TY - JOUR

T1 - Local likelihood tracking of fault lines and boundaries

AU - Hall, Peter

AU - Peng, Liang

AU - Rau, Christian

PY - 2001

Y1 - 2001

N2 - We suggest locally parametric methods for estimating curves, such as boundaries of density supports or fault lines in response surfaces, in a variety of spatial problems. The methods are based on spatial approximations to the local likelihood that the curve passes through a given point in the plane, as a function of that point. The local likelihood might be a regular likelihood computed locally, with kernel weights (e.g. in the case of support boundary estimation) or a local version of a likelihood ratio statistic (e.g. in fault line estimation). In either case, the local likelihood surface represents a function which is relatively large near the target curve, and relatively small elsewhere. Therefore, the curve may be estimated as a ridge line of the surface; we require only a numerical algorithm for tracking the projection of a ridge into the plane. This approach offers several potential advantages over alternative methods. First, the local (log-)likelihood surface can be graphed, and the degree of 'ridginess' assessed visually, to determine how the level of local smoothing should be varied in different spatial locations in order to emphasize the ridge and hence the curve adequately. Secondly, the local likelihood surface does not need to be computed in anything like its entirety; once we have a reasonable approximation to a point on the curve we may track it by numerically 'walking along' the ridge line. Thirdly, the method is appropriate without change for many different types of spatial explanatory variables - gridded, stochastic or otherwise. Three examples are explored in detail; fault lines in response surfaces and in intensity or density surfaces, and boundaries of supports of probability densities.

AB - We suggest locally parametric methods for estimating curves, such as boundaries of density supports or fault lines in response surfaces, in a variety of spatial problems. The methods are based on spatial approximations to the local likelihood that the curve passes through a given point in the plane, as a function of that point. The local likelihood might be a regular likelihood computed locally, with kernel weights (e.g. in the case of support boundary estimation) or a local version of a likelihood ratio statistic (e.g. in fault line estimation). In either case, the local likelihood surface represents a function which is relatively large near the target curve, and relatively small elsewhere. Therefore, the curve may be estimated as a ridge line of the surface; we require only a numerical algorithm for tracking the projection of a ridge into the plane. This approach offers several potential advantages over alternative methods. First, the local (log-)likelihood surface can be graphed, and the degree of 'ridginess' assessed visually, to determine how the level of local smoothing should be varied in different spatial locations in order to emphasize the ridge and hence the curve adequately. Secondly, the local likelihood surface does not need to be computed in anything like its entirety; once we have a reasonable approximation to a point on the curve we may track it by numerically 'walking along' the ridge line. Thirdly, the method is appropriate without change for many different types of spatial explanatory variables - gridded, stochastic or otherwise. Three examples are explored in detail; fault lines in response surfaces and in intensity or density surfaces, and boundaries of supports of probability densities.

KW - Boundary estimation

KW - Changepoint

KW - Density estimation

KW - Edge detection

KW - Frontier analysis

KW - Image analysis

KW - Jump

KW - Locally parametric methods

KW - Productivity analysis

KW - Response surface

KW - Smoothing

UR - http://www.scopus.com/inward/record.url?scp=0035535703&partnerID=8YFLogxK

U2 - 10.1111/1467-9868.00299

DO - 10.1111/1467-9868.00299

M3 - Article

SN - 1369-7412

VL - 63

SP - 569

EP - 582

JO - Journal of the Royal Statistical Society. Series B: Statistical Methodology

JF - Journal of the Royal Statistical Society. Series B: Statistical Methodology

IS - 3

ER -