=Paper=
{{Paper
|id=Vol-1904/paper23
|storemode=property
|title=Forecasting models generation of the electronic means quality
|pdfUrl=https://ceur-ws.org/Vol-1904/paper23.pdf
|volume=Vol-1904
|authors=Roman O. Mishanov,Sergey V. Tyulevin,Mikhail N. Piganov,E.katerina S. Erantseva
}}
==Forecasting models generation of the electronic means quality ==
https://ceur-ws.org/Vol-1904/paper23.pdf
Forecasting models generation of the electronic means quality
R.O. Mishanov1, S.V. Tyulevin2, M.N. Piganov1, E.S. Erantseva1
1
Samara National Research University, 34 Moskovskoe Shosse, 443086, Samara, Russia
2
JSC SRC Progress, 18 Zemetsa street, 443009, Samara, Russia
Abstract
The article describes the results of forecasting models generation of quality and reliability indicators of the electronic means. In the learning
process variants of normalizing and centering of controlled parameters are described. Much attention is given to the methods of the Theory
of Pattern Recognition and extrapolation methods. This paper gives information about the advanced technique of the models generation and
individual forecasting of electronic means for the space equipment. The verification of derived models is investigated in detail. Special
emphasis is paid to the analysis of the models efficiency.
Keywords: forecasting model; electronic means; verification; learning; informative parameters; analysis
1. Introduction
A realization of increasing requirements to the quality and reliability of the radio-electronic means and electronic
components (EC) is ensured by the improvement of their design, manufacturing technology, controlling methods and testing. In
addition, some hidden defects are not detected by the existing system of technological control and testing methods. The
decisive influence on the reliability of hidden defects determines the development of works on the investigation of mechanisms
and the causes of failures. However, a special interest is caused by using methods and means of flaw detection and
physicochemical analysis.
Despite the effectiveness of work in this direction, the complexity and high cost of their implementation caused the
necessity to search for and develop methods and means to identify hidden defects of the EC, which correspond to the pace of
modern batch production. In addition, about 30% of defects and failures of EC cannot be controlled by these methods and
means [1].
Thus, methods of testing and forecasting reliability and other quality indicators based on the informative parameters are
being developed [2-8], which are reposed on the assumption of the existence of a stochastic connection between reliability and
initial values of the informative parameters set of the product. The choice of the informative parameters set has a decisive
influence on the validity of testing and forecasting. Ensuring the presence of informative parameters in the initial set is assigned
to the researcher and in most cases is a very difficult task.
Ensuring the quality and reliability of space electronics requires a wide implementation of new methods of diagnostic
nondestructive testing (NDT) [9-15]. For their development, it is necessary to establish the dependencies of the main reliability
indicators on the physical properties and parameters of the devices, on the physicochemical processes occurring in them, and
on the physical nature of the failures mechanisms [16].
One of the promising directions in the development of effective and economically acceptable methods for assessing the
quality and reliability is to forecast their future state.
Forecasting failures of the devices can be carried out at various stages of their life cycle (control, testing, application,
operation). The individual forecasting (IF) provides the greatest accuracy. Its meaning is to estimate the potential reliability of
each instance using the forecasting model and information about the value of the informative parameter or results of monitoring
the instances [17]. A structural IF model is required to generate an operator (mathematical model), an algorithm, an individual
forecasting technique, and a hardware quality management. Such a model is generated in the form of an enlarged technological
scheme with a description of the functions performed by the component parts [18].
A new structural forecasting model was proposed to increase the accuracy of the IF. It includes the following interrelated
steps:
- analysis of the IF methods;
- physical and technical analysis of the failures;
- preliminary selection of the informative parameters and selection of the forecasting parameters;
- development of the investigation test technique;
- learning experiment;
- final selection of the informative parameters;
- selection of the IF method;
- algorithm development;
- program development;
- evaluation of the software product quality;
- development of the forecast model (the IF operator);
- evaluation of the IF operator models quality;
- development of working technique;
- verification of the model;
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- attestation of the technique;
- operational forecasting;
- optimization of the model;
- refinement of the IF model;
- clarifying learning experiment;
- development or selection of new informative parameters;
- definition of levels;
- development of the recommendations;
- technological process (TP);
- parameter checkout of the radio-electronic means;
- change of the design and technology option;
- refinement of the technique;
- verification of the updated technique;
- heuristic forecasting or a rejection.
