Density Estimation Research Articles

On the cluster of the families of hybrid polynomial kernels in kernel density estimation

This study introduces a novel cluster of hybrid polynomial kernel families, designed to achieve significantly lower asymptotic mean integrated squared error compared to traditional kernels. These hybrid kernels are developed by heuristically combining classical polynomial kernels using probability axioms. An in-depth analysis of error propagation within these kernels is conducted, utilizing both simulation experiments and real-life datasets, including the Life Span of Batteries and COVID-19 datasets. The findings consistently demonstrate that the proposed hybrid kernels outperform their classical counterparts in various density estimation tasks across different distribution types and sample sizes. This research highlights the potential of hybrid polynomial kernels to enhance accuracy in density estimation, advocating for their adoption in statistical modelling and analysis.

Study of the application prospects of film detectors for estimation of α-radiation density

Study of the application prospects of film detectors for estimation of α-radiation density

Non-ergodic statistics and spectral density estimation for stationary real harmonizable symmetric α-stable processes

Non-ergodic statistics and spectral density estimation for stationary real harmonizable symmetric α-stable processes

Docking assistance method for autonomous berthing by backward-time imitation learning and kernel density estimation based on AIS data

Docking assistance method for autonomous berthing by backward-time imitation learning and kernel density estimation based on AIS data

Performance assessment of GIS-based spatial clustering methods in forest fire data

Abstract Forest fires are a significant global issue, devastating large forest areas each year. Effective prevention and control are essential. Geographic Information System (GIS)-based spatial clustering methods are commonly used to manage forest fire risks. However, these methods rely on different mathematical foundations and parameters, resulting in varied hotspot maps. Consequently, areas identified as hotspots by one method may not be significant or may even be classified as cold spots by another. This study utilized forest fire data from 2021 and 2022 in Türkiye to conduct spatial clustering analyses using three methods: Getis Ord Gi*, Anselin Local Moran's I, and Kernel Density Estimation. The aim was to identify high-risk forest fire areas. The effectiveness of these methods was evaluated based on Hit Rate (HR), Predictive Accuracy Index (PAI), and Recapture Rate Index (RRI). The study concluded which method was most suitable for detecting risky forest fire areas in the region. This research fills a gap in the literature by providing a comparative performance evaluation of spatial clustering methods for forest fire risk assessment, offering valuable insights for future studies in this field.

Habitat-specific seasonal densities of sympatric raccoons and opossums in the southeastern United States

Abstract Raccoon and opossum densities have implications for rabies management, but estimates of seasonal densities of both species are lacking for rural nonagricultural habitats of the southeastern United States, a core portion of their geographic range. Consequently, it remains unclear whether the densities of 1 species limit the other, which is possible considering their substantial niche overlap. We carried out a mark–recapture study of raccoons and opossums in 4 rural nonagricultural habitats (bottomland hardwood forest, riparian forest, upland pine forest, and isolated wetlands) in South Carolina, United States (2020 to 2022), and combined this with previous data from the same habitats (2017 to 2019) to estimate habitat-specific spring and fall densities. Raccoon densities ranged from 5.17 ± 0.96 animals/km2 (bottomland hardwood fall) to 1.63 ± 0.83 animals/km2 (upland pine spring) and were on average 19% higher in fall compared to spring. Opossum densities ranged from 10.35 ± 1.98 animals/km2 (bottomland hardwood fall) to 1.11 ± 1.55 animals/km2 (upland pine spring) with divergent seasonal patterns among habitats. These low densities across all habitats compared to other studies are likely the result of low resource availability, consistent with other habitats that have minimal anthropogenic influence. We observed a positive association between raccoon and opossum densities across trapping grids, suggesting that raccoons do not suppress opossum densities, but that densities of both species increase with increasing resource availability. Our results can be used to inform oral rabies vaccination efforts such as refining bait densities and timing of bait distribution in these habitats.

