Coverage Interval Research Articles

Heat Capacity of Fish Oil at High Temperatures and Pressures

This paper reports new results from studies of the isobaric heat capacities of OMEGA-3 “950” fish oil provided by SOLGAR INC (USA) at high temperatures and high pressures. The measurements were carried out on a differential scanning calorimeter (ITs-400) with an automatic data acquisition system in the temperature range from 298.15 to 473.15 K and at pressures from 0.098 to 39.2 MPa. The coverage factor for the 95% confidence level for a two-way coverage interval is assumed to be 2. The expanded measurement uncertainty of heat capacity is estimated to be 2.4%, pressure is estimated to be 0.05% and temperature is estimated to be 15 mK. Experimental data on the fish oil isobaric heat capacity as a function of temperature and pressure are approximated by the correlation equation proposed in the work. At atmospheric pressure, the deviations between those calculated by the correlation equation and the current measured data on isobaric heat capacity are within the range of average absolute deviations (AAD) = 0.22% (standard deviation St. Dev = 0.28% and maximum deviation Max. Dev = 0.39%). Additionally, a comparison was made of the state parameters obtained during the experiment and the literature data at the studied parameters.

Green space-building integration for Urban Heat Island mitigation: Insights from Beijing's fifth ring road district

In this research, we delve into the complex arrangement of urban landscapes, where green spaces and buildings are not merely co-existing but are interwoven into a cohesive fabric that shapes the thermal environment. Our approach transcends the conventional methods of analysis, which typically isolate the roles of greenery or built environments. Instead, we adopt a synergistic perspective that recognizes the collective influence of these landscape constituents on the urban thermal pattern. Key insights are: (1) A linear decrease in average land surface temperature with increasing green space coverage is observed. However, substantial temperature variations (up to 8 °C) within the same coverage interval highlight the significant impact of built-up pattern on thermal conditions; (2) High Building Height and Floor Area Ratio, and low Building Coverage Ratio and Sky View Factor, are linked to cooler temperatures in areas with up to 50 % green space; (3) The study suggests that low-temperature areas can inform the adjustment of built-up patterns in high-temperature areas, offering a strategy for thermal environment optimization within specific green space coverage intervals. This research contributes insights into the integrated planning of green spaces and buildings, with implications for urban development and renewal initiatives aiming to enhance the urban thermal environment.

An adaptive photovoltaic power interval prediction based on multi-objective optimization

Photovoltaic (PV) power interval prediction can provide a variation range of prediction results, which is of great significance to promoting the optimization of power grid dispatching and maintaining the stability of the power system. However, the existing PV interval prediction fails to accurately reflect error fluctuations in their construction process due to dependency on data distribution assumptions and unreasonable estimation of prediction error standard deviations (PEStds). To address these issues, we propose an adaptive interval prediction method based on multi-objective optimization. Firstly, K-means clustering is used to segment similar days. Subsequently, Convolutional Neural Network-Gated Recurrent Unit-Attention (CNN-GRU-Attention) is utilized for point prediction. Furthermore, Grey Relation Analysis (GRA) is used to screen dates similar to the target date under the same weather type, enabling a more precise estimation of prediction error fluctuations. Ultimately, the estimated values are multiplied by non-fixed parameters to construct intervals, which avoids the dependency on data distribution assumptions. Among them, the optimal values for non-fixed parameters are obtained through multi-objective optimization considering interval coverage probability, interval width, and deviation. To verify the effectiveness of the proposed model, we conduct comparative experiments on two datasets with different resolutions. The results demonstrate that the proposed model offers more flexible and higher-quality intervals. Not only does this research improve the accuracy of PV power generation interval prediction, but it also helps to promote the development of smart grid technology and improve the adaptive ability of power systems facing dynamic environments and complex data, which has an important impact on the future energy management and power market.

An Interval Neural Network Method for Identifying Static Concentrated Loads in a Population of Structures

During the design and validation of structural engineering, the focus is on a population of similar structures, not just one. These structures face uncertainties from external environments and internal configurations, causing variability in responses under the same load. Identifying the real load from these dispersed responses is a significant challenge. This paper proposes an interval neural network (INN) method for identifying static concentrated loads, where the network parameters are internalized to create a new INN architecture. Additionally, the paper introduces an improved interval prediction quality loss function indicator named coverage and mean square criterion (CMSC), which balances the interval coverage rate and interval width of the identified load, ensuring that the median of the recognition interval is closer to the real load. The efficiency of the proposed method is assessed through three examples and validated through comparative research against other loss functions. Our research findings indicate that this approach enhances the interval accuracy, robustness, and generalization of load identification. This improvement is evident even when faced with challenges such as limited training data and significant noise interference.

