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; 3rd International conference β€œInformation Technology and Nanotechnology 2017” 124 Mathematical Modeling / R.O. Mishanov, S.V. Tyulevin, M.N. Piganov, E.S. Erantseva - 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 𝑦̃ and π‘₯𝑖 is adopted the estimation of the forecasting parameter value of the jth element is defined by [19]: (𝑗) (𝑗) (𝑗) (𝑗) (𝑗) 𝑦 βˆ—(𝑗) (𝑑𝑓 ) = 𝐻π‘₯ [{π‘₯𝑖 }] = 𝐡0 + 𝐡1 π‘₯1 + 𝐡2 π‘₯2 + β‹― + 𝐡𝑖 π‘₯𝑖 + β‹― + π΅π‘˜ π‘₯π‘˜ , (1) (𝑗) where π‘₯𝑖 – the value of the ith attribute of the jth element; Bi – 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‘₯̃𝑖𝑐 , which were determined by: π‘₯̃𝑖 βˆ’ π‘€βˆ— [π‘₯̃𝑖 ] π‘₯̃𝑖𝑐 = . 𝐷 βˆ—1/2 [π‘₯̃𝑖 ] М*[xi] and D*[xi] are the estimates of the expected value and standard deviation of the random variable π‘₯̃𝑖 calculated from the learning experiment data: 𝑛 βˆ— [π‘₯ ] 1 (𝑗) 𝑀 ̃𝑖 = βˆ‘ π‘₯𝑖 ; 𝑛 𝑗=1 𝑛 1 (𝑗) 𝐷 βˆ—1/2 [π‘₯̃𝑖 ] = √ βˆ‘(π‘₯𝑖 βˆ’ π‘€βˆ— [π‘₯̃𝑖 ])2 . π‘›βˆ’1 𝑗=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 𝑦̃𝑖𝑐 and xΜƒi𝑐 while the equation (2) has meaning regardless of the distribution law of random variables. 𝑦̃𝑐 = 𝑏1 π‘₯Μƒ1𝑐 + 𝑏2 π‘₯Μƒ2𝑐 + β‹― + π‘π‘˜ π‘₯Μƒπ‘˜π‘ + βˆ†π‘¦Μƒ, (2) In this equation 𝑏𝑖 are the constant coefficients of the regression model with centered and normalized values of the random variables; βˆ†π‘¦Μƒ – a forecasting error. If the values of the coefficients 𝑏𝑖 are found, the estimation of the forecasting parameter value can be determined from the expression (2). The coefficients 𝑏𝑖 must be such that the error variance 𝐷[βˆ†π‘¦Μƒ] is minimal, and the expected value of the error 𝑀[βˆ†π‘¦Μƒ] equals zero, i. e. 𝐷[βˆ†π‘¦Μƒ] β†’ π‘šπ‘–π‘›, 𝑀[βˆ†π‘¦Μƒ] = 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‘š) (π‘š) (π‘š) 𝑦 βˆ—(π‘š) (𝑑𝑓 ) = 𝐡0 + 𝐡1 π‘₯1 + 𝐡2 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) βˆ†πΌπ‘™π‘ Sample β„–44 = βˆ’29,53 + 29,11𝑑𝑝+ βˆ’ 51,07π‘ˆπ‘  𝐼𝑙𝑐 Sample β„–45 βˆ†π‘ˆπ‘  = βˆ’46,94 + 42,04𝐾𝑇 + 0,096𝑅𝑑 3rd International conference β€œInformation Technology and Nanotechnology 2017” 125 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. βˆ†πΌπ‘™π‘ /𝐼𝑙𝑐 – a leakage current drift, 𝑑𝑝+ – a rise time of the signal, π‘ˆπ‘  – a supply voltage, βˆ†π‘ˆπ‘  – a stabilized voltage drift, 𝐾𝑇 – a temperature coefficient of stabilization, 𝑅𝑑 – 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; Consumer’s risk (Ξ²-Risk) Π cons. equals 0,18; Producer’s risk (Ξ±-Risk) Π prod. equals 0,13. The minimum value of the Π cons. equals 0 when P = 0…16, Π inc.d = 0,6…0,42; Π prod. = 0,63…0,54. The minimum value of the Π prod. equals 0 when P = 80…90, Π inc.d = 0,3; Π cons. = 0,32…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; Consumer’s risk (Ξ²-Risk) Π