Dr. Sergey Zagraevsky


Application of contemporary mathematical methods

of expert estimations to artists rating




Art-critics were interested in precise evaluation of art in all times. Many attempts of mathematical assessment of "quality", "artistic value", "social and humanitarian significance" of paintings, sculptures and other works of art were made.

This problem arose also before the makers of artists rating.

Before turning to the substantive part of our study, it should be stated: to date, neither art-critics, nor mathematics, nor any other scientific discipline has scientifically based and accurate time-tested methods of art works evaluation.

Numerous Western ratings are based on prices of works, but it is not fit for very chaotic Russian art market.

Assessment of professionalism in painting (sculpture, etc.) techniques has lost its universal scale in the twentieth century, and is inapplicable for such common modern forms of art, as abstractionism or conceptualism. The same applies to such indicators of art as construction of composition, nature of strokes, plasticity, etc.

As a result, evaluation of art is purely indicative.

If we need to assess creativity of an artist as a whole, as a phenomenon of art, the problem becomes even more complicated. Such indicators as the number of publications, exhibitions, catalogs, honors or awards, which can be measured precisely (at least theoretically), in modern conditions can serve as auxiliary information only.

Statistical methods of research, which primarily concern the analysis of public opinion polls, can help in assessing of the social significance of an artist and his works, but not their significance from the standpoint of art history. The main argument is well known: art is not politics, questions can not be solved by a majority vote there. Moreover – focusing exclusively on non-professional public opinion leads to a tremendous amount of speculations about "artists, loved by people". In addition, it is easy to fraud statistics.

Thus, creators of artists rating face the task, which can be solved neither by precise nor by statistical methods.

It is necessary to use mathematical-heuristic methods to solve this problem. A common characteristic of these methods is the use of mathematical tools of analysis of expert assessments in a particular area of expertise, not amenable to systematization by precise mathematical methods.

In our case, fine art is the area of knowledge, which requires a kind of systematization.

So, let us set the task: it is required to determine its category and level of each artist in accordance with “Statute of United Art Rating”, basing on available information about each artist. Rating methods may by realizable in the form of a computer program.




In order to understand, what mathematical methods can be used, we have to distract from the issues of art and consider the relatively recent time period – from the mid-seventies to early nineties.

At this time heuristic (expert) methods have been implemented in science, economy and technology with unprecedented intensity. We list only a few academic disciplines: psychology, meteorology, geology, management of economic systems, scheduling of air, rail and road transport, forecasting of development of scientific and technical potential of countries and regions... The list may be very long – almost no scientific discipline, no sector of economy has remained "aloof".

At that time so-called "automation of control systems" began – the introduction of automated control into state, financial, industrial and business institutions. As it is well known, at that time the computers had far fewer opportunities, and developers faced a serious shortage of so-called machine resources – performance, memory, disk capacity, etc. For comparison with our time let us say that the resources of even so-called "Mainframe", which took huge rooms, were many times lower than that of modern portable computers.

Such an acute shortage of computer resources did not allow solving the tasks of precise calculation of the parameters of any major economic or scientific problem. Existing mathematical methods theoretically allowed doing that, but in practice the calculation took from several hours to several days, and that made unrealistic flexible (online) recalculation with new parameters.

Speaking about precise calculation, we mean the methods of "linear programming", "dynamic programming", "branches and bounds", etc. All these methods require multiple conversion of extremely cumbersome matrices and digital arrays, with the increase of required computer resources in quadratic or even cubic dependence of the increase of dimension of the problem.

 During that period heuristic (expert) methods of calculation had the widest distribution in order to save computer resources. Formally speaking, the main objective of any heuristic method is to reduce the dimension and solving time of the problem by "cutting" deliberately unpromising steps. And definition of the perspectives of a step is made on the basis of formalized and pre-processed information from the experts on the subject.

