Principal Matrices Research Articles

Number of Relevant Directions in Principal Component Analysis and Wishart Random Matrices

We compute analytically, for large N, the probability P(N+,N) that a N×N Wishart random matrix has N+ eigenvalues exceeding a threshold Nζ, including its large deviation tails. This probability plays a benchmark role when performing the principal component analysis of a large empirical data set. We find that P(N+,N)≈exp[-βN2ψζ(N+/N)], where β is the Dyson index of the ensemble and ψζ(κ) is a rate function that we compute explicitly in the full range 0≤κ≤1 and for any ζ. The rate function ψζ(κ) displays a quadratic behavior modulated by a logarithmic singularity close to its minimum κ⋆(ζ). This is shown to be a consequence of a phase transition in an associated Coulomb gas problem. The variance Δ(N) of the number of relevant components is also shown to grow universally (independent of ζ) as Δ(N)∼(βπ2)-1 lnN for large N.

On the algebraic relation between principal components corresponding to two different sets of weights

In principal component analysis, it is sometimes useful to give weights to observations. When two different sets of weights are attached to the observations of a data table, we derive the algebraic relation between the corresponding two principal component matrices.

Interface crack problems for metallic–piezoelectric composite structures

AbstractWe investigate three‐dimensional interface crack problems (ICP) for metallic–piezoelectric composite bodies. We give a mathematical formulation of the physical problem when the metallic and piezoelectric bodies are bonded along some proper part of their boundaries where an interface crack occurs. By potential methods the ICP is reduced to an equivalent strongly elliptic system of pseudodifferential equations on manifolds with boundary. We study the solvability of this system in different function spaces and prove uniqueness and existence theorems for the original ICP. We analyse the regularity properties of the corresponding electric and mechanical fields near the crack edges and near the curves where the boundary conditions change. In particular, we characterize the stress singularity exponents and show that they can be explicitly calculated with the help of the principal homogeneous symbol matrices of the corresponding pseudodifferential operators. We present some numerical calculations that demonstrate that the stress singularity exponents essentially depend on the material parameters. Copyright © 2009 John Wiley & Sons, Ltd.

Catechin and Catechin Fractions as Biochemical Markers to Study the Diversity of Indian Tea (Camellia sinensis (L.) O. Kuntze) Germplasm

The heterogeneous Indian tea germplasm includes 'China', 'Assam', 'Cambod', and their hybrids which were evaluated using biochemical markers viz., total catechin and their fractions, for varietal identification and characterization. Principal component analysis (PCA) of biochemical characters showed that the total catechin and trihydroxylated catechin has higher eigenvalues. The first two principal components (PCs) could differentiate more than 90% of the clones studied. This grouping based on first two principal component matrices differentiated 'China', and their hybrids with 'Assam' and 'Cambod' variety. Morphologically indistinct large-leaved 'Cambod' variety and 'Assam' varieties could not be differentiated using biochemical markers, since both varietal types taxonomically belong to a single species. Clones of 'China' type showed low total catechin content and catechin ratio which are distinctly grouped. The 'China-Assam' and 'China-Cambod' hybrids formed intermediate groups between 'China' PC group and 'Cambod'/'Assam' PC groups, providing evidence for genetic control of catechin ratio variation. Tea clones which are differentially positioned in the PC group could be explained based on the genetic contribution by other varietal type as parents. This biochemical characterization will be a useful tool in the development of quality-tea clones with different proportion of total catechin and their fractions.

Principal Components Analysis Preprocessing for Improved Classification Accuracies in Pattern-Recognition-Based Myoelectric Control

Information extracted from multiple channels of the surface myoelectric signal (MES) recording sites can be used as inputs to control systems for powered upper limb prostheses. For small, closely spaced muscles, such as the muscles in the forearm, the detected MES often contains contributions from more than one muscle, the contribution from each specific muscle being modified by the dispersive propagation through the volume conductor between the muscle and the detection points. In this paper, the measured raw MES signals are rotated by class-specific principal component matrices to spatially decorrelate the measured data prior to feature extraction. This "tunes" the data to allow a pattern recognition classifier to better discriminate the test motions. This processing technique was used to significantly ( p < 0.01) reduce pattern recognition classification error for both intact limbed and transradial amputee subjects.

