Visual Bandwidth Selection for Kernel Density Maps

Photogrammetrie • Fernerkundung • Geoinformation 5/2009, S. 445–454

Jukka M. krisp, stefan peters,Christian e. Murphy & hongChao fan, München

Keywords: Kernel Density, Visualization, Visual Analytics, Geostatistics, Kernel Bandwidth, Cartography

Summary: Within this paper we investigate the Zusammenfassung: Visuelle Radius Festlegung challenge to find an appropriate bandwidth in ker- für Kernel Dichte Abschätzungen. In dieser Arbeit nel density estimation. Kernel density estimation untersuchen wir die Möglichkeiten, um einen ange- methods can be used in visualizing and analyzing messenen Radius in der Kernel-Dichte Schätzung spatial data, with the objective of understanding zu erhalten. Kernel-Dichte Schätzung Methoden and potentially predicting event patterns. Our aim können in der Visualisierung und Analyse räumli- is to provide a computational tool to visually select cher Daten zum Verständnis und zur Vorhersage the bandwidth parameter and to visually investi- von räumlichen Mustern dienen. Unser Ziel ist es, gate the effect of different parameters within the ein Computer-Werkzeug zur visuellen Parameter kernel density calculations. With this tool the user Bestimmung zu entwickeln. Die Radien verschie- is able to visually find the most appropriate band- dener Kernel werden visuell geprüft und können width for the particular dataset and scale. The slid- experimentell untersucht werden. Mit diesem er-tool includes a graphical user interface and can Werkzeug kann der Anwender den am besten ge- read point datasets. A specific number of kernel eigneten Radius für den jeweiligen Datensatz be- density maps are generated by using a range of stimmen. Das Schieberegler-Werkzeug beinhaltet bandwidths. In the graphical interface these kernel eine grafische Benutzeroberfläche und kann Punkt- density maps can be drawn near real-time while the datensätze einladen. Anhand statistischer Parame- user changes the respective bandwidth using a slid- ter des verwendeten Punktdatensatzes wird eine er tool. This helps to determine which bandwidth bestimmte Anzahl der Kernel Dichte Karten mit setting fits the needs and provide an appropriate verschiedenen Radien vorberechnet. In der grafi- visualization of a particular point dataset. schen Benutzeroberfläche lassen sich diese Kernel- Dichte-Karten „in real-time“ darstellen. Der Be- nutzer selektiert verschiedene Radien mit einem Slider-Tool, wobei die jeweiligen Kernel Dichte Karten nacheinander abgebildet werden. Dadurch lässt sich der optimale Radius zur zweckdienlichen Visualisierung von einem bestimmten Datensatz finden.

1Introduction specific clusters are located on the map and visualize alternatives to visually cluster spa- Geographic information systems use different tial data. methods to display spatial information, and From a theoretical perspective, this relates to within this paper we investigate new tools to the concept of Geovisualization, which provides enhance the visual clustering of data. On a theory, additionally methods and tools for the general level the aim is to detect spatial pat- visual exploration, analysis, synthesis and pres- terns in data and formulate hypotheses based entation of data that contain geographic infor- on the clusters of point data. Therefore we mation (MacEachrEn & KraaK 2001). From a need to link numerical and graphical proce- broader perspective it includes the concept of ex- dures with a map. We try to identify where ploratory spatial data analysis to detect spatial