We present the most well-known example – the chess program "Deep Blue", which beat Garry Kasparov. This program did not consider all possible moves in a given situation, all possible responses to them, then all possible next moves, all responses to them, etc. Such a "tree" would have taken many hours even for an ultra-modern computer. In fact, the program "Deep Blue" made an analysis of several thousands of games, which different Grand-masters played in a variety of times (from Lasker and Capablanca to Kasparov himself), and the computer made every move with regard to their experience. This is one of the methods of expert estimates.

Studies of 1980–90s showed that expert evaluation methods of large economic systems modeling give only 5–7% deviation from a theoretically possible optimal result, and the usage of computer and time resources is lower by several orders of magnitude. Similar results were obtained in all other disciplines, where large dimension and complexity of the tasks made their solving at the available computers by precise calculation methods impossible. The calculation error of expert method is negligible for most purposes. 5–7-percent range of output characteristics is comparable in real systems with the scatter of input data, i.e. in practice the quality of the final decision is not worse than the quality of a solution obtained by precise mathematical methods such as "dynamic programming".

In recent years, a reverse process began thanks to the unprecedented growth of performance of the computers: automatic control systems developers, not worrying about virtually inexhaustible machine resources, use expert methods less and less, and increasingly – precise ones, because highly qualified experts attracting is always associated with additional time and financial costs. But, as we have seen above at the examples of art market and the chess program "Deep Blue", there still are problems, intractable by precise mathematical methods.

Let us summarize our small historical review: there are modern mathematical-heuristic methods to produce high-quality solutions for applications, where precise mathematical methods for one or another reason can not be applied. As we shall see soon, the problem, which the developers of artists rating have faced, is not an exception.




The task of an art rating creation belongs to the class of problems of dynamic multi-criteria optimization, as is necessary to use multiple competing criterions and to take into account the dynamics of creativity of an artist in a wide time range.

It is required to construct a mathematical model of this problem. As we have seen, precise methods are not applicable to it.

 First of all, we consider the dynamic nature of the task of rating. The modern approach to automation of all areas of science and technology provides the transition from analog to digital (discrete) representation of the model. Digital representation is more versatile, easier for computer realization and, what is most important, – does not require cumbersome trigonometric formulas, differential equations or Fourier transforms for simulation.

Discretization of the dynamics of creativity of an artist for the rating problem solution lies in the allocation of significant creative periods. However, due to inability to collect objective information on all phases of creative work of each artist, selection of his work periods, coinciding with the most important periods of artistic development in the light of a country and an era, is appropriate. The question of periodization requires special expertise in the study of the method debugging.

Let us denote the number of each period as i, its time frame as T(i), a system "artist and his works" in the period T(i) as X(T(i)), and the whole set of artists, ratings objects, as CX.

Thus, we have to deal with the task of incremental simulation of a dynamical system X(T), whose aim is to determine a rating R – an artist's place in the set of CX.


R=F(X(Ò)), Õ belongs to ÑÕ


Prof. A.V.Efremov, supervisor of Ph.D. thesis of the author of this study, in the seventies developed so-called “Model-heuristic” method of solution of multiobjective optimization of problems of large dimension, which is applicable to many areas of science and economics.

Let us consider the nature of “Model-heuristic” method.

For optimal (or high quality) solution of incremental tasks it is enough to take the best (or high quality) solution at each step. Decision taking at each step is affected by a number of so-called partial criterions.

Let us denote the space of partial criterions in the form of an array K(j) and describe it for our rating task. Note that criterions are arranged in the array in random order, not in ascending or descending order of importance.

