Patent Application: US-201514795905-A

Abstract:
a process for quantifying , by means of soft modeling , the characteristics of an electroplating solution is provided . the process includes obtaining a sample set , wherein each sample comprises a plating solution of proper performance , obtaining an electrochemical response for each of the sample to produce a multi - way electrochemical response data set , obtaining a training set that comprises the sample set and corresponding the multi - way electrochemical response data set , analyzing the training set by soft modeling using multi - way decomposition method coupled with outlier - detection analysis methods to produce a outlier - detection parameters data set , and validating said training data set by soft modeling to produce the multi - way predictive data set for a predictive model .

Description:
for this application the rta , which is a universal analytical system , is applied as a second - order instrument which generates a matrix ( a second - order tensor ) for each data sample . each sample is characterized by a matrix of dependent variables ( measured potential ) recorded for a combination of two independent variables ( set current density and time ), x ( j × k ) , where j is the number of set current densities while k is the number of time points when the potential was sampled . a collection of 2 nd order data from each of many samples in a training set ( of the total of i ) creates a third order tensor ( three - way array ), x ( i × j × k ) that could be used to form 2 nd order analytical model estimated by 2 nd order decomposition method . such three - way array data sets are decomposed in factor analysis for the purpose of data compression and information extraction by multi - way chemometric techniques like generalized rank annihilation method ( gram ) [ 23 ], paralell factor analysis ( parafac ) [ 22 , 23 ], tri - linear decomposition ( tld ) [ 23 ], or tucker models [ 23 ]. out of these decomposition techniques the parafac is most commonly used , and therefore was selected as an example to explain the applied throughout this patent chemometric methodology . parafac is a trilinear decomposition technique . in the three - way analysis a decomposition of data is made into triads . instead of one score vector and one loading vector ( dyad ) as in bilinear principal component analysis ( pca ), each component consists of one score vector and two loadings vectors ( triad ). in contrast to pca , parafac does not require orthogonality to identify the model . the parafac model of a three - way array is given by a three loading matrices a , b , and c with typical elements a if , b jf , and c kf : where { circumflex over ( x )} ( i × jk ) is the three - way model unfolded to an i × jk matrix . unfolding is a way of rearranging a multi - way array to a matrix by concatenating matrices for different levels . the symbol { circle around (×)} denotes a kronecker tensor product , which for matrices x and y where x is of size i × j is defined as : the parafac model can be formulated in terms of the unfolded array as x ( i × jk ) = a ( c |{ circle around ( x )}| b ) t + e ( i × jk ) ( 4 ) where the operator |{ circle around ( 0 )}| denotes a khatri - rao product , which for the matrices x and y having the same number of columns , f , is defined as : x |{ circle around (×)}| y =[ x 1 { circle around (×)} y 1 x 2 { circle around (×)} y 2 . . . x f { circle around (×)} y f ] ( 5 ) the parafac model is an approximate solution of the loss function : where x ( i × jk ) is the three - way array x ( i × j × k ) unfolded to an i × jk matrix . the operator |{ circle around (×)}| denotes a khatri - rao product . usually when using an existing model on new data , one is interested in estimating the loading of the first mode of new , unknown samples . assuming that the first mode refers to samples the estimation of the scores a u ( i × f ) of the unknown sample x u ( j × k ) ( becoming after the unfolding a row vector x u ( i × jk ) is calculated via following expression : a u = x u [( c |{ circle around (×)} b ) + ] t ( 7 ) the vector of residuals e u ( i × jk ) of the unknown sample can be obtained employing following equation : e u = x u { 1 −[( c |{ circle around (×)} b )( c |{ circle around (×)} b ) + ] t } ( 8 ) application of the proper preprocessing is an important aspect of the multiway analysis . before implementing the parafac decomposition the three - way array of electrochemical data was single - centered across the first mode and subsequently scaled within the second mode . by implementing multi - way techniques , the presented method provides not only all parameters that could be obtained by landau [ 8 ], but also it provides directly an overall score characterizing the plating bath performance . the obtained scores can subsequently be subjected to various chemometric outlier detection methods ( for instance those based on versions of mahalanobis distance [ 24 ] or hotelling &# 39 ; s t 2 ) or pattern recognition by means of classification techniques ( e . g . soft independent modelling of class analogy [ simca ]). as an example of the outlier detection technique the mahalanobis distance was presented in greater details . the mahalanobis distance ( d ) [ 24 ] is a statistical measure of sample distance from the training set mean . the squared d coupled with parafac is defined by the following equation : the modification of mahalanobis distance by combining it with q - residuals leads in many cases to significant increase of outlier detection capability . application of residuals significantly improves the sensitivity of the determinant analysis as compared to analysis based purely on the parafac loadings . the electrochemical residual can be obtained by subtracting electrochemical signal reconstructed from parafac loadings from the original electrochemical signal . parafac reconstructs the unknown electrochemical signals using the loadings b and c obtained for