The method of multivariate statistical analysis of the time multivariate critical quality attributes of manufacture process with the data factorization Текст научной статьи по специальности «Строительство и архитектура»

UDC 004.94:658

THE METHOD OF MULTIVARIATE STATISTICAL ANALYSIS OF THE TIME MULTIVARIATE CRITICAL QUALITY ATTRIBUTES OF MANUFACTURE PROCESS WITH THE DATA FACTORIZATION

Havrylko Ye. V. - Dr. Sc., Associate Professor, Professor of Computer Sciences Department, State University of Telecommunications, Kyiv, Ukraine.

Kurchenko O. A. - PhD, Associate Professor, Head of the Department of Management of Information and Cyber Security, State University of Telecommunications, Kyiv, Ukraine.

Tereshchenko I. V. - PhD, Associate professor of the Department of info-communication engineering, Kharkiv National University of Radio Electronics, Kharkiv, Ukraine.

Tereshchenko A. I. - Post-graduate student of the Department of Management of Information and Cyber Security, State University of Telecommunications, Kyiv, Ukraine.

ABSTRACT

Context. This paper presents a method for solving the problem of product's quality assurance at the stage of the initial manufacture process design in accordance with the process-analytical technology for the design of modern certified manufacturing -QbD. The method uses the information technologies of multivariate statistical analysis (MSA) to evaluate the influence of time multivariate critical process parameters (CPPs) on the time product critical quality attributes (CQAs). Preparatory transformation of clusters of critical process (manufacture process) parameters into factors of product critical quality attributes was carried out.

Objective. To disclose the method of multivariate statistical analysis for assessing the character and features of the influence of time multivariate critical process parameters on time multivariate critical quality attributes at the design stage of the manufacture process.

Method. The method consistently uses: statistical procedures of exploratory multivariate data analysis; transformation the homogeneous observed values matrices of CPPs and product CQAs into data frame (table) with factorized data; construction the regression trees of multivariate CPPs with a multivariate responses (CQAs). The method is implemented the R language packages software.

Results. Factorized time multivariate CPPs make it possible to use methods of multivariate statistical analysis for evaluating the influence of CPPs factors on the time multivariate CQAs.

Conclusions. This method of statistical analysis, together with statistical multivariate canonical analysis, represents an up-to-date information technology for detailed estimation the influence of time multivariate CPPs objects and some CPPs components on CQAs.

KEYWORDS: quality-by-design, critical quality attributes, critical process parameters, design of experiment, multivariate statistical analysis.

ABBREVIATIONS

CCA is a canonical correspondence analysis;

CMAs is a critical material attributes;

CPPs is a critical process parameters;

CQAs is a critical quality attributes;

DoE is a design of experiment;

MBSE is a model based systems engineering;

MRT is a multivariate regression trees;

MSA is a multivariate statistical analysis;

NDI is a non-development item;

NMDS is a nonmetric multidimensional scaling;

PAT is a process analytical technology;

PCA is a principal component analysis;

PDCA is a plan-do-check-act cycle;

QbD is a quality-by-design;

QFD is a quality function deployment;

RDA is a redundancy analysis;

SEMMA is a sample, explore, modify, model, assess

TPQP is a target product quality profile;

VAT is a visual assessment of cluster tendency;

VS is a variables selection.

NOMENCLATURE

X is a observations array of random multivariate manufacture process variables (CPPs observations);

Y is a observations array of random multivariate manufacture responses (CQAs observations);

XJ is a m-dimensional vector of the J-th ( J = 1,2,..., n ) observation of m random CPPs;

YJ is a ^-dimensional vector of the j-th ( j = 1,2,..., n ) observation of krandom CQAs;

xm is a m-th component of XJ , X] = {,...,xm};

yk is a k-th component of YJ , YJ = {,..., yk } ;

XV is a point in the ¿--dimensional space of attributes: X1J, X2,..., X{;

Xcq is a q x m (q = 1,2,..., n ) sample from the multivariate population at the v-th attribute.

INTRODUCTION

Market conjuncture and increased rivalry objectivity set the task of developing and improving methods for ensuring high quality products in stage of the initial manufacture process design with further management of the entire production cycle (PDCA, [1]) in accordance with the TPQP [2].

One of the actual ways to solve this problem is the QbD methodology [2, 3]. This methodology shifts the emphasis of product quality assurance towards information technology of statistical research and attentive experiment planning at the stage of product development and manufacture process design. The QbD concept is a significant transformation in product quality regulation from an empirical process to a more scientific and risk-based approach. The QbD concept combines categories under the general term "critical quality attributes": CMAs, CPPs and CQAs with TPQP. In turn, TPQP determines the quantitative parameters of the product safety and efficiency [2] and allows them to be compared with the requirements of standards, as shown in Fig. 1.

