Karhunen-Loeve Expansion Research Articles

A Distributed Non-Intrusive Load Monitoring Method Using Karhunen–Loeve Feature Extraction and an Improved Deep Dictionary

In recent years, the non-invasive load monitoring (NILM) method based on sparse coding has shown promising research prospects. This type of method learns a sparse dictionary for each monitoring target device, and it expresses load decomposition as a problem of signal reconstruction using dictionaries and sparse vectors. The existing NILM methods based on sparse coding have problems such as inability to be applied to multi-state and time-varying devices, single-load characteristics, and poor recognition ability for similar devices in distributed manners. Using the analysis above, this paper focuses on devices with similar features in households and proposes a distributed non-invasive load monitoring method using Karhunen–Loeve (KL) feature extraction and an improved deep dictionary. Firstly, Karhunen–Loeve expansion (KLE) is used to perform subspace expansion on the power waveform of the target device, and a new load feature is extracted by combining singular value decomposition (SVD) dimensionality reduction. Afterwards, the states of all the target devices are modeled as super states, and an improved deep dictionary based on the distance separability measure function (DSM-DDL) is learned for each super state. Among them, the state transition probability matrix and observation probability matrix in the hidden Markov model (HMM) are introduced as the basis for selecting the dictionary order during load decomposition. The KL feature matrix of power observation values and improved depth dictionary are used to discriminate the current super state based on the minimum reconstruction error criterion. The test results based on the UK-DALE dataset show that the KL feature matrix can effectively reduce the load similarity of devices. Combined with DSM-DDL, KL has a certain information acquisition ability and acceptable computational complexity, which can effectively improve the load decomposition accuracy of similar devices, quickly and accurately estimating the working status and power demand of household appliances.

Groundwater inverse modeling: Physics-informed neural network with disentangled constraints and errors

This study combines a physics-informed neural network (PINN) and Karhunen-Loeve expansion (KLE) (i.e., KLE-PINN) to solve the groundwater inverse problem. The hydraulic head (u) distribution is approximated by a deep neural network (DNN), while the hydraulic conductivity (K) field is constructed by KLE with given prior geostatistical information. KLE-PINN is applied to investigate the inversion using data from a single pumping test, natural gradient flow (NG), and hydraulic tomography (HT). The results from these cases demonstrate that our inverse method is robust. Our error analysis endeavors to quantify the sources of error by using two custom reference indicators, eforward and ecoupling. Moreover, the study finds that the inversion using data from multiple pumping tests (HT) yields more accurate estimates, leads to faster training convergence, and maintains higher stability. In addition, by investigating cases with different outer boundary conditions (BCs), we find that KLE-PINN is more flexible. Precisely, in scenarios with missing BCs, our network still fits well with the observed data, and the estimates capture the approximate spatial pattern in the region where the observation wells are distributed. Even with incorrect BCs, our network still performs well because the observational data constraints are strongly enforced during training.

Random Responses of Shield Tunnel to New Tunnel Undercrossing Considering Spatial Variability of Soil Elastic Modulus

This paper investigates the effect of spatial variability of soil elastic modulus on the longitudinal responses of the existing shield tunnel to the new tunnel undercrossing using a random two-stage analysis method (RTSAM). The Timoshenko–Winkler-based deterministic method considering longitudinal variation in the subgrade reaction coefficient and the random field of the soil elastic modulus discretized by the Karhunen–Loeve expansion method are combined to establish the RTSAM. Then, the proposed RTSAM is applied to carry out a random analysis based on an actual engineering case. Results show that the increases in the scale of fluctuation and the coefficient of variation of the soil elastic modulus lead to higher variabilities of tunnel responses. A decreasing pillar depth and mean value of the soil elastic modulus and an increasing skew angle strengthen the effect of the spatial variability of the soil elastic modulus on tunnel responses. The variabilities of tunnel responses under the random field of the soil elastic modulus are overestimated by the Euler–Bernoulli beam model. The results of this study provide references for the uncertainty analysis of the new tunneling-induced responses of the existing tunnel under the random field of soil properties.

