Principal components analysis

The principal axis transformation (HAT ) is a method of linear algebra to represent and classify equations of so-called hyper- surfaces of the second order in a normal form can. In theoretical physics principal axis transformations are mainly used for the diagonalization of a matrix, as described in more detail in Section.

As hypersurfaces geometrical objects are called, whose dimension is exactly one less than the dimension of the space in which they are presented. Thus, a plane whose mathematical dimension 2, in three-dimensional ( Euclidean ) space of our intuition a hypersurface, as a curve with dimension 1 in a two-dimensional space ( for example, a line on a sheet of paper ). A hypersurface second-order or second order is a hyper- surface, the variables appear in the equation determining a maximum in the second power. In two-dimensional space, the second-order hyper- surfaces correspond to the conics.

Procedure

The general equation of a hypersurface in Euclidean spaces is second order

Theoretically, the main axis transformations in unitary spaces are carried out with Hermitian matrices over the field of complex numbers, but for simplicity and for reasons of clarity is not below to the analogous processes received.

A graphical representation of hypersurfaces is useful only up to the third dimension. In order to classify the hypersurfaces that are represented by the above equations, solely based on their defining equations, the equations must be converted into one of the following three normal forms:

The conversion into normal form is performed by moving and rotating the coordinate axes, that is, a transformation of the coordinate system.

To simplify the general equations they are often first rewritten with the help of the symmetric matrix into a matrix equation:

Multiplied out is the previous equation in sum notation:

Depending on the values ​​of the elements of the matrix A, the co-ordinates that is to occur in a square, as well as easier in mutually mixed form. Since A is a symmetric matrix, by definition, the following applies: each coordinate pair occurs in the sum of the mixed terms so only once.

Based on the elements of A certain statements about the outcome of classification may at this point already be made: Are these such that A does not have n full rank, then the equation can be attributed only to the normal forms ( 2) or (3) are, however, is equal to n, the present equation is a variation of equation (1).

Rotation

First, the rotation of the object to the ordinary (target) coordinate system, which is described by the mixed terms with, must be eliminated, so all be for 0. The matrix must be converted to a suitable transformation into a diagonal matrix, that is, all values ​​that do not lie on the main diagonal of the new matrix must be equal to 0.

Since a symmetric matrix, there exists a full set of eigenvectors, and thus an orthogonal matrix whose columns are the normalized eigenvectors of. It shall be from the spectral theorem:

The matrix is a diagonal matrix with the eigenvalues ​​of on the main diagonal.

First, then the eigenvalues ​​of and an associated orthonormal basis must be determined from eigenvectors and from these matrices and assembled. If in the determination equation no simple powers of xi exist, so the last sum in the ausmultiplizierten matrix equation (and thus the displacement of the object ) is equal to 0, the matrix is not required and the calculation of the eigenvectors can be omitted.

If you place a in the above matrix equation and defined and, one obtains

Thus, the (new) coordinates are found only in a simple and square, but not in mixed form, the hypersurface was filmed.

Shift

Now, one possible parallel displacement of the object to the origin of the coordinate system must be eliminated. For this, the equation of the rotated hypersurface is multiplied out:

Subsequently, grouped according to the basis vectors and brings the individual brackets by affine substitution (eg, completing the square ) in quadratic form. The brackets are the basis vectors zi of the new, shifted coordinate system:

Which is due to completing the square.

Is the so-called signum function that returns the sign of x.

For the case that the summand is, it is now normalized to 1, that is, the whole equation is divided by the amount of. This is now - attributed to one of the three normal forms mentioned above - depending on the rank of the matrix.

Classification

With reference to the normal form can then classify the hypersurface be made: Any quadric through the choice of a Cartesian coordinate system in two - or three-dimensional space to exactly one of said forms, are brought Euclidean their normal form. Allows you to any affine coordinate system, then let two quadrics belonging to the same type - with respect to an affine coordinate system chosen for each of the quadric - describe even by the same equation. This is called the affine normal form of the quadric and may in the Euclidean normal forms, which are mentioned in the following tables, select. Two different quadrics can be, however, generally give no respect to the same Cartesian or affine coordinate system on their respective Euclidean or affine normal form.

