Optimization is achieved by fitting the simulated spectrum () to the experimental one by using the minimization of the least-squares residuals, which are a function of the parameters of the Hamiltonian. The goodness of least-squares fitting for one spectrum is defined as . For a global fitting of multiple spectra with different experimental parameters, for the individual spectrum is weighted by its own noise level. The global goodness of fitting is defined as:
where and are the noise variance and number of points in spectrum k, respectively. We assume that the experimental envelope modulation is free of artifacts (for example microwave phase drift, “glitches” above the rms noise level), which is the case in our experiments [22; 23; 24], and that the noise variance of each point in the ESEEM waveform is identical (dead time points removed). In this normalized representation and with the two assumptions, the variance of the simulation error of all data points is estimated as follows:
where is the optimized minimum at the global minimum, N is the total number of experimental points, as given by , and L is the number of independent parameters for optimization. can be expanded at , as:
where are the optimal parameters when . Assuming that the terms with order higher than two are negligible, and that the variance of the measurement errors follow a normal distribution, then the term:
follows a distribution with a freedom of L [25]. The simultaneous confidence region of parameters can be determined by the following expression:
The simultaneous confidence region gives an estimation of the uncertainty of a selected optimization parameter with respect to the other parameters. The probability of the simultaneous confidence region containing the true values of is determined by and a, as a cumulative distribution of from to a. For example, when L = 2 and a = 12, 22, or 32, the simultaneous confidence probability is 39.3%, 86.5% or 98.9%.