Low Sample Size Data Research Articles

Clustering of high-dimensional observations

We present a novel clustering method for high-dimensional, low sample size (HDLSS) data. The method is distance-based, takes advantage of the distance concentration phenomenon and the limiting values of the dissimilarity indices to construct clusters. We describe an algorithm that orders each row of the dissimilarity matrix to estimate the change points, which define cluster boundaries. We construct an agreement matrix of the Rand indices of the row clusters. The minimum of the row sum of the agreement matrix provides us with the best clusters. We prove that the new method achieves perfect clustering as the number of features diverges for a fixed sample size. Several examples are presented to illustrate the proposed method. We compare the new method with four other clustering techniques, including high-dimensional k-means, minimal spanning tree and Hierarchical Scan. The clustering methods are applied to the Lymphoma data set.

Some clustering-based change-point detection methods applicable to high dimension, low sample size data

Detection of change-points in a sequence of high dimensional observations is a challenging problem, and this becomes even more challenging when the sample size (i.e., the sequence length) is small. In this article, we propose some change-point detection methods based on clustering, which can be conveniently used in such high dimension, low sample size situations. First, we consider the single change-point problem. Using k-means clustering based on a suitable dissimilarity measures, we propose some methods for testing the existence of a change-point and estimating its location. High dimensional behavior of these proposed methods are investigated under appropriate regularity conditions. Next, we extend our methods for detection of multiple change-points. We carry out extensive numerical studies and analyze a real data set to compare the performance of our proposed methods with some state-of-the-art methods.

Asymptotic behavior of some multicategory classification methods for high-dimensional data

We consider multicategory extensions of binary discrimination methods via one-versus-one (OVO) or one-versus-rest (OVR) methodologies, focusing on extensions of the binary classification by linear mean difference (MD), support vector machine (SVM), maximal data piling (MDP), and distance weighted discrimination (DWD) via OVO, and the multicategory extension of MD via OVR, in the context of high-dimensional and low sample size (HDLSS) data. The asymptotic behavior of OVO-MD, OVO-SVM, OVO-MDP and OVO-DWD is described when the dimension of the data increases and the sample size is fixed, in terms of the probabilities of correct classification of a new data point, finding sufficient conditions for the correct classification probabilities to converge to one as the dimension approaches infinity. As in the binary case, OVO-MD, OVO-SVM and OVO-MDP have the same asymptotic behavior while OVO-DWD could behave differently. We also consider the asymptotic behavior of the OVR-MD methodology providing necessary and sufficient conditions for a new data point of a given class to be correctly classified with probability tending to one. A simulation experiment is conducted to further compare the methodologies, and consider the four binary methods in the OVR case. We evaluate the performance of the considered methods using a microarray data set.

Biobjective gradient descent for feature selection on high dimension, low sample size data.

Even though deep learning shows impressive results in several applications, its use on problems with High Dimensions and Low Sample Size, such as diagnosing rare diseases, leads to overfitting. One solution often proposed is feature selection. In deep learning, along with feature selection, network sparsification is also used to improve the results when dealing with high dimensions low sample size data. However, most of the time, they are tackled as separate problems. This paper proposes a new approach that integrates feature selection, based on sparsification, into the training process of a deep neural network. This approach uses a constrained biobjective gradient descent method. It provides a set of Pareto optimal neural networks that make a trade-off between network sparsity and model accuracy. Results on both artificial and real datasets show that using a constrained biobjective gradient descent increases the network sparsity without degrading the classification performances. With the proposed approach, on an artificial dataset, the feature selection score reached 0.97 with a sparsity score of 0.92 with an accuracy of 0.9. For the same accuracy, none of the other methods reached a feature score above 0.20 and sparsity score of 0.35. Finally, statistical tests validate the results obtained on all datasets.

