=Paper=
{{Paper
|id=Vol-2753/paper6
|storemode=property
|title=Methodology of Constructing Statistical Models for Nonlinear Non-stationary Processes in Medical Diagnostic Systems
|pdfUrl=https://ceur-ws.org/Vol-2753/paper4.pdf
|volume=Vol-2753
|authors=Peter Bidyuk,Irina Kalinina,Aleksandr Gozhyj
|dblpUrl=https://dblp.org/rec/conf/iddm/BidyukKG20
}}
==Methodology of Constructing Statistical Models for Nonlinear Non-stationary Processes in Medical Diagnostic Systems==
https://ceur-ws.org/Vol-2753/paper4.pdf
Methodology of Constructing Statistical Models for Nonlinear
Non-stationary Processes in Medical Diagnostic Systems
Peter Bidyuk a, Irina Kalinina b and Aleksandr Gozhyj b
a
National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine
b
Petro Mohyla Black Sea National University, Nikolaev, Ukraine
Abstract
The article presents a methodology for analysis and modeling nonlinear and non-stationary
processes associated with medical diagnostics and solving medical problems. The
methodology is based on collecting and preliminary processing of statistical data, identifying
and accounting for possible uncertainties, building and estimating the structure of
mathematical model, and evaluating its parameters, estimating model based forecasts and
calculating statistical criteria for the model adequacy as well as quality of the forecasts. The
analysis of selected models for linear and nonlinear processes is presented. A scheme for
combining forecasts when estimating a diagnosis is proposed. An example of using a
diagnostic system for predicting a patient's condition using combined (linear + nonlinear)
model is given and the methods used are analyzed.
Keywords 1
Medical decision support systems, Nonlinear and non-stationary processes, Statistical model,
Preliminary processing, Uncertainties, Combining forecasts.
1. Introduction
Appropriately designed medical decision support system (DSS) can provide a substantial help
regarding estimation of medical diagnosis, analyzing the processes taking place during the period of
patient treatment, forecasting patient state, modeling complex processes and situations so that to
derive correct conclusions, perform necessary simulations, generate appropriate advices etc. [1]. DSS
can easily analyze available and newly coming statistical data and expert estimates regarding various
processes taking place in medical environment. DSS can perform in a very short time sophisticated
and cumbersome computations, and provide final results in appropriate, convenient for medical staff
form. The well-known example of successfully working medical DSS is Quick Medical Reference
(QMR) diagnostic system [2, 3] based upon Bayesian, neural, statistical, and other methods for data
analysis and generating probabilistic inference regarding diagnosis, situational analysis, drawing
conclusions etc.
Some of the process studied in medical applications, are nonstationary or piecewise stationary and
contain nonlinearities. It means that their statistical parameters may change in time what requires
special attention regarding modeling, analyzing and forecasting the processes being studied. The
processes are also characterized by availability of stochastic or deterministic trends (conditional
expectation varies in time) dependently on specific situations, random disturbances and factors
influencing them. Usually nonstationary processes exhibit nonlinearities of various kind (nonlinearity
regarding variables or parameters). The deterministic trend regarding patient state can be formally
described by the linear, quadratic or cubic function, exponent, spline or harmonic function.
Estimation of variance is also an important stage in data research in medical diagnostic systems [4
- 6]. This is a key statistical parameter for generating correct diagnosis based on available statistical
IDDM’2020: 3rd International Conference on Informatics & Data-Driven Medicine, November 19–21, 2020, Växjö, Sweden
EMAIL: pbidyuke_00@ukr.net (P. Bidyuk); irina.kalinina1612@gmail.com (I. Kalinina); alex.gozhyj@gmail.com (A. Gozhyj)
ORCID: 0000-0002-7421-3565 ((P. Bidyuk); 0000-0001-8359-2045 (I. Kalinina); 0000-0002-3517-580X (A. Gozhyj)
©️ 2020 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
data and forecasts of relevant conditional variance. One of important points in modeling dynamics of
various processes is identification and taking into consideration possible uncertainties related to the
data available and expert estimates. The uncertainties are considered here as the factors of negative
influence to the modeling process as a whole that result in various computational errors decreasing
quality of intermediate and final results. As an example of possible uncertainties could be mentioned
measurement errors produced by the clinical laboratory devices, and previously unknown
consequences resulting from prescribed medications. Another problem is created by the so called
structural uncertainties related to estimation of model structure.
