International Journal on Advances in Intelligent Systems, vol 8 no 1 & 2, year 2015, http://www.iariajournals.org/intelligent_systems/ 27 Spatio-Temporal Density Mapping for Spatially Extended Dynamic Phenomena - a Novel Approach to Incorporate Movements in Density Maps Stefan Peters Liqiu Meng Department of Geoinformation Department of Cartography Universiti Teknologi Malaysia Technische Universität München Johor Bahru, Malaysia München, Germany e-mail: [email protected] e-mail: [email protected] Abstract - The visualization of density information and its In the next section, the state of the art related to density changes is a crucial support for the spatio-temporal analysis of maps, in particular, an overview of approaches considering dynamic phenomena. Existing density map approaches mainly the dynamics of movement data in the density visualization is apply to datasets with two different moments of time and thus given. In the section afterwards, our own approach is do not provide adequate solutions for density mapping of described in detail, followed by implementation processes, dynamic points belonging to the same moving phenomena. The discussions of the results and a conclusion. proposed approach termed as Spatio-Temporal Density Mapping intends to fill this research gap by incorporating and II. DENSITY MAPS - STATE OF THE ART visualizing the temporal change of a point cluster in a 2D density map. At first either straight or curved movement trajectories One of the most straight forward ways to visualize point based on centroids of spatio-temporal point clusters are density is a scatter plot or a dot map. Graphic variables for detected. The traditional Kernel density contour surface is then point symbols, such as size, shape, color and transparency, can divided into temporal segments, which are visually be applied in relation with the attribute value. In order to distinguished from one another by means of a rainbow color discern the density distribution, these graphic variables can be scheme. Furthermore, several ideas for the investigation of the iteratively adapted to the given map scale, but still the usability of our approach are addressed. occlusion of neighboring points cannot be always desirably avoided. The density value of each point can be obtained by Keywords - spatio temporal density map; rainbow color counting all points within a buffer around the point or within scheme; visual analytics; dynamic phenomena. a grid cell the point is located in. In the following, the density estimation and map principles I. INTRODUCTION are shortly presented and the state of the art of density maps As stated in a previous work by Peters and Meng [1], with static or dynamic data is given. visualization helps to investigate and understand complex relationships in a spatial context. Maps account as one of the A. KDE most powerful visualization forms. They represent geographic KDE [6] is a classic method widely used to determine information in abstract ways that support the identification of densities of individual points that represent a continuous spatial patterns and the interpretation of spatial phenomena. surface. The KDE approach is described in detail in [6-8]. The Furthermore, the visual presentation and analysis of dynamic standard KDE, a normal distribution function, uses a Gaussian data and dynamic phenomena is currently a hot research topic kernel: [2]. Hence, in today ’s society, the need for data abstraction along with the growing amount of available digital geodata is 1 n ) (1) rapidly increasing. One reasonable way of abstracting data is fh ()x = ∑ K(u) provided by density maps [3]. Density maps can be applied n ⋅ h i=1 for point data in various fields, for instance, in physical or human geography, geology, medicine, economy or biology [4, X − X i 1 1 2 whereby u = and K G ()u = ⋅ exp − u 5]. How to present the density for dynamic data/phenomena h 2π 2 is, however, not yet adequately addressed. In this paper, we introduce a novel density mapping ) with: f h (x ) = general Kernel density function approach for spatially and temporally changing data. The K = Kernel function approach is based on [1], whereby in this work a different test KG = standard Gaussian function dataset was used, two different types of movement trajectory h = smoothing parameter (bandwidth) concepts are introduced, a verification of the used rainbow n = number of points color scheme is presented, investigations about wrongly X = point ( x,y ) for which assigned points are considered and a more detailed the density will be estimated comparison between STDmap and its alternative in form of X1, X 2, ..., X N = sample points, placed within the Kernel Density Estimation (KDE) maps for each temporal kernel radius h interval is provided. 