ERROR PROPAGATION IN OVERLAY ANALYSIS

Transcription

1 ERROR PROPAGATION IN OVERLAY ANALYSIS Ali A. Alesheikh Department of Geomatics and Geodesy Eng. K.N.T University of Technology, Vali-asr St., Tehran, IRAN Tel: Fax: Rahim A. Abbaspour Department of Geomatics Eng. Faulty of Engineering, University of Tehran, North Karegar St., Tehran, IRAN Tel: Fax: ABSTRACT GISs give users complete freedom to combine, overlay and analyze data from many different sources, regardless of scale, accuracy, resolution and quality of the original data. The mixing of geographical information from different map scales and sources is a key aspect of GIS functionality, but it does raise the question as to what effects the combination of different levels of data uncertainty has on both the output maps and on the data derived from spatial query and analysis. It must be recognized that there are many good reasons for wishing to combine data in these ways, but a major problem arises because GIS packages fail to offer any means of keeping track of the effects of error propagation and how it affects the results. This paper is concerned with developing methods able to estimate the confidence regions of GIS outputs by taking into account certain selected sources of uncertainty affecting spatial databases. A Monte Carlo simulation-based method is used as a general means of estimating the effects of input data uncertainty on the map outputs after an arbitrary sequence of overlay analysis. The objective is to identify and handle the effects of data uncertainty in GIS by defining uncertainty envelopes to create credibility regions around the results. This is considered the minimum need to allow a GIS to function in a mixed data environment. 1. INTRODUCTION One of the most powerful capabilities of GIS, particularly for the earth sciences applications, is that it permits the derivation of new attributes from attributes already held in the GIS database. The many basic types of function used for derivations of this kind of information are often provided as standard functions or operations in many GISs, under the name of map algebra (Burrough 1986; Tomlin 1990).

2 In practice, many GIS operations are used in sequence in order to compute an attribute that is the result of a computational model. Using GIS for the evaluation of computational models is defined here by the term spatial modeling within GIS. To date, most works on spatial modeling with GIS has been concentrated on the business of deriving computational models that operate on spatial data, on the building of large spatial databases, and on linking computational models with the GIS. However, there is an important additional aspect that has long received too little attention. This concerns the issue of data quality and how errors in spatial attributes propagate through GIS operations. It can safely be said that no map stored in a GIS is truly error-free. Note that the word 'error' is used here in its widest sense to include not only mistakes or blunders, but also to include the statistical concept of error meaning variations (Burrough 1986). When maps that are stored in a GIS database are used as input to a GIS operation, then the errors in the input will propagate to the output of the operation. Therefore, the output may not be sufficiently reliable for correct conclusion to be drawn from it. Moreover, the error propagation continues when the output from one operation is used as input to an ensuing operation. Consequently, when no record is kept of the accuracy of intermediate results, it becomes extremely difficult to evaluate the accuracy of final result. Although users may be aware that errors propagate through their analyses, in practice they rarely pay attention to this problem. Experienced users perhaps know that the quality of their data is not reflected by the quality of graphical output of the GIS, but they cannot truly benefit from this knowledge because the uncertainty if their data still remain unknown. No professional GIS currently in use can present the user with information about the confidence limits that should be associated with the result of an analysis (Burrough 1992; Forier and Canters 1996). 2. THE THEORY OF ERROR PROPAGATION IN GIS The error propagation problem can be formulated mathematically as follows. Let U(.) be the output of GIS operation g(.) on the m input attributes A i (.) : U (.) = g( A i (.),..., Am (.)) (1) The operation g(.) may be one of various types, such as a standard filter operation to compute gradient from a gridded DTM (Carter 1992). The objective of the error propagation analysis is to determine the error in the output U(.), given the operation g(.) and errors in the input attributes A i (.). 2 The output map U(.) also is a random field, with mean µ (.) and variance σ (.). From an error 2 propagation perspective, the main interest is in uncertainty of U(.), as contained in its variance σ (.) (Heuvelink 1999). It must first be observed that the error propagation problem is relatively easy when g(.) is a linear function. In that case, the mean and variance of U(.) can be directly and analytically derived. The theory on functions of random variables also provides several analytical approaches to the problem for nonlinear g(.), but few of these can be resolved by simple calculation (Helstrom 1991). Thus for the general situation analytical methods are not very suitable. In this context, two methods of uncertainty assessment have been applied to date: analytical equations generally based on first-order Taylor series variance approximations and empirical simulations based on Monte Carlo processes.