Machine Learning Approach to Identify Dysregulation in the Plasma Proteome of CADASIL

AbstractBackgroundThe plasma proteome provides information to identify dysregulated molecular pathways underlying disease. Cerebrovascular diseases often manifest a significantly different plasma proteomic signature. Cerebral‐Autosomal‐Dominant‐Arteriopathy‐Subcortical‐Infarcts‐and‐Leukoencephalopathy (CADASIL) is one such vascular brain disease. The CADASIL plasma proteome has yet to be investigated.MethodTo investigate the plasma proteome in the context of CADASIL, we used large‐scale proteomics, measuring over 7,000 proteins via aptamer‐based technology (SomaLogic) from 53 study participants (nCADASIL = 25; nControl = 28). We employed machine learning (ML) methods as an unbiased approach to uncover disease‐associated change in proteomic networks. This approach is based on the premise that protein sets that best classify disease state could be important biological drivers of disease. We developed a novel ML method: coupling recursive feature extraction with logistic regression (LR) and XGBoost evaluators, as well maximum‐relevance‐minimum‐redundancy with Random‐Forest and F‐Statistic evaluators. The results of all four models were selectively aggregated in order to delineate the CADASIL plasma proteome signature while minimizing overfitting to the low sample size data. We developed a 45 protein model of the CADASIL proteomic signature from these results. To evaluate the classification ability, we tested the Repeated‐Stratified‐10‐fold‐Cross‐Validation accuracy of a LR classifier. Next, we investigated whether the CADASIL proteomic signature was relevant for other neurodegenerative diseases with vascular components; the classifier was applied to an Alzheimer’s Disease (AD) dataset. Finally, the functional signature of the 45 proteins was investigated through gene ontology (GO).ResultThe LR classifier in CADASIL is 99.8% accurate. In AD, the accuracy is 71.2%. This implies the CADASIL signature shares commonalities as well as pertinent differences with other neurodegenerative diseases. The GO of the 45 proteins resulted in GO terms corresponding to extracellular matrix and collagen, cellular membrane lipoproteins and glycolipids, and proteasome biological pathways.ConclusionThe proposed method allows for an initial unbiased discovery of important proteomic changes associated with disease state. The identified proteins would provide starting points for mechanistic studies. In addition, a panel of proteins could be used for screening at population level for risk stratification and more in‐depth phenotyping. Next steps include investigating the biological relevance of findings in mechanistic models for therapeutic developments.

Machine Learning Approach to Identify Dysregulation in the Plasma Proteome of CADASIL

AbstractBackgroundThe plasma proteome provides information to identify dysregulated molecular pathways underlying disease. Cerebrovascular diseases often manifest a significantly different plasma proteomic signature. Cerebral‐Autosomal‐Dominant‐Arteriopathy‐Subcortical‐Infarcts‐and‐Leukoencephalopathy (CADASIL) is one such vascular brain disease. The CADASIL plasma proteome has yet to be investigated.MethodTo investigate the plasma proteome in the context of CADASIL, we used large‐scale proteomics, measuring over 7,000 proteins via aptamer‐based technology (SomaLogic) from 53 study participants (nCADASIL = 25; nControl = 28).We employed machine learning (ML) methods as an unbiased approach to uncover disease‐associated change in proteomic networks. This approach is based on the premise that protein sets that best classify disease state could be important biological drivers of disease. We developed a novel ML method: coupling recursive feature extraction with logistic regression (LR) and XGBoost evaluators, as well maximum‐relevance‐minimum‐redundancy with Random‐Forest and F‐Statistic evaluators.The results of all four models were selectively aggregated in order to delineate the CADASIL plasma proteome signature while minimizing overfitting to the low sample size data. We developed a 45 protein model of the CADASIL proteomic signature from these results. To evaluate the classification ability, we tested the Repeated‐Stratified‐10‐fold‐Cross‐Validation accuracy of a LR classifier. Next, we investigated whether the CADASIL proteomic signature was relevant for other neurodegenerative diseases with vascular components; the classifier was applied to an Alzheimer’s Disease (AD) dataset. Finally, the functional signature of the 45 proteins was investigated through gene ontology (GO).ResultsThe LR classifier in CADASIL is 99.8% accurate. In AD, the accuracy is 71.2%. This implies the CADASIL signature shares commonalities as well as pertinent differences with other neurodegenerative diseases.The GO of the 45 proteins resulted in GO terms corresponding to extracellular matrix and collagen, cellular membrane lipoproteins and glycolipids, and proteasome biological pathways.ConclusionThe proposed method allows for an initial unbiased discovery of important proteomic changes associated with disease state. The identified proteins would provide starting points for mechanistic studies. In addition, a panel of proteins could be used for screening at population level for risk stratification and more in‐depth phenotyping. Next steps include investigating the biological relevance of findings in mechanistic models for therapeutic developments.

