Local Subsets Research Articles

IDEEA: information diffusion model for integrating gene expression and EEG data in identifying Alzheimer’s disease markers

Abstract Understanding the genetic components of Alzheimer’s disease (AD) via transcriptome analysis often necessitates the use of invasive methods. This work focuses on overcoming the difficulties associated with the invasive process of collecting brain tissue samples in order to measure and investigate the transcriptome behavior of AD. Our approach called IDEEA (Information Diffusion model for integrating gene Expression and EEG data in identifying Alzheimer’s disease markers) involves systematically linking two different but complementary modalities: transcriptomics and EEG data. We preprocess these two data types by calculating the spectral and transcriptional sample distances, over 11 brain regions encompassing 6 distinct frequency bands. Subsequently, we employ a genetic algorithm approach to integrate the distinct features of the preprocessed data. Our experimental results show that
IDEEA converges rapidly to local optima gene subsets, in fewer than 250 iterations. Our algorithm identifies novel genes along with genes that have previously been linked to AD. It is also capable of detecting genes with transcription patterns specific to individual EEG bands as well as those with common patterns among bands. In particular, the alpha2 (10-13 Hz) frequency band yielded 8 AD associated genes out of the top 100 most frequently selected genes by our algorithm, with a p-value of
0.05. Our method not only identifies AD-related genes but also genes that interact with AD genes in terms of transcription regulation. We evaluated various aspects of our approach, including the genetic algorithm performance, band-pair association and gene interaction topology. Our approach reveals AD-relevant genes with transcription patterns inferred from EEG alone, across various frequency bands, avoiding the risky brain tissue collection process. This is a significant advancement toward the early identification of AD using non-invasive EEG recordings.

FTDZOA: An Efficient and Robust FS Method with Multi-Strategy Assistance.

Feature selection (FS) is a pivotal technique in big data analytics, aimed at mitigating redundant information within datasets and optimizing computational resource utilization. This study introduces an enhanced zebra optimization algorithm (ZOA), termed FTDZOA, for superior feature dimensionality reduction. To address the challenges of ZOA, such as susceptibility to local optimal feature subsets, limited global search capabilities, and sluggish convergence when tackling FS problems, three strategies are integrated into the original ZOA to bolster its FS performance. Firstly, a fractional order search strategy is incorporated to preserve information from the preceding generations, thereby enhancing ZOA's exploitation capabilities. Secondly, a triple mean point guidance strategy is introduced, amalgamating information from the global optimal point, a random point, and the current point to effectively augment ZOA's exploration prowess. Lastly, the exploration capacity of ZOA is further elevated through the introduction of a differential strategy, which integrates information disparities among different individuals. Subsequently, the FTDZOA-based FS method was applied to solve 23 FS problems spanning low, medium, and high dimensions. A comparative analysis with nine advanced FS methods revealed that FTDZOA achieved higher classification accuracy on over 90% of the datasets and secured a winning rate exceeding 83% in terms of execution time. These findings confirm that FTDZOA is a reliable, high-performance, practical, and robust FS method.

Zooplankton functional diversity as a bioindicator of freshwater ecosystem health across land use gradient

Zooplankton are critical indicators of pressures impacting freshwater ecosystems. We analyzed the response of zooplankton communities across different sub-catchment types—headwaters, natural, urban, urban-agricultural, and agricultural—within the Łyna river–lake system in Northern Poland. Using taxonomic groups and functional traits (body size, feeding strategies), we applied Partial Least Squares Regression (PLS-R) to elucidate the relationships between environmental conditions, land use, and zooplankton metacommunity structure. Two-Way Cluster Analysis (TWCA) identified local subsets with characteristic patterns, while Indicator Species Analysis (ISA) determined area-specific taxa. The natural river zone exhibited significant habitat heterogeneity and feeding niches, whereas urban areas created functional homogenization of zooplankton, dominated by small, broad-diet microphages. Agricultural areas promoted diversity among large filter feeders (Crustacea), active suctors (Rotifera), and amoebae (Protozoa). However, intensified agricultural activities, substantially diminished the zooplankton population, biomass, taxonomic richness, and overall ecosystem functionality. The impact of land cover change is more pronounced at small-scale sub-catchments than at the catchment level as a whole. Therefore, assessing these impacts requires detailed spatial and temporal analysis at the sub-catchment level to identify the most affected areas. This study introduces a new sub-catchment-based perspective on ecosystem health assessment and underscores the zooplankton's role as robust indicators of ecological change.

