=Paper=
{{Paper
|id=Vol-2236/paper-05-005
|storemode=property
|title=
Numerical Aspects of Statistical Pattern Recognition
|pdfUrl=https://ceur-ws.org/Vol-2236/paper-05-005.pdf
|volume=Vol-2236
|authors=Yurii N. Orlov ,Elvira R. Zaripova,Alexey V. Chukarin
}}
==
Numerical Aspects of Statistical Pattern Recognition
==
https://ceur-ws.org/Vol-2236/paper-05-005.pdf
40
UDC 621.39
Numerical Aspects of Statistical Pattern Recognition
Yurii N. Orlov *† , Elvira R. Zaripova* , Alexey V. Chukarin*
*
Department of Applied Probability and Informatics,
Peoples’ Friendship University of Russia (RUDN University),
6 Miklukho-Maklaya st., Moscow, 117198, Russian Federation
†
Department of Kinetic Equations,
Keldysh Institute of Applied Mathematics of Russian Academy of Sciences,
4 Miusskaya Sq., Moscow, 125047, Russian Federation
Email: yuno@kiam.ru,
zaripova_er@rudn.university, chukarin_av@rudn.university
The paper presents numerical restrictions of Bayesian method to a pattern recognition.
The maximum probability of local state correspondence to one of the basis patterns is defined
through the expansion of examined vector over the patterns. The practical recognition example
without errors is presented in the frame of nearest neighbor method for the case, when the
probability interpretation of the expansion coefficients is not valid. The spectral portraits of
solving matrices are constructed for the literature texts author identification problem. Also,
the statistical properties of letters frequencies in European literature texts are investigated.
The determination of logarithmic dependence of letters sequence for one-language and two
language texts are examined. The Voynich Manuscript structure was considered for numerical
analysis. After our numerical analysis, we suppose, that the Voynich Manuscript was written
in two languages having the same alphabet without vowel letters: one of the Germanic
languages (Danish or German) and one of the Romance languages (Latin or Spanish).
Key words and phrases: non-stationary time series, probabilities of letters combination,
basis patterns, pattern recognition, European language statistics, Voynich Manuscript.
Copyright © 2018 for the individual papers by the papers’ authors. Copying permitted for private
and academic purposes. This volume is published and copyrighted by its editors.
In: K. E. Samouylov, L. A. Sevastianov, D. S. Kulyabov (eds.): Selected Papers of the 1st Workshop
(Summer Session) in the framework of the Conference “Information and Telecommunication
Technologies and Mathematical Modeling of High-Tech Systems”, Tampere, Finland, 20–23 August,
2018, published at http://ceur-ws.org
Orlov Yurii N., Zaripova Elvira R., Chukarin Alexey V. 41
1. Introduction
Theory of pattern recognition concerns to various aspects of statistical analysis of
big data. The methods and principles of the theory are presented in many special books,
see [1–3]. The basic principles of characters recognition were investigated in [4]. Some
algorithm aspects were discussed in [5–7].
Let us consider statistical recognition problem of belonging of the sample distribution
function (SDF) to a certain class or a cluster. We suppose, that each class is characterized
by a certain pattern or etalon distribution function. In practice very often there are
situations, when patterns are given as frequencies of empirical estimated parameters. The
corresponding functions {𝜙1 , ..., 𝜙𝑚 } , 𝜙𝑠 ∈ 𝑅𝑛 , where m is a number of states (or basis
patterns) and 𝑛 is a dimension of parameter space are treated to be etalon probability
distributions. If, as a result of observation, it was found, that the system is characterized
by empirical discrete probability distribution (DPD) 𝑓 ∈ 𝑅𝑛 , 𝑓 ≡ (𝑓1 , ..., 𝑓𝑛 ), the
state, corresponding to this DPD, is recognized according to Bayesian method [3].
In mathematical sense the recognition problem is solved by the expansion of vector
f over the basis {𝜙1 , ..., 𝜙𝑚 }. The problem is that the projection operator on the
non-orthogonal (in general case) basis vector system may cause an unacceptably large
error.
In this paper the concrete example of very exact recognition will be considered for
the non-traditional case, when the scalar products between basis vectors 𝜙𝑠 are close to
unit, but nevertheless the recognition is appeared to be possible as a consequence from
Bayesian method.
