Generalization Of Principal Component Analysis Research Articles

Robust generalized PCA for enhancing discriminability and recoverability

The dependency of low-dimensional embedding to principal component space seriously limits the effectiveness of existing robust principal component analysis (PCA) algorithms. Simply projecting the original sample coordinates onto orthogonal principal component directions may not effectively address various noise-corrupted scenarios, impairing both discriminability and recoverability. Our method addresses this issue through a generalized PCA (GPCA), which optimizes regression bias rather than sample mean, leading to more adaptable properties. And, we propose a robust GPCA model with joint loss and regularization based on the ℓ2,μ norm and ℓ2,ν norms, respectively. This approach not only mitigates sensitivity to outliers but also enhances feature extraction and selection flexibility. Additionally, we introduce a truncated and reweighted loss strategy, where truncation eliminates severely deviated outliers, and reweighting prioritizes the remaining samples. These innovations collectively improve the GPCA model’s performance. To solve the proposed model, we propose a non-greedy iterative algorithm and theoretically guarantee the convergence. Experimental results demonstrate that the proposed GPCA model outperforms the previous robust PCA models in both recoverability and discrimination.

Multi-way modelling of oral microbial dynamics and host-microbiome interactions during induced gingivitis

Gingivitis—the inflammation of the gums—is a reversible stage of periodontal disease. It is caused by dental plaque formation due to poor oral hygiene. However, gingivitis susceptibility involves a complex set of interactions between the oral microbiome, oral metabolome and the host. In this study, we investigated the dynamics of the oral microbiome and its interactions with the salivary metabolome during experimental gingivitis in a cohort of 41 systemically healthy participants. We use Parallel Factor Analysis (PARAFAC), which is a multi-way generalization of Principal Component Analysis (PCA) that can model the variability in the response due to subjects, variables and time. Using the modelled responses, we identified microbial subcommunities with similar dynamics that connect to the magnitude of the gingivitis response. By performing high level integration of the predicted metabolic functions of the microbiome and salivary metabolome, we identified pathways of interest that describe the changing proportions of Gram-positive and Gram-negative microbiota, variation in anaerobic bacteria, biofilm formation and virulence.

Efficient PCA denoising of spatially correlated redundant MRI data

Abstract Marčenko-Pastur PCA (MPPCA) denoising is emerging as an effective means for noise suppression in MR imaging (MRI) acquisitions with redundant dimensions. However, MPPCA performance can be severely compromised by spatially correlated noise—an issue typically affecting most modern MRI acquisitions—almost to the point of returning the original images with little or no noise removal. In this study, we explore different threshold criteria for principal component analysis (PCA) component classification that enable efficient and robust denoising of MRI data even when noise exhibits high spatial correlations, especially in cases where data are acquired with Partial Fourier and when only magnitude data are available. We show that efficient denoising can be achieved by incorporating a-priori information about the noise variance into PCA denoising thresholding. Based on this, two denoising strategies developed here are: 1) General PCA (GPCA) denoising that uses a-priori noise variance estimates without assuming specific noise distributions; and 2) Threshold PCA (TPCA) denoising which removes noise components with a threshold computed from a-priori estimated noise variance to determine the upper bound of the Marčenko-Pastur (MP) distribution. These strategies were tested in simulations with known ground truth and applied for denoising diffusion MRI data acquired using pre-clinical (16.4T) and clinical (3T) MRI scanners. In synthetic phantoms, MPPCA denoising failed to denoise spatially correlated data, while GPCA and TPCA better classified components as dominated by signal/noise. In cases where the noise variance was not accurately estimated (as can be the case in many practical scenarios), TPCA still provides excellent denoising performance. Our experiments in pre-clinical diffusion data with highly corrupted by spatial correlated noise revealed that both GPCA and TPCA robustly denoised the data while MPPCA denoising failed. In in vivo diffusion MRI data acquired on a clinical scanner in healthy subjects, MPPCA weakly removed noised, while TPCA was found to have the best performance, likely due to misestimations of the noise variance. Thus, our work shows that these novel denoising approaches can strongly benefit future pre-clinical and clinical MRI applications.

Comparative study and relationship analysis between purine content, uric acid, superoxide dismutase, and growth traits in purebred and crossbred Thai native chickens.