The dependence of product CQAs on CPPs and CQAs is considered as a quality function: CQAs = F(CPPs). This quality function is structured according to the main stages of the QFD [4] manufacture process, for which critical quality attributes are determine and QbD principles are applied, as shown in Fig. 1.

As an important tool for QbD, PAT [5, 6] is considered, which actively uses MSA information technologies to provide and maintain critical quality attributes within the standards requirements in the design space. The results of the data multivariate statistical analysis of the multifactor experiment at the design stage make it possible to establish the permissible variability ranges of CPPs and CMAs based on the extent of influence on CQAs.

It should be noted that with the strict requirements of standards (ISO 9001 [7], ISO 22000 [8]) for CMAs, the influence of CPPs on CQAs is more often considered [5]. In this case, the time multivariate statistical data CPPs generally have a group/object influence on CQAs and therefore need a software factorization of the time multivariate computer format data.

The object of the study is the process of product's quality assurance at the stage of the initial manufacture process design.

In accordance with the abovementioned, the task of products quality assurance at the stage of the initial manufacture process design is concretized as the definition of CPPs and their objects, which significantly affect on CQAs.

The solution of this problem is proposed to implement by consistently application of the following MSA methods:

- Defining the structure and hierarchy of time multivariate data: CPPs and CQAs;

- Defining of qualitative and quantitative measures of relations between the formed objects (factors) CPPs and CQAs;

- Comparison of the results of factor and component analysis of the influence of time multivariate CPPs on time multivariate CQAs.

Together with computer-intensive information technologies of statistical canonical analysis [9], the factorized multivariate data of critical quality attributes provide additional principal opportunities for multivariate statistical analysis of the affect of CPPs on CQAs.

The method of Genichi Taguchi is known for solving this problem by fractional and factored planning of the experiment with subjective test sets of design parameters and noise factors. These data form the matrix experiments using orthogonal arrays [10]. According to the effects of noise factors, orthogonal design parameters are evaluated for robustness and analyze possible losses of the quadratic quality function.

In contrast to this decision, it was suggested to apply MSA and computer-intensive information technologies for carrying out a statistical full-factor experiment with direct factorization of each of the observation tables after the exploratory data analysis. At the same time, statistical estimates of multivariate factor analysis are adequately supplemented by estimates of the component affect of CPPs on CQAs.

The subjects of studies are informational technologies for estimating the factor influence of manufacture CPPs on product CQAs.

Figure 1 - The relationship between categories of critical quality attributes and TPQP according to the QbD concept

The proposed method of MSA of the time multivariate critical quality attributes of manufacture process with the data factorization is should be considered as the evolution of the QbD approach of product quality assurance at the design development stage. The QbD concept and method are aimed at an experimental study of the influence degree of CPPs on cQAs.

1 PROBLEM STATEMENT

The statistics of the experiment operates with data from object-oriented n observations formed into two arrays of random multivariate variables X and Y, X ^ Y . A random variable X potentially determines the

properties of the random variable Y: X 3 {XJ},

Y 3 {y J }. The category X ^ Y is described by the quality function of the manufacture process:

CQAs = F(CPPi), CQAs 3 {cQAk } .

Formally, the results of monitoring (sampling) of random multivariate values of X and Y are data of the QbD development space, but do not represent a categorized information space of relations X ^ Y .

To determine and characterize the causal relations in the information space of the variables X and Y, it is necessary to inquire into the structure and hierarchy of observational data X and Y, as well as the qualitative and quantitative measures of the relations between them.

The problem of solving this task is that empirical arrays of X and Y data have a computer format in the form of time interrelated matrices that are not ready for issue. To conduct multivariate statistical analysis, the X and Y data arrays must undergo a preparatory processing step in the form of statistical procedures of exploratory data analysis, clustering, factorization and interpretation of the results [11]. In practice, observations of exogenous variables are not offline, but are groups of related CPPs. These groups form clusters with an interpreted dependence of variables for which it is of interest to estimate the influence on the multivariate responses of the

manufacture process (YJ ). It is suggested to cluster the

time random multivariate data CPPs (XJ) and identify them with the influence factors on the multivariate

responses of the manufacture process CQAs (Y J ).

These operations make it possible to determine the potential trends and relationships between factorized observations of variables X and an array of multivariate responses Y, and also allow MSA to be used to research dependence X ^ Y as a quality function:

CQAs = F (CPP,).