3 2.1. ANALYTICAL METHOD OF UNCERTAINTY Analytical methods refer to obtaining the stochastic characteristics of functionally dependent variables, given the characteristics of the independent variables and the functional relationships relating to the two sets of variables. Let L = [ l1, l2,..., ln ] be a set of random variables with a known n-dimensional probability density function f = ( l1, l2,..., ln ). Let = [ x1, x2,..., xn ] be another set of random set of random variables that is related to L by F = [ f 1, f 2,..., f m ] such as: F (, L)=0 (2) The task is to determine the stochastic properties of from those of L, i.e., to determine the probability density function g ( x1, x2,..., xu ) (Krakiwsky 1992). In most local or global GIS operations, the general Equation (2) can be simplified to a set of continuous, differentiable functions F (Cressie 1991) such that =F(L) (3) Where is the output, L is the input of GIS operation, and F is the function that relates the input and output data. The distribution of the unknown parameters g ( x1, x2,..., xu ) can be uniquely identified by its first, second and joint moments if g's distribution follows a Gaussian function. This assumption is usually applied in surveying, photogrammetric and geodetic measurements (Mikhail 1976). The reason for this lies mainly in the central limit theorem, which roughly states that the averaging of a sufficiently large number of random of random variables yields a normal distribution, no matter what the distribution of an individual random variable is (Doughety 1990). Since GIS data are functions of several errors introduced in different stage, it is reasonable to assume they are following normal functions. Another reason for invoking the normality assumption is that it makes the computation simple (Vanicek 1992). In principle the problems of uncertainty transmission in GIS operations can be handled by using covariance propagation (Burrough 1986). The covariance propagation method relies on approximating the result F(,L) by a truncated Taylor series. In the case of first-order Taylor method, F is linearised by taking the tangent of F in initial value ( 0, L 0 ) as follows: 0 0 F 0 F 0 F(, L) = F(, L ) ( ) ( L L ) + remainder = 0 (4), L, L L Assuming the magnitude of reminder to be negligible, it can be written as: W + Aδ + Br = 0 (5) Where W = F( 0, L 0 ) F A = 0 0, L 0 δ = ( ) F B = 0 0, L L r = ( L 0 L )

4 The remainder of Equation (4) contains the higher-order Taylor series terms of F, whose contribution to the result of F is comparatively small in the neighborhood of ( 0, L 0 ). By neglecting the higher-order terms, the output covariance matrix C can be computed by: T T C = [ A ( BCL B ) A + C ] (6) Where CL is the covariance matrix of the input data and C 0 is the a priori covariance matrix of the output map MONTE CARLO SIMULATION METHOD The lack of single continuous differentiable functions renders the use of explicit equations for error propagation impossible. Instead, it is simpler and more general to use a universal solution based on Monte Carlo simulation approach. The Monte Carlo method (Openshaw 1989) uses an entirely different approach to determine the uncertainty of geospatial objects. In this method, the results of Equation (3) are computed repeatedly, with input value L = [ l1, l2,..., ln ] that are randomly sampled from their joint distribution. The outputs of the equation construct random samples of which parameters of its distribution, such as mean and variance, can be estimated. The basic algorithm as might be applied in GIS is as follows (openshaw 1994): (i) Decide what levels and types of error characterize each data set as input to a GIS (ii) Replace the observed data by a set of (M) random variables drawn from appropriate probability distributions assumed to represent the uncertainty in the data inputs. (iii) Apply a sequence of GIS operations to the step (ii) data. (iv) For this set of realization l i, store the result i. (v) Compute summary statistics. The Monte Carlo method is general and can be applied to spatial or attribute data. Heuvelink 2 (1993) showed that the standard deviation of output mean ( M ) and output variance ( S ) are approximately inversely related to the square root of the number of Monte Carlo runs. Therefore, the accuracy of the method can also be controlled. However, the method is computationally intensive. 3. EVALUTION AND COMPARISON OF METHODS The analytical method, which is based on covariance propagation, is modest in computational load. It is also attractive because it yields an analytical expression for the variances of the output errors, although it should be noted that the solution is approximate only. The means, variances and correlations of the input data explicitly appear in Equation (5), and these allow one to examine quickly how the output uncertainty changes under variations in the input error parameters. The disadvantage of the method is that it is an approximate method only. When the function F is non-linear, then the approximation error may become unacceptably large. Iteration may be required to reduce the error; however, it will be at the cost of computational load. On the other hand, the approximation error is zero when F is linear. Another disadvantage of first-order Taylor covariance propagation method is that the function F must be continuously differentiable. Covariance propagation may not respect the ease of computation if the function F is a complicated computational model of errors in GIS, since systematic errors may exist in the process (Alesheikh 1998). The most important advantage of the Mont Carlo method is that it can provide the entire distribution of output data at an arbitrary level of accuracy. Other advantages are that method is easily