Analysis of slope stochastic fields using a novel deep learning model with attention mechanism

This paper proposes a novel deep learning model incorporating attention mechanisms for the analysis of slope stochastic fields. Initially, a deep learning model is designed to digitally image the stochastic field features of soil strength variability. This is achieved by discretizing the slope soil stochastic field using the Karhunen-Loeve expansion method and transforming the discrete results into digital images. These images are then used to establish a Convolutional Neural Network (CNN) surrogate model that maps the implicit relationship between stochastic field images and slope functional function values, thus calculating the probability of slope failure. The precision of the CNN surrogate model is enhanced through Bayesian optimization and five-fold cross-validation. Moreover, to overcome the limitations of existing data-driven landslide stability prediction models, this study also introduces a Spatial-Temporal Attention (STA) mechanism. By combining the CNN with Long Short-Term Memory (LSTM) networks, the model can accurately approximate the actual results of slope stability calculations in scenarios of high-dimensional representation imaging of stochastic fields and low-probability slope instability. Consequently, this significantly improves the computational efficiency of slope reliability analysis considering stochastic field simulations.

Human Activity Detection Events Through Human Eye Reflection using Bystander Analyzer

ABSTRACT Automatic video-based human activity recognition has shown promise and is mostly used in video surveillance applications for diverse purposes. However, there are still substantial performance issues that make real-world implementation challenging. The main barrier to the accurate detection of human movement remains to the perspective problem, which results from the fact that video sequences are frequently shot from random camera angles. Therefore, the main focus of this study is on how human eyesight reflects in different camera views that are used to detect human activity or identify intruders. First, a gaussian mixture model that previously used the Minimal Spanning Tree for segmentation is used for pre-processing. Using the pixel-based Kruskal methodology and this method, the input data set’s minimal weight is precisely determined. Segmentation comes next. Independent discriminant features are extracted using transformation using Karhunen-Loeve expansion, which isolates human behavior based on the Person Correlation Coefficient. To effectively identify data, Deep Lens Classifier is also employed to look for any suspect human behavior. With an impressive 78.8774% accuracy, 28.6961% sensitivity, 98.50% specificity, 75.6734% precision, 48.781% recall, and 65.10% F-measure, this approach is unique in the field of detecting human activity. Finally, our proposed system’s MATLAB performance retrieves the accurate detection.

On usage of the neural network technologies in the it- structure components’ diagnosing.

The idea of using neural network technologes to prove electrophysical diagnostic methods based on the integral physical effects of IT structure components is considered. It is proposed to transform the received information using a discrete Karhunen-Loeve expansion, which gives the minimum root mean square error of packing a priory vectors in multidimensional space. The use of neural networks: MLP, self-organizing (Kohonen Maps) and RBF in MATLAB environment is verified. The best result for microcircuits was obtained using probabilistic RBF-neural networks. A new neural network approach to diagnostics made it possible to perform individual sorting of elements and ststistical evaluation of the IT structure components batch.

Random analysis method for nonlinear interaction between shield tunnel and spatially variable soil

The spatial variability of soil properties is likely to have a remarkable influence on the longitudinal responses of shield tunnels. This paper proposes a one-dimensional random analysis method (RAM) for the interaction between the shield tunnel with circumferential joints and the spatially variable soil with nonlinear elastoplastic behaviors. A deterministic short beam-spring model with longitudinal variation in the initial subgrade reaction coefficient and the ultimate soil resistance is developed and solved by the finite difference method. The random field of the soil properties discretized by the Karhunen-Loeve expansion method is used to establish the RAM. A hypothetical example is further presented to demonstrate the applicability of the proposed method. The results imply that the soil nonlinearity may lead to a negative effect on the reliability control of the tunnel structure and should be considered to obtain more reasonable probabilistic evaluations of the tunnel deformations. The increasing joint rotational stiffness or shearing stiffness results in the lower variabilities of the maximum tunnel settlement, and reduces the probabilities of the maximum joint opening or joint dislocation exceeding the allowable value, respectively.