The objective was to compare and analyze the relationship between growth, purine content, uric acid, and superoxide dismutase (SOD) in purebred and crossbred Thai native chickens. A total of 300 Thai native chickens were divided into 3 groups. Group 1 was purebred Thai native chickens (100%TN), Group 2 was 50% Thai native chickens (50%TN), and Group 3 was 25% Thai native chickens (25%TN). Data included the body weight (BW), average daily gain (ADG), and breast circumference (BrC). At 6, 8, and 10 weeks of age, 10 chickens from each group were randomly euthanized to collect breast meat, liver, and blood samples to analyze the purine content consisting of total purine, adenine, guanine, xanthine, and hypoxanthine, and uric acid, in breast meat and liver and SOD in blood. A general linear model, Pearson correlation and principal component analysis were used to analyze the significant differences and relationship between variables. The results showed the 25%TN group had the highest growth traits at every age, while the 100%TN group had the lowest (p < 0.05). Consistent with the analysis results of purine values, purine content and uric acid in breast meat and liver and SOD in blood decreased with age (p < 0.05). The correlations between purine content (total purine, adenine, guanine, xanthine, and hypoxanthine) and growth traits (BW, ADG, and BrC) ranged from moderate negative to moderate positive (-0.542 to 0.253)(p < 0.05). The correlations between uric acid and growth traits (0.348-0.760) and SOD and growth traits (0.132-0.516) were low to moderate positive with significant differences (p < 0.05). The principal component plot, which highlighted three principal components (PC 1, PC 2, and PC 3), explained 86.44 and 86.53% of the total information in breast meat and liver for selecting animals for optimal balance of the variation in the growth traits, purine content, uric acid, and SOD. Although purebred Thai native chickens showed the lowest growth traits, purine content, uric acid, and SOD were also lowest compared to crossbred Thai native chickens. Therefore, the development of genetics in Thai native chickens to produce healthy food could be possible.

Изменчивость лицевого скелета у носителей джетыасарской археологической культуры Восточного Приаралья по данным 3D геометрической морфометрии

Introduction. The origin of the Jetyasar culture’s population remains a subject of debate. It was generally accepted that the Xiongnu penetrated into the Aral Sea region and participated in the formation of the local population. Previous studies have shown a complex and admixed origin of the Jetyasar people. But the spread of the head deformation tradition in this population imposes serious limitations on the application of craniometric methods for studying Jetyasar samples. Those limitations can be potentially mitigated with the use of geometric morphometrics (GM). The purpose of this work was to assess the degree of variation in the facial skeleton of the Jetyasar people in comparison with populations of the Early Iron Age and the Hunno-Sarmatian period. Material and methods. Digital, three-dimensional models of the skulls were created. Ten landmarks were placed on each of the models and subjected to general Procrustes analysis (GPA) and principal component analysis (PCA). Results. The first PC of the analysis of the total male sample reflects variation in the height of the face, nasal bridge protrusion, and the relative width of the infraorbital region. The second component describes the height of the lateral part of the zygomatic process of the maxilla and the adjacent part of the zygomatic bone. The Kosasar 2 male sample is the most diverse morphologically. This burial ground, according to archeological data, belonged to a migrant population related to the Xiongnu. In the female total sample, high values of the first PC are associated with a tall face, a weak nasal bridge protrusion, and a long zygomatic-maxillary suture. The second PC describes the morphology of the zygomatic process of the upper jaw. The Jetyasar samples are morphologically distinct from the skulls from China, Mongolia, Western Siberia, and Southern Siberia. An exception are the crania from the burial grounds of Kosasar 2, 3, and Altyn-Asar 4t which are similar to the early Iron Age samples from China (inner Mongolia, Warring States period, 5 th – 3rd centuries BC), Mongolia (Xiongnu period), and Tuva (Arzhan-2). Discussion. The Xiongnu migration could become a catalyst for the admixture and ethnogenetic processes in the region. Conclusion. Our study shows that the population of the Jetyasar culture was morphologically heterogeneous at all stages of the culture’s development. The hypothesis about the migrant roots of a part of the population is confirmed by our data.

