For an entirely different 'structure function', also introduced by Kolmogorov, see Kolmogorov's turbulence theory.
Statistical function
In 1973, Andrey Kolmogorov proposed a non-probabilistic approach to statistics and model selection. Let each datum be a finite binary string and a model be a finite set of binary strings. Consider model classes consisting of models of given maximal Kolmogorov complexity.
The Kolmogorov structure function of an individual data string expresses the relation between the complexity level constraint on a model class and the least log-cardinality of a model in the class containing the data. The structure function determines all stochastic properties of the individual data string: for every constrained model class it determines the individual best-fitting model in the class irrespective of whether the true model is in the model class considered or not. In the classical case we talk about a set of data with a probability distribution, and the properties are those of the expectations. In contrast, here we deal with individual data strings and the properties of the individual string focused on. In this setting, a property holds with certainty rather than with high probability as in the classical case. The Kolmogorov structure function precisely quantifies the goodness-of-fit of an individual model with respect to individual data.
The Kolmogorov structure function is used in the algorithmic information theory, also known as the theory of Kolmogorov complexity, for describing the structure of a string by use of models of increasing complexity.
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classes consisting of models of given maximal Kolmogorov complexity. The Kolmogorovstructurefunction of an individual data string expresses the relation...
described by Kolmogorov's turbulence law KolmogorovstructurefunctionKolmogorov–Uspenskii machine model Kolmogorov's zero–one law Kolmogorov–Zurbenko filter...
information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest...
mathematics, a topological space X is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of X, at least...
statistic, although it is restricted to linear estimators. The Kolmogorovstructurefunction deals with individual finite data; the related notion there...
mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited...
between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular...
distributions going back to the 1920s. In a 1932 paper, Kolmogorov derived a characteristic function for random variables associated with Lévy processes....
{F}}_{n}-F\|_{\infty }>z{\Big )}\leq 2e^{-2z^{2}}.} In fact, Kolmogorov has shown that if the cumulative distribution function F is continuous, then the expression n ‖ F...
the "Kolmogorov −5/3 spectrum" is generally observed in turbulence. However, for high order structurefunctions, the difference with the Kolmogorov scaling...
A trophic function was first introduced in the differential equations of the Kolmogorov predator–prey model. It generalizes the linear case of predator–prey...
The Kolmogorov Medal is a prize awarded to distinguished researchers with life-long contributions to one of the fields initiated by Andrey Kolmogorov. The...
likelihood function (often simply called the likelihood) is the joint probability mass (or probability density) of observed data viewed as a function of the...
estimates depending on the regularity of the target function and of the activation function. The Kolmogorov–Arnold representation theorem is similar in spirit...
characteristic function of a subset of the natural numbers) is random or not by invoking a notion of randomness for finite objects. Kolmogorov complexity...
result is equivalent to a Fourier transform of Kolmogorov's 1941 result for the turbulent structurefunction. The pressure fluctuations in a turbulent flow...
finitary functions on the natural numbers is uncountable so most are not computable. Concrete examples of such functions are Busy beaver, Kolmogorov complexity...
possible functions (in the set-theoretic sense of "function") are Kolmogorov random, and hence the NFL theorems apply to a set of functions almost all...
completely arbitrary function can be expanded in Fourier series, even if its Fourier coefficients are well-defined. For example, Kolmogorov (1922) constructed...
regular spaces and Tychonoff spaces are related through the notion of Kolmogorov equivalence. A topological space is Tychonoff if and only if it's both...
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