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Modeling of Artificial Intelligence, 2017, 4(2)

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Modeling

Modeling

of Artificial Intelligence

Has been issued since 2014. ISSN 2312-0355, E-ISSN 2413-7200

2017. 4(2). Issued 2 times a year

EDITORIAL BOARD

Simonyan Arsen – Sochi State University, Sochi (Editor in Chief) Dreizis Yurii – Sochi State University, Sochi Makarova Irina – Sochi State University, Sochi Simavoryan Simon – Sochi State University, Sochi Ulitina Elena – Sochi State University, Sochi Belyavskii Grigorii – Southern Federal University, Rostov-on-Don Chitchyan Robert – Yerevan State University, Armenian National Agrarian University, Yerevan Ohanyan Viktor – Yerevan State University, Yerevan Popov Georgii – Astrakhan State Technical University, Astrakhan Ravindranath Cherukuri – Gyan Ganga Institute of Technology and Management, Gyan Ganga Saakyan Vladimir – Institute for Informatics and Automation Problems of the National Academy of

Sciences, Yerevan Yicong Zhou – University of Macau, Macau

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C O N T E N T S

Articles and statements

Pattern Recognition by Cross Sections N.G. Aharonyan, V.K. Ohanyan .......................................................................................

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The Methodology of Risk Analysis in Assessing Information Security Threats A.S. Kopyrin, S.Zh. Simavoryan, A.R. Simonyan, E.I. Ulitina ........................................

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Multidimensional Limit Theorems in Models with Categorized-Time Absolute Priorities

A.R. Simonyan ..................................................................................................................

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Forecasting in Decision-Making Support Systems O.V. Tikhanychev .............................................................................................................

93

Application of Monte Carlo Method for Calculation of the Comets' Area by the Photographic Pictures

V.Ya. Tsvetkov ..................................................................................................................

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Copyright © 2017 by Academic Publishing House Researcher s.r.o.

Published in the Russian Federation Modeling of Artificial Intelligence Has been issued since 2014. ISSN: 2312-0355 E-ISSN: 2413-7200 2017, 4(2): 72-77 DOI: 10.13187/mai.2017.2.72 www.ejournal11.com

Articles and statements Pattern Recognition by Cross Sections

Narine G. Aharonyan a, Victor K. Ohanyan a , * a Yerevan State University, Armenia

Abstract The goal of the present paper is to investigate covariograms of convex bodies (it is equivalent

to investigate the orientation dependent chord length distribution functions). The applications of these problems are known in both geometric and computer tomography. Algorithms to reconstruct convex bodies by its covariogram for finite number of directions (the same problem for orientation dependent chord length distribution function has the negative solution) is one of the main problem of stochastic geometry. In particular, find the covariograms for classes of three dimensional convex bodies. Covariogram problem for three dimensional case is an open problem, while in the planar case the problem has the positive solution and if dimensionality of space greater than or equal 4 it has negative solution. The formulation of the problems is accompanied by discussion of the existing tools and ways of their implementation.

Keywords: сovariogram, kinematic measure, orientation-dependent chord length distribution, convex body.

1. Introduction Complicated geometrical patterns occur in many areas of science. Their analysis requires

creation of mathematical models and development of special mathematical tools. The corresponding area of mathematical research is called Stochastic Geometry (see Gardner, 2006 and Schneider, Weil, 2008). Among more popular applications are Stereology and Tomography. The objective of stereology is to draw inferences about the geometrical properties of -dimensional structure, , when information is only available in some lower-dimensional form via linear probes, planar sections, or projections of thick slices. Its application arises in the study of geometrical structure of inclusions or pores in opaque bodies such as metals, minerals, synthetic materials, or biological tissues; in these cases the available information must come from linear probes or planar sections. The methods and formulae of stereology relate characteristics of -dimensional structures to quantities arising from measurements of planar sections . The step from spatial structures to their sections involves a great loss of information and so stereological methods commonly yield only ``global'' information of a statistical character.

At the Conference on Tomography at Oberwolfach, R. Gardner introduced the term geometric tomography. In the R. Gardner monograph (Gardner, 2006), the following definition is

* Corresponding author E-mail addresses: [email protected] (N.G. Aharonyan), [email protected] (V.K. Ohanyan)

http://www.ejournal11.com/




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offered: ``Geometric tomography is the area of mathematics dealing with the retrieval of information about a geometric object from data about its sections, or projections, or both''. The word projection is used in the sense of a shadow, that is, the usual orthogonal projection on a line. The parallel X-ray of in the direction gives the length of the chord of intersection of with the line through x parallel to . The sections of bodies by random planes and lines (X-rays, cracks) are considered in many mathematical models of modern physics (computer tomography, crack tessellations etc). Two important mathematical problems are arisen: 1) for given convex body to calculate the chord distribution, 2) for given chord length distribution to reconstruct the convex body. Although there are many recent results and investigations in these directions, some problems are open, in particular, the computer programs for calculation of chord length distributions are missing. We are considered the problem of investigation of chord length distribution in -dimensional space. Recognition of planar domains by means of random lines intersecting is one of the interesting problem of Stochastic Geometry.

Let be the -dimensional Euclidean space, be a bounded convex body with inner points, and be the -dimensional Lebesgue measure in .

2. Discussion Definition 1. (see Matheron, 1975, Schneider, Weil, 2008). The function

( ) (1)

is called the covariogram of the body . Here is called the set covariance of . The definition of the covariogram is given by G. Matheron, who formulated it for more

general sets, and even for functions. In (Matheron, 1975), G. Matheron conjectured that the covariogram of a convex body determines within the class of all convex bodies, up to translations and reflections. G. Averkov and G. Bianchi (Bianchi, Averkov, 2009), showed that every planar convex body is determined within all planar convex bodies by its covariogram, up to translations and reflections.

Very little is known regarding the covariogram problem when the space dimension is greater than 2. It is known that centrally symmetric convex bodies in any dimension, are determined by their covariogram up to translations. For the problem is open. Nevertheless, for the case of bounded convex polyhedron for Matheron's conjecture received a positive answer. In fact, the covariogram problem is equivalent to the problem of determining a convex domain from all orientation-dependent chord length distributions (see Bianchi, Averkov, 2009, Schneider, Weil, 2008).

The problem of finding the measure of the segments of a constant length that are contained in has no simple solution and depends on the shape of . It is known the explicit form for the kinematic measures of some planar domains: a disk, a rectangle, if the length of the segment is less than the smaller side of the rectangle (see Santalo, 2004) and for the equilateral triangle, the rectangle (for an arbitrary length of the segment) and regular pentagon (see Gasparyan, Ohanyan, 2013).

Let denote the -dimensional sphere of radius 1 centered at the origin in . We consider a random line which is parallel to and intersects , that is, an element from the set:

Let be the orthogonal projection of onto the hyperplane (here stands for the

hyperplane with normal u, passing through the origin). A random line which is parallel to and intersects has an intersection point (denoted by )

with . We can identify the points of and the lines which intersect and are parallel to

u, meaning that we can identify the sets and . Assuming that the intersection point

is uniformly distributed over the convex body , we can define the following distribution function.


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Definition 2. The function

(2)

is called orientation-dependent chord length distribution function of in direction at point , where is the line which is parallel to and intersects at point and

.

Observe that each vector can be represented in the form , where is the direction of , and is the length of .

Lemma 1. (see Matheron, 1975) Let and be such that contains inner points. Then is differentiable with respect to and the following equality holds:

( ) (3)

At the right-hand derivative exists, and the same equality holds. Let be a random segment of length , which is parallel to a given fixed direction

and intersects . Consider the random variable , where , and the set is defined as follows:

Observe that each random segment lying on a line can be specified by the

coordinates , where is the one-dimensional coordinate of the center of on the line . As the origin on the line we take one of the intersection points of the line with the boundary of domain . Using the above notation, we can identify with the following set:

{ [

]}

where . Note that the set does not depend on the choice of the origin of the line , and the choice of the positive direction follows from the explicit form of the range of variation of . Further, we set

and observe that the sets and are measurable subsets of .