cons. equals 0,14; Producer’s risk (Ξ±-Risk) Π prod. equals 0,14. The minimum value of the Π cons. equals 0 when P = 0…8, Π inc.d = 0,54…0,26; Π prod. = 0,55…0,37. The minimum value of the Π prod. equals 0 when P = 24…90, Π inc.d = 0,22…0,44; Π cons. = 0,29…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: 𝑔(π‘₯1 , π‘₯2 , … , π‘₯π‘˜ ) = 𝐡1 π‘₯1 + 𝐡2 π‘₯2 + β‹― + π΅π‘˜ π‘₯π‘˜ = 𝑃𝑑 , where 𝑷𝒅 , π‘©πŸ , π‘©πŸ , … , π‘©π’Œ – constant coefficients that define the position of the hyperplane in the k-dimensional space. Then the discriminant function takes the form: 𝑔(π‘₯1 , π‘₯2 , … , π‘₯π‘˜ ) = 𝐡1 π‘₯Μƒ1 + 𝐡2 π‘₯Μƒ2 + β‹― + π΅π‘˜ π‘₯Μƒπ‘˜ . In this function the dimension of the coefficients 𝐡𝑖 is inverse to the dimension of the corresponding characteristics π‘₯̃𝑖 . 3rd International conference β€œInformation Technology and Nanotechnology 2017” 126 Mathematical Modeling / R.O. Mishanov, S.V. Tyulevin, M.N. Piganov, E.S. Erantseva It was required to find those values of the coefficients 𝑃𝑑 and 𝐡𝑖 , 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 𝛽𝑖 were determined. The following approach was used to find the estimates of the coefficients 𝛽𝑖 . According to the learning experiment, the (𝑗) actual class is known, to which each of n copies belongs – 𝐾𝑠 . It is possible to find the estimates of conditional expected th value and conditional variance of each i attribute π‘₯𝑖 : 𝑛1 1 (𝑗) π‘€βˆ— [π‘₯̃𝑖 /𝐾1 ] = βˆ‘ π‘₯𝑖 , 𝑛1 𝑗=1 π‘—βˆˆπΎ1 𝑛1 1 (𝑗) 𝐷 βˆ— [π‘₯̃𝑖 /𝐾1 ] = βˆ‘ {π‘₯𝑖 βˆ’ 𝐷[π‘₯̃𝑖 /𝐾1 ]}2 , 𝑛1 βˆ’ 1 𝑗=1 π‘—βˆˆπΎ1 𝑛2 1 (𝑗) π‘€βˆ— [π‘₯̃𝑖 /𝐾2 ] = βˆ‘ π‘₯𝑖 , 𝑛2 𝑗=1 π‘—βˆˆπΎ2 𝑛2 1 (𝑗) 𝐷 βˆ— [π‘₯̃𝑖 /𝐾2 ] = βˆ‘ {π‘₯𝑖 βˆ’ 𝐷[π‘₯̃𝑖 /𝐾2 ]}2 . 𝑛2 βˆ’ 1 𝑗=1 π‘—βˆˆπΎ2 𝑛1 and 𝑛2 – number of the instances, which belong to the class 𝐾1 and 𝐾2 , respectively, so that 𝑛1 + 𝑛2 = 𝑛. Using theorems on the numerical characteristics of random variables, the estimates of the conditional expected values of random variable were determined as: 𝐺 = 𝑔(π‘₯Μƒ1 , π‘₯Μƒ2 , … , π‘₯Μƒπ‘˜ ). If the instance belongs to the class 𝐾1 : π‘€βˆ— [𝐺/𝐾1 ] = βˆ‘π‘˜π‘–=1 𝛽𝑖 π‘€βˆ— [π‘₯̃𝑖 /𝐾1 ] (3) and to the class 𝐾2 : π‘€βˆ— [𝐺/𝐾2 ] = βˆ‘π‘˜π‘–=1 𝛽𝑖 π‘€βˆ— [π‘₯̃𝑖 /𝐾2 ]. (4) If the attributes are not correlated the corresponding estimates of conditional variances are equal: 𝐷 βˆ— [𝐺/𝐾1 ] = βˆ‘π‘˜π‘–=1 𝛽𝑖 2 𝐷 βˆ— [π‘₯̃𝑖 /𝐾1 ]; (5) βˆ— [𝐺/𝐾 ] π‘˜ 2 βˆ— 𝐷 2 = βˆ‘π‘–=1 𝛽𝑖 𝐷 [π‘₯ ̃𝑖 /𝐾2 ]; (6) βˆ— [𝐺/𝐾 ] βˆ— [𝐺/𝐾 ] βˆ— [𝐺/𝐾 ] βˆ— [𝐺/𝐾 ] If the classes are well separated, then 𝑀 1 and 𝑀 2 will differ significantly, i.e. 