Let us consider a vector 𝑓 (𝑘), where 𝑘 is a number of letter in alphabetical ordering,
corresponding to empirical frequency of the letter 𝑘 in a given text. For texts in Russian
language 𝑘 = 1, 2, ..., 33. For texts in English language 𝑘 = 1, 2, ..., 26 etc. It appears,
that literature texts by the same author have similar functions of letter frequencies
distribution [8] and these functions are stable for each author. The error of recognition
by the method of nearest neighbors for 100 authors and 1000 texts is equal to 0,15. If
we construct the distribution of bigrams of the texts by the same author, the unknown
text can be recognized in the library of author etalons with the accuracy, equal to 0.04.
And if we consider a three letters composition distribution, the empirical recognition
is absolutely accurate. The dimension of basis vectors in the last case is equal to
333 = 35937 (for Russian language). If we have a library of ten (or even 1000) authors,
then the technical problem is to project a vector with large dimension (number of
letters compositions) in a space with small dimension (number of authors). According
to Bayesian method this operation is numerically incorrect, but the method of nearest
neighbors remains valid.
Theoretical aspects and practical examples of the method of nearest neighbors are
discussed below.
2. Bayesian Recognition
Let vectors {𝜙1 , ..., 𝜙𝑚 } , 𝜙𝑠 ∈ 𝑅𝑛 are histograms, which correspond to definite
author letter distribution etalons. Then
𝑛
∑︁
∀𝑠 ∈ {1, ..., 𝑚} 𝜙𝑠 (𝑘) = 1. (1)
𝑘=1
Let also vector 𝑓 ∈ 𝑅𝑛 belongs to the convex hull of vectors {𝜙1 , ..., 𝜙𝑚 }, so that
𝑚
∑︁
𝑓 (𝑘) = 𝑦𝑠 𝜙𝑠 (𝑘) , 0 ≤ 𝑦𝑠 ≤ 1. (2)
𝑠=1
∑︀𝑚
From (1) and (2) it follows, that 𝑠=1 𝑦𝑠 = 1.
42 ITTMM-WSS—2018
If the expansion (2) is obtained, then we can define a number 𝑠* , so that
𝑠* = arg max 𝑦𝑠 . (3)
Then the most probable solution is that 𝑓 = 𝜙𝑠* . In this case the following
consequence is valid:
𝑠* = arg min ‖𝑓 − 𝜙𝑠 ‖ . (4)
In general the projection problem is solved by QR-expansion method [9, 10], where
optimal in a sense of 2-norm expansion is given by formula
(︁ )︁
𝑓 − Φ𝑦 = 𝐼𝑓 − 𝑄𝑅𝑦 = 𝐼 − 𝑄𝑄𝑇 + 𝑄𝑄𝑇 𝑓 − 𝑄𝑅𝑦 =
(5)
(︁ )︁ (︁ )︁
= 𝑄 𝑄𝑇 𝑓 − 𝑅𝑦 + 𝐼 − 𝑄𝑄𝑇 𝑓
Here Φ𝑛×𝑚 is a matrix of basis vectors {𝜙1 , ..., 𝜙𝑚 }, 𝑄𝑛×𝑚 is a matrix, for which
𝑄𝑇 𝑄 = 𝐼𝑚×𝑚 , and 𝑅𝑚×𝑚 is an upper triangular matrix. Since the last two terms in
the expression (5) are orthogonal, the optimal expansion is given by a formula
𝑦𝑜𝑝𝑡 = 𝑅−1 𝑄𝑇 𝑓. (6)
The relative error of expansion (2), (6) is defined as (7):
⃦(︀ )︀ ⃦
𝛿 ⃦ 𝐼 − 𝑄𝑄𝑇 𝑓 ⃦
𝜀= = . (7)
‖𝑓 ‖ ‖𝑓 ‖
If we have a statistical inaccuracy in numerical estimations of basis patterns and
vector f, the error, defined by (7), can be bounded as (8) from [10]:
(︂ )︂
‖∆𝑦‖ 2𝜅(Φ) (︀ )︀
≤𝜉· + 𝜅2 (Φ)𝑡𝑔𝜃 + 𝑂 𝜉 2 .
‖𝑦‖ cos 𝜃 (8)
(︁ )︁
‖ΔΦ‖ ‖Δ𝑓 ‖
In (8) was used 𝜉 = max ‖Φ‖ , ‖𝑓 ‖ , and 𝜅(Φ) - is a condition number of the
matrix Φ and sin 𝜃 = 𝜀.
