Gene expression data classification using topology and machine learning models

BackgroundAttention-deficit/hyperactivity disorder (ADHD) is currently diagnosed by a diagnostic interview, mainly based on subjective reports from parents or teachers. It is necessary to develop methods that rely on objectively measureable neurobiological data to assess brain-behavior relationship in patients with ADHD. We investigated the application of a topological data analysis tool, Mapper, to analyze the brain functional connectivity data from ADHD patients.MethodsTo quantify the disease severity using the neuroimaging data, the decomposition of individual functional networks into normal and disease components by the healthy state model (HSM) was performed, and the magnitude of the disease component (MDC) was computed. Topological data analysis using Mapper was performed to distinguish children with ADHD (n = 196) from typically developing controls (TDC) (n = 214).ResultsIn the topological data analysis, the partial clustering results of patients with ADHD and normal subjects were shown in a chain-like graph. In the correlation analysis, the MDC showed a significant increase with lower intelligence scores in TDC. We also found that the rates of comorbidity in ADHD significantly increased when the deviation of the functional connectivity from HSM was large. In addition, a significant correlation between ADHD symptom severity and MDC was found in part of the dataset.ConclusionsThe application of HSM and topological data analysis methods in assessing the brain functional connectivity seem to be promising tools to quantify ADHD symptom severity and to reveal the hidden relationship between clinical phenotypic variables and brain connectivity.

Asthma is a complex respiratory disease characterised by spatial ventilation heterogeneity (VH) in the lung. Multiple breath washout (MBW) and forced oscillation technique (FOT) testing has been shown to quantify VH at the mouth. Methods to provide fresh insight into the mechanisms driving VH are urgently needed to support treatment development. Computed tomography (CT) with parametric response map (PRM) analysis provides spatial and functional lung data. Integrated data sets of imaging features and VH markers present the problem of extracting meaningful information from high dimensional objects; topological data analysis (TDA) has demonstrated recent success in solving this problem. In this thesis topological methods are applied to extract information about lung anatomy and function from imaging data to study association with VH in innately high dimensional integrated data sets. It was sought to describe the spatial distributions of voxel based metrics, visualise the lungs using TDA methods, and combine PRM with simulated airway tree models to improve computational modelling of FOT. Data was obtained from a clinical trial involving 52 subjects, 41 asthmatic (11 healthy). An algorithm was developed to quantify directional Hounsfield unit change (ΔHU) gradients. TDA was applied to integrated data sets and voxel sets with measures of airway tree depth. FOT modelling was augmented with ΔHU about simulated acinar airways.Inferior-to-superior ΔHU gradient best differentiated VH markers. Visualisations of the lungs illustrated gradient reversal and increased heterogeneity in ΔHU in asthmatics. FOT modelling using ΔHU parameterisation showed increased correlation between clinically measured and simulated FOT small airway resistance (R5-R20). In this thesis additional insight is presented into mechanisms affecting VH markers derived from MBW and FOT. Topological methods to analyse imaging data were developed and implemented, including workflows to visualise the lungs and lung function. Linking simulated airway tree geometry with ΔHU showed improvement in FOT modelling.

Single-cell technologies have revolutionized our ability to interrogate biological systems at unprecedented resolution, revealing complex cellular heterogeneity and dynamic processes that underlie development, disease, and immune responses. However, the high dimensionality and nonlinear structure of single-cell data present substantial analytical challenges. Topological data analysis offers a powerful mathematical framework for capturing the intrinsic shape of data, providing novel insights that complement and extend traditional statistical and machine learning methods. By leveraging tools such as persistent homology and the Mapper algorithm, TDA enables the detection of subtle, multiscale patterns – including rare cell populations, transitional states, and branching trajectories – that are often obscured by conventional approaches. In this review, we explore the theoretical foundations of topological data analysis and examine its emerging applications across single-cell transcriptomics, proteomics, and spatial biology. We highlight how this approach can unveil previously unrecognized biological phenomena, from alternative differentiation paths to complex tissue architectures, and discuss the growing ecosystem of computational tools that support its use. As single-cell datasets become increasingly large and multimodal, topological data analysis stands out as a uniquely robust and interpretable approach, with the potential to deepen our understanding of cellular identity and function in health and disease. TDA is specially suited for fields such as systems immunology since it can capture the complex, nonlinear structures inherent in high-dimensional immune data helping to identify distinct immune cell states, differentiation pathways, and dynamic responses to infection or therapy. This topological perspective complements traditional statistical approaches, providing a robust, scale-invariant framework for uncovering hidden organization within the immune system’s complexity.

