Multimodal Alzheimer's disease (AD) cohorts capture cognition, function, neuroimaging, and fluid biomarkers, yet overall disease severity remains difficult to summarize on a single clinically meaningful scale. The apolipoprotein E ε4 (APOE ε4) allele is the strongest common genetic risk factor for late-onset AD, but its association with "progression" has been inconsistent because earlier placement along the disease continuum is often conflated with faster within-stage decline. Using data from the Alzheimer's Disease Neuroimaging Initiative (ADNI), we analyzed an amyloid-positive baseline cohort (N = 1058) and a longitudinal subset (N = 932; ≥ 2 visits and ≥ 4 of 13 measures per visit). Measures included standardized cognitive and functional assessments, structural and functional neuroimaging, cerebrospinal fluid biomarkers of amyloid‑beta and tau pathology, and plasma neurofilament light protein as a marker of neuroaxonal injury. Magnetic resonance imaging (MRI) volumes were adjusted using amyloid-negative cognitively normal controls with quadratic age and intracranial volume terms. Probabilistic principal component analysis (PPCA) was used to derive a latent severity coordinate, defined as the first principal component (PC1). Hierarchical Bayesian random-intercept and random-slope models were used to estimate individual trajectories, partition APOE ε4 effects into baseline severity and within-stage rate, and generate genotype-stratified ages at prespecified severity landmarks. Axis stability was assessed with 100 bootstrap refits, and predictive performance was assessed with participant-level fivefold cross-validation. The PC1 explained 38.7% of baseline variance and produced a clinically interpretable multimodal severity axis. Stability was high across bootstrap refits, and residual association with age was minimal after MRI volume adjustment. Higher APOE ε4 dose was associated with greater baseline latent severity, whereas within-stage rate differences were smaller than the baseline severity-position effect. A latent symptomatic landmark was reached approximately 3.0-3.3years earlier per ε4 allele. Adding APOE improved out-of-sample prediction by about 10% without loss of calibration. Probabilistic principal component analysis provides a stable, multimodal, biologically informed severity axis for longitudinal modeling in amyloid-positive ADNI. Within this framework, APOE ε4 was associated primarily with latent severity position and model-implied timing along the continuum, whereas within-stage rate differences were smaller. These findings support stage-aware longitudinal inference and methodological applications within this cohort, while external clinical calibration and validation remain necessary.

Alzheimer’s disease (AD) is a neurodegenerative disorder and the leading cause of dementia. Early diagnosis and monitoring of disease progression are crucial for effective intervention. This study presents a novel disease progression model based on Variational Probabilistic Principal Component Analysis (VPPCA), which uses a Bayesian framework for dimensionality reduction and uncertainty quantification. By analyzing 1,021 amyloid-positive patients from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database, we extracted 25 features, including CSF (ABETA, TAU, PTAU), PET (FDG, AV45), and MRI volumetrics, along with cognitive and functional assessments. VPPCA compresses these multi-modal biomarkers into a single first principal component score (VPPCA1), which serves as a measure of disease progression. To ensure biological grounding and avoid circularity, we demonstrated that a VPPCA1 model using only non-cognitive features (CSF, PET, MRI, demographics) correlates strongly with cognitive decline (r = 0.658 with ADAS-Cog13), confirming that it captures genuine pathological progression rather than simply reflecting cognitive assessments. Block-wise feature ablation revealed that multi-modal integration is essential, with cognitive features showing the highest importance (0.1064), though all modalities contribute complementarily. In classification tasks, VPPCA exhibited strong performance with ROC-AUC values of 0.990 (CN vs Dementia), 0.774 (CN vs MCI), and 0.785 (MCI vs Dementia). A Bayesian hierarchical longitudinal model effectively captured patient-specific progression trajectories, offering personalized predictions of future disease states. VPPCA outperforms Probabilistic PCA (PPCA) by providing uncertainty quantification, with patient-specific confidence levels (σ = 0.086–0.136), which correlate with data quality, enabling automatic risk stratification. This work demonstrates that VPPCA offers a robust, biologically-grounded framework for modeling AD progression, providing actionable uncertainty quantification that improves clinical decision support and facilitates personalized care.