A sample list of individual criterions for each artist in each period i:

K (1): age;

K (2): availability of professional education;

K (3): solo exhibitions;

K (4): group exhibitions;

K (5): evaluation of art historians;

K (6): catalogs and brochures;

K (7): participation in major Russian and foreign auctions;

K (8): acquisition of works by leading museums;

K (9): acquisition of works through commercial galleries;

K (10): presence of distinguished (academic) titles;

K (11): creative innovation;

K (12): membership in Artists Union of the USSR;

K (13): membership in associations "OST", "Group of 13", etc.;

K (14): degree of subordination to art market conditions;

K (15): number of mentions in the press;

K (16): prices of works;

K (17): art level of works;

K (18): social significance of works,



Disadvantages of "artificial intelligence" in comparison with human decision-making are well known: inflexibility and no such thing as intuition. But there is a definite advantage: the model can consider such a wide range of criterions, which is simultaneously used by no expert. Thus, relative inflexibility of a model is compensated by calculating of a larger number of parameters.

 In the conditions of incomplete source data, partial criterial functions can be used not completely. But any "artificial intelligence" must have a potential to take into account all necessary criterions, which are used by experts in the decision, so the list of partial criterions is a subject to continuous expansion.

Furthermore, at each step i partial criterions are incorporated into the general criterion OK(i):



where V(j) – “weight” of the partial criterion, i.e. numeric expression of value of the criterion.


This is a universal form of the criterial function of “Model-heuristic” method, which was  developed by Prof. Efremov.

However, the rating task is a particular case of incremental optimization, as the highest OK(i) in one of the periods of the artist’s creativity is not a guarantee of a high rate of OK(i+1), i.e. in the next period.

Therefore it is required to calculate OK(i) for each period i, and then to re-apply “Model-heuristic” method for calculating of the final criterion function FK of the artist X:



where W(i) – weight of the general criterion for each time period i, i.e. the numeric expression of the value of a certain period of time.


It remains to divide the possible range of values of FK(X) into the levels and categories, which are listed in the "Statute of the Rating Center of Artists Trade Union”, and we obtain the desired R(X), ÕÎÑÕ.




The main problem of “Model-heuristic” method implementing for the problem of artists ratings is a nonlinearity of the function



where D – input data for each of the partial criterions of artist evaluation,


as well as of the function



which expresses the dependence of the "weight" (significance) of a time period of the artist’s creativity on the parameters of his work during this period.


The function K(j)=F(D) has for each j unique species, which are not representable by universal mathematical formulas. For example, K(3) and K(4) (number of exhibitions) are simple integers, K(2) (availability of professional education) has boolean type (1 or 0), K(5) (critics evaluation) may have form of scoring, and the partial criterion K(16) (prices) is itself a compound function, which takes into account many parameters.

But this problem is common to all applications of “Model-heuristic” method, and the standard approach, which was developed by Prof. Efremov, provides a very effective solution to this problem: a single (and easily solvable) challenge is to bring all the elements of an array K(j) to numeric type and to give them extremumless nature, so that the function K(j)=F(D) should either increase or decrease over the whole range of values.

As we have seen, partial criterions K(j) in criterial functions OK(i) have weights V(j), which allow to "flatten" all contradictions between different K(j) and mix them into a single formula. All questions related to the dimensionality, nonlinearity and the physical meaning of K(j), are taken into account in the next phase of “Model-heuristic” method – optimization of weights.

Let us turn to the nonlinear function W(i)=F(K(j)).

This function, which expresses the value of weights of the time periods of the artist’s creativity, as opposed to a weight of a partial criterion V(j), is unique and requires special investigation. For example, the work of avant-garde artists in the era of the early sixties until the early eighties was associated with additional difficulties, and for socialist realists – with certain preferences, and this implies the non-linear elements



where i is in the range of values, which correspond to the era since the early sixties until the early eighties.


Such a situation can occur in many cases.

This problem can lead to instability of the model, and it must be resolved. Let us represent the "classical" form of the final criterion function




in the general form:


FK(X)=S W(i)ÎÊ(i),


where S – the summation of all elements with index i.



In turn,


OK(i)=S V(j)K(j).



Hence, FK(X)=S W(i) S V(j)K(i,j).

                          i           j


The transition from one-dimensional array K(j) to two-dimensional array (matrix) K(i,j) is done due to the fact that in each time period i values of partial criterions K(j) are different.

