Dr. Sergey Zagraevsky
Application of contemporary mathematic methods of peer
Published in Russian:
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The following text was translated from the Russian original by the computer program
and has not yet been edited.
So it can be used only for general introduction.
Accurate estimates of works of art of all times was excited art. There have been many attempts to mathematically evaluate the "quality", "artistic value", "social and humanitarian significance" as paintings, sculptures and other works of art and creative work of an artist.
This problem arose, and before the creators of the rating of artists.
Before turning to the substantive part of our research, we have to state: to date, no art criticism, neither mathematics nor any other scientific discipline not have scientifically based and proven accurate methods of assessment of works of art.
Numerous Western ratings are based on the value of the artist's works, for a very chaotic Russian art market does not fit.
Assessment of professionalism from the perspective of engineering art (sculpture, and the like) in the twentieth century has lost its universal scale, and to common modern arts as abstract art or conceptualism, is inapplicable. The same thing applies to such sensitive indicators of art history as composition, character stroke, modeling, etc.
In the end, the assessment of works of art is purely indicative.
If necessary the evaluation of the artist as a whole, as a phenomenon in the art, the task becomes more complex. Such irreparable (at least theoretically) indicators as the number of publications, exhibitions, catalogs, honors or awards, in modern conditions can only serve as auxiliary information.
Statistical methods of research, which primarily concerns the analysis of public opinion polls, could help in assessing the social significance of the artist and his works, but their significance from the point of view of art history. The main argument is known: art is not politics, this matters by majority vote of not being solved. Moreover, focus solely on the judgment of the non-professional public leads to a tremendous amount of speculation about "artists, favorite people." Moreover, statistical data are the easiest to fraud.
Thus, before the creators of the rating task was impossible neither accurate or statistical methods.
To solve this problem it is necessary to use mathematical and heuristic methods. A common characteristic of these methods is the use of mathematical tools of the analysis of expert estimations in this or that field of knowledge, beyond the systematization using precise mathematical methods.
In our case, this area of knowledge, which requires systematization, a fine art.
So, set the task: required, based on the available information about each artist, to determine its category and level in accordance with the "Regulations on the Rating center of the Professional Union of artists". The method of ranking can be implemented in the form of computer programs.
In order to understand what mathematical methods we can use to distract from the issues of art and consider a relatively recent time period - from the mid 1970s until the early 1990-ies.
At this time, heuristic (expert) methods have been with unprecedented intensity implemented in diverse areas of science and technology. Here are just some of the scientific disciplines: psychology, meteorology, Geology, management of economic systems, dispatching air, rail and road transport, forecasting the development of scientific-technical potential of the country and regions... the List goes on for a very long time - almost no scientific discipline, no sector of the economy remained "on the side".
The matter is that at this time began the so-called "automation of control systems" - the introduction of ACS both at the national and industry levels, and in the majority of large enterprises and institutions. As is known, at that time the computer had much less opportunity, and developers of automated control system was faced with a serious shortage of so-called machine resources - speed, memory, drives, etc. (!) computers-"laptops".
Such an acute shortage of machine resources did not allow to solve problems of precise calculation of parameters of any major economic or scientific tasks. The existing mathematical methods in theory allow it to do, but in practice the calculation takes from several hours to several days, making unrealistic flexible (operational) converting the introduction of new parameters.
Speaking on the exact calculation, we mean the methods of linear programming, dynamic programming", "branch and bound," and so All of these methods require multiple conversion extremely cumbersome matrices and digital arrays, and increase the dimension of the problem requires an increase in the required machine resources in quadratic or cubic dependence.
It was during this period in order to save computer resources and have the widest distribution of heuristic (expert) methods of settlements. Formally speaking, the main objective of any heuristic method is to reduce the dimension of time and solve the problem by "clipping" deliberately unpromising steps. And the definition of the prospects of a move is made on the basis of formalized and pre-processed information from the experts on this issue.
Here are the most famous example is a chess program “Deep Blue”, won from Garry Kasparov. We should not think that it is on every move is applied the so-called "variants" - all is possible in this situation moves, answers, then all possible next moves, answers and other Such "search tree" would take many hours, even for a super modern computers. Actually in the “Deep Blue” an analysis of several thousands of games played by different GMS at different times (from Lasker and Capablanca to Kasparov), and each turn the computer makes the basis of their experience. This is one method of expert estimations.
Research 1980-90-h years showed that the modeling of large economic systems, expert evaluation methods give results that are only 5-7 % deviating from the theoretically possible optimal results, at the cost of machine resources by several orders of magnitude lower. Similar results were obtained in all other disciplines, where the effect of large dimensions and complexity of the tasks the available computers with the exact methods of calculation could not cope.
