Time space transformations of geographic space for exploring, analyzing and visualizing transportation systems

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1 Journal of Transport Geography 15 (2007) Time space transformations of geographic space for exploring, analyzing and visualizing transportation systems Nobbir Ahmed *, Harvey J. Miller Department of Geography, University of Utah, 260 S. Central Campus Dr. Room 270, Salt Lake City, UT , USA Abstract Transportation systems exist within at least two types of space. One is the apparent geographic space, but equally important is the time space implied by the travel time relations created by the system. Differences between the geographic and time spaces are properties induced by the transportation system. Methods for time space transformations of geographic space to explore, visualize and analyze transportation systems were initially developed in the 1960s and 1970s. However, these methods have not been pursued beyond this initial flurry of research activity, most likely due to the difficulties associated with handling and processing digital geographic data. The rise of geographic information systems (GIS), as well as continued development and wider availability of transformation techniques such as multidimensional scaling (MDS) and spatial analytical techniques such as bidimensional regression can allow the potential of time space transformation techniques to be realized. This paper presents a general methodological framework that exploits recent advances in GIS, MDS and spatial analytical techniques. Results from applying these techniques to the Salt Lake City metropolitan area illustrate the power of these techniques to reveal spatial patterns in the travel time relationships induced by a transportation system. The application also addresses fundamental issues in time space transformations, such as two-dimensional versus three-dimensional solutions, Euclidean versus non-euclidean solutions and symmetric and asymmetric solutions. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Transportation; Travel time; Time space; Cartographic transformations; GIS; Multidimensional scaling; Bidimensional regression 1. Introduction Understanding the travel time relationships induced by a transportation system can be crucial for assessing its performance. Transportation systems attempt to improve the efficiency of trading time for space when moving between geographic locations. Greater time efficiency for movement can enhance individuals accessibilities to activities and resources by freeing more time for travel and activity participation. Conversely, less time efficiency in geographic movement can reduce accessibility through the consumption of scarce temporal resources that could otherwise be used for travel and activity participation (Hägerstrand, 1970). Spatial variations and patterns in these travel time * Corresponding author. Tel.: ; fax: addresses: nahmed@esri.com (N. Ahmed), harvey.miller@ geog.utah.edu (H.J. Miller). relationships can help transportation analysts and planners understand relative differences in system performance, guiding the planning, design and deployment of transportation infrastructure and services towards efficient and equitable outcomes. The travel time relationships induced by a transportation system imply a time space where relative locations and proximity relationships can differ from those in geographic space. As with geographic space, mapping and spatial analysis of time spaces can be illuminating. Time space maps can provide a synoptic visual summary of the travel time relationships in a given environment, indicating areas where the transportation system is performing well and other areas where it is inefficient. Also, since induced travel time relations are central to transportation systems, spatial analysis of time space can be more meaningful than analysis of geographic space in understanding transportation system performance /$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi: /j.jtrangeo

2 N. Ahmed, H.J. Miller / Journal of Transport Geography 15 (2007) Mapping time spaces and other functional spaces based on generalizations of the concept of distance has a long history in spatial analysis. Research dates back to pioneering work in the 1960s by Waldo Tobler and William Bunge (Bunge, 1960; Tobler, 1961). Cartographic transformations to generate time spaces reached a peak in the 1970s with the work of researchers such as Marchand (1973), Ewing (1974), Forer (1974, 1978), Ewing and Wolfe (1977), Clark (1977) and Muller (1978). Despite the efforts of these and subsequent researchers, key issues surrounding time space mapping remain unresolved. Inconclusive results regarding the nature of time spaces and their structure probably result from the state of key transformation techniques such as multidimensional scaling (MDS) and map comparison techniques such as bidimensional regression: these techniques were not well developed or widely available until recently. Also, previous time space mapping studies could not benefit from powerful geographic information system (GIS) software that allow management and processing of large and detailed geographic databases as well as tools for spatial analysis of these data and their products. This paper explores the structure of time spaces implied by a transportation system in light of contemporary developments in transformation methods, spatial analytical techniques and GIS-based software platforms. Using the travel time relations in the Salt Lake City, Utah, USA, metropolitan area as an empirical example, we generate and analyze time spaces at two geographic scales. Smallscale time space maps of the entire study area provide a synoptic summary of the overall time space patterns induced by the transportation system as well as changes in these patterns over annual and daily time frames. The small-scale maps also identify locations where the time space is distorted strongly relative to geographic space. Large-scale time space maps explore these locations in greater detail to determine the empirical conditions that lead to these distortions. Critical time space transformation issues addressed in these analyses include: (i) Euclidean versus non-euclidean nature of time space; (ii) mapping asymmetric travel times; and, (iii) conditions where solutions are three-dimensional rather than twodimensional. Although Salt Lake City is only one