To link to this article:

Transcription

1 This article was downloaded by: [University of Waterloo] On: 17 January 2015, At: 07:20 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Geocarto International Publication details, including instructions for authors and subscription information: Quantitative relations between spatial similarity degree and map scale change of individual linear objects in multiscale map spaces Haowen Yan ab a Department of GIS, Lanzhou Jiaotong University, Lanzhou, China Click for updates b Department of Geography & Environmental Management, University of Waterloo, Waterloo, Canada Accepted author version posted online: 14 Apr 2014.Published online: 12 May To cite this article: Haowen Yan (2014): Quantitative relations between spatial similarity degree and map scale change of individual linear objects in multi-scale map spaces, Geocarto International, DOI: / To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &

2 Conditions of access and use can be found at

3 Geocarto International, Quantitative relations between spatial similarity degree and map scale change of individual linear objects in multi-scale map spaces Haowen Yan a,b * a Department of GIS, Lanzhou Jiaotong University, Lanzhou, China; b Department of Geography & Environmental Management, University of Waterloo, Waterloo, Canada (Received 20 January 2014; final version received 1 March 2014) Quantitative relations between spatial similarity degree and map scale change in multi-scale map spaces play important roles in map generalization and construction of spatial data infrastructure. Nevertheless, no achievements have been made regarding this issue. To fill the gap, this paper firstly proposes a model for calculating spatial similarity degrees between an individual linear object at one scale and its generalized counterpart at the other scale. Then psychological experiments are designed to validate the new model, taking four different individual linear objects at five different scales as test samples. The experiments have shown that spatial similarity degrees calculated by the new model can be accepted by a majority of the subjects. After this, it constructs a formula that can calculate spatial similarity degree using map scale change (and vice versa) for individual linear objects in multi-scale map spaces by the curve fitting method using the point data from the psychological experiments. Both the formula and the model can calculate quantitative relations between spatial similarity degree and map scale change of individual linear objects in multi-scale map spaces, which facilitates automation of map generalization algorithms for linear features. Keywords: spatial similarity degrees; map generalization; map scale change; individual linear objects 1. Introduction Quantitative expression of spatial similarity relations have aroused the interests of researchers in the communities of cartography (Yan 2010) and geographic information science (Rodríguez & Egenhofer 2003; Rodríguez & Egenhofer 2004) for years. It is an important component of the theory of spatial relations along with distance (Hong 1994), topological (Egenhofer & Franzosa 1991; Du et al. 2008) and direction (Peuquet 1986; Goyal 2000; Yan et al. 2006), and also an element in spatial retrieval and spatial inference (Markman 1997; Goldstone 1999) and plays a significant role in human spatial cognition (Li & Fonseca 2006). Most importantly, it is an important factor in automated map generalization (Wang 1993; Ruas 2001). Map generalization is a technique for producing maps at multiple smaller scales using the maps at a large scale. In map generalization, the similarity degree between a generalized map and the original map and the scale change from the original map to the generalized map are dependent on each other (Yan 2010). The more the original map is generalized, the larger the scale changes from the original map to the * h24yan@uwaterloo.ca 2014 Taylor & Francis