Principal component analysis (PCA) is one of the most widely used approaches for multivariate datasets. Biologists use PCA to visualize data, identify patterns in large datasets, determine independent axes of variation, and reduce dimensionality for further statistical analyses. Phylogenetic PCA is an extension of regular PCA that seeks to identify the major axes of variation independent of the phylogeny. We extend these methods by estimating PCA parameters using an explicit probability modeling framework. We implement multiple models of trait evolution (Brownian motion, Ornstein-Uhlenbeck, Early Burst, and Pagel's λ) and use the Akaike information criterion for model selection. We also introduce a probabilistic approach to select the number of principal components to retain from a PCA. We demonstrate the advantages of probabilistic PCA, such as incorporating the error, or noise, arising from dimensionality reduction, which is ignored in regular PCA. We use extensive simulations and an empirical dataset with 35 traits to show the method's performance. We implemented the new approach in the R package "do3PCA" available from the RCran repository.

Collecting multiple types of data on the same set of subjects is common in modern scientific applications including genomics, metabolomics, and neuroimaging. Joint and Individual Variation Explained (JIVE) seeks a low-rank approximation of the joint variation between two or more sets of features captured on common subjects and isolates this variation from that unique to each set of features. We develop an expectation-maximization (EM) algorithm to estimate a probabilistic model for the JIVE framework. The model extends probabilistic PCA to multiple datasets. Our maximum likelihood approach simultaneously estimates joint and individual components, which can lead to greater accuracy compared to other methods. We apply ProJIVE to measures of brain morphometry and cognition in Alzheimer’s disease. ProJIVE learns biologically meaningful sources of variation, and the joint morphometry and cognition subject scores are strongly related to more expensive existing biomarkers. Data used in preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database. Code to reproduce the analysis is available at https://github.com/thebrisklab/ProJIVE. Supplementary materials for this article are available online.

BackgroundAge-related degeneration of parotid glands impacts oral and systemic health, yet non-invasive assessment remains limited. This study characterizes parotid gland aging using computed tomography (CT)-based radiomics for the early detection of glandular dysfunction or diseases.MethodsThis retrospective study included 320 healthy individuals (aged 12–84 years), stratified into four age groups (adolescent: <25, young adult: 25–44, middle-aged: 45–59, elderly: >59 years) with balanced sex distribution. Axial medial-lateral diameter (AMLD) and mean Hounsfield unit (HU) attenuation were manually measured on axial CT slices. Three-dimensional (3D) radiomic features were extracted from automatically segmented glands via PyRadiomics platform. Dimensionality reduction was performed by probabilistic principal component analysis (PCA) and unsupervised K-means clustering was applied to shape-, first-order-, and texture-related features to identify age-associated patterns.ResultsAMLD increased significantly with age from 32.1±3.5 mm in the adolescent group to 40.3±4.1 mm in the elderly group (P<0.05). Similarly, gland volume enlarged significantly, with males exhibiting larger volumes than females across all groups (P<0.05). In contrast, mean HU declined with age from −2.5±6.8 to −15.7±10.2 HU (P<0.05), indicating reduced tissue density and homogeneity. Radiomic uniformity also decreased significantly in adults compared to adolescents (P<0.05). Unsupervised clustering revealed a marked age-dependent shift: Cluster 1 (small, spherical glands) comprised 55.6% of adolescent glands but only 8.8% of elderly glands (P<0.05), whereas Clusters 3 and 4 (large, irregular glands) increased from 10.9% to 48.8% (P<0.05).ConclusionsCT-based radiomics captured age-related morphological and structural changes in parotid glands. These findings support the application of CT as a non-invasive tool for monitoring parotid gland aging and may inform the early detection of age-associated glandular dysfunction.