The purpose of this paper is to introduce transductive inference with support vector machines (TSVM) as a powerful methodology for classification of gene expression data, using training and prediction data sets. The following classification problems will be considered: determination of cancer diagnosis categories and classification of genes from the budding yeast Saccharomyces cerevisiae in functional groups. In the case of training samples, experts have already classified the samples in their respective classes. So, given each prediction sample, the purpose is to determine its corresponding class. The main aspect of TSVM is that the classification task will be implemented in just one step, improving the generalization capability of the classifier. The TSVM will be compared with the traditional inductive method (SVM) in a series of experiments concerning the two classification problems, with promising results.

A growing area of mathematics topological data analysis (TDA) uses fundamental concepts of topology to analyze complex, high-dimensional data. A topological network represents the data, and the TDA uses the network to analyze the shape of the data and identify features in the network that correspond to patterns in the data. These patterns extract knowledge from the data. TDA provides a framework to advance machine learning’s ability to understand and analyze large, complex data. This paper provides background information about TDA, TDA applications for large data sets, and details related to the investigation and implementation of existing tools and environments.

Abstract: Neurodegenerative diseases, such as Alzheimer's and Parkinson's, are characterized by progressive neuronal deterioration, leading to cognitive and motor impairments. Early detection is crucial for effective intervention and management. This study proposes a novel framework utilizing Topological Data Analysis (TDA), particularly persistent homology, to identify early biomarkers from neuroimaging and physiological data. This study explores the application of TDA techniques—specifically persistent homology and topological invariants—for the early detection of neurodegenerative diseases such as Alzheimer's. By analyzing functional brain connectivity, gait dynamics, and molecular structures associated with disease progression, the research demonstrates that TDA can capture subtle, yet critical, topological signatures indicative of early neurodegenerative changes. Layer-wise topological complexity in neural networks and persistent homology in clinical datasets reveal distinct structural degradation patterns. The integration of TDA with machine learning models enhances diagnostic accuracy, interpretability, and generalization, offering a promising direction for non-invasive, data-driven diagnostics in neurology.By capturing the intrinsic topological features of brain networks, the framework aims to distinguish between healthy and early-stage neurodegenerative conditions, offering a robust tool for early diagnosis.Topological Data Analysis (TDA) offers a robust mathematical framework for extracting shape-based features from high-dimensional and noisy biomedical data. Keywords: Topological Data Analysis, Persistent Homology, Neurodegenerative Disease, Alzheimer's, Brain Connectivity, Early Diagnosis, Betti Numbers, Gait Analysis, Functional Networks, Machine Learning, Biomedical Data

Chapter 9 - Effective dimensionality reduction model with machine learning classification for microarray gene expression data

BOLD-based fMRI is the most widely used method for studying brain function. The BOLD signal while valuable, is beset with unique vulnerabilities. The most notable of these is the modest signal to noise ratio, and the relatively low temporal and spatial resolution. However, the high dimensional complexity of the BOLD signal also presents unique opportunities for functional discovery. Topological Data Analyses (TDA), a branch of mathematics optimized to search for specific classes of structure within high dimensional data may provide particularly valuable applications. In this investigation, we acquired fMRI data in the anterior cingulate cortex (ACC) using a basic motor control paradigm. Then, for each participant and each of three task conditions, fMRI signals in the ACC were summarized using two methods: a) TDA based methods of persistent homology and persistence landscapes and b) non-TDA based methods using a standard vectorization scheme. Finally, using machine learning (with support vector classifiers), classification accuracy of TDA and non-TDA vectorized data was tested across participants. In each participant, TDA-based classification out-performed the non-TDA based counterpart, suggesting that our TDA analytic pipeline better characterized task- and condition-induced structure in fMRI data in the ACC. Our results emphasize the value of TDA in characterizing task- and condition-induced structure in regional fMRI signals. In addition to providing our analytical tools for other users to emulate, we also discuss the unique role that TDA-based methods can play in the study of individual differences in the structure of functional brain signals in the healthy and the clinical brain.