What’s the situation with ocular inflammation? A cross-seasonal investigation of proteomic changes in ocular allergy sufferers’ tears in Victoria, Australia

BackgroundOcular allergy (OA) is a localized subset of allergy characterized by ocular surface itchiness, redness and inflammation. Inflammation and eye-rubbing, due to allergy-associated itch, are common in OA sufferers and may trigger changes to the ocular surface biochemistry. The primary aim of this study is to assess the differences in the human tear proteome between OA sufferers and Healthy Controls (HCs) across peak allergy season and off-peak season in Victoria, Australia.Methods19 participants (14 OA sufferers, 5 HCs) aged 18–45 were recruited for this study. Participants were grouped based on allergy symptom assessment questionnaire scoring. Proteins were extracted from human tear samples and were run on an Orbitrap Mass Spectrometer. Peaks were matched to a DIA library. Data was analyzed using the software MaxQuant, Perseus and IBM SPSS.Results1267 proteins were identified in tear samples of OA sufferers and HCs. 23 proteins were differentially expressed between peak allergy season OA suffers vs HCs, and 21 were differentially expressed in off-peak season. Decreased proteins in OA sufferers related to cell structure regulation, inflammatory regulation and antimicrobial regulation. In both seasons, OA sufferers were shown to have increased expression of proteins relating to inflammation, immune responses and cellular development.ConclusionTear protein identification showed dysregulation of proteins involved in inflammation, immunity and cellular structures. Proteins relating to cellular structure may suggest a possible link between OA-associated itch and the subsequent ocular surface damage via eye-rubbing, while inflammatory and immune protein changes highlight potential diagnostic and therapeutic biomarkers of OA.

Negative population trend of the Reed Bunting Emberiza schoeniclus in an important wetland in southeastern Norway

Cover photo: Singing Reed Bunting Emberiza schoeniclus at Årnestangen, Nordre Øyeren nature reserve. Photo: Øyvind Hagen. The Reed Bunting Emberiza schoeniclus is considered the most common of the five species of Emberiza buntings breeding in Norway and is currently listed as a species of Least Concern. However, a complete census of territorial male Reed Buntings conducted in the Nordre Øyeren nature reserve in 2019 showed a decline in the local breeding population of 67% compared to the last counts completed 40 years ago. Censuses were conducted in 1976 and 1981 in the same area and with the same methods. Analyses of citizen science data from the Norwegian Biodiversity Information Centre also show a similar decreasing trend (-66%) for the Reed Bunting breeding population in Nordre Øyeren from 1976 to 2019. Moreover, image analyses of the census maps show that that the number and density of Reed Bunting territories have decreased locally and probably has done so since early 1980s. Interestingly, a subset of localities within the survey area still have the same territory densities as the 1976 and 1981 censuses. We suggest that changes in habitat, vegetation and farming practice, especially grazing, could be the underlying cause of variation in bird numbers. Our results indicate a negative population trend for the Reed Bunting locally, a tendency that is also observed regionally in the Nordic countries. Taken together, local and regional declines raise concerns about the conservation status of the Reed Bunting in Norway.

A microscopic interpretation of hysteresis in the water retention curve of sand

The hysteresis observed in a water retention curve was experimentally studied through the use of X-ray micro-computed tomography (μCT). The CT images were acquired during the main drying and wetting processes of a water retention test for uniform sand. The porosity and degree of saturation (Sr) of pore-scale local subsets were calculated using images segmented into soil, water and air phases. The simultaneous distribution of the local porosity and Sr was also analysed to investigate the relationship between the pore size and retained water volume. At similar suction levels, only small pore spaces retained water during wetting, while relatively large pore spaces retained water during drying, demonstrating the ink-bottle effect. At similar saturation levels, the simultaneous distribution was independent of the wetting and drying processes, indicating no significant difference in the water retention state between the two processes. Another image analysis was performed to calculate the principal curvatures of the air–water interface. At similar saturation levels, the calculated curvature of the drying process was greater than that of the wetting process, indicating contact angle hysteresis. The contact angle between the soil particles and water is different at the microscopic level, whereas the water retention states of the two processes are similar.