Having brought W(i) inside the second summation sign, we obtain:


FK(X)=S S W(i)V(j)K(i,j).

              i  j


We see that in this formula there are “weights” W(i) and V(j) – values of the same nature. We call W(i)V(j) a generalized weight parameter of a partial criterion K(i,j).

At each step i the generalized weight parameter has different values, leading to instability of the model in the case of a standard application of “Model-heuristic” method, where the weight parameters must be identical at each step.




To solve this problem, A.V.Efremov developed the method of synthesis of expert assessments and their mathematical formalization.

At this stage it is necessary to attract highly qualified experts, as well as holding lengthy calculations by one of the existing exact optimization techniques (e.g., “linear programming” or “branches and bounds method”). But this "training" can be conducted only once at the stage of pilot operation of the mathematical model.

Further the mathematical model has almost complete "independence", high speed and accuracy that meet all the requirements of "artificial intelligence". The problem of "intellectual aging" of the model exists, but it is comparable with a similar problem for any human mind, and, naturally, requires a "refreshement" with some frequency. In the case of ratings of artists, this problem is simplified due to the presence of the permanently working Rating Center, which includes Russian leading art historians.

The problem of "artificial intelligence training" requires a separate statement in our task of  art rating: to find the numerical values of the generalized weights OV(i,j), which expresses the importance of private criterions K(i,j) in any given time period T(i).

Prof. A.V.Efremov developed a simple and efficient algorithm for their search.

Experts, involved in the stage of "training", have the task of modeling of a real object. Each specialist may have his own methods for solving it, but the model is based in any case on “Model-heuristic” method, and for its "training" only the result of the work of the experts is important.

In the rating task, after the experts have come to common or similar conclusions by a representative sample of artists M, we have start and end points of the simulation for every artist X: the input data D(X) and the final result FKM(X). We can ask experts to evaluate artists’ creativity in digital form (having pre-determined the range of FKM), or in the form of rating categories.

At the stage of "model training" it is very important to choose the most qualified experts and a representative sample of artists M. Those artists should be attracted, for whom the experts have the most complete set of input data D.

So, after the experts’ exploration of the issue, we have numerical values of FKM(X) and the source data D(X) for each artist X(m) in the framework of a representative sample M.

We can write the universal formula of “Model-heuristic” method, considering the matrix of generalized weights:


FK(X)=S S ÎV(i,j)K(i,j), mÎM.

             i  j


Having represented K(i,j) as a function F from the input data D, we obtain:


FK(X(m))=S S ÎV(i,j)F(D,i,j), mÎM.

                   i  j


We have received the task that is ready to be solved by one of the numerous precise mathematical methods (e.g., “dynamic programming”, or even the simple computer sorting of variants): it is necessary to define the matrix of generalized weight parameters OV, which provide (being provided with the given input data D and a given form of F) values of FK, coinciding with the values of FKM, which are identified by the experts for the entire sample of artists M.

Theoretically it is possible that there are no values of the matrix OV to tackle the problem of coincidence FK(X) and FKM(X) for the entire sample of M. In this case, there is the possibility of issuing 5–7-percent tolerance for difference of these quantities.

If the experts determine not FKM(X), but directly artist's rating R(X), it is adequate to the  similar tolerance, as in the set of artists CX, taken as 100%, there are 14 rating categories. Tolerance in this case is:

100 / 14 : 2 = 3.5, i.e. plus or minus 3.5 %.

If the issuance of tolerance also has not led to any positive result, it is a signal for developers about incorrect definition of partial criterial functions K(j), or a signal for the experts about non-objectivity of their judgments. In the latter case, the experts take new decisions about the values FKM in the sample M, and the "model training" starts anew.

A successful solution to the problem of "training" gives us a matrix of weight parameters OV, which is later used in a program that implements “Model-heuristic” method.

Thus, having once spent time and efforts to evaluate the representative sample of artists M and to calculate weight parameters OV by precise mathematical methods, we shall have a high-speed model, which works for each artist Õ(m), mÏM, in accordance with principles of "artificial intelligence".




