Statistic Definition of Semantic Axes

Besides the algebraic interpretation there can also be given an equivalent statistic interpretation of the main components. Let us perform it in relation to characteristics of the studied phenomena.

Let us take an elementary situation, when only two characteristics: and  are studied in objects .

Two -dimensional vectors univalently determine the selection out of the corresponding two-dimensional general population.

Therefore, each object  in semantic space is located as a point with coordinates . Let the characteristics distribution by objects comply with a regular law: - two-dimensional probability density:

Let us single out a subset of all that got into the domain, restricted by condition:. The respective ellipse of concentration  (see Fig. 3) will be calculated from the following equation:

Hence we receive a second-degree equation in a general form:

,  where and  are normalized to and are values of and :

.

Fig. 3. Ellipse of concentration obtained by the section of the “hill”

of normal distribution density  at the “altitude” of

The quadric form will be put in the matrix notation as follows:

The left part of the equation would not change if the signs of and  are simultaneously changed to the opposite. Consequently, points of the quadric form graph are coupled symmetrically to the origin of coordinates. That means that the second power line possesses a symmetry center that is located at the origin of coordinates.

Let us bring the quadric form to its canonical form. To do that, let us rotate coordinate axes and  so that an element with product would disappear from the new coordinates.

.    

The system's inverse transformations are related to the inverse matrix. However, the matrix in the orthogonal transpositional transformation coincides with the inverse matrix, therefore, the relation of the old and new coordinates is expressed by the formulae:

For   and   location, let us plot a unit vector  on the new x-axis (See Fig. 4). Its projections (coordinates) onto the old axes equal: , where -s an and axes rotation angle.

Fig. 4 Rotation of coordinate axes by angle .

Unit vector that determines the direction of a new x-axis equals:. By analogy, the unit vector that determines the direction of a new y-axis, is calculated as follows:

.

, coefficients possess the followinng characteristics:

The latter means that the axes rotation has been performed with the scale unaffected.

Thus, to reduce a quadric form to the canonical form, one should convert to the new coordinates so that they would lack the middle element: .  The left part can be rewritten in the matrix form:.  The right part will be written as follows:

From that issues:

Since for orthogonal matrices, then having multiplied both parts by

, we receive:

Let us multiply the right and left matrices:

Comparing the corresponding matrix elements we get:

Therefore, the equation system:

has roots and for  , and roots and for .

Thus, for calculating и it is necessary to solve the following equation system:

Unit vectors specify new directions for axes and .

For the system to have a nontrivial solution, it is necessary for its determinant to equal zero:

Therefore,  will be found from equation:

As long as determinant , consequently and are real numbers. They are called characteristic numbers and are coefficients for the unknowns, after reducing it to its canonical form. If and do not equal zero and have the same signs, the quadric form is called elliptical.

Let , then. Having substituted and into the equation system, we obtain two directions:and of the new coordinate axes., where that quadric form assumes a canonical form.  Since the change of for and for  in the new frame of reference does not alter the quadric form , consequently, the ellipse is symmetric to and coordinate axes, i.e. coordinate axes pass through the main directions of the ellipse (Fig. 5).

Fig. 5. Axes rotation that brings the quadric form to its canonical form.

For we obtain: and . When these values have been substituted into the system, it turns into an identity, and any major directions comply with it, the ellipse degenerated into the circle.

Thus, reducing the quadric form to its canonical form amounts to the solution of a characteristic equation, i.e. calculation of eigenvalues and eigenvectors, which perform the coordinate axes rotation in the direction of the major and minor ellipse semiaxes, which means, toward the maximum of general dispersion of and .

Performing the linear transformation discussed above in the -dimensional space, we will receive elliptical hyper-spaces, the main axes of those coinciding with the major directions after reducing them to the canonical form. All the major directions are mutually perpendicular. Every eignevalue has a corresponding eigenvector that coincides with the -th major direction.

The discussed procedure can be easily generalized in case of variables. In this case, probability density will be written as follows:

Here is a covariance matrix. After normalization of:

, we receive:

Let , where is a correlation coefficient, and  is a correlation matrix. By analogy, let

, therefore:

,

where is a quadric form.

Consequently, for reduction of the quadric form to its canonical form amounts to a characteristic equation:.

Thus, the columns of matrix composed of eigenvectors determine weights of characteristics in the factors, while the lines yield decomposition of each separate characteristic into factors:

.

Matrix specifies a linear transformation of converting to a new frame of reference that coincides with the major directions.

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