4 2 H. Yan generalized map. Nevertheless, no achievement has been made on quantitatively describing such relations, which hampers the automation of map generalization, because a map generalization system/software does not know to what extent an original map should be generalized to produce a resulting map at a given scale if the similarity degree between the original map and the resulting map is not known before map generalization. Hence, this paper focuses on approaches that can quantitatively express relations between spatial similarity degree and map scale change of individual linear features on multi-scale maps. Many achievements on calculating spatial similarity degrees between curves in the domains of computational geometry (Alt et al. 1998; Agarwal & Varadarajan 2000; Alt et al. 2003), computer science (Cohen & Guibas 1997) and pattern recognition (Arkin et al. 1991) have been addressed; nevertheless, the similarity degrees that they calculate are neither for curves on maps nor for curves at different scales. Hence, they are applicable to automated map generalization. To facilitate the following discussion, spatial similarity degree and map scale change are defined here. There is an individual linear map object A. It is represented as A m at scale m and is generalized as A k at scale k. Sim (A m, A k ) is named the spatial similarity degree of object A at scale m and scale k, i.e. the spatial similarity degree between A m and A k. Here, m >0, k > 0, and Sim ða m ; A k Þ2½0; 1Š. C m;k ¼ m k is called the map scale change of A from scale m to scale k. Indeed, measuring similarity between lines/curves is a fundamental issue in many fields, and achievements have been made regarding calculating similarity degrees between different lines at the same scale (Alt & Godau 1995; Alt et al. 1998; Yan2010). Nevertheless, little has been done on quantitatively describing the relations between spatial similarity degrees and map scale changes of individual linear features (e.g. roads, rivers, contours, boundaries, etc.) in map spaces, i.e. calculating spatial similarity degrees between a line at one scale and a generalized counterpart of this line at the other scales. This paper is organized as follows: after the introduction and brief review of existing achievements regarding the approaches for calculating similarity degrees between linear features on multi-scale maps, a model for calculating spatial similarity degrees of individual linear features in multi-scale map spaces is proposed (Section 2); then psychological experiments are designed to validate the model (Section 3); after this, a formula is constructed by the curve fitting method using the point data from the psychological experiments and it can calculate spatial similarity degree using map scale change (and vice versa) for individual linear objects in multi-scale map spaces (Section 4). Finally, some concluding remarks are made (Section 5). 2. Approach to calculating spatial similarity degrees among individual linear objects at multiple scales Shape is viewed as the most crucial factor for describing planar lines/curves (Douglas & Peucker 1973; Mokhtarian & Mackworth 1992), and similarity of shape of lines is usually used to substitute for the similarity of the lines A formula for calculating shape similarity between lines Calculation of spatial similarity degrees of the shapes of individual lines in multi-scale map spaces may be based on the concept coincidence summary used to assess the similarity between maps (Berry 1993). Coincidence summary uses the percentage of the map area in agreement (or disagreement) between the two maps to indicate the

5 Geocarto International 3 overall similarity. In vector analysis, maps are intersected to generate the areas of the son-and-daughter polygons to summarize the type of disagreement. Based on coincidence summary and human s intuition in similarity judgments, similarity between two lines on the map can be evaluated by comparing the common length of the two lines. After overlapping the two lines at two different scales and matching their corresponding endpoints, their common length may be easily calculated (Figure 1), and their similarity degree can also be obtained. The spatial similarity degree of line A in shape at scale k and scale m can be expressed by Formula (1). Sim ða m ; A k Þ¼ l L (1) where L is the length of A at scale m; and l is the common length of A at scale m and simplified A at scale k Improvement of the formula By Formula (1), three similarity degrees of the lines in Figure 1 can be obtained. Sim ða m ; A k Þ¼ l L ¼ 1:00 Figure 1. Overlapped individual lines.

6 4 H. Yan Sim ðb m ; B k Þ¼ l L ¼ 0:00 Sim ðc m ; C k Þ¼ l L ¼ 0:32 It is obvious that Sim (A m, A k ) is acceptable, while the other two similarity degrees are discrepant with human spatial cognition. The major reason for this is that Formula (1) only considers the full intersection of the line at two different scales but ignores the function of the proximity of the overlapped line segments. To compensate for the shortcoming, an improved formula is proposed here, taking into account the distance between the overlapped lines. SimðA m ; A k Þ¼ Xn i¼1 w i l i =L (2) where L is the length of the original line; n is the number of the line segments contained in the resulting line; l i is the length of the ith line segment of the resulting line; and w i is the weight of l i, which can be calculated by w i ¼ 1 d il i P n j¼1 d (3) jl j where d i is the mean distance between l i and the original line, and it is the distance from the midpoint of l i to the original line. Compared with Formulas (1) (3), consider not only full intersects of the lines at two different scales but also the proximity of the overlapped line segments and using weighted values to evaluate their importance in the calculation of similarity degrees. Using Formula (2), the three similarity degrees in Figure 1 can be obtained. SimðA m ; A k Þ¼ Xn i¼1 SimðB m ; B k Þ¼ Xn i¼1 SimðC m ; C k Þ¼ Xn i¼1 w i l i =L ¼ 1:00 w i l i =L ¼ 0:78 w i l i =L ¼ 0:55 In this example, Sim(B m, B k ) and Sim(C m, C k ) calculated by Formula (2) are greater than that by Formula (1), for Formula (2) considers both the intersection and proximity of the two lines while Formula (1) only considers the intersection. Hence, the results obtained by Formula (2) are obviously more reasonable than that by Formula (1). 3. Validity of the new model People are accustomed to taking spatial similarity relation as a qualitative factor to describe the geographic space (Guo 1997); therefore, whether quantitative values of spatial similarity relations calculated by the proposed models coincide with human spatial cognition needs to be validated.