Statistical Significance of Clustering with Multidimensional Scaling

Clustering is a fundamental tool for exploratory data analysis. One central problem in clustering is deciding if the clusters discovered by clustering methods are reliable as opposed to being artifacts of natural sampling variation. Statistical significance of clustering (SigClust) is a recently developed cluster evaluation tool for high-dimension, low-sample size data. Despite its successful application to many scientific problems, there are cases where the original SigClust may not work well. Furthermore, for specific applications, researchers may not have access to the original data and only have the dissimilarity matrix. In this case, clustering is still a valuable exploratory tool, but the original SigClust is not applicable. To address these issues, we propose a new SigClust method using multidimensional scaling (MDS). The underlying idea behind MDS-based SigClust is that one can achieve low-dimensional representations of the original data via MDS using only the dissimilarity matrix and then apply SigClust on the low-dimensional MDS space. The proposed MDS-based SigClust can circumvent the challenge of parameter estimation of the original method in high-dimensional spaces while keeping the essential clustering structure in the MDS space. Both simulations and real data applications demonstrate that the proposed method works remarkably well for assessing the statistical significance of clustering. Supplementary materials for this article are available online.

Fuzzy clustering based classifier for extraction of individualities from high dimension low sample size data

The target of this research is high dimension low sample size (HDLSS) data in which the number of variables (or dimensions) is much larger than the number of objects. For such a data, it is well known that mathematically (or statistically), we cannot obtain correct solutions as eigen values of the covariance matrix of variables, therefore, a lot of multivariate analyses cannot apply to this type of data. To overcome this problem, we have proposed a methodology for dimensional reduction, which is a fuzzy cluster scaled principal component analysis (fuzzy cluster scaled PCA). This paper presents a study of the applicability of the previously proposed fuzzy cluster scaled PCA for the discrimination of individual subjects observed by sensors worn on the body during several activities. The analysis of this data is useful for healthcare, considering the individuality of the history of activities, such as the implementation of a custom-made system for healthcare.

Graph convolutional network-based feature selection for high-dimensional and low-sample size data.

Feature selection is a powerful dimension reduction technique which selects a subset of relevant features for model construction. Numerous feature selection methods have been proposed, but most of them fail under the high-dimensional and low-sample size (HDLSS) setting due to the challenge of overfitting. We present a deep learning-based method-GRAph Convolutional nEtwork feature Selector (GRACES)-to select important features for HDLSS data. GRACES exploits latent relations between samples with various overfitting-reducing techniques to iteratively find a set of optimal features which gives rise to the greatest decreases in the optimization loss. We demonstrate that GRACES significantly outperforms other feature selection methods on both synthetic and real-world datasets. The source code is publicly available at https://github.com/canc1993/graces.

A machine learning approach for hierarchical classification of software requirements

Context:Classification of software requirements into different categories is a critically important task in requirements engineering (RE). Developing machine learning (ML) approaches for requirements classification has attracted great interest in the RE community since the 2000s. Objective:This paper aims to address two related problems that have been challenging real-world applications of ML approaches: the problems of class imbalance and high dimensionality with low sample size data (HDLSS). These problems can greatly degrade the classification performance of ML methods. Methods:The paper proposes HC4RC, a novel ML approach for multiclass classification of requirements. HC4RC solves the aforementioned problems through semantic-role based feature selection, dataset decomposition and hierarchical classification. We experimentally compare the effectiveness of HC4RC with three closely related approaches — two of which are based on a traditional statistical classification model whereas one using an advanced deep learning model. Results:Our experiment shows: (1) The class imbalance and HDLSS problems present a challenge to both traditional and advanced ML approaches. (2) The HC4RC approach is simple to use and can effectively address the class imbalance and HDLSS problems compared to similar approaches. Conclusion:This paper makes an important practical contribution to addressing the class imbalance and HDLSS problems in multiclass classification of software requirements.

