varPhi left( {vec{alpha }} right) = frac{1}{2xi }left| {vec{d}_{text{cal}} - vec{d}_{text{meas}} } right|_{2}^{2}, (1 where (vec{alpha }) is the vector of the unknown parameters (to be estimated by optimization or nonlinear regression), (vec{d}_{text{cal}}) is the vector of modeled pressure data, (vec{d}_{text{meas}}) is the vector of measured data and ξ is a scaling factor.
Given a complete mathematical model and sufficient accurate quantitative data, the parameters in the model can be estimated by optimization techniques, i.e., by fitting the model to the data.
In the outer level of optimization, Θ is estimated by optimizing a criterion.
In the inner level of optimization, c is estimated by optimizing a criterion U c| Θ), given any value of Θ.
So, coefficients in ARIMA models that include lagged errors must be estimated by nonlinear optimization methods ("hill-climbing") rather than by just solving a system of equations.
However, our method is more general, merely requiring that the parameters of the model be estimated by a numerical optimization.
The target parameters can be estimated by the following optimization equation: (hat{theta},hat{R})=argquad {max}_{theta,R}|w^{H} (theta,R y|^{2}, (23).
v i 0 = K eq a i 0 − b i 0 c 1 + c 2 a i 0 + c 3 b i 0, where i = 1, 2, ⋯, n where all the a i, b i and v i are experimental measurements, all the relative errors σ are known and a i 0, b i 0, v i 0 are latent variables, and c1, c2 and c3 are the parameters to be estimated by solving the optimization problem.
In general, the method is capable of providing the useful coarse translational motion compensation even if strong noise presents, and the residual motion can be estimated by the coordinate descent optimization.
Taking the ℓ 1,2 mixed norm minimization as an example, the matrix V k can be estimated by the following constrained optimization problem [12]: begin{aligned} & underset{boldsymbol{V}}{text{minimize}} & & sum_{i=1}^{L} left|boldsymbol{bar{v}}_{i}right| & text{s.t.} & & |boldsymbol{Y}-boldsymbol{A}boldsymbol{V}|leqvarepsilon.
For the discriminant analysis between gene expression and disease data blocks, the optimum of the slack variable m and the loading vector α4 can be estimated by solving the following optimization problem: (15) arg max α 4, m 1 2 χ 4 α 4 − υ 5 + b ⊙ m 2 s.t. α 4 ≤ ξ 1, α 4 2 ≤ ξ 2. The Lagrangian function of (15) is ℒ = (1/2)‖ χ4 α4 − υ5 − b⊙ m‖ + λ4 | α4 | +((1 − λ4)/2)‖ α4‖.