So, in practice the condition of non-negativity of coefficients
∑︀ 𝑦𝑠 and a probability
interpretation of the expansion according to the condition 𝑚 𝑠=1 𝑦𝑠 = 1 may be violated.
In the paper [5] it was exhibited, that for m = 4 authors the violation of probability
interpretation of Bayesian recognition in the form (3) is equal to 0.55 for n = 33 letters,
0.35 for n = 332 letters combinations and 0.30 for n = 333 letters combinations.
The spectral portrait [9] of matrix Φ for 4 author etalons is presented in Fig. 1. The
color regions correspond to the same accuracies of matrix element in terms of decimal
power in the legend. Contours correspond to the eigenvalues regions.
The condition number of this matrix Φ is equal to 3500.
It should be emphasized, that the recognition according the method [4] for the same
statistical experiment lead to the accuracy of recognition 0.15, 0.04 and 0.00 respectively.
So in practice we should use the most effective method. The practical example is given
below.
Orlov Yurii N., Zaripova Elvira R., Chukarin Alexey V. 43
Figure 1. Spectral portrait of matrix of author patterns
3. Statistical Properties of European Languages and Voynich Manuscript
It is possible to construct the etalons not only for authors, but also for various
languages. In the most languages the dependence of ordered frequencies is logarithmic
[11], its determination is more than 0.98. Parameters of the logarithmic dependence are
determined by the number 𝑛 of characters in an alphabet and allow the interpretation
of the redundancy or the failure. Namely, the frequency of the letter with number 𝑘 is
given by formula (︂ )︂
1 1 𝑛!
𝑓 (𝑘) = 1 + ln 𝑛 . (9)
𝑛 𝑛 𝑘
Would it be correct using statistics of the letters frequency to make sufficiently
reliable supposition about texts language? This question has arisen from paper authors’
Voynich Manuscript discussion. The Voynich Manuscript (MV) [12] – is a hand-written
codex, dating from the XVI c. It consists of over 170,000 characters referred to as letters,
which are united by transcriptioners in 22 distinct characters. These characters are not
elements of any known alphabet. At the present time the manuscript is kept in the
Beinecke Library and has the status of a cryptographic puzzle.
Numerous studies in order to decrypt the text carried out more than a hundred years
and are still unsuccessfully. Versions of the authorship, content and language of the
manuscript [13–16] are not supported enough by the full-fledged statistical studies. Here
we try to get the answer to the following question: Is the MV an encrypted meaningful
text (and in which language it is written), or it is a hoax, i. e. meaningless set of
characters?
We consider transcription of MV into Latin alphabet according to [16]. Our goal is
to study statistic properties of the Manuscript. Researchers have proposed numerous
hypotheses about the structure of the Manuscript. There are some known theories:
– it was written with permutation of letters;
– two letters of the well-known alphabet correspond to one character of the manu-
script;
– there is a key without which you can not read the text, because the same characters
in different parts of the manuscript correspond to different letters;
– the manuscript is an encoded two-language text;
– vowels have been removed from the originally meaningful text;
– the text contains false spaces between words.
44 ITTMM-WSS—2018
At the same time in various concepts (unproven in the statistical sense) for the role of
the original language are proposed: Hebrew, Spanish, Russian, Manchurian, Vietnamese
and much more (even Arabic or “something Indian”). At the same time we can consider
the existence of false spaces as a real component of Manuscript structure. In addition, if
the text contains no vowels, the vowel recovery is not uniquely.
For finding linguistic invariants of European languages the following statistics are
used:
– the distance between distributions of empirical frequencies of letter combinations
in norm L1;
– determination level of logarithmic approximations of one-letter distributions for
texts without vocalisation;
– Hurst index distribution for a series of the number of letters concluded between
the two most frequently encountered same letters;
– spectral matrix portrait of two-letter combinations.
These indicators allow to make the formal clusterization of languages from Indo-
European family. As result our clusters have coincided with groups formed on the basis
of studies in Historical Linguistics.
For modern languages of Indo-European family and the same group (for example
German) logarithmic dependence of letter frequency on its rang is typical with accuracy
0.93-0.98 (see Fig. 2).
Figure 2. Distributions of letter frequencies for German language group
Actual distribution of odered frequencies for texts written in the same language
differs from logarithmic approximation in L1 norm within 0.08–0.13.
Letter frequency for Danish, Serbian, Croatian and Romanian texts without vocal-
ization have a lower approximation accuracy (0.93).