of a multitude of transportation systems that exist in settings around the world, it is quite typical of the automobileoriented transportation systems in many mid-sized North American cities. Although the answers in this paper are not definitive, they are suggestive and can guide subsequent research in other locales. Section 2 briefly reviews previous research on the generation and analysis of time spaces. Section 3 presents the methodology and its implementation using a GIS platform. Section 4 provides results from applying the methodology to the travel-time relations in Salt Lake County. Section 4.1 presents small-scale maps of the entire study area for free-flow travel times as well as travel times for four time periods of a weekday. Section 4.2 examines the large-scale time space maps to address issues regarding Euclidean versus non-euclidean nature of time space, mapping asymmetric travel times and three-dimensional solutions. Section 5 discusses the results and Section 6 concludes with recommendations for future research. 2. Background Functional spaces are continuous regions generated based on proximity relationships that generalize the concept of physical distance. Examples of functional spaces include those based on psychological or cognitive distances, social differences, transportation flows as well as communication flows such as s or telephone calls. Spatial analytic tools applied to functional space rather than geographic space may be more powerful if functional spaces better capture the proximity relations between locations. For example, Cliff and Haggett (1998) find that functional spaces based on airline traffic provide greater explanatory and predictive power in analyses of infectious disease diffusion in Iceland. Plane (1984) and Worboys et al. (1998) reach similar conclusions in analyses of migration flows. Time spaces are a type of functional space where distances correspond to the realized travel times between locations. Time spaces can be viewed as transformations of geographic space, and understanding the structure and pattern in these transformations can illuminate the warping effect of transportation on geographic space with respect to travel time between geographic locations. Time space maps provide visual summaries of the spatial patterns of this warping effect. Attempts to generate and analyze time space maps date back to the 1960s. Bunge (1960) provides two conceptual time-maps of Seattle. The first map uses irregular isochrones (lines of equal travel time) on a map to portray travel times from a given origin. The second map distorts the geographic space in such a way that the isochrones from a given origin are concentric circles. Tobler (1961) incorporates the cartographic principles that project the curved surface of the Earth to the plane. This generates a map where the shortest time distance between a pair of locations is equal to the straight line connecting them. Marchand (1973) obtains a configuration of points representing the time space of the Venezuelan cities at a regional scale. This is among the first to use multidimensional scaling (MDS) in mapping travel times. Using average speeds of different types of roads and their lengths, he estimates the travel times between cities for four time periods of 1936, 1941, 1950 and Results indicate that the dimensionality of time space reduces with the development of the inter-urban transportation network; this implies that lower time space dimensionality indicates increasing homogeneity of the network. Using the airline travel times between 17 cities of New Zealand, Forer (1974) obtains

3 4 N. Ahmed, H.J. Miller / Journal of Transport Geography 15 (2007) 2 17 time space configurations for 1953 and Similar to Marchand (1973), Forer (1974) generates a transformed configuration of points: the embedded space is not transformed as well, meaning that travel time relations between locations not in this point set cannot be inferred. Ewing (1974) develops a time space configuration of Montreal, Canada and superimposes it on geographic space, drawing vectors whose bases are at geographic locations and heads are at the corresponding time space locations. This produces a set of displacement vectors that show the pattern in the transformed geographic space. Ewing (1974) also discusses the use of similarity transformations that enable comparisons between the time space and geographic space by aligning the time space with the original geographic space. Weir (1975) analyzes the impact of the proposed alternatives of a by-pass freeway around State College, Pennsylvania, USA, illustrating the use of time space maps in transportation planning. Explicit time space transformations of transportation networks include Clark (1977), Ewing and Wolfe (1977) and Muller (1978). Clark (1977) uses a graph-theoretical approach to transform a transportation network, but from a single origin only. Ewing and Wolfe (1977) interpolate time space locations of the transportation network as well as locations outside the network. After obtaining the set of displacement vectors they superimpose a uniform grid on the vectors and obtain a warped grid based on the influence of the vectors. They employ an inverse distance weighting method to interpolate time space locations of lattice points of the grid. Muller (1978) uses an apparently similar procedure based on transforming a uniform grid but does not provide methodological detail; the procedure may in fact be manual. In the book Distance and Space, Gaterall (1983) summarizes time space concepts and methodologies. He also presents an approach known as Q-analysis (Atkin, 1974, 1981; Johnson, 1981). This approach focuses on set relations. The set is represented as a graph based on the connectivity of the nodes of a network to other nodes, degree of incidence and travel times along the connections. However, this generates an abstract graph that is not directly comparable with geographic space. In time space transformations, the metric nature of time space is an issue since it is difficult to represent non-metric spaces as a two-dimensional map or analyze the space using spatial analytical techniques. The travel times in the origin destination matrix obtained from a transportation network may violate two of the metric space axioms, namely, symmetry (distances between two locations are the same regardless of direction) and triangular inequality (indirect routes cannot be