Dr. Sergey Zagraevsky
Application of contemporary
mathematical methods
of expert estimations to artists
rating
1.
Artcritics 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 artcritics, nor mathematics, nor any
other scientific discipline has scientifically based and accurate timetested
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 nonprofessional 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 mathematicalheuristic 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.
2.
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 midseventies 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 socalled "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 socalled machine resources – performance, memory, disk capacity,
etc. For comparison with our time let us say that the resources of even
socalled "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 preprocessed information from the experts on the
subject.
We present the most wellknown 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 ultramodern computer.
In fact, the program "Deep Blue" made an analysis of several
thousands of games, which different Grandmasters 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. 5–7percent 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 mathematicalheuristic methods to produce highquality 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.
3.
The task of an art rating creation belongs to the
class of problems of dynamic multicriteria 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 socalled “Modelheuristic”
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 “Modelheuristic”
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 socalled 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,
etc.
Disadvantages of "artificial intelligence"
in comparison with human decisionmaking 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):
OK(i)=V(1)K(1)+V(2)K(2)+…+V(j)K(j),
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
“Modelheuristic” 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 reapply “Modelheuristic” method for calculating of the
final criterion function FK of the artist X:
FK(Õ)=W(1)OK(1)+W(2)OK(2)+…+W(i)OK(i),
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.
4.
The main problem of “Modelheuristic” method
implementing for the problem of artists ratings is a nonlinearity of the
function
K(j)=F(D),
where D – input data for each of the partial
criterions of artist evaluation,
as well as of the function
W(i)=F(K(j)),
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
“Modelheuristic” 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 “Modelheuristic” 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 avantgarde 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 nonlinear
elements
W(i)=F(Ê(12),Ê(14)),
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
FK(Õ)=W(1)OK(1)+W(2)OK(2)+…+W(i)OK(i)
in the general form:
FK(X)=S
W(i)ÎÊ(i),
i
where S – the summation of all elements with index i.
i
In turn,
OK(i)=S
V(j)K(j).
j
Hence, FK(X)=S
W(i) S V(j)K(i,j).
i
j
The transition from onedimensional array K(j) to
twodimensional 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 “Modelheuristic” method, where the weight
parameters must be identical at each step.
5.
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 “Modelheuristic” 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 predetermined 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
“Modelheuristic” 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 5–7percent
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
nonobjectivity 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 “Modelheuristic” 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 highspeed model, which works for
each artist Õ(m), mÏM, in accordance with
principles of "artificial intelligence".
6.
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} .
i
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}.
i
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.
1999
© Sergey Zagraevsky
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International Art Rating and the content of this site:
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Prof. Dr. Sergey
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