Embedding Dimension Research Articles

Correlation dimension for temperature and other micrometeorological variables in different ground cover from six biomes of the Amazon basin

Correlation dimension () for embedding dimensions, varying between 3 to 11, were estimated for a 1- or 2-month time series of air temperature and other micrometeorological variables, such as energy fluxes, solar radiation and concentration, in six ecosystems in Amazon and the central region of Brazil. Dc remained within the 1.5 - 3.0 range, for variables locally coupled to ecosystems, and outside the range for variables not locally coupled, such as precipitation and wind velocity. Results indicate that the feature of the variables does not have an important stochastic component. There is a small amount of data available concerning estimates of correlation dimension () of micrometeorological variables. The measurement of for several variables from six Brazilian biomes indicates that this parameter is sensitive to environmental conditions as the rainfall. The obtained results also indicates that the natural processes in the ecosystems remain in a non-chaotic regime. In certain circumstances, as in conditions of high humidity, natural ecosystems may stay inside a limit which is close but lower than chaos. Moreover, an important stochastic component was not observed.

Knowledge Graph Embedding for Hierarchical Entities Based on Auto-Embedding Size

Knowledge graph embedding represents entities and relations as low-dimensional continuous vectors. Recently, researchers have attempted to leverage the potential semantic connections between entities with hierarchical relationships in the knowledge graph. However, existing knowledge embedding methods that model entity hierarchical structures face the following challenges: (1) Setting the same embedding dimension for entities at different hierarchical levels can lead to overfitting issues and the waste of storage and computational resources; (2) Manually searching the embedding dimension in discrete space makes it easy to miss the optimal parameters and increases training costs. Therefore, we propose a knowledge embedding method, HEAES, that automatically learns the embedding dimensions for entities based on their hierarchical structure. Specifically, we first propose a new modeling method to capture the hierarchical relationships of entities, and we then train the model on a triple classification task. Then, we adaptively learn the pruning threshold to trim the embedding vectors, automatically learning different embedding dimensions for entities at different hierarchical levels. Experiments on the YAGO26K and DB111K datasets verify that introducing entity hierarchical information helps boost the model performance.

Grain storage temperature prediction based on chaos and enhanced RBF neural network

Grain storage has very strict temperature requirements. Aiming at the problems of nonlinear characteristics and poor prediction accuracy of temperature parameters in grain storage, a combination of chaos theory and enhanced radial basis neural network is proposed as a temperature prediction model for grain storage (C-ERBF). The model first determines the embedding dimension and time delay of the grain storage temperature sequence using chaos theory. It then calculates the Lyapunov exponent to confirm its chaotic properties and reconstructs the sequence in the phase space to extract the hidden dynamic information and structure behind the sequence. Furthermore, the q-Normalized Least Mean Square Fourth (qXE-NLMF) algorithm is designed to enhance the radial basis function (RBF) neural network model for weight updating, to improve its prediction accuracy, and to accelerate the training speed of the model. As verified by the simulation experiments of Mackey–Glass chaotic time series prediction, the enhanced RBF (ERBF) network has faster convergence speed and lower steady-state error compared to the traditional RBF network. Finally, the optimized dataset from chaos theory is input into the model to achieve accurate predictions of grain storage temperature series. The experimental results show that the proposed C-ERBF model has higher prediction accuracy compared to other time series prediction methods. It can realize the grain pile temperature in advance, and take control measures in advance. This proactive approach significantly reduces the consumption of stored grain and prevents issues before they arise.

Correlation Fuzzy measure of multivariate time series for signature recognition.

Distinguishing different time series, which is determinant or stochastic, is an important task in signal processing. In this work, a correlation measure constructs Correlation Fuzzy Entropy (CFE) to discriminate Chaos and stochastic series. It can be employed to distinguish chaotic signals from ARIMA series with different noises. With specific embedding dimensions, we implemented the CFE features by analyzing two available online signature databases MCYT-100 and SVC2004. The accurate rates of the CFE-based models exceed 99.3%.

Legendre Polynomial Fitting-Based Permutation Entropy Offers New Insights into the Influence of Fatigue on Surface Electromyography (sEMG) Signal Complexity.