Distances between frequency distributions for texts in Cirillic for Slavic group show
that Russian, Bulgarian and Serbian are related: the closest are Russian and Bulgarian
(with a distance 0.06), Russian and Serbian as well as Bulgarian and Serbian have a
distance 0.12.
For texts with the Latin alphabet distances between frequency distributions form
clusters in accordance with language groups in sense of closeness between themselves in
norm L1. It was found that Indo-European languages united in German, Romance and
Orlov Yurii N., Zaripova Elvira R., Chukarin Alexey V. 45
Slavic groups subgroups have the same statistical properties. The distances in norm
L1 between frequencies from one language group vary quite narrow (0.08–0.13). The
distance between different groups belongs to interval 0.14–0.22.
Now let us compare the MV transcription symbol distribution with analogical
distributions in European languages (Fig. 3).
Figure 3. Distribution of letter frequencies for MV transcription
The distance between MV transcription distribution and logarithmic approximation
(9) in the norm L1 is equal to 0.17. The distance MV transcription distribution and
Danish language letters distribution is equal to 0,10. So MV language could be treated
as Danish without vowels, but the logarithmic approximation of the last language differs
from actual distribution on the value 0.12, that is sufficiently less, than 0.17 for MV. It
should be emphasized, the distributions of text symbols for the same language have the
same accuracy of approximation. Hence, Danish is not MV language.
The distributions of Hurst exponents for the Manuscript and ordinary texts are
completely different [11]. For MV it is shifted to the right and have much less acute
maximum compared to texts on one language. It means that statistics of the Manuscript
does not agree with statistics of texts written in one particular language. The symbols in
the Manuscript are placed “more randomly”. The main suggestion is that the Manuscript
is written in several languages. So we accept the following working hypotheses regarding
the MV:
1. The manuscript is a bilingual text with a common alphabet.
2. Vowels have been deleted from the text before the decoding.
3. Decoding was a bijective letter replacing by a symbol.
4. Spaces in the text are not considered as characters.
After that we need to find out which pairs of languages with a common alphabet and
in which proportion could be considered as the Manuscript languages, whether they have
the same or different linguistic groups and which groups exactly. To test the bilingual
hypothesis we join two texts without vowels with about equal volumes, each one written
in its own language, but with the same alphabets in both texts. It appears, that the
mixture of pairs of texts from one linguistic group has the same statistical properties, as
a group etalon. Languages with the same group not only have close ordered frequency
distribution in the texts without vowels, but also a mixture of these languages has the
same logarithmic approximation determination to its components.
46 ITTMM-WSS—2018
It appears, that the MV transcription is even closer to the mixture of Latin and
Danish languages in ratio 2:1, the distance between these distributions is 0.09. But as
it was mentioned above, all languages from the same linguistic group have the same
statistical properties. So we can consider the MV text in detail – by pages, each of
them contains approximately 1 thousand symbols. The page distribution is compared
with language etalons and its language is recognized by the method (4). The result is
presented in Fig. 4 as a corresponding language “coloring”.
Figure 4. The recognized languages of MV pages
As it was mentioned above, the accuracy of method for one-symbol distribution is
equal to 0.15. So approximately 15 percents of text pages are recognized incorrectly.
From Fig. 4 it follows, that the level of critical distance is equal to 0.13, and this value
coincides with upper level of distances between texts distributions for one language. So
our results are consistent with properties of recognition model.
However, this identification method is effective only when we have a complete set
of references. Otherwise, the identification will be incorrect. Proposed method of
identification is sufficiently accurate if there is a correct reference among the reference
distributions. If it is not the case, the most similar reference will be found, but it
is no guarantee of the correct recognition of course. Once again, we would like to
emphasize that we are talking about the European language that is the closest to the
MV transcription instead of discussing which language the MV is actually written on. If
the distance to the nearest reference distribution becomes too large, it’s possible, that
the intended distribution is missing in the library. It seems interesting that along with
the expected Dutch and Latin you can see the pair of German and Spanish. As it was
already mentioned, the distributions of the frequencies correspond to the language group,
therefore it makes sense to identify Spanish and Latin as a single Roman language group,
and German and Dutch as a single German group.
Nevertheless, an abundance of arguments support the fact that the text to be written
in two or more European languages.