2 REVIEW OF THE LITERATURE

The QFD process methodology [4, 12], the methods of assurance the TPQP with the detailed QbD [2, 5, 13] and

the risk-oriented process approach [4, 14, 3] - are the dominants of modern quality management systems [7].

High requirements to product quality and market competition - lead to the need of products quality assurance at the QbD development stage [13, 5] at an decomposition of the quality function (QFD) [4]. An adequate tool for QbD are the PAT methods [6, 15], which are based on information MSA technologies.

At the design stage, the PAT methods are focused on the research of multifactorial relationships between CPPs and CQAs, exogenous factors and their influence on the on the final product quality [12, 16].

In [17], an actual systematic review of a large number of references on the topics of applications of MSA methods is given. The known methods for determining the nominal values of CQAs, CPPs and permissible variation for them offer statistical solutions to the multivariate VS. The VS statistical methods (essentially MSA methods) research the contribution of exogenous factors to the variability of the quality function values (a multivariate endogenous variable).

Thus, MSA methods create an adequate reasoned basis for transformation functional requirements into design parameters and for justification of design parameters permissible variation [18]. The adequate CPPs and CMAs are determined on the basis of an reasonable (standards-oriented) change in the characteristics of the quality function and the sensitivity of this function to critical quality attributes.

This problem can be considered from the position of system engineering [19, 20, 21], when models play a key role at one or several stages of the development process and are part of the process design by the MBSE [22]. The main stages of research and design of the product/process from the general requirements to the system [21] within the background of robust design [18] inevitably lead to the architecture (structure + logic) of the experiment [16, 21, 23]. One of the general trends of this logic is the decomposition of the quality function (QFD) [3, 4, 18] using the DoE technology [19], similar to the methodology of careful experiment planning (QbD) [2, 3, 5] with PAT tools [6].

It is also important to note one of the features of modern process design, which is the use of NDI technologies (for example, software packages of MSA methods on R language) [24].

At the design and experimentation level, this direction is getting widespread use in connection with the intensive introduction of information technologies of object-oriented MSA and the high confidence of engineers in the developed and tested software packages.

Thus, the review of literature enables to emphasize the following features of the task:

- the definition of the requirements for methods of technological approximation to the specified values of CQAs attributes, as a some quality function of the CPPs values, is made at the stage of the initial manufacture process design;

- the guaranteed and certified product quality at the hierarchical level of the initial manufacture process design is achieved by computer-intensive information MSA technologies of manufacture process and products in the multivariate space of design parameters.

3 MATERIALS AND METHODS

To ensure the standardized quality of products on the initial manufacture process design, the task is to determine from the data of n observations the m CPPs -{x1v..,xm} and their objects (factors) that significantly

affect on k CQAs of product - {y1,...,yk} multivariate process responses.

The cluster analysis methods combine samples XJ of the observable data set into groups of objects that are in some sense homogeneous and called clusters or classes, as shown in Fig. 2.

Suppose that each of the observations XJ is given as a point - X1 in the ¿--dimensional space of attributes:

XJ, XJ,

, X . The set of these points can be treated as a

sample q x m - Xq (q = 1,2,

- the number of

observations corresponding to the v-th attribute) from the multivariate population at the v-th attribute.

A number of observations of the q-th attribute form an object (cluster), which is a factor of the homogeneous

affect of q observations on CQAs. Some observations YJ are grouped into objects by this attribute by conducting a constrained ordination. Based on the results of such an analysis, a diagram is constructed - an "ordination three plot", which shows the variability of the CQAs with relation to the two CCA1-CCA2 (canonical correspondence axis) projection axes and the statistical relationships with each critical process parameter -

{X1,..., Xm } .

The research used partitioning algorithms of the R language that decompose the data set of n observations into 5 clusters with previously unknown parameters. This algorithms searches for centroids, i.e. the centers of points

concentration with the minimum scatter within each cluster as far as possible from each other and with the verification of clustering quality [11].

The interrelation between the components of multivariate data of X and Y arrays was established under the multivariate statistical canonical analysis with the means of software packages of the R language [11]. In particular, the mathematical apparatus of multivariate canonical ordination was used: RDA, CCA and NMDS.

These methods are positioned as an extension of multivariate statistical regression analysis in modeling the

multivariate response of the manufacture process (YJ ) to the multivariate value of the critical process parameter

(XJ).

Thus, the sequence of performing statistical procedures for clustering and factorization of multivariate data CPPs and CQAs creates a model of a statistical full-factor experiment that is realized by methods of multivariate statistical causal analysis.

