=Paper= {{Paper |id=Vol-2556/paper16 |storemode=property |title=Mathematical Justification for the Information System for Predicting Dangerous Situation (short paper) |pdfUrl=https://ceur-ws.org/Vol-2556/paper16.pdf |volume=Vol-2556 |authors=Valentin G. Degtyarev,Lidiya A. Kuharenko,Ruslan S. Kudarov,Rustem S. Kudarov }} ==Mathematical Justification for the Information System for Predicting Dangerous Situation (short paper)== https://ceur-ws.org/Vol-2556/paper16.pdf

Mathematical Justification for the Information
System for Predicting Dangerous Situation
Valentin G. Degtyarev Lidiya A. Kuharenko
Emperor Alexander I Emperor Alexander I
St. Petersburg State St. Petersburg State
Transport University Transport University
vdegt@list.ru

Ruslan S. Kudarov Rustem S. Kudarov
Emperor Alexander I Emperor Alexander I
St. Petersburg State St. Petersburg State
Transport University Transport University
r.s.kudarov@gmail.com r.s.kudarov@gmail.com

Fn ( x) = F (an x + bn ) (1)
where - 𝐹" (𝑥) the distribution of the amount of
independent normally distributed random values, the
corresponding parameters of scale and distribution shift.
Abstract In this sense, we can talk about the stability (or, if you
will, conformity) of the form of normal law regarding the
The work focuses on some of the basics of the procedure of composition of distributions.
theory and practice of applying extreme value The various distributions, which are likely to be of some
statistics and the basics of the information model known law of distribution, form the area of attraction of
for predicting dangerous states. The task of
this law. In this sense, all distributions that meet the
predicting dangerous situations is presented and conditions of the Central Limit Theoreme (e.g., Lapunov's
solved on the basis of a statistical description of conditions) are part of the area of attraction of normal
the characteristics of the object and taking into
law.
account the probabilistic characteristics of the
The second property is that for an arbitrary infinite
extreme external conditions of the object.
sequence of distributions that meet the conditions of the
Central Limit Theorem, the amount of asymptomatically
1 Introduction. Sustainability and the area normal in the sense of convergence by probability
of attraction of some distributions measure. In other words, many distributions that meet the
conditions of the Central Limit Theoreme (e.g. Lapunov's
Let's present known distribution properties, using perhaps conditions) are part of the scope of the normal law. The
unusual terms in this context. The composition (or simplicity and ease of applying normal law in applications
distribution of the sum) of two independent normal largely determined by these properties.
distributions leads to a normal distribution. And in Naturally, the question is whether there are distributions
general, the sum of any number of normally distributed and procedures with similar properties. For example, it is
random values is a normally distributed random value. obvious that an exponential law is sustainable when
This property expressed by the solution of the functional selecting a minimum for an arbitrary number of random
equation:1 amounts subject to that law. However, the details of the
properties of this law, which is a private case of gamma-
distribution, discussed below.
Copyright c by the paper's authors. Use permitted under Creative
Commons License Attribution 4.0 International (CC BY 4.0). In the task of modeling extreme situations, important
In: A. Khomonenko, B. Sokolov, K. Ivanova (eds.): Selected Papers of to note that the two-parametric double exponential law
the Models and Methods of Information Systems Research has similar properties: 1) the stability of the form relative
Workshop, St. Petersburg, Russia, 4-5 Dec. 2019, published at
http://ceur-ws.org. to the distribution of extreme value from any number /

92
equally distributed values; 2) is the limit for a certain x max = max xi ,
class, the "3" of the original distributions in the sense of 1 𝑐𝜎, where 𝑥̅ - the average value of The possibilities of applying these statistics to relevant
estimates are significantly limited by the fact that known
distribution 𝜎 - its standard deviation, with с - a pre-
distributions from which the samples are derived are
selected constant. Otherwise, if ,𝑥("+) − 𝑥̅ , < 𝑐𝜎 , the
assumed. Universal in this sense is the approach based on
observation is not rejected. the theorem of B.V. Gnedenko [Gne43] . Here's the
Strictly speaking, the constant value is determined by the following formulation.
level of significance of some specially chosen criterion, Gnedenko's theorem. If the random value distributed by
and therefore in the decision-making possible errors law (4) has a limit distribution, it distributed under one of
known in mathematical statistics 1 and 2 genus. the following three laws after the corresponding rationing.
