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Home / Issues / № 2, 2013

Materials of the conference "EDUCATION AND SCIENCE WITHOUT BORDERS"

OPTIMUM MEAN-SQUARE ESTIMATE OF THE COEFFICIENTS IN THE MODEL OF SELF-ORGANIZATION OF THE LABOR MARKET
Semenchin E.A., Nevecherya A.P.

Mathematical model of self-organization of the labor market is of the form [1, 2]:

(1)

(2)

(3)

, , (4)

, . (5)

Here - the total number of workers employed in the i-th branch at time t; - the number of potential workers who can be hired to work in the i-th branch, and which at time t are unemployed; , - given numbers; - probability that at time t unemployed i-th industry can find a job in the j-th branch; - probability of dismissal to the employee at the time t i-th branch.

In frames of the model (1) - (5) lets form the problem: given , , , , , for all determine, , , .

The aim of this work - to explore the specified task at … .

The finite-difference analogue of task (1) - (5) is:

(6)

Obviously, the system (4) (or, equivalently, (5)) contains the unknowns , , wherein - the number of sectors of model under study and equations, . As far as , , it is always underdetermined. We extend definition of it.

We consider two cases: n - odd, n - is even.

Let n - odd. Suppose that in the interval stochastic parameters , of the system (6) are constant, and in this interval instead of (6), we consider the extended system:

(7)

(as far as assumption is: n - odd, that - natural.)

The meanings of elements , , , at each , in the right side (7) we find the meanings , , , using the interpolation formula

, , , , ; (8)

meanings of elements , , , the left side of (7) we find, using formulas

, (9)

, ,, .

It is easy to see that the system (7) is fully defined: contains equations and the same number of unknowns. It can be represented in vector-matrix form:

, (10)

where

, (11)

, , (12)

T - transpose operation.

As far as the meanings , , , , , have been found by interpolation, they obviously contain computational errors that can be considered as random. To eliminate these errors we use the method of optimal Kalman-Bucy filter [3]. Consider the system (10) as:

, (13)

where - the vector of errors. We assume that v(t) is a white Gaussian noise. So, according to the method of Kalman-Bucy mean square estimation vector , , satisfies the condition:

, (14)

where ; ; ; E - expectation (mean meaning). The initial approximations , can be found by using the Tikhonov regularization method [4]:

where , , I - the identity matrix. Following vector approximations are found from:

.

Let n - even. So is not natural. In this case, problem 2 can be solved in two stages:

1. Choose one of these n branches, for example, r-th, , and split it into two noncrossing branches (i.e., do not contain common elements - workers) the fictitious subbranches and to provide the equality:

,. (15)

We get the branch, - is odd. We find estimates of the probabilities , , , according to the latter scheme.

2. We unite the branches , in one r-th, using the ratios (which can be easily obtained by using the formula of total probability and the properties of conditional probabilities):

, ,

, ,

, , . (16)



References:
1. Semenchin E.A., Zaitseva I.V. Mathematical model of self-organization of the labor market for the two branches of the economy / / Economics and Mathematical Methods. - 2004. - T. 40. B. 4. - S. 137-140.

2. Semenchin E.A., Zaitseva I.V. Mathematical model of self-organization of the labor market for a number of branches / / Economics and Mathematical Methods. - 2007. - T. 43. B. 1. - S. 133-136.

3. Sizikov V.S. Mathematical methods of measurement results processing. - St. Petersburg: University of Technology, 2001, 240 p.

4. Tikhonov A.N., Arsenin V.Y. Methods for solving wrong-posed problems. - Moscow: Nauka. Home Edition Physical and Mathematical Literature, 1979. - 142 p.



Bibliographic reference

Semenchin E.A., Nevecherya A.P. OPTIMUM MEAN-SQUARE ESTIMATE OF THE COEFFICIENTS IN THE MODEL OF SELF-ORGANIZATION OF THE LABOR MARKET. International Journal Of Applied And Fundamental Research. – 2013. – № 2 –
URL: www.science-sd.com/455-24430 (29.03.2024).