shorter than direct routes). Spatial analytical techniques require at least the triangular inequality condition to be met (see Huriot et al., 1989; Smith, 1989). Tobler (1997) points out that parsing the matrix by keeping only the shortest paths can avoid violating the triangular inequality axiom. Muller (1982, 1984) produces time space maps with non-euclidean distances using a generalized Minkowskian distance model to represent the distances in a time space. He estimates the parameters of the distance function empirically and uses this distance function in the MDS analyses. The results show a slightly better fit, suggesting that non-euclidean models better capture the structure of time spaces. However, comparison of two maps by MDS goodness-of-fit measures such as stress (see Section 3 below) is inadequate to conclude on the superiority of the one over the other. Dorigo and Tobler (1983) map asymmetric migration flows from a single origin by representing the net flows as vectors. This approach does not transform geographic space, and can lead to inconsistencies between vectors showing the displacements of locations from their geographic positions. Another issue is the dimensionality of the resulting time space. Two-dimensional solutions are beneficial since this allows easy visualization in cartographic formats, as well as access to a large set of two-dimensional spatial analysis tools. Most previous studies assume a two-dimensional transformation: the objective is to minimize the distortions between geographic space and time space given a twodimensional time space map. Only a few studies have addressed this issue explicitly. Tobler (1961) and Ewing and Wolfe (1977) treat the time space as a surface with distances on the surface representing travel times. As mentioned above, Marchand (1973) suggests that higher dimensionality indicates higher heterogeneity in the transportation system. He also shows the non-conformality of three-dimensional time space with respect to geographic space. Kruskal and Wish (1978) propose visualizing a three-dimensional time space as orthographic projections on three perpendicular planes. As the preceding discussion suggests, issues such as the correspondence between time space and geographic space, the metric properties of the resulting time space, as well as the spatial structure and pattern of the time spaces are not resolved. Also, time space transformation as a topic of investigation seems to have declined after a burst of activity through the 1970s and 1980s. Part of the reason may be the bias towards analysis of discrete structures such as networks and flow matrices afforded by the digital computer (Puu and Beckmann, 1999). Another possible reason for this decline in interest is the difficulty associated with handling and processing the spatial data required to perform time space transformations. Until recently, transformation and spatial analysis techniques were not well developed. Digital geographic data were not as detailed and available, nor was GIS software available for handling, visualizing and analyzing time space maps. In this paper, we examine issues surrounding time space transformations using contemporary techniques and software tools applied to detailed travel time data for the Salt Lake City, Utah, USA metropolitan area. We also provide a generic methodological framework that encompasses the major issues and decisions in generating and analyzing time space maps.

4 N. Ahmed, H.J. Miller / Journal of Transport Geography 15 (2007) Methodology 3.1. Methodological framework The goal of this study is to construct and analyze time spaces from a set of travel-time relations obtained from the transportation system of a geographic space. Fig. 1 shows the flow of basic operations and decisions required to construct a time space map in a GIS environment. The transportation data reside in a geodatabase maintained within a GIS. The data require pre-processing in order to obtain the all-pairs shortest-paths matrix among a set of origins and destinations. The multidimensional scaling program takes the shortest paths matrix as input and generates another matrix containing the coordinates of points in time space. Diagnostic tools such as scatter plot, scree plot as well as the output point configuration help to determine whether a meaningful spatial structure exists in the resulting time space, and whether this space is two or three-dimensional (see below). Although we test for the possibility, we do not consider higher dimensional solutions in our procedure since we are interested in mapping and analyzing time spaces as low-dimensional spaces akin to geographic space. Higher dimensional solutions are possible, but these cannot be treated as geographic space. If the time space is two-dimensional it can be mapped and analyzed using standard cartographic and spatial analytic techniques. Another set of functions converts the twodimensional structure into a map that enables quantitative comparison with the geographic space. Since the time space configuration obtained from an MDS does not have a geographic reference, we must project and transform the solution to align the time space to the coordinate system of the original geographic space. We project the time space configuration to the same reference system as the geographic space and then pass it through similarity transformations. The similarity transformations do not change the fundamental geometry of the time space solution: rather, they translate, rotate, re-scale or reflect the time space so that arbitrary differences between the time space and geographic space are eliminated. The remaining differences reflect valid distortions in the time space relative to geographic space rather than superficial differences due to the coordinate systems. If the resulting space is threedimensional, it can be visualized and analyzed by treating it as a surface, or by projecting it to two-dimensional planes. Geodatabase Geo-referenced data Topology Spatial adjustments Geometric Network Geographic space Compare & analyze (e.g., bidimensional regression) Time-space map m 2D planes or 3D surface Travel-time matrix Similarity Transformations Visualization and Projection Run MDS m = 2 Dimensions m? m > 2 m = 3 Time-space configuration NO Data random? m > 3 YES Data not suitable for time-space analysis EXIT Fig. 1. Flowchart of basic operations and decisions required to construct time space map.