Distributed process monitoring for large-scale processes typically involves two key steps: process decomposition and decision fusion. Decision fusion entails combining multiple local monitoring indices from each process block into a unified statistic, thereby facilitating rapid online fault detection throughout the entire process. Although existing studies have presented useful decision fusion strategies, they often overlook the number of potentially faulty blocks, thus limiting their performance. To address this limitation, this paper introduces the distributed probabilistic PCA (DisPPCA) model, which probabilistically describes the entire process in terms of multiple blocks. Building upon DisPPCA, two efficient decision fusion strategies, “Sum” and “Max”, are proposed. The “Sum” strategy demonstrates greater efficiency in detecting faults when a moderate or large number of faulty blocks are present, whereas the “Max” strategy excels at fault detection when there is only one or a small number of faulty blocks. Founded on the likelihood ratio test, the proposed decision fusion strategies possess a sound theoretical basis. Furthermore, their efficiency is substantiated through numerical simulations and a comprehensive case study.

Adaptive chemical industrial processes fault detection model based on Sparse Filtering-based Improved Mixed-Gaussian Probabilistic Principal Component Analysis considering low-probability events

There are significant differences in the deformation patterns of different parts of arch dams, and there is a common situation of periodic data loss. To accurately analyze the deformation behavior of arch dams, this paper proposes a safety warning and anomaly diagnosis method for arch dam deformation based on the separation of environmental variable effects in different partitions and a knowledge-driven approach. This method combines various techniques such as an optimized ISODATA clustering method, probabilistic principal component analysis (PPCA), square prediction error (SPE) norm control chart, and contribution chart. By defining data forms and rules, existing engineering specifications and experience are transformed into “knowledge” and applied to the operation and management of arch dams, achieving accurate monitoring of arch dam deformation status and timely diagnosis of outliers. Through monitoring data verification of horizontal displacement in a certain arch dam partition, the results show that this method can accurately identify deformation anomalies in the arch dam and effectively separate the influence of environmental variables and noise interference, providing strong support for the safe operation of the arch dam. Accurate deformation monitoring of arch dams is essential for ensuring structural safety and optimizing operational management. However, conventional early warning indicators and empirical models often fail to capture the spatial heterogeneity of deformation and the complex coupling between environmental variables and structural responses. To overcome these limitations, this study proposes a knowledge-driven safety early warning and anomaly diagnosis model for arch dam deformation, based on spatiotemporal clustering and partitioned environmental variable separation. The method integrates the optimized ISODATA clustering algorithm, probabilistic principal component analysis (PPCA), squared prediction error (SPE) control chart, and contribution chart to establish a comprehensive monitoring framework. The optimized ISODATA identifies deformation zones with similar mechanical behavior, PPCA separates environmental influences such as temperature and reservoir level from structural responses, and the SPE and contribution charts quantify abnormal variations and locate potential risk regions. Application of the proposed method to long-term deformation monitoring data demonstrates that the PPCA-based framework effectively separates environmental effects, improves the interpretability of zoned deformation characteristics, and enhances the accuracy and reliability of anomaly identification compared with conventional approaches. These findings indicate that the proposed knowledge-driven model provides a robust and interpretable framework for precise deformation safety evaluation of arch dams.

Semi-supervised mixture of probabilistic principal component analyzers for modeling human behavior

Stochastic subspace via probabilistic principal component analysis for characterizing model error