Local Identifiability Analysis, Parameter Subset Selection and Verification for a Minimal Brain PBPK Model.

Physiologically-based pharmacokinetic (PBPK) modeling is important for studying drug delivery in the central nervous system, including determining antibody exposure, predicting chemical concentrations at target locations, and ensuring accurate dosages. The complexity of PBPK models, involving many variables and parameters, requires a consideration of parameter identifiability; i.e., which parameters can be uniquely determined from data for a specified set of concentrations. We introduce the use of a local sensitivity-based parameter subset selection algorithm in the context of a minimal PBPK (mPBPK) model of the brain for antibody therapeutics. This algorithm is augmented by verification techniques, based on response distributions and energy statistics, to provide a systematic and robust technique to determine identifiable parameter subsets in a PBPK model across a specified time domain of interest. The accuracy of our approach is evaluated for three key concentrations in the mPBPK model for plasma, brain interstitial fluid and brain cerebrospinal fluid. The determination of accurate identifiable parameter subsets is important for model reduction and uncertainty quantification for PBPK models.

AAformer: Auto-Aligned Transformer for Person Re-Identification.

In person re-identification (re-ID), extracting part-level features from person images has been verified to be crucial to offer fine-grained information. Most of the existing CNN-based methods only locate the human parts coarsely, or rely on pretrained human parsing models and fail in locating the identifiable nonhuman parts (e.g., knapsack). In this article, we introduce an alignment scheme in transformer architecture for the first time and propose the auto-aligned transformer (AAformer) to automatically locate both the human parts and nonhuman ones at patch level. We introduce the "Part tokens (PARTs)", which are learnable vectors, to extract part features in the transformer. A PART only interacts with a local subset of patches in self-attention and learns to be the part representation. To adaptively group the image patches into different subsets, we design the auto-alignment. Auto-alignment employs a fast variant of optimal transport (OT) algorithm to online cluster the patch embeddings into several groups with the PARTs as their prototypes. AAformer integrates the part alignment into the self-attention and the output PARTs can be directly used as part features for retrieval. Extensive experiments validate the effectiveness of PARTs and the superiority of AAformer over various state-of-the-art methods.

ImageNet classification with Raspberry Pis: federated learning algorithms of local classifiers

PurposeThis study aims to propose novel edge device-tailored federated learning algorithms of local classifiers (stochastic gradient descent, support vector machines), namely, FL-lSGD and FL-lSVM. These algorithms are designed to address the challenge of large-scale ImageNet classification.Design/methodology/approachThe authors’ FL-lSGD and FL-lSVM trains in a parallel and incremental manner to build an ensemble local classifier on Raspberry Pis without requiring data exchange. The algorithms load small data blocks of the local training subset stored on the Raspberry Pi sequentially to train the local classifiers. The data block is split into k partitions using the k-means algorithm, and models are trained in parallel on each data partition to enable local data classification.FindingsEmpirical test results on the ImageNet data set show that the authors’ FL-lSGD and FL-lSVM algorithms with 4 Raspberry Pis (Quad core Cortex-A72, ARM v8, 64-bit SoC @ 1.5GHz, 4GB RAM) are faster than the state-of-the-art LIBLINEAR algorithm run on a PC (Intel(R) Core i7-4790 CPU, 3.6 GHz, 4 cores, 32GB RAM).Originality/valueEfficiently addressing the challenge of large-scale ImageNet classification, the authors’ novel federated learning algorithms of local classifiers have been tailored to work on the Raspberry Pi. These algorithms can handle 1,281,167 images and 1,000 classes effectively.

Comparing the performance of a deep learning-based lung gross tumour volume segmentation algorithm before and after transfer learning in a new hospital.