In recent years, thanks to the unprecedented growth of productivity of computers began the reverse process: the developers of automated control, not caring about saving almost inexhaustible machine resources are rarely used expert methods and increasingly precise, so as to attract highly qualified experts are always involves additional time and financial costs. But, as we have seen on the example and the art market, and the chess program “Deep Blue”, and now there is a problem, insoluble exact mathematical methods.
Summarize our short historical overview: for applications where precise mathematical methods for some reason does not apply, there are modern mathematical and heuristic methods to obtain high quality solutions. As we shall soon see, is no exception and the task set by the developers of the rating of artists.
Objective ratings of art belongs to the class of problems of dynamic multi-criteria optimization, as it is necessary to use multiple competing criteria and taking into account dynamics of development of creativity of the artist in a wide time range.
To build the mathematical model of this problem. As we have shown that the exact solution methods are not applicable to it.
First of all, consider the dynamic nature of the task rating. Modern approach to automation of all areas of science and technology provides a transition from analogue to digital (discrete) representation of the model. A digital representation of a universal, easily implemented on computers, and the main thing - do not require modeling cumbersome trigonometric formulas, differential equations or Fourier transforms.
Discretization of the dynamics of development of the artist for the ratings of problem is to allocate significant periods of his creativity. However, due to the inability to collect objective information on all phases of creative work of every artist, it is reasonable allocation in his work periods that coincide with the most important periods in the development of art of the given countries and epochs. Question periods requires a special expertise in the study of debugging method.
Let us denote the number of each period as i, its time limits as T(i), system "the artist and his works" in period T(i) as X(T(i)), and all the many artists that are subject to rating, as SH.
Thus, we are dealing with the task of step-by-step simulation of a
dynamic system X (T), which aims to determine the rating of R - the place of
the artist in many of the artists
Professor Averremote, scientific supervisor of the candidate dissertation of the author of this study, in the seventies was developed so-called model-heuristic method step-by-step solution of multicriteria optimization problems of large dimension applicable to many areas of science and economy.
Consider the essence of the model-heuristic method.
For the best (quality) solutions step-by-step tasks enough to take the best (quality) of the solution at each step. On decisions at each step affects a number of so-called individual criteria.
Denote the space of partial criteria in an array For(j) and describe it for our problem rating. Note that the criteria in the array are arranged in a random order, not in ascending or descending order of importance.
A sample list of individual criteria for each artist in every period i:
To(2): the availability of vocational education;
To(3): personal exhibition;
To(4): group exhibition;
To(5): assessment of the art critics;
To(6): catalogues and booklets;
To(7): participation in large Russian and foreign auctions;
To(8): the acquisition of works by the leading museums;
To(9): buying works through commercial galleries;
To(10): the presence of honor (academic) titles;
To(12): membership in the Union of artists of the
To(13): membership in associations "OST", "Group of 13", and so on;
To(14): the degree of subordination of creativity market conditions;
To(15): the number of mentions in the press;
To(16): the price of work;
To(17): the artistic level of the works;
(18): public importance of the works,
Disadvantages of "artificial intelligence" in comparison with the adoption of decisions by the person well known: it is inflexible and no such thing as intuition. But there is a definite advantage: the model can account for a wide range of criteria, which also does not have any expert. Thus, the relative inflexibility of the model is compensated by calculating a larger number of parameters.
Incomplete data source or that particular criterion functions can be used not completely. But any "artificial intelligence" shall have the potential to cover all the necessary criteria used by experts in the decision, so the list of individual criteria is subject to continuous expansion.
Next on each step i of private criteria are reduced in one General criterion OK(i):
where V(j) - "weight" of individual criteria, i.e. numeric expression of the significance of the criterion.
It is a universal form of criterion function model-heuristic method developed by Professor Averremote.
However, the task of rating is a special case of the problem step-by-step optimization, as high indicator OK(i) in one of the periods of the artist is not a guarantee of high indicator OK(i+1), i.e. in the next period.
So you want to calculate OK(i) for each period i, and then re-apply the model-heuristic method for the calculation of the final criterion function FK on this artist's:
where W(i) - "weight" General criteria for each time period i, i.e. numeric expression of the significance of this or that period of time.
It remains to break the possible range of values FK(X) on the levels and categories contained in the Regulations on the Rating Center of the Professional Union of artists", and we get the desired R(X), XOffSH.
The main problem of implementation of the model-heuristic method for the task of ratings of artists is a non-linear function
where D - initial data for each of the individual criteria of evaluation of the artist,
and the functions
expressing dependence of the weights (importance) of this or that time period in the artist from the parameters of his work during this period.
The function K(j)=F(D) is for each j a unique view, unrepresentable no universal mathematical formula. For example, To(3) and(4) (number of exhibitions) have the form of a simple natural numbers, (2) (professional education) - type Boolean variable (1 or 0), (5) (assessment of art critics) can take the form of the scores, and the private criterion(16) (rates) by itself is a complex function that takes into account many parameters.