4 EXPERIMENTS

In the case under consideration, the experiment assumes the study of cause-effect or causal dependencies.

To determine the causal relationships of multivariate X j

and YJ variables, three conditions must be fulfilled [25]:

1. The events and facts related with a X j variable must precede events and facts related with a YJ variable.

2. The variation of the X j variable must be

accompanied by a variation of the YJ variable.

3. The influence of the third variable, which implies an alternative explanation to empirical observations and a

statistically confirmed relationship between X j and Y j should be excluded.

Thus, is proposed a scheme of the experiment in the form of a sequence of actions above multivariate observational data as critical attributes of the quality of the manufacture process:

Figure 2 - Formation of factors of time multivariate data X and Y

1. The carrying out an exploratory analysis of exogenous variables data: determination of correlated variables, estimation of clustering tendency.

2. The carrying out of clustering of the constrained

values (XJ ) and quality estimation of this clustering.

3. The transformation of random multivariate

homogeneous arrays (data frames) of variables XJ and

YJ into data frames with factor variables.

4. The carrying out and estimating of unconstrained

ordination of responses Y J and constrained ordination of

responses YJ by factors of exogenous variables XJ . 4a. The comparison of the results of unconstrained

and constrained ordination of multivariate responses Y J (without and with affect the factors of exogenous

variables XJ).

4b. The comparison of estimates the constrained

ordination of the responses Y J with the canonical analysis by CCA and NMDS methods.

4c. The formation of the MRT of an exogenous X variables array with a multivariate responses Y.

5. The estimation the MSA results of the relations between an array of exogenous variables X and an array of multivariate responses Y.

In general, this algorithm corresponds to the SEMMA methodology [26] for determining physical regularities at statistical analysis. SEMMA focuses on structuring processes (such as QFD), as well as applying statistical analysis and visualization of results at each stage of the design process and gives the researcher the freedom to choose the concept and implementation of explore.

5 RESULTS

The results correspond to the multivariate statistical analysis of time-data arrays of predictors X and responses Y specified in the form of uniformly distributed random

values of the design space within the ranges specified in standards.

The exploratory data analysis was used for descriptive statistics of sample properties. This analysis included: processing and systematization of empirical data, visual presentation of multivariate data in the form of tables and graphs, as well as quantitative characteristics of the basic statistical parameters. Estimation of the weight or variable

importance of predictors X J and their tendency to clustering, taking into account the low multicollinearity, was evaluated using the principle components analysis and VAT, as shown in Fig. 3 a.

The search for an optimal scheme for combining predictors into clusters was performed by permutation of multisets at various combinations of the groups number, distances metrics and clustering methods using 30 quality parameters (indices).

Based on the results of the calculations, the optimal number of clusters was chosen, as shown in Fig. 3b.

The operation of transforming homogeneous arrays of numerical variables X and Y into tables with factor (cluster) attributes was performed by identifying the

factor with the cluster of observations XJ and

corresponding group of multivariate responses YJ .

As a result, tables (data.frame) with factorized data arrays X and Y were formed, as shown in Fig. 2.

Unlike unconstrained ordination, constrained ordination allows to connect variability of multivariate

values of responses YJ with affect of predictors XJ .

In Fig. 4 shows the ordination of the multivariate response components Y on the principal components plane (unconstrained ordination) and the distribution of these values over the objects of the X array factors on the principal components plane (constrained ordination).

Optimal number of clusters - k - 2

Principal component X

>

£ SO

Comp.1 Comp.2 Comp3 Comp.4 Camp.5 Corap.6

3 4 S

Numtef of clusters k.

b

Figure 3 - The result of estimating the weight of the predictor components X - a. The result of determining the optimal number of clusters according to the estimates of different indices - b

a

An unconstrained ordination analysis of responses matrix Y makes it possible to obtain a mapping in the orthogonal coordinate system of only some particular structural features of the research data array in the form of graphic projections of the components of multivariate

response YJ on the principal components plane, as shown in Fig. 4a.

In the diagram of Fig. 4b shows the grouping of observations by factor characteristics on the background of the explanatory components xm (dashed lines)

associated with the eigenvectors of the X array (ordination three plot).

Note, that the use of PCA-ordinance is a method of smoothing random data fluctuations and contributes to the construction of more stable (robust) cluster structures. The practice of direct ordination allows combining the operations of hierarchical formation of objects on the separating predictors of the X array with the projection of multivariate data of responses array Y into the space of the principal components.

To assess the component relationship of two arrays of multivariate variables X and Y, used statistical canonical ordination procedures: CCA and NMDS.

