Row-variable

Measurement tables are the raw data that result from measurements on a set of objects. For the sake of simplicity we restrict our arguments to measurements obtained by means of instraments on inert objects, although they equally apply to sensory observations and to living subjects. By convention, a measurement table is organized such that its rows correspond to objects (e.g. chemical substances) and that its columns refer to measurements (e.g. physicochemical parameters). Here we adopt the point of view that objects are described in the table by means of the measurements performed upon them. Objects and measurements will also be referred to in a more general sense as row-variables and column-variables. 

Historically, a distinction has been made between PCA of column-variables and that of row-variables. These are referred to as R-mode or Q-mode PCA, respectively. The modem approach is to consider both analyses as dual and to unify the two views (of rows and columns) into a single display, which is called biplot and which will be discussed in greater detail later on. 

This bipolar axis defines a contrast between the row-variables i and i. Note that a contrast involves the difference of two quantities. 

Any data matrix can be considered in two spaces the column or variable space (here, wavelength space) in which a row (here, spectrum) is a vector in the multidimensional space defined by the column variables (here, wavelengths), and the row space (here, retention time space) in which a column (here, chromatogram) is a vector in the multidimensional space defined by the row variables (here, elution times). This duality of the multivariate spaces has been discussed in more detail in Chapter 29. Depending on the chosen space, the PCs of the data matrix... 

B Loading matrix with m rows (variables, features) and l columns (compo-... 

The numerical entry represents the power to which the row variable, varied alone, must be raised to be proportional to the column variable to the first power. For example, the column variable SNR is proportional to the row variable Rs1/2 in the signal-limited case and to Rs1 in the background-limited case. 

If we desire to study the effects of two independent variables (factors) on one dependent factor, we will have to use a two-way analysis of variance. For this case the columns represent various values or levels of one independent factor and the rows represent levels or values of the other independent factor. Each entry in the matrix of data points then represents one of the possible combinations of the two independent factors and how it affects the dependent factor. Here, we will consider the case of only one observation per data point. We now have two hypotheses to test. First, we wish to determine whether variation in the column variable affects the column means. Secondly, we want to know whether variation in the row variable has an effect on the row means. To test the first hypothesis, we calculate a between columns sum of squares and to test the second hypothesis, we calculate a between rows sum of squares. The between-rows mean square is an estimate of the population variance, providing that the row means are equal. If they are not equal, then the expected value of the between-rows mean square is higher than the population variance. Therefore, if we compare the between-rows mean square with another unbiased estimate of the population variance, we can construct an F test to determine whether the row variable has an effect. Definitional and calculational formulas for these quantities are given in Table 1.19. 

Inspection of Table 1.20 shows that we reject the hypotheses of no effect of the column variable or the row variable. Both type of catalyst and temperature seem to have an effect. Of course, we have made only a preliminary survey. We would now take more data to determine which catalyst was best and to evaluate a quantitative relationship on the temperature effect. 

The parameter p is the contribution of the grand mean, a is the contribution of the i-th level of the row variable, Pj is the contribution of the j-th level of the column variable, and is the random experimental error. The model in Eq. (1.128) does not contain what is usually referred to as row-column interaction that is, the row and... 

The two main ways of data pre-processing are mean-centering and scaling. Mean-centering is a procedure by which one computes the means for each column (variable), and then subtracts them from each element of the column. One can do the same with the rows (i.e., for each object). ScaUng is a a slightly more sophisticated procedure. Let us consider unit-variance scaling. First we calculate the standard deviation of each column, and then we divide each element of the column by the deviation. 

For example, the objects may be chemical compounds. The individual components of a data vector are called features and may, for example, be molecular descriptors (see Chapter 8) specifying the chemical structure of an object. For statistical data analysis, these objects and features are represented by a matrix X which has a row for each object and a column for each feature. In addition, each object win have one or more properties that are to be investigated, e.g., a biological activity of the structure or a class membership. This property or properties are merged into a matrix Y Thus, the data matrix X contains the independent variables whereas the matrix Ycontains the dependent ones. Figure 9-3 shows a typical multivariate data matrix. 

