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Scatter plots are widely recognized as fundamental tools for illustrating the relationship between two numerical variables. Despite this, based on solid theoretical foundations, scatter plots generated from pairs of continuous random variables may not serve as reliable tools for assessing dependence. Sklar’s theorem implies that scatter plots created from ranked data are preferable for such analysis, as they exclusively convey information pertinent to dependence. This is in stark contrast to conventional scatter plots, which also encapsulate information about the variables’ marginal distributions. Such additional information is extraneous to dependence analysis and can obscure the visual interpretation of the variables’ relationship. In this article, we delve into the theoretical underpinnings of these ranked data scatter plots, hereafter referred to as rank plots. We offer insights into interpreting the information they reveal and examine their connections with various association measures, including Pearson’s and Spearman’s correlation coefficients, as well as Schweizer–Wolff’s measure of dependence. Furthermore, we introduce a novel visualization ensemble, termed a d-plot, which integrates rank plots, empirical copula diagnostics, and traditional summaries to provide a comprehensive visual assessment of dependence between continuous variables. This ensemble facilitates the detection of subtle dependence structures, including non-quadrant dependencies, that might be overlooked by traditional visual tools.

Since light propagation in a multimode fiber (MMF) exhibits visually random and complex scattering patterns due to external interference, this study numerically models temperature and curvature through the finite element method in order to understand the complex interactions between the inputs and outputs of an optical fiber under conditions of temperature and curvature interference. The systematic analysis of the fiber’s refractive index and bending loss characteristics determined its critical bending radius to be 15 mm. The temperature speckle atlas is plotted to reflect varying bending radii. An optimal end-to-end residual neural network model capable of automatically extracting highly similar scattering features is proposed and validated for the purpose of identifying temperature and curvature scattering maps of MMFs. The viability of the proposed scheme is tested through numerical simulations and experiments, the results of which demonstrate the effectiveness and robustness of the optimized network model.

Multimodal visualizations are essential for identifying and interpreting complex relationships in diverse, high-dimensional biological datasets. However, existing visualization tools often lack native capabilities for embedding explicit statistical and computational annotations, hindering effective quantitative interpretation. We introduce MultiModalGraphics, an R package designed specifically for creating annotated scatterplots and heatmaps of multi-omics and high-dimensional biological data. The package allows seamless embedding of statistical summaries such as fold-changes, p -values, q-values, and standard deviations, facilitating direct quantitative comparisons. MultiModalGraphics interoperates with Bioconductor packages including MultiAssayExperiment, limma, voom, and iClusterPlus, streamlining workflows from data preprocessing and differential expression analysis to visualization. Case studies on three distinct real-world multimodal datasets illustrate its practical utility. Source code, documentation, and example datasets are available via GitHub ( https://github.com/famanalytics0/MultiModalGraphics ) and under review for inclusion into Bioconductor.

This article develops a generalization of the scatterplot matrix based on the recognition that most datasets include both categorical and quantitative information. Traditional grids of scatterplots often obscure important features of the data when one or more variables are categorical but coded as numerical. The generalized pairs plot offers a range of displays of paired combinations of categorical and quantitative variables. A mosaic plot, fluctuation diagram, or faceted bar chart may be used to display two categorical variables. A side-by-side boxplot, stripplot, faceted histogram, or density plot helps visualize a categorical and a quantitative variable. A traditional scatterplot is suitable for displaying a pair of numerical variables, but options also support density contours or annotating summary statistics such as the correlation and number of missing values, for example. By combining these, the generalized pairs plot may help to reveal structure in multivariate data that otherwise might go unnoticed in the process of exploratory data analysis. Two different R packages provide implementations of the generalized pairs plot, gpairs and GGally . Supplementary materials for this article are available online on the journal web site.

