Pseudo-inverse matrix

C CT] is known as the pseudo inverse of C. Since the product of a matrix and its inverse is the identity matrix, [C CT][C CT] disappears from the right-hand side of equation [32] leaving... 

Furthermore, the implementation of the Gauss-Newton method also incorporated the use of the pseudo-inverse method to avoid instabilities caused by the ill-conditioning of matrix A as discussed in Chapter 8. In reservoir simulation this may occur for example when a parameter zone is outside the drainage radius of a well and is therefore not observable from the well data. Most importantly, in order to realize substantial savings in computation time, the sequential computation of the sensitivity coefficients discussed in detail in Section 10.3.1 was implemented. Finally, the numerical integration procedure that was used was a fully implicit one to ensure stability and convergence over a wide range of parameter estimates. 

In the 3rd run the porosity of the ten zones was estimated by using an initial guess of 0.1. Finally, in the 4,h run the porosity of all fifteen zones was estimated by using the same initial guess (0.1) as above. In this case, matrix A was found to be extremely ill-conditioned and the pseudo-inverse option had to be used. 

In the same way, the parameters kij and hij are joined to form a unique parameter matrix H. With these definitions a linear problem may be written like that of equation 5. The matrix H can then be estimated either by direct pseudo-inversion or by PLS. It is worth noting... 

As stated earlier, Matlab s philosophy is to read everything as a matrix. Consequently, the basic operators for multiplication, right division, left division, power (, /,, A) automatically perform corresponding matrix operations (A will be introduced shortly in the context of square matrices, / and will be discussed later, in the context of linear regression and the calculation of a pseudo inverse, see The Pseudo-Inverse, p.117). 

Figure 4-13. Schematic representation of the matrix equations involving multiplication of y by the left pseudo-inverse F+=(FtF) 1Ft.

For the computation of the pseudo-inverse, it is crucial that the vectors f j are not parallel, or more correctly, that they are linearly independent. Otherwise, the matrix FlF is singular and cannot be inverted. Matlab issues a warning. We can gain a certain level of understanding by adapting Figure 4-10 ... 

In a strictly mathematical sense this matrix is not singular but numerically it is rank deficient and has effectively a rank of only 4. Calculation of its pseudo-inverse consequently is impossible, or at least numerically unsafe. What can we do about that ... 

The pseudo-inverse for the calculation of the shift vector in equation (4.67) has been computed traditionally as J+= (J Jp1 J. Adding a certain number, the Marquardt parameter mp, to the diagonal elements of the square matrix J J prior to its inversion, has two consequences (a) it shortens the shift vector 8p and (b) it turns its direction towards steepest descent. The larger the Marquardt parameter, the larger is the effect. In matrix formulation, we can write ... 

J is the derivative of a matrix with respect to a vector. What is the structure of such an object and more disturbingly, what is its pseudo-inverse ... 

For a linear fitting exercise, e.g. the calculation of the emission spectra A, we assume to know the lifetimes t and hence the matrix Csim, which we used for the generation of the measurement. The linear regression has to be performed individually at each wavelength. This is due to the fact that at each wavelength Xj the appropriate vector (Ty j is different and each weighted matrix Cw and its pseudo-inverse, needs to be computed independently. There is no equivalent of the elegant A=C Y notation. 

The spreadsheet in Figure 4-62 is heavily matrix based (see Chapter 2, for an introduction to basic matrix functions in Excel). It is the only way to keep the structure reasonably simple. The matrix C in cells A21 C31 is computed in the usual way, see equation (4.63) the parameters required to compute the concentration matrix are in cells Q4 S4, they include the initial concentration for species A and the two rate constants that are to be fitted. In cells E 16 018 the computation of the best absorptivity matrix A for any given concentration matrix C, is done as a matrix equation, as demonstrated in The Pseudo-Inverse in Excel (p.146). Similarly the matrix Ycaic in cells E21 031 is written as the matrix product CA. Even the calculation of the square sum of the residuals in cell R7 is written in a compact way, using the Excel function SUMXMY2, especially designed for this purpose. We refer to... 

The computation of the linear parameters b is easy, an orthonormal matrix is equal to its transposed as the pseudo-inverse of... 

The advantage is that there is no pseudo-inverse to be calculated in this way. The computation of Tu, which comprises linear parameters is easier than usual as U is an orthonormal matrix, U+=Ul. As mentioned before, in equation (5.29), it is advantageous to compute the residuals as Ru = C - U(Ul C) it is considerably faster. 

Show by direct calculation that the pseudo-inverse (X X) (X ) is equivalent to the transpose of the X matrix for the design and model represented by Table 14.3 (see Equation 14.6). 

With this background infonnation on the inverse methods, it is instructive to examine the calculations for the inverse model in more detail. In Equation 5-23, the key to the model-building step is the inversion of the matrix CR ). This is a squire matrix with number of rows and columns equal to the number of measurement variables (nvars). From theory, a number of independent samples in the calibration set greater than or equal to nvars is needed in order to invert this matrix. For most analytical measurement systems, nvars (e.g., number of wavelengths) is greater than the number of independent samples and therefore RTr cannot be directly inverted. However, with a transformation, calculating she pseudo-inverse of R (R is possible. How this transformation is accomplished distinguishes the different inverse methods. 

The DCLS method can be applied to simple systems where all of the pure-component spectra can be measured. To construct the DCLS model, the pure-component spectra are measured at unit concentration for each of the analytes in the mixture. Tliese are used to form a matrix of pure spectra (S) and the model is then constructed as the pseudo-inverse of this S matrix. This calibration model is used to predict the concentrations in unknown samples. 

Some valuable information about the importance of each term can be obtained under such circumstances. Note, however, the design matrix is no longer square, and it is not possible to use the simple approaches above to calculate the effects, regression using tlte pseudo-inverse being necessary. 

Set up die design matrix and calculate die coefficients. Do this widiout using the pseudo-inverse. 

Explain why the inverse of the design matrix can be used to calculate the terms in the model, rather than using the pseudo-inverse b = (D. D) l.D. y. What changes in the design or model would require using the pseudo-inverse in the calculations ... 

By constructing the design matrix and then using the pseudo-inverse, calculate die coefficients for die best fit model given by the equation... 

A 10 parameter model is to be fitted to the data, consisting of the intercept, all single factor linear and quadratic terms and all two factor interaction terms. Set up the design matrix, and by using the pseudo-inverse, calculate the coefficients of the model using coded values. 

Several authors have shown how to calculate the matrix A [58,60-63]. A is the pseudo-inverse of the matrix B of Eq. 60 B- A = 1, where 1 is the Ns x Ns unit matrix. Polo [64] appears to have been the first to express the matrix A directly by constructing, for each atom a and each internal coordinate t, the 3 x 1 vector ... 

The third case is realized when Ws is of rank three or less. As it was above, the optimal matrix inverse is the pseudo-inverse. However, in this case only three or less of the Stokes vector elements can be measured. The corresponding measurement apparatus is called an incomplete Stokes polarimeter. 

As in (4.3), the existence, rank, and nniqneness of inverse matrix plays a key role in the solntion of eqnation (4.6). If contains sixteen independent colnnms, then all sixteen elements of the Mueller matrix can be determined. When N = 16, then is unique. If A > 16, then is overdetermined. The optimal least squares estimation for m j is given by the pseudo-inverse of ... 

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