Row-mean

In order to be consistent with the concept of the two dual data spaces, we must also define the vector of row-means... 

Similarly as in eq. (29.62), subtraction of the row-means from the elements in the corresponding rows of the matrix X results in the matrix of deviations from row-means or row-centered matrix Y ... 

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. 

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 this case we find that the pattern of column-profiles in is centred about the origin, as can be seen by computing the weighted row-means m ... 

Row Mean Scores Differ For an ordinal column variable, a significant p-value here indicates that the mean CMH score differs across columns for at least one stratum. P CMHRMS... 

Make sure your stratification variable appears first and your ordinal response variable appears last so that your stratification is properly applied and so that your row mean score test will be meaningful. 

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

Solvent number Row means Solvent number Row means... 

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. 

Within a row, means not sharing a common superscript are significantly different (P < 0.05) by Student s two-tailed unpaired t-test. 

Let Y he N X K dimensional data N < K) with mean-centered rows and columns as it is for Yr. Although the only requirement for PCA application is mean-centering of columns, having the rows mean-centered as well due to CD profile removal provides better scaling to remaining data. Define a covariance or scatter matrix Z = YY and let U = [uiU2...ujv] with Uj = be the orthonormal eigenvectors of Z such that... 

The blank entries represent situations that are either impossible (e.g., deleting an entry in one model and adding it in another) or where no change is required (e.g., if the element stays the same in both models). The entries that specify row mean that the version of the element from the model represented in the row should be taken, and similarly column indicates that the column s version should be taken. So, for example, if a sequence were unchanged in model A and deleted in model B, the action would be dictated by the entry at (no change, delete element), which, in this case is to delete the sequence since that is the action performed in model B, the one represented by the column. 

In the first step, the data have to be reviewed with respect to completeness. Missing data do not hinder mathematical analysis. Of course, missing data should not be replaced by zeros. Instead, the vacancies should be filled up either by the column/row mean or, in the worst case, by generating a random number in the range of the considered column/row. Features and/or objects can be removed from the data set if they are highly correlated with each other, or if they are redundant or constant. 

FIGURE 2). THE "EDGE" ROW MEANS THE ROW OF TEMPLATE HOLES NORMALLY OCCUPIED BY THE FIRST ROW OF FUEL THE MIDDLE ROW. LIKEWISE, REFERS TO THE ROW OF HOLES IN THE ARRAY CENTER. 

Fig. 26. Temporal temperature evolution at the TCEs for series 1 (left) and 4 (right). Upper row origin data Middle row mean square deviation Last row time-temperature dependence with error bars.

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