The article discusses concepts such 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 position of decision making under conditions of uncertainty.
It is worth starting with the definition of sustainability. It characterizes the ability of the system to return to an equilibrium state at the end of the disturbance, which violated the previously formed equilibrium.
It is important to note that his opponent - an unstable system - is constantly moving away from his equilibrium state (oscillates around him) with a returning amplitude.
Sustainability criteria: definition, types
This is a set of rules that allows 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 (compilation of a specific characteristic equation of algebraic expressions using special rules that characterize the stability of self-propelled guns);
- frequency (the object of study - frequency characteristics).
Hurwitz stability criterion from an algebraic point of view
It is an algebraic criterion, implying the consideration of a certain characteristic equation in the form of a standard form:
A (p) = aᵥpᵛ + aᵥ₋₁pᵛ¯¹ + ... + a₁p + a₀ = 0 .
Through its coefficients, the Hurwitz matrix is formed.
Hurwitz matrix rule
In the top-down direction, all coefficients of the corresponding characteristic equation are written in order, starting from aᵥ₋₁ to a0. In all columns, downward from the main diagonal are the coefficients of increasing degrees of the operator p, then upward - decreasing. Missing elements are replaced by zeros.
It is generally accepted that the system is stable when all available diagonal minors of the matrix under consideration are positive. If the main determinant is zero, then we can talk about finding it on the stability boundary, and a = 0. If the remaining conditions are met, the system under consideration is located on the border of a new aperiodic stability (the penultimate minor is equal to zero). With a positive value of the remaining minors, it is already on the border of vibrational stability.
Decision-making in a situation of uncertainty: the criteria of Wald, Hurwitz, Savage
They are the criteria for choosing the most appropriate strategy variation. The Savage criterion (Hurwitz, Walda) is applied in a situation where there are uncertain a priori probabilities of the states of nature. Their basis is the analysis of a risk matrix or a 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. He acts as a criterion of extreme pessimism (cautious observer). This criterion can be formed for both pure and mixed strategies.
He got his name on the basis of the assumption of the statistician that nature can realize states in which the gain is equal to the smallest value.
This criterion is identical to the pessimistic one, which is used in solving matrix games, most often in pure strategies. So, you must first select from each row the minimum value of the element. Then the decision-maker strategy is selected, which corresponds to the maximum element among the already selected minimum.
The options chosen by the criterion under consideration are devoid of risk, since the decision maker does not face a worse result than the one that acts as a guide.
So, according to Wald’s criterion, the purest strategy is recognized as the most acceptable, since it guarantees the maximum marginal gain in the worst conditions.
Next, consider the Savage criterion. Here, when choosing the first of the available solutions in practice, as a rule, they stop at one that will lead to minimal consequences if the choice is still erroneous.
According to this principle, every solution is characterized by a certain amount of additional losses arising during its implementation, compared with the correct one with the existing state of nature. Obviously, the correct solution cannot suffer additional losses, as a result of which their value is equal to zero. So, in the role of the most appropriate strategy is adopted, the magnitude of the losses in which is minimal in the worst case scenario.
The criterion of pessimism-optimism
So differently called the Hurwitz criterion. In the process of choosing a solution, in assessing the current situation, instead of two extremes, they stick to the so-called intermediate position, which takes into account the probability of both favorable and worst behavior of nature.
This compromise option was proposed by Hurwitz. According to him, for any solution you will need to establish a linear combination of min and max, then choose a strategy that corresponds to their largest value.
When is the application of the criterion justified?
It is advisable to use the Hurwitz criterion in a situation characterized by the following features:
- There is a need to take into account the worst of the options.
- Lack of knowledge regarding probabilities of state of nature.
- Assume some risk.
- A fairly small number of solutions are implemented.
Conclusion
In conclusion, it will be useful to recall that the article examined the criteria of Hurwitz, Savage and Wald. The Hurwitz criterion is described in detail from various points of view.