7 Geocarto International 5 Correctness of models is often addressed through model validation (Schlesinger 1979; Carson 2002; Banks et al. 2010). Model validation is usually defined to mean substantiation that a computerized model within its domain of applicability possesses a satisfactory range of accuracy consistent with the intended application of the model (Naylor & Finger 1967; Schlesinger 1979). In the geographic space, validity of a model sometimes can be tested by human spatial cognition (Sargent 2010; Sargent 2011). In essence, judgment of spatial similarity degrees relies on human s spatial cognition. Hence, it is a natural thought to validate the new model by psychological experiments. The experiment was done on 20 October 2013 in Lanzhou Jiaotong University, China. The subjects are 50 students at undergraduate level majoring in geography and each of them has at least six-month work experience in making maps. Four samples are used in the experiment (from Figures 2 to 5). Each sample consists of the original linear feature at a larger scale and five generalized feature at different smaller scales. The samples are printed and distributed to the subjects along with the similarity degrees listed in Table 1. The subjects are required to tell if they agree/ disagree with the similarity degrees or have no idea about the similarity degrees. It should be noticed that in Table 1 Sim V a;i refers to the similarity degree between (a) and (i) in the corresponding figure calculated by the new model; DScale a,i refers to the map scale change from (a) to(i) in the corresponding figure. Here, i = b, c, d, e, f, and the figures include Figures 2 5. N Agree /N Disagree is the number of the subjects that agree/disagree with the similarity degrees calculated by the new model; and N Noidea is the number of the subjects that have no idea about the three similarity degrees calculated by the new model. It is obvious in Table 1 that the numbers of the subjects that agree with the similarity degrees calculated by the new models are 50, 50, 48 and 50 out of 50. Hence, the new model can be acceptable by 100, 100, 96 and 100% of subjects in the experiment, respectively. 4. Formula for calculating map scale change by spatial similarity degree The new model provides a method for calculating the spatial similarity degree between a linear feature at one scale and its generalized counterpart at the other scale if both of them are known. However, it is sometimes useful to calculate the spatial similarity degree if only the map scale change is known while the linear feature and its Figure 2. A road at different map scales.

8 6 H. Yan Figure 3. Figure 4. A segment of a boundary line at different map scales. A coastline at different map scales. Figure 5. A ditch at different map scales. generalized counterpart are unknown. In other words, it is necessary to determine the quantitative relations between spatial similarity degree and map scale change of individual linear features in multi-scale map spaces. The following presents a curve fitting method to achieve this goal.

9 Geocarto International 7 Table 1. Calculated similarity degrees and experimental results. Figure Sim V a;b, DScale a,b, N Agree, Sim V a;c, DScale a,c, N Disagree, Sim V a;d, DScale a,d, N Noidea Sim V a;e, DScale a,d, Sim V a; f DScale a,d , 0.64, 0.38, 0.38, , 4, 8, 16, 32 50, 0, , 0.78, 0.52, 0.44, , 4, 8, 16, 32 50, 0, , 0.55, 0.44, 0.35, , 4, 8, 16, 32 48, 0, , 1.00, 1.00, 1.00, , 4, 8, 16, 32 50, 0, 0 Twenty points can be obtained in Table 1, taking (DScale a,i, Sim V a;i ) as the coordinate pairs, where, i = b, c, d, e, f. Supposing, Sim a,i = f(dscale a,i ), the curve fitting approach can be employed to construct empirical formulae using the 20 points. To simplify the following discussion, y = f(x) is used as a substitute for Sim a,i = f(dscale a,i ) Formula construction Curve fitting is a process of constructing a curve or a mathematical function that has the best fit to a series of data points (Kolb 1984; Arlinghaus 1994). Fitted curves should capture the trend in the data across the entire range, and can be used as an aid for data visualization to infer values of the function where no data are available and to summarize the relationships among two or more variables. Here, the curve fitting comprises the following three steps. First, determine the data points that are used in the curve fitting. All of the 20 data points obtained from the experiments may be adopted. In addition, a special point (1.000, 1.000) can be added in the point set. This point refers the situation that a point cloud is totally similar to itself; thus, its similarity is 1.00 and its map scale change is 1.00, too. Therefore, the 21 points are as follows: (1, 1.00), (2, 0.87), (4, 0.64), (8, 0.38), (16, 0.38), (32, 0.38), (2, 0.91), (4, 0.78), (8, 0.52), (16, 0.44), (32, 0.36), (2, 0.75), (4, 0.55), (8, 0.44), (16, 0.35), (32, 0.26), (2, 1.00), (4, 1.00), (8, 1.00), (16, 1.00), (32, 1.00). Second, select candidate functions. An infinite number of generic forms of functions can be chosen as candidates for almost any shape curves. It is not easy to select an appropriate function from numerous candidates to fit a series of points, because an inappropriate candidate may be either under-fit or over-fit. Potential candidate functions usually used in curve fitting comprise polynomials, power functions, logarithmic functions and exponential functions. A candidate function here should be monotonically decreased due to the apparent relation between map scale change and spatial similarity degree so only first- and second-order polynomials can be considered, because the other polynomials (e.g. 3rd and 4th order polynomials) have (is the order of the polynomial) inflexion point(s) which indicates that the curve is not monotonic. To sum up, the candidate functions are as follows: y ¼ a 1 x þ a 0 (4)