A Semi-Automatic Method on a Small Italian Sample for Estimating Sex Based on the Shape of the Crown of the Maxillary Posterior Teeth

Teeth are known to be reliable substrates for human identification and are endowed with significant sexual dimorphism not only in the size but also in the shape of the crowns. In the preliminary phase of our study (already published in 2021), a novel sex estimation method based on dental morphometric geometric (GMA) analysis combined with the artificial neural network (ANN) was developed and validated on a single dental element (first upper premolar) with an accuracy rate of 80%. This study aims to experiment and validate the combination of GMA-ANN on the upper first and second left premolars and the upper left first molar to obtain a reliable classification model based on the sexual dimorphic traits of multiple maxillary teeth of Caucasian Italian adults (115 males and 115 females). A general procrustes superimposition (GPS) and principal component analysis (PCA) were performed to study the shape variance between the sexes and to reduce the data variations. The "set-aside" approach was used to validate the accuracy of the proposed ANN. As the main findings, the proposed method correctly classified 94% of females and 68% of males from the test sample and the overall accuracy gained was 82%, higher than the odontometric methods that similarly consider multiple teeth. The shape variation between male and female premolars represents the best dimorphic feature compared with the first upper molar. Future research could overcome some limitations by considering a larger sample of subjects and experimenting with the use of computer vision for automatic landmark positioning and should verify the present evidence in samples with different ancestry.

A Field Study of Coal Fire Areas Re-Burning Behavior Assessment and Related Carbon Emissions

The re-burning of coal seams still occurs in coal fire areas after firefighting projects. A large amount of spontaneous combustion gas products is released from the surface fissure, which can cause air pollution and even threaten the production of surrounding mining areas. However, the distribution characteristics of fissure channels in the re-burning coal fire areas are local and scattered. Therefore, we developed a system for the in situ monitoring of spontaneous combustion gas emissions from fissure channels to conduct field investigations on the re-burning behavior of coal seams to explore the degree of re-burning in the coal fire area. The results showed that re-burning of the coal fire area induced fissure channels with different shapes. The CO2 concentrations are always higher than the CO concentrations among the spontaneous combustion gas products discharged from the fissure channels. The degree of re-burning of coal seams at different study locations was comprehensively determined using simplified MCE and CO2/CO ratios. The comprehensive factor α was obtained using the generalized principal component analysis (GPCA) by taking into account spontaneous combustion gas, meteorological factors, and the thermal physical characteristics of fissures. A carbon emission model for local fissure channels in the coal fire area was established, and the daily average carbon emissions at the study location were 2.56 t. Therefore, this provides essential theoretical support for taking corresponding fire extinguishing measures according to the degree of re-burning of the coal fire area.

ALASCA: An R package for longitudinal and cross-sectional analysis of multivariate data by ASCA-based methods.

The increasing availability of multivariate data within biomedical research calls for appropriate statistical methods that can describe and model complex relationships between variables. The extended ANOVA simultaneous component analysis (ASCA+) framework combines general linear models and principal component analysis (PCA) to decompose and visualize the separate effects of experimental factors. It has recently been demonstrated how linear mixed models can be included in the framework to analyze data from longitudinal experimental designs with repeated measurements (RM-ASCA+). The ALASCA package for R makes the ASCA+ framework accessible for general use and includes multiple methods for validation and visualization. The package is especially useful for longitudinal data and the ability to easily adjust for covariates is an important strength. This paper demonstrates how the ALASCA package can be applied to gain insights into multivariate data from interventional as well as observational designs. Publicly available data sets from four studies are used to demonstrate the methods available (proteomics, metabolomics, and transcriptomics).

Cyclic Nonlinear Correlation Analysis for Time Series

Principal component analysis (PCA) and kernel PCA allow the decorrelation of data with respect to a basis that is found via variance maximization. However, these techniques are based on pointwise correlations. Especially in the context of time series analysis this is not optimal. We present a novel generalization of PCA that allows to imprint any desired correlation pattern. Thus the proposed method can be used to incorporate previously known statistical dependencies between input variables into the model which is increasing the overall performance. This is achieved by generalizing the projection onto the direction of maximum variance—as known from PCA—to a projection onto a multi-dimensional subspace. We focus on the use of cyclic correlation patterns, which is especially of interest in the domain of time series analysis. Beneath introducing the presented variation of PCA, we discuss the role of this method with respect to other well-known time series analysis techniques.