DOI: 10.1127/1432-8364/2009/00321432-8364/09/0032$2.50 © 2009 DGPF / E. Schweizerbart'sche Verlagsbuchhandlung, D-70176 Stuttgart 446 Photogrammetrie • Fernerkundung • Geoinformation 5/2009 properties of data. There are different types of sets that are continuous/in raster form; proba- users who use geovisualization systems differ- bility distribution estimation; interpolation or ently and bring different knowledge to the deci- hot spot detection. It is important to under- sion-making process. The use of these tools stand visualization techniques as a means to should support the interaction between different explore geographic data and to detect correla- actors, e. g., decision-makers, the public and ex- tions in spatial data. Methods for kernel den- perts. The communication between these actors sity estimates have a wide variety of applica- supports the development of models and the cre- tions. These include the estimation of proba- ation of task-specific maps. Visualization is a bility distributions and the interpolation of key point in the mediation of content and deci- point data for hot spot detection. Kernel den- sions. Therefore, the choice of visualization sity estimation can also be used in the visual- methods, the orientation of the visualization on ization and analysis of temporal patterns, such the geographic area, as well as the selection of as crime events at different times of day and / the geographic area is of high importance. The or over different time periods. The goal is an overall goal is an adequate communication for understanding and prediction of potential pat- example to gain political support for a certain terns in the available data point. Previous re- decision and enhance communication between search includes work done by LEvinE (2004), experts from various fields. It is essential to pro- who provides an excellent discussion of the vide visualization techniques as a means of ex- various techniques and options, including al- ploring spatial data in order to detect features of ternative methods for bandwidth selection interest contained in them as suggested by (chiu 1991), together with examples from hEarnShaw & unwin (1994). As virrantauS et crime analysis, health research, urban devel- al. (2009) state, the impact of the visualization opment and ecology (LEvinE 2004, SMith et al. on knowledge acquisition (does the map present 2008). Research by KriSp & ŠpatEnKová (2009) unknown information, or is it used to display investigated the possibility to visually review and confirm previously known information?), its the significance of a bandwidth selection for role as an investigative tool (is the map for pri- the fire & rescue services resource planning. vate study, or is it part of a more public decision Bandwidth selection, as explained in the fol- making process?), and its didactic capabilities (is lowing, is often more of an art than a science the map being used interactively or is being read as suggested by SMith et al. (2008). passively?) can be researched through models of Historically, Point Pattern Analysis was in- visualization. In this paper the density map is re- troduced for scientific applications in the early garded as a tool to confirm and depict previously 1920s. GLEaSon (1920) and SvEdbErG (1922) known information. used Point Pattern Analysis (PPA) to statisti- Point densities can be displayed as isarithmic cally describe their botanical and ecological maps. An isarithmic map is created by interpo- work. Since then many different science fields lating a set of isolines between sample points of such as epidemiology, criminology, archaeol- known values (SLocuM et al. 2001, SLocuM et ogy or seismology use PPA. For instance, one al. 2005). In the special case of handling the can conduct a “Hot spot” analysis to have a locations of events as true point data the term better understanding of locations of crime. A isometric map is used. Kernel density estima- Hot Spot is a concentration of point events tion (KDE) can supply point densities from a set within a limited area. This can help the police, of point locations. For that reason we choose to for example, to make decisions where to inten- display point densities as isometric maps using sify daily patrols. Point Pattern Analysis in- kernel density estimation (KDE) method. In volves the ability to compare and describe these isometric maps the isolines show a curve point patterns and test whether there is a sig- in which the point density value is constant. nificant difference to a random spatial point Within this paper we investigate kernel densi- pattern. A point pattern is a set of locations of ty estimation methods which are included in a point events (o’SuLLivan & unwin 2003). variety of applications: point data smoothing; There are three main types of point patterns creation of continuous surfaces from point which can be analyzed, clustered, evenly spaced data in order to combine these with other data- and random spatial point patterns. In a clustered J. M. Krisp et al., Visual Bandwidth Selection 447