In a recently published work, we introduced local Legendre polynomial fitting-based permutation entropy (LPPE) as a new complexity measure for quantifying disorder or randomness in time series. LPPE benefits from the ordinal pattern (OP) concept and incorporates a natural, aliasing-free multiscaling effect by design. The current work extends our previous study by investigating LPPE's capability to assess fatigue levels using both synthetic and real surface electromyography (sEMG) signals. Real sEMG signals were recorded during biceps brachii fatiguing exercise maintained at 70% of maximal voluntary contraction (MVC) until exhaustion and were divided into four consecutive temporal segments reflecting sequential stages of exhaustion. As fatigue levels rise, LPPE values can increase or decrease significantly depending on the selection of embedding dimensions. Our analysis reveals two key insights. First, using LPPE with limited embedding dimensions shows consistency with the literature. Specifically, fatigue induces a decrease in sEMG complexity measures. This observation is supported by a comparison with the existing multiscale permutation entropy (MPE) variant, that is, the refined composite downsampling (rcDPE). Second, given a fixed OP length, higher embedding dimensions increase LPPE's sensitivity to low-frequency components, which are notably present under fatigue conditions. Consequently, specific higher embedding dimensions appear to enhance the discrimination of fatigue levels. Thus, LPPE, as the only MPE variant that allows a practical exploration of higher embedding dimensions, offers a new perspective on fatigue's impact on sEMG complexity, complementing existing MPE approaches.

Fractal-based supervised approach for dimensionality reduction of hyperspectral images

Dimensionality reduction is one of the most challenging and crucial issues apart from data mining, security, and scalability, which have retained much traction due to the ever-growing need to analyze the large volumes of data generated daily. Fractal Dimension (FD) has been successfully used to characterize data sets and has found relevant applications in dimension reduction. This paper presents an application of the FD Reduction (FDR) Algorithm on geospatial hyperspectral data, examining its usefulness for data sets with a relatively high embedding dimension. We examine the algorithm at two levels. First is the conventional FDR approach (unsupervised) at the image level. Alternatively, we propose a pixel-level supervised approach for band reduction based on time-series complexity analysis. Techniques for determining an optimal intrinsic dimension for the dataset using these two techniques are examined. We also develop a parallel GPU-based implementation for the unsupervised image-level FDR algorithm, reducing the run-time by nearly 10 times. Furthermore, both approaches use a support vector machine classifier to compare the classification performance of the original and reduced image obtained.

Adaptive variable sampling T-SSD algorithm

Addressing the challenges of non-unique decomposition outcomes and prolonged decomposition durations in the fault feature adaptive extraction algorithm based on tensor decomposition, this paper presents a novel algorithm called the adaptive variable sampling tensor singular spectrum decomposition (T-SSD) algorithm. The proposed approach centers on decomposing multichannel time series with adaptive sampling frequency, leveraging tensor singular value decomposition. Initially, the embedding dimension and the number of resampling points were optimized by power spectral density analysis and adaptive sampling algorithm. Subsequently, a third-order tensor is constructed based on the principle of tensor-tensor-preserving order multiplication, combining trajectory tensor construction and embedding dimension. Finally, the signal decomposition and reconstruction of multichannel component signals are achieved through adaptive sampling, interpolation, and complementary back steps. Experimental signal analysis indicates that this algorithm can better extract fault characteristics from signals compared to common signal processing algorithms. In comparison with the traditional T-SSD algorithm, this method significantly improves decomposition efficiency, particularly for low-frequency components. It effectively tackles the efficiency challenge caused by data redundancy, enabling the organic fusion and adaptive decomposition of multichannel signals.

Analysis of human ambulation as a chaotic time-series: with nonlinear dynamics tools.