Definition 3. The function

(

)

( )

( )∫

(4)

is called orientation-dependent distribution function of the length of a random segment in direction .

Let be the space of all lines in . A line can be specified by its direction

and its intersection point in the hyperplane . The density is the volume element of the unit sphere and is the volume element on at . Let be a locally finite measure on , invariant under the group of Euclidian motions. It is well known that the element of up to a constant factor has the following form (see Santalo, 2004):

Denote by

the surface area of the unit sphere in . For each bounded convex body , we denote the set of lines that intersect by

We have (see Santalo, 2004)

A random line in is the one with distribution proportional to the restriction of to . Therefore, for any , we have

which is called the chord length distribution function of . Let be a random segment of length in and let be the kinematic measure of

(Santalo, 2004).


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If is the line containing and is the one-dimensional coordinate of the center of on the line , then the element of the kinematic measure up to a constant factor is given by

where is the one-dimensional Lebesgue measure on and is a motion element in ,

that leaves unchanged (see Santalo, 2004 and Schneider, Weil, 2008). The length of a random segment , provided that it hits the body }, has the following

distribution function:

Theorem 1. (see Gasparyan, Ohanyan, 2014). We establish a relationship between the distribution function of the random variable and the orientation-dependent chord length distribution function in , given by the following formula:

{

∫

(5)

Note that explicit forms for orientation-dependent chord length distribution function for triangles, ellipses, regular polygons and parallelograms were obtained in the papers (Gasparyan, Ohanyan, 2013; Gasparyan, Ohanyan, 2014). Hence substituting in (10) and the corresponding formulas for , we get explicit expressions for for the mentioned

planar convex domains. Theorem 2. (see Gasparyan, Ohanyan, 2014). The distribution function of the random

variable and the covariogram over the interval are related by the following formula:

[

] (6)

Theorem 3. (see Gasparyan, Ohanyan, 2014). The following relationship between the distribution function of the length of a random segment intersecting and the chord length distribution function of in :

{

∫

(7)

Denote by probability, that random segment (of fixed length and direction ) entirely lying in body .

Proposition 1. (see Aharonyan, Ohanyan, 2018) Probability in terms of distribution function has the following form:

∫

, (8)

while in the terms of the covarigramm of body has the form:

(9)

Denote by probability, that random segment of length in having a common point with body entirely lying in body (in this case the direction of the segment is arbitrary). Note, that probability can be obtain from probability by integration over all directions .

Proposition 2. Probability in terms of chord length distribution function has the following form:

(∫

– )

(10)

Since the ball is an isotropic body, then does not depend on direction . Therefore, we get

(11)

It is known that the volume of -dimensional ball of radius equals to


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( )

(

)

(12)

while is the projection of -dimensional ball of radius on hyperplane equals

(

)

Where ∫

is the Gamma function.

It is easy to see, that the covariogram of -dimensional ball of radius equals to twice the volume of -dimensional spherical cap of high . Using the formula for -dimensional spherical cap (see Gasparyan, Ohanyan, 2015) we get

( )

∫

(14)

where

.

Therefore, putting from (14) we get that the probability that the segment of the length entire lies in -dimensional ball of radius equals to (see Aharonyan, Ohanyan, 2018)

( )

√

(

)

∫

Obviously, for any dimension we have for and

for . We have a computer program for calculating the chord length distribution for an arbitrary

convex polygon. 3. Acknowledgements The present research of the second author was supported by the Mathematical Studies Center

at Yerevan State University. References Aharonyan et al., 2010 – Aharonyan N. G., Harutyunyan H. S., Ohanyan V. K. (2010).

Random copy of a segment within a convex domain, Journal of Contemporary Mathematical Analysis (Armenian Academy of sciences), 45(5), 348-356.

Aharonyan, Ohanyan, 2005 –Aharonyan N. G. and Ohanyan V. K. (2005). Chord lenght distribution functions for polygons, 40(4), 43-56.

Aharonyan, Ohanyan, 2011 –Aharonyan N. G., Ohanyan V. K. (2011).Kinematic measure of intervals lying in domains, Journal of Contemporary Mathematical Analysis (Armenian Academy of sciences), 46(5), 280-288.

Aharonyan, Ohanyan, 2018 – Aharonyan N. G., Ohanyan V. K. (2018). Calculation of geometric probabilities using Covariogram of convex bodies, Journal of Contemporary Mathematical Analysis (Armenian Academy of sciences), 53(3).

Bianchi, Averkov, 2009 – Bianchi G., Averkov G. (2009). Confirmation of Matheron's Conjecture on the covariogram of a planar convex body, Journal of the European Mathematical Society, 11, 1187-1202.

Gardner, 2006 – Gardner R. (2006). Geometric Tomography, Cambridge University Press, Cambridge, UK, 2nd ed.

Gasparyan, Ohanyan, 2013 – Gasparyan A., Ohanyan V.K. (2013). Recognition of triangles by covariogram, Journal of Contemporary Mathematical Analysis (Armenian Academy of sciences), 48 (3), 110-122.

Gasparyan, Ohanyan, 2014 – Gasparyan A., Ohanyan V.K. (2014). Covariogram of a parallelogram, Journal of Contemporary Mathematical Analysis (Armenian Academy of sciences), 49(4), 194-206.


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Gasparyan, Ohanyan, 2015 – Gasparyan A., Ohanyan V. K. (2015). Orientation-dependent distribution of the length of a random segment and covariogram, Journal of Contemporary Mathematical Analysis (Armenian Academy of sciences), 50(2), 90-97.

Harutyunyan, Ohanyan, 2014 – Harutyunyan H. S., Ohanyan V.K. (2014). Orientation-dependent section distributions for convex bodies, Journal of Contemporary Mathematical Analysis (Armenian Academy of sciences), 49 (3), 139-156.

Matheron, 1975 – Matheron G. (1975). Random Sets and Integral Geometry, Wiley, New York.

Ohanyan, Aharonyan, 2009 – Ohanyan V.K., Aharonyan N.G. (2009). Tomography of bounded convex domains, SUTRA: International Journal of Mathematical Sciences, 2(1), 1-12.

Santalo, 2004 – Santalo L.A. (2004). Integral Geometry and Geometric Probability, Addision-Wesley, Reading, MA.

Schneider, Weil, 2008 – Schneider R., Weil W. (2008). Stochastic and Integral Geometry, Springer.


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The Methodology of Risk Analysis in Assessing Information Security Threats Andrey S. Kopyrin a, Simon Zh. Simavoryan a , *, Arsen R. Simonyan a, Elena I. Ulitina a

а Sochi state university, Russian Federation Abstract Information security is not an IT problem and cannot be reduced to the IT department.

Effective preservation of confidentiality, integrity and availability must be anchored throughout the organization. To meet this challenge efficiently, a risk-based approach is required. First, the organizational context must be determined. When implementing the risk management process, the quality of the risk identification is crucial. Risks that are not identified here are missing in the subsequent risk analysis and valuation and thus also in the risk treatment. There are several approaches to methodological risk identification, two of which are presented: the predominantly impact-based event-based approach and the cause-based approach based on values, threats and vulnerabilities. In order for the implementation of risk identification to be successful in practice, various prerequisites must be fulfilled. The decisive factor is that the top management performs its leadership role effectively and effectively. The key challenge is to keep the scope of risk identification manageable. For this purpose, the procedures of focusing and coarsening have proven themselves in practice. Finally, through the process of continuous improvement, an initially crude but unambiguous image of information security risks can be refined step by step and adapted to current requirements and threats.