Tribodynamic analysis of spur gear drives with uncertain time-variant loads: An interval process approach

Gear drives are inherently subjected to time-variant loads, which always create an uncertainty and may degrade the vibration performance and system reliability. However, due to their complicated statistical properties, it is impractical to employ probability-based dynamic analysis methods. Additionally, most studies neglect the elastohydrodynamic lubrication behaviors between tooth surfaces under time-variant load conditions. A novel tribodynamic analysis method is proposed to evaluate system responses efficiently, in which the uncertain time-variant loads are described using interval processes. Firstly, a deterministic tribodynamic model coupling the system dynamics and elastohydrodynamic lubrication is established. To reduce the computational cost, the interval Karhunen-Loeve expansion is employed to simplify the interval processes using a small number of uncorrelated interval variables, while the Chebyshev subinterval decomposition is carried out to approximate actual responses using one-variable functions. The proposed method shares a comparable accuracy with the time-consuming Monte Carlo simulation and experiments. Parametric studies show that the uncertain time-variant loads significantly affect the uncertainty of vibration and induce a more hazardous lubricating condition. Additionally, the variation trend of uncertain tribodynamic responses is highly correlated with the rotating speed.

Three-dimensional reliability stability analysis of earth-rock dam slopes reinforced with permeable polymer

Polymer is a new repair and reinforcement material for earth-rock dams developed in recent years. Models are developed where spatial variability is incorporated into the analytical framework to evaluate the stability and reliability of reinforced three-dimensional dam slopes. The Karhunen-Loeve (K-L) expansion method is employed to generate three-dimensional random fields. The simplified Bishop method and Monte Carlo method are used to calculate the safety factors and failure probabilities of the dam slopes. With the implementation of polymer grouting reinforcement, the safety factors are improved and the failure probabilities are reduced. The reinforcement effect will be affected by the amount of polymer dosing, dam slope characteristics, and polymer grouting parameters. The impact of various longitudinal autocorrelation distances on reliability is influenced by the length of the dam slope itself. An increase of the safety factor by approximately 37% can be achieved through a polymer dosing of around 3%. Nevertheless, with the elevation of dam slope height and slope ratio, the effectiveness of polymer reinforcement is diminished.

Band Gap Properties in Metamaterial Beam with Spatially Varying Interval Uncertainties

First, this study proposed a metamaterial beam model with spatially varying interval density. The interval dynamic equation of this model could be established by incorporating the decomposition results of the interval field based on Karhunen–Loeve expansion into the finite element method. An interval perturbation finite element method was developed to evaluate the bounds of the dynamic response interval vector. Then, an interval vibration transmission analysis could be performed, and the frequency range of the safe band gap could be determined. Meanwhile, Monte Carlo simulations and the vertex method are also presented to provide reference solutions. By comparison, it was found that the calculation accuracy of the interval perturbation finite element method was acceptable. The numerical results also showed that the safe band gap range was significantly smaller than that of the deterministic band gap.

Modeling the biological growth with a random logistic differential equation

We modeled biological growth using a random differential equation (RDE), where the initial condition is a random variable, and the growth rate is a suitable stochastic process. These assumptions let us obtain a model that represents well the random growth process observed in nature, where only a few individuals of the population reach the maximal size of the species, and the growth curve for every individual behaves randomly. Since we assumed that the initial condition is a random variable, we assigned a priori density, and we performed Bayesian inference to update the initial condition’s density of the RDE. The Karhunen–Loeve expansion was then used to approximate the random coefficient of the RDE. Then, using the RDE’s approximations, we estimated the density f(p, t). Finally, we fitted this model to the biological growth of the giant electric ray (or Cortez electric ray) Narcine entemedor. Simulations of the solution of the random logistic equation were performed to construct a curve that describes the solutions’ mean for each time. As a result, we estimated confidence intervals for the mean growth that described reasonably well the observed data. We fit the proposed model with a training dataset, and the model is tested with a different dataset. The model selection is performed with the square of the errors.

Functional data clustering via information maximization

ABSTRACT A novel method for clustering functional data is introduced that utilizes information maximization. The method employs unsupervised learning to develop a probabilistic classifier that maximizes the mutual information or squared loss mutual information between data points and their corresponding cluster assignments. A significant advantage of this method is that it involves only continuous optimization of model parameters, which is simpler than discrete optimization of cluster assignments and avoids the drawbacks of generative models. Unlike existing methods, this method does not necessitate the estimation of probability densities of Karhunen-Loeve expansion scores under different clusters and does not require the common eigenfunction assumption. The efficacy of the proposed method is demonstrated through simulation studies and real data analysis. Additionally, the technique permits out-of-sample clustering, and its performance is comparable to that of supervised classifiers.