Identification of Water Flooding Advantage Seepage Channels Based on Meta-Learning

As the water injection oilfield enters into the high water cut stage, a large number of water flooding advantage seepage channels are formed in the local reservoir dynamically changing with the water injection process, which seriously affects the water injection development effect. In oilfield production, water injection and fluid production profile test data are direct evidence to identify advantage seepage channels. In recent years, some scholars have carried out research related to the identification of advantage seepage channels based on machine learning methods; however, the insufficient profile test data limit the quantity and quality of learning samples, leading to problems such as low prediction accuracy of learning models. Therefore, the author proposes a new method of advantage seepage channel identification based on meta-learning techniques, using the MAML algorithm to optimize the neural network model so that the model can still perform well in the face of training tasks with low data sample size and low data quality. Finally, the model was applied to the actual blocks in the field to identify the advantage seepage channels, and the identification results were basically consistent with the tracer monitoring results, which confirmed the feasibility of the method. It provides a new solution idea for the task of identifying advantage seepage channels and other tasks with low data quality, which has a certain guiding significance.

James–Stein for the leading eigenvector

Recent research identifies and corrects bias, such as excess dispersion, in the leading sample eigenvector of a factor-based covariance matrix estimated from a high-dimension low sample size (HL) data set. We show that eigenvector bias can have a substantial impact on variance-minimizing optimization in the HL regime, while bias in estimated eigenvalues may have little effect. We describe a data-driven eigenvector shrinkage estimator in the HL regime called "James-Stein for eigenvectors" (JSE) and its close relationship with the James-Stein (JS) estimator for a collection of averages. We show, both theoretically and with numerical experiments, that, for certain variance-minimizing problems of practical importance, efforts to correct eigenvalues have little value in comparison to the JSE correction of the leading eigenvector. When certain extra information is present, JSE is a consistent estimator of the leading eigenvector.

Fuzzy cluster-scaled principal component analysis for mixed data and its application of educational effect based on device selection

This paper proposes a fuzzy cluster-scaled principal component analysis (fuzzy cluster-scaled PCA) for mixed data, which consists of both numerical and categorical data with respect to quantitative and qualitative variables. The fuzzy cluster-scaled PCA has been proposed for high-dimension, low-sample size (HDLSS) data in which the number of variables (or dimensions) is much larger than the number of objects (or samples). In this case, the target data is only for the numerical data. The proposed fuzzy cluster-scaled PCA in this paper can be applied to the mixed type of HDLSS data by utilizing the feature of the fuzzy cluster-scaled correlation, which is decomposed into two parts: the first part is the correlation of classification structures between variables and the second part is the correlation between variables. Then, the first part can be adapted to the categorical data, and the second part can be used for the numerical data through the same objects. Several numerical examples used data from a survey concerned with the relationship between students’ choice of devices and scores of examination/class marks to measure the educational effect show a better performance of this proposed method, and insightful, important information on educational effectiveness is clarified.

Assessment of Projection Pursuit Index for Classifying High Dimension Low Sample Size Data in R

Analyzing “large p small n” data is becoming increasingly paramount in a wide range of application fields. As a projection pursuit index, the Penalized Discriminant Analysis ($\mathrm{PDA}$) index, built upon the Linear Discriminant Analysis ($\mathrm{LDA}$) index, is devised in Lee and Cook (2010) to classify high-dimensional data with promising results. Yet, there is little information available about its performance compared with the popular Support Vector Machine ($\mathrm{SVM}$). This paper conducts extensive numerical studies to compare the performance of the $\mathrm{PDA}$ index with the $\mathrm{LDA}$ index and $\mathrm{SVM}$, demonstrating that the $\mathrm{PDA}$ index is robust to outliers and able to handle high-dimensional datasets with extremely small sample sizes, few important variables, and multiple classes. Analyses of several motivating real-world datasets reveal the practical advantages and limitations of individual methods, suggesting that the $\mathrm{PDA}$ index provides a useful alternative tool for classifying complex high-dimensional data. These new insights, along with the hands-on implementation of the $\mathrm{PDA}$ index functions in the R package classPP, facilitate statisticians and data scientists to make effective use of both sets of classification tools.

