfitting problems in an ANN model. Based on optimum feature selection, the appropriate number of neurons chosen in the hidden layer is 20, which was applied for the proposed IG+PCM+Boosted-ANN model. The proposed model is applied on 2 regular datasets mini-Mammographic Image Analysis Society (mini-MIAS) and Digital Database for Screening Mammography (DDSM). The proposed model was superior to other exiting models and the model in this study achieves the accuracy of 99 and 98.80 % for mini-MIAS and DDSM datasets, respectively.

Optimal Power Sharing Control of the Hybrid Energy Storage System of an Electric Vehicle Along a Standard Driving Cycle

Abstract The paper presents a strategy of energy loss minimization within a hybrid energy storage system of an electrical vehicle, composed by a battery and a supercapacitor. The optimization of the power sharing between these energy storage devices is performed for the New European Driving Cycle, using the Particle Swarm Optimization algorithm. The minimum energy storage required to pass through the driving cycle is taken into account as a time-variable constraint during the optimization. The dimension of the search space increases with the dimension of the optimization vector, which has to be kept low in order to keep the complexity of the problem manageable. It is shown, that the subdivision, and piecewise optimization of the driving cycle improves the result by means of relaxation of the constraint represented by minimum level of the required energy storage.

Analysis and Design of Multivalued High-Capacity Associative Memories Based on Delayed Recurrent Neural Networks.

This article aims at analyzing and designing the multivalued high-capacity-associative memories based on recurrent neural networks with both asynchronous and distributed delays. In order to increase storage capacities, multivalued activation functions are introduced into associative memories. The stored patterns are retrieved by external input vectors instead of initial conditions, which can guarantee accurate associative memories by avoiding spurious equilibrium points. Some sufficient conditions are proposed to ensure the existence, uniqueness, and global exponential stability of the equilibrium point of neural networks with mixed delays. For neural networks with n neurons, m -dimensional input vectors, and 2k -valued activation functions, the autoassociative memories have (2k)n storage capacities and heteroassociative memories have min {(2k)n,(2k)m} storage capacities. That is, the storage capacities of designed associative memories in this article are obviously higher than the 2n and min {2n,2m} storage capacities of the conventional ones. Three examples are given to support the theoretical results.

Fast and Accurate Least-Mean-Squares Solvers for High Dimensional Data.

Least-mean-squares (LMS) solvers such as Linear / Ridge-Regression and SVD not only solve fundamental machine learning problems, but are also the building blocks in a variety of other methods, such as matrix factorizations. We suggest an algorithm that gets a finite set of n d-dimensional real vectors and returns a subset of d+1 vectors with positive weights whose weighted sum is exactly the same. The constructive proof in Caratheodory's Theorem computes such a subset in O(n2d2) time and thus not used in practice. Our algorithm computes this subset in O(nd+d4logn) time, using O(logn) calls to Caratheodory's construction on small but "smart" subsets. This is based on a novel paradigm of fusion between different data summarization techniques, known as sketches and coresets. For large values of d, we suggest a faster construction that takes O(nd) time and returns a weighted subset of O(d) sparsified input points. Here, a sparsified point means that some of its entries were set to zero. As an application, we show how to boost the performance of existing LMS solvers, such as those in scikit-learn library, up to x100. Generalization for streaming and distributed data is trivial. Extensive experimental results and open source code are provided.

Encoder Blind Combinatorial Compressed Sensing

In its most elementary form, compressed sensing studies the design of decoding algorithms to recover a sufficiently sparse vector or code from a lower dimensional linear measurement vector. Typically it is assumed that the decoder has access to the encoder matrix, which in the combinatorial case is sparse and binary. In this paper we consider the problem of designing a decoder to recover a set of sparse codes from their linear measurements alone, that is without access to encoder matrix. To this end we study the matrix factorisation task of recovering both the encoder and sparse coding matrices from the associated linear measurement matrix. The contribution of this paper is a computationally efficient decoding algorithm, Decoder-Expander Based Factorisation, with strong performance guarantees. Under mild assumptions on the sparse coding matrix and by deploying a novel random encoder matrix, we prove that Decoder-Expander Based Factorisation recovers both the encoder and sparse coding matrix at the optimal measurement rate with high probability and from a near optimal number of measurement vectors. In addition, our experiments demonstrate the efficacy and computational efficiency of our algorithm in practice. Beyond compressed sensing, our results may be of interest for researchers working in areas as diverse as linear sketching, coding theory, matrix compression and dictionary learning.