The effects of variability in catch effort on the precision of statistical population reconstruction

Statistical population reconstruction (SPR) models have emerged as a robust and versatile framework for estimating the demographic dynamics of harvested wildlife populations using commonly collected age‐at‐harvest and catch‐effort data. Although numerous studies have suggested that higher interannual variability in catch effort may improve the accuracy and precision of reconstructed estimates, particularly in the absence of auxiliary data on annual abundance or survival, the extent and magnitude of these effects has not been explored. We examined the influence of catch‐effort variability, as measured by the ratio between years of highest and lowest effort, on the relative absolute deviation of reconstructed estimates of population abundance, as well as on the actual percent coverage and width of the corresponding confidence intervals. We used a Monte Carlo simulation to generate catch‐effort data with different levels of variability for populations experiencing a wide range of demographic and harvest conditions. For similar amounts of age‐at‐harvest data, using catch‐effort data with higher interannual variability resulted in reconstructed estimates of annual abundance that had significantly lower deviations from reality, better coverage, and narrower confidence intervals (as measured by the margin of error). These improvements were consistent and linear at low to medium levels of catch‐effort variability, but leveled off and became substantially less pronounced at higher levels. We found that the inclusion of auxiliary data largely mediated this relationship, although higher catch‐effort variability still resulted in more accurate and precise estimates of annual abundance even when these data were included. Our research highlights the need to include a thorough investigation of the available catch‐effort data alongside the established practices of assessing the number of years of available data, the average number of animals harvested each year, and the availability of auxiliary data from radio‐telemetry studies or other sources.

Application of the adaptive Monte Carlo method for uncertainty evaluation in the determination of total testosterone in human serum by triple isotope dilution mass spectrometry

The measurement uncertainty is a crucial quantitative parameter for assessing the reliability of the result. The study aimed to propose a new budget for uncertainty evaluation of a reference measurement procedure for the determination of total testosterone in human serum. The adaptive Monte Carlo method (aMCM) was used for the propagation of probability distributions assigned to various input quantities to determine the uncertainty of the testosterone concentration. The basic principles of the propagation and the statistical analysis were described based on the experimental results of the quality control serum sample. The analysis of the number of Monte Carlo trials was discussed. The procedure of validation of the GUM uncertainty framework using the aMCM was also provided. The number of Monte Carlo trials was 2.974 × 106 when the results had stabilized. The total testosterone concentration was 16.02 nmol/L, and the standard uncertainty was 0.30 nmol/L. The coverage interval at coverage probability of 95% was 15.45 to 16.62 nmol/L, while the probability distribution for testosterone concentration was approximately described by a Gaussian distribution. The validation of results was not passed as the expanded uncertainty result obtained by the aMCM was slightly lower, about 7%, than that by the GUM uncertainty framework with consistent results of the concentration.Graphical

Bayesian estimation and prediction under progressive-stress accelerated life test for a log-logistic model

Accelerated life tests play a very critical role in reliability analysis because highly reliable products are being produced with recent advanced technologies to sustain the market demand and competition. A progressive-stress accelerated life test is one of the kinds of accelerated life tests that allows applied stress to change continuously. Considering the importance of progressive-stress accelerated life tests, this paper deals with progressive-stress accelerated life tests when the lifetimes of units follow the log-logistic distribution and are progressively type-II censored where the associated scale parameter conforms to the inverse power law. Different estimators of model parameters are derived using maximum likelihood and Bayesian methods. Interval estimation is also considered. In the sequel, approximate confidence, bootstrap, and Bayes credible intervals are constructed. Bayes estimators are obtained under squared error, LINEX, and entropy loss functions using proper and improper prior distributions. A simulation study is conducted based on various censoring schemes. The coverage percentages and interval widths are computed via Monte Carlo simulations. Bayes predictive estimates and intervals are also obtained. Finally, two different accelerated life test data are analyzed for illustration purposes.