Anisotropic Organised Eddy Simulation for the prediction of non-equilibrium turbulent flows around bodies

Abstract The unsteady turbulent flow around bodies at high Reynolds number is predicted by an anisotropic eddy-viscosity model in the context of the Organised Eddy Simulation (OES). A tensorial eddy-viscosity concept is developed to reinforce turbulent stress anisotropy, that is a crucial characteristic of non-equilibrium turbulence in the near-region. The theoretical aspects of the modelling are investigated by means of a phase-averaged PIV in the flow around a circular cylinder at Reynolds number 1.4 × 10 5 . A pronounced stress–strain misalignment is quantified in the near-wake region of the detached flow, that is well captured by a tensorial eddy-viscosity concept. This is achieved by modelling the turbulence stress anisotropy tensor by its projection onto the principal matrices of the strain-rate tensor. Additional transport equations for the projection coefficients are derived from a second-order moment closure scheme. The modification of the turbulence length scale yielded by OES is used in the Detached Eddy Simulation hybrid approach. The detached turbulent flows around a NACA0012 airfoil (2-D) and a circular cylinder (3-D) are studied at Reynolds numbers 10 5 and 1.4 × 10 5 , respectively. The results compared to experimental ones emphasise the predictive capabilities of the OES approach concerning the flow physics capture for turbulent unsteady flows around bodies at high Reynolds numbers.

Morphometrics ofManipora(Tabulata; Upper Ordovician; southern Manitoba, Canada)

Multivariate morphometric analysis was applied for differentiation of closely related species and evaluation of intra- and interspecific variation inManiporafrom the Selkirk Member, Red River Formation, in southern Manitoba. Seven morphological characters were quantified in transverse thin sections of 46 coralla and statistically tested for selecting effective characters in discriminating species. Cluster analysis was performed on a raw data matrix coordinated with 46 coralla by three selected characters. Two major clusters on the resulting dendrogram were regarded as morphospecies, following comparative examination of the coralla using serial sections. Cluster analyses were also conducted on principal component score matrices obtained from the raw data set coordinated with 46 coralla by all seven characters, and from an experimental data set including the 46 coralla plus two replicates of each and six of the characters. the results agree closely with the first cluster analysis, but discrimination of morphospecies was slightly degraded. the validity of two morphospecies recognized in the first cluster analysis was verified by discriminant analyses, descriptive statistics, and bivariate plots. the results show that tabularium area is the most meaningful character for distinguishing these morphospecies; ranges of variations of the other six characters overlap between morphospecies.Another cluster analysis like the first was performed, but with the addition of 11 type specimens and reference coralla ofManiporaspecies from the Upper Ordovician of southern and northern Manitoba and Texas. Based on this analysis, together with comparative examination of thin sections, the two morphospecies are identified as valid species:M. amicarumSinclair, 1955 andM. manitoba(Sokolov, 1955).Manipora magnaFlower, 1961 is considered to be a synonym ofM. amicarum, whileM. trapezoidalisFlower, 1961 andM. garsonensisCaramanica, 1992 are considered to be synonyms ofM. manitoba, and the hypotypes ofM. amicarumof Nelson (1963) are assigned toM. manitoba.

On nilpotent incline matrices

Inclines are additively idempotent semirings in which products are less than or equal to either factor. Thus they generalize Boolean algebra, fuzzy algebra and distributive lattice. This paper studies the nilpotent incline matrices in detail. It is proved that an incline matrix is nilpotent if and only if it has index and the zero vector is its unique standard eigenvector. The nilpotent matrices over an incline without nilpotent elements are characterized in terms of principal minors, main diagonals, nilpotent indices and adjoint matrices. Also some properties of the reduction of nilpotent matrices over an additively residuated incline without nilpotent elements are established. The results obtained here generalize the corresponding ones on fuzzy matrices and lattice matrices shown in the references.

Positivity of principal minors, sign symmetry and stability

The relation between positivity of principal minors, sign symmetry and stability of matrices is studied. It is shown that for sign symmetric matrices, having positive principal minors is equivalent to stability, to D-stability, and to having a positive scaling into a stable matrix. The relation between spectra of matrices some of whose powers have positive principal minors and matrices whose corresponding powers have positive sums of principal minors of each order is studied as well. It is shown that for matrices of order less than 4 these two classes share the same spectra. The relation of these classes and stability is studied, in particular for sign symmetric matrices and for anti-sign symmetric matrices.