This study is directed towards further refinement of mathematical model constructing
methodology for nonlinear non-stationary processes in medical applications to be further used in
specialized diagnostic decision support systems. It is touching upon improvement of data quality
before model constructing, as well as model structure and parameter estimation techniques using
alternative statistical data analysis procedures.
Problem statement. The paper is focused on solving the following problems: development of
systemic methodology for mathematical models constructing for linear and nonlinear process in
medical applications; development of decision support system structure and its functions for
application in medical diagnostics; providing a review of some mathematical models for possible
prospective applications in medical decision support systems; presenting an example for possible DSS
application.
2. Materials and Methods
This section discusses the methodology for analysis and modeling of nonlinear and non-stationary
processes associated with medical diagnostics and solving medical problems. The analysis of selected
models for linear and nonlinear processes is presented. A possible scheme for combining forecasts
when making a diagnosis is proposed. An example of using the diagnostic system for prediction of a
patient state and the efficiency of the data analysis methods are proposed.
2.1. DSS structure and functions
The DSS proposed (Fig. 1) includes the following functional blocks: user interface, central
computing subsystem generating required results of data analysis according to specific problem
statement, knowledge and database (KDB), as well as intermediate and final results representation
subsystem. The KDB contains all necessary computational procedures, sets of a model and forecasts
quality criteria, statistical data, relevant expert estimates and best model selecting rules. All
computations are performed within the central computing subsystem that generates intermediate and
final results of data analysis according to the user requests. The results representation subsystem
provides a user with necessary information regarding the computational procedures, and data
presentation in convenient formats.
The basic functions of the system are as follows: data collection from local and external sources;
pre-processing, i.e. preparing data for model constructing and subsequent processes forecasting;
model structure and parameter estimation; computing forecasts of patient state and combination of the
separate forecasts; generation of recommendations and diagnostic messages; retrospective analysis of
available former results (for example, retrieval from memory and analysis of available diagnostic
messages) etc. The functional possibilities of the system are easily modified and expanded thanks to
the modular system construction. At the core of the diagnostic subsystem are intellectual data analysis
procedures such as Bayesian networks, neural networks and fuzzy-neural models, decision trees etc.
The selected statistical procedures are also widely used for preliminary data processing, identifying
and fighting statistical uncertainties, correlation and variance analysis of data, estimation of resulting
models adequacy, and quality of the model based forecasts; generation of alternative decisions based
upon probabilistic and statistical procedures.
Figure 1: DSS structure
2.2. The methodology of modeling proposed
The methodology for modeling and analyzing medical data in solving diagnostic problems is based
on the following steps:
Collection and preliminary statistical processing of data for building a model (adding
missing data, normalizing data, filtering, accounting and processing possible outliers, etc.).
Identification, estimation and accounting for uncertainties in the data (estimation of data
values that cannot be measured, estimation of statistical parameters for observations
(mean, median, variance, covariance, etc.); determination of the required data structure;
analysis of statistical characteristics for the random samples (type of distribution and its
parameters) that may affect for the diagnosis.
Evaluation of mathematical models structure using statistical and probabilistic methods of
data analysis for the selection and further use of the best of them in solving the forecasting
and decision making problems. The following parameters are used as the main
characteristics of the model structure being constructed: the model dimension (total
number of equations that form the model); model order (differential equation order, or
autoregression and the moving average orders); the presence of nonlinearity of the process
and estimation of its type (nonlinearity in variables and / or in terms of model parameters);
estimation of the input time delay (lag) of the process, etc.
To solve the problem of identification and taking into consideration possible nonlinearities it is
recommended to construct separately the models for linear and nonlinear parts of a process being
studied using various possibilities (special nonlinear components) for formal description of the
nonlinear part. The acceptable quality results were achieved with application of combined linear and
nonlinear regression; linear regression and neural or Bayesian networks; linear regression and special
nonlinear functions like polynomials and nonparametric kernels etc.
