Eigenvector Spatial Filtering Research Articles

Reducing spatial autocorrelation in the dynamic simulation of urban growth using eigenvector spatial filtering

In cellular automata (CA) based modeling of dynamic urban growth, non-spatial statistical approaches are usually affected by spatial autocorrelation and fail to describe the spatially non-stationary relationships between urban growth and its drivers. The eigenvector spatial filtering (ESF) is a representative method that can well estimate the regression coefficients in the presence of spatial autocorrelation. In this study, we developed two CA models based on the linear ESF (LESF) and the non-stationary ESF (NESF), of which LESF is featured by spatially stationary coefficients while NESF is featured by spatially varying coefficients. These two models were compared to a conventional logistic regression (LOGR) CA model. The three models (i.e. CALOGR, CALESF and CANESF) were used to simulate rapid urban growth in Suzhou between 2009, 2014 and 2019. The statistics suggest significant reduction in the spatial autocorrelation of the residuals in the CALESF and CANESF land transition maps, as well as an increase in the model reliability. The results show that compared with CALOGR, CALESF and CANESF respectively improved the overall accuracy by 2.0% and 2.2% in 2014, and improved by 0.4% and 0.7% in 2019; they also hit more new-urban-cells, respectively increasing the figure-of-merit (FOM) by 3.8% and 4.3% in 2014, 0.3% and 0.9% in 2019. All three models performed better when integrated spatial heterogeneity into the CA neighborhoods, increasing overall accuracy by over 1% and FOM by over 2%. We concluded that the ESF-based CA models can effectively derive spatially varying land transition rules to capture the urban dynamics, and more accurately predict urban growth featured by spatial autocorrelation.

Modelling skilled and less‐skilled internal migrations in China, 2010–2015: Application of an eigenvector spatial filtering hurdle gravity approach

AbstractIn consideration of the issue of network autocorrelation and zero‐inflated migration data, the study constructs an eigenvector spatial filtering (ESF) hurdle gravity model (ESF HGM) to examine the determinants of China's skilled and less‐skilled internal migrations between 2010 and 2015. In our case, the ESF technique effectively reduces network autocorrelation bias, while the hurdle approach enhances the model prediction on the probability of zeros. Results from the ESF HGM have illustrated significant differences in the determinants between skilled and less‐skilled migrations. It is found that the gravity factors (population sizes at origins and destinations; migration distance), regional industrial structure, unemployment rate, and education level are more related to the migration of less‐skilled people. However, the migration of skilled people is more affected by the wage disparity, natural comforts, and medical services of a region. Our results have also highlighted the differences in some factors (employment rate; education level at origins) between the probability of having migration and the migration volume.

Maximum likelihood estimation of spatially varying coefficient models for large data with an application to real estate price prediction

In regression models for spatial data, it is often assumed that the marginal effects of covariates on the response are constant over space. In practice, this assumption might often be questionable. In this article, we show how a Gaussian process-based spatially varying coefficient (SVC) model can be estimated using maximum likelihood estimation (MLE). In addition, we present an approach that scales to large data by applying covariance tapering. We compare our methodology to existing methods such as a Bayesian approach using the stochastic partial differential equation (SPDE) link, geographically weighted regression (GWR), and eigenvector spatial filtering (ESF) in both a simulation study and an application where the goal is to predict prices of real estate apartments in Switzerland. The results from both the simulation study and application show that the MLE approach results in increased predictive accuracy and more precise estimates. Since we use a model-based approach, we can also provide predictive variances. In contrast to existing model-based approaches, our method scales better to data where both the number of spatial points is large and the number of spatially varying covariates is moderately-sized, e.g., above ten.

Bayesian estimation of spatial filters with Moran’s eigenvectors and hierarchical shrinkage priors

This paper proposes a Bayesian method for spatial regression using eigenvector spatial filtering (ESF) and Piironen and Vehtari (2017)’s regularized horseshoe (RHS) prior. ESF models are most often estimated using variable selection procedures such as stepwise selection, but in the absence of a Bayesian model averaging procedure variable selection methods cannot properly account for parameter uncertainty. Hierarchical shrinkage priors such as the RHS address the foregoing concern in a computationally efficient manner by encoding prior information about spatial filters into an adaptive prior distribution which shrinks posterior estimates towards zero in the absence of a strong signal while only minimally regularizing coefficients of important eigenvectors. This paper presents results from a large simulation study which compares the performance of the proposed Bayesian model (RHS-ESF) to alternative spatial models under a variety of spatial autocorrelation scenarios. The RHS-ESF model performance matched that of the SAR model from which the data was generated. The study also highlights that reliable uncertainty estimates require greater attention to spatial autocorrelation in covariates than is typically given. A demonstration analysis of 2016 U.S. Presidential election results further evidences robustness of results to hyper-prior specifications as well as the advantages of estimating spatial models using the Stan probabilistic programming language.