Topological Data Analysis (TDA) and its mainstay computational device, persistent homology (PH), has established a strong track record of providing researchers across the data-driven sciences with new insights and methodologies by characterizing low-dimensional geometric structures in high-dimensional data. When combined with machine learning (ML) methods, PH is valued as a discriminating-feature extraction tool. This work highlights many of the recent successes at the intersection of TDA and ML, introduces some of the foundational mathematics underpinning TDA, and summarizes the efforts to strengthen the bridge between TDA and ML. Thus, this document is a launching point for experimentalists and theoreticians to consider what can be learned from the shape of their data.

Breast cancer remains a leading cause of mortality among women worldwide, with early and accurate diagnosis being critical for effective treatment. Gene expressionprofiling has emerged as a powerful tool for understanding the molecular mechanisms of cancer and for developing predictive models. However, the high dimensionality and class imbalance inherent in gene expression data pose significant challenges for developing robust predictive models.This study aims to develop and evaluate a predictive model for classifying breast cancer subtypes using high-dimensional gene expression data, addressing the challenges of class imbalance. The objective is to improve the accuracy and reliability of breast cancer subtype prediction to facilitate better diagnostic and treatment strategies.A comprehensive dataset of breast cancer gene expression profiles was utilized, comprising numerous gene expression levels across multiple samples. To address class imbalance, resampling techniques such as Synthetic Minority Over-sampling Technique (SMOTE) and Random Under-sampling, were employed. The machine learning algorithms employed included Support Vector Machines (SVM), Random Forests, and Neural Networks. The algorithms were trained and evaluated using cross-validation to identify the most effective model. The performance of these models was assessed based on metrics such as accuracy, precision, recall, and F1-score. The results indicate that the use of SMOTE in combination with SVM provided the most balanced and accurate predictions, with an F1-score significantly higher than models without resampling. The Random Forest algorithm also showed promising results, particularly in handling the high-dimensionality aspect of the data.The study demonstrates that addressing class imbalance through advanced resampling techniques can significantly enhance the predictive accuracy of models trained on high-dimensional gene expression data. The findings underscore the potential of machine learning models, particularly SVMs and Random Forests, in improving breast cancer subtype classification.

Persistent homology is a powerful tool in topological data analysis (TDA) to compute, study and encode efficiently multi-scale topological features and is being increasingly used in digital image classification. The topological features represent number of connected components, cycles, and voids that describe the shape of data. Persistent homology extracts the birth and death of these topological features through a filtration process. The lifespan of these features can represented using persistent diagrams (topological signatures). Cubical homology is a more efficient method for extracting topological features from a 2D image and uses a collection of cubes to compute the homology, which fits the digital image structure of grids. In this research, we propose a cubical homology-based algorithm for extracting topological features from 2D images to generate their topological signatures. Additionally, we propose a score, which measures the significance of each of the sub-simplices in terms of persistence. Also, gray level co-occurrence matrix (GLCM) and contrast limited adapting histogram equalization (CLAHE) are used as a supplementary method for extracting features. Machine learning techniques are then employed to classify images using the topological signatures. Among the eight tested algorithms with six published image datasets with varying pixel sizes, classes, and distributions, our experiments demonstrate that cubical homology-based machine learning with deep residual network (ResNet 1D) and Light Gradient Boosting Machine (lightGBM) shows promise with the extracted topological features.

