Choropleth Mapping with Uncertainty: A Maximum Likelihood–Based Classification Scheme

BackgroundThere is evidence of geographic disparities in COVID-19 hospitalization risks that, if identified, could guide control efforts. Therefore, the objective of this study was to investigate Zip Code Tabulation Area (ZCTA)-level geographic disparities and identify predictors of COVID-19 hospitalization risks in the St. Louis area.MethodsHospitalization data for COVID-19 and several chronic diseases were obtained from the Missouri Hospital Association. ZCTA-level data on socioeconomic and demographic factors were obtained from the American Community Survey. Geographic disparities in distribution of COVID-19 age-adjusted hospitalization risks, socioeconomic and demographic factors as well as chronic disease risks were investigated using choropleth maps. Predictors of ZCTA-level COVID-19 hospitalization risks were investigated using global negative binomial and local geographically weighted negative binomial models.ResultsCOVID-19 hospitalization risks were significantly higher in ZCTAs with high diabetes hospitalization risks (p < 0.0001), COVID-19 risks (p < 0.0001), black population (p = 0.0416), and populations with some college education (p = 0.0005). The associations between COVID-19 hospitalization risks and the first three predictors varied by geographic location.ConclusionsThere is evidence of geographic disparities in COVID-19 hospitalization risks that are driven by differences in socioeconomic, demographic and health-related factors. The impacts of these factors vary by geographical location implying that a ‘one-size-fits-all’ approach may not be appropriate for management and control. Using both global and local models leads to a better understanding of geographic disparities. These findings are useful for informing health planning to identify geographic areas likely to have high numbers of individuals needing hospitalization as well as guiding vaccination efforts.

American Community Survey (ACS) data have become the workhorse for the empirical analysis of segregation in the U.S.A. during the past decade. The increased frequency the ACS offers over the 10-year Census, which is the main reason for its popularity, comes with an increased level of uncertainty in the published estimates due to the reduced sampling ratio of ACS (1:40 households) relative to the Census (1:6 households). This paper introduces a new approach to integrate ACS data uncertainty into the analysis of segregation. Our method relies on variance replicate estimates for the 5-year ACS and advances over existing approaches by explicitly taking into account the covariance between ACS estimates when developing sampling distributions for segregation indices. We illustrate our approach with a study of comparative segregation dynamics for 29 metropolitan statistical areas in California, using the 2010–2014 and 2015–2019. Our methods yield different results than the simulation technique described by Napierala and Denton (Demography 54(1):285–309, 2017). Taking the ACS estimate covariance into account yields larger error margins than those generated with the simulated approach when the number of census tracts is large and minority percentage is low, and the converse is true when the number of census tracts is small and minority percentage is high.

The Maximum Likelihood (ML) and Best Linear Unbiased (BLU) estimators of the location and scale parameters of an extreme value distribution (Lawless [1982]) are compared under conditions of small sample sizes and Type I censorship. The comparisons were made in terms of the mean square error criterion. According to this criterion, the ML estimator of σ in the case of very small sample sizes (n < 10) and heavy censorship (low censoring time) proved to be more efficient than the corresponding BLU estimator. However, the BLU estimator for σ attains parity with the corresponding ML estimator when the censoring time increases even for sample sizes as low as 10. The BLU estimator of σ attains equivalence with the ML estimator when the sample size increases above 10, particularly when the censoring time is also increased. The situation is reversed when it came to estimating the location parameter μ, as the BLU estimator was found to be consistently more efficient than the ML estimator despite the improved performance of the ML estimator when the sample size increases. However, computational ease and convenience favor the ML estimators. Keywords: maximum likelihood estimatorsbest linear unbiased estimatorslife-testingreliability

The Multiple Comparison Procedures with Modeling Techniques (MCP-Mod) framework has been recently approved by the U.S. Food, Administration, and European Medicines Agency as fit-for-purpose for phase II studies. Nonetheless, this approach relies on the asymptotic properties of Maximum Likelihood (ML) estimators, which might not be reasonable for small sample sizes. In this paper, we derived improved ML estimators and correction for their covariance matrices in the censored Weibull regression model based on the corrective and preventive approaches. We performed two simulation studies to evaluate ML and improved ML estimators with their covariance matrices in (i) a regression framework (ii) the Multiple Comparison Procedures with Modeling Techniques framework. We have shown that improved ML estimators are less biased than ML estimators yielding Wald-type statistics that controls type I error without loss of power in both frameworks. Therefore, we recommend the use of improved ML estimators in the MCP-Mod approach to control type I error at nominal value for sample sizes ranging from 5 to 25 subjects per dose.