Keywords: risk-analysis, data protection services, security threats. 1. Introduction It is undisputed that adequate and effective protection of the confidentiality, integrity and

availability of information about business processes, support processes, employees, customers, suppliers, etc. has a great importance to all organizations. It does not matter in what form the information is available or tangible: it can be printed or written on paper, stored electronically, sent by post or e-mail, transmitted by electronic means, shown on photos, in videos or films, and so on.

If information gets into unauthorized hands, it can have very far-reaching consequences for the business, such as reputational damage, loss of competitive advantage, loss of technology leadership, lawsuits and penalties, breach of contractual obligations in service level agreements, etc., all the way to complete business cessation.

This raises the question of how information can best be adequately and effectively protected. First of all, it has to be said that this is not an IT problem and certainly not a matter that can be restricted to the IT department, as it completely shuts out those responsible for business processes and support processes as well as the risk owners. The information security risks to which the organization is exposed must therefore be comprehensively and sufficiently identified throughout

* Corresponding author E-mail addresses: [email protected] (S.Zh. Simavoryan)



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the organization. Only then they can be analyzed in terms of their size and evaluated according to their importance to the organization.

If an organization has poorly identified its information security risks, it has a poor foundation for choosing appropriate information security measures. As a result, the proposed information security risk treatment may be ineffective or inefficient, and thus inappropriately expensive.

It is the aim of this article to present the identification of information security risks from a methodological point of view and to provide guidance on how to do so in practice.

2. Discussion ISO 31000: 2009 describes a risk management process. Its subprocesses have the following

content:

Communication and consulting: supports defining and adapting the context as well as the continuous improvement of the other sub-processes during all phases of the risk management process.

Defining the context: Defines the objectives of the organization for the implementation of the risk management process, determines the external and internal issues as well as the needs and expectations of external and internal interested parties

Risk identification: Identifies the information security risks associated with the loss of confidentiality, integrity and availability of information within the scope of the Information Security Management System (ISMS) and identifies the risk owners.

Risk analysis: estimates the possible consequences of the identified risks, estimates the realistic probabilities of occurrence of the identified risks and determines the risk levels.

Risk Assessment: compares the results of the risk analysis with defined risk criteria and prioritizes the risks analyzed for the risk treatment.

Risk treatment: selects one or more appropriate risk treatment options, taking into account the results of the risk assessment, and sets out all measures necessary to implement the chosen risk treatment option (s).

Monitoring and review: identifies changes in the external and internal issues, the needs and expectations of external and internal interested parties, scope and risk criteria; provides further information to improve risk identification, analysis and evaluation; analyzes and gains insights from events, changes, trends, successes and mistakes; identifies emerging risks; recognizes the effectiveness and efficiency of the entire risk management process.

Let's describe the process of preparation for risk identification according to HansPeter Königs, 2013; Decker, Karsten M., 2017:

1. Setting the context First, the context needs to be specified. This must include all topics relevant to the purpose of

the organization, the interested parties relevant to the ISMS and their information security requirements be determined. If these considerations are made too narrowly or too superficially, essential elements are disregarded and, as a consequence, risk identification will inevitably be incomplete.

1.1. Defining the external context The external context encompasses all relevant topics, interested parties and their

requirements outside the sphere of influence of the organization. Examples that apply to organizations of all kinds include legislators and legislators, suppliers, contractors, competitors, and customers.

1.2. Defining the internal context The internal context encompasses all relevant topics, interested parties and their

requirements within the sphere of influence of the organization. Examples that apply to organizations of all kinds include goals, strategies and policies for their achievement, structure and leadership, including roles and responsibilities, employees, processes and procedures, and skills in the form of resources and knowledge (capital, time, persons, Processes, systems, technologies).

2. Determining the scope On the basis of the considerations described above, it is necessary to determine exactly where

the risk management process should apply and where not. To do this the limits and applicability of the ISMS are determined. That concerns the:


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- functions of the organization (products and services); - processes of organization; - areas of the organization; - the interfaces and dependencies between the activities carried out by the organization itself

and activities carried out by other organizations are determined. The scope of the risk management process may alter from the scope of the ISMS. This is the

case, for example, when an external organization performs or carries out a function, process or activity completely or in part. This organization is outside the scope of the ISMS, although the outsourced function, outsourced process or outsourced activity is within the scope of the ISMS scope. However, as the associated responsibilities remain with the organization, the risks associated with the spin-off must be part of the risk management process.

3. Risk criteria The description of the possible consequences of security events for the organization's

business activities regarding the confidentiality, integrity and availability of information may in principle vary greatly between different persons and when repeating the risk identification process. In order to meet the requirements of consistency, comparability and reproducibility of the results of this process these descriptions must be structured and standardized. This is done with the help of follow-up criteria, which must be specified in detail and with sufficient accuracy before the risk identification process begins.

Some risk impacts are shown in Table 1. The follow-up criteria are of key importance to the risk analysis process, which follows the

risk identification process.

Table 1. Risk consequences

Consequences for confidentiality

Consequences for integrity Consequences for availability

Violation of the privacy of internal or external users

Incorrect delivery due to contradictory data

Service degradation of services

Loss of competitive advantages

Impossibility to produce a correct annual statement

Unavailability of services

Loss of technology leadership

Inability to fulfill legal obligations

Business interruption

4. Risk identification The next stage is risk identification. It is the process of finding, recognizing and describing

risks. Risk is defined as the effect of uncertainty on goals (ISO 31000:2009). The goal of risk identification is to create a comprehensive list of risks based on those events that cause, enhance, prevent, mitigate, accelerate or delay the achievement of goals.

The risk identification process must be based on sufficiently detailed methods and tools so that repeated risk identifications lead to consistent, comparable and reproducible results.

Repeated applications of the risk identification process can help to identify problems with the chosen methods and tools.

Regardless of the methods and tools have been chosen, the risk identification process must ensure that:

- all risks are considered with the required level of detail; - the results are consistent and reproducible, so third parties can understand them; - the results are the same when different people identify the risks in the same context; - the results of repeated risk identifications are comparable. Regardless of the chosen risk identification approach, the starting points are the information

security objectives, the context of risk identification and their scope of application. There are two approaches, which are commonly used to identify risks: - an event-based approach; - an approach based on the identification of values, threats and vulnerabilities.


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Both approaches are consistent with the principles and general guidelines for the ISO assessment of ISO 31000: 2009.

Other approaches can be used, but are only recommended if they can ensure the requirements mentioned above. If ISMS is the target that meets the requirements of ISO/IEC 27001:2013, ensuring these requirements are essential. Let's consider both approaches in more detail

a) The event-based approach The event-based approach identifies risks by looking at events and their consequences.

Considered events may have happened in the past or may be expected for the future. In the first case they can include historical data, in the second case they can be based on theoretical analyzes, expert opinions and expert opinions as well as needs of interested parties.

First, by considering the questions "Who?", "What?", "Where?", "When?" and “Why?" possible events are described. To support the determination of events, in practice event catalogs are used. For example, the Basel Committee on Banking Supervision drew up a detailed catalog for the banking sector (Basel, 2006).

It identifies the root causes of these events to gain a deeper understanding of the risks and to gain insight into the underlying threats and vulnerabilities.

Finally, the possible consequences are described for all events. Consequences that can not be attributed to the goals of the organization do not add to the risks and can therefore be ignored. However, if such consequences are perceived as actually contributing to risks, this indicates that there are omissions in the list of goals of the organization that should be corrected.

The advantage of this predominantly cost-oriented approach is that its use is associated with relatively little effort. This makes it suitable for creating a first, rough picture of information security risks. It can be argued that such a focus of risk identification on the critical risks is supported.

The disadvantage is that existing threats and vulnerabilities as possible causes of the events are not necessarily determined systematically. This complicates the goal-oriented selection of measures in the subsequent risk treatment process. Another disadvantage is that risks can be overlooked.

b) The value, threat and vulnerability based approach This approach identifies risks by looking at values, threats, vulnerabilities, and related

consequences. These consequences arise when threats exploit vulnerabilities in a value or group of such assets and thus cause harm to an organization.