Research on the Nested Structure and Substitution Elasticity of China’s Power Energy Sources

In alignment with China’s “carbon peak and carbon neutrality” goals, carbon reduction and energy structure transformation are central priorities. As a major emitter, the power industry plays a key role in this transition, and identifying effective pathways for its green energy transformation is essential to driving broader industrial green transformation and ensuring sustainable development. This article calculates the elasticity of substitution between clean and non-clean energy within China’s power sector from 1993 to 2021, employing the kernel density estimation method. By further comparing the goodness-of-fit across various nested structures of clean energy sources, the study identifies the optimal internal nested structure and examines the interactions among its components. The results underscore two key insights: on the one hand, a robust substitutive relationship exists between clean and non-clean energy, with the substitution elasticity of 1.646, exhibiting pronounced regional heterogeneity characterized as “weaker in the east and stronger in the west”; on the other hand, the optimal nested structure of clean energy is identified as (hydropower + nuclear power)—wind power—solar power. In this structure, the elements display a substitutive relationship in the Eastern Region, while in the Western Region, they exhibit a complementary relationship.

Frequency-wavenumber Analyses by Steered Response Power Phase Transform Algorithm

This paper is devoted to borrowing some knowledge of related field that deals with array data processing to seismology. Steered-response power with phase transform (SRP-PHAT) is a modern signal processing technique that is treated here for application of small-aperture seismic arrays. It was developed in acoustics for effective sound source localization under strong noise and reverberant conditions, which are absolutely similar to those found in seismological applications and even more complex due to wave propagation through nonhomogeneous media. Mathematical representation of this method is given to solve the problem of frequency-wavenumber spectrum density estimation (F-K analysis). The experiments with real seismic array data are presented for two different data sets: The first set belongs to SPITS array data and contains records of P and S phases of the well-known earthquake from 6 October 2006. The second set belongs to temporary array, installed for monitoring and tracking tremor sources. It was shown that SRP-PHAT is a worthy alternative to conventional broadband F-K analysis method involved in most of free and commercial seismological software.

P-1434. Geostatistical Analysis of Rocky Mountain Spotted Fever Outbreak in Mexican Pediatric Patients: A Retrospective Observational Study

Abstract Background Rocky Mountain Spotted Fever (RMSF) is a bacterial vector-borne disease transmitted through infected ticks. Northern Mexico is currently experiencing an alarming rise of RMSF cases associated with high mortality, affecting the pediatric population. Geographic Information Systems (GIS) characterize geotemporal trends of infectious diseases and offer valuable information for public health strategies. Accurately pinpointing high burden areas of RMSF and tracking its geographical behavior throughout time was crucial to mitigate the outbreak. Heat map of pediatric RMSF cases from August 2022 and September 2023 at the Pediatric Reference of Hospital of Nuevo León, Mexico Heat map was projected with the minor ratio obtained by Kernel distribution Methods Geographical and clinical data of patients < 16 yr with confirmed RMSF diagnosed at the Pediatric reference hospital (HRMI) in NL, Mx from Aug 2022 to Sep 2023 were included. HRMI admitted the majority of affected children during the RMSF outbreak. QGIS® was used to map the geographical distribution of cases. The nearest neighbor index (NNI) was calculated, as well as the Kernel density estimation (KDE) to assess the influence ratio. Regional marginalization indices were obtained from official authorities. Results were visualized with heat maps. Map of Distance of pediatric RMSF cases from August 2022 and September 2023 to the Pediatric Reference of Hospital (HRMI), Nuevo León, Mexico Distance was projected in kilometers, the mean distance case-to-hospital was 24.1 km with a SD ± 10.1km, the lowest in 1.5 km and the highest in 42.6 km Results Total of 23 subjects were included. Mean age: 8 years (± 3.3). Most (82%) had a positive history of contact with ticks. Time from symptom onset 4.9 ± 1.8 days. High mortality (65.2%) despite in-hospital treatment. NNI: 0.54 (z-score -4.1), showing a statistically significant cluster distribution (p < 0.05). Mapping was performed with a KDE ratio of 8755 m, demonstrating two main geographical hotspots of Ped-RMSF. Case-to-hospital distance analysis shows a mean of 24.1 km (± 10.1km). Most cases located in areas with medium (65%) and high (30%) marginalization indices. Majority (95%) of cases lived >10 km from HRMI, where the diagnosis was made. Map of Correlation of RMSF cases from August 2022 and September 2023 at the Pediatric Reference of Hospital of Nuevo León, Mexico and de Margination Index Margination Index was obtain from CONAPO official data (2022). Most cases were located in areas with medium (65%) and high (30%) marginalization indices Conclusion The dynamism of vector-borne diseases requires advanced epidemiological surveillance. Through GIS-based analyses we were able to locate high burden areas of Ped-RMSF sharing them with local health authorities for the implementation of vector control strategies. The clustering pattern demonstrated a non-random distribution of cases, highlighting the importance of known factors involved in the transmission of diseases, like marginalization indices. Our study highlights the relevance of GIS use in outbreak response strategies. Clinical, managment, outcomes and mortality of Pediatric RMSF Outbreak cases from August 2022 and September 2023 at the Pediatric Reference of Hospital of Nuevo León, Mexico Disclosures All Authors: No reported disclosures