Clustered Point PatternEvenly Spaced Point PatternRandom Point Pattern

Fig. 1: Types of Point Patterns in practice. point pattern the events are grouped in regions of tion function extends then in all directions to the study area. Because of the spatial dependen- infinity. This gives the advantage of a through- cy in a clustered pattern one can speak of a posi- out covered study area that shows the intensity tive auto-correlation. An evenly spaced point at all locations. Generally it is difficult to com- pattern is a point pattern with approximately pare point density values in different datasets, equal distances between neighboring points.A because the number of points and the size of point´s neighbor is the nearest other point located the study area may differ greatly. This influ- to itself. Approximately equal distances of all ences the results of the calculations. The defi- neighboring points imply that the pattern is neg- nition of “high” and “low” point densities atively correlated. The random spatial point pat- needs to be considered because those values tern is used as a theoretical point pattern to be differ greatly depending on the dataset, the compared with an empirical point pattern. The scale of the map and the object to be studied. events in a random spatial point pattern are nor- mally distributed. There is no interaction between the points and they are therefore not auto- correlated. One way to define an independently 2Problem Statement and Aim and uniformly set of points is using a homoge- nous Poisson distribution. We suggest that looking at a point dataset as a Many different analyzing methods have continuous surface by using kernel density en- been developed in the past.A popular PPA hances the identification of clusters is of par- method is kernel density estimation. For ker- ticular importance. The choice of the band- nel density estimation a series of estimations width affects the kernel density estimation are made over a grid which is placed on the strongly. When the kernel density estimation entire point pattern. Every estimation shows is carried out with a small bandwidth, the esti- the intensity at a certain location. Therefore mation interacts with individual events. For kernel density estimation detects the highs this reason a user has to experiment with dif- and lows of point densities of the pattern and ferent bandwidths to visually find out which is useful for detecting hot spots. To make a results make most sense. A smaller bandwidth kernel density estimation the user has to select will produce a finer density estimate with all the kernel function and a bandwidth for the little peaks and valleys.A larger bandwidth estimation. Of a number of different kernel will result into a smoother distribution of point functions the normal distribution function is densities (cf. Fig. 2). There are only few rules the most commonly used (diGGLE 2003). Quite considered how to choose a bandwidth by oth- logically, the function weighs closer points er methods than by visual rating (vEnabLES & more than distant points. The normal distribu- ripLEy 1997). 448 Photogrammetrie • Fernerkundung • Geoinformation 5/2009

Fig. 2: Kernel Density with different bandwidths.

In many cases a domain-expert has specific knowledge about the issue at hand. The expert has a certain idea of a density surface in mind and intends to depict this surface in a density map. The expert’s knowledge can be utilized Fig. 3: Flowchart to visually determine the ker- to determine the optimal parameters for a den- nel density bandwidth. sity calculation. The challenge is to find an appropriate bandwidth in KDE. For every KDE density bandwidth for density estimation for and its resulting kernel density map two pa- an exemplary point dataset. rameters are fundamental and have to be specified: cell size and bandwidth. In existing re- Input Data search publications the choice of bandwidth is As a study area we have selected an area in the taken predominantly as the result of experi- centre of Munich, Germany, containing 1412 mental studies by visually comparing differ- points in an area roughly 1 km×1.5 km. The ent bandwidth setting, e. g., in woLff & aSchE exemplary dataset includes points represent- (2009). However, the lack of rules and stand- ing vegetation locations, for example for trees ards concerning bandwidth parameterisation and bushes. This is a partly artificial point da- prompts SMith et al. (2006). Our aim is to pro- taset for a showcase purpose. It does serve the vide a computational tool to visually select the purpose of studying the process to find an op- bandwidth parameter and to visually investi- timal bandwidth and acquire the most favora- gate different parameters within the kernel ble density map for this study area. The entire density calculations. This will help to deter- data set is used for input. o’SuLLivan & unwin mine which bandwidth settings would fit the (2003) stated that it is required to include a needs and provide an appropriate visualiza- complete data set and not a sample data set to tion of a particular point dataset. Therefore a analyze point patterns. computational tool is needed to display the density map based on a dynamically changing Pre-Processing & Determining the bandwidth and output grid size. Relevant Bandwidth Interval To provide a reasonable performance while using the slider-tool, a pre-processing of the 3Applied Method and Test Data kernel density maps is required. On the one hand a range of relevant bandwidth recording The flowchart in Fig. 3 shows the methodo- to the input data set has to be calculated. From logical steps to visually determine the kernel each determined bandwidth then different J. M. Krisp et al., Visual Bandwidth Selection 449 kernel density maps with different resolutions For a visual determination of the kernel were processed and saved as .png image files. bandwidth SiLvErMan’s calculation could be This pre-processing is required, because the used as a starting point for defining a relevant calculation of kernel density is very time-con- bandwidth interval. But this idea does not suming in case of large dataset, especially on consider the point distribution or their dis- demand of a large bandwidth and high resolu- tance to the closest neighbor. tion for the output map. For example, a resolu- In some cases the majority of the points are tion of 400×400 pixel the processing of a ker- located densely in one or only some small sec- nel density-map containing circa 2000 points tors of the area of investigation. The remai- requires around 15 minutes computing time ning points are sparsely distributed. The cal- (Standard Windows PC: 16 GB RAM, CPU culation based on SiLvErMan (1986) gives a 2×1.87 GHz). In contrast to this, the pre-proc- good solution for areas with densely distribut- essed maps can be loaded and drawn on the ed points. Alternatively an adaptive kernel can screen near real time. Eventually technologi- be applied. The following Fig. 4 outlines the cal improvements may make this aspect of the main steps of determining the relevant band- application work obsolete in the near future as width-interval: the maps may be computed on-the-fly. The nearest neighbor distance turned out to Which bandwidths are relevant for varying be an adequate parameter to determine the point datasets? Which are the minimal and max- bandwidth. First the nearest neighbor distance imal bandwidths to determine the optimal one (spherical) of each point was calculated (nnd). by visual observation of the kernel density maps? In the next step the amount of all these cal- These questions were discussed in this work. In culated distances was classified in 50 clusters existing research literature of kernel density es- using the k-means-algorithm. K-mean-cluster- timation, SiLvErMan worked on the issue of de- ing was used, because it takes the value-dif- termining the optimal bandwidth for a certain ferences into account. The resulting 50 values case, before making the actual density computa- represent the range of the relevant kernel tion. He suggested the calculation formulated in bandwidths. Additionally the value half of the the following equation 1 (SiLvErMan 1986). minimal bandwidth was added to consider a smaller bandwidth than the minimal nearest neighbor distance, for the case of evenly Formula based on (Silverman 1986): spaced points. Furthermore the value quarter bw_ optimal = of the maximum nearest neighbor distance  IQR()P 1 was added, to consider a bandwidth larger =⋅106.minvar(P), ⋅n 5 (1)  134.  With: bw_optimal =optimal bandwidth P =dataset: point coordinates IQR(P)=interquartile range (the distance between the 1st and 3rd quartile) of P var(P)=variance of P n =number of points (length of P)

