Application of Total Least Squares for Spatial Point Process Analysis

The total-least-squares approach is a relatively new adjustment method of estimating parameters in linear models that include error in all variables. Specifically, given an overdetermined set of linear equations $\mathbf{y}\approx \mathbf{A}\mathbf{\xi }$ where y is the observation vector, A is a positive defined data matrix, and ξ is the vector of unknown parameters, the total-least-squares problem is concerned with estimating ξ providing that the number of observations n is larger than the number of parameters to be estimated and given that both the observation vector y and the data matrix A are subjected to errors and need to be adjusted. This model is different from the classical least-squares model where only the observation vector y is subjected to errors. This paper starts with a brief summary of the least-squares approach and then explains how one can modify the approach to include error in all variables using the generalized least-squares technique. Then the total-least-squares problem is presented along with its formulas and the procedures used to solve it. Finally, the total-least-squares approach is used to determine the trend in a spatial point process.