P-1433. Characterization and Georeferencing of Tuberculosis Cases in Pediatric Patients at a Reference Hospital in México in 13 years

Abstract Background The 2022 "TB Global Report" documents a rise of Tuberculosis (TB) cases worldwide and emphasizes the importance of surveillance strategies. Geographic Information Systems (GIS) characterize geo-temporal trends of communicable diseases and assist in decision-making, including resource allocation. The use of GIS in pediatric TB is limited in Mexico, where the incidence of TB among children is high. Mexico’s public health system lacks outpatient clinics (OC) for diagnosis, treatment and prevention of TB. Identifying strategic areas to implement these facilities may reduce the burden of TB. Geographic distribution of pediatric patients with TB (n=145) diagnosed at the Pediatric Reference Hospital (HRMI) from 2010-2023. Interpretation of NNI and z-score revealed clusttering pattern for pulmonary, lymph node ad meningeal pedaitric TB Methods Subjects < 18 years with TB diagnosed at the Pediatric reference hospital (HRMI) in NL Mexico from 2010-2023 were included. Regional marginalization indices were obtained from official local authorities. Distance analysis techniques, nearest neighbor index (NNI) and hierarchical clustering were applied using QGis®. Kernel density estimation was calculated, and results were visualized with heat maps. Map 1. Distribution and Heat Map pf Pediatric TB Cases Diagnosed at the Pediatric Reference Hospital (HRMI) of NL, Mex. Heat Map was projected by Kernel Density Estimation of 31736.8 meters (SD ± 25279.2 meters), showing hot spots at the Metropolitan Area of Nuevo Leon. Results Clinical and geographic data of 145 subjects were included. Three areas showed the highest case concentration. NNI from the total sample showed a dispersion pattern; however, a clustering pattern was observed for pulmonary (p< 0.05), meningeal and lymph node TB (Table 1). Before diagnosis, most patients had at least one visit to a public OC where TB was not suspected. Average distance to the nearest public OC was 0.7 km and to HRMI 21.3 km. Case-to-hospital distance analysis showed that 65% of cases lived >10 km from HRMI. The majority of cases occurred in areas with high (23.5%) and medium (44%) marginalization indices. Map 2. Heat Maps of Pedaitric TB Subtypes with Clustter Distribution by Kernel Density Estimation (KDE) A. Pulmonary Pediatric TB: KDE 6434,8 mts B. Meningeal Pediatric TB: KDE 12,828 mts C. Lymph Node Pediatric TB: KDE 9,756 mts Conclusion We identify pediatric TB hotspots in NL, Mexico. The clustering patterns identified for pulmonary, lymph node, and meningeal TB indicate a non-random distribution, possibly influenced by marginalization, population density, and healthcare access. Long case-to-hospital distances and failure of general OCs to suspect pediatric TB highlight the importance of implementing specialized TB OCs in strategic high burden areas. Increasing the visibility of pediatric TB and implementing public health strategies according to community needs is crucial. Our full report will be shared with local health authorities in order to propose the creation of TB OCs. A. Map of distance Case-To-Hospital, category was determine by 4 gropus: Very Low (0 - 2.9 km), Low (3 - 9.9 km), High ( 10 - 29.9 km) and Very High ( > 30 km) B. Map of distance Case-To-Nearest OC, it demostrates that every case, was less tan 3km from an OC C. Map of relation of each case location and the Marginalization Index obtained by national authorities, showing vast majority in medium and high index areas Disclosures All Authors: No reported disclosures

P-1307. AI-Optimized Assessment of Vaccination Strategies in Smallpox Reemergence: an LA Case Study