Modeling China’s Prefecture-Level Economy Using VIIRS Imagery and Spatial Methods

Nighttime light (NTL) data derived from the Visible Infrared Imaging Radiometer Suite (VIIRS), carried by the Suomi National Polar Orbiting Partnership (NPP) satellite, has been widely used to evaluate gross domestic product (GDP). Nevertheless, due to the monthly VIIRS data fluctuation and missing data (excluded by producers) over high-latitude regions, the suitability of VIIRS data for longitudinal city-level economic estimation needs to be examined. While GDP distribution in China is always accompanied by regional disparity, previous studies have hardly considered the spatial autocorrelation of the GDP distribution when using NTL imagery. Thus, this paper aims to enhance the precision of the longitudinal GDP estimation using spatial methods. The NTL images are used with road networks and permanent resident population data to estimate the 2013, 2015, and 2017 3-year prefecture-level (342 regions) GDP in mainland China, based on eigenvector spatial filtering (ESF) regression (mean R2 = 0.98). The ordinary least squares (OLS) (mean R2 = 0.86) and spatial error model (SEM) (mean pseudo R2 = 0.89) were chosen as reference models. The ESF regression exhibits better performance for root-mean-square error (RMSE), mean absolute relative error (MARE), and Akaike information criterion (AIC) than the reference models and effectively eliminated the spatial autocorrelation in the residuals in all 3 years. The results indicate that the spatial economic disparity, as well as population distribution across China’s prefectures, is decreasing. The ESF regression also demonstrates that the population is crucial to the local economy and that the contribution of urbanization is growing.

Comparing spatial patterns of sole proprietorship and corporate payday lenders in Seoul, Korea

The financial regulation lowering the interest rate cap has changed the business model of payday loan services, and as such, we expect that the location of payday lenders may change as well. In particular, because financial business conditions for sole proprietorship and corporate payday lenders differ, their target customers are also expected to vary. The aim of this study is to identify and compare the spatial patterns of corporate and sole proprietorship payday lenders after the implementation of the financial regulation. In a case study of Seoul, Korea, we use a negative binomial regression for the number of corporate and sole proprietorship payday lenders after controlling for demographic conditions. Eigenvector spatial filtering is utilized to take into account the spatial autocorrelation among administrative regions in Seoul. The results from the regression analyses reveal that only sole proprietorship payday lenders are sensitive to less expensive housing and larger traditional markets. Our findings suggest that regulations that decrease the maximum payday loan interest rate can cause financial asymmetry in that economically distressed citizens may be more exposed to unhealthy payday lenders.

House price estimation using an eigenvector spatial filtering approach

PurposeNumerous geo-statistical methods have been developed to analyse the spatial dimension and composition of house prices. Despite these advances, spatial filtering remains an under-researched approach within house price studies. This paper aims to examine the spatial distribution of house prices using an eigenvector spatial filtering (ESF) procedure, to analyse the local variation and spatial heterogeneity.Design/methodology/approachUsing 2,664 sale transactions over the one year period Q3 2017 to Q3 2018, an eigenvector spatial filtering approach is applied to evaluate spatial patterns within the Belfast housing market. This method consists of using geographical coordinates to specify eigenvectors across geographic distance to determine a set of spatial filters. These convey spatial structures representative of different spatial scales and units. The filters are incorporated as predictors into regression analyses to alleviate spatial autocorrelation. This approach is intuitive, given that detection of autocorrelation in specific filters and within the regression residuals can be markers for exclusion or inclusion criteria.FindingsThe findings show both robust and effective estimator consistency and limited spatial dependency – culminating in accurately specified hedonic pricing models. The findings show that the spatial component alone explains 14.6 per cent of the variation in property value, whereas 77.6 per cent of the variation could be attributed to an interaction between the structural characteristics and the local market geography expressed by the filters. This methodological step reduced short-scale spatial dependency and residual autocorrelation resulting in increased model stability and reduced misspecification error.Originality/valueEigenvector-based spatial filtering is a less known but suitable statistical protocol that can be used to analyse house price patterns taking into account spatial autocorrelation at varying (different) spatial scales. This approach arguably provides a more insightful analysis of house prices by removing spatial autocorrelation both objectively and subjectively to produce reliable, yet understandable, regression models, which do not suffer from traditional challenges of serial dependence or spatial mis-specification. This approach offers property researchers and policymakers an intuitive but comprehensible approach for producing accurate price estimation models, which can be readily interpreted.