Choropleth maps are fundamental tools for geographic data analysis, primarily relying on color to convey information. Consequently, the design of their color schemes is important in choropleth map production. However, the traditional coloring methods offered by GIS tools such as ArcGIS and QGIS are not user-friendly enough for nonprofessionals. These tools provide numerous color schemes, making selection difficult, and cannot also easily fulfill personalized coloring needs, such as requests for “summer-like” map colors. To address these shortcomings, we develop a novel system that leverages a large language model and map color design principles to generate contextually relevant and user-aligned choropleth map color schemes. The system follows a three-stage process: Data processing, which provides an overview and classification of the data; Color Concept Design, where color theme and mode are conceptualized based on data characteristics and user intentions; and Color Scheme Design, where specific colors are assigned to classes. Our system incorporates an interactive interface for choropleth map color design and allows users to customize color choices flexibly. Through user studies and evaluations, the system demonstrates acceptable usability, accuracy, and flexibility, with users highlighting its efficiency and ease of use.

For a balanced two‐way mixed model, the maximum likelihood (ML) and restricted ML (REML) estimators of the variance components were obtained and compared under the non‐negativity requirements of the variance components by Lee and Kapadia (1984). In this note, for a mixed (random blocks) incomplete block model, explicit forms for the REML estimators of variance components are obtained. They are always non‐negative and have smaller mean squared error (MSE) than the analysis of variance (AOV) estimators. The asymptotic sampling variances of the maximum likelihood (ML) estimators and the REML estimators are compared and the balanced incomplete block design (BIBD) is considered as a special case. The ML estimators are shown to have smaller asymptotic variances than the REML estimators, but a numerical result in the randomized complete block design (RCBD) demonstrated that the performances of the REML and ML estimators are not much different in the MSE sense.

Over-the-air computation (OAC) is a promising technique to realize fast model aggregation in the uplink of federated edge learning (FEEL). OAC, however, hinges on accurate channel-gain precoding and strict synchronization among edge devices, which are challenging in practice. As such, how to design the maximum likelihood (ML) estimator in the presence of residual channel-gain mismatch and asynchronies is an open problem. To fill this gap, this paper formulates the problem of misaligned OAC for FEEL and puts forth a whitened matched filtering and sampling scheme to obtain oversampled, but independent samples from the misaligned and overlapped signals. Given the whitened samples, a sum-product ML (SP-ML) estimator and an aligned-sample estimator are devised to estimate the arithmetic sum of the transmitted symbols. In particular, the computational complexity of our SP-ML estimator is linear in the packet length, and hence is significantly lower than the conventional ML estimator. Extensive simulations on the test accuracy versus the average received energy per symbol to noise power spectral density ratio (EsN0) yield two main results: 1) In the low EsN0 regime, the aligned-sample estimator can achieve superior test accuracy provided that the phase misalignment is not severe. In contrast, the ML estimator does not work well due to the error propagation and noise enhancement in the estimation process. 2) In the high EsN0 regime, the ML estimator attains the optimal learning performance regardless of the severity of phase misalignment. On the other hand, the aligned-sample estimator suffers from a test-accuracy loss caused by phase misalignment.

This article introduces constraint-based breakpoints, a technique for designing responsive visualizations for a wide variety of screen sizes and datasets. Breakpoints in responsive visualization define when different visualization designs are shown. Conventionally, breakpoints are static, pre-defined widths, and as such do not account for changes to the visualized dataset or visualization parameters. To guarantee readability and efficient use of space across datasets, these static breakpoints would require manual updates. Constraint-based breakpoints solve this by evaluating visualization-specific constraints on the size of visual elements, overlapping elements, and the aspect ratio of the visualization and available space. Once configured, a responsive visualization with constraint-based breakpoints can adapt to different screen sizes for any dataset. We describe a framework that guides designers in creating a stack of visualization designs for different display sizes and defining constraints for each of these designs. We demonstrate constraint-based breakpoints for different data types and their visualizations: geographic data (choropleth map, proportional circle map, Dorling cartogram, hexagonal grid map, bar chart, waffle chart), network data (node-link diagram, adjacency matrix, arc diagram), and multivariate data (scatterplot, heatmap).

A maximum likelihood (ML) estimator is developed for determining time delay between signals received at two spatially separated sensors in the presence of uncorrelated noise. This ML estimator can be realized as a pair of receiver prefilters followed by a cross correlator. The time argument at which the correlator achieves a maximum is the delay estimate. The ML estimator is compared with several other proposed processors of similar form. Under certain conditions the ML estimator is shown to be identical to one proposed by Hannan and Thomson [10] and MacDonald and Schultheiss [21]. Qualitatively, the role of the prefilters is to accentuate the signal passed to the correlator at frequencies for which the signal-to-noise (S/N) ratio is highest and, simultaneously, to suppress the noise power. The same type of prefiltering is provided by the generalized Eckart filter, which maximizes the S/N ratio of the correlator output. For low S/N ratio, the ML estimator is shown to be equivalent to Eckart prefiltering.

