The Model Space

The purpose of this chapter is to give an insight into engineering modeling activities by step-by-step explanation, in accordance with emphases in present industrial practice. Basic concepts are explained to prepare readers for Chapters 3-9 that discuss areas of modeling in detail. 

Shape-centered design, analysis, and manufacturing planning rely on shape description of parts as engineering objects. Shapes are 

Eigure 2-2 gives a survey of shape-centered models of a mechanical system. Objects Oj, O2, and O3 are combined into complex shape O4. O4, O5, and constitute an assembly. Cy in O2 is described as a parametric curve. Parts are in contact at their flat surfaces. Oj is allowed to move relative to O4 along curve 2-Similarly, is allowed to move relative to O4 along line Lj. Contacts of the flat surface pairs are not broken during allowed movements. A sequence of three positions of O4 in the model space is described in time by Erames 1-3. Each frame is a discrete position of the animated object at a given time. The appearance of surfaces of is shown by shading. 

The virtual world that is established by model based engineering is in close connection with the designed, stored, visualized, and documented worlds (Eigure 2-3). The designed world is the physical world to be modeled. Modeling experts sometimes should remember that the final purpose of engineering activities is to 

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Oui recommendation is that one should use n-leave-out cross-validation, rather than one-leave-out. Nevertheless, there is a possibility that test sets derived thus would be incompatible with the training sets with respect to information content, i.e., the test sets could well be outside the modeling space [8]. 

First, one can check whether a randomly compiled test set is within the modeling space, before employing it for PCA/PLS applications. Suppose one has calculated the scores matrix T and the loading matrix P with the help of a training set. Let z be the characteristic vector (that is, the set of independent variables) of an object in a test set. Then, we first must calculate the scores vector of the object (Eq. (14)). 

Now, if the error, e, is greater than that of the training stages, the object is outside the modeling space. Further details can be found in Chapter 9 and also in Ref. [8]. 

All predictions must be taken for what they are, namely, generalizations based on current knowledge and understanding. There is a temptation for a user to assume that a computer-generated answer must be correct. To determine whether this is in fact the case, a number of factors concerning the model must be addressed. The statistical evaluation of a model was addressed above. Another very important criterion is to ensure that a prediction is an interpolation within the model space, and not an extrapolation outside of it. To determine this, the concept of the applicability domain of a model has been introduced [106]. 

The principal component space does not have the redundancy issue discussed above, because the PCs are orthogonal to one another. In addition, because each PC explains the most remaining variance in the x data, it is often the case that fewer PCs than original x variables are needed to capture the relevant information in the x data. This leads to simpler classification models, less susceptibility to overfitting through the use of too many dimensions in the model space, and less noise in the model. 

The KNN method [77] is probably the simplest classification method to understand. Once the model space and distance measure are defined, its classification rule involves rather simple logic ... 

We may now stress some formal similarities and differences between both approaches when the model space used in (SC) CAS-SDCI and the source of corrections in ec-CCSD is the same (a more mathematically oriented comparison will be showed in the next section). 

Two important remarks are in order here. The first is that eq. (3) does not impose any limitation on the degree of excitation of ([) , so that can be calculated whatever the composition and symmetry of Sq. In this way, even if eqs. (1), (2) and (4) assume a single reference determinant, the SDCI space that constitutes Sq can be built as a MR-SDCI and the procedure described above can be applied. The second is that care must be taken to remove the possible redundancies that could be present if Dj( i belongs to the model space Spg. In such a case, the physical effects included in each term of eq. (3) would be included twice in the diagonalization of H + A. ... 

Projections of the exact wave functions onto the model space (sometimes denoted as model functions)... 

The inverse operator may exist even in the case when the corresponding power series in terms of Beffective Hamiltonian, acting within the model space, in the same way as in the multi-root theory, i.e. 

Turning now back to the single-root MR BWCC approach, we derive the basic equations for the effective Hamiltonian and cluster amplitudes in the spin-orbital form without the use of the BCH formula. We limit ourselves to a complete model space which implies that amplitudes corresponding to internal excitations (i.e. excitations within the model space) are equal to zero. In our derivation we shall work with the Hamiltonian in the normal-ordered-product form, i.e. 

On the one hand, we can strive for a single cluster operator T, defining the valence universal wave operator U, U = exp(T), which will transform all the model space states ]< > ) into some linear combinations of the exact states jfl i), f = 1,2, , M, which in turn span the target space M, i.e.. 

Then we can employ the same procedure (in fact the same codes) as in the SR case by considering the model space configuration Tj) as a new reference (or Eermi vacuum) for each I I j). 

The P operator projects out of any function the component that lies in the model space, and the Q operator projects out the component in the orthogonal space. They also obey relation (3.6). 

It operates entirely within the model space and satisfies the eigenvalue equation... 

Let us consider further the case where the model space is completely degenerate. We assume also that all the states of the model space correspond... 

In general, the corresponding energies are obtained by diagonalizing this effective Hamiltonian in the model space. If the model functions are already found, then the energies can be calculated directly starting with Eq. (3.29) and evaluating the matrix element, i.e. 

Here a complete model space (P-space), defined on eigenfunctions of Ho, representing all possible distributions of electrons between open shells, is utilized. In our case the model space consists of the ls22s22p4 and ls22p6 configurations for the even parity states and of the ls22s2p5 configuration for odd parity states. The wave-operator may be written as... 

Here P is a projection operator (Pip = i) onto the model space. We assume here that the residual interaction He > HBr and the second-order part of Heff can be given by... 

There exists another path of approaching the ZFS that is beyond the SH formalism but retains features of the model space. It is based upon the partitioning technique. 

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