Maximum Decentral Projection Margin Classifier for High Dimension and Low Sample Size problems

Compared with relatively easy feature creation or generation in data analysis, manual data labeling needs a lot of time and effort in most cases. Even if automated data labeling​ seems to make it better in some cases, the labeling results still need to be checked and verified by manual. The High Dimension and Low Sample Size (HDLSS) data are therefore very common in data mining and machine learning. For classification problems with the HDLSS data, due to data piling and approximate equidistance between any two input points in high-dimension space, some traditional classifiers often give poor predictive performance. In this paper, we propose a Maximum Decentral Projection Margin Classifier (MDPMC) in the framework of a Support Vector Classifier (SVC). In the MDPMC model, the constraints of maximizing the projection distance between decentralized input points and their supporting hyperplane are integrated into the SVC model in addition to maximizing the margin of two supporting hyperplanes. On ten real HDLSS datasets, the experiment results show that the proposed MDPMC approach can deal well with data piling and approximate equidistance problems. Compared with SVC with Linear Kernel (SVC-LK) and Radial Basis Function Kernel (SVC-RBFK), Distance Weighted Discrimination (DWD), weighted DWD (wDWD), Distance-Weighted Support Vector Machine (DWSVM), Population-Guided Large Margin Classifier (PGLMC), and Data Maximum Dispersion Classifier (DMDC), MDPMC obtains better predictive accuracy and lower classification errors than the other seven classifiers on the HDLSS data.

Classification for high-dimension low-sample size data

High-dimension and low-sample-size (HDLSS) data sets have posed great challenges to many machine learning methods. To deal with practical HDLSS problems, development of new classification techniques is highly desired. After the cause of the over-fitting phenomenon is identified, a new classification criterion for HDLSS data sets, termed tolerance similarity, is proposed to emphasize maximization of within-class variance on the premise of class separability. Leveraging on this criterion, a novel linear binary classifier, termed No-separated Data Maximum Dispersion classifier (NPDMD), is designed. The main idea of the NPDMD is to spread samples of two classes in a large interval in the respective positive or negative space along the projecting direction when the distance between the projection means for two classes is large enough. The salient features of the proposed NPDMD are: (1) The NPDMD operates well on HDLSS data sets; (2) The NPDMD solves the objective function in the entire feature space to avoid the data-piling phenomenon. (3) The NPDMD leverages on the low-rank property of the covariance matrix for HDLSS data sets to accelerate the computation speed. (4) The NPDMD is suitable for different real-word applications. (5) The NPDMD can be implemented readily using Quadratic Programming. Not only theoretical properties of the NPDMD have been derived, but also a series of evaluations have been conducted on one simulated and six real-world benchmark data sets, including face classification and mRNA classification. Experimental results and comprehensive studies demonstrate the superiority of the NPDMD in terms of correct classification rate, mean within-group correct classification rate and the area under the ROC curve.

IDF21-0476 Macrovascular complications in type 1 diabetes mellitus patients with over 25 years of disease