Abstract Background The resilience of Los Angeles (LA) to smallpox was evaluated using a modified Susceptible, Exposed, Infectious, Recovered, Hospitalized, Deceased (SEIRHD) model, which simulated its spread via an artificial intelligence (AI) mobility matrix and assessed different vaccination regimens (one dose primary series vs a two dose primary series) for outbreak control. Methods The matrix combined data from travel surveys, transportation authorities, GPS caches, and population density estimates from LA County. Fifteen LA government districts plus a 30-mile radius city limit were considered for differential force of infection and mobility matrix variables. Transmission parameters, derived from a review of 20th-century epidemics, assigned 90% transmission to households and 80% to workplaces. Residual immunity against smallpox was set at 8.8%. Intensive care unit (ICU) capacity within 30 miles of LA city limits was set as 2,500 beds per the California Health and Human Services Open Data Portal. One and two dose ring vaccine regimens initiated at day 7 were assessed in 16 districts and as a dispersed city model to understand zonal resilience in eradication and if contagion exceeded ICU capacity limits via an R-programed scenario. Results The naïve general scenario had an infection peak at day 73, with a hospitalization peak of 5,600,227 (day 80) of a susceptible population of 18,833,640. With a 50% effective one dose vaccine, contagion remained below the ICU threshold in the basic 30-mile radius model. However, in specific districts (1, 6-9, and 13-15), even a 95% effective one dose vaccine couldn’t maintain contagion below the ICU threshold according to the AI-assisted zonal model. Hypothetical one dose primary series vaccination regimens outperformed two dose regimens in both city-wide and zonal scenarios (P< 0.05). Zonal analysis based on housing conditions and mobility revealed that even with the most effective hypothetical one dose vaccine (95% efficacy after 7 days), 7 out of 15 LA district ICU thresholds were exceeded, indicating the importance of geospatial modelling. Conclusion The model highlights the importance of zone-based evaluations in preparedness and the need for high-efficacy single-dose vaccinations in ring vaccination. Disclosures Tristan Learoyd, MPharm MA MSc MBA PhD MRPharmS, Emergent: Employee John E. Pittman, PharmD, Emergent: Employee

Measurement of population agglomeration, dynamic change characteristics, and motivations in metropolitan agglomerations-A case study of the Xi'an metropolitan area.

This article compares the population agglomeration characteristics of the Xi'an metropolitan area in western China with those of metropolitan areas in other regions officially approved by the Chinese government. The kernel density estimation method and Markov chain model were used to conduct the study. The results revealed that from 2010 to 2020, the population agglomeration level of the Xi'an metropolitan area showed a trend of first increasing and then decreasing. The absolute gap in the population agglomeration level between cities within the metropolitan area gradually narrowed, and the polarization phenomenon of population agglomeration was not obvious. Compared with metropolitan agglomerations such as Nanjing, Wuhan, Fuzhou, Changsha-Zhuzhou-Xiangtan, Chongqing, and Chengdu, the Xi'an metropolitan agglomeration had a lower population agglomeration level, with a significant gap. Moreover, there was an obvious "club convergence" phenomenon in the population agglomeration levels of different urban agglomerations. The probability of the population agglomeration level remaining stable was at least 53.85%, indicating that there was a "Matthew effect" in which the rich become richer and the poor become poorer. Through the convergence models of α and β, the analysis suggested that there was no significant α convergence between the population agglomeration level of the Xi'an metropolitan agglomeration and that of other metropolitan agglomerations. Instead, there was a significant β divergence, indicating that the gap between the Xi'an metropolitan agglomeration's population agglomeration level and that of other metropolitan agglomerations is gradually widening. An integrated theoretical framework of population agglomeration was constructed from three dimensions: producers, consumers, and social people. An empirical analysis was conducted on the causes of population agglomeration in the Xi'an metropolitan area and other metropolitan areas. The multiple regression results showed that the income level, public consumption expenditure level, education level, comfortable living environment, and educational level were important factors leading to differences in population agglomeration. The geographic detector results showed that factors in the consumer dimension were the main reasons for population agglomeration in metropolitan areas.

Use of cross-validation Bayes factors to test equality of two densities

We propose a nonparametric, two-sample Bayesian test for checking whether or not two data sets share a common distribution. The test makes use of data splitting ideas and requires only simple priors for the bandwidths of two kernel density estimates. Importantly, it does not require priors for high- or infinite-dimensional parameter vectors, as do other nonparametric Bayesian procedures. We provide evidence that the new procedure leads to more stable Bayes factors than do methods based on Pólya trees. Somewhat surprisingly, the behaviour of the proposed Bayes factors when the two distributions are the same is usually superior to that of Pólya tree Bayes factors. We showcase the effectiveness of the test by proving its consistency, conducting a simulation study and applying the test to Higgs Boson data.