Radiation therapy for lung cancer requires a gross tumour volume (GTV) to be carefully outlined by a skilled radiation oncologist (RO) to accurately pinpoint high radiation dose to a malignant mass while simultaneously minimizing radiation damage to adjacent normal tissues. This is manually intensive and tedious however, it is feasible to train a deep learning (DL) neural network that could assist ROs to delineate the GTV. However, DL trained on large openly accessible data sets might not perform well when applied to a superficially similar task but in a different clinical setting. In this work, we tested the performance of DL automatic lung GTV segmentation model trained on open-access Dutch data when used on Indian patients from a large public tertiary hospital, and hypothesized that generic DL performance could be improved for a specific local clinical context, by means of modest transfer-learning on a small representative local subset. X-ray computed tomography (CT) series in a public data set called "NSCLC-Radiomics" from The Cancer Imaging Archive was first used to train a DL-based lung GTV segmentation model (Model 1). Its performance was assessed using a different open access data set (Interobserver1) of Dutch subjects plus a private Indian data set from a local tertiary hospital (Test Set 2). Another Indian data set (Retrain Set 1) was used to fine-tune the former DL model using a transfer learning method. The Indian data sets were taken from CT of a hybrid scanner based in nuclear medicine, but the GTV was drawn by skilled Indian ROs. The final (after fine-tuning) model (Model 2) was then re-evaluated in "Interobserver1" and "Test Set 2." Dice similarity coefficient (DSC), precision, and recall were used as geometric segmentation performance metrics. Model 1 trained exclusively on Dutch scans showed a significant fall in performance when tested on "Test Set 2." However, the DSC of Model 2 recovered by 14 percentage points when evaluated in the same test set. Precision and recall showed a similar rebound of performance after transfer learning, in spite of using a comparatively small sample size. The performance of both models, before and after the fine-tuning, did not significantly change the segmentation performance in "Interobserver1." A large public open-access data set was used to train a generic DL model for lung GTV segmentation, but this did not perform well initially in the Indian clinical context. Using transfer learning methods, it was feasible to efficiently and easily fine-tune the generic model using only a small number of local examples from the Indian hospital. This led to a recovery of some of the geometric segmentation performance, but the tuning did not appear to affect the performance of the model in another open-access data set. Caution is needed when using models trained on large volumes of international data in a local clinical setting, even when that training data set is of good quality. Minor differences in scan acquisition and clinician delineation preferences may result in an apparent drop in performance. However, DL models have the advantage of being efficiently "adapted" from a generic to a locally specific context, with only a small amount of fine-tuning by means of transfer learning on a small local institutional data set.

Soft Sensor Modeling Method for the Marine Lysozyme Fermentation Process Based on ISOA-GPR Weighted Ensemble Learning.

Due to the highly nonlinear, multi-stage, and time-varying characteristics of the marine lysozyme fermentation process, the global soft sensor models established using traditional single modeling methods cannot describe the dynamic characteristics of the entire fermentation process. Therefore, this study proposes a weighted ensemble learning soft sensor modeling method based on an improved seagull optimization algorithm (ISOA) and Gaussian process regression (GPR). First, an improved density peak clustering algorithm (ADPC) was used to divide the sample dataset into multiple local sample subsets. Second, an improved seagull optimization algorithm was used to optimize and transform the Gaussian process regression model, and a sub-prediction model was established. Finally, the fusion strategy was determined according to the connectivity between the test samples and local sample subsets. The proposed soft sensor model was applied to the prediction of key biochemical parameters of the marine lysozyme fermentation process. The simulation results show that the proposed soft sensor model can effectively predict the key biochemical parameters with relatively small prediction errors in the case of limited training data. According to the results, this model can be expanded to the soft sensor prediction applications in general nonlinear systems.

Disentangling environmental drivers of subarctic dinocyst assemblage compositional change during the Holocene