Binomial Confidence Intervals for Rare Events: Importance of Defining Margin of Error Relative to Magnitude of Proportion

Confidence interval performance is typically assessed in terms of two criteria: coverage probability and interval width (or margin of error). In this article, we assess the performance of four common proportion interval estimators: the Wald, Clopper-Pearson (exact), Wilson and Agresti-Coull, in the context of rare-event probabilities. We define the interval precision in terms of a relative margin of error which ensures consistency with the magnitude of the proportion. Thus, confidence interval estimators are assessed in terms of achieving a desired coverage probability whilst simultaneously satisfying the specified relative margin of error. We illustrate the importance of considering both coverage probability and relative margin of error when estimating rare-event proportions, and show that within this framework, all four interval estimators perform somewhat similarly for a given sample size and confidence level. We identify relative margin of error values that result in satisfactory coverage while being conservative in terms of sample size requirements, and hence suggest a range of values that can be adopted in practice. The proposed relative margin of error scheme is evaluated analytically, by simulation, and by application to a number of recent studies from the literature.

MbDecoda: a debiased approach to compositional data analysis for microbiome surveys.

Potentially pathogenic or probiotic microbes can be identified by comparing their abundance levels between healthy and diseased populations, or more broadly, by linking microbiome composition with clinical phenotypes or environmental factors. However, in microbiome studies, feature tables provide relative rather than absolute abundance of each feature in each sample, as the microbial loads of the samples and the ratios of sequencing depth to microbial load are both unknown and subject to considerable variation. Moreover, microbiome abundance data are count-valued, often over-dispersed and contain a substantial proportion of zeros. To carry out differential abundance analysis while addressing these challenges, we introduce mbDecoda, a model-based approach for debiased analysis of sparse compositions of microbiomes. mbDecoda employs a zero-inflated negative binomial model, linking mean abundance to the variable of interest through a log link function, and it accommodates the adjustment for confounding factors. To efficiently obtain maximum likelihood estimates of model parameters, an Expectation Maximization algorithm is developed. A minimum coverage interval approach is then proposed to rectify compositional bias, enabling accurate and reliable absolute abundance analysis. Through extensive simulation studies and analysis of real-world microbiome datasets, we demonstrate that mbDecoda compares favorably with state-of-the-art methods in terms of effectiveness, robustness and reproducibility.

An efficient Burrows–Wheeler transform-based aligner for short read mapping

Read mapping as the foundation of computational biology is a bottleneck task under the pressure of sequencing throughput explodes. In this work, we present an efficient Burrows–Wheeler transform-based aligner for next-generation sequencing (NGS) short read. Firstly, we propose a difference-aware classification strategy to assign specific reads to the computationally more economical search modes, and present some acceleration techniques, such as a seed pruning method based on the property of maximum coverage interval to reduce the redundant locating for candidate regions, redesigning LF calculation to support fast query. Then, we propose a heuristic verification to determine the best mapping from amounts of flanking sequences. Incorporated with low-distortion string embedding, most dissimilar sequences are filtered out cheaply, and the highly similar sequences left are just right for the wavefront alignment algorithm’s preference. We provide a full spectrum benchmark with different read lengths, the results show that our method is 1.3–1.4 times faster than state-of-the-art Burrows–Wheeler transform-based methods (including bowtie2, bwa-MEM, and hisat2) over 101bp reads and has a speedup with 1.5–13 times faster over 750bp to 1000bp reads; meanwhile, our method has comparable memory usage and accuracy. However, hash-based methods (including Strobealign, Minimap2, and Accel-Align) are significantly faster, in part because Burrows–Wheeler transform-based methods calculate on the compressed space. The source code is available: https://github.com/Lilu-guo/Effaln.

A novel combined wind speed forecasting system based on fuzzy granulation and multi-objective optimization

With the increasing application of wind energy, reliable wind speed prediction has become imperative. However, prior studies predominantly concentrated on single-model predictions, disregarding the inherent uncertainty in wind speed. This oversight resulted in inadequate deterministic and probabilistic forecasting outcomes across varying scenarios. To make up for these shortcomings, a novel forecasting system combining a data preprocessing technique, a sub-model selection method, and a modified multi-objective integrate optimization strategy is designed in this paper. According to the data obtained from China's wind farm, the forecasting efficiency of this system is verified from multiple perspectives. The findings show that the system takes advantage of each model to boost the precision and stability of point prediction successfully. Furthermore, it achieves higher interval coverage and narrower interval width under distinct confidence levels. These results highlight the system's potential as a reliable technical support for efficient dispatching of the entire power system.

