Geometrical Interpretations

In a system, consisting of only one component, there are two intensive variables sufficient to describe it. This is usually temperature and pressure. In general, this is not true. For example, if the system consists of large surfaces, then more variables are needed. 

However, in most common situations two variables can describe the behavior of a system. For this reason, we can take a two-dimensional plot for a one-component system, as shown in Fig. 7.1. All the information can be shown on a two-dimensional 

The plane is divided into two parts by a line, with geometrical dimension one. Assume that on one side of the plane is the solid state, and on the other side is the liquid state. If we want to establish a state, where both phases are present, then it is very natural that we can move along the line with the variables temperature and pressure. This line provides a functional dependence of the variables in question, i.e., pressure and temperature 

We can hardly imagine another situation, for geometrical reasons. If there is still another region besides solid and liquid, then we have to place another line on the plane to separate these regions. Along this new line, the same is true as for the solid -liquid border. To establish the existence of two phases, we can move along the new line. 

Now two lines, if they are not parallel, are cutting in a point. Here we think of the triple point. To establish the existence of three phases for a one-component system, we have the equations of the two lines, in the two-dimensional space 

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Figure Al.6.32. (a) Initial and (b) final population distributions corresponding to cooling, (c) Geometrical interpretation of cooling. The density matrix is represented as a point on generalized Bloch sphere of radius R...

A mapping is said to be symplectic or canonical if it preserves the differential form dp A dq which defines the symplectic structure in the phase space. Differential forms provide a geometric interpretation of symplectic-ness in terms of conservation of areas which follows from Liouville s theorem [14]. In one-degree-of-freedom example symplecticness is the preservation of oriented area. An example is the harmonic oscillator where the t-flow is just a rigid rotation and the area is preserved. The area-preserving character of the solution operator holds only for Hamiltonian systems. In more then one-degree-of-freedom examples the preservation of area is symplecticness rather than preservation of volume [5]. 

In order to clarify these definitions, a geometrical interpretation of FXt0 t(x1,x2) is given in Pig. 3-7. It should be clear from this picture that is the fraction of the time that, simultaneously, X(t) 5... 

This is the fundamental differential equation. The reader who is acquainted with the rules for transforming the variables in a surface integral will observe that it has the geometrical interpretation that corresponding elements of area on the (v, p) and (s, T) diagrams are equal (cf. 43). 

Fig. 29.1. Geometrical interpretation of the scalar product of x y as the projection of the vector x upon the vector y. The lengths of x and y are denoted by 11 xl I and 11 yl I, respectively, and their angular separation is denoted by i9.

This relationship is of importance in multivariate data analysis as it relates distance between endpoints of two vectors to distances and angular distance from the origin of space. A geometrical interpretation is shown in Fig. 29.2. 

Fig. 29.5. Geometrical interpretation of an nxp matrix X as either a row-pattern of n points P" in p-dimensional column-space S (left panel) or as a column-pattern of p points / in n-dimensional row-space S" (right panel). The p vectors Uy form a basis of 5 and the n vectors v, form a basis of 5".

This is an alternative way of computing determinants which, unlike the usual one described in Section 9.3.4, allows for a geometrical interpretation. 

Singularity of the matrix A occurs when one or more of the eigenvalues are zero, such as occurs if linear dependences exist between the p rows or columns of A. From the geometrical interpretation it can be readily seen that the determinant of a singular matrix must be zero and that under this condition, the volume of the pattern P" has collapsed along one or more dimensions of SP. Applications of eigenvalue decomposition of dispersion matrices are discussed in more detail in Chapter 31 from the perspective of data analysis. 

The geometrical interpretation of column-standard deviations is in terms of distances of the points representing the columns of Y from the origin of 5" (multi-... 

Fig, 29.10. Geometrical interpretation of multiple linear regression (MLR). The pattern of points in S representing a matrix X is projected upon a vector b, which is imaged in 5" by the point y. The orientation of the vector b is determined such that the distance between y and the given y is minimal. 

In a general way, we can state that the projection of a pattern of points on an axis produces a point which is imaged in the dual space. The matrix-to-vector product can thus be seen as a device for passing from one space to another. This property of swapping between spaces provides a geometrical interpretation of many procedures in data analysis such as multiple linear regression and principal components analysis, among many others [12] (see Chapters 10 and 17). 

The geometrical interpretation of MLR is given in Fig. 29.10. The n rows (objects) of X form a pattern P" of points (represented by x,) which is projected upon an (unknown) axis b. This causes the axis b in S to be imaged by X in the dual space S" at the point y. The vector of observed measurements y has dimension n and, hence, is also represented as a point in 5". Is it possible then to define an axis b in S " such that the predicted y coincides with the observed y Usually this will not be feasible. One may propose finding the best possible b such that y comes as close to y as possible. A criterion for closeness is to ask for the distance between y and y, which is equal to the normlly - yll, to be as small as possible. 

Fig. 29.11. Geometrical interpretation of a rotation of as a change of the frame of coordinate axes to new directions which are defined by the columns in the rotation matrix V (left panel). Likewise, a rotation of S" can be interpreted as a change of the frame of coordinate axes to new directions which are defined by the columns in the rotation matrix U (right panel).

Summarizing, we have obtained two sets of principal components S and L (for a particular a and P) from one and the same data table X. In the next section on the geometrical interpretation of principal components, we will show in a more general way that S contains the coordinates of the rows of X in factor space and that L contains the coordinates of the columns of X in the same factor space. Sometimes one refers to U and V as the principal components of X. This is also legitimate when one refers to the special case where a = P = 0, as can be readily seen from eqs. (31.9) and (31.11). 

In the previous section we have developed principal components analysis (PCA) from the fundamental theorem of singular value decomposition (SVD). In particular we have shown by means of eq. (31.1) how an nxp rectangular data matrix X can be decomposed into an nxr orthonormal matrix of row-latent vectors U, a pxr orthonormal matrix of column-latent vectors V and an rxr diagonal matrix of latent values A. Now we focus on the geometrical interpretation of this algebraic decomposition. 

Procrustes analysis is a method for relating two sets of multivariate observations, say X and Y. For example, one may wish to compare the results in Table 35.1 and Table 35.2 in order to find out to what extent the results from both panels agree, e.g., regarding the similarity of certain olive oils and the dissimilarity of others. Procrustes analysis has a strong geometric interpretation. The... 

Figure 3. Number K of independent parameters in a projector geometrical interpretation of Pecora s vs CGM s formulae.

A geometrical interpretation of the differential is represented in Fig. 8. It is apparent that in general dy < Ay or dy > Ay, as the curve is concave upward or downward, respectively. 

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