A value is anything that is valuable to an organization and therefore needs protection. Two types of values can be distinguished: primary values and supporting values.

The primary values are made up of business processes and activities, as well as information that is central to the organization's purpose. These are the values that must be considered first in risk identification.

The supporting values can be classified in hardware, software, network, personnel, location and organization. They may be considered containers in a broader sense to process, store, archive, or otherwise process or handle the primary values. Supporting values typically have vulnerabilities that can be exploited by threats designed to harm the primary values.

For each value, should be named a person who is responsible for the handling of the asset as well as its maintenance and safety. This person is often the most suitable one to estimate the value.

A threat is one possible cause of an incident that can cause harm to a system or organization. Threats can exploit the vulnerabilities of one or more values. They can be based on natural

phenomena, accidental, accidental or intentional and therefore conscious and deliberate origin. Threats can be classified by type. ISO/IEC 27005: 2011 suggests e.g. the following

classification: - physical damage; - natural events; - loss of utilities; - disturbance due to radiation; - compromising information; - technical failures;


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- unauthorized actions; - compromise tasks.

Fig. 1. Multidimensional model of classification of Threats

On another hand this classification can be sufficient for a stable environment (a small

organization) where security threats are relatively stable, but in an ever-changing environment, organizations do not protect themselves from insider threats (Geric, Hutinski Z, 2007). In fact, organizations are subject to several threats that affect their reputation, and it is important that they identify all characteristics of threats in order to mitigate their risks. Classification allows an organization to know the threats that affect their assets, and the areas in which each threat can affect and therefore protect their assets in advance. In addition, it helps managers create


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information systems of their organizations with fewer vulnerabilities. In addition to this, the main problems can be identified in the work of existing threats. In fact, existing classifications do not support the principles of classification (Lindqvist, Jonsson, 1997; Gordon, Lawrence et al, 2005; Tang et al, 2012). At this point, the usual solution is to combine different classifications and create a hybrid. Because of the above results, we propose a hybrid model of classification of threats to the security of the information system, which we called the multidimensional model of classification of threats, intending to respect all principles of threat classification. The main idea of our model is to combine the criteria of threat classification and show their potential impact. The model is shown on the fig. 1.

When determining threats, this classification scheme can ensure that important threats are not forgotten. According to Simavoryan, 2017 malicious actions can be searched by automated means using specialized software. To do this, you need to build a specialized database. The structure of such a database was developed by the authors on the basis of FSTEC, 2015 is shown in fig. 2.

A vulnerability is a weakness of a value or measure that can be exploited by one or more threats.

Weaknesses are thus essential properties of values or measures. These properties do not have to be exclusively negative. For example, the great agility of mobile devices of all kinds (Laptops, netbooks, tablets, smartphones, etc.). On the negative side, however, this agility makes these devices easier to exploit for threats such as theft, eavesdropping or remote espionage. The same applies to values such as removable media.

Analogously, in the case of measures, e.g. be called a weak access control. While these facilitate access to premises and access to systems or applications, on the negative side are the exploitability of threats such as theft or destruction of data media, documents or equipment, the unauthorized use of equipment, the falsification of data or the denial of activities to call.

It is consistent with the logic of the value, threat and vulnerability-based approach to risk identification that a vulnerability itself can not cause harm because a threat must exist to exploit this vulnerability. However, such vulnerabilities need to be identified and monitored because changes in the internal or external environment can re-emerge appropriate threats.

When the vulnerability of a value is exploited by a threat, an immediate impact on information security is first caused. For example, confidential information is disclosed to unauthorized persons, falsified documents are circulated or an information system fails. Such events may eventually trigger consequences for the organization's operations. In the examples mentioned, these may be e.g. breach of contractual agreements or applicable laws, performance degradation or unavailability of services or a complete interruption of operations.

The distinction between immediate information security implications and consequences for the organization's business causes loss of confidentiality, integrity or availability of information. The effects and consequences of each event where a threat exploits a vulnerability of a value must be identified and described with sufficient accuracy.

Vulnerabilities can be classified as threats by type. ISO/IEC 27005: 2011 classify them according to supporting values:

- hardware; - software; - network; - staff; - location; - organization. The most important advantage is that this cause-based approach allows the consequences of

events to be systematically linked to the weaknesses of values and measures. This creates the prerequisites for the measures to be selected in a very specific and targeted manner in the subsequent risk treatment process, and for the required scope of implementation and depth to be precisely determined. Thus, both the effectiveness and the efficiency of the ISMS is ensured. Also, this approach can better ensure that all relevant risks are taken into account.

The disadvantage is the potentially great effort for this approach. The identification of relevant values is not always easy. Each value can have one or more vulnerabilities, each of which


84

can be exploited individually by one or more threats. That is why the number of events can increase very quickly in a combinatorial way.

5. Identification of the risk owner For the process steps following the risk identification, it is important to identify a risk owner

for each identified risk. These individuals are responsible for managing these risks, which can also be cross-process. In order to fulfill this responsibility, the necessary resources must be assigned to them by the process owners.

Fig. 2. Database logical model

3. Results The importance of methodological risk identification for the effective management of

information security risks is not recognized in most cases. However, it is essential for an effective risk management process. Without one that is tailored to the specific needs of the organization, the whole process has little value. Any risk that is not identified in this process step is ignored in the subsequent process steps. There are different approaches to risk identification whose respective strengths and weaknesses must be taken into account when defining the process.

In the practical implementation, the fulfillment of various conditions is crucial for the success. However, if risk identification is carried out with due diligence, there are at least two major benefits immediately. On the one hand, the prerequisites are created to prepare and manage not only the selection of measures of any kind in the subsequent processes of risk analysis, assessment and treatment, but in particular, also to determine their scope and depth, to the extent that they are responsible for the Organization is required. The cost-effectiveness of the measures implemented in this way can be (significantly) higher than simply relying on the frequently


85

advertised good practices without further consideration and differentiation. On the other hand, sound risk identification also creates the prerequisite for a discussion with hardware and software suppliers on an equal footing. This can help to save unnecessary costs.

4. Acknowledgement The reported study was funded by Russian Foundation for Basic Research (RFBR) according

to the research project No. 16-01-00527. References Basel, 2006 – Basel Committee on Banking Supervision, International Convergence of

Capital Measurement and Сapital Standards; A Revised Framework, comprehensive Version, June 2006, Bank for International Settlements.

Decker, Karsten, 2017 – Decker, Karsten M. (2017). Informationssicherheit–ohne methodische Risikoidentifizierung ist alles Nichts." HMD Praxis der Wirtschaftsinformatik 54.1: 21-36.

FSTEC, 2015 – FSTEC of Russian Federation "Methodology for determining threats to information security in information systems", 2015.

Geric, Hutinski, 2007 – Geric S, Hutinski Z. (2007). Information system security threats classifications. Journal of Information and Organizational Sciences; P. 31-51.

Gordon, Lawrence A., et al, 2005 – Gordon, Lawrence A. et al. 2005 CSI/FBI computer crime and security survey. Computer Security Journal 21.3 (2005): 1.

HansPeter Königs, 2013 – HansPeter Königs (2013). ITRisikomanagement mit System. Praxisorientiertes Management von Informationssicherheits und IT Risiken. Springer Verlag, 4. Auflage.

ISO 31000:2009 – ISO 31000:2009, Risk management – Principles and guidelines. ISO/IEC 27001:2013 – ISO/IEC 27001:2013, Information technology – Security techniques

– Information security management systems – Requirements. ISO/IEC 27005:2011 – ISO/IEC 27005:2011, Information technology – Security techniques

– Information security risk management. Lindqvist, Jonsson, 1997 – Lindqvist, Ulf, Erland Jonsson (1997). How to systematically

classify computer security intrusions." Security and Privacy, Proceedings., 1997. IEEE Symposium on. IEEE, 1997.