SiLvErMan’s formula for calculating an optimal bandwidth has two outcomes, one optimal radius for the x-coordinates and another one for the y-coordinates. Thereby it takes either the variance (var) or the interquartile range (IQR) of the coordinates to the basis, depend- Fig. 4: Flowchart showing the calculation of ing on which one is smaller. relevant bandwidth-interval. 450 Photogrammetrie • Fernerkundung • Geoinformation 5/2009

than the maximum nearest neighbor distance, example color or shape. Image classifications for the case of clustered points in only one sec- basically differ depending on parametric or tion of the study area. This gives the possibil- non-parametric, hard or soft, pixel by pixel, ity of a wider range for the relevant kernel section by section; monitored or unsupervised bandwidth shown in equation 2. classification. Parametric classification meth- Formula to calculate the amount of the rel- ods assume a parameterized data distribution, evant kernel bandwidth: for example a Gaussian-distribution. In case = of unsupervised classification methods the []bw user merely determines the number of the  50  1   1 classes in which the data have to be divided. min,()nnd k_mean ()nnd,max() nnd , dist 2 i1=  4 There is no need for samples. The cluster algo- (2) rithms lead to the separation of classes based With: on statistical parameters. The assignment of bw =bandwidth spectral classes to information classes results nnd=nearest neighbor distance after the classification. The conditional inter- k_mean =50values resulting from a k-mean vention often leads to undesirable overlap be- clustering of all nnd-values tween the information classes. In case of un- dist =diagonal distance of study area supervised classification methods the user determines the number of the classes as well as Kernel Density Map Output and Classifi- their spectral occurrence in the multispectral cation feature space. Thereby an analysis of samples The calculation of the pixel kernel density val- leads to pre-determined features. The separa- ues is based on a bimodal Gaussian distribution of these spectral classes is done on the tion. The density value for every pixel results basis of spectral-geometric or statistical clas- from the addition of all kernels laying over sifiers. Based on the definition of the spectral this particular pixel. The estimation is shown classes before the classification, an automated in equation 3 based on Scott (1992): assignment by the classification algorithm is done (Köhn 1996). Formula based on (Scott 1992): There are numerous methods and algorithms for image classifications. One simple ! 1 N xx− fx()= ∑Ku() with u = i, method to separate a raster image into regions h ⋅ Nhi=1 h is the use of a grey-tone histogram for determining the regions (aLbErtz & wiGGEnhaGEn 1 1 Gaussian-Kernel: =⋅−2(3) 2008). For the assignment of the different KuGexp  2π2 grey-scales different algorithms are used, With: among others the natural-breaks algorithm ! () fxh =general Kernel density (also referred to as Jenks classification), the function Quantile- algorithm or the separation into KG =standard Gaussian function equal intervals. h=smoothing parameter (band- An approved algorithm for image classifica- width) tion is the k-means clustering. But there is a