Online Generalized Eigenvectors Extraction Via a Fixed-Point Approach

Generalized principal component analysis (GPCA) has been an active area of research in statistical signal processing for decades. It is used, e.g., for denoising in subspace tracking as the noise of different nature is incorporated into the procedure of maximizing signal-to-noise ratio (SNR). This paper presents a fixed-point approach concerning the principal generalized eigenvector extraction. It is based on the basis iteration for maximizing the generalized Rayleigh quotient (GRQ) with a given matrix pencil. The proposed approach extracts multiple generalized eigenvectors of a matrix pencil by exploiting the orthogonal complement structure of its estimation. It has no requirement to choose the commonly used step size. This enhances its practical applicability, as selecting an appropriate step size is a bottleneck for most gradient flow based algorithms. Our approach is more suitable for online processing because of its easy implementation and low computational complexity. To show the efficacy, efficiency and practical applicability of the proposed algorithm, we conduct several experiments, two of which concern smart antenna and blind source separation applications. Our simulation results show that the proposed algorithm outperforms several existing algorithms in terms of convergence speed as well as computational time.

A Note on Exploratory Item Factor Analysis by Singular Value Decomposition

We revisit a singular value decomposition (SVD) algorithm given in Chen et al. (Psychometrika 84:124–146, 2019b) for exploratory item factor analysis (IFA). This algorithm estimates a multidimensional IFA model by SVD and was used to obtain a starting point for joint maximum likelihood estimation in Chen et al. (2019b). Thanks to the analytic and computational properties of SVD, this algorithm guarantees a unique solution and has computational advantage over other exploratory IFA methods. Its computational advantage becomes significant when the numbers of respondents, items, and factors are all large. This algorithm can be viewed as a generalization of principal component analysis to binary data. In this note, we provide the statistical underpinning of the algorithm. In particular, we show its statistical consistency under the same double asymptotic setting as in Chen et al. (2019b). We also demonstrate how this algorithm provides a scree plot for investigating the number of factors and provide its asymptotic theory. Further extensions of the algorithm are discussed. Finally, simulation studies suggest that the algorithm has good finite sample performance.

Phenotyping circulating endothelial cells as a marker for transplant kidney damage

Background & Aim The diagnosis of transplant rejection is mostly based on late and insensitive markers that do not reliably discriminate from other causes of kidney transplant injury. Early, specific and minimally-invasive biomarkers may improve rejection diagnosis. Endothelial cell (EC) layer damage is one of the reasons for the functional deterioration of rejecting kidney transplants. Bioassays quantifying circulating EC numbers and its characteristics as a read-out parameter is an attractive method to monitor kidney transplant integrity. Methods, Results & Conclusion We examined the association between the count and phenotypic characteristics of circulating EC from peripheral blood of kidney transplant recipients and clinical outcome, including allograft function (eGFR using creatinine). A cohort of 52 kidney recipients transplanted in 2019 was studied. Circulating EC were measured 1 day before transplantation and 3 and 7 days after transplantation. In addition, blood was sampled and analyzed whenever acute rejection was suspected throughout the first three months after transplantation. Flow cytometry assessed the number of circulating EC by staining for CD31, CD34, CD45, CD105, CD133, and CD146. This panel also included, the kidney injury molecule (KIM1). Data analysis was performed using a non linear generalization of principal component analysis (PCA), unsupervised clustering, and tSNE visualization for unbiased selection of circulating EC (Fig 1). Paired analysis revealed that the number of circulating EC increased 2.6 [0.5-5] fold at the third day after transplantation, recovering to preoperative levels at day 7 post-transplant (Fig 2a). Patients who had a biopsy proven acute rejection and recorded a higher amount of circulating EC in comparison to the previous measured time-point, showed a significant (2-fold) increase in the MFI for the damage marker KIM-1 (p

Modified Subspace Constrained Mean Shift Algorithm

A subspace constrained mean shift (SCMS) algorithm is a non-parametric iterative technique to estimate principal curves. Principal curves, as a nonlinear generalization of principal components analysis (PCA), are smooth curves (or surfaces) that pass through the middle of a data set and provide a compact low-dimensional representation of data. The SCMS algorithm combines the mean shift (MS) algorithm with a projection step to estimate principal curves and surfaces. The MS algorithm is a simple iterative method for locating modes of an unknown probability density function (pdf) obtained via a kernel density estimate. Modes of a pdf can be interpreted as zero-dimensional principal curves. These modes also can be used for clustering the input data. The SCMS algorithm generalizes the MS algorithm to estimate higher order principal curves and surfaces. Although both algorithms have been widely used in many real-world applications, their convergence for widely used kernels (e.g., Gaussian kernel) has not been sown yet. In this paper, we first introduce a modified version of the MS algorithm and then combine it with different variations of the SCMS algorithm to estimate the underlying low-dimensional principal curve, embedded in a high-dimensional space. The different variations of the SCMS algorithm are obtained via modification of the projection step in the original SCMS algorithm. We show that the modification of the MS algorithm guarantees its convergence and also implies the convergence of different variations of the SCMS algorithm. The performance and effectiveness of the proposed modified versions to successfully estimate an underlying principal curve was shown through simulations using the synthetic data.

