approach to finding a suitable answer; however, it has the severe limitations that it is both
dependent on an original model and a pre-determined form.
x ) a -(x -b)/(2c2)
f (x)=ae A
0 1 2 3 20 30 40 50
t (s) Distance (A)
Figure 2-22. Pictorial representation of the Monte Carlo analysis.
Tikhonov regularization
A third approach, and the one used for all DEER analysis described in this dissertation, is
called Tikhonov regularization (TKR) (Tikhonov 1943; Hansen 1998; Chiang, Borbat et al.
2005). TKR uses the function in Equation 2-11 to find the best answer to the ill-posed problem
by balancing the quality of fit with the smoothness of the solution by varying the regularization
parameter X (often referred to as a). The first term represents the quality of the fit and the
second term represents the smoothness of the solution; P is probability of the spin-spin distance,
K and L are operators, S is the experimental data vector. Following the TKR process, the log of
rq() is plotted against the log of p(X) to produce an L-curve which is analyzed in order to
determine the optimal regularization parameter. Each individual point on the plot is given by
Equation 2-12, and rq() and p(X) are given in Equations 2-13 and 2-14.
D[P] =1 KP-S 112 +i2 II LP 2 (2-11)
G,(P) = S(t) -D(t) 1 2 2 l P(r) 12
ar (2.12)
p(A) =1 S(t)- D(t) 12 (2.13)