Hurwitz criterion. Stability criteria of Wald, Hurwitz, Savage

2024 Author: Henry Conors | [email protected]. Last modified: 2024-02-12 02:55

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

It's worth starting with a definition of sustainability. It characterizes the ability of the system to return to the equilibrium state after the end of the perturbation, which violated the previously formed equilibrium.

It is important to note that its opponent - an unstable system - is constantly moving away from its equilibrium state (oscillates around it) with a returning amplitude.

Sustainability criteria: definition, types

This is a set of rules that allow you to judge the existing signs of the roots of the characteristic equation without looking for its solution. And the latter, in turn, provide an opportunity to judge the stability of a particular system.

As a rule, they are:

• algebraic (drawing up algebraic expressions according to a specific characteristic equation using specialrules that characterize the stability of the ACS);
• frequency (object of study - frequency characteristics).

Hurwitz stability criterion from an algebraic point of view

It is an algebraic criterion, which implies consideration of a certain characteristic equation in the form of a standard form:

A(p)=aᵥpᵛ+aᵥ₋₁pᵛ¯¹+…+a₁p+a₀=0.

Using its coefficients, the Hurwitz matrix is formed.

The rule for compiling the Hurwitz matrix

In the direction from top to bottom, all the coefficients of the corresponding characteristic equation are written out in order, starting from aᵥ₋₁ to a0. In all columns down from the main diagonal indicate the coefficients of increasing powers of the operator p, then up - decreasing. Missing elements are replaced with zeros.

It is generally accepted that the system is stable when all available diagonal minors of the considered matrix are positive. If the main determinant is equal to zero, then we can talk about its being on the stability boundary, and aᵥ=0. If the other conditions are met, the system under consideration is located on the border of a new aperiodic stability (the penultimate minor is equated to zero). With a positive value of the remaining minors - on the border of already oscillatory stability.

Decision making in a situation of uncertainty: criteria of Wald, Hurwitz, Savage

They are the criteria for choosing the most appropriate variation of the strategy. The Savage (Hurwitz, Wald) criterion is used in situations where there are uncertain a priori probabilities of the states of nature. Their basis is the analysis of the risk matrix or payment matrix. If the probability distribution of future states is unknown, all available information is reduced to a list of its possible options.

So, it's worth starting with Wald's maximin criterion. It acts as a criterion for extreme pessimism (cautious observer). This criterion can be formed for both pure and mixed strategies.

It got its name on the basis of the statistician's assumption that nature can realize states in which the amount of gain is equated to the smallest value.

This criterion is identical to the pessimistic one, which is used in the course of solving matrix games, most often in pure strategies. So, first you need to select the minimum value of the element from each row. Then the decision maker strategy is selected, which corresponds to the maximum element among the already selected minimum ones.

The options selected by the criterion under consideration are risk-free, since the decision maker does not face a worse result than the one that acts as a guideline.

So, according to the Wald criterion, the pure strategy is recognized as the most acceptable one, since it guarantees the maximum maximum gain in the worst conditions.

Next, consider Savage's criterion. Here, when choosing one of the available solutions, in practice, as a rule, they stop at the one that will lead to minimal consequences in the event thatif the choice still turns out to be wrong.

According to this principle, any decision is characterized by a certain amount of additional losses arising in the course of its implementation, compared with the correct one in the existing state of nature. Obviously, the correct solution cannot incur additional losses, which is why their value is equated to zero. Thus, the most expedient strategy is the one in which the amount of losses is minimal under the worst set of circumstances.

Criterion of pessimism-optimism

This is another name for the Hurwitz criterion. In the process of choosing a solution, in the course of assessing the current situation, instead of two extremes, they adhere to the so-called intermediate position, which takes into account the likelihood of both favorable and worst behavior of nature.

This compromise was proposed by Hurwitz. According to him, for any solution, you need to set a linear combination of min and max, then choose a strategy that corresponds to their largest value.

When is the criterion in question justified?

It is advisable to use the Hurwitz criterion in a situation characterized by the following features:

1. There is a need to take the worst case into account.
2. Lack of knowledge regarding the probabilities of the states of nature.
3. Let's take some risk.
4. A fairly small number of solutions are implemented.

Conclusion

Finally, it would be useful to recall that the articleHurwitz, Savage and Wald criteria. The Hurwitz criterion is described in detail from various points of view.