Sparse polynomial chaos expansion for nonlinear finite element simulations with random material properties

AbstractThis contribution deals with the uncertainty quantification for applied nonlinear structural engineering problems, including high stochastic dimensions. A finite element problem with different material models is investigated. The efficiency, accuracy and convergence of sparse PCE are studied numerically and compared with Monte‐Carlo Simulation (MCS) for non‐linear structural analysis including elasto‐plastic and damage models. In both models, the Young's modulus is considered as random fields discretised by Karhunen Loeve Expansion (KLE). In the provided studies, sparse PCE converges fast and is highly efficient for linear elastic and elasto‐plastic material models. However, sparse PCE loses its effectiveness and exhibits lower accuracy for the damage material model.

Bayesian inversion of log-normal eikonal equations

We study the Bayesian inverse problem for inferring the log-normal slowness function of the eikonal equation, given noisy observation data on its solution at a set of spatial points. We contribute rigorous proofs on the existence and well-posedness of the problem. We then study approximation of the posterior probability measure by solving the truncated eikonal equation, which contains only a finite number of terms in the Karhunen–Loeve expansion of the slowness function, by the fast marching method (FMM). The error of this approximation in the Hellinger metric is deduced in terms of the truncation level of the slowness and the grid size in the FMM resolution. It is well known that the plain Markov chain Monte Carlo (MCMC) procedure for sampling the posterior probability is highly expensive. We develop and justify the convergence of a multilevel MCMC method. Using the heap sort procedure in solving the forward eikonal equation by the FMM, our multilevel MCMC method achieves a prescribed level of accuracy for approximating the posterior expectation of quantities of interest, requiring only an essentially optimal level of complexity. Numerical examples confirm the theoretical results.

Robust Topology Optimization of Continuum Structures under the Hybrid Uncertainties: A Comparative Study

Due to the inevitable involvement of multisource uncertainties related to the load, material property and geometry in practical engineering designs, robust topology optimization (RTO) has recently attracted increasing attention to account for these uncertain effects. However, the majority of the existing RTO works are concerned with single source uncertainty, and very few studies have considered the multisource (hybrid) uncertainties simultaneously. To this end, a comparative study on the hybrid uncertainties (HU), i.e., material-loading, geometric-loading, material-geometric, and material-geometric-loading uncertainties, for RTO of continuum structures is presented in this paper. A truncated Karhunen-Loeve expansion is adopted for uncertainty representation and a sparse grid collocation method for uncertainty propagation of the objective function and constraints. Effects of the various HU on the compliance and robust design are comprehensively investigated and compared with the RTO models under individual component uncertainty using two continuum benchmarks. An important observation from the results is that the hybrid uncertainty model is a conservative state, and the resulting RTO designs tend towards those with loading uncertainty only.

Uncertain Dynamic Characteristic Analysis for Structures with Spatially Dependent Random System Parameters.

This work presents a robust non-deterministic free vibration analysis for engineering structures with random field parameters in the frame of stochastic finite element method. For this, considering the randomness and spatial correlation of structural physical parameters, a parameter setting model based on random field theory is proposed to represent the random uncertainty of parameters, and the stochastic dynamic characteristics of different structural systems are then analyzed by incorporating the presented parameter setting model with finite element method. First, Gauss random field theory is used to describe the uncertainty of structural material parameters, the random parameters are then characterized as the standard deviation and correlation length of the random field, and the random field parameters are then discretized with the Karhunen-Loeve expansion method. Moreover, based on the discretized random parameters and finite element method, structural dynamic characteristics analysis is addressed, and the probability distribution density function of the random natural frequency is estimated based on multi-dimensional kernel density estimation method. Finally, the random field parameters of the structures are quantified by using the maximum likelihood estimation method to verify the effectiveness of the proposed method and the applicability of the constructed model. The results indicate that (1) for the perspective of maximum likelihood estimation, the parameter setting at the maximum value point is highly similar to the input parameters; (2) the random field considering more parameters reflects a more realistic structure.