The following methodology is proposed for determining and describing possible nonlinearities of
the data being analyzed:
Correct estimation of model parameters based on alternative methods (NLS, ML, MCMC
and other methods) allows for calculating unbiased parameter estimates, determining the
type of distribution for variables and its parameters, estimating alternative structures of the
model. The use of such alternative estimation methods in decision-making procedures
allows for further comparison of estimates and the selection of the best model.
Computing statistical parameters that are quality criteria and characterize adequacy of
mathematical models. On their basis, the most adequate model is selected. Usually there
exist several candidate models that can be constructed using different methods. The final
choice of the model is performed after its application to solve the specific problem stated.
Construction of forecasts (usually short-term or medium-term) and evaluation of their
quality to determine the best predictive model. For this purpose the following statistical
quality criteria are used: Theil coefficient, MAPE (mean absolute percentage error), MAE
(mean absolute error), etc.
Testing the models constructed using the processes data with similar statistical
characteristics (model calibration).
Usually in the model constructing procedures it is necessary to take into account the following
types of uncertainties: data uncertainties, model structure uncertainties, and parametric uncertainties.
The data uncertainties include: missing measurements, the presence of short uninformative data
samples, possible extreme values (outliers), distortions due to availability of observation and external
noise processes, etc. Basically, these types of uncertainties are easily handled using filtering
procedures. Uncertainties regarding the structure of the model are due to poor data structure, which
does not contain enough necessary information for estimating the model structure and its parameters.
Parametric uncertainties depend on the quality of the statistical data available. Usually they are
observed in the form of biased parameter estimates. Elimination of the bias is performed by applying
several methods of parameter estimation, such as: OLS, maximum likelihood (ML) algorithms, and
Monte Carlo algorithms with Markov chain [6, 7].
To monitor the software implementation of the methodology it is necessary to consider at least
three sets of statistical quality criteria during the modeling process: data quality parameters, model
adequacy, and forecast quality statistics. It is necessary to provide for the software implementation of
the methodology for the analysis of alternative solutions developed on the basis of the calculated
forecasts.
The practical application of the proposed modeling methodology lies in solving the problem of
predicting the patient's condition, monitoring this condition, as well as making decisions regarding
diagnostics, and modeling complex situations with the use of specific computational procedures.
2.3. Some widely used models of linear and nonlinear processes
Today, there are many statistical methods for analyzing and studying the processes associated with
clinical treatment and diagnosis identification based upon regression analysis results. One of the
approaches is based on the state space (SS) representation of the models constructed. Based on this
approach, the patient's condition is assessed and predicted [8 - 14]. Another approach to developing
linear and nonlinear models for diagnosis is based on data mining (IDA) and machine learning (ML)
methods like following: neural networks, fuzzy sets, neural fuzzy models, Bayesian networks (static
and dynamic), complex multivariate probability distributions, models describing interactions of
various factors, non-parametric and semi-parametric models, decision trees, etc.
The structure of a time series mathematical model can be described by the following expression:
𝑆 = {𝑟, 𝑝, 𝑚, 𝑛, 𝑑, 𝑤, 𝑙},
where r is model dimension (number of model equations); p is model order (maximum order of
differential or difference equation used for specific process description); m is a number of
independent variables (regressors); n is nonlinearity and its type (with respect to variables or
parameters); d is input delay time (lag or medication transport delay); w is external (or possibly
internal) stochastic disturbance and type of its probability distribution; l represents possible
constraints on variables and/or parameters [15].
In medical practice it is possible to use the models based on nonlinear regression. For example,
similar models can be used to describe two interrelated processes, 𝑦1 (𝑘), and 𝑦2 (𝑘) when predicting
the patient's condition and identifying diagnosis:
𝑦1 (𝑘) = 𝑎0 + 𝑎1 𝑦1 (𝑘 − 1) + 𝑏12 𝑒𝑥𝑝(𝑦2 (𝑘)) + 𝑎2 𝑥1 (𝑘)𝑥2 (𝑘) + 𝜀1 (𝑘),
𝑦2 (𝑘) = 𝑐0 + 𝑐1 𝑦2 (𝑘 − 1) + 𝑏21 𝑒𝑥𝑝(𝑦1 (𝑘)) + 𝑐2 𝑥1 (𝑘)𝑥2 (𝑘) + 𝜀2 (𝑘),
where 𝑦1 (𝑘) is a principal state variable for the first process under study; 𝑦2 (𝑘) is a principal state
variable for the second process; 𝑥1 (𝑘) is a level of the first medication being used; 𝑥2 (𝑘) is a level of
the second medication. Sometimes, it is also possible to apply the following generalized linear model:
𝑝 𝑞 𝑚 𝑠
𝑦(𝑘) = 𝑎0 + ∑ 𝑎𝑖 𝑦(𝑘 − 𝑖) + ∑ 𝑏𝑗 𝑣(𝑘 − 𝑗) + ∑ ∑ 𝑐𝑖,𝑗 𝑦(𝑘 − 𝑖)𝑣(𝑘 − 𝑗) + 𝜀(𝑘),
𝑖=1 𝑗=1 𝑖=1 𝑗=1
where p, q, m and s are positive numbers that represent the model order [15].
