Identification of typical GDV-grams on the basis of analysis of their fractal dynamics

Identification of typical GDV-grams on the basis of analysis of their fractal dynamics

4. Evaluation of parameters of the model of power spectral envelope.
The simplest model of such class is the following:
M (n) = k .(1)
where n is the component's number in the power spectrum proportional to its frequency,
k and ß - energy and information parameters subject to evaluation.

Let us take into consideration a simple formula [1] linking the fractal degree d and the fractal index ß:
d = (2)
It is also advisable to remember that digitization frequency fd of the process and frequency of the corresponding n-spectral constituent fn are connected by the following relation:
fn = . fd , (3)
where m is the number of components in A0 , B0 vector. In our case we have m=128, this is why if fd =128 is chosen, fn = n.

For the estimation of k and ß parameters it is necessary to solve the task of non-linear regression which can be easily reduced to the linear at the first stage. Let us demonstrate the procedure. As usual, comparing the object (here it is the corresponding power spectrum) to the model we receive the system of equations:

CM n = k . , n = 0..63, (4)

where CM n is the n-component of CM vector. Applying logarithm to the both parts of (4) we get:
ln(k) - ß. ln(n) = ln(CM n) (5)

XA0 = , U =, VA0 = ,
then in the vector-matrix form we receive the following systems of equations:
U. XA0 = VA0 , i = 0..7 , (6)

for which we would have the next solutions:

For the computational convenience let us assume:

i.e. VA0 is a matrix of 28 x 8 in size, then the decision matrix is given by :

Having transposed it (upper index T) we receive matrix, where the column with zero number includes the evaluation logarithms of parameter k:

<0> = , (9)

and, accordingly, the column with number one is the evaluation vector of parameter ß:

= (10)

Taking into account that in (9) by definition each component is

it is not difficult to get the evaluation vector:
K0 = (12)

Thus, for A0 matrix all the estimates of parameters of model (1) are received in the form of vectors (10) and (12). In the same way, using the relations (6)-(12) it is possible to get the estimates of the required parameters k and b for the rest matrixes A1…A7, B0…B7. Arranging the received evaluation vectors into the corresponding matrixes, let us find the next four matrixes of evaluation of the parameters:
ßA = , (13)

KA = , (14)

ßB = , (15)

KB = (16)

It is worth recalling that the family of matrixes A (A0…A7) is generated by the data matrix F1, and the family B (B0…B7) - by the data matrix F2. The four matrixes found (13)-(16) contain valuable information about the fractal dynamics of the processes studied - about genesis of GDV-images of fingers. This is why these matrixes may be called the information matrix.

5. Conversions of the information matrixes
The goal of these conversions is the necessity to find some integral estimates which could serve as the most informative features of the investigated GDV-image of finger and could help to solve the problem of identification of this image's type. Such estimates have been found. There are three of them for each information matrix (13) - (16):

integral average,
normalized standard deviation,
maximal singular number.

Let us study these estimates in more detail in terms of ßA matrix. If we determine a mean value for each column of this matrix:

we would receive a vector of means for all the columns:
ßAm = (18)

For this vector (18), in its turn, it is possible to find a mean value, which can be called "integral mean":
ßAint.m = mean (ßAm), (19)

where mean is the operator defining the estimate of mean value of ßAcp. vector components. The normalized standard deviation is determined as the ratio of standard deviation to integral mean:

where the operator stdev determines the standard deviation of ßAm vector. Any information matrix may be presented in terms of ßA matrix as:

where Q and W are the unitary matrixes, and matrix is diagonal, moreover its diagonal elements are non-negative square roots of the characteristic values of ßA. matrix, and, hence, are defined uniquely. The columns of Q matrix are the characteristic values of ßA. matrix, and the columns of W matrix - the characteristic values of . ßA matrix. Both systems of characteristic vectors are organized in accordance with the distribution of the characteristic values (for instance, in ascendancy). Diagonal elements of S matrix - sii are called singular numbers of ßA matrix. The columns of Q and W matrixes are named, respectively, left and right singular vectors and the decomposition (21) is called singular decomposition. The latter possesses [6] a remarkable property: any matrix is conditioned ideally relative to the problem of calculating singular numbers, however, at that it can be poorly conditioned relative to the problem of evaluation its characteristic values. The vector of diagonal elements (singular numbers) of ßA matrix in the MathCad pack is determined through the operator svds:
ZßA = svds (ßA) (22)

and, hence, the maximal value of singular number is found in the following way:
ZßAmax = max (ZßA) (23)

Drawing some conclusions we may say that as a result of the analysis of fractal dynamics, compact integral estimates brought together in Table 1 are obtained.

Table 1. Integral fractal estimates of the GDV-grams.

EnergyEstimates (F1)	InformationEstimates (F1)	Informationestimates (F2)	Energyestimates (F2)
KAint.m	ßAint.m	ßBint.m	Kßint.m

ZKAmax	ZßAmax	ZßBmax	ZKBmax

The estimates placed in the 1st and the 3rd columns, as shown by the practice of their application, represent the GDV-image in the best way when the problem of identification of its type according to the classification of types proposed by K.G.Korotkov is being solved. Therefore, therein after the following evaluation vector is used:

O I = [ßBint.m , , ZßBmax , KAint.m , , ZKAmax ] (24)

This fact states that from the total number of combinations of 12 to 6, equal to 924, one optimal combination is chosen, meaning minimum of errors of the image type identification.

The method of identification of the images type»

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