Автор неизвестен - Бионика интелекта информация язык интеллект№ 3 (77) 2011научно-технический журналоснован в октябре 1967 г - страница 35

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C

21

C

22

C

2B

CN1 Ci\

where columns correspond to cases and rows correspond to components (N components and B cases).

We propose an algorithm that reveals the system relationships of above mentioned type, on the base of transition functions Fj , j = 1,2,...,N and the observa­tion table MM .

This algorithm allows us to determine between- and intra-components relationships, which are as close as possible relationships that form matrix (2) in a certain mean. Assume, that a number K and transition func­tions are given. In this case, for initial will be (A1(0), A2 (0) , ..., AN (0))T є жN and the given sets L1 (u, s), L2 (u, s) , ..., LN (u,s) , u є {-,0, +}, s є {-,0, +} , which makes it possible to calculate the matrix (1) or the minor (2). Let

P

1

21

r12

1

'1N

'2 N

1

Л

be a Pearson correlation matrix between rows of minor (2). Now for the matrix MM let us calculate Pearson cor­relation matrix of its rows

'   1       p12         p1N Л p =    p21       1      —    P 2N

v pN1    p N 2          1 J

Let us introduce the measure of distance between correlation matrices P and P

N-1 N 2

D(P, P) = Z   Z (rj - Pj f. (7)

j=1 j=i+1

Consider the task of minimization D(P, P) by all possible vectors of initial states (A1 (0), A2(0), —, AN (0))T є жN and all allowable sets Lj (s,u), s,u є W for all j

D(P, P) ь-» min

by all initial states and by all allowable Lj (s, u).

Now, we can explain the mean of the stated task. Suppose, that a process in some real system is described by cyclical trajectory (2). One cannot the possibility to observe the dynamic of this trajectory, i. e. a full length cycle. The observation are taken from the system at ran­dom instants of time t from s to s + T -1 with equal probability. When an observation is fixed, the column (A1 (t), A2(t), , AN(t))T from (2) is attached to table of observations. In other words, the columns of table of observations MM are obtained from (2) by an equiproba-ble choice of columns. The stated task means a search of

such relationships between components, that the minor (2) is to be as close as possible to the table of observa­tions in the mean of the measure (7).

The following theorem shows, that this task is well-grounded in some mean.

If the table of observations MM is obtained from the minor (2) by equiprobable choice of columns, then the correlation matrix of the observations table P converg­es to the correlation matrix of minor P (in probability)

lim pj =   , і

B->oo

1,2, —,N, j = 1,2, —.,N.

Proof. Since the Pearson coefficient is a pairwise characteristics (between two variables), it is enough to prove the theorem for the case N =2. Let

4,1

VX2,1

4,2 x2,2

X1,T X2,T

be the minor (2), where we use the notation X ,j instead A (j + s -1) for convenience.Let хг

1

1,2

T Z j=ix j

are to be the means of rows. If a sample variance of both rows is not 0, the Pearson correlation coefficient (be­tween rows) is equal to

Z (x1, j

j

x0(x2,j - x2)

R0

Now we can present the observation matrix MM as a frequency table

x1,1

x1,2

x1,T

x2,1

x2,2 .

x2,T

ПІ1

ПІ2 ■

піт

where mj is a frequency of the column (x1y, x2y )T , which is taken from the minor (2) and placed into the observation matrix MM.

The means of the rows of the observation matrix MM

are

B =1

1 T

в ZmJxi,j Bj=1

T m,

j=1 B

xj,j, j

1,2.

m, 1

According to the Bernoulli theorem [10],     

B T

(in probability, when B -co) for all j = 1,2, , T. From this it follows Q x, (in probability, B — <x>),

j=1,2.

The Pearson correlation coefficient (between rows) of the observation matrix MM equals

Z X7 (x1,k

=1 B

x1)(x2,k - x2)

Z X7 (x1,k

=1 B

x1)2

Z X7 (x2,k

=1 B

x2)2

Then,

lim Rb

B—x

lim

B—x

ZT=1 mt(x1,k - C1^(x2,k - C2) B

Z Г mk (x C )2 ZT mk (x C )2 Zk=1~B~(x1,k - C1) \Zk=1~B(x2,k - C2)

lim

B—x     t mk

Z 1k=\mt(x1,k - x1")(x2,k - x2) B

B (x1,k - x1)2JZ T=1m (x2,k - x2))

T    1 - -

Zk=1^p (x1,k - x1)(x2,k - x2 )

■x2)2

2. Results

By using digital photography, we captured separate walking phases (which by no means form a complete cy­cle) of a two year-old male Emys orbicularis along the bottom of an enameled pool located in a terrarium. In the case of using a stressor, a turtle was kept lying on its back for two minutes. Then another turtle was imme­diately put into the pool in a back up position (without using a stressor).Using the images, we calculated the ra­tio of the following distances to the length of the turtle's shell: from the tail head to the ankle of each of the four legs the right foreleg (rf), the right hand leg (rb), the left foreleg (lf), the left hand leg (lb). We considered such a distance to reflect the degree of a leg straightening.

Using DMDS, the structures of relationships and sets of states of the four-component system were ob­tained for all four parameters rf, rb, lf, lb (the compo­nents of the system are the turtle's legs). The results correspond to the observations both for the stressed and non-stressed cases in the outmost degree (in the con­ventional sense for DMDS).

For modeling with DMDS, we used the approach based of the principles of the von Liebig law and pro­posed that K = 3 (three levels of components' values). For these two cases, the system trajectories which showed the sequence of walking phases (the cycle including dif­ferent combinations of contracting and straightening of each of four legs) were constructed. On these trajecto­ries, the system factors analyzed the body's stability for the stressed and the non-stressed turtles walk.

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