2015, © Copyright by authors, Published under agreement with IARIA - www.iaria.org International Journal on Advances in Intelligent Systems, vol 8 no 1 & 2, year 2015, http://www.iariajournals.org/intelligent_systems/ 28 C. Dynamic data and density information Beside a Gaussian kernel, also other kernel types can be In the following sections, an overview is given about applied, such as Triangular, Biweight, Epanechnikov or existing works related to density maps of dynamic points. Uniform kernel [7]. A certain bandwidth (search radius) is 1) Sequence of KDE maps for dynamic points defined for the kernels, located around each point. For each A straightforward way of visualizing the density of cell of an underlying grid (defined by a certain resolution) a dynamic points would be a sequence of density surfaces (one density value is calculated as shown in equation (1) and hence per time interval). The change of the density in time could be a smooth surface is provided [9]. The kernel bandwidth value better discernable by means of an animation of these density strongly effects the resulting density surface [10]. A formula maps. We could also arrange the local density contours of for an optimal bandwidth is offered by Silverman [7] as shown each time interval on the same map. Transparency and a in equation (2). unique color scheme for each time interval could be applied in order to distinguish different density contours. However, the tinted intervals may spatially overlap and make the map 1 IQR(P) − reading a difficult endeavor. bw_optimal = .1 06∗min var()P , ∗n 5 (2) .1 34 2) Dual KDE Jansenberger and Staufer-Steinocher [18] analyzed two with: bw_optimal = optimal bandwidth different point datasets recorded within the same area, but at P = point dataset (coordinates) two different moments of time. The authors suggest a Dual- IQR (P) = interquartile range KDE approach, which results in a map illustrating the spatio- var (P) = variance of P temporal density difference of the two datasets. The absolute n = number of points difference is used, that is, the absolute density of the second point dataset subtracted from that of the first one. In order to detect clusters, KDE has been applied in in 3) DKDE various applications, such as crime analysis and population The approach called Directed Kernel Density Estimation analysis. Kwan [11] used KDE and 3D visualization to (DKDE) that takes the dynamics of moving points inside investigate spatio-temporal human activity patterns. The density maps into account was suggested in previous works author applied the density estimation as a method of [19-24]. The DKDE is applicable for discrete moving points geovisualization to find patterns in human activities related to and it considers two moments of time. Instead of an upright other social attributes. The classic KDE was investigated in kernel as in the KDE method, a tilted kernel is used, as [12-14], and thereby defined as a visual clustering method. In illustrated in Figure 1. these works, KDE maps were created in order to visually provide a better overview and insight into the given data. B. Contour lines and intervals A common technique to map point densities calculated using KDE are isopleth maps with filled contour intervals. The term “isopleth map” refers to one of two types of isoline maps (also called isarithmic or contour maps). In the first type of isoline maps each contour line indicates a constant rate or ratio derived from the values of a buffer zone or kernel area. In this sense, the continuous density surface is derived from an Figure 1. Linear kernel (left) and Directed linear kernel taking point speed and movement direction into account (right), source: [24]. originally discrete surface. In the other type of isoline maps (commonly referred to “isometric map”), contour lines (isometers) are drawn through points with directly measurable The tilt depends on the movement direction vector of the equal value or intensity such as terrain height or temperature respective point. The resulting DKDE-map shows the so- [15]. It is assumed that the data collected for enumeration called “ripples”, which can be interpreted as an indicator for units are part of a smooth, inherently continuous phenomenon the movement direction and density change of points that are [16]. In our context, we only use contour lines to delimit the located closely to each other with very similar movement intervals (the areas between contour lines). speeds and directions. These ripples are visible among Furthermore, Langford and Unwin [17] provided a good overlapped contour lines. The tinted contour intervals do not overview of density surfaces used in Geographic Information contain the information about movement or density change. Systems (GIS) as choropleth population density maps, Peters and Krisp [24], for instance, used 2D airplane population density on grids, population density surfaces, and positions in the area of Germany at two moments of time with pseudo-3D population density surfaces. In several works as a time lag of five minutes. The resulting DKDE-map is shown [4, 5], the KDE concept is adapted for the 3D space density in Figure 2.