It is known that any intelligence in terms of lack of information can take decisions, basing on previous experience, although the quality of such decisions will be reduced depending on the degree of lack of information. Let us see if we can implement this principle in our model – an "artificial intelligence" of artists rating.

So, we have an incomplete set of input data ND by an artist X. Incompleteness may consist either in complete absence of data for a period i, or in incomplete data, making impossible to count one of partial criterions K(i,j).

In this case, one or more elements of the summation is zero:


FK(X)=S S ÎV(i,j)K(i,j),

              i  j


That does not lead to the impossibility of artist’ rating calculating, but poses a serious problem in the final result obtaining.

 The problem is that the criterion FK has applicable character, so zeroing of an element of the summation leads to decreasing of the sum and, hence, to unreasonably low ranking of the artist. And requiring of absolutely complete source of input data on each artist is unrealistic.

Thus, we have found necessary to solve the problem of the model achievement of completeness of the input data D, basing on available incomplete set of input data ND, NDÎD.

This problem belongs to a class of problems of experimental data approximation.

We represent the values of K(j)=F(D) at each step i in the form of a matrix K(i,j), in which each element K(i,j)=F(D):


          1            2     ......    j

1    Ê(1,1)   Ê(1,2)  ......  ......

2    Ê(2,1)   Ê(2,2)  ......  ......

…    ….       …..      ...... …..

 i      ......      .......     .....  K(i,j)


In the case of an incomplete set of initial data ND we have a number of zero elements in this matrix. A situation, when an entire line i will be zero because of the lack of data for a particular period of the artist's creativity, should be considered as a typical.

Linear approximation of the columns may be conducted by the method of least squares.

The method is as follows: the values of K(i,j) in a column with a fixed number j is represented as an array of expert data on the artist’s periods i. Basing on this data, a linear approximation function is analytically described, and by the formula of this function we can calculate the value of the model data for any period, for which we have no real data.

Graphically, this can be represented as follows:


| K(i,j)

|                                         * KÅ(i,j)

|                   *        *

|         *                   

|    *             


0   1   2   3   4   5   6 ………….. i


Icons "*" are the values of partial criterions KE(i,j), which are calculated on the basis of available input data ND. We see that in the periods with numbers 3 and 5 the values are assumed to be zero.

It is required to describe analytically the linear function, whose graph will be as close as possibly to all points "*". Then the values of this function, when i = 3 and i = 5, will turn out to be approximated values of K(3,j) and K(5,j).

We use the method of least squares to obtain this function.

The general form of a linear function:


y = ax + b.


In our case:


K(i,j) = ai + b.


We need to find values a and b, so that the sum of deviations of all values of the function from "*" would be minimal. Since deviations can be expressed both by positive and negative numbers, we square the summation of their values, hence the name of the method – the least squares.

Thus, for each j the values for a and b should provide


min S (KÅ(i,j) K(i,j))2 .



Known values KE(i,j) act as an array constant c(i) for the method of least squares. Let us rewrite the function, for which we want to find the values of a and b, which ensure its minimum:


min S (ñ(i) – ài b)2.



This problem is solved by one of the existing exact methods (from “linear programming” to a simple computer sorting of a and b), because the dimension of this problem is small.

After that, having substituted any values i into the function K(i,j) = ai + b, we obtain approximated values of all elements of the column j, and that is what we needed.

The size of the column j (the number of periods of the creativity of the artist i) depends on the age, the first exhibition date and a number of other factors. In any case, the principles of linear approximation dictate the following theoretical limit: the number of nonzero elements in each column should not be less than two. Otherwise, the program operation is interrupted, and it is necessary to input additional data.

If novice artist’s creativity fits into one or two time periods, we have a "degenerate" matrix, and in this case any approximation is unlawful. We need a maximum full set of input data for objective rating of such artists.



© Sergey Zagraevsky




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