The reverberation power spectral density (PSD) is often required for dereverberation and noise reduction algorithms. In this work, we compare two maximum likelihood (ML) estimators of the reverberation PSD in a noisy environment. In the first estimator, the direct path is first blocked. Then, the ML criterion for estimating the reverberation PSD is stated according to the probability density function of the blocking matrix (BM) outputs. In the second estimator, the speech component is not blocked. Instead, the ML criterion for estimating the speech and reverberation PSD is stated according to the probability density function of the microphone signals. To compare the expected mean square error (MSE) between the two ML estimators of the reverberation PSD, the Cramer–Rao Bounds (CRBs) for the two ML estimators are derived. We show that the CRB for the joint reverberation and speech PSD estimator is lower than the CRB for estimating the reverberation PSD from the BM outputs. Experimental results show that the MSE of the two estimators indeed obeys the CRB curves. Experimental results of multimicrophone dereverberation and noise reduction algorithm show the benefits of using the ML estimators in comparison with another baseline estimators.

Cooperative communication combats the effects offadingwithoutthe need for multiple antennas.The cooperative networks use multiple distributed nodes that virtually behave like multiple antennas each having their own local oscillators. But the oscillators cause multiple Carrier Frequency Offsets (CFO), unknown channel gains and Multiple Timing Offsets (MTO). For a good performance of the system perfect synchronization is needed which is achieved by estimation and compensation of multiple offsets. Also, the coverage and capacity can be improved by employing intermediate nodes called relays within cooperative networks. These cooperative relay networks can perform in a better manner even when considering MCFOs, MTOs and unknown channel gains by having more computationally efficient decoders that can use Maximum Likelihood (ML) estimation subject to the Cramer-Rao Lower Bound (CRLB) on the variance of any impartial estimate. The simulation results show that frequency offset still exist after Amplify and Forward (AF) relaying. The performance of a conventional cooperative network seems to have improved with respect to Bit Error Rate of the system when an AF relay is hosted. The estimate of the frequency offset is chosen such that it gives the signal, a high probability of occurrence using ML estimator. In ML estimate, since the bias and variance is difficult to compute, a lower bound called the CRLB on the variance of any unbiased estimate is derived which is satisfied by the ML estimator.Keywords: AF, Cooperative Communication, CFO, ML Estimation and Compensation, OFDM

In estimation theory, the asymptotic (in the number of samples) efficiency of the Maximum Likelihood (ML) estimator is a well known result [1]. Nevertheless, in some scenarios, the number of snapshots may be small. We recently investigated the asymptotic behavior of the Stochastic ML (SML) estimator at high Signal to Noise Ratio (SNR) and finite number of samples [2] in the array processing framework: we proved the non-Gaussiannity of the SML estimator and we obtained the analytical expression of the variance for the single source case. In this paper, we generalize these results to multiple sources, and we obtain variance expressions which demonstrate the non-efficiency of SML estimates.

Choropleth maps are a common and effective way to visualize geographic thematic data. Although cartographers have established many principles about map design, data binning and color usage, less is known about how mapmakers make individual decisions in practice. We interview 16 cartographers and geographic information systems (GIS) experts from 13 government organizations, NGOs, and federal agencies about their choropleth mapmaking decisions and workflows. We categorize our findings and report on how mapmakers follow cartographic guidelines and personal rules of thumb, collaborate with other stakeholders within and outside their organization, and how organizational structures and norms are tied to decision-making during data preparation, data analysis, data binning, map styling, and map post-processing. We find several points of variation as well as regularity across mapmakers and organizations and present takeaways to inform cartographic education and practice, including broader implications and opportunities for CSCW, HCI, and information visualization researchers and practitioners.

Latent growth models (LGMs) are an application of structural equation modeling and frequently used in developmental and clinical research to analyze change over time in longitudinal outcomes. Maximum likelihood (ML), the most common approach for estimating LGMs, can fail to converge or may produce biased estimates in complex LGMs especially in studies with modest samples. Bayesian estimation is a logical alternative to ML for LGMs, but there is a lack of research providing guidance on when Bayesian estimation may be preferable to ML or vice versa. This study compared the performance of Bayesian versus ML estimators for LGMs by evaluating their accuracy via Monte Carlo (MC) simulations. For the MC study, longitudinal data sets were generated and estimated using LGM via both ML and Bayesian estimation with three different priors, and parameter recovery across the two estimators was evaluated to determine their relative performance. The findings suggest that ML estimation is a reasonable choice for most LGMs, unless it fails to converge, which can occur with limiting data situations (i.e., just a few time points, no covariate or outcome, modest sample sizes). When models do not converge using ML, we recommend Bayesian estimation with one caveat that the influence of the priors on estimation may have to be carefully examined, per recent recommendations on Bayesian modeling for applied researchers.

Laser pointing systems for small targets are mainly confronted with two pointing errors, jitter and boresight, arising due to atmospheric turbulence, mechanical vibrations and errors in optical alignment. Knowledge of these parameters provides information about the quality of the pointing and tracking system. In the past, several techniques have been investigated to estimate these parameters from returned laser signals. These include the key ratio technique, the chi-squared method and the maximum likelihood (ML) estimation technique. These techniques have been studied in the literature for Gaussian irradiance profiles. In particular, the ML estimation technique has been found to provide excellent results. In this paper, we extend the ML estimation technique from Gaussian profiles to near-Gaussian irradiance profiles. Our results show that the modified estimator performs much better than a direct application of the original ML estimator.