To represent the results of NMDS, when the least distorted distance metric between the two-dimensional

projections of the Y J values on the principal component axis is determined. This distance metric is determined by the Spearman correlation coefficient for each of the

variants of the distance metric in the Y

j

response spaces

and predictors XJ .

The NMDS results: the projections of the response

YJ values on the ordinate axis NMDS1-NMDS2 and the relations with the components of predictor variables xm

are shown in Fig. 5.

At constructing of regression trees of an exogenous variables array X with multivariate responses Y (MRT), the task of estimating the affect of the components of exogenous variables xm on the common variability of the

components of the multivariate response yk is solved, as shown in Fig. 6.

Variables factor map (PCA)

Dim 1 (49 83%)

d = 1

a b

Figure 4 - Diagrams of unconstrained ordination - a and constrained ordination - b of the array of responses Y on principal

components plane (PCA)

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0.0 NMDS1

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Figure 5 - The ordinance results of the response values Y on the NMDS1-NMDS2 axis and the relations

with the predictors components x

Table 1 - The expected contribution of components xm to the common variation of the X data array

Comp. Com.1 Com.2 Com.3 Com.4 Com.5 Com.6

Param.

Standard deviation 1.15 1.02 0.99 0.97 0.937 0.84

Prop-n of variance 0.22 0.17 0.16 0.15 0.14 0.12

Cum-ve prop-n 0.22 0.40 0.57 0.73 0.878 1.00

Figure 6 - MRT with multivariate response yk for clusters of an array of exogenous variables X

MRTs are formed as a result of a recursive operation of dividing the rows of the responses data array Y into homogeneous subsets using an array of exogenous variables X as the separating parameters.

MRTs provide visualization of the effect of the components of exogenous variables xm on the formation

of an array of multivariate responses Y and show which components of the array initiate the decomposition of the researched population Y in the tree nodes and determine the composition of the formed objects Y.

The results demonstrate the great potential of MSA at the analysis of time multivariate data for a practical application of products quality assurance at the stage of the initial manufacture process design.

6 DISCUSSION

The article presents the results of the research at the design stage of the manufacture process, during which a multivariate statistical analysis of the cause-effect relationships between the variability of the critical quality attributes and the action of certain factors identified with the critical process parameters. The study contains elements of exploratory, descriptive and analytical statistical analysis.

The evaluation of the x predictors variable

m r

importance and the tendency to clustering the X array, taking into account the possible multicollinearity of the components xm , were estimated using the PCA-method

and visualization of the trends formation of clusters (VAT).

The PCA-method yields the relative expected contribution of each component xm (Com.m) to the

common data variation, as shown in Table 1.

Note that all components of a multivariate X array are characterized by an almost equal contribution (proportions of variance 16.6%) to the expected distribution and together they give a common contribution of 100%. The clustering tendency available in this test is characterized by Hopkins statistics (0.5087), which shows a moderate tendency to form clusters at a random distribution of data.

The criterion for the effectiveness of clusterization is based on the metric hypothesis of compactness. If it is possible to find such a division of observations into groups that the distances between observations from one group (intra-cluster distances) will be less than a definite value s > 0, and between observations from different groups (cross-cluster distance) - bigger than e and therefore clustering is successful.

According to this criterion, the results of assessing the quality of clustering according to the Zagoruiko test for clusters of an array of multivariate data X were obtained (Fig. 3b), as shown in Table. 2. In Table 2 on the main diagonal are located average intra-cluster distances, which are smaller than the cross-cluster distances (off-diagonal elements of the table), that indicates the possibility of division the value space of an X array into two clusters.

Table 2 - The quality data of clustering estimation by the Zagoruiko test

Cl. dist Cl. dist. 1 Cl. dist. 2

1 3.044665 3.623451

2 3.623451 3.009617

The operation of transforming a homogeneous table of endogenous variables observations Y into a table with factor variables was carried out after research the characteristics of this array.

The results of the projection in the principal components orthogonal coordinate system of some features of the internal relation of the multivariate response array components Y (unconstrained ordination) are shown in Table 3.

Table 3 - The results of unconstrained ordination of the data Y array on the principal components axis

Dim. 1 Dim. 2 Dim. 3

y1 43.56287 10.7237730 45.71336

y2 8.30943 88.6417708 3.04880

y3 48.12770 0.6344562 51.2378

From Table 3 that the components of the responses array yk in value of their contribution to the components

of the decomposition Dim1, Dim2, Dim3, the factors do not form and have a weak correlation. The unconstrained ordination is limited only to an estimate of the distribution

of the elements y>k on the principal components plane

and the interpretation of the contribution of each component. The constrained ordination allows to establish correspondence the variability of multivariate values of

responses YJ with affect of exogenous variables xm .