An eigenvector spatial filtering based spatially varying coefficient model for PM2.5 concentration estimation: A case study in Yangtze River Delta region of China

Ordinary interpolation using PM2.5 ground monitoring observations can seldom reveal the PM2.5 concentration distribution characteristics due to the uneven distribution of monitoring stations and because ordinary linear regression often neglects the spatial autocorrelation among geographical locations. In this study, we developed an eigenvector spatial filtering based spatially varying coefficient (ESF-SVC) model to estimate ground PM2.5 concentration. To generate and analyze the spatiotemporal distribution of PM2.5 concentration in the China's Yangtze River Delta (YRD) region, ESF-SVC model which uses a set of satellite remote sensing data, factory locations, and road networks, was fitted at different time scales from December 2015 to November 2016. Comparisons among the ESF-SVC, eigenvector spatial filtering (ESF) and geographically weighted regression (GWR) models suggest that the ESF-SVC model with an average annual and seasonal adjusted R2 of 0.684, is 10.3 and 13.8% higher than the GWR and ESF models, respectively. The average annual and seasonal cross validation root mean square error (RMSE) of the ESF-SVC models lower than the GWR and ESF models. PM2.5 concentration distribution maps for annual and seasonal were produced to illustrate YRD region's spatiotemporal characteristics. In summary, an ESF-SVC model offers a reliable approach for PM2.5 concentrations estimation in large area.

Flood Risk Evaluation in the Middle Reaches of the Yangtze River Based on Eigenvector Spatial Filtering Poisson Regression

A Poisson regression based on eigenvector spatial filtering (ESF) is proposed to evaluate the flood risk in the middle reaches of the Yangtze River in China. Regression analysis is employed to model the relationship between the frequency of flood alarming events observed by hydrological stations and hazard-causing factors from 2005 to 2012. Eight factors, including elevation (ELE), slope (SLO), elevation standard deviation (ESD), river density (DEN), distance to mainstream (DIST), NDVI, annual mean rainfall (RAIN), mean annual maximum of three-day accumulated precipitation (ACC) and frequency of extreme rainfall (EXE) are selected and integrated into a GIS environment for the identification of flood-prone basins. ESF-based Poisson regression (ESFPS) can filter out the spatial autocorrelation. The methodology includes construction of a spatial weight matrix, testing of spatial autocorrelation, decomposition of eigenvectors, stepwise selection of eigenvectors and calculation of regression coefficients. Compared with the pseudo R squared obtained by PS (0.56), ESFPS exhibits better fitness with a value of 0.78, which increases by approximately 39.3%. ESFPS identifies six significant factors including ELE, DEN, EXE, DIST, ACC and NDVI, in which ACC and NDVI are the first two main factors. The method can provide decision support for flood risk relief and hydrologic station planning.

Modelling interprovincial migration in China from 1995 to 2015 based on an eigenvector spatial filtering negative binomial model

AbstractInterregional migration is a key issue affecting China's future pattern of urbanisation and regional development. In response to the phenomenon of network autocorrelation (NA) commonly found in migration networks, this paper combines eigenvector spatial filtering (ESF) with a negative binomial gravity model based on four‐stage panel data derived from censuses and population sampling surveys, and it analyses the factors that influenced China's interprovincial migration from 1995 to 2015. The results showed that (a) there is a significant spatial spillover effect in the interprovincial migration network. ESF can effectively capture NA in the data to reduce the model's estimation bias. The top 1.4% eigenvectors can extract high NA. (b) There is overdispersion in the interprovincial migration flows. A negative binomial regression gravity model is more appropriate for estimating the driving mechanism for migration than other models. (c) As with the initial variables of the gravity model, population size still exerts a great impact on both outflows and inflows. After considering the influence of NA, the effect of spatial distance is weakening. Additionally, economic, employment, social security, and educational factors are the main forces that are shaping the pattern of interprovincial migration. If the regional unemployment rate and average wages increase by 1%, the outflows and inflows increase by 0.351% and 0.502%, respectively; the coefficient of regional sex ratio at origin is also high, which has a close relationship with the migration motivations and gender differences in the employment market.

