Scalar data , multivariate

Multivariate data are represented by one or several matrices. Variables (scalars, vectors, matrices) are written in italic characters scalars in lower or upper case (examples n, A), vectors in bold face lower case (example b). Vectors are always column vectors row vectors are written as transposed vectors (example bv). Matri ces are written in bold face upper case characters (example X). The first index of a matrix element denotes the row, the second the column. Examples x,- - or x(i, j) is an element of matrix X, located in row i and column / xj is the vector of row i xy is the vector of column j. 

Multiplication of two matrices is the most important operation in multivariate data analysis. It is not performed element-wise, but each element of the resulting matrix is a scalar product (see Section A.2.3). A matrix A and a matrix B can be multiplied by A B only if the number of columns in A is equal to the number of rows in B this... 

Analysis of full sheet data is useful for process performance evaluations and product value calculations. For feedback control or any other on-line application, it is necessary to continuously convert scanner data into a useful form. Consider the data vector Y ,k) for scan number k. It is separated into its MD and CD components as Y( , A ) = yM )( )+Yc )( , k) where Ymd ) s the mean of Y ,k) as a scalar and YcD -,k) is the instantaneous CD profile vector. MD and CD controllers correspondingly use these calculated measurements as feedback data for discrete time k. Univariate MD controllers are traditional in nature with only measurement delay as a potential design concern. On the other hand, CD controllers are multivariate in form and must address the challenges of controller design for large dimensional correlated systems. 

Vectors A series of scalars can be arranged in a column or in a row. Then, they are called a column or a row vector. If the elements of a column vector can be attributed to special characteristics, e.g., to compounds, then data analysis can be completed. The chemical structures of compounds can be characterized with different numbers called descriptors, variables, predictors, or factors. For example, toxicity data were measured for a series of aromatic phenols. Their toxicity can be arranged in a column arbitrarily Each row corresponds to a phenolic compound. A lot of descriptors can be calculated for each compound (e.g., molecular mass, van der Waals volume, polarity parameters, quantum chemical descriptors, etc.). After building a multivariate model (generally one variable cannot encode the toxicity properly) we will be able to predict toxicity values for phenolic compounds for which no toxicity has been measured yet. The above approach is generally called searching quantitative structure - activity relationships or simply QSAR approach. 

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