The complete model of nonlinear processes can be based on linear combination of linear and
nonlinear components like follows:
𝑝
𝑦(𝑘) = 𝛽 𝑇 𝐳(𝑘) + ∑ 𝛼𝑖 𝜑𝑖 (𝜃𝑖𝑇 𝐳(𝑘)) + 𝜀(𝑘),
𝑖=1
where z(k) is a vector of time delayed values of basic dependent variable y(k), as well as former and
current values of independent explaining variables x(k) with appropriate values of time delay. Here,
𝜑𝑖 (𝑥),, is a set of linear and possibly nonlinear functions that may include the following components:
power function 𝜑𝑖 (𝑥) ≡ 𝑥 𝑖 ; the harmonic trigonometric functions like 𝜑𝑖 (𝑥) = sin 𝑥 or 𝜑𝑖 (𝑥) =
cos 𝑥, etc. If necessary, this equation can be expanded with quadratic form of the type; 𝐳 𝑇 (𝑘)𝐀𝐳(𝑘);
𝜑𝑖 (𝑥) = 𝜑(𝑥), 𝑖 , where 𝜑(𝑥) is a suitable link function, for example appropriate probability
density function or logistic function of the type:
1
𝜑(𝑥(𝑘, 𝑧)) = ,
1 + exp(−𝑥(𝑘, 𝑧))
𝑥(𝑘) = 𝛼0 + 𝛼1 𝑧1 (𝑘) + ⋯ + 𝛼𝑚 𝑧𝑚 (𝑘) + 𝜀(𝑘),
where 𝑧𝑖 (𝑘), 𝑖 = 1,2, … , 𝑚 are explaining variables for the intermediate principal variable 𝑥(𝑘), and
𝜑(𝑥(𝑘, 𝑧)), respectively.
Another general class of nonlinear models (suitable for modeling and forecasting patient state) can
be presented as follows:
𝑝
𝐲(𝑘) = ∑ 𝜑𝑗 (𝐱(𝑘 − 1))𝐲(𝑘 − 𝑗) + 𝜇(𝐱(𝑘 − 1)) + 𝜀(𝑘),
𝑗=1
where 𝐲(𝑘) is [n1] vector of dependent variables; 𝐱(𝑘) = [𝐲(𝑘), 𝐲(𝑘 − 1), … , 𝑦(𝑘 − 𝑛 + 1] is a
vector of state variables; here dynamics of the state variables can be described by the following state
space model:
𝐱(𝐤) = ℎ(𝐱(𝑘 − 1)) + 𝐅(𝐱(𝑘 − 1))𝐱(𝑘 − 1) + 𝑣(𝑘).
When constructing patient state forecasting model usually there are developed several candidates
with subsequent selection of the best one using a set of model adequacy criteria, say determination
coefficient, Student t-statistics, Durbin-Watson statistics, Akaike information criteria and others
suitable for specific case. The separate criteria can be used for constructing a single combined criteria
that enables for automatic selection of the best model.
Bayesian networks. Mathematical models in the form of Bayesian networks (BN) are very useful
for creating clinical diagnostic systems as well as diagnostic systems for engineering and economic
applications. Generally BN is defined as a directed acyclic graph the vertices (nodes) of which are
variables selected to characterize behavior of a system (or multivariate processes) under study, and
the arcs indicate to existing causal relations between the variables. To each daughter node of BN is
assigned conditional probability table that is used for computing probabilistic inference. The model
has the following advantages: dimension of the model can be very high, and the variables hired can be
discrete or continuous; the separate facts generated by experts can also be taken into consideration.