모런 고유벡터 공간 필터링 기반 공간 가변 계수를 이용한 서울시 아파트 가격 결정요인의 공간적 이질성 탐색

공간 가변 계수 모형은 회귀 분석 기반의 공간적 이질성 탐색을 위한 도구로 활용되어 왔다. 그 중 하나인 공간 가중회귀 모형(GWR)은 모형의 설명력 부족과 계수 간의 다중공선성과 같은 한계에도 불구하고 대다수의 주택 가격 결정요인의 공간적 이질성 탐색 연구에서 널리 이용되었다. 본 연구에서는 GWR 모형의 한계 극복을 위한 대안으로 모런 고유벡터 공간 필터링(MESF) 기반 모형을 서울시 아파트 매매가격의 결정요인을 탐색하기 위하여 적용하고 그 결과를 GWR, 그리고 전역적 MESF 모형과 비교하였다. 본 연구 결과에서는 모형의 설명력, 잔차의 공간적 자기상관 유무, 그리고 추정된 공간 가변 계수 간의 다중공선성 유무를 기준으로 MESF가 GWR보다 향상된 결과를 보였다. 또한 국지적으로 매우 다른 양상을 보이는 각 결정요인들의 아파트 매매가격 대한 효과는 전역적 수준에서의 결정요인의 효과가 하위지역에서 동일하지 않는 점을 보여주며 공간적 이질성 탐색이 필수적임을 시사한다

Eigenvector Spatial Filtering-Based Logistic Regression for Landslide Susceptibility Assessment

Logistic regression methods have been widely used for landslide research. However, previous studies have seldom paid attention to the frequent occurrence of spatial autocorrelated residuals in regression models, which indicate a model misspecification problem and unreliable results. This study accounts for spatial autocorrelation by implementing eigenvector spatial filtering (ESF) into logistic regression for landslide susceptibility assessment. Based on a landslide inventory map and 11 landslide predisposing factors, we developed the eigenvector spatial filtering-based logistic regression (ESFLR) model, as well as a conventional logistic regression (LR) model and an autologistic regression (ALR) model for comparison. The three models were evaluated and compared in terms of their prediction capability and model fit. The ESFLR model performed better than the other two models. The overall predictive accuracy of the ESFLR model was 90.53%, followed by the ALR model (76.21%) and the LR model (74.76%), and the areas under the ROC curves for the ESFLR, ALR and LR models were 0.957, 0.828 and 0.818, respectively. The ESFLR model adequately addressed the spatial autocorrelation of residuals by reducing the Moran’s I value of the residuals to 0.0270. In conclusion, the ESFLR model is an effective and flexible method for landslide analysis.

Implementing Moran eigenvector spatial filtering for massively large georeferenced datasets

ABSTRACTMoran eigenvector spatial filtering (MESF) furnishes an alternative method to account for spatial autocorrelation in linear regression specifications describing georeferenced data, although spatial auto-models also are widely used. The utility of this MESF methodology is even more impressive for the non-Gaussian models because its flexible structure enables it to be easily applied to generalized linear models, which include Poisson, binomial, and negative binomial regression. However, the implementation of MESF can be computationally challenging, especially when the number of geographic units, n, is large, or massive, such as with a remotely sensed image. This intensive computation aspect has been a drawback to the use of MESF, particularly for analyzing a remotely sensed image, which can easily contain millions of pixels. Motivated by Curry, this paper proposes an approximation approach to constructing eigenvector spatial filters (ESFs) for a large spatial tessellation. This approximation is based on a divide-and-conquer approach. That is, it constructs ESFs separately for each sub-region, and then combines the resulting ESFs across an entire remotely sensed image. This paper, employing selected specimen remotely sensed images, demonstrates that the proposed technique provides a computationally efficient and successful approach to implement MESF for large or massive spatial tessellations.