Machine Learning algorithms have been widely used for gene expression data classification, despite the fact that these data have often intrinsic limitations, such as high dimensionality and a small number of examples. Few studies try to characterize to which extent these aspects can influence the performance of the classification models induced. In this paper we compute different measures characterizing the complexity of gene expression data sets for cancer diagnosis. We then investigate how these measures relate to the classification performances achieved by support vector machines, a popular Machine Learning technique usually employed in the analysis of gene expression data. The results obtained indicate that some of the complexity indices utilized are indeed successful in explaining the difficulty involved in the classification of cancer gene expression data.

Classification of gene expression data is the common denominator of various biomedical recognition tasks. However, obtaining class labels for large training samples may be difficult or even impossible in many cases. Therefore, semi-supervised classification techniques are required as semi-supervised classifiers take advantage of the unlabeled data. Furthermore, gene expression data is high dimensional which gives rise to the phenomena known under the umbrella of the curse of dimensionality, one of its recently explored aspects being the presence of hubs or hubness for short. Therefore, hubness-aware classifiers were developed recently, such as Naive Hubness Bayesian k-Nearest Neighbor (NHBNN). In this paper, we propose a semi-supervised extension of NHBNN and show in experiments on publicly available gene expression data that the proposed classifier outperforms all its examined competitors.

Topological data analysis (TDA) has exploded as a tool for analyzing and making sense of high dimensional datasets across a variety of fields. Mapper is a tool from TDA that captures low-dimensional structure from high-dimensional data, precisely the approach needed to capture relevant information from high-dimensional neural time series. Electrical potential scalp recording, or electroencephalography (EEG), is routinely used in clinical applications and research studies thanks to its noninvasive nature, relatively inexpensive equipment, and high temporal resolution. But, it is prone to contamination, exhibits low spatial resolution, and has a non-stationary nature. Thus, it requires advanced signal processing and mathematical analysis methods for tasks requiring unsupervised brain state clustering. We introduce MapperEEG, an approach to unsupervised brain state clustering that uses tools from classical EEG analysis combined with Mapper to cluster and connect brain states. We show that MapperEEG can serve as a clustering algorithm in the spectral domain and provide additional information about the underlying brain state connectivity in a tapping task. Additionally, we use a go/no-go shooting task to explore how MapperEEG can still provide insight into the underlying structure and clusters of brain states even when it and other clustering methods fail. We demonstrate that it outperforms six other clustering algorithms such as hierarchical clustering, Hidden Markov Models, and basic autoencoders on identifying states in a tapping task. MapperEEG offers a novel and effective approach to analyzing EEG data, showing promise for brain state clustering and analysis.

• A novel foundation for topological pattern recognition of point cloud data, through persistent homology of graph approximations. • An extension of persistent homology to point cloud data of metric trees, including theoretical and experimental verification. • The first charting of cell trajectory data sets that explains some of the difficulties this field is confronted with. The rising field of Topological Data Analysis (TDA) provides a new approach to learning from data through persistence diagrams , which are topological signatures that quantify topological properties of data in a comparable manner. For point clouds, these diagrams are often derived from the Vietoris-Rips filtration—based on the metric equipped on the data—which allows one to deduce topological patterns such as components and cycles of the underlying space. In metric trees these diagrams often fail to capture other crucial topological properties, such as the present leaves and multifurcations. Prior methods and results for persistent homology attempting to overcome this issue mainly target Rips graphs, which are often unfavorable in case of non-uniform density across our point cloud. We therefore introduce a new theoretical foundation for learning a wider variety of topological patterns through any given graph . Given particular powerful functions defining persistence diagrams to summarize topological patterns, including the normalized centrality or eccentricity , we prove a new stability result, explicitly bounding the bottleneck distance between the true and empirical diagrams for metric trees. This bound is tight if the metric distortion obtained through the graph and its maximal edge-weight are small. Through a case study of gene expression data, we demonstrate that our newly introduced diagrams provide novel quality measures and insights into cell trajectory inference.