Today there exist numerous procedures for estimating BN structure and computing probabilistic
inference on their basis. Besides, the models are suitable for fighting probabilistic uncertainties of the
type: “is the event going to happen or not, and what is the probability of occurring the event?” The
known successful applications of BN today are numerous and their number continues to grow.
Generally the BN constructing procedure includes the following steps:
research problem statement;
a thorough analysis of a system (processes) under consideration aiming to revealing
specific features of its functioning, as well as selection of parent and daughter variables;
identification of existing system models and determining the possibilities for their usage in
the frames of the DSS constructed;
estimating existing causal relations between the variable selected using appropriate set of
statistics;
possible reduction of the model dimensionality;
scaling and (possibly) discretization of the model variables selected;
determining semantic (logical) constraints for the model;
structure estimation for candidate models using appropriate optimization procedures and
model quality criteria;
model adequacy analysis and selection of the best one(s);
application of the model(s) constructing to solving the problem stated (step 1); comparison
of the results obtained with other possible models;
the final model selection.
Structural uncertainties. According to the methodology proposed the following procedures are
used to cope with possible structural uncertainties of a model: refinement of model order by applying
recursive adaptive approach to modeling and automatic search for the “best” structure using combined
statistical criteria; adaptive estimation of input delay time, and the type of statistical data probability
distribution with its parameters; describing detected nonlinearities with alternative analytical forms,
and with subsequent quality estimation of the forecasts generated [12, 13]. Another wide class of
nonlinear heteroscedastic processes exists today, and is described by the models of conditional
variance dynamics. Usually studying of such processes includes constructing of the two model types:
model for the process amplitude, and the model for time changing variance. As far as formal
description of variance is based on quadratic variables and functions, heteroscedastic processes are
nonlinear by definition [11].
An important stage in the constructing of statistical diagnostic models, including predictive ones,
is the assessment of their quality. Usually, two or more state prediction methods are used to compute
forecast scores in order to be able to combine predictions to further improve the state prediction score.
The forecasts combining scheme with equal or different weighting coefficients used in the approach
proposed is explained by Fig. 2.
The model based forecasts can be computed, for example, with six selected techniques as shown in
Fig. 2. Regression model (autoregression (AR) or AR with moving average (ARMA)) is used for
generating forecast as well as its transformed version into state space (SS) form is necessary for
further application of optimal Kalman filter (KF). Adaptive version of KF is interesting from the point
of view that it provides a possibility for forecasting and on-line (or off-line) estimation of state
disturbance and measurement noise covariance. An alternative approach to Kalman filtering is
application of Bayesian probabilistic (particle) filter providing for the forecasts in the form of
probability distribution that can be selected of necessary type. The distribution can be further used for
estimating its parameters showing future patient state as well as possible span of the state values in
selected space.
Figure 2: The principle of combining alternative forecasts
Some other advantages of using the probabilistic approach to filtering are as follows: it helps to
take into consideration some uncertainties relevant to conditional probabilities of events hidden in the
data being processed; and it helps to generate possible future paths for the states under consideration
using appropriate mathematical models. Also the Bayesian filtering approach represents useful
instrument for data processing and events simulation in parallel with optimal Kalman filter on the
purpose of comparison of the results achieved and computing alternative state estimates. Generally
the diagnostic system under development should contain a set of digital, optimal and probabilistic
filters providing the possibilities for filtering and smoothing statistical\experimental data, imputation
of missing measurements, preform short term prediction of states and estimate possible available non-
measurable variables or parameters.
The well-known group method of data handling and modeling procedure (GMDH) provides the
possibility for constructing models in the general form of Kolmogorov-Gabor polynomials, and the
last three methods mentioned in the figure are related to the popular today intellectual data analysis
techniques. The GMDH approach to modeling is very convenient from the point of view that it
estimates model structure by its internal procedures.
Thus, here we propose the combination of classic regression (statistical) approach with the
intellectual data analysis methodology. The best result of combining the forecasts with respect to
enhancement the quality of final forecast is achieved when variances of forecasting errors for selected
forecasting techniques do not differ substantially (say, not by an order). Some other possibilities for
hiring possible linear and nonlinear models to describe and forecast patient state are given in the
Table 1 below.