Prediction of the Water-Bearing Properties of Weathered Bedrock Aquifers Based on Kernel Density Estimator–Bayes Discriminant

The weathered bedrock aquifer in the Jurassic coalfield of northern Shaanxi Province is a direct water-bearing aquifer, and accurately predicting its water-bearing properties is essential for preventing and controlling water hazards in mining operations. Traditional Bayes discriminant methods have limitations in predicting water-bearing properties, particularly because not all primary factors influencing water-bearing properties meet the criteria for multivariate normal distribution. In this paper, the southern flank of the Ningtiaota Minefield is taken as an example, with the weathered bedrock aquifer as the research object. Six main controlling factors are selected: weathered bedrock thickness, core recovery rate, degree of weathering, lithological combination, elevation of the weathered bedrock surface, and sand-to-base ratio. A kernel density estimator–Bayes (KDE–Bayes) discriminant method for predicting water-bearing properties is presented. The kernel density estimation was carried out on the three main controlling factors that do not conform to a normal distribution—weathered bedrock thickness, core recovery rate, and sand-to-base ratio—and, in conjunction with other primary factors, a KDE–Bayes model was constructed for predicting the water-bearing properties in the southern flank of the Ningtiaota Minefield, based on which a detailed prediction of the water-bearing properties of the south flank of the Ningtiaota Minefield was conducted. By analyzing the actual dewatering data from the S1231 working face and past water inrush (or outburst) incidents, the feasibility and accuracy of this prediction method are demonstrated, providing valuable insights for predicting the water-bearing properties of weathered bedrock aquifers in the Ningtiaota Coal Mine and similar mining conditions.

Selective Scale-Aware Network for Traffic Density Estimation and Congestion Detection in ITS

Traffic congestion detection in surveillance video is crucial for road traffic condition monitoring and improving traffic operation efficiency. Currently, traffic congestion is often characterized through traffic density, which is obtained by detecting vehicles or using holistic mapping methods. However, these traditional methods are not effective in dealing with the vehicle scale variation in surveillance video. This prompts us to explore density-map-based traffic density detection methods. Considering the dynamic characteristics of traffic flow, relying solely on the spatial feature of traffic density is overly limiting. To address these limitations, we propose a multi-task framework that simultaneously estimates traffic density and dynamic traffic congestion. Specifically, we firstly propose a Selective Scale-Aware Network (SSANet) to generate a traffic density map. Secondly, we directly generate a static congestion level from a traffic density map through a linear layer, which can characterize the spatial occupancy of traffic congestion in each frame. In order to further describe dynamic congestion, we simultaneously consider the dynamic characteristics of traffic flow, using the overall traffic flow velocity integrated with static congestion estimation for a dynamic assessment of congestion. On the collected dataset, our method achieves state-of-the-art results on both congestion detection and density estimation task. SSANet also obtains 99.21% accuracy on the UCSD traffic flow classification dataset, which outperforms other state-of-the-art methods.

Vertical kinematics of the young Galactic clusters

Abstract The young disc vertical phase is paramount in our understanding of Galaxy evolution. Analysing the vertical kinematics at different galactic regions provides important information about the space-time variations of the Galactic potential. The vertical phase snail-shell structure found after Gaia DR2 release encompasses a wide range of ages. However, the structure of the VZ vs Z diagram appears linear when the analysis is limited to studying objects younger than 30 Ma. Based on the vertical velocity and height-over-disc maps obtained for a sample of young open clusters, this method also allows the matter density in the Solar neighbourhood to be estimated using a completely different approach than previously found in the literature. We use two different catalogues of star clusters to confirm the previous result and study new age ranges. The linear pattern between VZ and Z shows different slopes, ∂VZ/∂Z, for various age groups. The results fit a simple model (harmonic oscillator) of in-plane decoupled vertical dynamics up to a certain age limit, corresponding to ∼ 30 Ma. This work also analyses the relationship between the local volumetric density of matter (ρ0) and the disc vertical kinematics for different age ranges, all below 50 Ma. The best estimates of the effective volumetric mass density in the Solar neighbourhood, 0.09-0.15 M⊙ pc−3, agree with those given by other authors, assessing the reliability of the proposed dynamical model. These values are a minorant of the actual matter density in the region.