5 6 N. Ahmed, H.J. Miller / Journal of Transport Geography 15 (2007) Multidimensional scaling Multidimensional scaling (MDS) takes the origin destination travel time matrix as input and generates a matrix of coordinates of a point configuration representing the time space. For a given set of proximities (e.g., travel times) {d ij }, MDS attempts to find a set of points such that distances {d ij } between these points correspond as closely as possible to these proximities. This involves solving the following objective function: MIN fd ijg r ¼ X ½d ij f ðd ij ÞŠ 2 where f( ) is an hypothesized proximity function. A zero value for r means a perfect fit between the proximities and the distances, with increasingly positive values indicating increasingly poor fit, meaning that it is difficult to find a set of consistent distance relationships that capture the proximity relations. In the MDS literature a value of less than is acceptable (Kruskal and Wish, 1978). However, there are other factors to consider when judging the fitness of a solution; we review these in conjunction with discussing results from the Salt Lake City application in a subsequent section of this paper. Eq. (1) is difficult to solve exactly; heuristic methods are available. The applications in this paper use the PROXimity SCALing (PROXSCAL) technique (Commandeur and Heiser, 1993) for producing most of the time space maps (all but two cases in this paper; these will be noted). PROX- SCAL is a recent technique that implements the Iterative Majorization (IM) algorithm (Commandeur and Heiser, 1993). While most MDS heuristics conduct steepest descent or similar neighborhood searches, IM improves on these techniques through calculating, at each step, a linear auxiliary function that summarizes the local neighborhood of the current solution. Solving the auxiliary function minimization problem is easier than solving the original objective function, thus improving the search efficiency. IM almost guarantees a local minimum very close to the global one. The asymmetric travel time case requires the Alternating Least squares SCALing (ALSCAL) technique (Takane et al., 1977) while cases involving non-euclidean solutions requires the MINI-Smallest-Space-Analysis (MINISSA) technique (Lingoes and Roskam, 1973). Both are steepest descent methods. MDS models vary depending on the level of data measurement; the major types being ordinal, interval and ratio. Since travel times are ratio data, the ratio model appears at first glance to be the most suitable for constructing time spaces. However, higher data accuracy is required compared to the ordinal model. Although ordinal MDS yields better results by requiring only the rank order of the data, this measurement model increases the risk of obtaining a degenerate solution. A degenerate solution is one where the derived configuration is a set of points randomly scattered in a circular area. The goodness-of-fit of a degenerate solution is very high although it does not represent the ð1þ underlying time space. Another possibility is the spline model that treats travel-times as curvilinear relationships. Several MDS models can be estimated to determine the best fit to the data. As the dimensionality of the configuration of points representing time space is unknown to the analyst initially, it is necessary to produce configurations for each of the hypothesized dimensions. Generally, lower dimensional solutions have higher stress. Increasing dimensionality improves the goodness-of-fit, however, at a certain point raising dimensionality does not decrease stress substantially. This is the most appropriate dimensionality and the configuration at this dimensionality is the solution Bidimensional regression Bidimensional regression is a technique to compare two or more two-dimensional surfaces such as geographic maps or images. Although originally developed in 1978, it was not widely known until the technique was published in the 1990s (Tobler, 1994). Bidimensional regression is an extension of linear regression where each variable is a pair of values representing a location in two dimensional space. Bidimensional regression numerically compares the similarity among two-dimensional surfaces through an index called bidimensional correlation (Tobler, 1994) P n R 2 i¼1 ¼ 1 ðx i ^x i Þ 2 þ P n i¼1 ðy i ^y i Þ 2 P n i¼1 ðx i xþ 2 þ P n i¼1 ðy ð2þ i yþ 2 where (x,y) is the location of a point in the dependent plane, ^x i and ^y i are predicted values of x i and y i,andx and y are averages of x i and y i (the centroid of the original map). This index condenses a large amount of information into a single number and thus warrants some precaution when taken as the only measure to compare maps. A visual representation may be more revealing. For example, displacement vectors (see Section 4 below) can show the spatial distribution of similarity of two maps and reveal locations with good or poor fit. In time space mapping it is hypothesized that, given several time space maps, the map that is more similar to the original geographic space better represents the time space. This can be viewed as an extension of the principle of parsimony: the time space that is most similar to the original geographic space is the simplest time space that captures the travel time relations within that space GIS implementation The overall time space transformation procedure works within the ArcGIS Ò software environment. Fig. 2 shows the framework of the system architecture used to construct and analyze time spaces. There are five modules in the system. The travel time module pre-processes the data and generates the travel time matrix for next step. The transformation module is the core of the system and performs all