Use of benthic macroinvertebrates to assess the biological status of Pampean streams in Argentina

Macro-invertebrate communities and environmental variables were assessed seasonally for two years in seven streams in North-Eastern of Buenos Aires province (Argentina) in order to analyse changes in their structure and composition in relation with the quality of the water. The study includes pristine streams and others affected by urban and industrial effluents with high conductivity, nitrates, nitrites, phosphates and low oxygen content. Organisms with well-known pollution tolerance were identified to assess biological water quality, using a new Biotic Index (IBPAMP: Biotic Index for PAMPean rivers and streams) in comparison with other existing biotic indices. The usefulness of principal component analysis (PCA) and correlation matrices were examined to evaluate the efficiency of the method to assess disturbances. In general IBPAMP did well correlate with several classical measures of biological water quality (taxon richness, diversity and several biotic indices). The El Gato stream was the most disturbed ecosystem among all studied sites. It was characterised by low dissolved oxygen levels, high turbidity in the middle course, high BOD5 (>30 mg l−1) and COD (>40 mg l−1) values. The Bunirigo stream has a bad quality in the industrial area, but varying according to the dry and wet periods. In general, in the mountainous areas the water quality of streams was good with the exception of the stations located downstream of cities like Ayacucho on the Tandileofu stream.

Mechanics and dynamics of B1 domain of protein G: role of packing and surface hydrophobic residues.

The structural organization of the B1 domain of streptococcal protein G (PGA) has been probed using molecular dynamics simulations, with a particular emphasis on the role of the solvent exposed Ile6 residue. In addition to the native protein (WT-PGA), three single-mutants (I6G-PGA, I6F-PGA, and I6T-PGA), one double-mutant (I6T,T53G-PGA), and three isolated peptide fragments (corresponding to the helix and the two beta-hairpins) were studied in the presence of explicit water molecules. Comparative analysis of the various systems showed that the level of perturbation was directly related to the hydrophobicity and the size of the side chain of residue 6, the internal rigidity of the proteins decreasing in the order I6T-PGA > I6G-PGA > WT-PGA > I6F-PGA. The results emphasized the importance of residue 6 in controlling both the integrity of the sheet's surface and the orientation of the helix in relation to the sheet by modulation of surface/core interactions. The effects of mutations were delocalized across the structure, and glycine residues, in particular, absorbed most of the introduced strain. A qualitative structural decomposition of the native fold into elementary building-blocks was achieved using principal component analysis and mechanical response matrices. Within this framework, internal motions of the protein were described as coordinated articulations of these structural units, mutations affecting mostly the amplitude of the motions rather than the structure/location of the building-blocks. Analysis of the isolated peptidic fragments suggested that packing did not play a determinant role in defining the elementary building-blocks, but that chain topology was mostly responsible.

Total energies of alkanes in terms of through-space and through-bond interactions. Analysis of the one-electron density matrix

The total energy of alkanes has been expressed in terms of two principal matrices G( 1) , and G( 2) introduced previously [V. Gineityte, J. Mol. Struct. (Theochem), 343 (1995)183] and describing the direct (through-space) and indirect (through-bond) interactions of bond orbitals (BOs). As a result, the stabilization energy of an alkane versus the respective set of isolated bonds has been related to sum of squares of the through-space interactions over the whole molecule. The common one-electron density matrix (DM) of alkanes containing the same matrices G( 1) and G( 2) has been transformed into the basis of sp 3-hybrid AOs of carbon atoms and ls HAOs of hydrogen atoms. Analysis of this new representation of the DM allowed the stabilization energy to be alternatively expressed as spur of the so-called rebonding matrix. This matrix is among the building blocks of the transformed DM and describes redistribution of bond orders when making up an alkane molecule. In this connection, stabilization of alkanes has been concluded to be due to the rebonding effect. The extent of additivity of the stabilization energy with respect to contributions of separate bonds also has been studied.

Speech recognition using dynamic features

A speech recognition technique utilizes a set of N different principal discriminant matrices. Each principal discriminant matrix is associated with a distinct class. The class is an indication of the proximity of a speech segment to neighboring phones. A technique for speech encoding includes arranging speech signal into a series of frames. A feature vector is derived which represents the speech signal for a speech segment or series of speech segments for each frame. A set of N different projected vectors are generated for each frame, by multiplying the principal discriminant matrices by the vector. This speech encoding technique is capable of being used in speech recognition systems by utilizing models, in which each model transition is tagged with one of the N classes. The projected vector is utilized with the corresponding tag to compute the probability that at least one particular speech port is present in said frame.