A simple consequence of this fact:

This conclusion is so far weaker than the statements of affine normal forms, as the different types of "empty" quadrics by any bijective mapping of the space are transferred as point sets with each other.

Application

Apart from the purely mathematical and geometrical meaning of the principal axis transformation for determining the type of hypersurfaces of the second order, it is used in many disciplines of theoretical physics and in computer science and geosciences.

In theoretical physics the main axis transform is used in classical mechanics for describing the kinematics rigid body: this can have a principal axis transformation of the inertia tensor, which indicates the moments of inertia of the body with respect to rotations about different axes, may - for example with a centrifugal - presence of inertia for disappearance are brought.

A product of inertia is a measure of the tendency of a rigid body, to vary its rotational axis. Of inertia are combined with the moments of inertia in inertia tensors, the inertia moments are on the main diagonal of the tensor, the products of inertia on the secondary diagonals. As shown above, the balanced inertia tensor can be placed on a diagonal form. The set by the principal axis transformation axes of the new, custom coordinate system is called the principal axes of inertia, the new coordinate system as the principal axis system. The diagonal elements of the transformed tensor are consistently called the principal moments of inertia.

Clearly fall after a main axis transformation with symmetrically -built bodies, the coordinate axes of the principal axis system with the symmetry axes of the rigid body together, so mixed terms vanish from the equation.

Also available in other branches of classical mechanics, the principal axes transformation is used, for example, in the strength of materials used to calculate the principal stresses acting on a body. Often applied principal axis transformations continue in relativistic mechanics to the base representation of space-time in four-dimensional Minkowski space.

In addition, the principal axis transformation is part of a Karhunen- Loeve transform, referred to in the image processing as the main component analysis in particular. Sometimes the terms are used synonymously, but both are not identical transformations.

Practically, the main component analysis is used as part of the principal component analysis to reduce the size of large data sets without any significant loss of data. Leveraging existing relationships between statistical variables by converting it into a new, linearly independent problem- adapted coordinate system can be reduced as much as possible. For example, the required number of signal channels can be reduced by this order of variance and the variance of the lowest channel, if necessary, be removed without loss of data from the relevant data set. As a result, the efficiency and result of a later analysis of the data is improved.

In the electronic image processing, the reduction of the data set size by principal component analysis, especially in remote sensing by satellite images and the associated scientific disciplines of geodesy, geography, cartography and climatology used. Here the quality of the satellite images can be improved significantly by suppressing the noise by means of principal component analysis.

In computer science, the principal axes transformation is mainly applied in pattern recognition and in neural net theory, ie the creation of artificial neural networks, a branch of artificial intelligence for data reduction.

Example

Consider the equation

With the main axis transform to this equation are now transferred to a normal shape and the type of the hyper area shown by the equation to be determined.

In matrix notation the equation is:

The rank of A is equal to 3, that is, the matrix has full rank, and discussed equation is a variation of equation (1). The eigenvalues ​​of A are with algebraic multiplicity 1 and algebraic multiplicity 2; the corresponding eigenvectors are:

Yet provide v1, v2 and v3 no orthonormal basis of the vector space, since only the eigenvectors corresponding to different eigenvalues ​​are orthogonal to each, but v2 and v3 are eigenvectors for the same eigenvalue. We therefore need the vectors by an appropriate method - for example, the Gram -Schmidt orthogonalization 's - first orthogonalize and then normalize their sums to 1. We then get:

Now the matrices Q and D from the eigenvectors and eigenvalues ​​of A are assembled:

It arises for the rotated hypersurface with grouping by the basis vectors x ', y ' and z ', and supplement square:

Or transformed into basis vectors x ', y ' and z ':

The equation thus describes an elliptical cone. In this specific case the two half axes of the cone base is (as described by the pre-factors and to ) is equal, so it is a circular cone.