10 8 H. Yan y ¼ a 2 x 2 þ a 1 x þ a 0 (5) y ¼ a 2 e a1x þ a 0 (6) y ¼ a 1 ln ðxþþa 0 (7) y ¼ x a (8) Third, calculate the coefficient(s) of each function and determine the best fit function. The least square method (Lanczos 1988), a widely used method, is used to pick the coefficient(s) of each function that best fits the curve to the data points. R 2, i.e. R-squared, is usually used to compare the candidate functions. The greater the R 2, the better its corresponding curve. Thus, the curve with the greatest R 2 among all of the candidates is the best curve fitting the point set. R 2 can be calculated by the following method. For y = f(x), its dependent variable y has n modelled/predicted values ^y i and n observed values y i. Here, i = 1, 2,, n. y is the mean of the observed data: y ¼ 1 P n n i¼1 y i, where n is the number of observations. The variability of the data-set is measured through different sums of squares: SS Total ¼ P n i¼1 ðy i yþ 2 : the total sum of squares (proportional to the sample variance); SS Regression ¼ P n i¼1 ð^y i yþ 2 : the regression sum of squares; and SS Residual ¼ P n i¼1 ðy i ^y i Þ 2 : the sum of squares of residuals. The most general definition of the coefficient of determination is R 2 1 SS Regression (9) SS Total R 2 is a statistic that gives some information about the goodness of fit of a model. In regression, the R 2 coefficient of determination is a statistical measure of how well the regression line approximates the real data points. R 2 of 1 indicates that the regression line perfectly fits the data. Here, the Microsoft Excel (v10.0) was employed to construct the candidate curves and calculate their (Figure 6). It is obvious that the resulting function should be y ¼ 1; if the original line is a straight line, else 1:0164x 0:343 (10) because its corresponding R 2 = is the greatest in the five R 2 of the candidate curves, which indicates that it is the best-fitted curve of the 21 points used in the curve fitting procedure. In conclusion, Sim a;i ¼ where DScale a;i 2½1; 1Þ; Sim a;i 2½0; 1Š. 1; if the original line is a straight line, else 1:0164DScale 0:343 (11) 4.2. Discussion Some insights can be gained from Formula (11).

11 Geocarto International 9 (a) (b) (c) (d) (e) Figure 6. Curve fitting for obtaining the formula. First, Formula (11) can be used to calculate similarity degree using map scale change as the only independent variable and vice versa. Second, Formula (11) means that if the map scale change of an individual linear feature is given, the spatial similarity degree between the original feature and the generalized one is determined. Third, it should be noted that in Formula (11) DScale a;i 2½1; 1Þ. Hence, the formula cannot be applicable if the resulting map scale is greater than the original map scale. Fourth, Formula (11) is an empirical function. Thus, the credibility of the results obtained by the formula depends on the accuracy of the points used in curve fitting. The more points are used and the more accurate the points are, the more accurate the resulting formula should be. Last, the formula can be used in automated map generalization to calculate spatial similarity degrees if the original map scale and target map scale are known. This may facilitate the automation of map generalization softwares.