6 N. Ahmed, H.J. Miller / Journal of Transport Geography 15 (2007) Travel-time module (Network Analyst) Graphical User Interface ArcGIS (ArcMap) Transformation modules (MDS and custom) Geodatabase ArcGIS (ArcCatalog) Visualization module (ArcGIS, MatLab) Fig. 2. The GIS-based platform showing interactions among different modules. operations of time space mapping using the travel time matrix as input. ArcCatalog Ò is the engine of the database module and helps create, update and maintain the database. The travel time module, with the help of the geometric network wizard of ArcMap Ò, uses a distance matrix calculator (adapted from Brochu and Dussault, 2003) custom routine to obtain the all-pairs shortest-paths matrix. The transformation module works within ArcMap Ò which provides programming tools that allow development of custom routines using ArcObjects Ò. The transformation module uses several custom routines including bidimensional regression, a spatial interpolation algorithm adapted from Ewing and Wolfe (1977) and a built-in spatial adjustment tool for similarity transformations. ArcMap also helps create, edit and visualize the maps. The core of the transformation toolkit, the MDS techniques, is exogenous to ArcGIS Ò. Statistical Software for Social Sciences (SPSS, 2003) implements PROXSCAL Ò and ALSCAL algorithms, and NewMDSX Ò (NewMDSX, 2004) implements the MINISSA algorithm. The GIS and MDS programs exchange input/output via text files. SPSS also produces all the diagnostic results. MatLab Ò and ArcScene Ò make possible the visualization of threedimensional space. The GIS environment also supports queries and calculations required for the input data, as well as analysis of the MDS solution itself. MDS programs require a travel time matrix as input. Earlier studies use the centroids of traffic analysis zones (TAZs) as control points since these were the data most available in the past. In contrast, this study uses a set of major nodes at network intersections; this reduces the chances of introducing artifacts imposed by the arbitrary TAZ system. The geometric network wizard of ArcCatalog Ò uses this data layer and a street data layer to create the network, a custom routine takes the matrix as input to generate the all-pairs shortest travel time matrix. From the travel times matrix the MDS program yields an n m matrix representing a configuration of points, where n is the number of points and m the dimensionality. The appropriate dimensionality can be selected by examining summary statistics and graphics such as stress values and the scree plot (see Section 4 below). This matrix represents the time space in an arbitrary coordinate system. ArcCatalog Ò converts this matrix into an ArcGIS shapefile with the same geographic coordinate system as the transportation network layer. Importing the shapefile into Arc- Map Ò allows use of a spatial adjustment tool(available in ArcMap) to adjust the MDS result to a final time space configuration. The basic purpose of the tool is to convert data from one coordinate system to another; in this case, the arbitrary coordinates of the MDS solution to the coordinate system of the original geographic space. As discussed above, we accomplish this using the similarity transformation methods in the spatial adjustment tool. In some cases a reflection transformation may also be required. The spatial adjustment tool does not have a reflection method, but this can be accomplished by mirroring the configuration about a suitable axis in ArcMap and then re-running the similarity transformation. After the data are integrated within the GIS environment, we generate displacement vectors represented as arrows from geographic points to time space points. These vectors show the pattern of deviations of time space from the geographic space. They can also serve as a basis for warping of a dense uniform grid imposed on the space to show the overall trend in time space distortions. Another set of vectors can be drawn between the intersection points of grid lines of the uniform grid and the warped grid the bases of the vectors being at the uniform grid points. This second set of vectors act as a mapping function to support time space warping of the transportation network itself. 4. Results 4.1. The study area and data This paper presents time space maps and their analyses of the road transportation system of the Salt Lake City, Utah, USA, metropolitan area. Three limited access highspeed highways, I-15, I-215 and I-80, run through the area (Fig. 3). Besides some parks and small rivers, there are no large natural barriers within the metropolitan area, although the metropolitan area itself is surrounded by mountains on the west, south and east, and a large lake on the north. The Salt Lake City International Airport is located at the northwest of the area. The metropolitan area exhibits urban sprawl: low-density residential land use is distributed widely throughout the area. This study uses transportation network data for the time periods of 1992 and 2001 obtained from Wasatch Front Regional Council (WFRC), a local metropolitan planning organization. In addition to the geographic location of network nodes, attribute data include travel times for street segments for four time periods of the day morning, midday, afternoon and evening. WFRC calculates these times from free-flow