Dimensionality reduction is critical across various domains of science including neuroscience. Probabilistic Principal Component Analysis (PPCA) is a prominent dimensionality reduction method that provides a probabilistic approach unlike the deterministic approach of PCA and serves as a connection between PCA and Factor Analysis (FA). Despite their power, PPCA and its extensions are mainly based on linear models and can only describe the data in a Euclidean coordinate system around the mean of data. However, in many neuroscience applications, data may be distributed around a nonlinear geometry (i.e., manifold) rather than lying in the Euclidean space around the mean. We develop Probabilistic Geometric Principal Component Analysis (PGPCA) for such datasets as a new dimensionality reduction algorithm that can explicitly incorporate knowledge about a given nonlinear manifold that is first fitted from these data. Further, we show how in addition to the Euclidean coordinate system, a geometric coordinate system can be derived for the manifold to capture the deviations of data from the manifold and noise. We also derive a data-driven EM algorithm for learning the PGPCA model parameters. As such, PGPCA generalizes PPCA to better describe data distributions by incorporating a nonlinear manifold geometry. In simulations and brain data analyses, we show that PGPCA can effectively model the data distribution around various given manifolds and outperforms PPCA for such data. Moreover, PGPCA provides the capability to test whether the new geometric coordinate system better describes the data than the Euclidean one. Finally, PGPCA can perform dimensionality reduction and learn the data distribution both around and on the manifold. These capabilities make PGPCA valuable for enhancing the efficacy of dimensionality reduction for analysis of high-dimensional data that exhibit noise and are distributed around a nonlinear manifold, especially for neural data.

Due to urbanization and modern lifestyle, most of women in today’s world are prone to Polycystic Ovarian Syndrome (PCOS), which is a hormonal disorder. Though the symptoms shown by this disease are often uncared, it seriously affects the reproductive health of women. Early detection of PCOS helps in managing several other attributes that are closely related to it. This article aims to study the impact of Vitamin D3 in PCOS and non-PCOS individuals. The goal is attained by building a tailored dataset with 1368 records and 43 attributes. Initially, the acquired dataset is pre-processed by handling missed values, outlier detection and data balancing by employing Probabilistic Principal Component Analysis (PPCA), Interquartile Range (IQR), Z-score standardization and SMOTE respectively. The significant features are selected by exploring different approaches such as filter based (Chi-Square, ANOVA), wrapper based (Electric Eel Foraging Optimization Algorithm) and embedded methods (LASSO, XGBoost). The selected features are utilized to train classifiers such as Random Forest (RF), k-Nearest Neighbour (k-NN), Decision Tree (DT) and Support Vector Machine (SVM). The experimental results show that the performance of EEFOA with RF prove the best accuracy rates of 98.8% with a F-measure of 98.19%. Explainable Artificial Intelligence (XAI) techniques such as SHAP and LIME are then employed to showcase the feature importance. It is observed that over 40% of PCOS patients are affected by deficiency and insufficiency of vitamin D3.

The lack of sensory feedback coupled with mechanical constraints due to the prosthetic leg increases walking instability and the risk of falling in lower limb amputees. We investigated kinematic regularity and stability of different body segments in lower limb amputees during walking to identify possible altered dynamics leading to compensatory movements. We measured the three-dimensional acceleration and angular velocity of 15 body segments during two minutes of treadmill walking at three different velocities. The maximal Lyapunov exponents and fuzzy entropy were calculated from these data to assess local dynamic stability and regularity. Probabilistic principal component analysis (PPCA) was used to select the body segments that showed the highest variability between amputees and able-bodied individuals. Amputees exhibited increased instability in acceleration patterns, particularly at low walking velocity (1 km/h), regardless of body segment and direction. Angular velocity patterns were more unstable in amputees, especially on the amputated side. Altered regularity adaptation was observed with higher velocity in amputees, with the intact side showing less adaptive patterns than controls. These results further suggest that amputees have a holistically disrupted gait and balance system. Our analysis of non-linear gait dynamics provides new insights into the complex challenges faced by amputees during walking, particularly in adapting to different gait velocities.