Representation, synthesis, variability and data preprocessing of a three-way data set

In the first section of this paper we describe the structures (vectors and matrices) on which a three-way data set X can be organized, and the information we can point out when using these structures. Many three-way analyses are based on pooled representations of X, that are systematically studied. The information given by a three-way data set can be synthesized according to the structures utilized to represent X. In the second section we define one-way, two-way and three-way syntheses of X. Also the variability of a three-way data set is evaluated, in section three, according three different levels: one-way or fiber variability, two-way or slab variability and three-way variability. The syntheses and the variability indices of X can be used for data preprocessing of X, which is here discussed in section four. Furthermore we discuss, in Section 5, the Principal Matrices Analysis on the base of three-way variability indices.

Norms of principal axis realizations: Application of abstract realization theory

Principal axis realizations are an important class of implementations, associated with good numerical properties. The spectral (l/sub 2/) norm of the principal axis realization matrices of arbitrary transfer functions is studied. In particular, it is shown that if the transfer function is contractive, then its principal axis realization matrices have a norm less than square root 2, while the internally balanced realization is conjectured to have a norm of less than 1. Furthermore, if the transfer function is an all-pass transfer function, then the three principal axis realizations become identical and equal to an orthogonal matrix. Abstract realization theory is used to prove these results. >

Clustering per le matrici a tre vie

Consider N statistical units (U), p variables (X), and K occasions or sources of data (O). Cluster analysis for three way and three mode data matrices can concern itself with many aspects. We can group the K matrices in m homogeneous clusters following the scheme { O; (U; X) } . In the scheme { U; (X; O) } we seek m clusters of matrices (variables per occasion) of homogeneous units. In the scheme { X; (U; O) } the matrices (units per occasion) are grouped in function of homogeneous variables. For two mode and three way matrices the schemes are {O; (X; X)} and {O; (U; U) }. In this paper, after recalling the INDCLUS and ADCLUS algorithms, two more recent proposals by Bellacicco-Ronzoni and Chiodi are illustrated. At the end a new algorithm, PRINCLUS, based on the individuation of m < K principal matrices is presented.

A Statistical Analysis of Mathematical Measures for Linear Simplification

The implementation of automated cartography has resulted in the digitization of linear data and the development of simplification algorithms for generalizing these data. Some algorithms, such as the nth point and random point methods are simple in both practice and operation. Others, such as polynomial reconstruction, appear to be conceptually overly complex and computationally time consuming. This study presents a method for the evaluation of simplification algorithms. Thirty mathematical measures are developed for this purpose, including both single attribute measurements, which may be applied to a single line, and measures of displacement for evaluating differences between a line and its simplification. These measures are used to compare thirty-one unsimplified naturally-occurring lines with two simplifications of each. Using principal components analysis, correlation matrices, and cartographic judgment, the thirty measures are reduced to six by eliminating statistical redundancy. The resulting six mea...

Principal component analysis of trace substance concentrations in rainwater samples

A total of 197 bulk precipitation samples from different rural and suburban stations in Bavaria, F.R.G., were analysed for the following trace substance concentrations: PAH (fluoranthen, benzo-ghiperylene, benzo-a-pyrene), chlorinated hydrocarbons (α- and γ-BHC, HCB, PCB) and trace metals (Zn, Cd, Pb, Cu, Fe). Data analysis was carried out with multivariate principal component analysis separately for suburban and rural sites, and for summer and winter events. The resulting principal component loadings matrices are characterized by typical patterns indicating differences of the emission of the single pollutants in their temporal and spatial distribution. PAH and Pb are concentrations parameters for the interpretation of the data structure of suburban sites. While Pb concentrations are elevated in the rainwater samples of these stations during all seasons, PAH indicate space-heating by showing elevated levels during winter. Rural sites are characterized by a lower interrelation between the various parameters and chlorinated BHC pesticides as further pollution variables. These substances are highly enriched in rainwater samples from spring and summer events.

The principal matrices of a Riemann matrix

The principal matrices of a Riemann matrix