7 8 N. Ahmed, H.J. Miller / Journal of Transport Geography 15 (2007) 2 17 Table 1 Stress values for different MDS models (dimensionality = 2) Measure MDS model Ordinal Ratio Interval Spline Normalized raw stress Stress-I Stress-II S-Stress Fig. 3. Street network of Salt Lake City metropolitan area. Circles show the specific areas for larger-scale analyses: D for downtown, W for west, S for south. White smaller circles represent control points. travel times and other variables such as speed limits, volume/capacity ratio, number of lanes and other factors. This section presents results at two cartographic scales. Small-scale time space maps of the entire study area provide overall, synoptic solutions. Large-scale maps of regions highlighted by the large circles in Fig. 3 examine in greater detail locations where distortions and irregularities appear in the overall map. The large-scale maps address issues such as Euclidean versus non-euclidean solutions, two-dimensional versus three-dimensional solutions, and dealing with asymmetric travel times Small-scale time space maps Benchmark solution: 2001 free-flow travel times This section presents, first, the overall time space map of the Salt Lake County area with free-flow travel times and then four maps with travel times for different time periods of the day, and finally compares the maps to identify how these maps vary from free-flow traffic levels. Seventy nine major nodes of the transportation network act as control points (small circles in Fig. 3). These control points act as origins and destinations to yield the travel time matrix. Using the same control points for all geographic spaces of the same scale facilitates comparisons among time space maps of different time periods. Since there is no algorithm or procedure for selecting control points, this requires careful judgment by the analyst. We chose control points corresponding to major network intersections that uniformly blanket the study area. The free-flow travel times from 2001 serve as a benchmark for analyzing other small-scale maps. We will report detailed diagnostics for this solution as an illustration. For brevity, we will not report diagnostics for other solutions although similar tests were applied. Table 1 provides goodness-of-fit statistics for a twodimensional solution with ordinal, interval, ratio or spline MDS models. Stress-I is the most common measure found in the literature (Kruskal, 1964) P ðdij d ij Þ 2 r 1 ¼ P ð3þ d 2 ij where d ij is the travel time between two points, d ij is the corresponding distance in the resulting MDS space. Changing the denominator of (3) to P d 2 ij gives the normalized raw stress (Borg and Groenen, 1997) andto P ðd ij dþ 2 gives Stress-II, where d is the average distance in MDS space. (Kruskal, 1964). With no denominator (3) corresponds to the raw stress. Selection of the best fitting model is not straightforward. Table 1 suggests the ordinal model yields the best result since it has lowest stress. However, each solution should also be checked for degeneracy. A degenerate solution will have a very low stress value (near zero), but the bidimensional correlation with the original geographic space is also very low. In addition, similarity transformations cannot bring it closer to the original geographic space. We have encountered degeneracy in several instances. In these cases, the MDS algorithm can be run again: since it is a heuristic, it may find another non-degenerate local solution. If a nondegenerate solution cannot be obtained, the particular model must be discarded. Another consideration is spatial similarity with the original geographic space. As discussed above, we seek the simplest time space that captures the travel time relationships: this is the time space that is most similar to the original geographic space. We examine all acceptable results (i.e., those with stresses below the cut-off value of 0.100), produce cartographic representation of these solutions and select the one that is most similar by comparing the geographic control points with their corresponding locations in the time space. Note that if the solution is two-dimensional, it is always possible to find a direct one-to-one correspondence between the control points in

8 N. Ahmed, H.J. Miller / Journal of Transport Geography 15 (2007) geographic space and time space. One way to check spatial similarity is to draw displacement vectors between locations in geographic space and time space. The map with the smaller vectors has a better fit. This can be conducted visually. Also, the overall lengths of the vectors can be summarized using the root-mean squared error (RMSE) statistic (see Mahling, 1989). Bidimensional correlation is another way of assessing the match between geographic space and time space. Superior solutions have higher bidimensional correlations with geographic space. A final consideration is comparability: if we wish to compare several solutions they should be based on the same measurement model. Based on the criteria of nondegeneracy, correspondence to geographic space and comparability, in most cases discussed below we have used the ratio model. Exceptions to the ratio model will be noted where needed. Figs. 4 and 5 show the diagnostic plots for a two-dimensional ratio MDS solution with free-flow travel time matrix based on the 2001 data. The scatter matrix (Fig. 4) shows Dimension 1 Dimension 2 Dimension 3 Dimension 1 Dimension 2 Dimension 3 Fig. 4. Visualization and analysis of higher dimensional time space configurations by orthographic projections. 0.2 Stress Distance in MDS (a) Dimensionality (b) Travel times Fig. 5. Diagnostic plots of MDS analysis: (a) scree plot and (b) Shepard diagram.