Spatially Filtered Multilevel Analysis on Spatial Determinants for Malaria Occurrence in Korea.

Since its re-emergence in 1993, the spatial patterns of malaria outbreaks in South Korea have drastically changed. It is well known that complicated interactions between humans, nature, and socio-economic factors lead to a spatial dependency of vivax malaria occurrences. This study investigates the spatial factors determining malaria occurrences in order to understand and control malaria risks in Korea. A multilevel model is applied to simultaneously analyze the variables in different spatial scales, and eigenvector spatial filtering is used to explain the spatial autocorrelation in the malaria occurrence data. The results show that housing costs, average age, rice paddy field ratio, and distance from the demilitarized zone (DMZ) are significant on the level-1 spatial scale; health budget per capita and military base area ratio are significant on the level-2 spatial scale. The results show that the spatially filtered multilevel model provides better analysis results in handling spatial issues.

Estimating bicycle trip volume for Miami-Dade county from Strava tracking data

Sports and fitness apps on GPS enabled cell phones and smart watches have become a rich source of GPS tracking data for nonmotorized traffic, including walking, running, and cycling. These crowd-sourced data can be analyzed to better understand the cycling behavior of a large user community. Using Strava tracking data from the Miami-Dade County area, this study identifies which transport network measures, characteristics of the built environment, and sociodemographic factors are associated with increased or decreased bicycle ridership in census block groups. For this purpose, a set of linear regression models are estimated to predict non-commute and commute bicycle kilometers travelled per block group, as well as bicycle kilometers travelled on weekends and weekdays. Eigenvector spatial filtering is applied to explicitly model spatial autocorrelation and to avoid parameter estimation bias. Results suggest that Strava data, due to its high spatial resolution and coverage, can identify in detail how the influence of explanatory variables on estimated bicycle trip volume varies between different trip purposes and days of the week. Based on the regression results, the paper presents a set of guidelines for practical design detailing which groups of cyclists would benefit most from specific bicycle infrastructure improvements.

The Importance of Scale in Spatially Varying Coefficient Modeling

Although spatially varying coefficient (SVC) models have attracted considerable attention in applied science, they have been criticized as being unstable. The objective of this study is to show that capturing the “spatial scale” of each data relationship is crucially important to make SVC modeling more stable and, in doing so, adds flexibility. Here, the analytical properties of six SVC models are summarized in terms of their characterization of scale. Models are examined through a series of Monte Carlo simulation experiments to assess the extent to which spatial scale influences model stability and the accuracy of their SVC estimates. The following models are studied: (1) geographically weighted regression (GWR) with a fixed distance or (2) an adaptive distance bandwidth (GWRa); (3) flexible bandwidth GWR (FB-GWR) with fixed distance or (4) adaptive distance bandwidths (FB-GWRa); (5) eigenvector spatial filtering (ESF); and (6) random effects ESF (RE-ESF). Results reveal that the SVC models designed to capture scale dependencies in local relationships (FB-GWR, FB-GWRa, and RE-ESF) most accurately estimate the simulated SVCs, where RE-ESF is the most computationally efficient. Conversely, GWR and ESF, where SVC estimates are naïvely assumed to operate at the same spatial scale for each relationship, perform poorly. Results also confirm that the adaptive bandwidth GWR models (GWRa and FB-GWRa) are superior to their fixed bandwidth counterparts (GWR and FB-GWR). Key Words: flexible bandwidth geographically weighted regression, Monte Carlo simulation, nonstationarity, random effects eigenvector spatial filtering, spatial scale.

Tree species richness predicted using a spatial environmental model including forest area and frost frequency, eastern USA.

