### Abstract

This is the second paper of a four-part series of papers on the development of a general framework for error analysis in measurement-based geographic information systems (MBGIS). In this paper, we discuss the problem of point-in-polygon analysis under randomness, i.e., with random measurement error (ME). It is well known that overlay is one of the most important operations in GIS, and point-in-polygon analysis is a basic class of overlay and query problems. Though it is a classic problem, it has, however, not been addressed appropriately. With ME in the location of the vertices of a polygon, the resulting random polygons may undergo complex changes, so that the point-in-polygon problem may become theoretically and practically ill-defined. That is, there is a possibility that we cannot answer whether a random point is inside a random polygon if the polygon is not simple and cannot form a region. For the point-in-triangle problem, however, such a case need not be considered since any triangle always forms an interior or region. To formulate the general point-in-polygon problem in a suitable way, a conditional probability mechanism is first introduced in order to accurately characterize the nature of the problem and establish the basis for further analysis. For the point-in-triangle problem, four quadratic forms in the joint coordinate vectors of a point and the vertices of the triangle are constructed. The probability model for the point-in-triangle problem is then established by the identification of signs of these quadratic form variables. Our basic idea for solving a general point-in-polygon (concave or convex) problem is to convert it into several point-in-triangle problems under a certain condition. By solving each point-in-triangle problem and summing the solutions, the probability model for a general point-in-polygon analysis is constructed. The simplicity of the algebra-based approach is that from using these quadratic forms, we can circumvent the complex geometrical relations between a random point and a random polygon (convex or concave) that one has to deal with in any geometric method when probability is computed. The theoretical arguments are substantiated by simulation experiments.

Original language | English (US) |
---|---|

Pages (from-to) | 355-379 |

Number of pages | 25 |

Journal | Journal of Geographical Systems |

Volume | 6 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 2004 |

Externally published | Yes |

### Fingerprint

### Keywords

- Algebra-based probability model
- Approximate covariance-based error band
- Point-in-polygon
- Point-in-triangle
- Quadratic form

### ASJC Scopus subject areas

- Geography, Planning and Development
- Earth-Surface Processes

### Cite this

**A general framework for error analysis in measurement-based GIS Part 2 : The algebra-based probability model for point-in-polygon analysis.** / Leung, Yee; Ma, Jiang Hong; Goodchild, Michael.

Research output: Contribution to journal › Article

*Journal of Geographical Systems*, vol. 6, no. 4, pp. 355-379. https://doi.org/10.1007/s10109-004-0142-3

}

TY - JOUR

T1 - A general framework for error analysis in measurement-based GIS Part 2

T2 - The algebra-based probability model for point-in-polygon analysis

AU - Leung, Yee

AU - Ma, Jiang Hong

AU - Goodchild, Michael

PY - 2004/12/1

Y1 - 2004/12/1

N2 - This is the second paper of a four-part series of papers on the development of a general framework for error analysis in measurement-based geographic information systems (MBGIS). In this paper, we discuss the problem of point-in-polygon analysis under randomness, i.e., with random measurement error (ME). It is well known that overlay is one of the most important operations in GIS, and point-in-polygon analysis is a basic class of overlay and query problems. Though it is a classic problem, it has, however, not been addressed appropriately. With ME in the location of the vertices of a polygon, the resulting random polygons may undergo complex changes, so that the point-in-polygon problem may become theoretically and practically ill-defined. That is, there is a possibility that we cannot answer whether a random point is inside a random polygon if the polygon is not simple and cannot form a region. For the point-in-triangle problem, however, such a case need not be considered since any triangle always forms an interior or region. To formulate the general point-in-polygon problem in a suitable way, a conditional probability mechanism is first introduced in order to accurately characterize the nature of the problem and establish the basis for further analysis. For the point-in-triangle problem, four quadratic forms in the joint coordinate vectors of a point and the vertices of the triangle are constructed. The probability model for the point-in-triangle problem is then established by the identification of signs of these quadratic form variables. Our basic idea for solving a general point-in-polygon (concave or convex) problem is to convert it into several point-in-triangle problems under a certain condition. By solving each point-in-triangle problem and summing the solutions, the probability model for a general point-in-polygon analysis is constructed. The simplicity of the algebra-based approach is that from using these quadratic forms, we can circumvent the complex geometrical relations between a random point and a random polygon (convex or concave) that one has to deal with in any geometric method when probability is computed. The theoretical arguments are substantiated by simulation experiments.

AB - This is the second paper of a four-part series of papers on the development of a general framework for error analysis in measurement-based geographic information systems (MBGIS). In this paper, we discuss the problem of point-in-polygon analysis under randomness, i.e., with random measurement error (ME). It is well known that overlay is one of the most important operations in GIS, and point-in-polygon analysis is a basic class of overlay and query problems. Though it is a classic problem, it has, however, not been addressed appropriately. With ME in the location of the vertices of a polygon, the resulting random polygons may undergo complex changes, so that the point-in-polygon problem may become theoretically and practically ill-defined. That is, there is a possibility that we cannot answer whether a random point is inside a random polygon if the polygon is not simple and cannot form a region. For the point-in-triangle problem, however, such a case need not be considered since any triangle always forms an interior or region. To formulate the general point-in-polygon problem in a suitable way, a conditional probability mechanism is first introduced in order to accurately characterize the nature of the problem and establish the basis for further analysis. For the point-in-triangle problem, four quadratic forms in the joint coordinate vectors of a point and the vertices of the triangle are constructed. The probability model for the point-in-triangle problem is then established by the identification of signs of these quadratic form variables. Our basic idea for solving a general point-in-polygon (concave or convex) problem is to convert it into several point-in-triangle problems under a certain condition. By solving each point-in-triangle problem and summing the solutions, the probability model for a general point-in-polygon analysis is constructed. The simplicity of the algebra-based approach is that from using these quadratic forms, we can circumvent the complex geometrical relations between a random point and a random polygon (convex or concave) that one has to deal with in any geometric method when probability is computed. The theoretical arguments are substantiated by simulation experiments.

KW - Algebra-based probability model

KW - Approximate covariance-based error band

KW - Point-in-polygon

KW - Point-in-triangle

KW - Quadratic form

UR - http://www.scopus.com/inward/record.url?scp=10444227320&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=10444227320&partnerID=8YFLogxK

U2 - 10.1007/s10109-004-0142-3

DO - 10.1007/s10109-004-0142-3

M3 - Article

AN - SCOPUS:10444227320

VL - 6

SP - 355

EP - 379

JO - Journal of Geographical Systems

JF - Journal of Geographical Systems

SN - 1435-5930

IS - 4

ER -