Background: Macrovascular complications (MvC) are the most common cause of death in people with diabetes. Intensive glycemic control has proven to reduce its incidence in diabetes mellitus type 1 (T1DM). However, as for the general population, there are other factors influencing the development of cardiovascular disease. Aim: In a sample of T1DM patients with 25 years of disease: to evaluate the prevalence of MvC; to analyze the relationship between the development of MvC and several factors; to analyze the relationship between macrovascular and microvascular complications. Method: We performed a retrospective study based on the clinical files of T1DM patients with over 25 years of disease. Information was gathered regarding years of disease, average A1C per patient for each five year time period (A-A1C), types of treatment and complications. Statistical analysis was performed with SPSSv23, using: qui-square test for categorical variables, t-test (if normal distribution) or Mann-Whitney (if non-normal distribution) for continuous variables and Kaplan-Meier curves. The results were considered statistically significant if p<0,05. Results: The sample was constituted by 43 patients, 60,5% female, average age 52,6±1,5 years and average duration of T1DM 35,1±1,0 years. Arterial hypertension was present in 60,5% and dyslipidemia in 83,7%. The prevalence of MvC was 39,5% (average time for first MvC 29,4±2 years after diagnosis): 23,3% coronary heart disease; 20,9% heart failure; 20,9% peripheral artery disease and 9,3% cerebrovascular disease. All the patients were treated with intensive insulin therapy (median beginning time 1±7,5 years after diagnosis); 88,4% used functional insulin therapy (average start 19,5±2,0 years after diagnosis); 46,5% used subcutaneous continuous insulin perfusion (average start 21,7±1,5 years after diagnosis). A-A1C analysis was difficulted by several missing values, specially from the initial years of disease. However we observed that patients with MvC tended to have greater A-A1C values in most time periods, without statistical significance. The group with MvC had a longer course of disease (p=0,036) and predominance of the male gender (p=0,036). The patients with arterial hypertension had more (p=0,003) and more precocious (p=0,022) MvC. There were also more MvC in the patients with dyslipidemia (p=0,019). There was a tendency towards more prevalence of nephropathy, retinopathy and neuropathy in the group with MvC (p=0,068, 0,056 e 0,084, respectively). The group with MvC had more time of evolution of microvascular complications in average (p=0,019). Discussion: This work stands out the importance of dyslipidemia and arterial hypertension as contributors to MvC. There was a longer time of evolution of microvascular complications in the MvC group and a tendency towards a higher prevalence of each type of microvascular complications in this group. This data is likely explained by the importance of glycemic control in both types of complications. Although higher A-A1C values in the group with McV did not attain statistical significance, this is probably due to low sample size and several missing data.

Multiclass Classification of Software Requirements with Imbalanced, High Dimensional and Low Sample Size Data

Multiclass Classification of Software Requirements with Imbalanced, High Dimensional and Low Sample Size Data

Some clustering-based exact distribution-free [formula omitted]-sample tests applicable to high dimension, low sample size data

Testing homogeneity of k(≥2) multivariate distributions is a challenging problem in statistics, especially when the dimension of the data is much larger than the sample size. Most of the existing tests often perform poorly in this high dimension, low sample size (HDLSS) regime, and many of them cannot be used at all. In this article, we propose some nonparametric tests for this purpose. These tests have the distribution-free property in finite sample situations. They are based on a high dimensional clustering algorithm that makes a partition of the data to form a contingency table. Using the cell frequencies of that table, we construct the test statistics. We can develop tests based on a k-partition of the data or estimate the number of partitions from the data and construct tests based on it. Under appropriate regularity conditions, we prove the consistency of these tests in the HDLSS asymptotic regime. We also consider a multiscale approach, where the results for different number of partitions are aggregated judiciously. Extensive simulation study and analysis of some benchmark datasets illustrate the superiority of the proposed tests over some existing methods.

Multi-task multi-modality SVM for early COVID-19 Diagnosis using chest CT data

In the early diagnosis of the Coronavirus disease (COVID-19), it is of great importance for either distinguishing severe cases from mild cases or predicting the conversion time that mild cases would possibly convert to severe cases. This study investigates both of them in a unified framework by exploring the problems such as slight appearance difference between mild cases and severe cases, the interpretability, the High Dimension and Low Sample Size (HDLSS) data, and the class imbalance. To this end, the proposed framework includes three steps: (1) feature extraction which first conducts the hierarchical segmentation on the chest Computed Tomography (CT) image data and then extracts multi-modality handcrafted features for each segment, aiming at capturing the slight appearance difference from different perspectives; (2) data augmentation which employs the over-sampling technique to augment the number of samples corresponding to the minority classes, aiming at investigating the class imbalance problem; and (3) joint construction of classification and regression by proposing a novel Multi-task Multi-modality Support Vector Machine (MM-SVM) method to solve the issue of the HDLSS data and achieve the interpretability. Experimental analysis on two synthetic and one real COVID-19 data set demonstrated that our proposed framework outperformed six state-of-the-art methods in terms of binary classification and regression performance.