5 implemented and generally applicable. The method merely treats in function F as a black box, whose response to the perturbed inputs is studied from the resulting outputs. The main disadvantage of the Monte Carlo method is that it is computationally intensive. Another disadvantage is that any sensitivity analysis requires the repeated execution of the entire process. In most practical situations, N will take a value between 50 and 200, but it may occasionally be as 100, METHODOLOGY Much of the functionality of GIS lies with their ability to overlay one or more digital maps for the purpose of Boolean or network analyses. In this study, we use Monte Carlo simulation method to estimate the levels of uncertainty in the output of an overlay analysis, so a simulation is made by generating random numbers from their probability distributions (which assumed Normal) which define by covariance matrix of each polygon, and adding them to the coordinates of vertices of two measured polygons (Figure 1). This process is repeated 20 times. Two groups of polygons generated. Each of the randomized input polygons is then overlaid on another one. The final results were then saved. The entire process is repeated 20 times. Then the total set of output maps were used to draw confidence intervals or credibility regions that were overlaid on the deterministic results as a visual indication of effects of data uncertainty. The nature of simulation methodology is outlined in Figure 2.b with respect to normal procedure (figure 2.a). The GIS software used in this case study is the widely used ARC/INFO package as well as MATLAB. Figure 1. Polygons, their overlays and vertices with error ellipses (ellipses are not in scale) To analyze the resulted polygons, they rasterised in 0.1 m cell size. As covariance matrix shows, every measured point is to some degree uncertain, as are the straight-line segments generated by connecting the points. Hence, modeling the observational uncertainty of a line segment may be determined by considering (a) the observational uncertainty coordinates of endpoints, and (b) the correlation among the points that construct the line, i.e. a full covariance matrix should be considered (Alesheikh 1998).

6 Data input Data input Perturb data Sequence of GIS Operation Repeat M times Sequence of GIS Operation Output stored Identify confidence region (a) Figure 2. Simulation approach to error propagation in GIS: (a) Monte Carlo Methodology; (b) Normal GIS procedure (Openshaw et al. 1991) Data output(s) A commonly used method to visualize uncertainty is using gray value. However, due to the limitation of the human eye in Distinguishing grey values, the grey-level coding may not work very well in some cases, so the method, which involves three dimensional and color representation techniques (Shi and et. al.) is used for the visualization of uncertainty in this study as well as former method to compare. (b) Figure 3. 3D view of the Polygon Density Function

7 To visualize the area of constant probability a mesh has been superimposed on top of the random polygons. Each generated polygon intersects a number of pixels. These intersections are summed up for each resulting in a lookup table of intersections, which can then be represented in 3D (Figure 3). The cells and the relative frequencies show the probability density of the random polygon. Pixels of constant probability density can now be constructed in the same way as counter lines in digital terrain models (Figure 4.a, 4.b). The probability associated with the contours of (e.g. 20%) can be determined by the summation of the relative frequency of pixels (f (, Y)) located inside the contour (Figure 6). This area presents the confidence interval of a polygon (Alesheikh 1999). (a) (b) Figure 4. The visualization of the polygon uncertainty: (a) A polygon boundary; (b) part of a polygon boundary The results of our simulation test illustrate that just 98.85% of overlaid region is definitely in and the area of uncertainty are 1.15% of total. The data of this test are seen. The number of pixels for each probability value is seen in Figure 5. Figure 5. Number of pixels (in percent) with respect to the density