Simavoryan, 2017 – Simon Zh. Simavoryan, Arsen R. Simonyan, Elena I. Ulitina, Rafik A. Simonyan, Elina A. Pilosyan, Nadezhda A. Kornienko. (2017). Search Fuzzy Image of the Attacker Based on the Use of Automatic Classification Methods // Modeling of Artificial Intelligence, 4(1): 29-38.

Tang et al., 2012 – Tang J, Wang D, Ming L, Li X. (2012). A Scalable Architecture for Classifying Network Security Threats. Science and Technology on Information System Security Laboratory.


86



Multidimensional Limit Theorems in Models with Categorized-Time Absolute Priorities Arsen R. Simonyan a , *

a Sochi state university, Russian Federation

Abstract In the queuing theory, there are single-channel models with a Poisson incoming flow. Among

these models, parametric models are considered to be the most suitable for use when priority among incoming flows is specified by functions that depend on one or more parameters. Such models are called parametric models and their research is rarely found in modern scientific literature, since a complex apparatus of the theory of random processes is used.

The article describes the class of all possible limit distributions for a random vector of waiting time in the same parametric system mass service with absolute priorities. In the process of the results, obtained limit theorems for univariate and multivariate characteristics of the system related to the timeouts.

Keywords: queueing system, waiting times, random vector, limit distributions, the period of employment, length of queue, random process, distribution function.

1. Introduction The parametric model of queueing system considered in this paper is based on the principle

of the quantification of the time axis (Bronshtein, 1976). In a single-server queueing system with waiting there arrive independent Poisson flows of

customers, … , customers with parameters , respectively. The service times are independent in their totality, do not depend on the arrival process and for the customers ( ) they have the distribution functions . There are no customers in the model at time . The time axis is divided into intervals of fixed length, called "quanta": ,…; an customer has absolute priority before a customer ( ) if both arrive in the system in the same quantum. In the zones of all the flows, arriving at different quanta, the customers are served in the order of arrival. The indicated model is called a model with categorized-time absolute priorities or a ( ) scheme (Simonyan, 2014; Danielyan, 1980). The quantity T is the parameter of the model ( ).

We introduce notations: ∑ is the load of the model by the customers (

customers, … , customers), where ∫

is the

under-load of the model by the – customers; is the conditional virtual waiting time

of a customer at the moment under the condition that the accessibility of the customers in the model ceases starting from the moment .

* Corresponding author E-mail addresses: [email protected] (A.R. Simonyan)




87

The investigation is carried out the following conditions: if , then for

∫

one has the expansions

, (1.1)

where constants. For a given , we represent time in the form

is an integer).

For and an "arbitrary variation" of , the limit distributions for

depend in an essential manner on the relation between and and on the ratio of the loads and we are interested in the case . For the correct formulation of the problem one requires additional explanations. We turn for to the ratio ⁄ . The limit points of this ratio fill out completely the interval at an arbitrary variation of . If for ⁄ when one considers the limit point 0, then one has to take the limit points of

another ratio

⁄⁄ , where ( ) .

The limit points of the last ratio, under the condition that one considers the limit point 0 of the ratio ⁄ fills out completely the semiline [0, ). In the usual sense, for , the limit distribution for

does not exist. But if we take a sequence of moments when

such that there exist the limits (

⁄ ) and (

⁄⁄ ) , then there

exists a limit distribution for ).

Consequently, we assume , and thus, that there exists the limit

⁄⁄ (1.2)

Condition (1.2) appears at the investigation of the limiting DF for when

. In (Danielyan, 1982) one solves a series of problems, one of which is formulated in the following manner.

Under the condition of the existence of the limit (1.2), it is necessary to describe the class of all possible limit laws for

. We formulate the results of (Danielyan, 1982) for

. We introduce notations. Let be the DF of a nonnegative random variable (RV), defined by its Laplace-Stieltes transform (LST)( Re is Euler's Gamma function):

∫

{

( ⁄ )∫

}

Let be the stationary DF of the waiting time of a customer in the scheme We set

{ } ∑

⁄⁄

Table 1.

Load Limit point Norma-lization

Cen- ter- ing

Scheme Limit distribution

1 0

⁄ 0

Under a unit load there exists the limit

P {

}


88

where the functions occurring in (1.3) are given by Table I. We note that for and the condition of type (1.2) is not required. As shown in

(Danielyan, 1981), there exists the limit (1.3), where , . The Formulation of the Problem. Assume that in the scheme we have , the

loads are fixed, conditions (1.1) hold, and for , the quantity can vary in an arbitrary manner. Under the condition of the existence of the limit (1.2), one has to describe the class of all possible limit laws for the vector

.

In Sac. 2 we give auxiliary results, some of which will be proved. In Sac. 3 we formulate and prove the fundamental results of the paper. 2. Auxiliary Results The analysis of the scheme is based on the relation between the processes

.

and the processes

where is the total service time of the customers, arriving over a time interval of length relative to is a process with independent increments); w(u) is the virtual waiting time at the moment u of the model M|G|1|∞ (Simonyan, 2004) with entrance intensity and with DF of the service time of the customers (hare ); is the busy

period in the servicing of the customers with lag in the scheme ( ) , i.e. the time interval starting with the lag , at the beginning of which there are no customers, and ending first time after the gap when the servicing device is free of the customers relative to is a process with independent increments). Here by the lag we mean the interval time in which the customers accumulate but are not served. The relation between the above described processes in the "terminology of RV" is established in (Danielyan, 1982) and is given by the following statement.

LEMMA 2.1. Let . Then for we have the

relations

{

if ( )

( ( ))

if ( )

The symbol d indicates the equality of the DF of both sides of the random equality on the set indicated in the rlght-hand side of the equality;

is the conditional virtual waiting time of a

customer at the moment t in the scheme ( ), i.e. By virtue of formulas

(2.1), for the proof of the limit theorems in the case for , one has to have

available the corresponding limit statements for the processes for for . We denote ( )

⁄ {

⁄

Here is the total service time of the customer, arriving in an interval of time .

LEMMA 2.2. Assume that conditions (1.1) hold. Then for there exist limits ( )

P

P{ }

P

where

∫ ((

) ⁄

(

) ⁄

)


89

∫

Here is a stable law with parameter , defined by its characteristic function ( is the imaginary unit, is a real number):

∫

The proof of (2.2) is given in (Danielyan, 1982) and (2.3) is proved similary to (2.2). We processed to the proof of the relations (2.4) – (2.7).

We perform the computations ( ; t>0): P

∫P {

⁄

⁄

⁄

⁄

⁄ }

( ∫

∫

)P{

⁄

⁄[

⁄

⁄]

}

In this case for every the number is select sufficiently large so that we should have the inequality

. Then by virtue of (2.2) we have (

∫ P{

⁄

⁄

[

⁄

⁄]

}

where

∫

uniformly with respect to y o Making use of the statement of Theorem 1 from (Shilov, 1965) in the second term of the right-hand side of (2.8) we change the sign of the integral and we interchange the limits; then we let go to zero (then ), which by virtue of (2.9) yields (

∫

P {

⁄

⁄

[

⁄

⁄]

}

By (2.10), the problem reduces to the computation of the limit (

P {

⁄

⁄

[

⁄

⁄]}

We consider the mutually exclusive cases, mentioned in Lemma 2.2: a) Obviously, . Then by virtue of (2.3), we obtain

((

) ⁄

(

) ⁄

)

for where there follows (2.5); b) Obviously, and . Then

⁄

⁄


90

⁄

⁄

Consequently, , from where we o,tain (2.6); c) Obviously, and . Then

⁄

⁄

⁄

⁄

and, consequently, , from where there follows (2.7).

