364
M.A. Chapman et al.
EVALUATION OF THE PROPOSED UNCERTAINTY MODEL
Under the assumption that the uncertainty of a line segment is affected only by the
errors of its two endpoints, the above uncertainty model has been derived. The
proposed model contributes to the improvement of existing boundary uncertainty
models in the following ways:
1) it provides the error distribution of the uncertainty (unlike the deterministic
interpretation of the epsilon band model)
2) the model is analytically derived (contrary to Dutton’s simulation-based
model)
3) the model respects the correlation among the contributing parameters, and
finally
4) the model is a superset of the currently used ones. All other models can be
assumed as special cases of this proposed model
Moreover the model has undergone several tests and proved its superiority. In
practice, the initial assumption of the model can be realized when the function (linear
in this case) is assumed perfect or the line segment is long, such as the railroad or
township street, where modeling error is minimized. Because of this assumption, the
uncertainty of the points located along the line should not exceed the uncertainty of the
endpoints.
However, if the modeling error is under question or the modeling error is of a
considerable magnitude, then the line uncertainty model should be improved. To
accompany modeling error, Equation 2 is generalized to;
and the covariance matrix of point U will be:
where Equation 6 is what we considered in the above uncertainty model and, Equation
7 is the contribution of the modeling error. Indeed, if there is a correlation between
modeling error and endpoints error, the summation will add a new term indicating the
correlation. The modeling error in this study is nothing but the error attributed to the
slope and intercept of line segments. It can easily be found that based on the magnitude
of the modeling error the shape of the line uncertainty model may well be represented
by an ellipse rather than a hyperbola. Fig. 6 demonstrate the differences between line
uncertainty models. The modeling error in this case is estimated by a Gauss-Markov
process.
M.A. Chapman et al.
EVALUATION OF THE PROPOSED UNCERTAINTY MODEL
Under the assumption that the uncertainty of a line segment is affected only by the
errors of its two endpoints, the above uncertainty model has been derived. The
proposed model contributes to the improvement of existing boundary uncertainty
models in the following ways:
1) it provides the error distribution of the uncertainty (unlike the deterministic
interpretation of the epsilon band model)
2) the model is analytically derived (contrary to Dutton’s simulation-based
model)
3) the model respects the correlation among the contributing parameters, and
finally
4) the model is a superset of the currently used ones. All other models can be
assumed as special cases of this proposed model
Moreover the model has undergone several tests and proved its superiority. In
practice, the initial assumption of the model can be realized when the function (linear
in this case) is assumed perfect or the line segment is long, such as the railroad or
township street, where modeling error is minimized. Because of this assumption, the
uncertainty of the points located along the line should not exceed the uncertainty of the
endpoints.
However, if the modeling error is under question or the modeling error is of a
considerable magnitude, then the line uncertainty model should be improved. To
accompany modeling error, Equation 2 is generalized to;
and the covariance matrix of point U will be:
where Equation 6 is what we considered in the above uncertainty model and, Equation
7 is the contribution of the modeling error. Indeed, if there is a correlation between
modeling error and endpoints error, the summation will add a new term indicating the
correlation. The modeling error in this study is nothing but the error attributed to the
slope and intercept of line segments. It can easily be found that based on the magnitude
of the modeling error the shape of the line uncertainty model may well be represented
by an ellipse rather than a hyperbola. Fig. 6 demonstrate the differences between line
uncertainty models. The modeling error in this case is estimated by a Gauss-Markov
process.