The constrained ordination of the responses array (Fig. 4b) is shown against the background of the components (dotted lines) and corresponds to relation the internal

features of responses YJ with the factorized effects of the critical process parameters XJ . The location of responses Y J in both factor objects indicates a weak correlation of the components yk and the predominance of random

affects of the components xm on the response

components variability YJ . This fact makes it necessary the research of the component influence of predictors xm on the variability of the components of a multivariate

dependent random variable Y J . The predominance result of the influence the random constituent of the predictors variability xm on the components of multivariate

responses YJ is confirmed by the NMDS of projections of the array values Y on the ordinate axis NMDS1-NMDS2.

The NMDS method is most applicable for data arrays with a significant influence of random or unaccounted factors, as in the case under consideration.

The results of application of the NMDS method are summarized in Table 4.

Table 4 - The NMDS results of the response array Y on the NMDS1-NMDS2 axis and the relations with the predictors x

"---NMDS^ NMDS1 NMDS2 r2 Pr(>r)

x1 0.48665 0.87360 0.0329 0.201

x2 -0.33369 -0.94268 0.0292 0.239

x3 0.97049 -0.24114 0.0213 0.327

x4 0.06534 -0.99786 0.0087 0.665

x5 0.99269 -0.12071 0.0010 0.960

x6 -0.71850 -0.69552 0.0024 0.900

From Table 4 it follows that with a low statistical significance of the influence of exogenous variables xm

on the responses Y J , there is a high correlation of some components xm with the ordinate axes (for example, x5

with NMDS1 and x4 with NMDS2).

At interpreting (Fig. 5), it should be borne in mind that the result of the nonmetric scaling of multivariate

responses Y J is imposed on the components of exogenous variables xm , which situated depending on

their calculated correlation coefficients with the ordinate axes NMDS1-NMDS2.

From Fig. 5 that the NMDS model, even with a small number of multivariate data components X and Y, gives only a conditional idea of the affect of exogenous

variables x on the responses Y J .

m

The detailing impact of the components xm is usually performed by projection on the constrained ordinance of the responses Y J of isolines contours of the smoothed components xm of the data array X. Note, that with

increasing number of components, as well as with a weak tendency to clustering of X and Y array values, the analysis becomes more complicated.

At the final stage of the analysis, an MRT of an array of exogenous variables X with a multivariate response Y was formed, as shown in Fig. 6. For this MRT, "leaves" are clusters of objects Y with minimal differences between points in a multivariate space within each cluster. Note, that in order to explain the effect of variables x on

m

responses Y J the MRT software package has the option of selecting a various number of clusters. For this MRT (Fig. 6), the dividing variable is the component x6, which forms equivalent clusters with an almost equal number of observations (47 and 53) at the absence of explicit dominance of the components yk .

The discussion shows that the experimental data of the computer format require processing, analysis and interpretation procedures that determine the potential trends and dependencies between unit components and objects of the time multivariate data for the purpose of practical interpretation at the stage of the initial manufacture process design.

CONCLUSIONS

The idea of design a quality function of the manufacture process at each of QFD stages in order to obtain predictable values of the attributes of the product critical quality at variability of the critical process parameters values within the tolerances is realized using the multivariate statistical analysis methods.

The scientific novelty of obtained results is that the MSA method uses the preparatory transformation of clusters of critical process parameters of manufacture process into factors of product critical quality attributes is carried out.

This method is used under statistical experiments with multivariate arrays of critical quality attributes of computer format to products quality assurance at the stage of the initial manufacture process design.

The researches results are consequence of combining the causal methods of multivariate statistical analysis that allow to obtaining the numerical characteristics with an adequate practical interpretation of the influence the critical process parameters on the product critical quality attributes.

The prospect of further research consists in the synthesis or application of robust design methods to provide standard values of product critical quality attributes that are resistant to the affects of various environmental factors and to the variability of critical process parameters.