The structure of models No. 1-8, presented in Table 1, is partially determined and can be changed
(refined) in the process of adaptation using specific statistical data. Model 1 can be used to describe
various trends along with deviations from the conditional mean. Models 2 and 4 describe bilinear and
exponential nonlinearity. Model 3 describes nonlinearity with saturation. Models 5 and 6 are used to
describe the changes in conditional variance in the study and modeling of heteroscedastic processes.
Model No. 6 shows the best results for short-term forecasting of conditional variance. Models 7, 8,
and 9 can be used to describe arbitrary nonlinearities with high-order model members. The use of
fuzzy sets in modeling involves the construction of a set of rules that describe processes or systems
and carry out logical inference under conditions of information uncertainty. The models based on
neural networks and fuzzy neural networks are used to model complex nonlinear functions under
conditions where some variables are unobservable. Bayesian networks (static and dynamic) are
statistical and probabilistic models, with the help of which it is possible to model complex
multidimensional processes with obtaining the final result of their application in the form of
probabilistic inference (conditional probabilities) characterizing the patient's condition [16].
Table 1
Some linear and nonlinear models for describing process dynamics
No. Model description Formal model structure
𝑝
1 AR + polynomial of
(𝑘) = 𝑎0 + ∑ 𝑎𝑖 𝑦(𝑘 − 𝑖) + 𝑏1 𝑘 + ⋯ + 𝑏𝑚 𝑘 𝑚 + 𝜀(𝑘),
time 𝑖=1
𝑘 = 0,1,2, … is discrete time; 𝑡 = 𝑘𝑇𝑠 ; 𝑇𝑠 is sampling time.
𝑝 𝑞
2 Generalized bilinear
𝑦(𝑘) = 𝑎0 + ∑ 𝑎𝑖 𝑦 (𝑘 − 𝑖) + ∑ 𝑏𝑗 𝑣(𝑘 − 𝑖) =
model 𝑖=1 𝑗=1
𝑚 𝑠
=∑ ∑ 𝑐𝑖𝑗 𝑦(𝑘 − 𝑖)𝑣(𝑘 − 𝑗) + 𝜀(𝑘)
𝑖=1 𝑗=1
3 Logistic regression 1
𝜑(𝑥(𝑘, 𝑧)) = ,
1 + exp(−𝑥(𝑘, 𝑧))
𝑥(𝑘) = 𝛼0 + 𝛼1 𝑧1 (𝑘) + ⋯ + 𝛼𝑚 𝑧𝑚 (𝑘) + 𝜀(𝑘)
4 Nonlinear extended 𝑦1 (𝑘) = 𝑎0 + 𝑎1 𝑦1 (𝑘 − 1) + 𝑏12 exp(𝑦2 (𝑘)) + 𝑎2 𝑥1 𝑥2 + 𝜀1 (𝑘),
econometric 𝑦2 (𝑘) = 𝑐0 + 𝑐1 𝑦2 (𝑘 − 1) + 𝑏21 exp(𝑦1 (𝑘)) + 𝑐2 𝑥1 𝑥2 + 𝜀2 (𝑘)
autoregression
𝑞 𝑝
5 Generalized
2 (𝑘
autoregression with ℎ(𝑘) = 𝛼0 + ∑ 𝛼𝑖 𝜀 − 𝑖) + ∑ 𝛽𝑖 ℎ(𝑘 − 𝑖)
conditional 𝑖=1 𝑖=1
heteroscedasticity
(GARCH)
𝑝 𝑝
6 Exponential |𝜀(𝑘 − 𝑖)| 𝜀(𝑘 − 𝑖)
generalized log[ℎ(𝑘)] = 𝛼0 + ∑ 𝛼𝑖 + ∑ 𝛽𝑖 +
autoregression with 𝑖=1
√ℎ(𝑘 − 𝑖) 𝑖=1
√ℎ(𝑘 − 𝑖)
𝑞
conditional
heteroscedasticity + ∑ 𝛾𝑖 log[ℎ(𝑘 − 𝑖)] + 𝑣(𝑘)
(EGARCH) 𝑖=1
𝑝
7 Nonparametric
𝑦(𝑘) = ∑ {𝛼𝑖 + (𝛽𝑖 + 𝛾𝑖 𝑦(𝑘 − 𝑑)) ∙ exp(−𝜃𝑖 𝑦 𝑚 (𝑘 − 𝑑))} +
model with 𝑖=1
functional +𝜀(𝑘)
coefficients
8 Radial basis function 𝑀
(𝑥(𝑘) − 𝜇𝑖 )2
𝑓𝜃 (𝑥(𝑘)) = ∑ 𝑖 exp (− ) + 𝜀(𝑘),
2𝜎𝑖2
𝑖=1
𝜃 = [𝜇𝑖 , 𝜎𝑖 , 𝑖 ]𝑇 ; 𝑀 = 2,3, …
9 State-space 𝐱(𝑘) = 𝐅[𝐚(𝑘), 𝐱(𝑘 − 1)] + 𝐁[𝐛(𝑘), 𝐮(𝑘 − 𝑑)] + 𝐰(𝑘
representation
10 Neural networks Selected (constructed) network structures
11 Fuzzy sets and neuro- Combination of fuzzy variables and neural network model
fuzzy models
12 Dynamic Bayesian Probabilistic Bayesian network structure constructed with data
networks and/or expert estimates
13 Multivariate Say, copula application for describing multivariate distribution
distributions
14 Immune systems Immune algorithms and combined models
2.4. Example of the medical DSS application
The example is touching upon patient state forecasting using combined (linear + nonlinear part)
model. The combined model proposed includes optimal and digital filters, linear regression models
and nonlinear logit model (Fig. 3).
Figure 3: Combined model: filtering + linear regression + nonlinear regression