Convolutional PCA for Multiple Time Series

We study a fundamental generalization of principal component analysis (PCA) that looks for a small set of common time series, i.e., principal components (PCs), whose filtered versions can explain most of the variances of multiple observed time series. This problem boils down to PCA in the frequency domain, in principle. But, frequency domain processing suffers from aliasing, and brings inconveniences when handling certain time domain properties like nonstationarity, and sparsity. We propose novel time, and $z$ -domain costs for such PCA, and study its properties in detail with setting of either finite or diverging filter lengths. We further discuss its implementations, and possible extensions, and present numerical results for empirical performance study. Convolution is used to extract these PCs, thus the name convolutional PCA.

NPSA: Nonorthogonal Principal Skewness Analysis.

Principal skewness analysis (PSA) has been introduced for feature extraction in hyperspectral imagery. As a thirdorder generalization of principal component analysis (PCA), its solution of searching for the local maximum skewness direction is transformed into the problem of calculating the eigenpairs (the eigenvalues and the corresponding eigenvectors) of a coskewness tensor. By combining a fixed-point method with an orthogonal constraint, the new eigenpairs are prevented from converging to the same previously determined maxima. However, in general, the eigenvectors of the supersymmetric tensor are not inherently orthogonal, which implies that the results obtained by the search strategy used in PSA may unavoidably deviate from the actual eigenpairs. In this paper, we propose a new nonorthogonal search strategy to so lve this problem and the new algorithm is named nonorthogonal principal skewness analysis (NPSA). The contribution of NPSA lies in the finding that the search space of the eigenvector to be determined can be enlarged by using the orthogonal complement of the Kronecker product of the previous eigenvector with itself, instead of its orthogonal complement space. We also give a detailed theoretical proof on why we can obtain the more accurate eigenpairs through the new search strategy by comparison with PSA. In addition, after some algebraic derivations, the complexity of the presented algorithm is also greatly reduced. Experiments with both simulated data and real multi/hyperspectral imagery demonstrate its validity in feature extraction.

Phenetic analysis of Curcuma spp. in Yogyakarta, Indonesia based on morphological and anatomical characters

Abstract. Sungkawati M, Hidayati L, Daryono BS, Purnomo. 2019. Phenetic analysis of Curcuma spp. in Yogyakarta, Indonesia based on morphological and anatomical characters. Biodiversitas 20: 2340-2347. Curcuma spp., also known as ginger (Zingiberaceae), has economic value in traditional medicine. However, its many morphological variations cause difficulties in identification and classification. Therefore, observation of its morphological and anatomical characteristics, and of the phenetic relationships between Curcuma species, is important. This research aims to determine the specific characteristics of Curcuma spp. and to examine its phenetic relationships based on morphological and anatomical characteristics. The research was conducted in July 2018-February 2019 in the Bantul, Gunungkidul, Sleman and Karanganyar regions. Observation of the anatomical characteristics was conducted on the leaves and rhizomes and the data analyzed according to descriptive and quantitative/numerical methods. Clustering analysis with the Gower General Similarity Coefficient and Principal Component Analysis (PCA) was performed to determine the role of each character in groupings. The results of the research found seven species from 23 OTUs observed (C. aeruginosa, C. domestica, C. manga and C. xanthorrhiza, each with four OTUs; C. soloensis and C. zedoaria, with three OTUs each; and C. heyneana with one OTU) and showed that specific morphological characteristics were found in the flesh color rhizome, pseudostem color and midrib color. Specific anatomical characteristics were evident in the secretion cell color and the presence of trichomes on the leaves and rhizomes. The dendrogram shows a 0.70 phenon line consisting of two groups, group A (C. soloensis and C. domestica) fused in a 0.760 similarity index, and group B (C. aeruginosa, C. mangga, C. heyneana, C. soloensis, C. xanthorrhiza and C. zedoaria) fused in a 0.654 similarity index, which means that C. soloensis and C. domestica have a close phenetic relationship. The 0.80 phenon line consisted of five groups: C. domestica, C. soloensis, C. xanthorrhiza, C. zedoaria-C. mangga-C. heyneana and C. aeruginosa.