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14. ISO 31000 Risk management. [Electronic resource]. Access mode: https://www.iso.org/iso-31000-risk-management.html

15. Guidance for Industry. PAT - A Framework for Innovative Pharmaceutical Development, Manufacturing, and Quality Assurance. [Electronic resource]. Access mode: https://www.fda.gov/downloads/drugs/guidances/ucm070305.p df

16. Mohamed I. Progressive Modeling: The Process, the Principles, and the Applications, Procedia Computer Science, 2013, Vol. 16, pp. 39-48. https://doi.org/10.1016/j.procs.2013.01.005

17. Araujo F., Pimentel P., Fogliatto F. S. Variable selection methods in multivariate statistical process control: A systematic literature review, Department of Industrial Engineering, Federal University of Rio Grande do Sul, 90035-190 Porto Alegre, RS, Brazil, Computers & Industrial Engineering. January 2018, Vol. 115, pp. 603-619. https://doi.org/10.1016Zj.cie.2017.12.006

18. Gohler S. M., Husung S., Howard T. J. The Translation between Functional Requirements and Design Parameters for Robust Design, Procedia College International pour la Recherche en Productique (CIRP), 2016, Vol. 43, pp. 106-111. https://doi.org/10.1016/j.procir.2016.02.028

19. Gausemeier J., Gaukstern T., Tschirner C. Systems Engineering Management Based on a Discipline-Spanning System Model, Procedia Computer Science, 2013, Vol. 16, pp. 303-312. https://doi.org/10.1016/j.procs.2013.01.032

20. MacCalman A., Kwak H., McDonald M., Upton S. Capturing Experimental Design Insights in Support of the Model-based System Engineering Approach, Procedia Computer Science, 2015, Vol. 44, pp. 315-324. https://doi.org/10.1016/j.procs.2015.03.030

21. Kaiser L., Bremer C., Dumitrescu R. Exhaustiveness of Systems Structures in Model-Based Systems Engineering for Mechatronic Systems, Procedia Technology, 2016, Vol. 26, pp. 428-435. https://doi.org/10.1016/j.protcy.2016.08.055

22. Mac Calman A., Lesinski G., Goerger S. Integrating External Simulations Within the Model-Based Systems Engineering Approach Using Statistical Metamodels, Procedia Computer Science, 2016, Vol. 95, pp. 436-441. https://doi.org/10.1016/j.procs.2016.09.309

23. Lemazurier L., Chapurlat V., Grossetête A. // An MBSE Approach to Pass from Requirements to Functional Architecture, International Federation of Accountants (IFAC)-Papers OnLine, July 2017, Vol. 50, Issue 1, pp. 7260-7265. https://doi.org/10.1016/j.ifacol.2017.08.1376

24. Clement S., Register A., Wise R. Key Enablers for Leveraging Non-Development Items in a System, Procedia Computer Science, 2015, Vol. 44, pp. 164-173. https://doi.org/10.1016/j.procs.2015.03.009

25. Cohen J., Cohen P., West S. G., Aiken L. S. Applied multiple regression/correlation analysis for the behavioral sciences. Mahwah, New Jersey, US: Lawrence Erlbaum Associates Publishers, Third Edition, 2003, 703 p. https://doi.org/10.1177/019394598000200320

26. Azevedo A., Santos M. F. KDD, SEMMA and CRISP-DM: a parallel overview, IADIS European Conference on Data Mining, Amsterdam, the Netherlands 22-27 July 2008, Proceedings of the IADIS European Conference on Data Mining, 2008, pp. 182-185. [Electronic resource]. Access mode:

https://pdfs.semanticscholar.org/7dfe/3bc6035da527deaa72007a 27cef94047a7f9.pdf

Received 14.06.2018.

Accepted 01.10.2018.

УДК 004.94:658

МЕТОД БАГАТОВИМ1РНОГО СТАТИСТИЧНОГО АНАЛ1ЗУ ЧАСОВИХ БАГАТОВИМ1РНИХ КРИТИЧНИХ АТРИБУТ1В ЯКОСТ1 ПРОЦЕСУ ВИРОБНИЦТВА З ФАКТОРИЗАЦ16Ю ДАНИХ

Гаврилко €. В. - д-р техн. наук, професор кафедри комп'ютерно1 шженери, Державний ушверситет телекомушкацш, Ки1в, Укра1на.

Курченко О. А. - канд. техн. наук, доцент, завщувач кафедри управлшня шформацшною та юбернетичною безпекою, Державний ушверситет телекомунгкацгй, Кшв,Укра1на.

Терещенко I. В. - канд. техн. наук, доцент, доцент кафедри шфокомунжацшно! шженери, Харювський нацюнальний ушверситет радюелектронжи, Харкв, Укра1на.

Терещенко А. I. - астрант кафедри управлшня шформацшною та юбернетичною безпекою, Державний унгверситет телекомушкацш, Кшв, Украша.