The purpose of using the two alternative filters is to perform data smoothing (suppressing
undesirable high frequency components often contained in measurements) and this way prepare it for
modeling and state forecasting. The two filters provide two alternatives for subsequent constructing
linear regression models. Besides, application of the optimal Kalman filter additionally provides a
possibility for solving the following problems: estimation of non-measurable state vector components;
variance/covariance estimation for observations hired for model constructing; and short-term state
forecasting when necessary. In this specific example the following hourly measurements were used:
arterial blood pressure, heart rate, skin resistance, and body temperature. The model purpose was to
provide a forecast for evolution of a patient state to “better” (indicated by “1”), and to “worse”
(indicated by “0”). Table 2 contains quality of forecasting direction of the patient state evolution.
Table 2
Results of forecasting direction for evolution of patient state
Model type Probability of correct direction
forecast
Logistic regression 72.75%
Classification tree 69.61%
Logistic regression + extra forecast by linear model 77.69%
Classification tree + extra forecast by linear model 75.54%
Thus, in both cases (logistic regression and classification tree) the best state forecasting results
were achieved with the use of extra state forecast by the linear regression model. The statistical
quality characteristics of the forecasts achieved show high quality of the forecasts and possibility of
their use in estimation of patient state. Certainly there are possibilities for further improvement of the
preliminary results obtained.
3. Conclusions
The article proposed the technique for modeling and forecasting nonlinear non-stationary
processes based on the analysis and processing of statistical data in medical applications. The
technique is based on general systemic (system analysis) principles and represents a hierarchical
structure in the analysis of medical data, taking into account possible statistical uncertainties. With the
help of the systemic approach, the development of adaptive schemes for assessing the structure and
parameters of the model, the use of statistical and probabilistic criteria for the construction and
selection of the best model for solving medical problems is implemented. The procedures for optimal
and digital filtering of data, methods for filling in the missing measurements, methods for estimating
model parameters and Bayesian programming were proposed as methods for overcoming possible
uncertainties.
On the basis of the methodology developed, it is possible to construct combined models, including
statistical regression and probabilistic models in the form of Bayesian networks, decision trees,
logistic regression, etc. This approach has shown its high effectiveness in constructing short-term
forecasts. The presented application example demonstrated the adequacy of the model constructed
and high quality of short-term patient states predictions.
Further development of the proposed methodology will be aiming at the development of more
advanced model structures for nonlinear non-stationary processes encountered in medical applications
to improve diagnostic and therapeutic processes. The technique is implemented in a diagnostic
medical decision support system and can be used for short-term prediction of a patient's condition and
diagnosing its state.
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