Functional principal component analysis

Di CZ, Crainiceanu CM, Caffo BS, Punjabi NM, 2009. Multilevel functional principal component analysis. Annals of Applied Statistics 3, 458-88. 

Green s Function/Principal Component Analysis and Essential Dynamics... 

The idea of the Green s function/principal component analysis is closely related to the essential dynamics approach recently introduced into biomolecular simulations. Other similar works include those by Garcia, Ichiye and Karplus, Go and coworkers,and developers of the quasi-harmonic method. " The basic idea of the essential dynamics approach is to diagonalize a covariance matrix a whose elements are given by the formula... 

The method of principal component analysis of matrix S (PCAS) was discussed in Sect. 5.3. The PCAS method allows the identification of the most important parameters related to selected simulation results. Therefore, if the objective function includes the concentrations of the important and necessary species (see Sect. 7.2) and the investigated parameters are the rate coefficients (or A-factors) of the reaction steps (Vajda et al. 1985 Vajda and Tur yi 1986 Turanyi 1990b Xu et al. 1999 Liu et al. 2005), it is also applicable for the generation of a reduced mechanism containing less reaction steps. A further development of the PCAS method is functional principal component analysis (fPCA) (Gokulakrishnan et al. 2006). This method facilitates the investigation of temporal and spatial changes in the importance of reaction steps in reaction—diffusiOTi systems. 

RL Doty, R Smith, DA McKeown, J Raj. Tests of human olfactory function Principal components analysis suggests that most measure a common source of variance. Percect Psychophys 56 701-707, 1994. 

Grubmiiller described a method to induce conformational transitions in proteins and derived rate constants for these ([Grubmiiller 1994]). The method employs subsequent modifications of the original potential function based on a principal component analysis of a short MD simulation. It is discussed in more detail in the chapter of Eichinger et al. in this volume. 

The field points must then be fitted to predict the activity. There are generally far more field points than known compound activities to be fitted. The least-squares algorithms used in QSAR studies do not function for such an underdetermined system. A partial least squares (PLS) algorithm is used for this type of fitting. This method starts with matrices of field data and activity data. These matrices are then used to derive two new matrices containing a description of the system and the residual noise in the data. Earlier studies used a similar technique, called principal component analysis (PCA). PLS is generally considered to be superior. 

One of the main attractions of normal mode analysis is that the results are easily visualized. One can sort the modes in tenns of their contributions to the total MSF and concentrate on only those with the largest contributions. Each individual mode can be visualized as a collective motion that is certainly easier to interpret than the welter of information generated by a molecular dynamics trajectory. Figure 4 shows the first two normal modes of human lysozyme analyzed for their dynamic domains and hinge axes, showing how clean the results can sometimes be. However, recent analytical tools for molecular dynamics trajectories, such as the principal component analysis or essential dynamics method [25,62-64], promise also to provide equally clean, and perhaps more realistic, visualizations. That said, molecular dynamics is also limited in that many of the functional motions in biological molecules occur in time scales well beyond what is currently possible to simulate. 

Because protein ROA spectra contain bands characteristic of loops and turns in addition to bands characteristic of secondary structure, they should provide information on the overall three-dimensional solution structure. We are developing a pattern recognition program, based on principal component analysis (PCA), to identify protein folds from ROA spectral band patterns (Blanch etal., 2002b). The method is similar to one developed for the determination of the structure of proteins from VCD (Pancoska etal., 1991) and UVCD (Venyaminov and Yang, 1996) spectra, but is expected to provide enhanced discrimination between different structural types since protein ROA spectra contain many more structure-sensitive bands than do either VCD or UVCD. From the ROA spectral data, the PCA program calculates a set of subspectra that serve as basis functions, the algebraic combination of which with appropriate expansion coefficients can be used to reconstruct any member of the... 

In the previous examples and figures we indicated that functions for two independent variables can be selected. When three (or more) independent variables occur, advanced analysis tools, such as experimental design (see Section 2.4) or principal component analysis (Jackson, 1991), are required to determine the structure of the model. 

One of the Important functions of principal component analysis Is the reduction of dimensionality so that an overview or graphical... 

Some of the linear combinations will be well defined and others poorly defined. The latter may be eliminated in a filtering procedure, referred to in the literature under the names characteristic value filtering, eigenvalue filtering, and principal component analysis. If the parameter set is not homogeneous, but includes different types, relative scaling is important. Watkin (1994) recommends that the unit be scaled such that similar shifts in all parameters lead to similar changes in the error function S. 

Principal components analysis can be best understood using a simple m o-variable example. With only two variables it is possible to plot the row space without the need to reduce the number of variables. Although this docs not fully present the utilit> of PCA. it is a good demonstration of how it functions. A two-dimensional plot of the row space of an example data set is shown in Figure 4.23. The data matrix consists of two columns, representing the two measurements, and 40 rows, representing the samples. Each row of the matrix is represented as a point (O) on the graph. 

As seen from Fig. 5.3, the substrate concentration is most sensitive to the parameters around t = 7 hours. It is therefore advantageous to select more observation points in this region when designing identification experiments (see Section 3.10.2). The sensitivity functions, especially with respect to Ks and Kd, seem to be proportional to each other, and the near—linear dependence of the columns in the Jacobian matrix may lead to ill-conditioned parameter estimation problem. Principal component analysis of the matrix STS is a powerful help in uncovering such parameter dependences. The approach will be discussed in Section 5.8.1. 

Ochre is very common in the Terminal Archaic-Early Formative archaeological site of Jiskairumoko, (Rio Have, Lake Titicaca Basin, southern Peru). Within the site, ochre was found on tools, palettes, and in burials and soil deposits within structures in several contexts, suggesting both symbolic and functional uses of ochre. Variations in the color and contexts imply possibilities for different uses of ochre.. Instrumental neutron activation analysis was used to analyze the ochre samples found in Jiskairumoko. Multivariate analysis of the elemental data by principal components analysis suggests trends in the data related to the compositional variation of ochres on the site. Further analysis of the ochre will lead to conclusions about the variation in composition of the ochres from Jiskairumoko and possible archaeological conclusions about ancient technologies and uses of ochre on the site. 

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