8 Figure 6. 2D view of the Polygon Density Function 5. CONCLUSION The error model used here is an obvious over-simplification of reality. Each vertex in the polygons was perturbed by selecting random numbers from a Normal distribution, based on the given covariance matrix. Analytical results to the error propagation problem can be obtained in a few special cases, such as when the GIS operation is a linear function. Numerical solutions are not very attractive either, and so two alternative approaches are suggested instead. The analytical method approaches the problem by approximating the GIS operation by a linear function. The linearization is done in a way such that approximation errors are relatively small for input values that have a larger probability of occurrence. The linearization makes the problem analytically tractable, so that the method yields analytical results that can provide useful insight into the error propagation process. The Monte Carlo method uses a simulation approach to analyze the propagation of error in GIS operations. It repeatedly draws realizations from the joint probability distribution of the input attributes, each time substituting these realizations into the operation, computing the result and storing it. In this way a random sample from the output distribution is obtained, which is analyzed by using techniques from classical sampling theory. The main problem with the analytical method is that the results are approximate only. It will not always be easy to determine whether the approximations involved using these techniques are acceptable. The Monte Carlo method does not suffer from this problem, because it can reach an arbitrary level of accuracy. The Monte Carlo method brings along other problems, though. High accuracies are reached only when the number of runs is sufficiently large, which may cause the

9 method to become extremely time consuming. Another disadvantage of the Monte Carlo method is that the results do not come in an analytical form. 6. RECOMMENDATION Although this paper runs the overlay operation many times, the question of how many Monte Carlo runs are needed to obtain a given level of accuracy is remained. Some attention has been given to this problem, but this particular point deserves more attention. In line with this problem, a variable option seems to replace point estimates by interval estimates (Casella &Beger 1990). Interval estimation yields wider intervals when the number of Monte Carlo runs is small, and so it implicitly incorporates the confidence gain resulting from an increase of runs. It was also shown that the efficiency of the Monte Carlo method is proportional to the square root of the number of runs. This implies that it is extremely costly to obtain results with a high level of accuracy. It seems worthwhile to examine how the crude Monte Carlo method can be improved by employing various variance reduction techniques (Hammersley & Handscomb 1979). One possible would be to use the result of the analytical method as a control variable, after which the remainder may be evaluated using the crude Monte Carlo method. Another option is to employ importance or stratified sampling (Stein 1987). Although the problem with variance reduction method is that the complexity increases, they are certainly worth investigating, regardless of the awaited emergence of parallel GIS machines (Openshaw et al. 1991). 7. REFERENCES [1]Alesheikh, A.A., Blais, J.A.R.,Chapman, M.A.,Karimi, H. (1999). "Rigorous Geospatial Data Uncertainty Models For GIS" Eds. K.Lowell and A.Jaton, "About Spatial Accuracy Assessment", chapter 24, Ann press, Michigan, USA [2] Alesheikh, A.A. (1998). Modeling and Managing Uncertainty in Object-Based Geospatial Information System. Ph.D. Thesis, University of Calgary, Canada [3] Burrough, P.A. (1986). Principles of Geographic Information Systems for Land Resources Assessment, Claredon Press, Oxford, UK [4] Congalton, R.G. (1994). Proceedings, International Symposium on the Spatial Accuracy of Natural Resource Databases, [5] Heuvelink, G. (1993). Error Propagation in Quantitative Spatial Modeling. Ph.D. Thesis, Universiteit Utrecht, Netherlands [6] Krakiwsky, E.J. (1992). The Method of Least Squares: A Synthesis of Advances. Department of Geomatics Engineering, The University of Calgary, Canada [7] Openshaw, S. (1994). "Learning to live with errors in spatial databases" Eds.M. Goodchild and S. Gopal, Accuracy of Spatial Databases, Taylor and Francis press, London, pp [8] Openshaw, S., Charlton, M., Carver, S. (1991). Error propagation: a Monte Carlo simulation, Handling Geographical Information, Methodology and Practical Applications, Scientific and Technical, pp [9] Stein, M. (1987). Large sample properties of Simulations using Latin hypercube sampling. Techno metrics 29, pp [10] Vanicek, p. and E.J.Krakiwsky (1986). Geodesy: The concept, Second Rev., North Holland, Amsterdam Bethesda, MD: ASPRS, pp