Parameter estimation for a class of time‐varying systems with the invariant matrix

AbstractThis article is concerned with the identification of time‐varying systems. Differently from the conventional polynomial approximation approaches, the changing laws of the time‐varying parameters are considered to build the identification model for the time‐varying systems. Specifically, the concept of the invariant matrix is put forward to characterize the time‐varying parameters and to establish the state‐space model with regard to the system parameters. Then this article proposes a stacked state estimation algorithm to achieve the time‐varying parameter estimation. Moreover, for the purpose of enhancing the computational efficiency, a detached state estimation algorithm is proposed by reducing the dimension of the state vector to reconstruct the state equation. Finally, a numerical simulation example is employed to demonstrate the effectiveness of the proposed algorithms.

Naïve Bayes dan Support Vector Machine Berbasis PSO untuk Seleksi Fitur pada Sentiment Analysis

Sentiment analysis is a process that aims to determine the content of the dataset in the form of positive, negative and neutral text. Currently the opinion of the general public is an important source of decision making. Social media is a place to express public opinion on an object, problem or event. Such as the government's policy regarding the relocation of the capital city of Indonesia, which was originally in Jakarta to Kalimantan, did not escape the attention of the public, especially Twitter users. One of the problems in sentiment analysis is the high number of attributes and dimensions in the dataset. In this study, sentiment analysis was carried out on the relocation of the national capital using the Naïve Bayes method, and the Support Vector Machine based on Particle Swarm Optimization (PSO). The advantages of the Support Vector Machine are High dimensional space and Vector document space. Feature selection greatly affects the performance of the classification, the use of PSO as feature selection to improve accuracy. The results of this study obtained the best accuracy value of 96.45% and the AUC value of 0.920 from the application of PSO on the Support Vector Machine. This result has increased when compared to the experimental results using Naïve Bayes and Support Vector Machine without PSO. The application of PSO on the Support Vector Machine is proven to get better accuracy results in predicting sentiment analysis on the dataset of moving the State Capital.

Stern-Gerlach effect of vector light bullets in a nonlocal Rydberg medium.

We show that stable (3+1)-dimensional vector light bullets with ultraslow propagating velocity and extreme low generation power can be realized in a cold Rydberg atomic gas. They can also be actively controlled by using a nonuniform magnetic field; especially, trajectories of their two polarization components can have significant Stern-Gerlach deflections. The results obtained are useful for revealing the nonlocal nonlinear optical property of Rydberg media and for measuring weak magnetic fields.

Musical Emotions Recognition Using Entropy Features and Channel Optimization Based on EEG.

The dynamic of music is an important factor to arouse emotional experience, but current research mainly uses short-term artificial stimulus materials, which cannot effectively awaken complex emotions and reflect their dynamic brain response. In this paper, we used three long-term stimulus materials with many dynamic emotions inside: the "Waltz No. 2" containing pleasure and excitement, the "No. 14 Couplets" containing excitement, briskness, and nervousness, and the first movement of "Symphony No. 5 in C minor" containing passion, relaxation, cheerfulness, and nervousness. Approximate entropy (ApEn) and sample entropy (SampEn) were applied to extract the non-linear features of electroencephalogram (EEG) signals under long-term dynamic stimulation, and the K-Nearest Neighbor (KNN) method was used to recognize emotions. Further, a supervised feature vector dimensionality reduction method was proposed. Firstly, the optimal channel set for each subject was obtained by using a particle swarm optimization (PSO) algorithm, and then the number of times to select each channel in the optimal channel set of all subjects was counted. If the number was greater than or equal to the threshold, it was a common channel suitable for all subjects. The recognition results based on the optimal channel set demonstrated that each accuracy of two categories of emotions based on "Waltz No. 2" and three categories of emotions based on "No. 14 Couplets" was generally above 80%, respectively, and the recognition accuracy of four categories based on the first movement of "Symphony No. 5 in C minor" was about 70%. The recognition accuracy based on the common channel set was about 10% lower than that based on the optimal channel set, but not much different from that based on the whole channel set. This result suggested that the common channel could basically reflect the universal features of the whole subjects while realizing feature dimension reduction. The common channels were mainly distributed in the frontal lobe, central region, parietal lobe, occipital lobe, and temporal lobe. The channel number distributed in the frontal lobe was greater than the ones in other regions, indicating that the frontal lobe was the main emotional response region. Brain region topographic map based on the common channel set showed that there were differences in entropy intensity between different brain regions of the same emotion and the same brain region of different emotions. The number of times to select each channel in the optimal channel set of all 30 subjects showed that the principal component channels representing five brain regions were Fp1/F3 in the frontal lobe, CP5 in the central region, Pz in the parietal lobe, O2 in the occipital lobe, and T8 in the temporal lobe, respectively.

Prediction of Buckwheat Maturity in UAV-RGB Images Based on Recursive Feature Elimination Cross-Validation: A Case Study in Jinzhong, Northern China

Buckwheat is an important minor grain crop with medicinal and edible functions. The accurate judgment of buckwheat maturity is beneficial to reduce harvest losses and improve yield. With the rapid development of unmanned aerial vehicle (UAV) technology, it has been widely used to predict the maturity of agricultural products. This paper proposed a method using recursive feature elimination cross-validation (RFECV) combined with multiple regression models to predict the maturity of buckwheat in UAV-RGB images. The images were captured in the buckwheat experimental field of Shanxi Agricultural University in Jinzhong, Northern China, from September to October in 2021. The variety was sweet buckwheat of "Jinqiao No. 1". In order to deeply mine the feature vectors that highly correlated with the prediction of buckwheat maturity, 22 dimensional features with 5 vegetation indexes, 9 color features, and 8 texture features of buckwheat were selected initially. The RFECV method was adopted to obtain the optimal feature vector dimensions and combinations with six regression models of decision tree regression, linear regression, random forest regression, AdaBoost regression, gradient lifting regression, and extreme random tree regression. The coefficient of determination (R2) and root mean square error (RMSE) were used to analyze the different combinations of the six regression models with different feature spaces. The experimental results show that the single vegetation index performed poorly in the prediction of buckwheat maturity; the prediction result of feature space "5" combined with the gradient lifting regression model performed the best; and the R2 and RMSE were 0.981 and 1.70 respectively. The research results can provide an important theoretical basis for the prediction of the regional maturity of crops.

Machine Learning Approach for Fake News Detection

A vital role is played by the media in the circulation of public information about the happenings. A fast spread of information using social media and social networks or websites is possible due to the quick development of the internet. With no worry about the reliability of the information, the misinformation is spread via social websites or networks and reaches multiple users. It is a very big challenge for society to deal with the spread of misinformation. The present research interest is the analysis of instinctive credibility of the news articles. For linguistic modeling, LSTM (Long Short-Term Memory), RNN (Recurrent Neural Network), etc. are widely utilized deep learning models. This paper presents a Bidirectional LSTM (BiLSTM) based sequential model for the detection of fake news. The proposed model uses word embeddings for the representation of words in high dimensional vector format. These word embeddings and Bidirectional LSTM tries to incorporate more semantic information of the content than that of RNN and unidirectional LSTM. 3 different publicly available datasets from Kaggle dataset repository are combined together for training, testing and validation of the proposed model. Along with this, the performance of different word embeddings is analyzed in this paper.

Structure-embedding network for predicting the transmission spectrum of a multilayer deep etched grating.

This Letter presents a structure-embedding network (SEmNet) to predict the transmission spectrum of a multilayer deep etched grating (MDEG). Spectral prediction is an important procedure in the MDEG design process. Existing approaches based on deep neural networks have been applied to spectral prediction to improve the design efficiency of similar devices, such as nanoparticles and metasurfaces. Due to a dimensionality mismatch between a structure parameter vector and the transmission spectrum vector, however, the prediction accuracy decreases. The proposed SEmNet can overcome the dimensionality mismatch problem of deep neural networks to increase the accuracy of predicting the transmission spectrum of an MDEG. SEmNet consists of a structure-embedding module and a deep neural network. The structure-embedding module increases the dimensionality of the structure parameter vector with a learnable matrix. The augmented structure parameter vector then becomes the input to the deep neural network to predict the transmission spectrum of the MDEG. Experiment results demonstrate that the proposed SEmNet improves the prediction accuracy of the transmission spectrum in comparison with the state-of-the-art approaches.

Leveraging Self-Attention Mechanism for Attitude Estimation in Smartphones.

Inertial attitude estimation is a crucial component of many modern systems and applications. Attitude estimation from commercial-grade inertial sensors has been the subject of an abundance of research in recent years due to the proliferation of Inertial Measurement Units (IMUs) in mobile devices, such as the smartphone. Traditional methodologies involve probabilistic, iterative-state estimation; however, these approaches do not generalise well over changing motion dynamics and environmental conditions, as they require context-specific parameter tuning. In this work, we explore novel methods for attitude estimation from low-cost inertial sensors using a self-attention-based neural network, the Attformer. This paper proposes to part ways from the traditional cycle of continuous integration algorithms, and formulate it as an optimisation problem. This approach separates itself by leveraging attention operations to learn the complex patterns and dynamics associated with inertial data, allowing for the linear complexity in the dimension of the feature vector to account for these patterns. Additionally, we look at combining traditional state-of-the-art approaches with our self-attention method. These models were evaluated on entirely unseen sequences, over a range of different activities, users and devices, and compared with a recent alternate deep learning approach, the unscented Kalman filter and the iOS CoreMotion API. The inbuilt iOS had a mean angular distance from the true attitude of 117.31∘, the GRU 21.90∘, the UKF 16.38∘, the Attformer 16.28∘ and, finally, the UKF-Attformer had mean angular distance of 10.86∘. We show that this plug-and-play solution outperforms previous approaches and generalises well across different users, devices and activities.

A Preserving Precision Mixed Finite Element Dimensionality Reduction Method for Unsaturated Flow Problem

The unsaturated flow problem is of important applied background and its mixed finite element (MFE) method can be used to simultaneously calculate both water content and flux in soil, which is the most ideal calculation method. Nonetheless, it includes many unknowns. Thereby, herein we will employ the proper orthogonal decomposition (POD) to lower the dimension of unknown solution coefficient vectors in the MFE method for the unsaturated flow problem. Thus, we first examine the MFE method for the unsaturated flow problem and the existence and convergence of the classical MFE solutions. We then take advantage of the initial L MFE solution coefficient vectors to generate a set of POD basis vectors and utilize the most POD basis vectors to create the preserving precision MFE reduced-dimension (PPMFERD) format. Under the circumstances, the PPMFERD format has the same basis functions as the classical MFE format so that it can maintain the same accuracy as the classical MFE format, but it only includes a few unknowns, so it greatly lightens the calculating load, retards the accumulation of computing errors, saves CPU runtime, and improves the accuracy of the real-time calculation in the computational process. Next, we employ the analysis of matrices to demonstrate the existence and convergence of the PPMFERD solutions such that the theoretical analysis becomes very simple and elegant. Finally, we take advantage of some numerical simulations to check on the correctness of the PPMFERD method. It shows that the PPMFERD method is effective and feasible for simulating both water content and flux in unsaturated flow soil.

Deep autoencoder based hybrid dimensionality reduction approach for classification of SERS for melanoma cancer diagnostics

Melanoma, a kind of fatal skin cancer, originates in melanin secreting cells of the dermis. Disease identification in the early stages assures a high survival rate for the patient. Most of the existing techniques retard the cancer detection phase. Surface-Enhanced Raman Spectroscopy (SERS) can capture fine details from the specimens that machine learning models can utilize to discriminate between healthy and diseased individuals rapidly. Our research work proposes a deep autoencoder based hybrid dimensionality reduction approach with a machine learning model on SERS spectrums of human skin fibroblast for melanoma cancer diagnostics. SERS measurements of 307 samples in total, belonging to two different classes, such as normal (157 samples) and malignant melanoma (150 samples), are used in this study. The SERS spectra measurements for both the samples lie between 100cm-1 and 4278cm-1. The variations in the intensity of Raman bands between both classes are intrinsically subtle. Neighborhood Component Analysis (NCA) technique has been exerted to transform 2090 dimensional spectral features into 2090 dimensional vectors and then the Deep Autoencoder (DAE) model is used to handle the nonlinearity in the data and produce the latent space, while Linear Discriminant Analysis (LDA) classifier have been employed for discriminating the normal and cancer cells. The k-fold cross-validation technique with a k value of 10 is implemented to assess the metrics of the model. The stated hybrid (NCA and DAE) model with 10-dimension latent space achieves an accuracy of 98%, the sensitivity of 99% and specificity of 97%, respectively. Due to the high-intensity nature of the SERS spectrum, the existing linear dimensionality reduction based discriminating model fails if the class label (Normal or Cancer) gets distributed on the low variance side. The proposed methodology captures both linear and nonlinear underlying structures present in the spectrums, resulting in better classification compared to the standard dimensionality reduction techniques.

Joint Characterization of Sentinel-2 Reflectance: Insights from Manifold Learning

Most applications of multispectral imaging are explicitly or implicitly dependent on the dimensionality and topology of the spectral mixing space. Mixing space characterization refers to the identification of salient properties of the set of pixel reflectance spectra comprising an image (or compilation of images). The underlying premise is that this set of spectra may be described as a low dimensional manifold embedded in a high dimensional vector space. Traditional mixing space characterization uses the linear dimensionality reduction offered by Principal Component Analysis to find projections of pixel spectra onto orthogonal linear subspaces, prioritized by variance. Here, we consider the potential for recent advances in nonlinear dimensionality reduction (specifically, manifold learning) to contribute additional useful information for multispectral mixing space characterization. We integrate linear and nonlinear methods through a novel approach called Joint Characterization (JC). JC is comprised of two components. First, spectral mixture analysis (SMA) linearly projects the high-dimensional reflectance vectors onto a 2D subspace comprising the primary mixing continuum of substrates, vegetation, and dark features (e.g., shadow and water). Second, manifold learning nonlinearly maps the high-dimensional reflectance vectors into a low-D embedding space while preserving manifold topology. The SMA output is physically interpretable in terms of material abundances. The manifold learning output is not generally physically interpretable, but more faithfully preserves high dimensional connectivity and clustering within the mixing space. Used together, the strengths of SMA may compensate for the limitations of manifold learning, and vice versa. Here, we illustrate JC through application to thematic compilations of 90 Sentinel-2 reflectance images selected from a diverse set of biomes and land cover categories. Specifically, we use globally standardized Substrate, Vegetation, and Dark (S, V, D) endmembers (EMs) for SMA, and Uniform Manifold Approximation and Projection (UMAP) for manifold learning. The value of each (SVD and UMAP) model is illustrated, both separately and jointly. JC is shown to successfully characterize both continuous gradations (spectral mixing trends) and discrete clusters (land cover class distinctions) within the spectral mixing space of each land cover category. These features are not clearly identifiable from SVD fractions alone, and not physically interpretable from UMAP alone. Implications are discussed for the design of models which can reliably extract and explainably use high-dimensional spectral information in spatially mixed pixels—a principal challenge in optical remote sensing.

Fault Recognition for Mechanical Arm by Using Relative Margin SVM

Monitoring and detecting faults during the operation of the manipulator is the prerequisite for fault recognition and safe operation. Accurate classification of mechanical arm faults can support to effectively eliminate mechanical arm faults. In this paper, the authors utilize a relative margin support vector machine (RMSVM) to classify and monitor the faults for mechanical arm. First, the status of mechanical arm are represented a high dimensional vector which consists of the mean, variance, correlation coefficient of the residual momentum signal in time domain, and the wavelet packet energy spectrum in frequency domain. The collected feature vectors for mechanical arm status are used to train RMSVM. A virtual prototype of mechanical arm is used to analyze the changes in the features of the residual momentum caused by fault and evaluate the RMSVM model for future mechanical arm status. The simulation results show that RMSVM can effectively detect the faults during the operation of manipulator.