АНОТАЦ1Я

Актуальнiсть. У статт запропоновано метод розв'язання задач1 забезпечення якоста продукцп на етат первинного проектування процесу виробництва вщповщно до актуально1 процесно-аналпично1 технологи конструювання сучасних сертифшзваних виробництв - «яюсть через дизайн» (QbD). Об'ектом дослщження е процес забезпечення якоста продукту на рантх етапах проектування. Метод використовуе шформацшн технологи багатовим1рного статистичного анал1зу MSA (Multivariate Statistical Analysis) для оцшки впливу часових багатовим1рних критичних параметргв процесу виробництва на часов1 атрибути критичного якоста продукту. Проводиться трансформацш кластергв критичних параметр1в процесу виробництва в чинники критичних атрибута якоста продукту.

Метод. Метод послщовно використовуе: статистичн процедури розвщувального анал1зу багатовим1рних даних; трансформащю однорщних матриць значень спостережень критичних параметр1в виробничого процесу i критичних атрибута якостi продукту в таблицi з фактс^зованими даними; побудову дерев регресй багатовимiрних критичних параметргв процесу виробництва з багатовимiрним вщгуком. Метод реалiзований за допомогою програмних пакетв мови R.

Результати. Фак1ч^зоваш часовi багатовимiрнi критичнi атрибути якостi процесу виробництва надають додатковi ступенi свободи використання методгв багатовимiрного статистичного аналiзу для оцшювання впливу факторгв критичних параметргв виробничого процесу на часовi багатовимiрнi критичнi атрибути якостi продукту.

Висновки. Запропонований метод статистичного аналiзу представляе актуальну iнформацiйну технологта детального оцшювання впливу об'екта часових багатовимiрних даних критичних параметргв процесу виробництва i окремих складових на критичнi атрибути якосп продукту.

КЛЮЧОВ1 СЛОВА: яюсть через дизайн, атрибути критично1 якоста, критичнi параметри процесу, дизайн експерименту, багатовимiрний статистичний аналiз.

УДК 004.94:658

МЕТОД МНОГОМЕРНОГО СТАТИСТИЧЕСКОГО АНАЛИЗА ВРЕМЕННЫХ МНОГОМЕРНЫХ КРИТИЧЕСКИХ АТРИБУТОВ КАЧЕСТВА ПРОЦЕССА ПРОИЗВОДСТВА С ФАКТОРИЗАЦИЕЙ ДАННЫХ

Гаврилко Е. В. - д-р техн. наук, профессор кафедры компьютерной инженерии, Государственный университет телекоммуникаций, Киев, Украина.

Курченко О. А. - канд. техн. наук, доцент, заведующий кафедрой управления информационной и кибернетической безопасностью, Государственный университет телекоммуникаций, Киев, Украина.

Терещенко И. В. - канд. техн. наук, доцент, доцент кафедры инфокоммуникационной инженерии, Харьковский национальный университет радиоэлектроники, Харьков, Украина.

Терещенко А. И. - аспирант кафедры управления информационной и кибернетической безопасностью, Государственный университет телекоммуникаций, Киев, Украина.

АННОТАЦИЯ

Актуальность. В статье предложен метод решения задачи обеспечения качества продукции на этапе первоначального проектирования процесса производства в соответствии с актуальной процессно-аналитической технологией конструирования современных сертифицированных производств - «качество через дизайн» (QbD). Объектом исследования является процесс обеспечения качества продукта на ранних этапах проектирования. Метод использует информационные технологии многомерного статистического анализа MSA (Multivariate Statistical Analysis) для оценки влияния временных многомерных критических параметров процесса производства на временные атрибуты критического качества продукта. Проводится трансформация кластеров критических параметров процесса производства в факторы критических атрибутов качества продукта.

Метод. Метод последовательно использует: статистические процедуры разведывательного анализа многомерных данных; трансформацию однородных матриц значений наблюдений критических параметров производственного процесса и критических атрибутов качества продукта в таблицы с факторизованными данными; построение деревьев регрессии многомерных критических параметров процесса производства с многомерным откликом. Метод реализован с помощью программных пакетов языка R.

Результаты. Факторизованные временные многомерные критические атрибуты качества процесса производства предоставляют дополнительные степени свободы использования методов многомерного статистического анализа для оценки влияния факторов критических параметров производственного процесса на временные многомерные критические атрибуты качества продукта.

Выводы. Предложенный метод статистического анализа представляет актуальную информационную технологию детальной оценки влияния объектов временных многомерных данных критических параметров процесса производства и отдельных составляющих на критические атрибуты качества продукта.

КЛЮЧЕВЫЕ СЛОВА: качество через дизайн, атрибуты критического качества, критические параметры процесса,

дизайн эксперимента, многомерный статистический анализ.

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