Column-mean

No entry in this column means data are from N spectra. 

In the DWASH column of the output, NO indicates that no downwash is included, HS means that Huber-Snyder downwash is included, SS means that Schulman-Scire downwash is included, and NA means that downwash is not applicable since the downwind distance is less than SL. A blank in the DWASH column means that no calculation was made for that distance because the concentration was so small. 

The vector of column-means m of an nxp matrix X is defined as follows ... 

Usually, the raw data in a matrix are preprocessed before being submitted to multivariate analysis. A common operation is reduction by the mean or centering. Centering is a standard transformation of the data which is applied in principal components analysis (Section 31.3). Subtraction of the column-means from the elements in the corresponding columns of an nxp matrix X produces the matrix of... 

Finally, we can center the matrix X simultaneously by rows and by columns, which yields the matrix of deviations from row- and column-means or the double-centered matrix Y ... 

A vector of column-standard deviations provides for each column of X a measure of the spread of the elements around the corresponding column-mean ... 

Preprocessing is the operation which precedes the extraction of latent vectors from the data. It is an operation which is carried out on all the elements of an original data table X and which produces a transformed data table Z. We will discuss six common methods of preprocessing, including the trivial case in which the original data are left unchanged. The effects of each of these six types of preprocessing will be illustrated numerically by means of the small 4x3 data table from the study of trace elements in atmospheric samples which has been used in previous sections (Table 31.1). The various effects of the transformations can be observed from the two summary statistics (mean and norm). These statistics include the vector of column-means m and the vector of column-norms of the transformed data table Z ... 

The vector of column-means nip defines the coordinates of the centroid (or center of mass) of the row-pattern P" that represents the rows in column-space Sf . Similarly, the vector of row-means m defines the coordinates of the center of mass of the column-pattern that represents the columns in row-space S". If the column-means are zero, then the centroid will coincide with the origin of SP and the data are said to be column-centered. If both row- and column-means are zero then the centroids are coincident with the origin of both 5" and S . In this case, the data are double-centered (i.e. centered with respect to both rows and columns). In this chapter we assume that all points possess unit mass (or weight), although one can extend the definitions to variable masses as is explained in Chapter 32. 

Column-centering is a customary form of preprocessing in principal components analysis (Section 17.6.1). It involves the subtraction of the corresponding column-means from each element of the table X ... 

After this transformation we find that the column-means nip are zero as shown in Table 31.4. 

In this case it is required that the original data in X are strictly positive. The effect of the transformation appears from Table 31.6. Column-means are zero, while column-standard deviations tend to be more homogeneous than in the case of simple column-centering in Table 31.4 as can be seen by inspecting the corresponding values for Na and Cl. 

It is assumed that the original data in X are strictly positive. As is evident from Table 31.7 both the row-means m and the column-means of the transformed table Z are equal to zero. 

In 5 the row-profiles are centred about the origin, as can be shown by working out the expression for the weighted column-means m ... 

The pattern of points produced by Z is centred in both dual spaces and S, since the weighted row- and column-means m and are zero ... 

The mean of each column, and the difference of each column mean from the grand mean (this estimates the influence of the values of the factor corresponding to the columns)... 

Table 10-5 Part B - RESIDUALS for ANOVA from Table 10-4 after correcting for row and column means...

The answer to this question is in the residuals. While the residuals might not seem to bear any relationship to either the original data or the errors (which in this case we know because we created them and they are listed above), in fact the residuals contain the variance present in the errors of the original data. However, the value of the error sum of squares is reduced from that of the original data, because of the subtraction of some fraction of the error variation from the total when the row and column means were subtracted from the data itself. This reduction in the sum of squares can be compensated for by making a corresponding compensation in the degrees of freedom used to calculate the mean square from the sum of squares. In this data the sum of squares of the residuals is 5.24 (check it out). 

For PCA, it is generally recommended to use mean-centered data. Note that there are different possibilities for mean-centering. One could subtract arithmetic column-means from each data column, but also more robust mean-centering methods can be applied (see Section 2.2.2). 

No entry in this column means data are from 15N spectra. d (85LA1732, 86JCS(PI)I249, 88JCS(PI)1509). 

In the least squares regression of y on a constant and X, in order to compute the regression coefficients on X, we can first transform y to deviations from the mean, y, and, likewise, transform each column of X to deviations from the respective column means second, regress the transformed y on the transformed X without a constant. Do we get the same result if we only transform y What if we only transform X ... 

The cij is the column contribution (the contribution that arises if the column population means are different so that each column mean would be different from the grand population). 

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. 

We note from Table 1.19 that the sums of squares between rows and between columns do not add up to the defined total sum of squares. The difference is called the sum of squares for error, since it arises from the experimental error present in each observation. Statistical theory shows that this error term is an unbiased estimate of the population variance, regardless of whether the hypotheses are true or not. Therefore, we construct an F-ratio using the between-rows mean square divided by the mean square for error. Similarly, to test the column effects, the F-ratio is the be-tween-columns mean square divided by the mean square for error. We will reject the hypothesis of no difference in means when these F-ratios become too much greater than 1. The ratios would be 1 if all the means were identical and the assumptions of normality and random sampling hold. Now let us try the following example that illustrates two-way analysis of variance. 

Derivatizations have been employed to enhance detection in CE separations. Derivatization can be achieved either by post-column or pre-column means. 

The literature on iron (II) compounds exhibiting spin crossover has been tabulated according to the keywords given below. R in a column means that the corresponding physical method has been applied at room temperature only. 

Table 11.1 gives the results from the application of PCA on column mean-centered data, on column autoscaled data, and on log-transformed column mean-centered data. Using five PCs, the amount of variance explained was 84.0, 46.1, and 69.6%, respectively. The results for the column mean-centered data were nearly identical to those obtained for the nonmean-centered raw data. The reason for this is that the means of... 

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