Phoenix's deconvolution function determines the optimal smoothing without actually measuring the degree of smoothing. The degree of smoothing is not quantified but is controlled through the dispersion function to optimize the “consistency” between the data and the estimated drug level curve. Consistency is defined here according to the cross validation principles. Let rj denote the differences between the predicted and observed concentration at the j-th observation time when that observation is excluded from the dataset. The optimal cross validation principle applied is defined as the condition that leads to the minimal predicted residual sum of squares (PRESS) value, where PRESS is defined as:
For a given value of the smoothing parameter d, PRESS is a quadratic function of the wavelet scaling parameters x. Thus, with the non-negativity constraint, the minimization of PRESS for a given d value is a quadratic programming problem. Let PRESS (d) denote such a solution. The optimal smoothing is then determined by finding the value of the smoothing parameter d that minimizes PRESS (d). This is a one variable optimization problem with an embedded quadratic-programming problem.