Estimating and Assessing Monthly Water Level Changes of Reservoirs and Lakes in Jiangsu Province Using Sentinel-3 Radar Altimetry Data

Generating accurate monthly estimations of water level fluctuations in reservoirs and lakes is crucial for supporting effective water resource management and protection. The dual-satellite configuration of Sentinel-3 makes it possible to monitor water level changes with great coverage and short time intervals. However, the potential of Sentinel-3’s Synthetic Aperture Radar Altimetry (SRAL) data to enable operational monitoring of water levels across Jiangsu Province on a monthly basis has not yet been fully explored. This study demonstrated and validated the use of Sentinel-3’s SRAL to generate accurate monthly water level estimations needed to inform water management strategies. The monthly water levels of lakes and reservoirs from 2017 to 2021 were produced using Sentinel-3 level-2 land products. Results showed that, compared with in situ data across eight studied lakes, all lakes presented R (Pearson correlation coefficient) values greater than 0.5 and Root Mean Square Error (RMSE) values less than 1 m. Notably, water level estimates for Tai Lake, Gaoyou Lake, and Luoma Lake were particularly accurate, with R above 0.9 and RMSE below 0.5 m. Furthermore, the monthly water level estimates derived from the Sentinel-3 data showed consistent seasonal trends over the multi-year study period. The annual water level of all lakes did not change significantly, except for Shijiu Lake, of which the difference between the highest and lowest water level was up to about 5 m. Our findings confirmed the water level observation ability of Sentinel-3. The accuracy of water level monitoring could be influenced by internal water level differences, terrain features, as well as the area and shape of the lake. Larger lakes with more altimetry sampling points tended to yield higher accuracy estimates of water level fluctuations. These results demonstrate that the frequent, wide-area coverage offered by this satellite platform provides valuable hydrological information, especially across remote regions lacking in situ data. Sentinel-3 has immense potential to support improved water security in data-scarce regions.

Optimal Scheme Selection Model of Distributed Photovoltaic Consumption in Smart City Based on Gaussian Mixture Model

In an effort to guarantee the stable action and photovoltaic absorption capability of the spread mesh after a macro digit of distributed solar panels are related to the mesh, this paper puts forward the definitive scheme selection model of distributed photovoltaic absorption in smart cities based on Gaussian mixture model. The principal constituent analysis arithmetic is used to process the chronicle data of distributed photovoltaic action, and the processed data are transmitted to the Gaussian compound model. According to this model, the short-term power situation of distributed photovoltaic is predicted, and the external ability and restrict requirements of the definitive programme of distributed photovoltaic consumption are determined according to the prediction fruits. The definitive programme of distributed photovoltaic consumption is selected by the approximate ideal solution sorting method. The measurement fruits show that the model has a good short-term power prediction effect of distributed photovoltaic, and the prediction fruits are in high agreement with the actual fruits, and the coverage rate and interval average bandwidth fruits are above 0.918 and below 0.11 respectively. The photovoltaic consumption of the definitive programme is above 130%.

Asymmetric morphological filter for roughness evaluation of multifunctional surfaces

The scope of multifunctional surfaces in mechanical engineering and instrumentation is constantly expanding. Examples are products with multilayer coatings, products made of composite materials or using additive manufacturing technologies. A feature of multifunctional surfaces is two levels of texture with different parameter values. Various types of filters are used to decomposition and then analyze texture components. Traditionally, for such problems, a robust Gaussian regression filter and a morphological filter are used. The advantage of the morphological filter lies in the lower computational complexity and the ease of removing the shape component from the profile. This research presents generalized analysis of various types of an asymmetric morphological filter based on simulation. The asymmetric filter differs from the standard morphological filter by using different nesting indexes for combinations of morphological opening and closing operations. We have designed a compositional model of profile, which is superposition of two Gaussian distributions with different parameters and waviness. Relative filtering error of the arithmetic mean heights Ra was chosen as evaluation parameter. Based on the Monte Carlo experiments technique, the relative errors of the filters are determined. The main result of the research is algorithm for choosing nesting indexes depending on the type of primary profile. Thus, we give scientifically justified choice of the filter type and its task-specific parameters for applied use. The choice of filter type and two different nesting indexes in accordance with the developed algorithm provide relative error with a coverage interval less 3 % for 95 % coverage probability. The asymmetric morphological filter is effective for scale space analysis of surfaces with significant shape errors and waviness. The results obtained can be used to analyze the tribological properties of multifunctional surfaces of products.