2. If the anomaly of the observation is confirmed, it is For maximums:
necessary to create an adequate model to plan
observations for the use of factor dispersion or, if FI = exp( - exp( - x )), - ¥ < x < +¥.
possible, regression analysis. You can't limit yourself to a
ìexp( -( - x )a , x £ 0, a > 0,
one-dimensional sample.
3. As we can see, this rule of abnormality testing,
FII ( x ) = í
proposed in n 1, solves the issue only at some, satisfying î 1, x > 0.
researcher, the level of significance of one or another
ì 0, x £ 0,
criterion. Objectively speaking, necessary to recognize the
FIII ( x ) = í -a
possibility of implementing elements of the sample,
anomalous in the sense of p.1, in the same observational îexp( - x ), x > 0, a > 0.
conditions. These distributions called extreme distributions of the
3 Distribution of extreme values type I, II and III respectively.
The distribution of the first type, under some additional
In any case, it is advisable, on the basis of the final conditions, caused, for example, by the task of ejecting
number of one-dimensional samples, located in a non- the values of the stationary random process. The
decreasing order, to turn to the distribution of statistics distribution of the second type, as we can see, is the
suspicious of abnormality. We are talking about statistics distribution of Weibull, in particular, convenient for
of extreme values. describing the strength of the object on the break.
The sample is considered {𝑥" }"(* "() , with equally The first distribution, called the Double Exponential Law,
distributed by law with the function of distribution F(x) describes a wide class of original distributions that make
elements 𝑥3 ,𝑖 = 1,2, … , 𝑁. The extreme elements of the up the area of its attraction. The second and third
sample represented by statisticians: distributions refer to random values limited to the left and
right, respectively. These distributions can brought to
the first [Dav79].

93
5 Extreme Situation Forecast
The task of forecasting strength, taking into account
4 Some properties of double exponential
external conditions, can lead to the composition of the
law two distributions in the following sense.
The annexes look at two forms of recording the double The strength reserve characterized by a different
exponential law distribution function: for maximum characteristic of the strength of the study object X and the
values real load (in the same units of measurement) that occurs
F ( x; q, a ) = 1 - exp(- exp(a ( x - q)) (5) during its operation Y. If 𝑋 ≤ 𝑌, the object is in working
or extreme condition, otherwise, if 𝑋 < 𝑌, the object does
and minimum values
not work (suffers failure).
F ( x; q, a ) = exp(- exp(-a ( x - q)). (6) Thus, we will be interested for an arbitrary probability
One feature is given to another replacementxon -x. Here event 𝑋 − 𝑌 < 𝑥, or composition of distributions 𝑋 +
q- the shear parameter α- the scale parameter. The (−𝑌).
parameters are determined by the first two moments of It is clear that the margin of safety, for the sake of
distribution: mathematical expectation and standard simplicity in practice, determined unequivocally by the
deviation: ratio of these values 𝑋/𝑌 (at best - the ratio of their
g average values) does not stand up to criticism, as a
q = m1 + a , a = sp , (7) characteristic of the strength of the object. In this sense, it
is natural to turn to the marginal distributions of statistics
whereγ - Euler's constant. The moments of the higher
(2) taking into account the first two points of the original
order of this distribution have been received. Analysis of
distributions.
the moments in particular.points out that it is unacceptable
Suppose each of the component of the sum 𝑋) = 𝑋, 𝑋B =
to replace these distributions for the sake of simplicity
−𝑌 distributed under a double exponential law with the
with a normal law. This form obtained accordingly for
appropriate parameters:
maximum and minimum values by substitution by / (for
simplicity do not change the designations):
F ( x) = 1 - exp(- exp x), F ( x; q1 , a1 ) = 1 - exp(- exp(a1 ( x - q1 ))) (11)
(8)
F ( x) = exp(- exp(- x)) F ( x; q2 , a 2 ) = 1 - exp(- exp(a 2 ( x - q2 )))(12)
Distributions (6) have, as mentioned above, known
and let's turn to the distribution of the amount of
properties: 1) the stability of the form relative to the
independent random amounts x1+x2. Note at once that the
transition to the distribution of extreme value from any
distribution of this amount of distributions (11) and (12)
number /same distributions respectively (5) or (6); 2)
does not retain the original distribution form. This
Distribution is the limit (after appropriate rationing) for a
complicates the technique in comparison with the
certain class of original distributions in the sense of
simplest model based on normal distribution of
convergence by probability. There's a little
characteristics x, y which is incorrect here.
bit more to do with that. The first means that each
As you know, it is possible to determine the distribution
function (8) is the solution to the functional equation
of the amount of independent values by the characteristic
Fn ( x) = F (an x + bn ) (9) function expressed through characteristic functions
component of composition: / In relation to distributions
where 𝐹" (𝑥) - distribution of extreme value from the same
we receive (6):
distributions (8), corresponding parameters of scale and
distribution shift (8).
For the convenience of using the second property, let us
j (t ) = ei (q1 + q2 )t G(1 + ait )G(1 + ait ) (13)
1 2
give a sufficient condition of belonging to the distribution
of the area of attraction of the first limit law 𝐹< (𝑥) Where Г(𝑡) – gamma function defined by Euler's
integrative:
[Gne43], [Dav79]. . It is enough that the distribution
function, at least for large modules, has the appearance: ¥
F ( x) = 1 - exp(-h( x)), (10) G( z ) = ò e - x x z -1dx, Re z > 0 . (14)
where the function h(x) increases monotonously and 0
indefinitely. This condition is satisfied with symmetrical However, the reverse transition from a characteristic
distributions of exponential type, including - normal law. function/ to an appropriate function of the distribution of
the amount is very problematic precisely because the law

94
(8) does not retain forms in the composition of In general, 𝛼𝜖[0, +∞). But tables are sufficient for the
distributions (not stable), as will be shown below. proposed methodology, 𝛼𝜖[0,1]. This follows from the
Therefore, we have directly turned to the roll-out of commutativeness of the bundle of distributions. However,
distribution densities and the known formula for the the distribution of the option for all 𝛼𝜖[0, +∞): is
function of distributing the amount x1+x2 that can be associated with assigning indexes 1.2 for the original
presented in the form of: distributions (11), (12). Therefore, it is preferable for a
+¥ specially untrained user to complete limit distribution
tables 𝐻H (𝑥), 𝛼𝜖[0, +∞).
H ( x; q1, q2 ,a1,a 2 ) = ò F ( z - x, q2 ,a 2 )dF ( x, q1,a1 )
-¥
(15)
The immediate calculation of this formula leads to results
that can formulated in the form of the following theorem. 6 Table of probabilities of dangerous
Theorem 1. The composition of two independent states
distributions (9) and (10) has a distribution 𝑞) , 𝑞B , 𝛼) , 𝛼B :
a The dangerous state here discussed as approximating the
a = a 2 , a = a1 , b = q2 - q1. object parameter to an unacceptable value, taking into
2 2 account external influences during its operation. The
Theorem 2. The function 𝐻H (𝑎𝑥 + 𝑏) if 𝑎 = 1, operational characteristics of the system are known to be
determined by the ratio of different characteristics, among
𝑏 = 0 has a view: which it is necessary to distinguish 1) the own
characteristics of the object, and 2) the characteristics of
¥
its operating conditions. Considered, a pair of one-
H a ( x ) = 1 - ò exp( -e x u -a - u)du (16) dimensional characteristics 1 and 2 preferably of the same
0 dimension: strength and active load. Their attitude 𝑋) <
Private cases: 𝐻L (𝑥) = 1 − exp (−𝑒 Q ). 𝑋B , 𝑋) = 𝑋B or 𝑋) > 𝑋B really determines the assessment
¥ of the quality of the system. Simple Risk Assessment
x -1 Procedure is based on the probability of the probability of
H ( x ) = 1 - ò exp( -e u - u )du.
1 the different characteristics 𝑋) − 𝑋B . In this case, the
0 distribution function 𝐹(𝑥) of the relevant criterion 𝑡 =
Thus, the prediction of dangerous states of an object can 𝑋) − 𝑋B determined by the composition of the respective
based on the distribution: distributions.
¥ ¥
ax +b -a
H a (ax + b) = 1 - ò exp(-e u - u )du (17) Ka ( x) = 1 - ò exp(-e xu -a - u )du .
0 0
H )
Where is 𝛼 = HR , 𝑎 = H , 𝑏 = 𝑞B − 𝑞) . (21)
R R
A special standard single-parametric family of Normalized distribution (4) linear transformation with
distributions, generated by the composition (1): is shear and scale parameters (𝑎 𝑎𝑛𝑑 𝑏) determines the
introduced: desired probability distribution (19).
F1 ( x), F2 ( x ) : The risk of an error associated with accepting a
satisfactory criterion t assessment result justifies the
+¥ choice [Dav79] of a relatively difficult one for the
F ( z ) = ò F1 ( z - x)dF2 ( x) (18) distribution of extreme performance distributions 𝑋) , 𝑋B .
-¥ This is a double exponential law, depending on two
distribution for which function value tables 𝐻H (𝑥) parameters 𝑎, 𝑞, which in turn are clearly determined by
depending on 𝛼, 𝑥, a snippet of which is below, should be the first two moments of the respective distributions. This
available. A reference to these existing tables does not is an interest in the worst-case scenario: the least strength
mean a complete solution to this issue. We need such is the greatest load. This complexity is due to the fact that
tables in a form that corresponds to modern information the distribution composition (1) does not retain the shape
tools, and are more convenient for the user, who does not of the distribution component 𝑋) , 𝑋B . In the case when the
have special training. distributions of independent random values are subject to
the limit distribution of extreme values of type 1, their

95
composition results after the corresponding rationing 𝑧 =
[0; 1] is sufficient to calculate probabilities by
𝑎𝑥 + 𝑏 to distribution [Kuk75], [Deg03], [Kuk96], formula (4) at all values 𝛼.
depending on the shape setting 𝛼: However, for values 𝛼, 𝛼 > 1, the proposed method
¥ of assessing probabilities is somewhat complicated.
x -a
К a (ax + b) = 1 - exp(-e u - u )du This
ò
follows from the fact that the parameters of the
shift and the scale in formula (4), defined by the first
0 two moments of input distributions, depend on the
(19) order of their chosen statistics: 𝑋) − 𝑋B or 𝑋B − 𝑋) .
Here are the distribution options 𝛼; 𝑎, 𝑏 are uniquely This complication is surmountable after selecting the
determined by the parameters 𝛼) , 𝑞) ; 𝛼B , 𝑞B initial appropriate table column and recalculating the scale
HR and shift parameters 𝑎, 𝑏.
distributions and 𝛼 = H . At the same time 𝛼 and 𝑎 not
[
independent, which presents some difficulties for the
distribution application (2) in the task of assessing the
probability of dangerous states. Overcoming these
difficulties in practice provided by the appropriate
function tables 𝐻H (𝑥):
¥
Ha (x) = ò exp(-e x u -a - u )du (20)
0
Developed [Kuk96], [Kry17] Detailed feature table
𝐻H (𝑥): and a probability calculation methodology
based on this table. Below is a snippet of the
abbreviated version of this table (table.1).
