PCA component

The selection of relevant effects for the MLR in PCR can be quite a complex task. A straightforward approach is to take those PCA scores which have a variance above a certain threshold. By varying the number of PCA components used, the... 

PCA components with small variances may only reflect noise in the data. Such a plot looks like the profile of a mountain after a steep slope a more flat region appears that is built by fallen, deposited stones (called scree). Therefore, this plot is often named scree plot so to say, it is investigated from the top until the debris is reached. However, the decrease of the variances has not always a clear cutoff, and selection of the optimum number of components may be somewhat subjective. Instead of variances, some authors plot the eigenvalues this comes from PCA calculations by computing the eigenvectors of the covariance matrix of X note, these eigenvalues are identical with the score variances. 

The cumulative variance, vCumul of the PCA scores shows how much of the total variance is preserved by a set of PCA components (Figure 3.5, right). As a rule of thumb, the number of considered PCA components should explain at least 80%, eventually 90% of the total variance. 

For autoscaled variables, each variable has a variance of 1, and the total variance is in, the number of variables. For such data, a mle of thumb uses only PCA components with a variance >1, the mean variance of the scores. The number of PCA components with variances larger than 0 is equal to the rank of the covariance matrix of the data. 

Cross validation and bootstrap techniques can be applied for a statistically based estimation of the optimum number of PCA components. The idea is to randomly split the data into training and test data. PCA is then applied to the training data and the observations from the test data are reconstmcted using 1 to m PCs. The prediction error to the real test data can be computed. Repeating this procedure many times indicates the distribution of the prediction errors when using 1 to m components, which then allows deciding on the optimal number of components. For more details see Section 3.7.1. 

The nonlinear iterative partial least-squares (NIPALS) algorithm, also called power method, has been popular especially in the early time of PCA applications in chemistry an extended version is used in PLS regression. The algorithm is efficient if only a few PCA components are required because the components are calculated step-by-step. 

After a PC has been calculated the information of this component is peeled off from the currently used X-matrix (Figure 3.12b). This process is called deflation, and is a projection of the object points on to a subspace which is orthogonal to p, the previously calculated loading vector. The obtained X-residual matrix Xres is then used as a new X-matrix for the calculation of the next PC. The process is stopped after the desired number of PCs is calculated or no further PCA components can be calculated because the elements in Xres are very small. 

Calculate the residual matrix of X. Stop if the elements of Xres are very small because no further PCA components are reasonable. 

The NIPALS algorithm is efficient if only a few PCA components are required. Because the deflation procedure increases the uncertainty of following components, the algorithm is not recommended for the computation of many components (Seasholtz et al. 1990). The algorithm fails if convergence is reached already after one cycle in this case another initial value of the score vector has to be tried (Miyashita et al. 1990). 

PCA transforms a data matrix X(n x m)—containing data for n objects with m variables—into a matrix of lower dimension T(n x a). In the matrix T each object is characterized by a relative small number, a, of PCA scores (PCs, latent variables). Score ti of the /th object xt is a linear combination of the vector components (variables) of vector x, and the vector components (loadings) of a PCA loading vector/ in other formulation the score is the result of a scalar product xj p. The score vector tk of PCA component k contains the scores for all n objects T is the score matrix for n objects and a components P is the corresponding loading matrix (see Figure 3.2). 

Determination of the optimum complexity of a model is an important but not always an easy task, because the minimum of measures for the prediction error for test sets is often not well marked. In chemometrics, the complexity is typically controlled by the number of PLS or PCA components, and the optimum complexity is estimated by CV (Section 4.2.5). Several strategies are applied to determine a reasonable optimum complexity from the prediction errors which may have been obtained by CV (Figure 4.4). CV or bootstrap allows an estimation of the prediction error for each object of the calibration set at each considered model complexity. 

A standard cut-off percentage error can be used, for example, 1 %. Once the error has reduced to this cut-off, ignore later PLS (or PCA) components. 

Note that unlike the autoprediction error, this term is always divided by 7 because each sample in the original dataset represents an additional degree of freedom, however many PCA components have been calculated and however the data have been preprocessed. Note that it is conventional to calculate this error on the V block of the data. 

These equations are the same for both PLS-DA and PC-LDA. The difference lies in the decomposition of X. In PC-LDA, T and P correspond to the scores and loadings, respectively, from PCA. That is, the class of the samples is completely ignored, and the only criterion is to capture as much variance as possible from X. In PLS-DA, on the other hand, the scores and loadings are taken from a PLS model and the decomposition of X does take into account class information the first PLS components by definition explain more, often much more, variance of y than the first PCA components. 