A matrix can be defined as a two-dimensional arrangement of elements (numbers, variables, vectors, etc.) set up in rows and columns. The elements a are indexed as follows ... 

The ordered set of measurements made on each sample is called a data vector. The group of data vectors, identically ordered, for all of the samples is called the data matrix. If the data matrix is arranged such that successive rows of the matrix correspond to the different samples, then the columns correspond to the variables as in Figure 1. Each variable, or aspect of the sample that is measured, defines an axis in space the samples thus possess a data stmcture when plotted as points in that / -dimensional vector space, where n is the number of variables. 

X andjy are data matrices in row format, ie, the samples correspond to rows and the variables to columns. Some mathematical Hterature uses column vectors and matrices and thus would represent this equation as T = X. The purpose of rotation in general is to find an orientation of the points that results in enhanced understanding of the underlying chemical behavior of the system. 

Thus, the transpose of is a complete B-matiis. Since there are seven variables involved in the phenomenon and the tank of Dis 4, from Theorem 1 there are three dimensionless products in a complete set of B-numbers, each of which corresponds to a row of... 

Suppose that the problem is to find a B-matris of D such that the variables C, and E each occur in one and only one of the B-vectors. Since the submatris Af of Cconsisting of the first three rows corresponding to the variables C, and E is nonsingular, according to Theorem 6 there exists a B-matrix with the desired property. Let Af be the adjoint matrix of M. Then (eq. 52) ... 

There is one term for each Auj in the row vector which is in the curly braces (]. These terms are caUed constrained derivatives, which tehs how the object func tion changes when the independent variables Uj are changed while keeping the constraints satisfied (by varying the dependent variables Xi). 

The table of results is laid out in a column, and a second column is constructed in which in the hrst four rows the results would be added sequentially in pahs, e. g. Xi + X2, xj, + X4, x + jcg etc., and the lower four rows are calculated by subuacting the second value from dre preceding value thus, JC2 — JCi, JC4 — JC3 etc., a thh d column is prepared from these results by canying out the same sequence of operations. The process is continued until there are as many columns as the number of variables. Thus in the present tluee-variable, two level-study the process is repeated tluee times (Table 15.1), and in the general -variable, two-level case it is repeated n times. (The general description of uials of this kind where tlrere are n variables and two levels, is 2 factorial uials ). 

Each value in the hnal column of the table consuiicted above is now divided by 4, which is the number of additions or subuactions made in each column. The results of tlris division show the numerical effects of each variable and the interaction between variables. The value opposite tire second row shows the effect of the temperature, the third shows the effect of the slag phase composition, and, the hfth the effect of the metal composition. The interaction terms then follow the symbols of each row, dre fourth showing the effect of... 

Differential Pressure Variable Area Uagmeter Mass Row Vortex Shedding Turbine. 3.1.2 Level... 

In general, because the noise in the concentration data is independent from the spectral noise, each optimum factor, W, will lie at some angle to the plane that contains the spectral data. But we can find the projection of each W, onto the plane containing the spectral data. These projections are called the spectral factors, or spectral loadings. They are usually assigned to the variable named P. Each spectral factor P, is usually organized as a row vector. 

Because the preceding factor experiment suggests a, b, c, and abc as independent variables, cf. bottom row in Table 3.2, the data table would take on the form ... 

Variables Row found and Row control the flow of execution of the lookup procedure and calculations within this procedure /... 

Rowe RC, Roberts RJ. The effect of some formulation variables on crack propagation in pigmented tablet film coatings using computer simulation. Int J Pharm 1992 86 49-58. 

The second time variable, is the so called real-time variable, representing the time spent in data acquisition, while ti is the evolution interval between the two pulses during the experiment. The effect of can only be observed indirectly by noting its influence on It is therefore necessary to carry out F2 transformation first, in order to generate a series of spectra (rows of the matrix), which is then used as a pseudo-FID for F] transformation in the second step. 

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