Background: Prostate cancer (PCa) remains one of the most prevalent malignancies in men, with diagnostic challenges arising from the limited specificity of current biomarkers, like PSA. Improved stratification tools are essential to reduce overdiagnosis and guide personalized patient management. Objective: This study aimed to identify and validate clinical and hematological biomarkers capable of differentiating PCa from benign prostatic hyperplasia (BPH) and precancerous lesions (PL) using univariate and multivariate statistical methods. Methods: In a cohort of 514 patients with suspected PCa, we performed a univariate analysis (Kruskal-Wallis and ANOVA) with preprocessing via adaptive Box-Cox transformation and missing value imputation through probabilistic principal component analysis (PPCA). LASSO regression was used for variable selection and classification. An ROC curve analysis assessed diagnostic performance. Results: Five variables-age, PSA, Index %, hemoglobin (HGB), and the International Index of Erectile Function (IIEF)-were consistently significant across univariate and multivariate analyses. The LASSO regression achieved a classification accuracy of 70% and an AUC of 0.74. Biplot and post-hoc analyses confirmed partial separation between PCa and benign conditions. Conclusions: The integration of multivariate modeling with reconstructed clinical data enabled the identification of blood-based biomarkers with strong diagnostic potential. These routinely available, cost-effective indicators may support early PCa diagnosis and patient stratification, reducing unnecessary invasive procedures.

In this review article, the term “hierarchy” is related to constrained-ness, but not to superiority. Procedures A and B forming a hierarchy means that A is a constrained variant of B or vice versa. A goal of this article is to present a hierarchy of principal component analysis (PCA) and factor analysis (FA) procedures, which follows from a comprehensive FA (CompFA) model. This model can be regarded as a hybrid of PCA and prevalent FA models. First, we show how a non-random version of the CompFA model leads to the following hierarchy: PCA is a constrained variant of completely decomposed FA, which itself is a constrained variant of matrix decomposition FA. Then, we prove that a random version of the CompFA model leads to minimum rank FA (MRFA) and constraining MRFA leads to random PCA (RPCA), so as to present the following hierarchy: Probabilistic PCA is a constrained variant of prevalent FA, and the latter is a constrained variant of RPCA, which is itself a constrained variant of MRFA. Finally, this hierarchy and the above hierarchy following from the non-random version are unified into one. We further utilize the unified hierarchy to present a strategy for selecting a procedure suitable to a data set.

Dimensionality reduction is critical across various domains of science including neuroscience. Probabilistic Principal Component Analysis (PPCA) is a prominent dimensionality reduction method that provides a probabilistic approach unlike the deterministic approach of PCA and serves as a connection between PCA and Factor Analysis (FA). Despite their power, PPCA and its extensions are mainly based on linear models and can only describe the data in a Euclidean coordinate system around the mean of data. However, in many neuroscience applications, data may be distributed around a nonlinear geometry (i.e., manifold) rather than lying in the Euclidean space around the mean. We develop Probabilistic Geometric Principal Component Analysis (PGPCA) for such datasets as a new dimensionality reduction algorithm that can explicitly incorporate knowledge about a given nonlinear manifold that is first fitted from these data. Further, we show how in addition to the Euclidean coordinate system, a geometric coordinate system can be derived for the manifold to capture the deviations of data from the manifold and noise. We also derive a data-driven EM algorithm for learning the PGPCA model parameters. As such, PGPCA generalizes PPCA to better describe data distributions by incorporating a nonlinear manifold geometry. In simulations and brain data analyses, we show that PGPCA can effectively model the data distribution around various given manifolds and outperforms PPCA for such data. Moreover, PGPCA provides the capability to test whether the new geometric coordinate system better describes the data than the Euclidean one. Finally, PGPCA can perform dimensionality reduction and learn the data distribution both around and on the manifold. These capabilities make PGPCA valuable for enhancing the efficacy of dimensionality reduction for analysis of high-dimensional data that exhibit noise and are distributed around a nonlinear manifold, especially for neural data.

Probabilistic latent variable models (PLVMs), such as probabilistic principal component analysis (PPCA), are widely employed in process monitoring and fault detection of industrial processes. This article proposes a novel deep PPCA (DePPCA) model, which has the advantages of both probabilistic modeling and deep learning. The construction of DePPCA includes a greedy layer-wise pretraining phase and a unified end-to-end fine-tuning phase. The former establishes a hierarchical deep structure based on cascading multiple layers of the PPCA module to extract high-level features. The latter builds an end-to-end connection between the raw inputs and the final outputs to further improve the representation of the model to high-level features. After constructing the model structure of DePPCA, we first present the detailed training processes of the pretraining and fine-tuning stages, then clarify the theoretical merits of the proposed model from the perspective of variational inference. For process monitoring purposes, we develop two statistics based on the established DePPCA. The monitoring performance of these two statistics can remain superior even if the features extracted by DePPCA are significantly compressed to univariate. This makes the feature extraction process and online monitoring procedure of DePPCA quite fast. In other words, the proposed DePPCA can achieve accurate and efficient process monitoring by only extracting one feature for each sample. Finally, the effectiveness of DePPCA is evaluated on the Tennessee Eastman (TE) process and the multiphase flow (MPF) facility.