Lidar is an active remote sensing technology that has many advantages, but the echo lidar signal is extremely susceptible to noise and complex atmospheric environment, which affects the effective detection range and retrieval accuracy. In this paper, a wavelet transform (WT) and locally weighted scatterplot smoothing (LOWESS) based on ensemble empirical mode decomposition (EEMD) for Rayleigh lidar signal denoising was proposed. The WT method was used to remove the noise in the signal with a signal-to-noise ratio (SNR) higher than 16 dB. The EEMD method was applied to decompose the remaining signal into a series of intrinsic modal functions (IMFs), and then detrended fluctuation analysis (DFA) was conducted to determine the threshold for distinguishing whether noise or signal was the main component of the IMFs. Moreover, the LOWESS method was adopted to remove the noise in the IMFs component containing the signal, and thus, finely extract the signal. The simulation results showed that the denoising effect of the proposed WT-EEMD-LOWESS method was superior to EEMD-WT, EEMD-SVD and VMD-WOA. Finally, the use of WT-EEMD-LOWESS on the measured lidar signal led to significant improvement in retrieval accuracy. The maximum error of density and temperature retrievals was decreased from 1.36% and 125.79 K to 1.1% and 13.84 K, respectively.

The monitoring of crop phenology informs decisions in environmental and agricultural management at both global and farm scales. Current methodologies for crop monitoring using remote sensing data track crop growth stages over time based on single, scalar vegetative indices (e.g., NDVI). Crop growth and senescence are indistinguishable when using scalar indices without additional information (e.g., planting date). By using a pair of normalized difference (ND) metrics derived from hyperspectral data—one primarily sensitive to chlorophyll concentration and the other primarily sensitive to water content—it is possible to track crop characteristics based on the spectral changes only. In a two-dimensional plot of the metrics (ND-space), bare soil, full canopy, and senesced vegetation data all plot in separate, distinct locations regardless of the year. The path traced in the ND-space over the growing season repeats from year to year, with variations that can be related to weather patterns. Senescence follows a return path that is distinct from the growth path.

One of the most important data that forest management planners need for effective decisions is the data related to the forest structure. The aim of the study is to investigate and analyse the structure of protective forests in Irano-Turanian phytogeographical regions, Iran. Since there are many shrubs in this phytogeographical region, it is very difficult to measure the stand diameter at any height (breast or root height). For this reason, it is necessary to analyse the parameters of height and crown cover to investigate and analyse forest structure. For that purpose, two study plots were selected, and basic data were analysed by using statistical distributions, scatter plots and R 2 coefficients. With EasyFit software and Anderson‒Darling test, it was found that the Weibull (3P) and Pearson 6 (4P) distributions for the crown cover factor and the Gen-Pareto and Pert distributions for the height factor have the best goodness-of-fit for the distribution of the different crown cover classes and heights in the studied forest. Moreover, the results confirm that there is a very weak R 2 coefficient between crown cover and root collar diameter, with R 2  = 0.513 and 0.369 in plots 1 and 2, respectively. Therefore, the combination of crown cover and height parameters is more suitable for use when analysing stand structure in such forests, although the values of R 2 are still low (0.673 and 0.524 in plots 1 and 2, respectively). The results of this study show that in protective forests with many shrubs, it is better to focus on the height and crown cover of ​trees and of shrubs rather than on parameters related to stand/tree diameter when analysing stand structure.

Scatterplots require different symbols for different purposes. For presentation, aesthetically pleasing symbols are popular. For analysis, highly discriminable symbols aid pattern detection. This study identifies a default symbol set suitable for both presentation and analysis. This is achieved by using popular symbols with preattentive differences. Supplemental materials for this article are available online.

Applying the panel data of 16 districts in Beijing, China from 2009 to 2020 as the research object, this study measures and analyzes the carbon emission intensity and the level of industrial structure upgrading. Based on the above results, a spatial econometric model is established to analyze the spatial spillover effect of industrial structure upgrading on carbon emission intensity. Conclusions are drawn as follows: (a) In 2009, 2015 and 2020, the carbon emission intensity in most districts of Beijing has decreased, and in some areas even decreased significantly. The upgrading of industrial structure in all districts has been improved (b). According to the results of spatial autocorrelation, the carbon emission intensity in Beijing shows significant positive spatial autocorrelation in 2009 and 2020, while negative spatial autocorrelation in 2015; The upgrading of industrial structure in Beijing shows significant positive spatial autocorrelation in 2009, 2015 and 2020 (c). The regression results of the spatial econometric model show that industrial structure upgrading not only reduces the carbon emission intensity of the region, but also decreases the carbon emission intensity of the surrounding areas.