12 10 H. Yan 5. Conclusions Spatial similarity relation is of great importance in map generalization, and calculation of spatial similarity degree between individual linear features on maps at different scales can facilitate the automation of map generalization algorithms and systems. This paper proposes a model and a formula regarding this issue. The model can calculate spatial similarity degrees between an individual linear feature at one scale and its generalized counterpart at the other scale. The formula can calculate spatial similarity degrees taking map scale changes as the only independent variable and vice versa. Our future study will focus on the automation of the linear feature generalization algorithms that are dependent on spatial similarity degree using the proposed model and formula. Acknowledgements The work described in this paper is partially funded by the Natural Science Foundation Committee, China (No ), partially funded by the National Key Technologies R&D Program of China (No. 2013BAB05B01), and partially funded the NSERC, Canada. References Agarwal PK, Varadarajan K Efficient algorithms for approximating polygonal chains. Discrete Comput Geom. 23: Alt H, Godau M Computing the Fréchet distance between two polygonal curves. Int J Comput Geom Appl. 5: Alt H, Efrat A, Rote G, Wenk C Matching planar maps. J Algorithms. 49: Alt H, Fuchs U, Rote G, Weber G Matching convex shapes with respect to the symmetric difference. Algorithmica. 21: Arkin EM, Chew LP, Huttenlocher DP, Kedem K, Mitchell JSB An efficiently computable metric for comparing polygonal shapes. IEEE Trans Pattern Anal Mach Intell. 13: Arlinghaus SL PHB practical handbook of curve fitting. Boca Raton (FL): CRC Press. Banks J, Carson JS, Nelson BL, Nicol DM Discrete-event system simulation. 5th ed. Upper Saddle River: Pearson Education. Berry JK Beyond mapping: concepts, algorithms and issues in GIS. Hoboken (NJ): Wiley; 246p. Carson J Model verification and validation. In: Yücesan E, Chen C-H, Snowdon JL, Charnes JM, editors. Proceedings of the 2002 Winter Simulation Conference. wsc02papers/008.pdf Cohen SD, Guibas LJ Partial matching of planar polylines under similarity transformations. In: Eighth Annual ACM-SIAM Symposium on Discrete Algorithms; Philadelphia (PA); p Douglas D, Peucker T Algorithms for the reduction of the number of points required to represent a digitized line or its caricature. Cartographica: Int J Geogr Inf Geovisual. 10: Du SH, Qin QM, Wang Q, Ma HJ Reasoning about topological relations between regions with broad boundaries. Int J Approximate Reasoning. 47: Egenhofer M, Franzosa R Point-set topological spatial relations. Int J Geogr Inf Syst. 5: Goldstone RL Similarity. In: Wilson RA, Keil FC, editors. MIT encyclopedia of the cognitive sciences. Cambridge (MA): MIT Press; p Goyal RK Similarity assessment for cardinal directions between extended spatial objects [PhD thesis]. Orono (ME): The University of Maine. Guo RZ Spatial analysis. Wuhan: Press of Wuhan Technical University of Surveying and Mapping (in Chinese). Hong J Qualitative distance and direction reasoning in geographic space [PhD thesis]. Orono (ME): University of Maine. Kolb WM Curve fitting for programmable calculators. Carlisle (PA): Syntec; p. 94.

13 Geocarto International 11 Lanczos C Applied analysis (Reprint of 1956 Prentice-Hall edition). New York (NY): Dover Publications; p Li B, Fonseca FT TDD: a comprehensive model for qualitative spatial similarity assessment. Spat Cogn Comput. 6: Markman AB Constraints on analogical inference. Cogn Sci. 21: Mokhtarian F, Mackworth AK A theory of multiscale, curvature-based shape representation for planar curves. IEEE Trans Pattern Anal Mach Intell. 14: Naylor TH, Finger JM Verification of computer simulation models. Manage Sci. 14: B92 B101. Peuquet D The use of spatial relationships to aid spatial database retrieval. In: The Proceedings of the 2nd International Symposium on Spatial Data Handling (SDH); Zurich, Switzerland. Rodríguez A, Egenhofer M Determining semantic similarity among entity classes from different ontologies. IEEE Trans Knowl Data Eng. 15: Rodríguez A, Egenhofer M Comparing geospatial entity classes: an asymmetric and context-dependent similarity measure. Int J Geogr Inf Sci. 18: Ruas A Automating the generalization of geographical data. In: Proceedings of the 20th International Cartographic Conference. China: Beijing; p Sargent RG A new statistical procedure for validation of simulation and stochastic models, Technical Report SYR-EECS Syracuse, NY: Department of Electrical Engineering and Computer Science, Syracuse University. Sargent RG Verification and validation of simulation models. In: Jain S, Creasey RR, Himmelspach J, White KP, Fu M, editors. Proceedings of the 2011 Winter Simulation Conference. Baltimore (MD): IEEE; p Schlesinger S Terminology for model credibility. Simulation. 32: Wang JY Principles of general map generalization. Beijing: Surveying and Mapping Press (in Chinese). Yan HW Fundamental theories of spatial similarity relations in multi-scale map spaces. Chine Geogr Sci. 20: Yan HW, Chu YD, Li ZL, Guo RZ A quantitative description model for direction relations based on direction groups. Geoinformatica. 10: