Bayesian data analysis in modeling and forecasting nonlinear nonstationaryfinancial and economic processes
The article provides a brief overview of modern Bayesian data analysis methods, highlighting the use of generalized linear models (GLMs) in the analysis of nonlinear non-stationary processes, emphasizing their capabilities and application to processes of various types. The article presents an exampl...
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| description | The article provides a brief overview of modern Bayesian data analysis methods, highlighting the use of generalized linear models (GLMs) in the analysis of nonlinear non-stationary processes, emphasizing their capabilities and application to processes of various types. The article presents an example of using GLMs to forecast financial losses in insurance and proposes the use of Bayesian data analysis in a specialized decision-support intelligent system, which enhanced the quality of the computation results.
У статті представлено короткий огляд сучасних байєсівських методів аналізу даних, наведено особливості застосування узагальнених лінійних моделей (УЛМ) в аналізі нелінійних нестаціонарних процесів, підкреслено їхні можливості та особливості застосування до процесів різної природи. У статті наведено приклад застосування УЛМдля прогнозування фінансових втрат у страхуванні, запропоновано використання байєсівського аналізу даних у спеціалізованій інтелектуальній системі підтримки прийняття рішень, що дозволило підвищити якість результатів обчислень.
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© O. TROFYMCHUK, P. BIDYUK, T. PROSYANKINA-ZHAROVA, O. TERENTIEV, 2023
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2023, № 4 71
СТОХАСТИЧНІ СИСТЕМИ, НЕЧІТКІ МНОЖИНИ
UDC 004.89:519.22](043.3)
O. Trofymchuk, P. Bidyuk, T. Prosyankina-Zharova, O. Terentiev
BAYESIAN DATA ANALYSIS IN MODELING
AND FORECASTING NONLINEAR NONSTATIONARY
FINANCIAL AND ECONOMIC PROCESSES
Oleksandr Trofymchuk
Institute of Telecommunications and Global Information Space of the National Acade-
my of Sciences of Ukraine, Kyiv,
orcid: 0000-0003-3358-6274
Trofymchuk@nas.gov.ua
Petro Bidyuk
National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute»,
orcid: 0000-0002-7421-3565
pbidyuke_00@ukr.net
Tetyana Prosyankina-Zharova
Institute of Telecommunications and Global Information Space of the National Acade-
my of Sciences of Ukraine, Kyiv,
orcid: 0000-0002-9623-8771
t.pruman@gmail.com
Oleksandr Terentiev
Institute of Telecommunications and Global Information Space of the National Acade-
my of Sciences of Ukraine, Kyiv,
orcid: 0000-0002-4288-1753
o.terentiev@gmail.com
The study focuses on some aspects of modeling and forecasting the nonlinear
nonstationary processes (NNP) of applying the modern Bayesian methods of da-
ta, in particular, generalized linear model (GLM) that are popular in analysis of
NNP. All Bayesian techniques of data analysis are very popular today thanks to
their flexibility, high quality of results, availability of possibilities for structural
and parametric optimization and adaptation to new data and conditions of func-
tioning. The structural and parametric adaptation of Bayesian generalized linear
models supposes taking into consideration the following elements: number of
equations that are necessary for adequate formal description of the processes un-
der study; availability of nonlinearity and nonstationarity; type of random dis-
turbance — its probability distribution and corresponding parameters; order of
model equations, and some other structural elements. Such approach to model-
ing improves model adequacy and quality of final result of their application. Pa-
rameter estimation of the models can be performed by making use of rather wide
mailto:o.terentiev@gmail.com
72 ISSN 2786-6491
set of methods, more precisely the following: ordinary LS (OLS), nonlinear LS
(NLS), maximum likelihood (ML), the method of additional variable (MAV),
and Monte Carlo for Markov Chain (MCMC). The last method is distinguished
by universality of application to estimation of linear and nonlinear models. Be-
sides, each of Bayesian approaches to data analysis is well supported by appro-
priate sets of statistical criteria that make it possible thorough quality analysis of
intermediate and final results of computations. Illustrative examples are present-
ed the usage of the Bayesian approach for analysis and forecasting of NNP, in
particular, in specialized intellectual decision support system.
Keywords: nonlinear nonstationary processes, Bayesian methods, modeling,
forecasting, generalized linear models.
Introduction
Many studies today are related to modeling and forecasting evolution of processes
in various areas; they are mostly touching upon widely spread nonlinear nonstationary
processes (NNP). To be more exact, definition of NNP means that such processes ex-
hibit at least one type on nonstationarity (regarding trend or integration, and variance or
heteroscedasticity) as well as nonlinearity regarding variables or model parameters.
Such processes create majority in ecology, economy, finances, industrial technologies,
engineering systems, hydrology, climate studies, in the problems of technical, medical
and economic diagnostics, physical experiments of various type etc. For example, many
processes in economy show availability of low (first or second) order trend, but transi-
tion to second order of integration automatically shifts the process from the class of lin-
ear to the nonlinear ones because quadratic and higher order dependence indicates rela-
tion to the class of nonlinear characteristics.
When we talk about process analysis we mean, as a rule, solving first of all the two
most often met problems such as constructing the selected process model and forecast-
ing. The widely used models of NNPs include, at least, the following types: differential
equations, nonlinear regression, combination of linear and nonlinear regression, nonlin-
ear autoregression, semi- and nonparametric methods, kernel based models, vector par-
ametric models and methods, vector semi- and nonparametric approaches, state space
and frequency-domain models, generalized linear models (GLM), and some others. To-
day high popularity acquired the methods of intellectual data analysis such as neural
networks, Bayesian networks, decision trees and forests, immune, genetic and molecu-
lar algorithms. A separate subclass of nonlinear models belongs to models that are non-
linear in parameters such as logit and probit (logistic regression), ecological function, ir-
rational and hyperbolic functions, Tornquist functions, and many others [1, 2]. The
models nonlinear in parameters require application of special nonlinear estimation tech-
niques for parameter estimation such as nonlinear LS, maximum likelihood (ML), Mon-
te Carlo for Markov Chains (MCMC) and others.
One of the most widely met in practice subclass of nonlinear processes (and their
models) is created by heteroscedastic processes, for example, they are very often con-
sidered in financial analysis. This direction of modeling and forecasting is widely
known and practically used due to the fact that variance is considered in this case as dy-
namic variable which is described by dynamic model and its value is predicted to solv-
ing many problems. For example, predicted standard deviation (volatility) and variance
itself are used for financial risk estimation and short-term forecasting of process vola-
tility. Existing today variety of variance models and their estimation techniques is very
high due to wide possibilities for their practical applications [3, 4].
Very popular approach to modeling NNP today is based upon Bayesian data analy-
sis. Besides well-known static and dynamic Bayesian networks includes Bayesian re-
gression, structural equation models, probabilistic filtering techniques, complex distri-
bution analysis etc. The Bayesian approach to analysis of data and expert estimates has
such positive features as structural flexibility of the models, possibility for taking into
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2023, № 4 73
consideration uncertainties that are available practically in every area and case of stud-
ies, availability of model parameter estimation procedures for the cases of linear and
nonlinear modeling, forecasting of complex distributions etc. [5–8].
This paper consider uncertainties as the factors of negative influence to the compu-
tational procedures used for model structure and parameter estimation, forecasts and
control actions computing etc. Influence of the factors results in lower quality of inter-
mediate and final results of data analysis, i.e. model adequacy, forecast estimates, con-
trol actions and decision alternatives.
The main goals of the study are as follows:
— to provide a review of Bayesian data processing and model constructing meth-
ods for their further use in intellectual decision support system for modeling and fore-
casting nonlinear nonstationary processes in economy and finances;
— to present illustrative examples of Bayesian techniques application to solve the
problems mentioned;
— to stress the necessity of development intellectual decision support system
(IDSS) for high quality solving the problems.
The review of Bayesian methods for modeling NNP illustrates some applications,
and highlight the methods that could be used for reaching high quality results of data
analysis. For example, it is often useful to stress development and application of a spe-
cialized IDSS and apply it to solving specific complicated problem. Then we applied
this to building the GLM model of actuarial process and to building the Bayesian net-
work to model the socio-economic processes taking place in regions of Ukraine.
Methodologies.
Bayesian methods for modeling and forecasting
Today there exists a wide set of Bayesian methods that are often used for prelimi-
nary data processing, model constructing, forecasting of future processes evolution, risk
estimation, control in various spheres, decision support, classification and solving some
other practical problems. Among others, the following Bayesian methods and tech-
niques are actively used in practice, and should be mentioned [7–13]:
— generalized linear models; the set of exponential distribution laws used in the
case of GLM application are as follows: normal, Poisson, binomial, Gamma, inverse
Gaussian; such approach enhances substantially number of process that can be formally
described with GLM;
— structural equation models (SEM); the models of this type make it possible con-
structing mathematical models for another class of random process that exhibit specific
structure that can be adequately described by Bayesian techniques;
— static and dynamic Bayesian networks (BN); BN represent powerful probabilis-
tic instrumentation capable formally describe sophisticated stochastic process and ex-
pert estimates, and generate probabilistic inference;
— Bayesian filtering of data and recursive parameter estimation; filtering touches
upon the problem of data processing, i.e. reducing influence of noise components spoil-
ing collected observation; parameter estimation relates to parameters of distributions
and various models, first of all regression models we have to construct in multiple ap-
plications;
— Bayesian maps; trajectory synthesis and control of robotic systems;
— Markov localization models; the problem of robot localization and control is
considered;
— multivariate distribution constructing and analysis; forecasting multivariate dis-
tributions, parameter estimation;
— decision trees and forests;
— combining Bayesian and statistical techniques into single model; model com-
bining provides a possibility for modeling and forecasting complicated NN processes.
74 ISSN 2786-6491
The methods listed above are distinguished with their high flexibility and possibili-
ties for taking into consideration possible data uncertainties and generating alternative
final results directed to support of decisions according to specific problem state-
ment [14–19]. Consider details of some methods.
Generalized linear models
Generalized linear models is a class of regression models that allow for taking into con-
sideration interaction between model factors, specific distribution law of dependent variable
and possible nonlinearity [12, 13]. GLM consists of the three basic components: systematic,
stochastic, link function and can be formally represented as follows:
( )1[ ] ,i i ij i ijE y g X−µ = = β + ξ∑ (1)
( )
[ ] ,i
i
i
V
Var y
φ µ
=
ω
where yi is a vector of observations for dependent variable; g(x) is a link function Xij; is
a matrix of observations for a model factors; βj is a vector of parameters estimated on
factor observations; ξi is a vector of stochastic residuals; ϕ is a vector of scale parame-
ters for the function of V(x); ωi — prior weights of confidence level.
Thus, GLM is characterized by the following elements: distribution law for de-
pendent variable, Y; parameters and specific features of a link function g(.); features of
the linear predictor, η = Xβ.
Usually it is reasonable to make the following suggestions regarding GLM:
• all the components of dependent variable Y are independent and their distribution
law belongs to the family of exponential distributions;
• the suggestion regarding systematic feature of a model is treated as follows: р
predictors are combined into single «linear predictor» η;
• the suggestion regarding link function: mutual dependence between suggestions
of stochastic and systematic features is expressed by the link function that is supposed
to be differentiable and monotonic and has an inverse:
1[ ] g ( ).E y −= µ = η (2)
The set of possible distribution laws for GLM and parameters of the distributions
for dependent variable are presented in Table 1 [17].
Table 1
Component
Normal distribution Poisson Binomial Gamma
distribution Inverse Gaussian
N(µ, σ2) P(µ) G(µ, v) IG(µ, σ2)
Variance
param., ϕ ϕ = σ2 1 1
n
ϕ = v–1 ϕ = σ2
Cumulant
function, b(θ)
2
2
θ exp(θ) log(1+eθ) – log(– θ) – (–2θ)1/2
c(y, ϕ)
2
2
2
1 log (2 )
2
y − + πσ
σ
– log(y!) log n
y
vlog(vy) –
– log(y) –
– log(Γ(ν))
31 1log(2 )
2
y
y
− πφ +
φ
µ(θ) = E(Y, θ) θ exp(θ)
1
e
e
θ
θ+
1
−
θ
(– 2θ)1/2
Canonic link,
θ(µ) identity log logit reciprocal 2
1
µ
Variance func,
V(µ) 1 M µ(1–µ) µ2 µ3
Var (µ) σ2 nµ(1–µ) µ
2µ
ν
2
3
σ
µ
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2023, № 4 75
The link function links linear predictor η to the estimate μ related to Y. In a classic
linear model mean the value of dependent variable and linear predictor are identical, and
identity link (η and μ) are selected arbitrarily but from the set of real numbers. GLM is
distinguished from linear model of general form (special case of such model is multiple
regression) by the following:
— distribution of dependent variable (system reaction) can be non-Gaussian and
not necessarily continuous, for example, binomial;
— dependent variable predictions are computed as linear combination of predictors
that are linked to the dependent variable with a link function.
Dependently on the distribution law of dependent variable and type of the link
function there exist types of GLM [17, 18] presented in Table 2.
Table 2
Model type Link function Dependent variable distribution
Linear model of general type g(µ) = µ Normal distribution
Log-linear model g(µ) = ln(µ) Poisson distribution
Logit-model g(µ) = ln(µ/(1 – µ) Binomial distribution
Probit-model g(µ) = Φ – 1µ Binomial distribution
«Survival» analysis g(µ) = µ – 1 Gamma distribution, exponential distribu-
tion
Thus, GLM represents quite general class of statistical models that includes linear
regression, variance and covariance analysis, Log-linear models for analysis of random
tables, logit/probit models, Poisson regression and many others.
Each distribution has its specific link function for each exists substantiated equality
statistic regarding parameter vector β of linear predictor, .i ixη = β∑ Such canonic link
comes to being when, θ = η, where θ is canonic parameter which is determined when
likelihood function is introduced. Specific distribution type corresponding to each link
function is given in Table 2, and generalized form of distribution is given below:
( )( , , ) exp ( , )
( )
y bf y c y θ − θ
θ φ = + φ α φ
, (3)
where a, b, c — are functions that correspond to specific distribution law; y is depend-
ent variable; θ is canonic parameter or a function of some parameter of a specific distri-
bution; ϕ is variance parameter.
Function b(.) has a special meaning in generalized linear models because it de-
scribed relation between mean value and variance in a distribution. When ϕ is known
then we have exponential model with canonic parameter θ. When ϕ is unknown, then
exponential distribution can be of two-parameter type. Thus, canonic link for a set of
exponential distributions has the form presented in Table 3.
Table 3
Distribution type Canonic link
Normal distribution η = µ
Poisson η = log(µ)
Binomial distribution log
1
n π =
− π
Gamma distribution η = µ – 1
Inverse Gaussian distribution η = µ – 2
A substantiated statistic for canonical forms is the vector XTy:
,ij ii x y∑ j = 1…p. (4)
76 ISSN 2786-6491
It should be noted that canonical link results in desirable features of a model, espe-
cially for short samples, for example existence of systematic effects. And here are ap-
proaching the problem of selecting statistic criteria for model estimating. As a result of
analysis of various criteria for model selection the conclusion was made that Akaike in-
formation criterion fits better to solving the problem together with the maximum likeli-
hood approach to estimation.
Quality estimation for generalized linear models
Usually in the process of analyzing quality (adequacy) of a model several different
criteria are applied. In some cases these criteria coincide with the same statistical crite-
ria used in constructing linear and/or nonlinear regression. When we use the models on
the purpose of classification, among them there are the following: common accuracy of
a model; I-st and II-nd type errors; ROC-curve, and Gini index.
Common accuracy (CA) statistic is determined by the expression:
_ ,Correct ForecastСА
N
= (5)
where, Correct Forecast is a number of correctly forecasted cases (examples); N is a to-
tal number of cases under investigation.
This criterion is somewhat subjective measure of a model quality because it de-
pends on a number of defaults in the dataset, and the level of the cut-off threshold. Dif-
ferent levels of the threshold result in different values of common accuracy.
ROC-curve (receiver operation characteristic) shows dependence of a number of
correctly classified positive examples on the number of incorrectly classified negative
examples. The first set of examples is called true positive, and the second set are re-
ferred to as false positive ones. Here it is suggested that the classifier has some parame-
ter that can be varied to reach necessary division into classes. This parameter is called
cut-off point or threshold. Depending on the parameter value different values of the I-st
and II-nd type errors will be achieved.
Most often the following statistics (in percentages) are used for determining quality
of a model:
— a part of true positive examples (true positives rate) [16]:
;TPTPR
TP FN
=
+
— a part of false positive examples (false positives rate):
.FPFPR
TN FP
=
+
Usually for completeness of the analysis two more characteristics are applied: sen-
sitivity and specificity.
Model sensitivity is defined as a part of true positive cases [16, 17]:
.TPRSe TPR
TP FN
= =
+
(6)
Model specificity is defined as a part of true negative cases that were correctly
classified by a model [16, 17]:
.TNSp
TN FP
=
+
(7)
Now, it is obvious that [16, 17]:
1 1 .TN FP FP FPSp FPR
TN FP TN FP
− +
= = − = −
+ +
(8)
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2023, № 4 77
The model that exhibits high sensitivity provides for a true result when number of
positive cases is also high (i.e., it reveals well positive cases). And the model that exhib-
its high specificity usually provides for better (true) results when the number of negative
cases is high (i.e., the model discovers better negative cases). The ROC curve (or Lo-
rentz curve) graph uses the axes Y for sensitivity Se, and axis Х for a part of false posi-
tive cases FPR or (1-Sp).
The graph of ROC-curve for an ideal classifier tends to the left upper corner where
part of true positive cases tends to 1 (the case of ideal sensitivity), and the part of false
positive samples tends to zero. Thus, the closer approaches ROC-curve to the left upper
corner the better is the model regarding prediction of a true value. As a consequence,
straight diagonal line corresponds to the classifier that is unable to distinguish between
the two classes. As far as visual comparison of different ROC-curves not always allows
selection of better model, there is a numerical criterion in the form of area under curve
(AUC) that makes the model selection easier. It is computed using the trapeze method
as follows [18]:
1
1( ) ( ).
2
i i
i ii
x x
AUC f x dx Y Y+
+
+ = = ∗ −
∑∫ (9)
As alternative model quality measure is used Gini index that is computed as an ar-
ea between diagonal and Lorentz curve divided by the whole area under the diagonal.
The index is widely used for resolution analysis of a system developed for credit risk es-
timation. In this case the model is used for dividing the clients into two groups: those
who are inclined to default, and those who are not. Usually, probability is hired as a
measure of inclination to default.
Application of Bayesian approach to estimation
In most cases of solving the problems of process modeling, forecasting and decision
support based on statistical data we meet uncertainties of various types. As an example can
serve structural, parametric and statistical uncertainties available in development and practical
application of identification procedures for analysis of processes of various nature. Structural
uncertainties are linked to the uncertainties of developed model structure, and parametric are
referred to the model parameter estimates. Statistical uncertainties are mostly related to obser-
vations, for example to the difficulties of determining true data distribution, influence of ex-
ternal random factors corrupting the measurements, missing observations, extreme values etc.
Theoretical studies of such problems are mostly related to analysis of reasons for emergence,
classification and influence estimation of the uncertainties as well as level and probabilities of
respective risks.
Very often we lack statistical data to solve the problems of risk analysis and deci-
sion making. Such problems cannot be solved with traditional statistical frequency ap-
proach because the data available cannot provide necessary information. Moreover, the
situations related to decision making may change substantially and lack the results of
preliminary analysis. Such particular features lead to complications with decision mak-
ing and may influence negatively final results. That is why the Bayesian approach be-
comes in such cases useful and highly effective instrument of modeling, forecasting and
decision support.
A distinctive feature of Bayesian approach to data and expert estimates processing
is that researcher considers the level of his belief to possible models and forecasts be-
fore receiving data and represents his view in the form of a probability. As soon as the
necessary data is received the Bayes theorem provides a possibility for computing an-
other set of probabilities representing with them refined beliefs to the candidate models.
The key advantage of Bayesian approach to data expert estimate analysis is in the
possibility of using any prior information related to system under study state and its
78 ISSN 2786-6491
model (for example, model structure and its parameters). Such information is presented
in the form of prior probabilities and/or density distribution. The prior probabilities are
recalculated (improved) further on using the data that are used to determine posterior
distribution of parameter estimates or respective output variables.
Consider as example application of conjugate distributions what means that prior
and posterior distributions are of the same type.
Let experimental data X1,…, Xn is normal random sample with mean value µ and
variance σ2. Suggest that prior distribution is also normal with mean µ0 and variance
2
0.σ Then posterior distribution for µ with known sample X1,…, Xn and known prior
distribution will also be normal with mean *µ and variance 2
*σ that are computed as
follows [8, 10]:
2 2
0 0
* 2 2
0
,
n X
n
σ µ + σ
µ =
σ + σ
2 2
2 0
* 2 2
0
,
n
σ +σ
σ =
σ + σ
where 1
n
ii X
X
n
−=
∑ is sample mean.
Bayesian analysis often uses parameter characterizing quality of results, and de-
termined as inverse to variance: η = 1/σ2. Thus, we have for the prior distribution, η0 =
= 1/σ2 and for posterior distribution * 2
*
1 .η =
σ
Now the expressions for posterior pa-
rameter will take the form [8, 10]:
* 0 ,nη = η + η 0
* 0
* *
.n X
η η
µ = µ +
η η
Thus, for the case of normal sample information about µ is contained in the sample
mean X that is complete statistic for µ. The quality of distribution identification is de-
termined by the relation: X:
2 .nη
= η
σ
Quality of posterior distribution description is de-
termined by the quality of two following components: prior distribution representation,
and statistical characteristics of sample data. The posterior mean is weighted prior mean
and sample mean with weighting coefficient that is proportional to the quality parame-
ter. It means that influence of prior distribution is decreasing with growing size of data
sample n.
Consider the problem of forecasting random value х on the basis of historical ob-
servations y using Bayesian approach. That is, it is necessary to determine type of dis-
tribution for the future values of х given values of y. From the probabilistic point of
view the problem is in determining forecast density ( )x yπ that describes possible
changes with known values of y.
Most often, the model for which the forecast density is needed does not exist. But
known is the probabilistic model for х that is expressed in terms of distribution ( )g x θ
depending on unknown parameter θ that is based on the model describing observations y.
If we denote posterior density of θ given y as ( )P yθ then forecasting density for х can
be written as follows [8, 10]:
( )) ( ) ( ) .x y g x P y dθπ = θ θ θ∫ (10)
If a point forecast is required, it is enough to use the point estimate of ( ).x y To
find an interval forecast it is enough to compute interval estimates for ( ).x y
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2023, № 4 79
Results & discussion.
The application of GLM for modeling the actuarial processes
As an example of actuarial process modeled by GLM the problem of estimating
(forecasting) financial loss in car insurance had been selected. The experimental data
includes one dependent variable, «Loss», reflecting paid volume of insurance among the
cars of the following brands: VAZ, Mitsubishi and Toyota. The regions of selling (dis-
tribution) of the car policies include the cities of Kyiv, Donetsk and Odesa. The data re-
lates to the years starting from 2006 with the sample size of 9546 examples. The results
of GLM constructing for different distribution laws are presented in Tables 4–6.
Table 4
Model
Distribution of dependent variable Link function
Gamma LOG
Normal LOG
Poisson LOG
Normal Identity
Table 5
Total loss,
bln. UAH Mean Std.
deviance Min Max Mean of stan-
dard error Variance, %
102,01 11805,69 15358,118 6273,867 18549,819 0,075 130,091
18,11 1897,46 939,910 4010,98 634,054 0,120 49,535
17,92 1877,53 1027,57 4234,95 558,35 0,176 54,73
17,92 1877,53 999,30 3535,4 118,0 224,35 53,22
Table 6
Total loss Log-likelihood Difference Risk
102008320,905 – 15742,754 84087288,32 1,301
18111231,380 – 98700,167 190198,799 0,495
17921032,574 – 42173677,24 0,007 0,547
17921032,589 – 98700,167 0,009 0,532
102008320,905 – 15742,754 84087288,32 1,301
Table 6 shows that the risk of financial loss for the models constructed varies ap-
proximately between 40–60% what is marginally acceptable but requires undertaking of
some measures regarding its minimization.
Usage of Bayesian methods for modeling the socio-economic processes
As a part of the study, computational experiments using various socio-economic
indicators were performed the quality of results obtained are quite acceptable. In partic-
ular, the paper presents the use of a continuous Bayesian network to study the effective-
ness of the decentralization reform [20, 22].
The time series of indices that are relevant to the reform period: 2015-2021 [20–22],
were considered, which characterize the socio-economic component of the reform —
the growth of local budget revenues per resident of the community due to the increase in
tax revenues of local budgets as a result of the activation of the community economy.
These processes are sophisticated for studying: they are nonlinear and nonstationary.
Table 7 shows the values of the estimates of mathematical expectation and standard de-
viation, according to the normal distribution of the regressors and the target variable.
80 ISSN 2786-6491
Table 7
Variable Indicator, % Mathematical expecta-
tion of the variable
Standard deviation
of the variable
U1 Index of the volume of industrial products sold – 2,414 5,845
U2 Agricultural production volume index 1,428 8,625
U3 Index of construction products 5,314 12,232
U4 Increase in revenues to the general fund – 30,214 43,204
U5 Growth rate of land fee revenues 19,457 21,464
U6 Basic grant – 0,771 0,969
U7 Index of physical change of gross regional
product 0,128 5,055
X Dynamics of local budget income per inhabitant
of community 3,891 12,474
A continuous time Bayesian network was developed and used to model the studied
processes. Continuous Time Bayesian networks are used to model stochastic processes
in continuous time state space. The Gaussian distribution is used in this problem what
was proved by appropriate statistics. The topology of the Bayesian network developed
is presented in figure.
The target variable X has multiple parents 1 2 5 6{ , , , }.U U U U U=
( ) ( ; * , ) ,i x i i xf X U N x b= µ + µ σ
2( ( * , ))1 1( ; * , ) exp *
22
x i i x
x i i x
xx
x b
N x b
− µ + µ σ
µ + µ σ = σπσ
,
where μ — is the mathematical expectation, σ — is the variance, and is the coefficient
1
2 xπσ
normalization constant ensuring that ( ; * , ) 1,x i i x
x
N x bµ + µ σ =∫ bi — coeffi-
cient characterizing the relationship between X and its i-th parent (it is also called a
weight coefficient) [23].
The relationship between specific variable X and its parents (U1, U2, ..., Un) is
formally described by a linear regression model:
U7
U4
U3
U1 U5 U2 U6
X
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2023, № 4 81
1 1 2 2 ... ,n n xX b U b U b U Q= + + + +
where Qx is a noise component that can be written in the form of a Gaussian distribution
with zero mathematical expectation, and b1, b2,..., bn are regression coefficients show-
ing the relationship between the variable X and U1, U2,..., Un — its parents.
Regression coefficients of the model were estimated by the method of least
squares. The structure of the regression model is as follows:
1 2 51,23 3,34 0,39 9,15 .n xX U U U U Q= − + − + +
The greatest impact on the target variable X is exerted by U6 (base subsidy, % of
the schedule), which indicates insufficient independence of local budgets.
The predictive characteristics of the model are characterized by the following sta-
tistics: RMSE = 3,4, MAPE = 13,5 % — they are quite acceptable.
Conclusion
Thus, it is presented short review of modern data processing Bayesian techniques. An
example of modeling and forecasting of financial actuarial process is presented. It was shown
that generalized linear models can serve as effective instrument analyzing financial and eco-
nomic processes that helps to take into consideration actual complicated factor interactions
and their influence on dependent variable. There also exists a possibility for model and fore-
casts quality analysis (of the results achieved) using a set of appropriate statistical criteria.
Further research in this direction should be focused on the problems touching upon
the following issues: refinement of the forecasting model; more profound analysis of
factors influencing dependent variable; more active use of Bayesian and neural net-
works and other methods of intellectual data analysis to modeling and forecasting actu-
arial processes; development and use of commercial intellectual decision support sys-
tem. The DSS will help to construct and study more sophisticated combined model con-
sisting of linear and nonlinear parts, to reach higher quality forecasts of dependent
variable, and consequently improve estimates of possible financial loss. This approach
to financial processes analysis will help to minimize financial risks in insurance as well
as in many other spheres of human activities. Finally such studies will positively influ-
ence macro economy as a whole.
О.М. Трофимчук, П.І. Бідюк,
Т.І. Просянкіна-Жарова, О.М. Терентьєв
БАЙЄСІВСЬКИЙ АНАЛІЗ ДАНИХ
У МОДЕЛЮВАННІ ТА ПРОГНОЗУВАННІ
НЕЛІНІЙНИХ НЕСТАЦІОНАРНИХ
ФІНАНСОВО-ЕКОНОМІЧНИХ ПРОЦЕСІВ
Трофимчук Олександр Миколайович
Інститут телекомунікацій і глобального інформаційного простору НАН України,
м. Київ,
Trofymchuk@nas.gov.ua
Бідюк Петро Іванович
Національний технічний університет «Київський політехнічний інститут імені
Ігоря Сікорського»,
pbidyuke_00@ukr.net
mailto:Trofymchuk@nas.gov.ua
mailto:pbidyuke_00@ukr.net
82 ISSN 2786-6491
Просянкіна-Жарова Тетяна Іванівна
Інститут телекомунікацій і глобального інформаційного простору НАН України,
м. Київ,
t.pruman@gmail.com
Терентьєв Олександр Миколайович
Інститут телекомунікацій і глобального інформаційного простору НАН України,
м. Київ,
o.terentiev@gmail.com
У статті представлено короткий огляд сучасних байєсівських методів ана-
лізу даних, наведено особливості застосування узагальнених лінійних мо-
делей (УЛМ) в аналізі нелінійних нестаціонарних процесів, підкреслено
їхні можливості та особливості застосування до процесів різної природи.
Усі байєсівські методи аналізу даних сьогодні дуже популярні завдяки сво-
їй гнучкості, високій якості результатів, можливості структурно-пара-
метричної оптимізації та адаптації до нових даних і умов функціонуван-
ня. Структурно-параметрична адаптація байєсівських УЛМ передбачає
врахування кількості рівнянь, які необхідні для адекватного опису дослі-
джуваних процесів; наявності нелінійності та нестаціонарності; типу випа-
дкового збурення — його розподілу ймовірностей; порядку використаних
рівнянь та деяких інших елементів структури. Все це сприяє підвищенню
адекватності моделей, що будуються, і якості остаточного результату їх за-
стосування. Для оцінювання параметрів цих моделей можна примінити
досить широку множину методів, зокрема таких: звичайний метод най-
менших квадратів (МНК), нелінійний МНК (НМНК), метод максимальної
правдоподібності (ММП), метод допоміжної змінної (МДП) і метод Мон-
те–Карло для марковських ланцюгів (МКМЛ), який відрізняється універса-
льністю застосування до оцінювання параметрів лінійних та нелінійних
моделей. Крім того, кожен з байєсівських методів аналізу даних добре
підтримується відповідними множинами статистичних критеріїв, які роблять
можливим ретельний якісний аналіз проміжних і кінцевих результатів.
У статті наведено приклад застосування GLM для прогнозування фінансо-
вих втрат у страхуванні, запропоновано використання байєсівського аналі-
зу даних у спеціалізованій інтелектуальній системі підтримки прийняття
рішень, що дозволило підвищити якість результатів обчислень.
Ключові слова: нелінійні нестаціонарні процеси, байєсівські методи, мо-
делювання, прогнозування, узагальнені лінійні моделі.
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Introduction
Methodologies.
Bayesian methods for modeling and forecasting
Generalized linear models
Quality estimation for generalized linear models
Application of Bayesian approach to estimation
Results & discussion.
The application of GLM for modeling the actuarial processes
Usage of Bayesian methods for modeling the socio-economic processes
Conclusion
|
| id | nasplib_isofts_kiev_ua-123456789-211049 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0572-2691 |
| language | English |
| last_indexed | 2026-03-14T16:21:13Z |
| publishDate | 2023 |
| publisher | Інститут кібернетики ім. В.М. Глушкова НАН України |
| record_format | dspace |
| spelling | Trofymchuk, O. Bidyuk, P. Prosyankina-Zharova, T. Terentiev, O. 2025-12-23T03:32:20Z 2023 Bayesian data analysis in modeling and forecasting nonlinear nonstationaryfinancial and economic processes / O. Trofymchuk, P. Bidyuk, T. Prosyankina-Zharova, O. Terentiev // Проблеми керування та інформатики. — 2023. — № 4. — С. 71-83. — Бібліогр.: 24 назв. — англ. 0572-2691 https://nasplib.isofts.kiev.ua/handle/123456789/211049 004.89:519.22](043.3) 10.34229/1028-0979-2023-4-6 The article provides a brief overview of modern Bayesian data analysis methods, highlighting the use of generalized linear models (GLMs) in the analysis of nonlinear non-stationary processes, emphasizing their capabilities and application to processes of various types. The article presents an example of using GLMs to forecast financial losses in insurance and proposes the use of Bayesian data analysis in a specialized decision-support intelligent system, which enhanced the quality of the computation results. У статті представлено короткий огляд сучасних байєсівських методів аналізу даних, наведено особливості застосування узагальнених лінійних моделей (УЛМ) в аналізі нелінійних нестаціонарних процесів, підкреслено їхні можливості та особливості застосування до процесів різної природи. У статті наведено приклад застосування УЛМдля прогнозування фінансових втрат у страхуванні, запропоновано використання байєсівського аналізу даних у спеціалізованій інтелектуальній системі підтримки прийняття рішень, що дозволило підвищити якість результатів обчислень. en Інститут кібернетики ім. В.М. Глушкова НАН України Проблеми керування та інформатики Стохастичні системи, нечіткі множини Bayesian data analysis in modeling and forecasting nonlinear nonstationaryfinancial and economic processes Байєсівський аналіз даних у моделюванні та прогнозуванні нелінійних нестаціонарних фінансово-економічних процесів Article published earlier |
| spellingShingle | Bayesian data analysis in modeling and forecasting nonlinear nonstationaryfinancial and economic processes Trofymchuk, O. Bidyuk, P. Prosyankina-Zharova, T. Terentiev, O. Стохастичні системи, нечіткі множини |
| title | Bayesian data analysis in modeling and forecasting nonlinear nonstationaryfinancial and economic processes |
| title_alt | Байєсівський аналіз даних у моделюванні та прогнозуванні нелінійних нестаціонарних фінансово-економічних процесів |
| title_full | Bayesian data analysis in modeling and forecasting nonlinear nonstationaryfinancial and economic processes |
| title_fullStr | Bayesian data analysis in modeling and forecasting nonlinear nonstationaryfinancial and economic processes |
| title_full_unstemmed | Bayesian data analysis in modeling and forecasting nonlinear nonstationaryfinancial and economic processes |
| title_short | Bayesian data analysis in modeling and forecasting nonlinear nonstationaryfinancial and economic processes |
| title_sort | bayesian data analysis in modeling and forecasting nonlinear nonstationaryfinancial and economic processes |
| topic | Стохастичні системи, нечіткі множини |
| topic_facet | Стохастичні системи, нечіткі множини |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211049 |
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