Abstract. Analysis of compositional changes in fossil organic-walled dinoflagellate cyst (dinocysts) assemblages is a widely used tool for the quantitative reconstruction of past environmental parameters. This approach assumes that the assemblage composition is significantly related to the reconstructed environmental parameter, which requires an independent correlation between the assemblage and the variable, meaning that the variable explains a dimension of the assemblage variance that is not also explained by other parameters. Theoretically, dinocyst assemblages can be used to reconstruct multiple environmental variables. However, there is evidence that primary and subordinate drivers for assemblage compositions regionally differ, and it remains unclear whether a significant relationship to specific parameters in the present ocean always implies that this relationship is significant in fossil assemblages, questioning if past changes in these multiple parameters can be reconstructed simultaneously from fossil assemblages. Here, we analysed a local subset of the Northern Hemisphere dinocyst calibration dataset (n=1968), including samples from the Baffin Bay area (n=421), and evaluated the benefits of a local versus a more regional or global calibration for the environmental reconstruction of Baffin Bay oceanography during the Holocene. We determined the dimensionality of the dinocyst ecological response and identified environmental drivers in the Baffin Bay area for the modern dataset. We analysed four existing Holocene records along a north–south transect in the area and evaluated the statistical significance of downcore reconstructions by applying the local and global datasets with different techniques: the modern analogue technique (MAT), the weighted average partial least square (WA-PLS) and maximum likelihood (ML). We evaluated reconstructions tested as significant in light of the existing state of knowledge about West Greenland's Holocene palaeoceanography. Our analyses imply that present-day and Holocene dinocyst assemblages in the Baffin Bay area are primarily driven by salinity changes; other parameters were less important in driving assemblage compositions, and their contribution differed among the studied records. We show that the objectively occurring and temporally coherent shifts in dinocyst assemblages in the Holocene of Baffin Bay can be robustly interpreted only by transfer functions that are locally calibrated. Transfer functions based on the broad Northern Hemisphere calibration yielded many insignificant environmental reconstructions. At the same time, we show that even in the local calibration, not all parameters that appear to significantly affect dinocyst assemblages in the calibration dataset can be meaningfully reconstructed in the fossil record. A thorough evaluation of the calibration dataset and the significance of downcore applications is necessary to reveal the region-specific information contained in dinocyst assemblage composition.

Enhancing Diversity by Local Subset Selection in Evolutionary Multiobjective Optimization

Enhancing Diversity by Local Subset Selection in Evolutionary Multiobjective Optimization

Granule-specific feature selection for continuous data classification using neighborhood rough sets

Neighborhood rough set theories are commonly used in global feature selection to achieve high performance in continuous data classification. However, selecting a single feature subset to represent the entire dataset may degrade the performance when there are intra-class dissimilarities among objects. Therefore, this paper proposes a novel feature-selection method, Granule-specific Feature Selection (GFS) to select local feature subsets for continuous data classification. The feature selection approach constitutes a novel feature selection algorithm and a novel feature evaluation function and uses existing approaches for granule identification and classification with some adjustments. The neighborhood rough set theories are used in granule (subclass) identification within each class when there are no subclass label information available in the training data, while an improved k-Nearest Neighbors approach is used in classification with granule-specific feature subsets. Experimental results show GFS outperforms most of the global, class-specific, and local feature selection baselines in terms of classification performance.

Stationary local random countable sets over the Wiener noise

The times of Brownian local minima, maxima and their union are three distinct examples of local, stationary, dense, random countable sets associated with classical Wiener noise. Being local means, roughly, determined by the local behavior of the sample paths of the Brownian motion, and stationary means invariant relative to the Lévy shifts of the sample paths. We answer to the affirmative Tsirelson’s question, whether or not there are any others, and develop some general theory for such sets. An extra ingredient to their structure, that of an honest indexation, leads to a splitting result that is akin to the Wiener–Hopf factorization of the Brownian motion at the minimum (or maximum) and has the latter as a special case. Sets admitting an honest indexation are moreover shown to have the property that no stopping time belongs to them with positive probability. They are also minimal: they do not have any non-empty proper local stationary subsets. Random sets, of the kind studied in this paper, honestly indexed or otherwise, give rise to nonclassical one-dimensional noises, generalizing the noise of splitting. Some properties of these noises and the inter-relations between them are investigated. In particular, subsets are connected to subnoises.

MMNet: A Mixing Module Network for Polyp Segmentation.

Traditional encoder-decoder networks like U-Net have been extensively used for polyp segmentation. However, such networks have demonstrated limitations in explicitly modeling long-range dependencies. In such networks, local patterns are emphasized over the global context, as each convolutional kernel focuses on only a local subset of pixels in the entire image. Several recent transformer-based networks have been shown to overcome such limitations. Such networks encode long-range dependencies using self-attention methods and thus learn highly expressive representations. However, due to the computational complexity of modeling the whole image, self-attention is expensive to compute, as there is a quadratic increment in cost with the increase in pixels in the image. Thus, patch embedding has been utilized, which groups small regions of the image into single input features. Nevertheless, these transformers still lack inductive bias, even with the image as a 1D sequence of visual tokens. This results in the inability to generalize to local contexts due to limited low-level features. We introduce a hybrid transformer combined with a convolutional mixing network to overcome computational and long-range dependency issues. A pretrained transformer network is introduced as a feature-extracting encoder, and a mixing module network (MMNet) is introduced to capture the long-range dependencies with a reduced computational cost. Precisely, in the mixing module network, we use depth-wise and 1 × 1 convolution to model long-range dependencies to establish spatial and cross-channel correlation, respectively. The proposed approach is evaluated qualitatively and quantitatively on five challenging polyp datasets across six metrics. Our MMNet outperforms the previous best polyp segmentation methods.