The aim of the present study is to investigate the complexity and stability of human ambulation and the implications on robotic prostheses control systems. Fourteen healthy individuals participate in two experiments, the first group run at three different speeds. The second group ascended and descended stairs of a five-level building block at a self-selected speed. All participants completed the experiment with seven inertial measurement units wrapped around the lower body segments and waist. The data were analyzed to determine the fractal dimension, spectral entropy, and the Lyapunov exponent (LyE). Two methods were used to calculate the long-term LyE, first LyE calculated using the full size of data sets. And the embedding dimensions were calculated using Average Mutual Information (AMI) and the False Nearest Neighbor (FNN) algorithm was used to find the time delay. Besides, a second approach was developed to find long-term LyE where the time delay was based on the average period of the gait cycle using adaptive event-based window. The average values of spectral entropy are 0.538 and 0.575 for stairs ambulation and running, respectively. The degree of uncertainty and complexity increases with the ambulation speed. The short term LyEs for tibia orientation have the minimum range of variation when it comes to stairs ascent and descent. Using two-way analysis of variance we demonstrated the effect of the ambulation speed and type of ambulation on spectral entropy. Moreover, it was shown that the fractal dimension only changed significantly with ambulation speed.

Classification of BCI-EEG Based on the Augmented Covariance Matrix.

Electroencephalography signals are recorded as multidimensional datasets. We propose a new framework based on the augmented covariance that stems from an autoregressive model to improve motor imagery classification. From the autoregressive model can be derived the Yule-Walker equations, which show the emergence of a symmetric positive definite matrix: the augmented covariance matrix. The state-of the art for classifying covariance matrices is based on Riemannian Geometry. A fairly natural idea is therefore to apply this Riemannian Geometry based approach to these augmented covariance matrices. The methodology for creating the augmented covariance matrix shows a natural connection with the delay embedding theorem proposed by Takens for dynamical systems. Such an embedding method is based on the knowledge of two parameters: the delay and the embedding dimension, respectively related to the lag and the order of the autoregressive model. This approach provides new methods to compute the hyper-parameters in addition to standard grid search. The augmented covariance matrix performed ACMs better than any state-of-the-art methods. We will test our approach on several datasets and several subjects using the MOABB framework, using both within-session and cross-session evaluation. The improvement in results is due to the fact that the augmented covariance matrix incorporates not only spatial but also temporal information. As such, it contains information on the nonlinear components of the signal through the embedding procedure, which allows the leveraging of dynamical systems algorithms. These results extend the concepts and the results of the Riemannian distance based classification algorithm.

Multi-Frequency Entropy for Quantifying Complex Dynamics and Its Application on EEG Data.

Multivariate entropy algorithms have proven effective in the complexity dynamic analysis of electroencephalography (EEG) signals, with researchers commonly configuring the variables as multi-channel time series. However, the complex quantification of brain dynamics from a multi-frequency perspective has not been extensively explored, despite existing evidence suggesting interactions among brain rhythms at different frequencies. In this study, we proposed a novel algorithm, termed multi-frequency entropy (mFreEn), enhancing the capabilities of existing multivariate entropy algorithms and facilitating the complexity study of interactions among brain rhythms of different frequency bands. Firstly, utilizing simulated data, we evaluated the mFreEn's sensitivity to various noise signals, frequencies, and amplitudes, investigated the effects of parameters such as the embedding dimension and data length, and analyzed its anti-noise performance. The results indicated that mFreEn demonstrated enhanced sensitivity and reduced parameter dependence compared to traditional multivariate entropy algorithms. Subsequently, the mFreEn algorithm was applied to the analysis of real EEG data. We found that mFreEn exhibited a good diagnostic performance in analyzing resting-state EEG data from various brain disorders. Furthermore, mFreEn showed a good classification performance for EEG activity induced by diverse task stimuli. Consequently, mFreEn provides another important perspective to quantify complex dynamics.

Understanding and mitigating dimensional collapse of Graph Contrastive Learning: A non-maximum removal approach

Graph Contrastive Learning (GCL) generates graph-level embeddings by maximizing Mutual Information between different augmented views of the same graph (positive pairs), and shows promising performance in graph representation learning (GRL) without the supervision of manual annotations. However, GCL suffers from a dimensional collapse problem, i.e., embedding vectors reside in a restricted low-dimensional subspace, curtailing the expressiveness of certain embedding dimensions. In this paper, we present a theoretical analysis identifying the smoothing effect of graph pooling and graph convolution’s implicit regularization as principal causes of dimension collapse in graph contrastive learning. To mitigate the above effects, we propose a Non-Maximum Removal Graph Contrastive Learning (nmrGCL) approach, which removes “prominent” dimensions (those significantly contributing to the similarity measure) for positive pairs within the pretext task. Comprehensive experiments on multiple benchmark datasets are conducted and the results show that the proposed nmrGCL outperforms the state-of-the-art methods.