9 10 N. Ahmed, H.J. Miller / Journal of Transport Geography 15 (2007) 2 17 the point configurations for pair-wise dimensionality and helps determine whether a third dimension is significant. In this case, the scatter of points across pair-wise dimensions shows that the third dimension is unnecessary since it has a strong linear relationship with the other dimensions and is therefore redundant. Hence, only the first two dimensions are sufficient to map the time space. However, it is interesting to note that a few points look like outliers in the cells of dimension 3 (Fig. 4). These are the points at the airport area (northwest) and contribute to higher stress. The scree plot (Fig. 5a) shows a clear elbow at dimensionality 2 indicating a two-dimensional solution. The Shepard diagram (Fig. 5b) that relates travel times with distances in the output configuration shows a linear relationship. This linearity indicates a strong, two-dimensional metric quality of the time space. These diagnostics clearly suggest a twodimensional solution is appropriate. Fig. 6 shows the time space map after the similarity transformations to maximize correspondence with the original geographic space. The white small circles in Fig. 6 are the control points and the black circles are their corresponding time space locations. Vectors drawn from the geographic locations to time space locations show the displacement pattern. The high shrinkage in northwest is due to highway I-80 and other relatively high-capacity and lowtraffic roads. Thus, the area near the Salt Lake airport, although far away in geographic space, is closer to the central area in time space than other peripheral locations. Shrinkage also occurs in the north and up to the intersections of highways I-15, I-215 and I-80 (the central area). Fig. 6. Time space map for free-flow travel time of Salt Lake City for 2001 data. Shrinkage is low through the middle of the area although I-15 passes through this corridor, suggesting that it is not highly efficient relative to the rest of the system, most likely due to congestion. From the downtown to the south and most of the east have little warping, indicating close correspondence between geographic space and time space. The probable reason is the high residential density and traffic volumes in these areas blunting the efficiency of the transportation system. The most distorted part of the area lies in the west and southwest where there is considerable stretching relative to geographic space. As residential density and traffic volume is not particularly high in this area, a possible reason is the heterogeneity of the network: some street nodes are not well connected to the rest of the network. Detailed time space maps of those areas (see below) can reveal the causes of this stretching Network changes over time: 1992 and 2001 solutions Fig. 7 portrays the general warping of the geographic space for both free-flow travel time data of 1992 and These maps show both the displacement vectors and warped grids interpolated from those vectors. Based on diagnostics similar to those discussed above, we confirmed that the 1992 data can also be mapped in two dimensions. The displacement vectors for the 1992 map and 2001 map (Fig. 7a and b) show trends that vary mostly in degree. Warped grids for both sets of vectors also look similar except the grid for the 1992 data has more warping (stretching) in the south and southeast of the county. This indicates a general time space convergence from 1992 to 2001 for the Salt Lake County transportation system, implying an overall improvement in the transportation network performance in south and southeast Time space maps of daily travel flows This section compares four maps for the time periods within a day to assess the time spaces of daily travel time variations on the Salt Lake time space in 1992 and The time periods are morning (AM; 6:00 am to 9:00 am), mid-day (MD: 9:00 am to 3:00 pm), afternoon (PM: 3:00 pm to 6:00 pm) and evening (EV: 6:00 pm to 6:00 am). Fig. 8 shows configurations of points corresponding to the five time spaces (free-flow travel times and the four daily time periods) superimposed on the geographic space. The general trend for maps of 1992 and 2001 is similar stretching in southwest and west and shrinkage in northwest and north while the central areas are having very little distortion. However, the control point clusters in 1992 maps are more spread than in 2001 maps. This implies more within-day travel time variability in 1992 than in Table 2 shows correlations between morning, midday, afternoon and evening time space maps for 1992 and 2001 respectively. The correlations for 2001 data are higher than those for 1992 data, confirming the visual pattern in Fig. 8. This evidence suggests the spread of congestion from peak hours to non-peak hours over the intervening time period between 1992 and 2001.

10 N. Ahmed, H.J. Miller / Journal of Transport Geography 15 (2007) Fig. 7. Warped grids showing time space pattern of Salt Lake City for the free-flow travel times of (a) 1992 and (b) Fig. 8. Cluster of points showing each geographic point s (gray circle) time space locations for four time periods and free-flow time for (a) 1992 data and (b) 2001 data.

11 12 N. Ahmed, H.J. Miller / Journal of Transport Geography 15 (2007) 2 17 Table 2 Correlation among time space maps of different time periods of day MD PM EV AM MD PM AM MD MD PM PM EV Table 3 Bidimensional correlations of time space maps with geographic space Year Time period AM MD PM EV Free-flow Table 3 shows the bidimensional regression correlations for the corresponding time space maps relative to geographic space. The general trend is slightly higher correlations for 2001 data, suggesting that the transportation system is less efficient, probably due to more uniform congestion. However the differences in regression correlations are not significant, suggesting that correspondence between geographic space and the daily travel-time spaces has remained stable Large-scale time space maps Fig. 3 identified three areas of Salt Lake County for more detailed analysis. The Western Valley (W) illustrates three-dimensional versus two-dimensional solutions induced by changes in network connectivity. The Southern Valley (S) is a region where relatively large differences in travel time with direction within a network link raise the possibility of asymmetric solutions. The gridiron street pattern in the Downtown (D) area means that non-euclidean solutions may be required Western valley: 2D versus 3D solutions Large-scale time space maps of the Western Valley reveal the impact new road construction between 1992 and 2001, as well as the possibility of a three dimensional solution for