Latent value

If we only include the first r latent variables we have to redefine our relationships between data, latent vectors and latent values ... 

In the previous section we have developed principal components analysis (PCA) from the fundamental theorem of singular value decomposition (SVD). In particular we have shown by means of eq. (31.1) how an nxp rectangular data matrix X can be decomposed into an nxr orthonormal matrix of row-latent vectors U, a pxr orthonormal matrix of column-latent vectors V and an rxr diagonal matrix of latent values A. Now we focus on the geometrical interpretation of this algebraic decomposition. 

We consider the special biplot in which both rows and columns are represented in a single display of latent variables subjected to the constraint that a + P equals 1. As we have seen above, this constraint allows us to reconstruct the original data X from which the latent variables U, V and the latent values A have been computed (eq. (31.22)). 

The corresponding latent value X, is then defined by means of ... 

The power algorithm [21] is the simplest iterative method for the calculation of latent vectors and latent values from a square symmetric matrix. In contrast to NIPALS, which produces an orthogonal decomposition of a rectangular data table X, the power algorithm decomposes a square symmetric matrix of cross-products X which we denote by C. Note that Cp is called the column-variance-covariance matrix when the data in X are column-centered. 

Most manifest and latent values we share can be magnified by manufacturers advertising campaigns [7]. For example, if a homemaker perceives that s/he is taking better care of the children by having them use antimicrobial soaps (a manifest value), and if s/he feels more valuable, more loveable, and/or more needed by his/her family, etc. (latent values), s/he will be motivated to purchase the product. Finally, much of what consumers believe to be true is not grounded in objective reality [8]. Most of these beliefs are formed from their interpretation of mass media reports, opinions of experts and explanations of phenomena from various notorieties [6,9]. 

This category includes the personal interpretation of cultural and subcultural world views. Relative to antimicrobials, these include subjective likes and dislikes of characteristics such as the fragrance and feel of the product, the perceived quality of the product, and other aesthetic considerations [7]. As with cultural attributes, there are manifest and latent values in this category. Hence, if one likes the springtime fragrance of a consumer body wash product (manifest), the latent or deeper value may be that it makes one feel younger and, therefore, more physically attractive and desirable. 

The maximum number of latent variables is the smaller of the number of x values or the number of molecules. However, there is an optimum number of latent variables in the model beyond which the predictive ability of the model does not increase. A number of methods have been proposed to decide how many latent variables to use. One approach is to use a cross-validation method, which involves adding successive latent variables. Both leave-one-out and the group-based methods can be applied. As the number of latent variables increases, the cross-validated will first increase and then either reach a plateau or even decrease. Another parameter that can be used to choose the appropriate number of latent variables is the standard deviation of the error of the predictions, SpREss ... 

The industrial value of furfuryl alcohol is a consequence of its low viscosity, high reactivity, and the outstanding chemical, mechanical, and thermal properties of its polymers, corrosion resistance, nonburning, low smoke emission, and exceUent char formation. The reactivity profile of furfuryl alcohol and resins is such that final curing can take place at ambient temperature with strong acids or at elevated temperature with latent acids. Major markets for furfuryl alcohol resins include the production of cores and molds for casting metals, corrosion-resistant fiber-reinforced plastics (FRPs), binders for refractories and corrosion-resistant cements and mortars. 

Estimates of the latent heat of vaporization of a pure Hquid at any temperature from a known value at a single temperature are possible by several methods. The known value may be experimental, or it may be estimated by equation 16. Of the methods proposed, the Watson correlation (1), which is both simple and reHable, has found the greatest acceptance ... 

Values for many properties can be determined using reference substances, including density, surface tension, viscosity, partition coefficient, solubihty, diffusion coefficient, vapor pressure, latent heat, critical properties, entropies of vaporization, heats of solution, coUigative properties, and activity coefficients. Table 1 Hsts the equations needed for determining these properties. 

Calculation of Liquid-to-Gas Ratio The minimum possible liquid rate is readily calculated from the composition of the entering gas and the solubility of the solute in the exit liquor, saturation being assumed. It may be necessaiy to estimate the temperature of the exit liquid based on the heat of solution of the solute gas. Values of latent and specific heats and values of heats of solution (at infinite dilution) are given in Sec. 2. 

Lower Heating Value. The lower heating value of the gas is one in whieh the H2O in the produets has not eondensed. The lower heating value is equal to the higher heating value minus the latent heat of the eondensed water vapor. 

The latent heat of vaporization in Figure 28 is expressed in units of Btu per pound of water evaporated. The value is taken from the steam water vapor at various temperatures (i.e., from the steam tables). 

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