COROLLARY 2.1. The limit relation (2.2) is a consequence of the relation (2.4). Namely (

,

. (2.11)

The proof of the first of the equalities (2.11) follows trivially from (2.5)-(2.7), while the proof of the second one is obvious except in the case . On the basis of (2.5) we have

∫ ((

) ⁄

) ((

) ⁄

)

((

) ⁄

) ((

) ⁄

)

where * is the convolution sign, while the symbol above the convolution sign indicates the variable with respect to which the convolution is taken. The right-hand side of (2.12) is the DF of the random variable

(

) ⁄

(

) ⁄

where the RV and in our case we have has a stable DF . Sinc in our case we have

, on the basis of Theorem 2 (Feller, 1950), we conclude that the RV has a stable DF

this proves the second of the equalities (2.11).

It is known (Feller, 1950) that the DF has density It turns out that the DF has density

COROLLARY 2.2. a) For we have

(

) ⁄

((

) ⁄

(

) ⁄

)

b) for we have

,

where

{ if

if x y.

c) for we have

COROLLARY 2.3. Assume that conditions (1.1) hold. Then there exists the limit

P

where has the multidimensional density

∫

∫

Moreover,


91

∏

∏

The proof is similar to the proof of Corollary I from (Grigoryan, 1982). LEMMA 2.3. Assume that in the scheme of the model Mr |Gr|1|∞ we have the

loads are fixed, and conditions (1.1) are satisfied. Then there exists the limit

P {

⁄

}

We consider the vector function (

( ) (

)+

(

)+

+( ⏟

). (2.13)

Here the processes

( ) and are independent and identically distributed.

IN the right-hand side of (2.13) the vectors are added component wise. Assume that the following conditions hold: for ther exist the limits

Then a consequence of a theorem of E. A. Danielyan (Danielyan, 1980) holds LEMMA 2.4. . Assume that in the scheme of the model Mr |Gr|1|∞ we have

the loads are fixed, and conditions (1.1) and (2.14) are satisfied. Then there exists the limit

P {

}

where

{

otherwise.

3. Fundamental Results

We describe the class of all possible limit DF of the vector process (

) for

, fixed loads and for an arbitrary variation of . TEOREM 3.1. Let , assume that the loads are fixed, conditions (1.1) are satisfied, and

for there exists the limit (1.2). a) If then there exist the limit

P {

⁄

}

where is the limit distribution for and jf the vector

(

);

b) If then there exist the limit

P{ }

(

)

PROOF. The proof of the theorem follows immediately from Lemmas 2.1-2.4. References Bronshtein, 1976 – Bronshtein O.I., Dukhovnyi I.M. (1976). Modeli prioritetnogo

obsluzhivaniya v informatsionnovychislitel'nykh sistemakh [Models of priority maintenance in information and computing systems]. M.: Nauka, 220 p.

Simonyan, 2014 – Simonyan A.R., Simonyan R.A. (2014). Modern Trends in the Study of Single-channel parametric Models of Queueing // Modeling of Artificial Intelligence. № 4 (4). pp. 184-188.

Simonyan, 2005 – Simonyan A.R., Ulitina E.I. (2005). O parametricheskikh modelyakh massovogo obsluzhivaniya [On parametric queuing models]. // Obozrenie prikladnoi i

https://elibrary.ru/item.asp?id=22789956


https://elibrary.ru/contents.asp?issueid=1359913

https://elibrary.ru/contents.asp?issueid=1359913&selid=22789956


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promyshlennoi matematiki. T. 12. S. 184. Danielyan, 1980 – Danielyan E.A. (1980). Waiting time in a model with categorized-time

priorities // Kibernetika, No. 6, pp. 103-109. Danielyan, 1982 – Danielyan E.A., Khostikyan P.T. (1982). Limit theorems in the models Mr

|Gr |1| ∞ with time-categorized absolute and relative priorities for fixed // Intercollegiate Transactions in Mathematics, No. 1, Yerevan , pp. 134-164.

Danielyan, 1981 – Danielyan E.A. (1981). "The mathematical theory of the priority models Mr |Gr |1| ∞. Abstract of Doctoral Dissertation, Moscow State Univ.

Simonyan, 2004 – Simonyan A.R, Ulitina E.I. (2004). A Theorem on the convergence to a stableaw in the M|G|1|∞ model // Russian Mathematical Surveys. Т. 59. № 3. С. 589-590.

Shilov, 1965 – Shilov G.E. (1965). Mathematical analysis. Oxford, New York, Pergamon Press, 485 p.

Feller, 1950 – Feller W. (1950). Introduction to Probability Theory and its Applications (Volume 2). Published by John Wiley and Sons, 626 p.

Grigoryan, 1982 – Grigoryan G., Danielyan E.A. (1982). On a method of analysis of priority systems // Intercollegiate Transactions in Mathematics, No. 1, Yerevan, pp.165-196.



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93



Forecasting in Decision-Making Support Systems Oleg V. Tikhanychev a

a Research Institute of Control, Information and Modeling Academy of Military Sciences, Russian Federation

Abstract To improve management efficiency, different methods of support tools for decision makers

may be used at all stages of the management cycle. The paper addresses the application of mathematical modeling in decision making support systems. In modern conditions it is the main mathematical modeling tools produce forecasts. The new work analyses the main trends of decision making support systems, in the first place, in terms of application of prediction tools as part of such systems.

Keywords: forecasting, prediction, mathematical simulation, decision-making support system, mathematical modelling, control automation

1. Introduction Management theory and practice often uses “a standard management cycle” concept.

A standard management cycle includes a number of main stages: goal setting (or verification of a task set), situation evaluation, decision making, planning, targets setting and control of their implementation. Any decision-making support systems (DMSS) implements this very management cycle, providing decision making support at all stages.

An integral part of any management cycle is predicting the consequences of implementing the decisions made. To improve the reliability of the forecasts made at different times, different mathematical tools have been used, each of which has certain advantages and disadvantages, when applied in different situations and within certain limits.

2. Discussion To improve management efficiency, different methods of support tools for decision makers

may be used at all stages of the management cycle. Existing approaches to implementation of the formation of management decisions involve the use of DMSS or expert systems. In terms of solution of the most complicated issues, when every mistake comes at a high cost, and when a decision maker is personally responsible, the DMSS is the basic tool providing more opportunities to the manager to make a competent decision.

An integral part of the management cycle, including implemented with the use of DMSS, is the prediction of the consequences of the decisions being made. It is true for every process: activity planning, management of processes and systems, design and development of systems.

To improve the reliability of the forecasts made at different times various mathematical tools have been used: expert evaluations, formal logic, theory of probability, games theory, trend extrapolation methods, mathematical modeling, etc. (Bruce, 1996; Vypasnyak, 2014). Mathematical modeling is commonly used to predict the behavior of complex systems at a high cost



94

of decision errors as one of the most reliable means of prediction (Vypasnyak, 2015; Tikhanychev, 2012; Tarnovskiy, 2016).

These methods are usually divided into two basic groups of predictive estimation methods: intuitive (expert) methods dealing with the subjective judgements, and formal methods using calculation methods and mathematical models. These models are implemented through application of various mathematical tools: starting with the expert assessment methods and up to complex mathematical models implemented in factual approaches. Heuristic methods use both formal and subjective approaches (Fig. 1).

As noted previously, each of the methods in Figure 1 has its limits to applicability with their own advantages and disadvantages.

Expert methods allow prediction in non-algorithmic situations, but they are less suited to automation and have no such good operational efficiency. Heuristic methods are based on the construction of analogies. Factual methods based on time series models are simpler and more effective, but can give serious errors in case of an abrupt change of parameters, especially if these changes were not previously known. Factual methods based on problem domain models and logical and probabilistic models provide a detailed and fairly accurate prediction, but they are demanding of computing resources and less effective, especially in terms of data input.

Fig. 1. Generalized classification of forecasting methods

Selecting a certain prediction tool in DMSS is determined by the conditions of implementing a certain management cycle and specific features of each prediction method.