But this problem is common for all applications model-heuristic method, and the standard approach developed Averremote, provide a very effective solution of this question: the only (and easily solvable) problem is to bring all elements of the array K(j) to numeric mind and giving them betaxtreme nature, that is, the function K(j)=F(D) must either increase or decrease on the whole range of values.
The matter is that, as we have seen, in the criterion functions OK(i) private criteria K(j) have a "weight" V(j)that allow you to "smooth" all contradictions between K(j) and aggregate them into a single formula. All issues related to the dimension, nonlinearity and "physical sense" K(j), accounted for the next phase model-heuristic method - optimization "weights".
Move on to the nonlinear function W(i)=F(K(j)).
This function, which expresses the values of "balance" time periods of the artist, in contrast to "balance" partial criteria V(j)that is unique and requires special investigation. For example, for the avant-garde artists creativity in the era since the beginning of the sixties to the early eighties was associated with additional difficulties, as for realism - with certain preferences, and hence the nonlinearity elements
where i is in the range of values corresponding to the era from the early sixties to the early eighties.
This situation can occur in many cases.
This problem can lead to instability of the model, and it must be resolved.
Imagine a "classical" form of final criterion function
in the General form:
where S is the sum of all elements with index i.
So, FK(X)=S W(i) S V(j)K(i,j).
The transition to our records from a one-dimensional array K(j) a two-dimensional array (matrix) K(i,j) due to the fact that in each time period i values of individual criteria K(j) different.
Making W(i) inside the second sign of summation, we get:
FK(X)=S S W(i)V(j)K(i,j).
We see that in this formula there was "weight" W(i) and V(j) - value of the same nature. We will work W(i)V(j) the generalized weight parameter of private criterion K(i,j).
At each step, i generalized the weight parameter has different values, leading to instability of the model in the case of standard model-heuristic method, where at each step, the weights should be the same.
But we can successfully solve the problem by entering a new variable for the generalized weight parameter:
Array OV(i,j) turned out to be two-dimensional.
At first glance, the task becomes more complicated. But actually debugging weight parameters "manual" methods was unreal and V(j), we in any case will require the use of computers and modern mathematical methods and software implementations, a slight increase in the dimension of the array (with one-to two-dimensional) does not create problems.
Most importantly, we managed from complex nonlinear functions W(i) go to the numerical matrix of OV(i,j).
And so we come to the main element model-heuristic method is to identify the specific values of generalized weight indicators OV(i,j), which depends on the values and common criteria OK(i), and the final criterion FK(X), and, therefore, the rating of the artist - R(X).
To solve this problem Averremote method was developed for the synthesis of expert assessments and their mathematical formalization.
At this stage it is necessary to attract highly qualified experts, and conducting cumbersome calculations by one of the existing exact optimization techniques (e.g. linear programming or method of branches and borders). But this "training" is needed once during the trial operation of a mathematical model.
Further, the mathematical model is almost complete "independence", high speed and precision that meets all the requirements of the "artificial intelligence". The problem of "intellectual aging model" exists, but it is quite comparable with a similar problem for any human mind, and, of course, with any degree of frequency requires a "refresher". In the case of ratings of artists, this task is simplified due to the presence of a permanent Rating Centre, which includes leading Russian art.
The problem of "learning artificial intelligence requires a separate statement as part of our problem rating: it is necessary to find the numerical values of generalized weight indicators OV(i,j)expressing the significance of one or another private criterion K(i,j) at one or another time period T(i).
Prof. Avetisov has developed a simple and effective algorithm of their search.
Before the experts involved at the stage of "education", the task of modeling the real object. Each specialist can have their own methods of solution, but for us it is not important, as the model in any case, based on the model-heuristic method, and for her "training" is only important outcome of the work of experts.
The task of rating, after specialists come to the same or very similar conclusions on a fairly representative sample of artists, M), we have a start and end points of the simulation for each artist X: original data D(X) and the end result FKM(X). You can ask experts to evaluate the work of artists both in numeric form (first identifying the range of values FKM), and in the form of rating categories.
At the stage of "education" model is of great importance selection of the most qualified experts and a representative sample of artists , M. As a rule, are those artists which experts have the most complete set of initial data D.
So, after consideration of the issue by the experts we have in the framework of representative sample M on each artist X(m) numeric values FKM(X) and original data D(X).
Let's write a universal formula model-heuristic method taking into account the matrix of the generalized weight indicators:
FK(X)=S S OV(i,j)K(i,j), mOffM.
Presenting K(i,j) as a function F from the original data D, get
FK(X(m))=S S OV(i,j)F(D,i,j), mOffM.
We received a task that is ready to resolve computer on one of the numerous exact mathematical methods (for example, dynamic programming, or even a simple computer variants): it is necessary to determine the values of a matrix of the generalized weight parameters OfV, which (given initial data D and given functions F) values FK, coincide with values FKM, certain experts, for the whole sample artists M.