Confidence interval comparison: Precision of maximum likelihood estimates in LLOQ affected data.

When data is derived under a single or multiple lower limits of quantification (LLOQ), estimation of distribution parameters as well as precision of these estimates appear to be challenging, as the way to account for unquantifiable observations due to LLOQs needs particular attention. The aim of this investigation is to characterize the precision of censored sample maximum likelihood estimates of the mean for normal, exponential and Poisson distribution affected by one or two LLOQs using confidence intervals (CI). In a simulation study, asymptotic and bias-corrected accelerated bootstrap CIs for the location parameter mean are compared with respect to coverage proportion and interval width. To enable this examination, we derived analytical expressions of the maximum likelihood location parameter estimate for the assumption of exponentially and Poisson distributed data, where the censored sample method and simple imputation method are used to account for LLOQs. Additionally, we vary the proportion of observations below the LLOQs. When based on the censored sample estimate, the bootstrap CI led to higher coverage proportions and narrower interval width than the asymptotic CI. The results differed by underlying distribution. Under the assumption of normality, the CI's coverage proportion and width suffered most from high proportions of unquantifiable observations. For exponentially and Poisson distributed data, both CI approaches delivered similar results. To derive the CIs, the point estimates from the censored sample method are preferable, because the point estimate of the simple imputation method leads to higher bias for all investigated distributions. This biased simple imputation estimate impairs the coverage proportion of the respective CI. The bootstrap CI surpassed the asymptotic CIs with respect to coverage proportion for the investigated choice of distributional assumptions. The variety of distributions for which the methods are suitable gives the applicant a widely usable tool to handle LLOQ affected data with appropriate approaches.

Uncertainty Analysis of Friction and Wear-Rate of Cylinder Liner–Piston Ring Tribo Pair Under Boundary Lubrication Conditions

Abstract Uncertainty analyses can improve the reliability and validity of the assessment of friction and wear-rate of tribosystems. This work analyses the various sources of uncertainty in the estimation of friction and wear-rate of liner-ring pairs using a linear reciprocating tribometer (LRT) as per the analytical method suggested by Guide to the Expression of Uncertainty in Measurement (GUM) and simulation approach using Monte Carlo simulation (MCS) method. The uncertainty analysis was conducted by performing sliding wear experiments using a hypereutectic Al–Si cylinder liner specimen against the chrome-coated piston ring as the counter specimen. The experiments were performed to mimic the engine’s condition under a boundary layer lubrication regime. The type A and type B uncertainty components of the tribosystem, such as uncertainty involved in the measurement of mass, linear dimensions of the specimen, radius of the specimen, normal force, stroke length, and frictional force, were evaluated to study their influence on the assessment of the friction and wear-rate. The probability density function of all these uncertainty sources was simulated using the MCS approach to compute the 95% coverage interval for friction and wear-rate directly. The variation in absolute value between the coverage interval limits computed by the GUM framework and predicted by the Monte Carlo method for wear-rate was 11.83%, and for friction coefficient, it was 12.005%.

Content proliferation and narrowcasting in the age of streaming media

Streaming media companies have changed how contents are consumed, produced, and delivered. This research develops a theoretical model on optimal content policies for streaming media companies in order to maximize customer engagement. We have the following interesting findings. First, in contrast to the results in prior literature that firms produce just enough programs and coverage intervals of the programs do not overlap, we show that placing programs closer can be a better policy for engagement‐based firms. Second, more contents are produced under engagement‐based model when the customer value of the content is high. Third, on learning the distribution of the customers, the media firm will always place programs closer when the distribution density is higher such that neighboring programs always have overlapped target audience. Additionally, in facing the tradeoffs of content quality and quantity, the firm should use a high‐quality and low‐variety policy for crowded clusters but a low‐quality and high‐variety policy for niche clusters. Furthermore, when customers consume multiple shows in a period, a good policy is to produce TV shows or series with multiple episodes, whereas individual movies are more suitable for an infrequent watcher market. Our research contributes to the literature on digital media, and the results provide interesting and insightful implications for streaming companies.

Car Bumper Effects in ADAS Sensors at Automotive Radar Frequencies.