Homotopy based stochastic finite element model updating with correlated static measurement data

This paper focuses on studying the impaction of the correlation between the measurement data on structural model updating. A new homotopy based stochastic finite element model updating frame is constructed to cope with the correlated static measurement data. To judge whether considering the correlation of the measurement data, the sensitivity analysis of structural responses about structural parameters is implemented firstly. Then the discrete Karhunen–Loeve expansion is utilized to transform the correlated measurement data into a linear combination of multiple independent random variables. Furthermore, a novel stochastic model updating equation about the correlated static measurement data is set up, and is solved by the homotopy stochastic finite element method. It is significant to check the consistence of the correlation coefficients of the responses of the updated model with those of the measured data when the correlation actually exists. In this case, the proposed approach can effectively update the structural model.

An AI-based spectral data analysis process for recognizing unique plant biomarkers and disease features

This study introduces a novel method using reduced datasets for extracting plant reflectance signatures. Plant reflectance data from hyperspectral imaging sensors were used to develop biomarker signatures for plant identification and early disease detection purposes. The Karhunen-Loeve Expansion (KLE) of spectral reflectance data, taken from healthy and diseased plants (two case studies of citrus canker in citrus and Laurel wilt in avocado), were used to identify a basis set of functions that represents the distribution of the reflected signal energy. By spectral decomposition, the eigenvalues were related to the KLE basis set. The eigenvalues can be used to identify the KLE eigenvectors, which comprise the highest variation in the data. These components can be interpreted as the weighted variables which carry with them most of the information on the reflectance spectrum of the plant. From indications presented by this multivariate KLE analysis, a frequency reconstruction was adapted to convert the eigenvector information to a wave function. This reconstruction via KLE and frequency transformation formed the signature identification process (i.e., unique biomarker). These frequency spectra can be used as average signature reflectance patterns for plant identification, classification, and disease biomarkers. Defining these spectral identification biomarkers or signatures is purposeful since it could lead to less invasive classification and disease diagnostics techniques. The techniques used to determine these reflectance spectra require a unique spectrum reconstruction method, using Fourier transform methods and KLE for data reduction.

A coupling analysis method of foundation soil dynamic responses induced by metro train based on PDEM and stochastic field theory

Foundation soil is an important medium for vibration wave propagation in the running of metro train. The spatial variability of soil properties has an important influence on vibration response. The parameters of foundation soil are often determined. Based on the probability density evolution method (PDEM) and stochastic field theory, a coupling model of the metro train-track-shield tunnel-foundation soil is established. The stochastic field is expressed by the series of stochastic variables and deterministic space functions by the Karhunen-Loeve Expansion (KLE) method. The stochastic parameters of the cross-section of tunnel-foundation soil are specified to simulate the stochastic field of foundation soil. Based on PDEM, the environmental vibration caused by metro trains in shield tunnels is studied. The results of ground surface vibration displacement under different metro train speeds are analyzed, and the upper and lower limits and probability density information of ground surface vibration displacement are obtained. The results show that the probability density distribution of ground vibration displacement at each time is approximately normal due to the stochastic field of foundation soil. The peak and valley of the vibration displacement curve at the inverted arch of the tunnel are caused by the wheel distance of the metro train. The probability density distribution of velocity and acceleration is more sensitive to the stochastic field of velocity foundation soil and the change in metro train speed. The maximum Z vibration level results under different metro train speeds all meet the requirements of vibration allowable limits in the specification.

Influence of Spatially Distributed Out-of-Plane CFRP Fiber Waviness on the Estimation of Knock-Down Factors Based on Stochastic Numerical Analysis

The presence of waviness defects in CFRP materials due to fiber undulation affects the structural performance of composite structures. Hence, without a reliable assessment of the resulting material properties, the full weight-saving potential cannot be exploited. Within the paper, a probabilistic numerical approach for improved estimation of material properties based on spatially distributed fiber waviness is presented. It makes use of a homogenization approach to derive viable knock-down factors for the different plies on the laminate level for reference material and is demonstrated for a representative tension loadcase. For the stochastic analysis, a random field is selected which describes the complex inner geometry of the plies in the laminate model and is numerically discretized by the Karhunen–Loeve expansion methods to fit into an FE model for the strength analysis. Conducted analysis studies reveal a substantial influence of randomly distributed waviness defects on the derived knock-down factors. Based on a topological analysis of the waviness fields, the reduction of the material properties was found to be weakly negatively correlated related to simple geometrical properties such as maximum amplitudes of the waviness field, which justifies the need for further subsequent sensitivity studies.