Hurwitz criterion.

The article deals with concepts such as the Hurwitz criterion, Savage and Wald.The focus is mainly on the first.Hurwitz criterion is described in detail both from the algebraic point of view and from the point of decision making under uncertainty.

should start with the definition of sustainability.It describes the ability of the system to return to equilibrium at the end of the disturbance which disrupted preformed balance.

important to note that his opponent - unstable system - permanently removed from its equilibrium state (oscillates around it) with the returning amplitude.

Sustainability criteria: definition, types

This set of rules that allow us to judge the existing signs of the roots of the characteristic equation without searching for its decision.And the latter, in turn, provide an opportunity to judge the sustainability of a particular system.

As a rule, they are:

  • algebraic (drawing on a specific characteristic equation algebraic expressions with the use of special rules that characterize the stability of ACS);
  • frequency (object of study - the frequency characteristics).

Hurwitz stability criterion from the algebraic point of view

them to perform algebraic criterion, implying the consideration of a particular characteristic equation in the form of a standard form:

A (p) = aᵥpᵛ + aᵥ₋₁pᵛ¯¹ + ... + a₁p + a₀ = 0.

Through its coefficients form a matrix Hurwitz.

Rule compilation matrix Hurwitz

from top to bottom in the order written out all the coefficients of the corresponding characteristic equation, from aᵥ₋₁ to a0.All columns below the main diagonal ratios indicate increasing degrees of operator p, then up - decreasing.The missing elements are replaced with zeros.

accepted that the system is stable when all the diagonal minors of the matrix are positive.If the main determinant is zero, then we can talk about finding her on the stability boundary, and aᵥ = 0.In the case of compliance with the other conditions the system in question is located on the border of the new aperiodic stability (penultimate Minor is set to zero).With positive values ​​remain minors - on the border already vibrational stability.

Decision making under uncertainty: Wald test, Hurwitz Savage

They are the criteria for selecting the most appropriate strategy variations.Criterion Savage (Hurwitz, Wald) is applied in a situation where we have the a priori probabilities of uncertain states of nature.Their basis - Analysis of the risk matrix or payoff matrix.In the case of the unknown probability distribution of future states of all available information is limited to list its options.

So, we should start with the maximin criterion Wald.It is a criterion of extreme pessimism (careful observer).This criterion can be formed for the net, and mixed strategies.

It got its name on the basis of assumptions regarding the extras that can realize the nature of the state in which the value of winning equated to the lowest value.

This test is identical to the pessimistic, which is used in the course of solving matrix games, often in pure strategies.Thus, first select each row of the minimum value of the element.Then it released the strategy decision-makers, which is the maximum element among the already selected minimum.

selected by the criterion of the options under consideration without risk, as the decision maker faces no worse result than the one that serves a guide.

So the most suitable, according to Wald test, recognized net strategy, as it is in the worst conditions ensures a maximum limit of winnings.

Next Savage criterion worth considering.Here, in the selection of the 1st of the available solutions in practice they tend to stop at this, which will lead to minimal consequences if the choice would still be wrong.

According to this principle, every solution is characterized by a certain quantity of additional losses that arise in the course of its implementation, in comparison with the best available in the state of nature.It is obvious that a correct decision can not bear the additional losses due to which their value is nil.So, as the most appropriate strategy is adopted, the amount of loss which is minimal at worst set of circumstances.

criterion pessimism-optimism

so differently called Hurwitz criterion.In the process of selection decisions, the evaluation of the situation instead of two extremes adhere to the so-called intermediate position, which takes into account the likelihood of both favorable and the worst behavior of nature.

This Hurwitz suggested a compromise.According to him, for any solution will need to install a linear combination of the min and max, then choose a strategy that fits their highest value.

When justified by the application of the reporting criteria?

Use Hurwitz criterion advisable in a situation characterized by the following features:

  1. There is a need to take into account the worst option.
  2. Lack of knowledge about the probabilities of states of nature.
  3. Assume some risk.
  4. implemented sufficiently small number of decisions.

Conclusion Finally

is worth mentioning that in the article examined the Hurwitz criterion, Savage and Wald.Hurwitz criterion is described in detail from different perspectives.