A class of integer 3 × 3 matrices related to numerical semigroups with embedding dimension 4

The aim of this paper is to present a class of integer [Formula: see text] matrices related to numerical semigroups with embedding dimension 4, generated by four integers [Formula: see text] and [Formula: see text] such that [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. For a matrix [Formula: see text] of this class, we describe the Apéry set with respect to [Formula: see text], of the numerical semigroup related to [Formula: see text].

Estimation of Carleman operator from a univariate time series.

Reconstructing a nonlinear dynamical system from empirical time series is a fundamental task in data-driven analysis. One of the main challenges is the existence of hidden variables; we only have records for some variables, and those for hidden variables are unavailable. In this work, the techniques for Carleman linearization, phase-space embedding, and dynamic mode decomposition are integrated to rebuild an optimal dynamical system from time series for one specific variable. Using the Takens theorem, the embedding dimension is determined, which is adopted as the dynamical system's dimension. The Carleman linearization is then used to transform this finite nonlinear system into an infinite linear system, which is further truncated into a finite linear system using the dynamic mode decomposition technique. We illustrate the performance of this integrated technique using data generated by the well-known Lorenz model, the Duffing oscillator, and empirical records of electrocardiogram, electroencephalogram, and measles outbreaks. The results show that this solution accurately estimates the operators of the nonlinear dynamical systems. This work provides a new data-driven method to estimate the Carleman operator of nonlinear dynamical systems.

Bounds on injective dimension and exceptional complete intersection maps

We prove that if f : R → S is a local homomorphism of noetherian local rings, and M is a non-zero finitely generated or artinian S-module whose injective dimension over R is bounded by the difference of the embedding dimensions of R and S, then M is an injective S-module and f is an exceptional complete intersection map.

Deep network embedding with dimension selection

Network embedding is a general-purpose machine learning technique that converts network data from non-Euclidean space to Euclidean space, facilitating downstream analyses for the networks. However, existing embedding methods are often optimization-based, with the embedding dimension determined in a heuristic or ad hoc way, which can cause potential bias in downstream statistical inference. Additionally, existing deep embedding methods can suffer from a nonidentifiability issue due to the universal approximation power of deep neural networks. We address these issues within a rigorous statistical framework. We treat the embedding vectors as missing data, reconstruct the network features using a sparse decoder, and simultaneously impute the embedding vectors and train the sparse decoder using an adaptive stochastic gradient Markov chain Monte Carlo (MCMC) algorithm. Under mild conditions, we show that the sparse decoder provides a parsimonious mapping from the embedding space to network features, enabling effective selection of the embedding dimension and overcoming the nonidentifiability issue encountered by existing deep embedding methods. Furthermore, we show that the embedding vectors converge weakly to a desired posterior distribution in the 2-Wasserstein distance, addressing the potential bias issue experienced by existing embedding methods. This work lays down the first theoretical foundation for network embedding within the framework of missing data imputation.

A Conjecture for the Clique Number of Graphs Associated with Symmetric Numerical Semigroups of Arbitrary Multiplicity and Embedding Dimension

A subset S of non-negative integers No is called a numerical semigroup if it is a submonoid of No and has a finite complement in No. An undirected graph G(S) associated with S is a graph having V(G(S))={vi:i∈No∖S} and E(G(S))={vivj⇔i+j∈S}. In this article, we propose a conjecture for the clique number of graphs associated with a symmetric family of numerical semigroups of arbitrary multiplicity and embedding dimension. Furthermore, we prove this conjecture for the case of arbitrary multiplicity and embedding dimension 7.

Characterization of perfect numerical semigroups in terms of pseudo-Frobenius numbers

In the theory of numerical semigroups, characterizing numerical semigroups in terms of pseudo-Frobenius numbers is one of the fundamental problems, which is very difficult to achieve in general. This article's main purpose is to answer this problem in the case of perfect numerical semigroups having a Maximal Embedding Dimension (MED).