ASSESSING COVID-19 SPREAD AND VACCINATION EFFECTIVENESS IN CHAD THROUGH MEAN KERNEL DENSITY ESTIMATION

This study introduces an innovative approach to assess the impact of COVID-19 vaccination on the spread of the virus in Chad by employing mean kernel density (MKD) estimation for non-parametric analysis. Utilizing epidemiological data collected between January 2021 and June 2023, we explore the correlation between vaccination rates and confirmed cases across various regions of Chad. Our analysis demonstrates that the application of MKD provides a smoother and more accurate visualization of infection trends before and after vaccination campaigns. The results reveal a significant reduction in confirmed cases post-vaccination, particularly in areas with higher vaccination rates. Correlation analysis further highlights the effectiveness of vaccination campaigns, showing a strong association between vaccination rates and the decrease in infections. However, the data also suggest a reactive vaccination strategy in regions with initially high infection rates. This innovative application of mean KDE provides new insights into the dynamics of vaccination efforts and their crucial role in reducing the spread of COVID-19 in Chad.

Exploring the role of electrode density in capturing spatiotemporal dynamics of resting-state networks with EEG

Abstract This study investigates the role of electrode density in capturing resting-state brain activity, an area of significant clinical relevance, where electroencephalography (EEG) is favored for its cost-efficiency. We analyze how different electrode configurations affect the precision of cortical current density estimation in EEG recordings. Using exact Low-Resolution Electromagnetic Tomography (eLORETA), we estimated the cortical current density in regions of interest linked to Resting State Networks (RSN). Point process analysis was employed to identify regions of high activity over time, revealing dynamic brain salient activity patterns, or brain maps. We evaluated the impact of electrode density by comparing 64-channel and 20-channel configurations and found that both configurations yielded similar and consistent brain maps. To confirm the robustness of our approach, we assessed the Berger effect in eyes-closed (EC) versus eyes-open (EO) conditions, observing that the functional differences between EC and EO states remained stable regardless of electrode density, aligning with previous research. Conversely, randomization of data or the use of non-homogeneous electrode configurations disrupted the resulting patterns, highlighting the physiological relevance of our methodology. Overall, our results demonstrate that this approach reliably captures the spatiotemporal dynamics of brain activity, even with fewer electrodes, and holds promise for broader clinical applications.

First density estimates of the Endangered Claire's mouse lemur Microcebus mamiratra and recommendations for its conservation

Abstract Mouse lemurs Microcebus spp. are small, nocturnal primates endemic to Madagascar. The genus is extraordinarily diverse, with 25 extant species, several of which have been described recently. The Endangered Claire's mouse lemur Microcebus mamiratra was first described in 2006, but, similarly to other newly described mouse lemurs, remains understudied, and estimates of its population size are unavailable, hampering effective conservation management. We conducted line transect distance sampling surveys of M. mamiratra across several habitat types in and around Lokobe National Park on the island of Nosy Be in north-western Madagascar. Using a systematic random design we surveyed 15 transects over a 6-week period in 2023, recording 92 detections from a total survey effort of 46.5 km. We estimate the density of M. mamiratra on Nosy Be to be 125.1 individuals/km2, which extrapolates to an estimate of c. 4,700 individuals across the forested areas of its range on the island. Our results indicate that Nosy Be harbours moderately high densities of M. mamiratra, with the highest encounter rates in the unprotected secondary and degraded forests around Lokobe National Park. Our population estimate will inform future conservation status assessments and conservation planning for this range-restricted species and provide a baseline for monitoring population changes over time. We present recommendations for the conservation of M. mamiratra and highlight the potential for lemur watching, sustained by the strong tourism industry on Nosy Be, to help protect lemur habitat and generate economic opportunities for local communities.

On the sign changes of 𝜓(𝑥)-𝑥

We improve the lower bound for V ( T ) V(T) , the number of sign changes of the error term ψ ( x ) − x \psi (x)-x in the Prime Number Theorem in the interval [ 1 , T ] [1,T] for large T T . We show that \[ lim inf T → ∞ V ( T ) log ⁡ T ≥ γ 0 π + 1 60 , \liminf _{T\to \infty }\frac {V(T)}{\log T}\geq \frac {\gamma _{0}}{\pi }+\frac {1}{60}, \] where γ 0 = 14.13 … \gamma _{0}=14.13\ldots is the imaginary part of the lowest-lying non-trivial zero of the Riemann zeta-function. The result is based on a new density estimate for zeros of the associated k k -function, over 4 ⋅ 10 21 4\cdot 10^{21} times better than previously known estimates of this type.