2. Development of the IF operators based on the regression models
The IF task including the value estimation of the forecasting parameter with a large number of the informative parameters
was solved using the regression models. A problem statement was reduced to the determination of the operator Hx.
When the linear model of the connection between π¦Μ and π₯π is adopted the estimation of the forecasting parameter value of
the jth element is defined by [19]:
(π) (π) (π) (π) (π)
π¦ β(π) (π‘π ) = π»π₯ [{π₯π }] = π΅0 + π΅1 π₯1 + π΅2 π₯2 + β― + π΅π π₯π + β― + π΅π π₯π , (1)
(π)
where π₯π β the value of the ith attribute of the jth element; Bi β constant coefficients.
To find the coefficients Bi in a linear regression model, it is more convenient to turn the initial data to the centered and
normalized values π₯Μππ , which were determined by:
π₯Μπ β πβ [π₯Μπ ]
π₯Μππ = .
π· β1/2 [π₯Μπ ]
Π*[xi] and D*[xi] are the estimates of the expected value and standard deviation of the random variable π₯Μπ calculated from
the learning experiment data:
π
β [π₯ ]
1 (π)
π Μπ = β π₯π ;
π
π=1
π
1 (π)
π· β1/2 [π₯Μπ ] = β β(π₯π β πβ [π₯Μπ ])2 .
πβ1
π=1
The idea of representing the connection between the forecasting parameter and informative parameters in the form of a
regression model is as follows [20].
The coefficients bi always can be found for any centered and normalized values π¦Μππ and xΜiπ while the equation (2) has
meaning regardless of the distribution law of random variables.
π¦Μπ = π1 π₯Μ1π + π2 π₯Μ2π + β― + ππ π₯Μππ + βπ¦Μ, (2)
In this equation ππ are the constant coefficients of the regression model with centered and normalized values of the random
variables; βπ¦Μ β a forecasting error.
If the values of the coefficients ππ are found, the estimation of the forecasting parameter value can be determined from the
expression (2). The coefficients ππ must be such that the error variance π·[βπ¦Μ] is minimal, and the expected value of the error
π[βπ¦Μ] equals zero, i. e.
π·[βπ¦Μ] β πππ, π[βπ¦Μ] = 0.
If the error variance does not exceed the allowable value, the forecasting operator can be recommended to estimate the
value of the forecasting parameter of new instances. In this case, having measured the values of its characteristics for the mth
instance and substituting them into expression (1), we obtain the estimate:
(π) (π) (π)
π¦ β(π) (π‘π ) = π΅0 + π΅1 π₯1 + π΅2 π₯2 + β― + π΅π π₯π .
The estimation of the forecasting error will be more accurate than the larger sample size is used in learning experiment. In
this case the estimates of the expectation value, the standard deviation and the correlation coefficient will be found more
accurately. For CMOS chips and stabilitrons the forecasting operators were obtained (Table 1).
Table 1. The forecasting models of study samples.
Number of sample Forecasting model (IF operator)
βπΌππ
Sample β44 = β29,53 + 29,11π‘π+ β 51,07ππ
πΌππ
Sample β45 βππ = β46,94 + 42,04πΎπ + 0,096π
π
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Mathematical Modeling / R.O. Mishanov, S.V. Tyulevin, M.N. Piganov, E.S. Erantseva
Fig. 1. The dependence of the probabilistic characteristics on the threshold P of the regression function of the CMOS chips.
βπΌππ /πΌππ β a leakage current drift, π‘π+ β a rise time of the signal, ππ β a supply voltage, βππ β a stabilized voltage drift, πΎπ β
a temperature coefficient of stabilization, π
π β a differential resistance.
Figure 1 shows the influence of the threshold P on the forecasting efficiency of the CMOS chips.
The analysis of this model have shown that the forecasting operator for the CMOS chips provides the optimal value of the
forecasting indicators at the threshold P = 35. In this case the risk of the incorrect decision Π inc.d equals 0,22; Consumerβs risk
(Ξ²-Risk) Π cons. equals 0,18; Producerβs risk (Ξ±-Risk) Π prod. equals 0,13. The minimum value of the Π cons. equals 0 when P =
0β¦16, Π inc.d = 0,6β¦0,42; Π prod. = 0,63β¦0,54. The minimum value of the Π prod. equals 0 when P = 80β¦90, Π inc.d = 0,3; Π cons. =
0,32β¦0,33.
Figure 2 shows the influence of the threshold P on the forecasting efficiency of the stabilitrons.
