Details
| Original language | English |
|---|---|
| Pages (from-to) | 110-123 |
| Number of pages | 14 |
| Journal | Journal of Geodetic Science |
| Volume | 10 |
| Issue number | 1 |
| Publication status | Published - 31 Dec 2020 |
Abstract
B-spline curves are a linear combination of control points (CP) and B-spline basis functions. They satisfy the strong convex hull property and have a fine and local shape control as changing one CP affects the curve locally, whereas the total number of CP has a more general effect on the control polygon of the spline. Information criteria (IC), such as Akaike IC (AIC) and Bayesian IC (BIC), provide a way to determine an optimal number of CP so that the B-spline approximation fits optimally in a least-squares (LS) sense with scattered and noisy observations. These criteria are based on the log-likelihood of the models and assume often that the error term is independent and identically distributed. This assumption is strong and accounts neither for heteroscedasticity nor for correlations. Thus, such effects have to be considered to avoid under-or overfitting of the observations in the LS adjustment, i.e. bad approximation or noise approximation, respectively. In this contribution, we introduce generalized versions of the BIC derived using the concept of quasi-likelihood estimator (QLE). Our own extensions of the generalized BIC criteria account (i) explicitly for model misspecifications and complexity (ii) and additionally for the correlations of the residuals. To that aim, the correlation model of the residuals is assumed to correspond to a first order autoregressive process AR(1). We apply our general derivations to the specific case of B-spline approximations of curves and surfaces, and couple the information given by the different IC together. Consecutively, a didactical yet simple procedure to interpret the results given by the IC is provided in order to identify an optimal number of parameters to estimate in case of correlated observations. A concrete case study using observations from a bridge scanned with a Terrestrial Laser Scanner (TLS) highlights the proposed procedure.
Keywords
- AR(1), B-splines approximation, BIC, Control Points, correlation, Generalized BIC, Information Criteria, quasi-likelihood, Terrestrial Laser Scanning, Variance Inflation Factor
ASJC Scopus subject areas
- Earth and Planetary Sciences(all)
- Earth and Planetary Sciences (miscellaneous)
- Earth and Planetary Sciences(all)
- Geophysics
- Earth and Planetary Sciences(all)
- Computers in Earth Sciences
- Mathematics(all)
- Applied Mathematics
- Physics and Astronomy(all)
- Astronomy and Astrophysics
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In: Journal of Geodetic Science, Vol. 10, No. 1, 31.12.2020, p. 110-123.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - On the BIC for determining the number of control points in B-spline surface approximation in case of correlated observations
AU - Kermarrec, Gaël
AU - Alkhatib, Hamza
PY - 2020/12/31
Y1 - 2020/12/31
N2 - B-spline curves are a linear combination of control points (CP) and B-spline basis functions. They satisfy the strong convex hull property and have a fine and local shape control as changing one CP affects the curve locally, whereas the total number of CP has a more general effect on the control polygon of the spline. Information criteria (IC), such as Akaike IC (AIC) and Bayesian IC (BIC), provide a way to determine an optimal number of CP so that the B-spline approximation fits optimally in a least-squares (LS) sense with scattered and noisy observations. These criteria are based on the log-likelihood of the models and assume often that the error term is independent and identically distributed. This assumption is strong and accounts neither for heteroscedasticity nor for correlations. Thus, such effects have to be considered to avoid under-or overfitting of the observations in the LS adjustment, i.e. bad approximation or noise approximation, respectively. In this contribution, we introduce generalized versions of the BIC derived using the concept of quasi-likelihood estimator (QLE). Our own extensions of the generalized BIC criteria account (i) explicitly for model misspecifications and complexity (ii) and additionally for the correlations of the residuals. To that aim, the correlation model of the residuals is assumed to correspond to a first order autoregressive process AR(1). We apply our general derivations to the specific case of B-spline approximations of curves and surfaces, and couple the information given by the different IC together. Consecutively, a didactical yet simple procedure to interpret the results given by the IC is provided in order to identify an optimal number of parameters to estimate in case of correlated observations. A concrete case study using observations from a bridge scanned with a Terrestrial Laser Scanner (TLS) highlights the proposed procedure.
AB - B-spline curves are a linear combination of control points (CP) and B-spline basis functions. They satisfy the strong convex hull property and have a fine and local shape control as changing one CP affects the curve locally, whereas the total number of CP has a more general effect on the control polygon of the spline. Information criteria (IC), such as Akaike IC (AIC) and Bayesian IC (BIC), provide a way to determine an optimal number of CP so that the B-spline approximation fits optimally in a least-squares (LS) sense with scattered and noisy observations. These criteria are based on the log-likelihood of the models and assume often that the error term is independent and identically distributed. This assumption is strong and accounts neither for heteroscedasticity nor for correlations. Thus, such effects have to be considered to avoid under-or overfitting of the observations in the LS adjustment, i.e. bad approximation or noise approximation, respectively. In this contribution, we introduce generalized versions of the BIC derived using the concept of quasi-likelihood estimator (QLE). Our own extensions of the generalized BIC criteria account (i) explicitly for model misspecifications and complexity (ii) and additionally for the correlations of the residuals. To that aim, the correlation model of the residuals is assumed to correspond to a first order autoregressive process AR(1). We apply our general derivations to the specific case of B-spline approximations of curves and surfaces, and couple the information given by the different IC together. Consecutively, a didactical yet simple procedure to interpret the results given by the IC is provided in order to identify an optimal number of parameters to estimate in case of correlated observations. A concrete case study using observations from a bridge scanned with a Terrestrial Laser Scanner (TLS) highlights the proposed procedure.
KW - AR(1)
KW - B-splines approximation
KW - BIC
KW - Control Points
KW - correlation
KW - Generalized BIC
KW - Information Criteria
KW - quasi-likelihood
KW - Terrestrial Laser Scanning
KW - Variance Inflation Factor
UR - http://www.scopus.com/inward/record.url?scp=85099123136&partnerID=8YFLogxK
U2 - 10.1515/jogs-2020-0110
DO - 10.1515/jogs-2020-0110
M3 - Article
AN - SCOPUS:85099123136
VL - 10
SP - 110
EP - 123
JO - Journal of Geodetic Science
JF - Journal of Geodetic Science
IS - 1
ER -