the predominant variances within training set . such reconstruction is not efficient in case there are other sources of variance in unknown electrochemical signals absent in the training set . this in turn results in substantial residuals by calculating sum of squares of the electrochemical signal residuals across all the selected j points of k electrochemical signals ( known as q - residuals ) an additional representative value can be generated for each i - th sample of the training set : the i values of q - residuals constitute a column vector q ( 1 × 1 ) which is then i meancentered by subtracting q defined as the column vector of meancentered q residuals { tilde over ( q )} ( 1 × 1 ) is appended as a f + 1 column to the matrix of loadings of the first mode of the training set a ( i × f ) to form a residual augmented matrix t ( i ×( f + 1 ) . the matrix t is used to calculate — the mahalanobis matrix with q residuals m q defined as : by analogy to equation 9 for d , the mahalanobis distance with q residuals ( d q ) for the unknown sample is calculated via following expression : d qu 2 = t u m q − 1 t u t ( 15 ) where t u ( i ×( f + 1 )) is the unknown sample vector of loadings of the first mode a u appended by q residual centered with the parameters of the training set , { tilde over ( q )} u : analogously to equation 11 for the training set , the elements of the vector of residuals for the unknown sample e u ( 1 &# 39 ; jk ) ( obtained via equation 8 ) are employed to calculate q residuals corresponding to that unknown sample : this treatment can also be conducted for constant potential experiment results and / or for combination of constant - current and constant - potential . this way , a comprehensive quantitative characterization of plating solution can be delivered . this example presents the organization of several physicochemical parameters characterizing bath performance including : current efficiency and slope and intercept of polarization curve into a multivariate training set subjected subsequently to factor analysis and outlier detection techniques for defining a cluster in eigenvector space of properly performing baths . a bath solution having the following concentration of technic copper of constituents is used : training set data was collected for eight solutions composed according to two - level , 5 - component fractional factorial [ 25 ], with levels determined by calibration ranges . additionally , the data recorded for a ninth solution of a target composition composed of : was augmented to the training set . each solution was analyzed 10 times , therefore the training set consisted of i = 9 × 10 = 90 samples . current efficiency was measured by dividing the integrated - over - time current of anodic voltammetric peak by the total cathodic charge value of the immediately preceding coulometric measurement . the coulometric measurements were conducted for six different current densities , each producing a j - th variable for the training set data . additionally , the j variables of the training set are augmented by slope and intercept data of polarization curves drawn for different times . for brevity of presentation a limited number of variables ( j = 17 ) was chosen , therefore it is justified to use for data compression and information extraction a two - way decomposition technique principal component analysis ( pca ) [ 25 , 26 ] rather than multi - way decomposition techniques . however , the presented method is general can also chemometrically process multi - way data arrays . the autoscaled training set matrix x ( i × j ) is decomposed by pca into a matrix of scores a ( i × f ) and loads bp × f ) for the number of factors f = 3 . the scores of the training set form the cluster presented in fig3 . in order to distinguish the subset of the training set corresponding to the data of target solution , the symbol of square was used while the remaining data corresponding to eight solutions was presented symbolized by pentagrams . the scores corresponding to the target solution &# 39 ; s data are located centrally within the cluster . the entire training set cluster is surrounded by an ellipsoid consisting of equidistant points in the sense of mahalanobis distance [ 24 ] ( equation 9 ) corresponding to the maximum value obtained by take - one - out cross - validation within the training set [ 28 ] of 2 . 9 . all solutions , whose corresponding scores are located within the training set ellipsoid , are considered proper in terms of their plating performance ( like all the data of the training set ). the solutions , whose corresponding scores are positioned beyond the training set ellipsoid , are considered outliers , and characteristics of their plating performance can differ from that of the training set . the scores for unknown samples are obtained by projecting their data of the eigenvector space of the training set . all patents , publications and other references cited herein are hereby incorporated by reference . although the invention has been particularly described with reference to certain embodiments , skilled artisans appreciate that changes in form and detail may be made without departing from the scope of the appended claims . k . j . wikiel , a . jaworski , and h . wikiel , method and apparatus for real time monitoring of electroplating bath performance and early fault detection , u . s . pat . no . 7 , 124 , 120 ( 2006 ). r . o . hull , apparatus and process for the study of plating solutions , u . s . pat . no . 2 , 149 , 344 ( 1939 ). r . o . hull and j . b . winters , apparatus for the determination of plating characteristics of plating baths , u . s . pat . no . 2 , 801 , 963 ( 1957 ). r . o . hull jr . and j . a . zelmder , analytical electroplating apparatus , u . s . pat . no . 3 , 121 , 053 ( 1961 ). j . a . abys , and i . v . kadija , hydrodynamically modulated hull cell , u . s . pat . no . 5 , 288 , 976a ( 1993 ). 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