Assessing geographic patterns of species richness is essential to develop biological conservation as well as to understand the processes that shape these patterns. We aim to improve geographic prediction of tree species richness (TSR) across eastern USA by using: 1) gridded point-sample data rather than spatially generalized range maps for the TSR outcome variable, 2) new predictor variables (forest area FA; mean frost day frequency MFDF) and 3) regression models that account for spatial autocorrelation. TSR was estimated in 50 km by 50 km grids using Forest Inventory and Analysis (FIA) point-sample data. Eighteen environmental predictor variables were employed, with the most effective set selected by a LASSO that reduced multicollinearity. Those predictors were then employed in Generalized linear models (GLMs), and in Eigenvector spatial filtering (ESF) models that accounted for spatial autocorrelation. Models were evaluated by model fit statistics, spatial patterns of TSR predictions, and spatial autocorrelation. Our results showed gridded TSR was best-predicted by the ESF model that used, in descending order of influence: precipitation seasonality, mean precipitation in the driest quarter, FA, and MFDF. ESF models, by accounting for spatial autocorrelation, outperformed GLMs regardless of the predictors employed, as indicated by percent deviance explained and spatial autocorrelation of residuals. Small regions with low TSR, such as the Midwest prairie peninsula, were successfully predicted by ESF models, but not by GLMs or other studies. Gridded TSR in Florida was only correctly predicted by the ESF model with FA and MFDF, and was over-predicted by all other models.

A Segmented Processing Approach of Eigenvector Spatial Filtering Regression for Normalized Difference Vegetation Index in Central China

A segmented processing approach of eigenvector spatial filtering (ESF) regression is proposed to detect the relationship between NDVI and its environmental factors like DEM, precipitation, relative humidity, precipitation days, soil organic carbon, and soil base saturation in central China. An optimum size of 32 × 32 is selected through experiments as the basic unit for image segmentation to resolve the large datasets to smaller ones that can be performed in parallel and processed more efficiently. The eigenvectors from the spatial weights matrix (SWM) of each segmented image block are selected as synthetic proxy variables accounting for the spatial effects and aggregated to construct a global ESF regression model. Results show precipitation and humidity are more influential than other factors and spatial autocorrelation plays a vital role in vegetation cover in central China. Despite the increase in model complexity; the parallel ESF regression model performs best across all performance criteria compared to the ordinary least squared linear regression (OLS) and spatial autoregressive (SAR) models. The proposed parallel ESF approach overcomes the computational barrier for large data sets and is very promising in applying spatial regression modeling to a wide range of real world problem solving and forecasting.

Eigenvector Spatial Filtering Regression Modeling of Ground PM2.5 Concentrations Using Remotely Sensed Data

This paper proposes a regression model using the Eigenvector Spatial Filtering (ESF) method to estimate ground PM2.5 concentrations. Covariates are derived from remotely sensed data including aerosol optical depth, normal differential vegetation index, surface temperature, air pressure, relative humidity, height of planetary boundary layer and digital elevation model. In addition, cultural variables such as factory densities and road densities are also used in the model. With the Yangtze River Delta region as the study area, we constructed ESF-based Regression (ESFR) models at different time scales, using data for the period between December 2015 and November 2016. We found that the ESFR models effectively filtered spatial autocorrelation in the OLS residuals and resulted in increases in the goodness-of-fit metrics as well as reductions in residual standard errors and cross-validation errors, compared to the classic OLS models. The annual ESFR model explained 70% of the variability in PM2.5 concentrations, 16.7% more than the non-spatial OLS model. With the ESFR models, we performed detail analyses on the spatial and temporal distributions of PM2.5 concentrations in the study area. The model predictions are lower than ground observations but match the general trend. The experiment shows that ESFR provides a promising approach to PM2.5 analysis and prediction.

Eigenvector Spatial Filtering for Large Data Sets: Fixed and Random Effects Approaches

Eigenvector spatial filtering (ESF) is a spatial modeling approach, which has been applied in urban and regional studies, ecological studies, and so on. However, it is computationally demanding, and may not be suitable for large data modeling. The objective of this study is developing fast ESF and random effects ESF (RE‐ESF), which are capable of handling very large samples. To achieve it, we accelerate eigen‐decomposition and parameter estimation, which make ESF and RE‐ESF slow. The former is accelerated by utilizing the Nyström extension, whereas the latter is by small matrix tricks. The resulting fast ESF and fast RE‐ESF are compared with nonapproximated ESF and RE‐ESF in Monte Carlo simulation experiments. The result shows that, while ESF and RE‐ESF are slow for several thousand samples, fast ESF and RE‐ESF require only several seconds for the samples. It is also suggested that the proposed approaches effectively remove positive spatial dependence in the residuals with very small approximation errors when the number of eigenvectors considered is 200 or more. Note that these approaches cannot deal with negative spatial dependence. The proposed approaches are implemented in an R package “spmoran.”