Application of prediction tools in automated DMSS that ensure management of complex man-machine systems has certain features (Tikhanychev, 2016; West, Harrison, 1997; Xu, Zhou, 2011; Xu, 2013):

- high cost of error decisions that require a high prediction accuracy; - automation of initial data collection, their processing, formation of aggregated output

model data significantly reduces the respective requirements for the system components, including mathematical models;


95

- prediction efficiency shall meet the requirements for the duration of management cycle, which in its turn is determined by the responsiveness of the controlled system;

- DMSS is usually designed for addressing ill-defined problems. In order to obtain acceptability assessment, these features are associated to the

characteristics of certain prediction methods. The use of expert approaches requires involvement of a representative expert group for each specific problem, while this can take quite a long time. Reducing the number of experts impairs the accuracy of prediction. Moreover, it is theoretically possible, when no experts in a specific problem can be involved in the process, the work of the whole DMSS prediction subsystem can be wiped out. These drawbacks of the expert approach reduce the possibility of their use in the automated DMSS (Tarnovskiy, 2016; Zhang et al., 2015; Lototsky, 2016).

At the same time, the main shortcomings of mathematical models can be fended off due to their diversity, allowing the user to select the type of a model for a specific task, as well as use the automation software and equipment to improve the efficiency of data input and analysis of simulation results. Herewith, the accuracy of simulation is generally not impaired, ensuring prediction validity while maintaining the efficiency of its obtaining. Working with non-algorithmic problems with the use of models in automated DMSS has no special problems, as automated systems by definition handle the formal data, even when describing non-algorithmic situations.

3. Conclusion The analysis of management practices shows that prediction of the results of the decisions

made in a changing environment is one of the most important stages of decision making. Thus, developing DMSS shall require implementation of prediction tools designed, in the first place, on the basis of mathematical modeling.

Thus, based on the analysis of the requirements for prediction means in DMSS and considering the specific features of prediction methods, it is appropriate to apply mathematical modeling as one of the most reliable and efficient prediction tool to predict the behavior of complex systems under automated control.

References Bruce, 1996 – Bruce W. (1996). Fowler De Physical Belly. In Introduction to Lanchestrial

Attrition Mechanic, DMSTTIAC IIT Research Institute. Part I-II, 1996. Lototsky, 2016 – Lototsky V.L. (2016). Spatial Information Modeling. Modeling of Artificial

Intelligence, Vol.(10), Is. 2, pp. 94-103. Tarnovskiy, 2016 – Tarnovskiy D.A. Actual Questions of Mathematical Modeling. //

Modeling of Artificial Intelligence, Vol.(10), Is. 2, pp. 117-124. Tikhanychev, 2012 – Tikhanychev O.V. Decision-Making Support Systems: Prospects for

Troops Control Automation. Military Thought. Vol. 21 No.3, pp. 74-83. Tikhanychev, 2016 – Tikhanychev O.V. (2016). Modelling in Decision-Making Support

Systems. Мoscow: Editus Publ., 74 p. Vypasnyak, 2014 – Vypasnyak V.I. and etc. Combat Simulation: Past, Present and Future.

Military Thought. Vol. 23 No.3, 2014, pp.30-41. Vypasnyak, 2015 – Vypasnyak V.I. and etc. (2015). A Decision-Making Support System as a

Virtual HQ. Military Thought. Vol. 24 Number 1, pp. 129-136. West, Harrison, 1997 – West M., Harrison J. Bayesian Forecasting and Dynamic Models. Inc.

Springer-Verlag New York, 680 p. Xu, 2013 – Xu Z. (2013). Intuitionistic Preference Modeling and Interactive Decision Making.

Springer, 2013, 250 p. 13 illus., 8 illus. in color. Xu, Zhou, 2011 – Xu J., Zhou X. (2011). Fuzzy-Like Multiple Objective Decision Making.

Springer, 461 p. Zhang et al., 2015 – Zhang G., Lu J., Gao Y. (2015). Multi-Level Decision Making: Models,

Methods and Applications. Springer, 396 p.


96



Application of Monte Carlo Method for Calculation of the Comets' Area by the Photographic Pictures V.Ya. Tsvetkov a , * a Research and Design Institute of design information, automation and communication on railway transport, Russian Federation

Abstract The article describes application of statistical method for definition of cross-sectional area of

comets and other moving astronomical bodies. Application conditions of this method are described. The simulation modeling which form the basis of this method, is defined. Spatial restrictions which are imposed on the application of this method, are described. The article describes technical implementation of the method. Analysis of method errors is provided. Fields of method application are described. The method is applicable for areal objects estimate by means of thermal pictures and radar images.

Keywords: space researches, comets, celestial bodies, statistical computations, Monte Carlo method, cross-sectional area.

1. Introduction Comet research represents an important field of space researches (Anders, 1989, Weissman,

2002, Bryant, 2014). One of characteristics of small celestial bodies, meteorites and comets is their cross-sectional area (Jenniskens, 1997). This characteristic is conditional considering that the cross section is defined by choice of the cross-sectional plane. The cross section is defined by the observation point. In the event of using the camera the cross section is determined by the position of the camera view point and obliquity of the principal optical axis in relation to direction towards the object. However, for all its disadvantages, the cross section provides quite complete information on the size and weight of the object. From the perspective of space geoinformatics (Bondur, 2015) the cross section of the comet represents an areal object. Areal is commonly referred to as a certain space domain which has an area.

The main toolkit for areal objects identification is space monitoring (Black, 2002). The main toolkit for research of the objects which has a cross section are space images (Aldrich, 1971). During last decades analogue photographic pictures were replaced by digital ones (Vislotskii, 2000). Digital pictures are inferior to analogue ones when imaging of high-speed processes and bodies traveling at high speed. However, the simple acquisition and the promptness of the data transfer have determined a migration to digital imaging of the astronomical objects. The remote sensing allows acquisition of various images – photogrammetric, thermal, and radar ones. The suggested method makes it possible to process images obtained in different spectrums of electromagnetic waves: optical, infra-red, ultra-violet, radar and X-ray one. This actualizes the development of

* Corresponding author E-mail addresses: [email protected] (V.Ya. Tsvetkov)




97

methods for area determination of arbitrary shaped objects according to the photographic pictures and models.

2. Material and methods of the research The existing works in the field of Solar system structure and works in the field of the units of

measurement were used as material. The statistical and correlation analysis were used as a research methodology.

3. Results The method is based on using of the well-known Monte Carlo model (Mooney, 1997, Robert,

2004, Earl, 2008) for the statistical analysis. The source material is obtained on the basis of the remote sensing. Proposed method comprises of conditions and limitations. The basis of the method is an availability of the space picture containing two images of one moving astronomical object, obtained at time interval T. In this case the background on which the moving object is depicted is not considered for calculations. The other version is an availability of two comparable images. In this event the background or the background image is of importance for images identification.

Let us consider the first version according to which there are two pictures of one moving objects are depicted on one image at different time points (Fig. 1).

X

b

C C

L = V t

a

Fig. 1. Information situation (Tsvetkov, 2012) of the object movement

The underlying situation of the object (comet) fixation is identified with the letter а at Fig. 1.

In time interval t the repetitive imaging is carried out and object is fixed in a position which is marked with letter b. The speed of an object V is considered to be known. Besides, the boundaries of the object should be recognized. From the perspective of the comet it is a core (C) and an exterior contour. The conventional coordinate system XY is set on the image.

Considering the known speed V and time interval t between the imaging it is possible to gauge the distance L which the object has covered during the imaging.

L= Vt.(1) The value L is descriptive of the actual distance. The distance at the model l which can be

measured according to the coordinates of the image will correspond to the actual distance. Thus the scale of the imaging mx can be determined along Х axis as

mx= L / l.(2) Parameters of the camera for scale determination are not to be considered inasmuch as

during imaging of the moving objects the anamorphic images are obtained. The anamorphic images are images which scale along X-axis – mx and scale along Y-axis – mY are different.

mxmY. Fig. 2 shows auxiliary geometric constructions, which provide the basis for the future

calculations. The condition of the geometric constructions is fixation of the object (comet) boundaries at the image. It is quite complicated condition for the comets which contour changes its shape. However, if the time interval between two imagings is not long the comet contour is not reshaped considerably and the boundaries of two situations of object fixation can be compared.


