Percent variation described

In this example, the first principal component describes 98% of the variation. Successive PCs can be estimated that describe a portion of the remaining variation in this example, the second PC contains 2%. This illustnites another property of PCs, that is, successive PCs describe decreasing amounts of variation. Knowing the percent variation described is veiy important when interpreting the plots. For example, if close to 100% of the variation is described using two PCs, a two-dimensional plot can effectively be used to study the variation in the data set. However, a two-dimensional plot will not be adequate... 

In this example, the percent variation indicates that the data form a two-dimensional object (plane) in the measurement space. Figure 4.29a shows the residuals after removing the contribution from the first principal component. The magnitude and shape of the residual spectra are examined to determine if one PC is sufficient to describe the data. If the noise in the measurements is approximately 2 response units, one might conclude that one principal component is sufficient to describe the data. If the noise is known to be much smaller, more PCs should be used to describe the relevant variation. The structure in the residuals in Figure 4.29a is also an indication that additional PCs are required to describe the systematic variation in the data set. Figure 4.29 reveals considerably smaller residuals remaining after the contribution of the second PC is removed. 

When the average centred variable matrix (X — X) is used, the matrix (X - X) (X - X) contains the sums of squares and the crossproducts of the variables. Since the means of each variable have been subtracted, the elements in (X - X) (X - X) are related to the variances and the covariances of the variables over the set of N compounds. The total sum of squares is equal to the sum of the eigenvalues. The variation described by a component is proportional to the sum of squares associated with this component, and this sum of squares is equal to the corresponding eigenvalue. It is usually expressed as a percentage of the total sum of squares and is often called "explained variance", although this entity is not corrected for the number of degrees of freedom. Percent "explained variance" by component "j" is therefore obtained as follows ... 

Coefficient of Variation One of the problems confronting any user or designer of crystallization equipment is the expected particle-size distribution of the solids leaving the system and how this distribution may be adequately described. Most crystalline-product distributions plotted on arithmetic-probability paper will exhibit a straight line for a considerable portion of the plotted distribution. In this type of plot the particle diameter should be plotted as the ordinate and the cumulative percent on the log-probability scale as the abscissa. 

More than 93.5 percent of the variance is explained by the first two components, which tells us that two degrees of freedom describe most of the natural isotopic variation with the five chronometers. This observation has led to the concept of the Mantle Plane of Zindler et al. (1982), since a plane is defined by only two independent variables, and has been extensively discussed by Allegre et al. (1987). 

The first principal component explains the maximum amount of variation possible in the data set in one direction. Stated another way. it is the direction that describes the maximum spread of data points. Furthermore, the percent of the total variation in the data set described b) any principal component can be precisely calculated. 

PCA of Class B—Percent Variance Plot (Model Diagnostic) The first principal component describes 99.15% of the variance, the second describes 0.85%, and the third describes less than 0.01%. Assuming the noise in the data is measured to be greater than or equal to 0.01% of the variation, one would infer that these data lie on a two-dimensional plane. 

PCA of TEA—Percent Variance Plot (Model Diagnostic) For the TEA class the first through fourth PCs describe 97.3%. 2.2%. 0.2%, and 0.1% of the variation, respectively. This suggests that a rank of m o is appropriate, assuming the noise in the data is more than 0.2% of the variance. 

PCA of MEK—Percent Variance (Model Diagnostic) The first through fourth PCs of the MEK data describe 99.0%, 0.4%, 0.3%, and 0.1% of the variation, respectively. Assuming that the noise is greater than 0.4% of the variation, a one-component PCA model may be appropriate. 

In the previous section we have described the three types of phase behavior observed in the low-molecular-weight PMMA/PS system and reviewed the four types observed in the low-molecular-weight PS/PMMA system. These various phase relationships have been studied in terms of their dependence on the molecular weight (Mn) and weight percent (W) of the initial polymer present. Further, we have presented quantitative data concerning the sizes of the dispersed particles, again correlated to variations in Mn and W. In this section we will discuss the results in terms of the poly (methyl methacrylate )/polystyrene/styrene and poly-styrene/poly( methyl methacrylate)/methyl methacrylate ternary phase diagrams, whichever is appropriate. 

The relevance of Fig. 5.11 to the problem of setting control chart limits on means is that if one is furnished with a description of the typical pattern of variation in y, sensible expectations for variation in y follow from simple normal distribution calculations. So Shewhart reasoned that since about 99.7 percent (most) of a Gaussian distribution is within three standard deviations of the center of the distribution, means found to be farther than three theoretical standard deviations (of y) from the theoretical mean (of y) could be safely attributed to other than chance causes. Hence, furnished with standard values for /x and a (describing individual observations), sensible control limits for y become... 

The form of the dissolved sulfur has not been characterized properly yet. While stable at ambient temperatures, a substantial amount can be converted to crystalline sulfur at elevated temperatures or by solvent separation. This observation led to the development of a rapid liquid chromatography method to determine elemental sulfur in SA binders. The procedure which has been described previously by Cassidy (17) is based on gel permeation principle and uses a Styragel column and a uv detector. Results showed that 2-14% of the elemental sulfur added reacted chemically with the asphalt. Petrossi (18) and Lee (19), who determined free sulfur by extraction with sodium sulfite followed by titration with iodine, calculated a higher percent of bonded sulfur in sulfur-asphalt compositions. The observed differences are most likely caused by variations in the asphalt composition with regard to polar aromatics and naphthene components as well as by reaction temperature and contact time. 

Of these absorptions the latter two produce most of the emission intensity so that we are concerned mainly with the v = 32 excited vibrational level. Detailed studies of single vibrational-rotational states show only slow variation of the relaxation constants with upper state J. Thus in this experiment it will be assumed that a single decay constant is sufficient to describe the average relaxation. This assumption has been validated by using a Nd YAG laser that had a single-frequency output, which was tunable to any of the three transitions the decay times vary by less than 10 percent among these three upper states. ... 

Equation (12) describes the variation in and 6, as a function of parameters 6r and and a function of Be (percent localization of C). This relationship is reasonable in terms of the description of restricted-access delocalization given earlier. That is, a significant change in... 

Foraminifera from the sediments of the Cariaco Trench (Fig. 7.6) (Hughen et al, 2004). Since the sediments of this anoxic basin are varved, the age filter applied to most sediment cores by bioturbation is not an issue. Calendar ages of the varves in the sediments of this basin were determined by matching the percent reflectance (a measure of the color of the sediments) with 5 0 variations in the ice of a Greenland ice core (described later in Fig. 7.19). Since the latter record is precisely dated back to 40 000 years by actual counting of annual ice layers, and the two records are undeniably correlated, it was possible to determine an accurate calendar age for the Cariaco Trench sediment core by using variations in the percent reflectance record. The results in Fig. 7.6 indicate offsets of up to 5 ky between C age and calendar age at about 30 ky BP and an abrupt shift at 40 calendar kiloyears (cal. ky) BP in which 7000 C years elapsed in only 2000 y. The results have been explained as variations in the source function and the ventilation of the deep sea and are now used to correct C dates back to more than 40 cal. ky BP. 

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