Characterization of Drought Propagation in the Han River Basin Based on Chaos Theory and Nonlinear Dynamic System

This study characterized drought propagation in the Han River basin based on chaos theory and nonlinear dynamic systems. The nonlinear complexity of drought propagation, specifically between meteorological and hydrological droughts, was identified using convergent cross-mapping and phase-space reconstruction techniques. The analysis results on the causality of drought propagation between meteorological and hydrological droughts revealed that among the 24 middle-sized watersheds of the Han River basin, 11 watersheds had bidirectional causality, and the other 13 watersheds had one-directional causality. This study also confirmed that there were large correlations between the two drought types when 12 of the 13 watersheds with one-directional causality had a lag time of 4 months and the remaining watershed had a delay time of 7 months. The complexity of the interaction between different droughts was confirmed through adjustments of key parameters, such as the embedding dimension, time delay, correlation dimension, and Lyapunov exponent. The approach proposed in this study made it possible to analyze the nonlinear dynamic characteristics of drought propagation, which have not been analyzed in previous studies.

A novel method of nonuniform phase space reconstruction for multivariate prediction of daily runoff

Phase space reconstruction is crucial for predicting chaotic hydrological time series. However, traditional multivariate phase space reconstruction methods, such as high-dimensional and low-dimensional phase space reconstruction (HDPSR and LDPSR) neglect variable interdependence, resulting in redundant information in the reconstructed phase space and affecting the model prediction accuracy. To address this limitation, this study proposes a new nonuniform phase space reconstruction (NUPSR) method that combines the minimum redundancy maximum relevance criterion and conditional entropy. Then, using HDPSR, LDPSR, NUPSR, particle swarm optimization (PSO), support vector regression (SVR), and bidirectional long- and short-term memory neural networks (BiLSTM), the models HDPSR-PSO-SVR, LDPSR-PSO-SVR, NUPSR-PSO-SVR, HDPSR-PSO-BiLSTM, LDPSR-PSO-BiLSTM, and NUPSR-PSO-BiLSTM are built to evaluate the performance of the NUPSR method. To demonstrate the effectiveness of the NUPSR method in multistep daily runoff prediction, the models PSO-NUPSR-BiLSTM, LDPSR-PSO-BiLSTM, and HDPSR-PSO-BiLSTM are also built. The multifactor daily runoff data from five hydrological stations in the Weihe River Basin are used to evaluate the prediction performance of all models. The results show that the state variables obtained by NUPSR are more independent and less redundant than those obtained by HDPSR and LDPSR, as evidenced by the embedded state variables and lower embedding dimension of NUPSR and the performance of the model based on NUPSR outperforms based on HDPSR and LDPSR. Particularly, the PSO-NUPSR-BiLSTM model exhibits a slower decrease in prediction accuracy compared to the other two models as the number of prediction steps increases. In summary, the NUPSR method effectively selects and reconstructs predictors, significantly improves the prediction accuracy of the coupled models, and has broad application prospects in multivariate prediction in fields such as hydrology.

An in-depth investigation of global sea surface temperature behavior utilizing chaotic modeling.

Sea surface temperature (SST), with its complex and dynamic behavior, is a major driver of ocean-atmosphere interactions. The purpose of this study is to investigate the behavior of SST and its prediction using a chaotic approach. Average mutual information (AMI) and Cao methods were used to reconstruct the phase space. The Lyapunov exponent and correlation dimension were used to investigate chaos. The Lyapunov exponent index was used to predict SST with a 5-year average prediction horizon using the local prediction method between 2023 and 2027. The results showed a 3-month delay time for the Pacific and Antarctic Oceans, and a 2-month delay time for the Atlantic, Indian, and Arctic Oceans. The optimal embedding dimension for all oceans is between 6 and 7. Our analysis reveals that the dynamics of SST in all oceans exhibit varying degrees of chaos, as indicated by the correlation dimension. The local prediction method achieves relatively accurate short-term SST predictions due to the clustering of SST points around specific attractors in the phase space. However, in the long term, the accuracy of this method decreases as the points in the phase space of SST can spread randomly. The model performance ranking with a Percent Mean Relative Absolute Error shows that the Indian Ocean has the best performance compared to other oceans, while the Atlantic, Pacific, and Antarctic and Arctic Oceans are in the next ranks. This study contributes to understanding the dynamics of SST and has practical value for use in the development of climate models.