98

DC

l = k D

L = m lX

Y

Fig. 2. Information situation of geometric constructions

According to the image the particular size D is selected, which characterizes cross-sectional

dimension of the object along Y-axis. This value can be measured by using the image and is known. The particular rectangle is selected at the image (indicated by crosshatch at Fig. 2). One side of the rectangle is set by cross sectional dimension D at the object image (Fig. 2). The second side of the rectangle is set by l parameter, which is also measured according to the image as the distance between two positions of the object (Fig. 1, Fig. 2).

The picture of boundary points at the image should be sharp to make it possible to form the correct geometric area using numerical intervals.

The condition of area calculation comprises of using random number generator with homogeneous distribution which covers the set interval of the rectangle of the geometric field. The rectangular area is necessary for simple setting of the rate for the random number generator based along two coordinates. The situation of modeling and area calculation is shown at Fig. 3.

The particular rectangle lD is indicated by crosshatch on which the object researched is laid BA (BigArea). Instead of BA comet core С can be researched.

D

l

BA

X

Fig. 3. Information situation of the area calculation

This method applies the simulation modeling, according to which "point" with "random"

coordinates is "dropped" on the picture of a rectangular area in accordance with Monte Carlo

method. The field of dropping is set by the boundaries of the particular rectangle lD. Along Х axis the interval of random numbers is set by the value l. Along Y axis the interval of random numbers is set by the value D.

The essence of Monte Carlo method application is generating coordinates xm,ym, in a random manner which set the random point М(xm,ym) at the known rectangle – standard A. In the event of ramdom dropping М(xm,ym) always falls into the area of the standard. This provides reasons to introduce the first relation – the relation of the actual standard area to the number of random dots on it. As this takes place the number of points at the standard is always proportional to its area, but depending on the point density the relation will be different. The point density is determined by the number of droppings. The larger is the number of droppings the higher is the density.

Another areal BA which area at the planet is unknown is situated in the standard areal А (cross section of the moving object with sharp boundaries). In case of random dropping the point М(xm,ym) in some cases falls into the object area on the image. During series of droppings a part of


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points fall into the object area BA meanwhile the part fall outside the object. But every point falls on the standard. In the event of large number of tests the number of points on object area BA becomes proportional to its area. It's possible to introduce the second relation – actual area of the object (area on the surface) to the number of random points on it. In accordance with the law of large numbers, as the number of tests is increased the limits of two marked relation of the area to the number of point become equal.

The following parameters are used for the calculation. SA is standard А area in the field, SPA is the model area (the pictures are in image) of the standard А in image, SВА is object О area in the field, SPBA is area of the object model BA in image, N is a total number of points during tests, NBA is a number of points fallen into the zone of the object BA on model, NE is a number of points outside the object on standard A model. In the initial state the point counters

N=NBA=NE = 0. As a result of modeling

N≠ 0, NBA 0, NE 0, N= NBA + NE.

Every point M(x,y) has random coordinates xm,ym according to which it is determined

whether this point is situated in the field of an object or outside of it. Mathematically boundaries of the object BA at the model is defined by two conditions depending on the selected Information situation. The conditions are read as follows: "above – below" are vertical boundaries and "left – right" are horizontal boundaries.

Y1(x)

Y2(x)

BAXminXmax

X

Y

Fig. 4. Information situation of boundaries definition of the object researched

For information situation on fig. 4 the boundaries are defined as follows: The condition

"above – below".

Y1(x)>Ym(xm)>Y2(x) (3).

Condition "left – right" xmin<xm<xmax(4).

The functions Y1(x),Y2(x) are determined on the basis of the boundary approximation BA.

The values xmin, xmax are determined on the basis of the image measuring. If a random point falls into BA

NBA= NBA+1 If for random "dropping" and obtaining coordinates of the points xm,ym at least one of

conditions (3) и (4) is not met or both of them are not met, the point is outside the object. In this situation the point counter outside the object area increases by one.

NЕ= NE+1

In the event of every test

N= N + 1.


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During the process of modeling N= NBA+NЕ >1000 tests are carried out. Modern computers

are able to perform this operation in several seconds. After tests completion the weight or "conventional area" is determined. It is estimated on assumption that

SA SPA N (5)

The area of the standard A on the surface of the comet which is proportional to the area of the

standard in image and also proportional to the total number of points which were used for modeling. In addition to the expression (5) another proportionality exists.

SBA SPBA NBA (6)

The value in the event of large numbers N is a constant for two relations.

= SA/ NA= SO/ NAO (7)

It is called the weight of point during random test and/or "point area". It follows from here

that unknown area of the object BA will be determined as follows

SBA=NBA (8)

Let us estimate the deviation of the method. The method is iterative, therefore its deviation is

associated with the notion of convergence. For estimate of the calculation deviation it is required to perform additional K<<NA tests. It is possible to determine the difference between estimated area of the object SBA(N) in the event of tests number N of points and the object area SBA(N+ K) in case of tests number N + K of points.

SO= [SBA- SBA (N+ K)] (9)

In addition, a part of points KOA fall into area of the object. Simple arithmetic calculations

using expressions (7), (8), (9) provide the formula of relative error which is calculated by means of formula

= {(NBAK - NKBA)/(N[NBA+ KBA] )} (10) A numerical illustration 1) The number of tests NA=1000, the number of points fallen into the object NAO= 500.

The number of additional tests K=20, the number of additional points fallen into the object KO= 10.

= 0. 2) The number of tests NA=1000, the number of points fallen into the object NAO= 500.

The number of additional tests K=20, the number of additional points fallen into the object KO = 5.

Relative deviation = 0.009 or 0.9 % of the measured object's area (but not a standard object). The value NA can be increased by times, but the basis should be formed by a reasonable

sufficiency. Provided that the accuracy/deviation does not exceed the tolerance calculations can be stopped and a series can be ceased.

4. Discussion This method is conditional but in a number of instances it is the only one that allows

obtaining of the result. An idea of comet's cross section construction is related to the estimation method of the middle section (Wang, 2002) of an arbitrary shaped body well-known in ballistics. The essence of the method is that if we find an arbitrary shaped object (meteorite fragment, bomb or shell splinter), three of its mutually perpendicular diameters are measured. According to them the volume and a specific weight of the body is conventionally measured. By means of them the penetrative impact of the object is estimated. Therefore this method can be considered as middle section estimate of the comet in plane of parallel plane of the photographic picture.


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5. Conclusion An offered method makes it possible to determine the cross-sectional area of moving objects

by means of different images. The main condition is the presence of the object contour at the background of the other bodies or outer space. It can be used for estimation of areals' areas by thermal pictures and radar images. Considering the simplicity of an algorithm, it is possible to build it in any software system for calculations of the areas.

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Geoinformatics // European Journal of Technology and Design, 4 10(4), 118-126. Bryant, 2014 – Bryant, E. (2014). Comets and Asteroids. In Tsunami. Springer International

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outburst: the cross section of a comet dust trail. Planetary and space science, 45(12), 1649-1652. Mooney, 1997 – Mooney, C. Z. (1997). Monte carlo simulation (Vol. 116). Sage Publications. Robert, 2004 – Robert, C. P. (2004). Monte carlo methods. John Wiley & Sons, Ltd. Tsvetkov, 2012 – Tsvetkov, V.Y. (2012). Information Situation and Information Position as a

Management Tool. // European Researcher. Series A. 36(12-1), 2166-2170. Vislotskii, 2000 – Vislotskii, A.I., Vintaev, V.N., Konstantinov, I.S., & Ushakova, N.N.

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