Finally let us consider the spectral portraits of bigram matrix for MV and any Euro-
pean language of Germanic and Romanic groups without vowels. Using the calculation
procedure, described in [9], it is possible to construct domains of eigenvalues. The areas
with the same color have eigenvalues of the matrices if the elements of these matrices
Orlov Yurii N., Zaripova Elvira R., Chukarin Alexey V. 47
are known with the precision noted in the legend. The typical comparison is presented
in Fig. 5.
Figure 5. The MV spectral portrait (on the left)
and spectral portrait of the text in English (on the right).
For all European languages the spectrum area is approximately limited by the circle
with the radius 0.2. According to [8] the area of the spectrum for the texts with the
full alphabet has the form not of a circle but of an ellipse with semimajor axis equal to
approximately 0.5 and semiminor axis still equal to 0.2.
Comparing pictograms in Fig. 5, we see that the regions with equal accuracy are
markedly different in finding eigenvalues of the matrices for MV and conventional texts.
For MV transcription the circle (it is not an ellipse!) of the location of the eigenvalues has
approximately two times larger radius than for natural languages. It has fundamental
importance.
It is important to emphasize, all discussed arguments are fundamentally different,
i.e. they express the peculiar properties of independent statistics, indicating that the
interpretation of the MV as part of the composite manuscript is acceptable.
It is necessary to indicate the accuracy of the results presented in this paper. We work
on statistical pattern recognition by comparison with the standard. The critical point
is the accuracy with known standards. In this case standard refers to the probability
distribution of text characters. If the text is made up of 𝑁 signs and written by alphabet
of 𝑛 signs, the distribution of these characters in the text is determined with a precision
𝜀 that find numerically from equation, obtained in [8]:
√ 𝑛 √︀
𝑢1−𝜀/2 𝑁 ∑︁
= , Σ𝑁 (𝑛) = 𝑓𝑁 (𝑗) · (1 − 𝑓𝑁 (𝑗)). (10)
𝜀 Σ𝑁 (𝑛) 𝑗=1
Here 𝑢 is a quantile of the normal distribution. The quantile of Student’s distribution
with large values of 𝑁 is approximated by corresponding 𝑢, and 𝑓𝑁 (𝑗) is the empirical
frequency of symbol 𝑗 in this text with length 𝑁 . In particular, for the logarithmic
ordering model (9) when 𝑛 = 20 the value of the sum in (10) is equal to 3.93; in relation
to the MV with the number of signs 𝑁 = 170 thousands its actual distribution leads to
Σ𝑁 (𝑛) = 3.65. The right side of the equation (10) with respect to 𝜀 for a theoretical
model is equal to 105, and for the MV is equal to 113. These values correspond to similar
accuracies 𝜀 = 0.02, which differ in the third decimal place. It is similarly found out
that the accuracy of the frequency distribution for a single page (1500 characters) is 0.1.
Consequently, the differences between the distributions of the fragments at the level of
48 ITTMM-WSS—2018
0.08–0.13, and sheets of fragments at the level of 0.20–0.40 are not caused by statistical
noise of samples, they caused by objective reasons. Thus, the difference between the
samples with the specified accuracy of statistical estimations is well defined.
4. Conclusions
The results of presented statistical investigations can be summarized as follows.
First of all, It should be noted, the Bayesian recognition may be incorrect in statistical
sense, because the probability interpretation of projections over system of non-orthogonal
patterns may be violated. Nevertheless the recognition with the use of consequence of
Bayesian method, i.e. the nearest neighbours method, appeared to be very exact in
some practical cases.
With the use of this recognition method the classification of the Indo-European
languages into distinct groups can be performed very accurately according to a formal
statistical procedure [17, 18], i.e., pairwise clusterization of symbol frequencies distribu-
tions in texts without vowel letters. Within these sub-groups languages can be mixed
together without changes in the corresponding frequency distribution. Concerning the
Voynich Manuscript, it seems most plausible that it was written in two languages having
the same alphabet without vowel letters: 30% of the text is written in one of the
Germanic languages (Danish or German) and the rest 70% – in one of the Romance
languages (Latin or Spanish).
As our further research, it will be very interesting to apply methods for data analysis
from application areas [19–22], where new quality indicators, such as as traditional
indicators, i.e. SIR, are analyzed. Another methods of analysis, such as a combination of
linear topology and automated control [23,24], or information description cluster method
and multidimensional approache [25, 26] are also very actual and could be applied to
our current tasks.
Acknowledgments
The publication has been prepared with the support of the “RUDN University
Program 5-100” and funded by RFBR according to the research project No 17-07-00845.
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