𝐷 1 and 𝐷 2 are small. Therefore, as an optimization criterion for finding estimates of the coefficients 𝛽𝑖 , we used an expression of the form: π‘€βˆ— [𝐺/𝐾1 ]βˆ’π‘€βˆ— [𝐺/𝐾2 ] β†’ extr. (7) √𝐷 βˆ— [𝐺/𝐾1 ]+𝐷 βˆ— [𝐺/𝐾2 ] 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: βˆ‘π‘˜ βˆ— π‘˜ βˆ— 𝑖=1 𝛽𝑖 𝑀 [π‘₯̃𝑖 /𝐾1 ]βˆ’βˆ‘π‘–=1 𝛽𝑖 𝑀 [π‘₯̃𝑖 /𝐾2 ] 𝑉(𝛽1 , … , π›½π‘˜ ) = | |. (8) 2 βˆ— 2 βˆ— βˆšβˆ‘π‘˜ π‘˜ 𝑖=1 𝛽𝑖 𝐷 [π‘₯̃𝑖 /𝐾1 ]βˆ’βˆ‘π‘–=1 𝛽𝑖 𝐷 [π‘₯̃𝑖 /𝐾2 ] Taking partial derivatives πœ•π‘‰/πœ•π›½π‘– and equating them to zero, we obtain a system of k algebraic equations with k unknown coefficients 𝛽1 , 𝛽2 ,..., π›½π‘˜ for finding optimal estimates 𝛽𝑖 π‘œπ‘π‘‘ . The obtained coefficients 𝛽𝑖 π‘œπ‘π‘‘ will determine the best slope of the hyperplane in the feature space. Then we find the threshold value 𝑃𝑑 for the discriminant function 𝑔(π‘₯1 , π‘₯2 , … , π‘₯π‘˜ ), which specifies the best position of the separating hyperplane. Obviously, the following condition must be satisfied: π‘€βˆ— [𝐺/𝐾1 ] > 𝑃𝑑 > π‘€βˆ— [𝐺/𝐾2 ] or π‘€βˆ— [𝐺/𝐾1 ] < 𝑃𝑑 < π‘€βˆ— [𝐺/𝐾2 ]. 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 𝑃𝑑 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‘₯𝑖 (π‘š) of the new mth instance are measured and the discriminant function has the form: π‘˜ (π‘š) (π‘š) (π‘š) (π‘š) 𝐺 = 𝑔(π‘₯1 , π‘₯2 , … , π‘₯π‘˜ ) = βˆ‘ 𝛽𝑖 π‘₯𝑖 (π‘š) . 𝑖=1 3rd International conference β€œInformation Technology and Nanotechnology 2017” 127 Mathematical Modeling / R.O. Mishanov, S.V. Tyulevin, M.N. Piganov, E.S. Erantseva If π‘€βˆ— [𝐺/𝐾1 ] > π‘€βˆ— [𝐺/𝐾2 ] and 𝐺 (π‘š) β‰₯ 𝑃𝑑 , then a decision is to relegate the mth instance to the class 𝐾1 , 𝐺 (π‘š) < 𝑃𝑑 , then a decision is to relegate it to the class 𝐾2 . 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) βˆ†πΌπ‘™π‘ Sample β„–44 𝑃𝑑 = + 0,76𝑑𝑝+ + 0,5π‘ˆπ‘  𝐼𝑙𝑐 Sample β„–45 𝑃𝑑 = βˆ†π‘ˆπ‘  + 0,75𝐾𝑇 + 0,28𝑅𝑑 Figure 3 and 4 show the dependencies of the probabilistic characteristics on the discriminant function threshold 𝑃𝑑 for the CMOS chips and stabilitrons. Fig. 3. The influence of the threshold 𝑃𝑑 on the performance characteristics of the IF operator for the CMOS chips. Fig. 4. The influence of the threshold 𝑃𝑑 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 𝑃𝑑 = 44. In this case the risk of the incorrect decision Π inc.d = 0,17; Consumer’s risk (Ξ²-Risk) Π cons. = 0,27; Producer’s risk (Ξ±-Risk) Π prod. = 0,13. The minimum value of the Π cons. equals 0,27 when 𝑃𝑑 = 44. The minimum value of the Π prod. equals 0 when 𝑃𝑑 = 57; Π inc.d = 0,21; Π cons. = 0,37. The optimal values of the forecasting indicators for the stabilitrons are at the threshold 𝑃𝑑 = 16. In this case the risk of the incorrect decision Π inc.d = 0,18; Consumer’s risk (Ξ²-Risk) Π cons. = 0,25; Producer’s risk (Ξ±-Risk) Π prod. = 0,13. The minimum value of the Π cons. equals 0,25 when 𝑃𝑑 = 16. The minimum value of the Π prod. equals 0 when 𝑃𝑑 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. 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