Table1 Distribution table
𝑥 α= α= α= α= α= α=
0,1 0,3 0,5 0,7 0,9 1 Fig. 1 Schedule Distributions
-4 0.929 0.93 0.931 0.932 0.933 0.933
-3,5 0.897 0.898 0.9 0.901 0.903 0.903 References
[Gne43] B. V. Gnedenko (1943) Sur la distribution
-3 0.853 0.855 0.857 0.859 0.861 0.862 limite du terme maximum
-2,5 0.794 0.797 0.8 0.802 0.805 0.806 d'uneseriealeatoiire//Ann. Math. Sttist. 44.
[Gum65] E. Gumbel (1965) Extreme stats. M.: The
-2 0.719 0.722 0.725 0.729 0.732 0.734 World.
-1,5 0.625 0.63 0.634 0.638 0.642 0.643 [Dav79] G. David (1979) Serial statistics.-M.: Science.
[Kra60]. G. Kramer, Leadbetter (1960) Stationary random
-1 0.517 0.521 0.526 0.53 0.535 0.537 processes. Properties of selective features and
-0,5 0.398 0.403 0.408 0.412 0.417 0.42 their applications. M.: Mir.
[Kuk75]. L.A. Kukharenko (1975) Study of electrical
0 0.28 0.284 0.289 0.293 0.298 0.3 strength insulation Monolith / Informelectro
30, sulfur. Electrical materials, 9.
[Deg03]. V.G. Degtyarev, L.A. Kukharenko, V.A.
Pictured (Figure 1) shows the function 𝐻H (𝑥): for Khodakovsky (2003) Stochastic transport
parameter values 𝛼, 𝛼𝜖[0; 1] with long intervals system tasks/ PSUPS Herald. Issue 1 - St.
∆𝛼 = 0,2. Given the properties of the Petersburg.
commutativeness of the roll-up operation, as well as [Kuk96]. L.A. Kukharenko (1996) On the composition of
the fact that determined by the ratio of input distributions of extremes. Proceedings of the
_ international scientific and methodological
variances 𝛼 = ^_[, we've developed a table 𝛼 ∈
R
conference Mathematics at the university. St.
Petersburg.

96
[Kuk02]. L.A. Kukharenko, T.S. Moiseenko (2002) Methodical Conference Mathematics e
About one model of extreme situations. University. St. Petersburg, PSUPS.
Materials of the scientific and practical [Bor15] T.P. Borovkova, L.A. Kukharenko, E.V. Runev
conference Topical economic problems. St. (2015) About the mathematical model of
Petersburg, IBI. quality control. Proceedings of the
[Kuk10]. L.A. Kukharenko, T.S. Moiseenko, O.I. International Scientific and Methodical
Januszewski (2010) Stochastic models in the Conference Mathematics e University and
mathematical training of an engineer. Materials School. St. Petersburg, PSUPS.
of the Scientific and Methodical Conference [Xen18]. V.A. Xenophontova, L.A. Kukharenka, E.V.
Problems of Mathematical Training in Runev (2018) About the moments and instant
Engineering Education. St. Petersburg, PSUPS. characteristics of one limiting law of extreme
[Kry17]. A.S. Kryukov, V.A. Xenophontova, L.A situations. System analysis and analytics. No5
Kukharenko (2017) Table of probabilities of (6). p. 15-23.
dangerous states. Mathematics at university [Kuk16]. L.A.Kukharenko, V.A. Xenophontova, A.S.
and at school. A collection of works of the Kryukov (2016) A mathematical model of
national scientific and methodological predicting dangerous situations. A collection
conference. Pskov. 2017. p. 112-113. 96-99 of works IV international scientific and
methodological conference Problems of
[Kho08]. V.A. Khodakovsky, L.A. Kukharenko, L.P.
mathematical and naturally scientific training
Pirozerskaya (2008) Probability theory. Ch.2:
in engineering education. St. Petersburg,
study. allowance - St. Petersburg, PSUPS.
PSUPS.
[Kuk11] L.A. Kukharenko, E.V. Runev, O.I. Januszewski
.
(2011) About strength and variance.
Proceedings of the International Scientific and