Fig. 2. The dependence of the probabilistic characteristics on the threshold P of the regression function of the stabilitrons.
The analysis of this model have shown that the forecasting operator for the stabilitrons provides the optimal value of the
forecasting indicators at the threshold P = 16. In this case the risk of the incorrect decision Π inc.d equals 0,15; Consumerβs risk
(Ξ²-Risk) Π cons. equals 0,14; Producerβs risk (Ξ±-Risk) Π prod. equals 0,14. The minimum value of the Π cons. equals 0 when P =
0β¦8, Π inc.d = 0,54β¦0,26; Π prod. = 0,55β¦0,37. The minimum value of the Π prod. equals 0 when P = 24β¦90, Π inc.d = 0,22β¦0,44;
Π cons. = 0,29β¦0,44.
3. The models verification
The method of discriminant functions was used for the models verification.
In general terms the problem formulation of such forecasting reduces to find the operator Hxcl. It is desirable to have the
simplest model, when the hyperplane is a surface that divides the space into two regions.
The equation of the (k-1)-dimensional hyperplane in the k-dimensional feature space has the form:
π(π₯1 , π₯2 , β¦ , π₯π ) = π΅1 π₯1 + π΅2 π₯2 + β― + π΅π π₯π = ππ ,
where π·π
, π©π , π©π , β¦ , π©π β constant coefficients that define the position of the hyperplane in the k-dimensional space.
Then the discriminant function takes the form:
π(π₯1 , π₯2 , β¦ , π₯π ) = π΅1 π₯Μ1 + π΅2 π₯Μ2 + β― + π΅π π₯Μπ .
In this function the dimension of the coefficients π΅π is inverse to the dimension of the corresponding characteristics π₯Μπ .
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It was required to find those values of the coefficients ππ and π΅π , which in the best way (in the sense of a misclassifications
minimum) would specify the position of this hyperplane in the feature space. Since the sample size is limited the estimates π½π
were determined.
The following approach was used to find the estimates of the coefficients π½π . According to the learning experiment, the
(π)
actual class is known, to which each of n copies belongs β πΎπ . It is possible to find the estimates of conditional expected
th
value and conditional variance of each i attribute π₯π :
π1
1 (π)
πβ [π₯Μπ /πΎ1 ] = β π₯π ,
π1
π=1
πβπΎ1
π1
1 (π)
π· β [π₯Μπ /πΎ1 ] = β {π₯π β π·[π₯Μπ /πΎ1 ]}2 ,
π1 β 1
π=1
πβπΎ1
π2
1 (π)
πβ [π₯Μπ /πΎ2 ] = β π₯π ,
π2
π=1
πβπΎ2
π2
1 (π)
π· β [π₯Μπ /πΎ2 ] = β {π₯π β π·[π₯Μπ /πΎ2 ]}2 .
π2 β 1
π=1
πβπΎ2
π1 and π2 β number of the instances, which belong to the class πΎ1 and πΎ2 , respectively, so that π1 + π2 = π.
Using theorems on the numerical characteristics of random variables, the estimates of the conditional expected values of
random variable were determined as:
πΊ = π(π₯Μ1 , π₯Μ2 , β¦ , π₯Μπ ).
If the instance belongs to the class πΎ1 :
πβ [πΊ/πΎ1 ] = βππ=1 π½π πβ [π₯Μπ /πΎ1 ] (3)
and to the class πΎ2 :
πβ [πΊ/πΎ2 ] = βππ=1 π½π πβ [π₯Μπ /πΎ2 ]. (4)
If the attributes are not correlated the corresponding estimates of conditional variances are equal:
π· β [πΊ/πΎ1 ] = βππ=1 π½π 2 π· β [π₯Μπ /πΎ1 ]; (5)
β [πΊ/πΎ ] π 2 β
π· 2 = βπ=1 π½π π· [π₯ Μπ /πΎ2 ]; (6)
β [πΊ/πΎ ] β [πΊ/πΎ ] β [πΊ/πΎ ] β [πΊ/πΎ ]
If the classes are well separated, then π 1 and π 2 will differ significantly, i.e. π· 1 and π· 2 are
small. Therefore, as an optimization criterion for finding estimates of the coefficients π½π , we used an expression of the form:
πβ [πΊ/πΎ1 ]βπβ [πΊ/πΎ2 ]
β extr. (7)
βπ· β [πΊ/πΎ1 ]+π· β [πΊ/πΎ2 ]
After substituting in the expression (7) the estimates of the conditional expected values and conditional variances of the
random variable G, determined by the expressions (3) - (6), we obtain the function:
βπ β π β
π=1 π½π π [π₯Μπ /πΎ1 ]ββπ=1 π½π π [π₯Μπ /πΎ2 ]
π(π½1 , β¦ , π½π ) = | |. (8)
2 β 2 β
ββπ π
π=1 π½π π· [π₯Μπ /πΎ1 ]ββπ=1 π½π π· [π₯Μπ /πΎ2 ]
Taking partial derivatives ππ/ππ½π and equating them to zero, we obtain a system of k algebraic equations with k unknown
coefficients π½1 , π½2 ,..., π½π for finding optimal estimates π½π πππ‘ . The obtained coefficients π½π πππ‘ will determine the best slope of
the hyperplane in the feature space.