ABSTRACT Big Area Additive Manufacturing (BAAM) of composites requires significant time, energy, and material, so it is critical to reduce production inefficiencies to make functional parts without multiple iterations. Statistical process control coupled with Principal Component Analysis (PCA) is a powerful technique that provides a quick, computationally inexpensive, and intuitive way for operators to detect defects that form in a manufacturing process without massive datasets. Recently, a combined index that is a weighted sum of the Hotelling's T 2 and squared residual error statistics has been proposed that can be monitored in one chart, improving interpretation accuracy and simplicity. However, the literature does not offer a formal method to optimise the weights. Here, we introduce two new approaches to the traditional weight selection approach using simulated and BAAM image data. Approach 1 uses a theoretically motivated optimum inspired by probabilistic principal component analysis. Approach 2 systematically varies the ratio of the weights to find the optimum. We show that approach 1 delivers optimal anomaly detection performance in select cases while approach 2 fares better in practice. Surprisingly, we also show that choosing a more complex PCA model has a minimal negative impact on anomaly detection performance compared to a more simplistic model.

Purpose Owing to the growing evidence of crypto asset connectedness and correlation with traditional financial assets, this study sought to determine if there is a time-varying correlation and/or connectedness between the stablecoin market and the currencies of emerging market and developing economies (EMDEs) with significant cryptocurrency penetration. Design/methodology/approach This study uses a probabilistic principal component analysis (PPCA) to create stablecoin and EMDEs currency returns and volatility indices for EMDEs with significant cryptocurrency penetration. We then employ a time-varying correlation and time-varying parameter vector autoregressive (TVP-VAR) connectedness measures to document the time-dependent correlation and connectedness between the EMDE currencies and the stablecoin market. Findings The result points to a spillover of return shocks from the EMDE currencies to the stablecoin market prior to and after the COVID-19 pandemic. This is indicative of a flight-to-safety role of stablecoins for EMDE currencies. This calls for increased attention to the stablecoin market by money market investors and monetary authorities. Originality/value The paper contributes to the growing cryptocurrency and finance literature by empirically examining the level of connectedness between stablecoins and emerging market currencies. Knowing the relationship (correlation) and shock spillover (connectedness) between the stablecoins and the EMDE currencies will be valuable to currency investors’ diversification and hedging strategies, and to macroeconomic policymakers in designing and implementing regulation.

In recent years, deep learning has shown significant progress for image compression compared to traditional image compression methods. Although conventional standard-based methods are still used, they are limited in handling repetitive patterns and complex calculations, which can lead to image reconstruction issues. In this study, we propose a novel learning-based image compression method that integrates both channel attention (CA) and probabilistic principal component analysis (PPCA) blocks as core components to enhance encoding efficiency. PPCA is used to focus on essential features and manage noise. Unlike traditional PCA, PPCA’s probabilistic approach better preserves meaningful data structure, enhancing compression and robustness. The CA mechanism in our model emphasizes significant image features by prioritizing dominant pixel values, allowing the compression process to retain essential details while minimizing less relevant information. Furthermore, a foveated image quality assessment metric is proposed, prioritizing visually significant regions to enhance the evaluation of dominant information guided by attention mechanisms and to assess the impact of the CA and PPCA blocks on image reconstruction. Experimental results demonstrate that the proposed method obtained significant coding efficiency across various metrics on the Kodak and Tecnick datasets compared with state-of-the-art methods.