Dr. Sergey Zagraevsky
Application of
contemporary mathematic methods of peer
to the ratings of artists
Published in Russian:
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Attention!
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.
1.
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 nonprofessional 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.
2.
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 1990ies.
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 scientifictechnical 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 socalled "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 socalled
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 preprocessed 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 socalled "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 198090h years showed that the modeling of large economic
systems, expert evaluation methods give results that are only 57 %
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.
3.
Objective ratings of art belongs to the class of problems of dynamic
multicriteria 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 stepbystep 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
R=F(X(T)), XOFFCX
Professor Averremote, scientific supervisor of
the candidate dissertation of the author of this study, in the seventies was
developed socalled modelheuristic method stepbystep solution of multicriteria optimization problems of large dimension
applicable to many areas of science and economy.
Consider the essence of the modelheuristic method.
For the best (quality) solutions stepbystep tasks enough to take the
best (quality) of the solution at each step. On decisions at each step affects
a number of socalled 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(1): age;
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(11): innovation;
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,
etc.
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):
OK(i)=V(1)K(1)+V(2)K(2)+...+V(j)K(j),
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 modelheuristic method
developed by Professor Averremote.
However, the task of rating is a special case of the problem
stepbystep 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 reapply the modelheuristic method for
the calculation of the final criterion function FK on this artist's:
FK(X)=W(1)OK(1)+W(2)OK(2)+...+W(i)OK(i),
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.
4.
The main problem of implementation of the modelheuristic method for the
task of ratings of artists is a nonlinear function
K(j)=F(D),
where D  initial data for each of the individual criteria of evaluation
of the artist,
and the functions
W(i)=F(K(j)),
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 modelheuristic 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
modelheuristic 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
avantgarde 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
W(i)=F(K(12)To(14)),
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
FK(X)=W(1)OK(1)+W(2)OK(2)+...+W(i)OK(i)
in the General form:
FK(X)=S W(i)OK(i),
i
where S is the sum of all
elements with index i.
i
In turn,
OK(i)=S V(j)K(j).
j
So, FK(X)=S W(i) S V(j)K(i,j).
i
j
The transition to our records from a onedimensional array K(j) a
twodimensional 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).
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 modelheuristic 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:
OV(i,j)=
W(i)V(j).
Array OV(i,j) turned out to be
twodimensional.
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 oneto twodimensional) 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 modelheuristic 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).
5.
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 modelheuristic 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 modelheuristic method taking into
account the matrix of the generalized weight indicators:
FK(X)=S S OV(i,j)K(i,j), mOffM.
i j
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.
i j
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 57% 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 modelheuristic 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 highspeed model running on each artist X(m), m : M, in full
accordance with the principles of "artificial intelligence".
6.
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),
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 underrated 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:
 K(i,j)
 * 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:
y=ax+b.
In our case:
K(i,j)=a(i+b.
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} .
i
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)andib)^{2}.
i
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