the time space. In 1992, the northern part of this area was not well connected. Construction of some new roads since 1992 has improved the connectivity. Fig. 9 illustrates the transportation network in this area, with bold lines representing the 1992 streets and the dashed thin lines showing roads added by Circles indicate the control points. Stress values for the two-dimensional MDS solutions for 1992 and 2001 data of the western part are and respectively. Both values are higher than the maximum acceptable ceiling of Fig. 10 shows the warped grids of the two-dimensional solutions for the time periods of 1992 and In the warped grid of 1992 (Fig. 10a) the parallel grid lines overlap each other in several instances, Fig. 9. Part of street network of the Western Valley for detailed analysis. indicating non-metric properties of the time space. This can be explained by the relative inaccessibility of some locations. For example, point A G in geographic space is connected only from one location C G (Fig. 9). Going to point A G from all other surrounding points takes considerably more time compared to the geographic distances. That is why the time space location of this point (A T ) is displaced to the north (Fig. 10a). This is also true for point B G (Fig. 9). The lack of connectivity has disrupted the metric properties of the time space. Compared to the map of 1992, the two-dimensional time space map of the same area with 2001 data shows smoother, metric properties (Fig. 10b). Stress for three-dimensional solutions for 1992 and 2001 are and respectively, suggesting a three-dimensional space is more appropriate for this sub-area. This space can be visualized as a surface by considering the values of the third dimension as elevations. Fig. 11 shows a uniform surface generated from the 28 control points in this sub-area. Comparison with Fig. 10 is possible by noting the locations of A T and B T. Highly distorted regions of the 1992 two-dimensional solution correspond to peaks on the surface, confirming the space-distorting effect of poor network connectivity. These results confirm Marchand s, (1973) suggestion that high-dimensionality indicates a more heterogeneous network Southern valley: symmetric versus asymmetric solutions The Southern Valley (S in Fig. 3) has a large number of streets with asymmetric travel time (defined as directional travel time variation of 10% or greater), suggesting that asymmetric transformations may be required. However, if asymmetric time space solutions are not substantially different from the symmetric solutions, than simpler symmetric transformations are sufficient. This section compares

12 N. Ahmed, H.J. Miller / Journal of Transport Geography 15 (2007) Fig. 10. The warping of uniform grid (a) in the time space with 1992 data shows significant distortions and (b) smoother warping with 2001 data. 15% but still found no significant additional warping due to this additional asymmetry. Fig. 11. Three-dimensional surface of the western part of Salt Lake County for 1992 data. symmetric and asymmetric solutions for this subarea. The ALSCAL MDS technique generated these solutions since it can handle asymmetric proximity relationships. Since asymmetric solutions assume ordinal proximity measures, both solutions correspond to this measurement model. Visual comparison of the symmetric (Fig. 12a) and asymmetric solutions (Fig. 12b) suggest that the asymmetry does not generate substantially different warping than the symmetrized data. Bidimensional correlations of symmetrized and asymmetric maps with geographic space are and respectively, and the bidimensional correlation between the time-maps is 0.996, meaning they are nearly identical. These results suggest a symmetric solution is sufficient for these data. As a hypothetical experiment, we magnified the directional travel-time variations by Downtown: Euclidean versus non-euclidean solutions This section examines time space with non-euclidean distances. The streets in the downtown area of Salt Lake City follow a strict city-block pattern; therefore, we have selected this area to explore the nature of time space produced by Euclidean versus city-block or Manhattan distances (see Richardson, 1981). We use the MINISSA MDS algorithm (Lingoes and Roskam, 1973) to generate these time spaces since it can accommodate non-euclidean distances. Both solutions correspond to the ordinal measurement model. Fig. 13 illustrates the two-dimensional solutions using the city-block and Euclidean metrics. Stress values for time space configurations for the Euclidean and city-block of and (respectively) are both acceptable, although the city-block solution has slightly better fit. Bidimensional correlations of both spaces with geographic space are and respectively, which are not conclusive. Since stress is a broad measure, visual inspection of warping of the grid and the pattern of displacement vectors (Fig. 13) can shed more light on this issue. Visually the overall warping of uniform grids looks similar but local variations can be identified by the displacement vectors which have higher values at the west edge and northeastern corner, indicating higher local distortions in the case of the Euclidean model. This suggests that the city-block model better represents the travel time relations, although only marginally. 5. Discussion All of the time spaces generated in this study clearly show the presence of spatial pattern. In all but two cases the spatial structure is two-dimensional. Stress values for

13 14 N. Ahmed, H.J. Miller / Journal of Transport Geography 15 (2007) 2 17 Fig. 12. Comparison of time spaces with (a) symmetric and (b) asymmetric travel time relations. Fig. 13. Two-dimensional time space solutions from MINISSA-N: (a) using city-block distance metric and (b) using Euclidean distance metric. the two-dimensional solutions range between and 0.090, meaning that the distances in the time spaces explain over 90% of the travel-time relations. In the MDS literature any result with a stress of less than is acceptable (Kruskal and Wish, 1978). Thus, although the solutions are not perfect, they are attainable via MDS with a high degree of accuracy. Some researchers have proposed analytical time space transformation methods such as finding mathematical projection functions; e.g., Tobler (1994) proposes differential geometry techniques that map the curved surface of the Earth to a two-dimensional map. However, analytical solutions require stringent assumptions and conditions that may only be attainable if the travel time relationships are very well-behaved and comprise a strictly consistent set of distance relations. These strict requirements mean that the resulting space may not be metric and/or have