Theoretically it is possible that there are no values of the matrix OfV, which provides a solution of the problem of coincidence FK(X) and FKM(X) on the entire sample M. In this case, there is the possibility of issuing the 5-7% tolerance on the difference of these values.
If experts determine not FKM(X), and directly to the artist's rating R(X), is adequately grant similar access, as in a number of artists SC, taken for 100 %, is allocated 14 rating levels and categories. Tolerance in this case will be:
100 / 14 : 2 = 3,5, i.e. plus or minus 3.5 percent.
If the issue of tolerance has not led to a positive result, it is a signal to developers about the incorrect definite form of private criterion functions K(j), a signal to the experts about the partiality of their judgments. In the latter case, experts are adjusting their decisions on values FKM sample M, and the program "learning model" starts anew.
The successful solution of the problem of "training" gives us a matrix of weight parameters OfV, which is later used in the implementation of model-heuristic method.
Thus, once having spent time and energy on expert evaluation representative sample of artists M and cumbersome calculation of weight parameters OfV exact mathematical methods on the computer, we get debugged high-speed model running on each artist X(m), m : M, in full accordance with the principles of "artificial intelligence".
As you know, any intelligence in terms of lack of information to make decisions based on previous experience, although the quality of decisions is reduced depending on the degree of lack of information. Let's see if we can implement this principle in terms of our model - "artificial intelligence" ratings of artists.
Let the artist X we have an incomplete set of initial data ND. Incompleteness can be concluded in the complete absence of data for the period iand incomplete data, which does not allow to count one of the individual criteria K(i,j).
In this case, reset one or more elements of summation
FK(X)=S S OV(i,j)K(i,j),
that does not lead to the impossibility of calculating the rating of the artist, but creates a serious problem in obtaining the final result.
The fact that the criterion FK has applicative character, so zeroing one of the elements of summation leads to a decrease in the amount and therefore unnecessarily under-rated artist. Submission to the program entry requirements absolutely complete source of data on each artist is unrealistic.
Thus, we came to the necessity to solve the task model complete set of initial data Dbased upon a partial set of initial data ND, NDOFFD.
This problem belongs to the class of problems of approximation of experimental data.
We represent the values of K(j)=F(D) at each step i in the matrix K(i,j), in which each element K(i,j)=F(D):
1 2 ...... j
1 K(1,1) TO(1,2) ...... ......
2 K(2,1) TO(2,2) ...... ......
... .... ..... ...... .....
i ...... ....... ..... K(i,j)
In the case of an incomplete set of initial data ND we get in the matrix, the number of zero elements. Typical you should consider the situation when due to the absence of data for a particular period of the artist zero will be the entire line i.
A linear approximation of the columns using the method of least squares.
The essence of the method consists in the following: the values of K(i,j) in the column with a fixed number j is represented as an array of expert data on the periods i creativity of the artist. On the basis of these data is analytically described by a linear function approximation, and the formula of this function, we can calculate the value of data at any period for which actual data are not available.
Graphically, this can be presented as follows:
| * KF(i,j)
| * *
0 1 2 3 4 5 6 .............. i
Badges "*" denotes values of individual criteria KE(i,j), calculated on the basis of initial data ND. It is seen that in periods with numbers 3 and 5, these values are missing and are assumed to be zero.
Required to analytically describe the linear function whose graph will be as close as possible to all points "*". Then the values of this function at i=3 & i=5 and will be approximated values(3,j) and K(5,j).
To obtain the formula for this function and apply the method of least squares.
General view any line of the function:
In our case:
It is necessary to find such values a and b, so that the sum of the variances of all the values of a function from the "*" was minimal. Since deviations can be expressed both positive and negative numbers, before we constructed by summing their values in the square, from where the name came method of least squares.
So, for each j values a and b should provide
min S (K,E(i,j) - K(i,j))2 .
The known values KE(i,j) for the method of least squares act as an array of constants C(i). Let's rewrite the function to which you want to find the values of a and b, providing its minimum:
min S (s(i)-andi-b)2.
This problem is solved one of the existing exact methods from linear programming to simple computer variants a and b, since the dimension of this problem is small.
Then, by substituting all values of i in the function K(i,j)=a(i+b, we obtain the approximated values of any elements of column j, that we required.
The size of the column j (number of periods of the artist i) depends on age, date of the first exhibition and a number of other factors. In any case, the principles of linear approximation dictated by the following theoretical limit: the number of nonzero elements in each column should not be less than two. Otherwise, you must enter additional source data.
If creativity novice artist fits into one or two time periods, we have a "degenerate" matrix and an approximation in this case is unlawful. For such artists for objective ratings need a full set of original data.
© Sergey Zagraevsky