Then we find the threshold value ππ for the discriminant function π(π₯1 , π₯2 , β¦ , π₯π ), which specifies the best position of the
separating hyperplane. Obviously, the following condition must be satisfied:
πβ [πΊ/πΎ1 ] > ππ > πβ [πΊ/πΎ2 ]
or
πβ [πΊ/πΎ1 ] < ππ < πβ [πΊ/πΎ2 ].
When the threshold is changed, the risk of the incorrect decisions will change. The value of the threshold was found by
several recalculations of the probability of incorrect decisions from the data of the learning experiment for various ππ and by
choosing one of them at which the risk of incorrect decisions turned out to be the least.
If the obtained risk does not exceed the permissible value, the previously found operator can be used forecast the class of
new instances (which not participating in the learning experiment). For this, the values of the attributes π₯π (π) of the new mth
instance are measured and the discriminant function has the form:
π
(π) (π) (π) (π)
πΊ = π(π₯1 , π₯2 , β¦ , π₯π ) = β π½π π₯π (π) .
π=1
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If πβ [πΊ/πΎ1 ] > πβ [πΊ/πΎ2 ] and πΊ (π) β₯ ππ , then a decision is to relegate the mth instance to the class πΎ1 , πΊ (π) < ππ , then a
decision is to relegate it to the class πΎ2 .
The method of discriminant functions made it possible to obtain the forecasting operators (Table 2):
Table 2. The forecasting models of study samples.
Number of sample Forecasting model (IF operator)
βπΌππ
Sample β44 ππ = + 0,76π‘π+ + 0,5ππ
πΌππ
Sample β45 ππ = βππ + 0,75πΎπ + 0,28π
π
Figure 3 and 4 show the dependencies of the probabilistic characteristics on the discriminant function threshold ππ for the
CMOS chips and stabilitrons.
Fig. 3. The influence of the threshold ππ on the performance characteristics of the IF operator for the CMOS chips.
Fig. 4. The influence of the threshold ππ on the performance characteristics of the IF operator for the stabilitrons.
The optimal values of the forecasting indicators for the CMOS chips are at the threshold ππ = 44. In this case the risk of the
incorrect decision Π inc.d = 0,17; Consumerβs risk (Ξ²-Risk) Π cons. = 0,27; Producerβs risk (Ξ±-Risk) Π prod. = 0,13. The minimum
value of the Π cons. equals 0,27 when ππ = 44. The minimum value of the Π prod. equals 0 when ππ = 57; Π inc.d = 0,21; Π cons. =
0,37.
The optimal values of the forecasting indicators for the stabilitrons are at the threshold ππ = 16. In this case the risk of the
incorrect decision Π inc.d = 0,18; Consumerβs risk (Ξ²-Risk) Π cons. = 0,25; Producerβs risk (Ξ±-Risk) Π prod. = 0,13. The minimum
value of the Π cons. equals 0,25 when ππ = 16. The minimum value of the Π prod. equals 0 when ππ β₯ 36; Π inc.d = 0,52; Π cons. =
0,57.
4. Conclusion
The method of regression models was chosen for the forecasting models generation of the spacecraft electronic means. The
CMOS chips and the stabilitrons were used as the electronic means. The forecasting models allow to provide the IF with the
probability of correct decisions Pcor.d = 0,78 for the chips and Pcor.d = 0,85 for the stabilitrons. The method of discriminant
functions was used to verify obtained models. They gave close to the initial models probabilities of the incorrect decisions: for
the chips Π inc.d = 0,22 and 0,17; for the stabilitrons Π inc.d = 0,15 and 0,18.
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Consequently, these models can be used at the stage of operational forecasting.
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