x1, x2, ..., xN =points, placed within the number of other classification methods like kernel radius h Maximum-Likelihood classification, the Group-average clustering or the classification Input values of this function are the point co- method based on the minimum distance. More ordinates resolution and the kernel bandwidth. detailed information is given by JähnE (2002). The function performs a bivariate Gaussian Which classification method is finally chosen kernel estimation. The result is a kernel den- depends on the purpose to be achieved and on sity map stored in a raster image which can be the characteristics of the data. There is no classified differently according to the values “right” or “correct” decision for choosing a in each raster cell. Image classifications (clus- certain method. With every of the 52 pre-de- tering) are based on feature characteristics, for fined kernel bandwidths in each case 5 kernel J. M. Krisp et al., Visual Bandwidth Selection 451

1slider tooltochange the bandwidth 2displayed currentbandwidth 3scale information 4colour scale 5image window 6data inputbutton 7resolution adjustment via button-group 8Axonometric depiction of kernel densityvalues

Fig. 5: Graphical User-Interface: kernel-slider-tool. density maps with different resolutions were density estimation can be demonstrated clear- processed and saved, altogether 260 maps. ly and an appropriate bandwidth can be deter- The lowest resolution is 50 ×50pixel, whereby mined visually. In the case of our test data in 1 pixel refers to 22 m. The following contains the background of the kernel density map an 100×100pixel,then200×200pixel,300×300 Open-Street-map of the associated area was pixel up to the maximum of 400×400 pixel, drawn. The kernel density map was shown in whereby 1 pixel refers to 2.75 m. 20 colour- the same screen-frame for each resolution. classes were used for the colour-classification Furthermore the color range and the scale in- of the created kernel density maps. A green formation are shown in the interactive user colour scale was applied. Dark green pixels interface. refer to very high density values and light- Additionally in Fig. 5 a “3D-print” button is green coloured pixel stand for low density val- to see on the top-left corner. It activates an ax- ues. onometric depiction of kernel density values using the current settings of bandwidth and

4Resulting Tool for Visual Determination of an Adequate Kernel Bandwidth

A visual determination of an adequate kernel bandwidth for kernel density estimations can be done best using a Slider-tool. In this work a slider-tool was developed using Matlab GUI (graphical user interface) as shown in Fig. 5. By changing the bandwidth slider interactively the related pre-processed kernel density map will be drawn on the image-window. Likewise the resolution can be changed using the resolution-button-group, while the kernel density map will be actualized. This way the Fig. 6: Axonometric depiction of kernel density influence of the kernel bandwidth to the kernel values. 452 Photogrammetrie • Fernerkundung • Geoinformation 5/2009