Data clustering based on principal curves

In this contribution we present a new method for data clustering based on principal curves. Principal curves consist of a nonlinear generalization of principal component analysis and may also be regarded as continuous versions of 1D self-organizing maps. The proposed method implements the k-segment algorithm for principal curves extraction. Then, the method divides the principal curves into two or more curves, according to the number of clusters defined by the user. Thus, the distance between the data points and the generate curves is calculated and, afterwards, the classification is performed according to the smallest distance found. The method was applied to nine databases with different dimensionality and number of classes. The results were compared with three clustering algorithms: the k-means algorithm and the 1-D and 2-D self-organizing map algorithms. Experiments show that the method is suitable for clusters with elongated and spherical shapes and achieved significantly better results in some data sets than other clustering algorithms used in this work.

Morfometrik tubuh kambing Peranakan Ettawa pada berbagai paritas di balai Pembibitan dan Budidaya Ternak Terpadu Kabupaten Kendal

The objective of this study was to determine the relation between morphometric of different parity and litter size in Ettawa Grade goat. The material of this study was 47 heads of Ettawa Grade goat. Purposive sampling was applied to determine sample based on population and breeding policy. The Ettawa Grade used were does from1-4 parities and having litter size records from first kidding. Morphometric characteristics measured were chest depth, chest width, hip width, hip height, heart girth, body length, and withers height. The data obtained were analyzed using General Linear Model (GLM) and Principal Component Analysis (PCA) of Statistycal Analysis System (SAS) Ver 6.12. The results showed that morphometric of Ettawa Grade goat was vary and different in various parity. The PC1 of body length and heart girth were 0.685 and 0.530 respectively, indicating the most discriminant variabel to determine the differences among parity. Body index of Ettawa Grade goat does did not affect the litter size. In conclusion, body length and heart girth can be used as the differentiation parameter among parity. Litter size will increase with increasing parity.

Principal symmetric space analysis

Principal Geodesic Analysis is a statistical technique that constructs low-dimensional approximations to data on Riemannian manifolds. It provides a generalization of principal components analysis to non-Euclidean spaces. The approximating submanifolds are geodesic at a reference point such as the intrinsic mean of the data. However, they are local methods as the approximation depends on the reference point and does not take into account the curvature of the manifold. Therefore, in this paper we develop a specialization of principal geodesic analysis, Principal Symmetric Space Analysis, based on nested sequences of totally geodesic submanifolds of symmetric spaces. The examples of spheres, Grassmannians, tori, and products of two-dimensional spheres are worked out in detail. The approximating submanifolds are geometrically the simplest possible, with zero exterior curvature at all points. They can deal with significant curvature and diverse topology. We show that in many cases the distance between a point and the submanifold can be computed analytically and there is a related metric that reduces the computation of principal symmetric space approximations to linear algebra.

Direct feature extraction from multi-electrode recordings for spike sorting

Information from extracellular action potentials (EAPs) of individual neurons is of particular interest in experimental neuroscience. It advances the understanding of brain functions and is essential in the emerging field of brain-machine interfaces. As EAPs from distinct neurons are generally not recorded individually, a process to separate them from the multi-unit recordings, referred to as spike sorting, is required. For spike sorting, the feature extraction step is crucial. Starting from acquired data, the task of feature extraction is to find a set of derived values or “features” that are informative and non-redundant to facilitate efficient and accurate sorting, compared with using the raw data directly. It not only reduces the dimensionality of the data but also the impact of noise. In this paper, two novel feature extraction algorithms for sorting multi-electrode EAPs are proposed. These algorithms can be seen as generalizations of principal component analysis and linear discriminant analysis, but the features that match the dominant subspaces observed in the multi-electrode data are obtained without the need for vectorizing a multi-electrode EAP or breaking it into separate EAP channels. These algorithms require no construction of EAP templates and are applicable to multi-electrode recordings regardless of the number of electrodes. Clustering using both simulated data and real EAP recordings taken from area CA1 of the dorsal hippocampus of rats demonstrates that the proposed approaches yield features that are discriminatory and lead to promising results.