Radars in the W-band are being integrated into car bumpers for functionalities such as adaptive cruise control, collision avoidance, or lane-keeping. These Advanced Driving Assistance Systems (ADAS) enhance traffic security in coordination with Intelligent Transport Systems (ITS). This paper analyzes the attenuation effect that car bumpers cause on the signals passing through them. Using the free-space transmission technique inside an anechoic chamber, we measured the attenuation caused by car bumper samples with different material compositions. The results show level drops lower than 1.25 dB in all the samples analyzed. The signal attenuation triggered by the bumpers decreases with the frequency, with differences ranging from 0.55 dB to 0.86 dB when comparing the end frequencies within the radar band. Among the analyzed bumper samples, those with a thicker varnish layer or with talc in the composition seem to attenuate more. We also provide an estimation of the measurement uncertainty for the validation of the obtained results. Uncertainty analysis yields values below 0.21 dB with a 95% coverage interval in the measured frequency band. When comparing the measured value with its uncertainty, i.e., the relative uncertainty, the lower the frequency in the measured band, the more accurate the measurements seem to be.

Uncertainty Evaluation for the Quantification of Urinary Amphetamine and 4-Hydroxyamphetamine Using Liquid Chromatography-Tandem Mass Spectrometry: Comparison of the Guide to the Expression of Uncertainty in Measurement Approach and the Monte Carlo Method with R.

Estimating the measurement uncertainty (MU) is becoming increasingly mandatory in analytical toxicology. This study evaluates the uncertainty in the quantitative determination of urinary amphetamine (AP) and 4-hydroxyamphetamine (4HA) using a liquid chromatography-tandem mass spectrometry (LC-MS/MS) method based on the dilute-and-shoot approach. Urine sample dilution, preparation of calibrators, calibration curve, and method repeatability were identified as the sources of uncertainty. To evaluate the MU, the Guide to the Expression of Uncertainty in Measurement (GUM) approach and the Monte Carlo method (MCM) were compared using the R programming language. The MCM afforded a smaller coverage interval for both AP (94.83, 104.74) and 4HA (10.52, 12.14) than that produced by the GUM (AP (92.06, 107.41) and 4HA (10.21, 12.45)). The GUM approach offers an underestimated coverage interval for Type A evaluation, whereas the MCM provides an exact coverage interval under an abnormal probability distribution of the measurand. The MCM is useful in complex settings where the measurand is combined with numerous distributions because it is generated from the uncertainties of input quantities based on the propagation of the distribution. Therefore, the MCM is more practical than the GUM for evaluating the MU of urinary AP and 4HA concentrations using LC-MS/MS.

Spatial Clustering of Tobacco Retailers Near US Public Schools.

Previous studies have found that tobacco retailers cluster near schools. However, all retail outlets may be located near each other and near schools due to existing infrastructure and zoning policies. We assessed whether tobacco retailers cluster near schools in the United States more than expected when accounting for existing retail locations. We identified 322 056 probable tobacco retailers, 95 110 public schools, and more than 3.8 million businesses comparable to tobacco retailers in land use and business type. We created 500 simulated tobacco retailer datasets by randomly selecting from the larger list of businesses. For each simulated dataset, we calculated the distance from schools to the nearest tobacco retailer (proximity) and the count of tobacco retailers within 800 m of schools (density). Observed proximity and density values were compared to 95% coverage intervals from the 500 simulations. We stratified analyses by urbanicity, percentage of students in the free and reduced-priced lunch program (FRLP), and percentage of Hispanic/Latino, non-Hispanic Black, and non-Hispanic white students. Tobacco retailers were closer to schools in rural areas, cities, and towns and more dense around schools in rural areas, cities, and suburbs compared to random locations in potential retail space. Schools with more students receiving FRLP had higher density than expected while schools with fewer students receiving FRLP had lower density than expected. Within rural areas, clustering did not vary across sociodemographic groups. Within non-rural areas, there were inequities in clustering by racial, ethnic, and socioeconomic school composition. Tobacco retailers cluster near schools after accounting for existing business patterns. There are inequities in clustering by sociodemographic school composition. This study provides compelling evidence that tobacco retailers cluster near US public schools and that there are racial, ethnic, and socioeconomic inequities in clustering, even when accounting for overall retail location patterns. Given that public schools tend to reflect neighborhood demographics, policies to limit tobacco retailers near schools may reduce both school-based and neighborhood-based inequities.