Fig. 7: Screenshots-series with different bandwidth-settings in the slider-tool. resolution for the slider-tool. The plot is shown for each resolution. For very small bandwidths in Fig. 6. a rough structure results in the kernel density An interactive zoomand navigation tool al- map. On the other side a quite strong smooth- lows theusertostudy thethree dimensional ing can be seen in the kernel density maps by kernel density map from different perspec- using a large bandwidth. tives. Hencethe 2D-and the3D-visualisation of thekerneldensity mapcan be drawnsideby side and compared directly with each other. 5Discussion and Conclusions Fig. 7 illustrates the advantage of the direct interactively print output by changing the band- Kernel Density Estimations are well estab- widthusing aslider. So thedifferent resulting lished in the visual analysis of point densities. maps can be compared easily and the appropri- To find an appropriate bandwidth does take ate bandwidth can be chosen visually better. experience with the point data at hand or ex- An advantage for using resolutions under perimentation is required to achieve a satis- 100×100 pixel is the short image processing factory output. Therefore the bandwidth slider time. But for an adequate kernel density a map tool lets the user select and browse through a resolution higher than 300×300 pixel is nec- large number of pre-processed density compu- essary. For values over 500×500 pixel no vis- tations, significantly eases the process of find- ible differences can be recognized in the re- ing an appropriate bandwidth setting. sulting map. It is to add on that the kernel den- The choice of the size of the bandwidth sity map was shown in the same screen-frame plays a decisive role for the kernel density es- J. M. Krisp et al., Visual Bandwidth Selection 453 timation. In some references the bandwidth is kinds of densities will be investigated, so that also called smoothing-parameter, meaning the representation can give users a good visual when the bandwidth is set too large, important impression. Additionally further research will information may be lost. In case of a small explore the possibility to use an asynchrony bandwidth local data information has a more kernel to calculate densities for dynamic point significant impact on the kernel density esti- data. The visual analysis of these moving mation map. The kernel slider tool supports points, for example individual mobile phone the interactive process to select the optimal locations, will require research into advanced bandwidth. dynamic visual clustering methods. Modifications on the bandwidth slider tool may include an adapted zoom tool to change the scale of the map window and a pan tool to References move the map in the display area. However, the idea is to keep the tool as simple as possi- aLbErtz, J. & wiGGEnhaGEn, M., 2008: Taschen- ble and to let the user focus on the bandwidth buch zur Photogrammetrie und Fernerkundung: finding process. Guide for Photogrammety and Remote Sensing. – Herbert Wichmann Verlag. chiu, S. 1991: Bandwidth Selection for Kernel Den- sity Estimation. – The Annals of Statistics 19 6Further Research (4): 1883–1905. diGGLE, p.J., 2003: Statistical Analysis of Spatial The bandwidth slider tool will be evaluated Point Patterns. – Academic Press, London, UK, further and further research will include test Second Edition. on users from specific domains. In many cases GLEaSon, a.c., 1920: Some applications of the the bandwidth setting for a specific point data- quadrat method. – Bulletin of the Torrey Botani- set is not clear. By visually identifying the op- cal Club: 21–33. timal bandwidth setting for a more specific hEarnShaw, h.M., wood, M., & brodLiE, K., 1994: dataset and scale, for example mobile phone VISC and GIS: Some Fundamental Consider- base stations in the centre of Munich, we may ations. – Visualization in Geographic Informa- tion Systems, Wiley, New York, USA: 3–8. draw more general conclusions on a most ad- JähnE, b., 2002: Digitale Bildverarbeitung. – vantageous KDE bandwidth setting. An alter- Springer. native method to modify the visual output of Köhn, c., 1996: Bildanalayse und Bilddatenkom- the estimation is to use an adaptive rather than pression. – Hanser Verlag. a fixed bandwidth, meaning a dynamic search KriSp, J.M. & ŠpatEnKová, o., 2009: Kernel density radius with a fixed amount of points included. estimations and their application in visualizing The current tool uses a rather time consum- mission density for fire & rescue services. – Car- ing pre-processing stage. This pre-processing tography and Geoinformatics for Early Warning may be computed faster by an alternative KDE and Emergency Management: Towards Better Solutions. algorithm or by advanced hardware. We sug- LEvinE, n., 2004: CrimeStat: A Spatial Statistics gest that future developments in more power- Program for the Analysis of Crime Incident Lo- ful hardware technology will make the pre- cations (v 3.0). – Ned Levine & Associates, Na- processing time for the bandwidth selection tional Institute of Justice, Washington, DC. tools significantly faster or obsolete. www.icpsr.umich.edu/NACJD/crimestat.html/, In this work the colors from light green to Houston, USA. dark green were used to represent the corre- MacEachrEn, a. &KraaK, M.J., 2001: Research sponding densities of trees. These selected Challenges in Geovisualization. – Cartography colors are quite appropriate for our specified and Geoinformation Science: 3–12. case, as the representation can reflect the natu- o’SuLLivan, d. & unwin, d.J., 2003: Geographic ral phenomenon quite well. However, this set Information Analysis. – Wiley, New Jersey, USA. of colors is not suitable for representing other Scott, d.w., 1992: Multivariate Density Estima- densities, e. g., the density of street lamps. In tion. – Wiley. the nearest future, the matching between different color sets, transparencies and different 454 Photogrammetrie • Fernerkundung • Geoinformation 5/2009

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