The present study was designed to assess the validity of the PCA method. The first approaeh to assessing the validity was to determine whether the autonomic space created by the PCA components was consistent with the autonomic space created by PEP and RSA. Because PEP and RSA are considered to be more direct sympathetic and parasympathetic measures, respectively (e.g., Cacioppo et al., 1994), similarity of the autonomic spaces would indicate that PCA is a valid method of obtaining cardiac autonomic information. The second approach was to examine correlations between the sympathetic component and PEP and between the parasympathetic component and RSA. If the PCA components are valid, then they should correlate significantly with the more direct measures of autonomic fimction. 

Differenee seores from baseline-to-task were computed for the PCA components, PEP, and RSA. A positive difference score means that a task elicited increased heart period (i.e., slower heart rate) that could be caused by sympathetic withdrawal and/or parasympathetic activation. A negative difference score means that a task elieited deereased heart period (i.e., faster heart... 

Autonomic modes of control are represented graphically in Figs. 7.1 through 7.6 by a vector in autonomic space from the origin (i.e., zero alter baseline correction) to a point determined by the sympathetic and parasympathetic change fiom baseline-to-task. PEP and RSA have been converted to z-scores in the figures to make the display more directly comparable with the PCA components. 

The autonomic space created by the PCA components was consistent with the autonomic space created by PEP and RSA for the exercise tasks (see Fig. 7.1). Both the PCA component (7 [2, 18] = 86.32 and 217.91, p <. 001) and the PEP and RSA P- [2, 18] = 82.82 and 185.74, p <. 001) vectors were significantly different from the origin for low and high intensity exercise. In addition, heart period, the sympathetic and parasympathetic components, PEP, and RSA were each significantly different from baseline for the two exercise tasks (Table 7.3). The vectors indicate that the exercise tasks elicited reciprocally coupled sympathetic activation and parasympathetic withdrawal. This consistency between vectors indicates that the PCA component scores are valid indicators of cardiac autonomic information for the exercise task. 

The autonomic space created by the PCA components was consistent with the autonomic space created by PEP and RSA in the illusion task and in the low memory load condition of the memory task (see Pig. 7.5). For these tasks, however, neither vector was significantly different from the origin, and heart period, the PCA components, PEP, and RSA were not significantly different from base-... 

PCA component scores are valid indicators of cardiac autonomic information for the illusion task and low memory load condition of the memory task. 

Finally, the only task for which the autonomic spaces created by the PCA components and PEP and RSA were not consistent was the cold pressor (see Fig. 7.6). Although heart period was significantly shorter than baseline (i.e., faster heart rate, see Table 7.3), only the PCA component vector was signifi-... 

The goal of the present study was to examine the validity of PCA of multiple psychophysiological measures computed tiom heart period for use in mental workload assessment. Validity of the PCA components was examined in two ways (a) by comparing the autonomic space created tiom the PCA component scores to that ereated from PEP and RSA and (b) by the correlations between the sympathetic component and PEP and between the parasympathetic component and RSA. Both approaches provided some support for PCA component validity. 

Comparison of the autonomic space derived from the sympathetic and parasympathetic components to that derived fiom PEP and RSA suggests that the PCA components were valid for all task conditions except the cold pressor (Table 7.4). The discrepancy for the cold pressor task illustrates the disadvantage of using residual heart period as the sympathetic marker variable. Residual heart period is the variance in heart period that is unaccounted for by RSA and is affected by all other fectors that determine heart period. Eor cognitive tasks like memory and tracking, where the primary determinants of heart period are probably neurogenic in origin, the PCA components led to the same conclusions as PEP and RSA. However, for tasks like the cold pressor, where heart period may be primarily determined by responses to vasoconstriction, the PCA components dissociate from PEP and RSA. 

Changes in autonomic space must be consistent with changes in heart period. Comparison of the autonomic space derived fiom the sympathetic and parasympathetic components to heart period change fiom resting baseline also suggests that the PCA components were valid (Table 7.4). All task conditions. 

Consistencies (+) and Inconsistencies (—) Between the PCA Components and PEP and RSA, Heart Period, and the Within-Task Correlations... 

In contrast, the correlation results provided mixed support for the validity of the PCA components. The within-task correlations indicated that only the parasympathetic component was valid (Table 7.4). However, the within-task correlations are not as important to the conclusions about the validity of the PCA components for the purpose of mental workload assessment as are the within-subject correlations. Because mental workload will typically be assessed within an individual (e.g., a pilot), the psychological-physiological relation need only hold within that individual. The correlation between the sympathetic component and PEP across subjects (within a task) is much less important than the correlation within subjects. Therefore, the correlation results support the validity of the PCA method when viewed fk>m the perspective of whether the PCA components would be valuable for mental workload assessment. 

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