Decentralized Stochastic Multi-Player Multi-Armed Walking Bandits

Multi-player multi-armed bandit is an increasingly relevant decision-making problem, motivated by applications to cognitive radio systems. Most research for this problem focuses exclusively on the settings that players have full access to all arms and receive no reward when pulling the same arm. Hence all players solve the same bandit problem with the goal of maximizing their cumulative reward. However, these settings neglect several important factors in many real-world applications, where players have limited access to a dynamic local subset of arms (i.e., an arm could sometimes be ``walking'' and not accessible to the player). To this end, this paper proposes a multi-player multi-armed walking bandits model, aiming to address aforementioned modeling issues. The goal now is to maximize the reward, however, players can only pull arms from the local subset and only collect a full reward if no other players pull the same arm. We adopt Upper Confidence Bound (UCB) to deal with the exploration-exploitation tradeoff and employ distributed optimization techniques to properly handle collisions. By carefully integrating these two techniques, we propose a decentralized algorithm with near-optimal guarantee on the regret, and can be easily implemented to obtain competitive empirical performance.

A locally unsupervised hybrid learning method for removing environmental effects under different measurement periods

Environmental effects induce deceptive variability in unlabeled vibration data for structural health monitoring (SHM). Although unsupervised learning is an effective solution to this issue, some new challenges such as the size of training data in different measurement periods and the type of learning between local and global frameworks seriously affect overall performance of this technique. To tackle these issues, we propose a locally unsupervised hybrid learning method based on an innovative discriminative reconstruction-based dictionary learning (DRDL) algorithm. The proposed method initially uses a Gaussian mixture model to provide local information for the DRDL algorithm by clustering entire training data into local subsets. Subsequently, this algorithm computes sub-dictionaries and sparse coefficients to reconstruct local training subsets. Using these subsets, an anomaly detector developed from the Mahalanobis-squared distance is used to determine damage indices for SHM. Real data from two bridges are incorporated to verify the proposed method with some comparisons.

On the Risks of Collecting Multidimensional Data Under Local Differential Privacy

The private collection of multiple statistics from a population is a fundamental statistical problem. One possible approach to realize this is to rely on the local model of differential privacy (LDP). Numerous LDP protocols have been developed for the task of frequency estimation of single and multiple attributes. These studies mainly focused on improving the utility of the algorithms to ensure the server performs the estimations accurately. In this paper, we investigate privacy threats (re-identification and attribute inference attacks) against LDP protocols for multidimensional data following two state-of-the-art solutions for frequency estimation of multiple attributes. To broaden the scope of our study, we have also experimentally assessed five widely used LDP protocols, namely, generalized randomized response, optimal local hashing, subset selection, RAPPOR and optimal unary encoding. Finally, we also proposed a countermeasure that improves both utility and robustness against the identified threats. Our contributions can help practitioners aiming to collect users' statistics privately to decide which LDP mechanism best fits their needs.

Local Sparse Principal Component Analysis for Exploring the Spatial Distribution of Social Infrastructure

The primary purpose of this study is to develop a method that can assist in exploring infrastructure-related multidimensional data. The spatial distribution of social infrastructure, including housing and service facilities, is usually uneven across a nation. The underlying reasons behind the spatial configuration of infrastructure vary, and its comprehensive examination is crucial to understanding the true implications of their skewed distribution. However, simultaneous examination of all social infrastructure is not always straightforward due to the volume of data. The presence of strong correlations between the facilities may further impede the finding of meaningful patterns. To this end, we present an extension of PCA that constructs sparse principal components for local subsets of the data. To demonstrate its strengths and limitations, we apply it to a dataset on housing and service facilities in Korea. The results exhibit clear geographic patterns and offer valuable insights into the spatial patterns of social infrastructure, which the standard PCA only partly addressed. It provides empirical evidence that the proposed method can be an effective alternative to the traditional dimension reduction techniques for exploring spatial heterogeneity in massive multidimensional data.