Auto-association by multilayer perceptrons and singular value decomposition

The multilayer perceptron, when working in auto-association mode, is sometimes considered as an interesting candidate to perform data compression or dimensionality reduction of the feature space in information processing applications. The present paper shows that, for auto-association, the nonlinearities of the hidden units are useless and that the optimal parameter values can be derived directly by purely linear techniques relying on singular value decomposition and low rank matrix approximation, similar in spirit to the well-known Karhunen-Loève transform. This approach appears thus as an efficient alternative to the general error back-propagation algorithm commonly used for training multilayer perceptrons. Moreover, it also gives a clear interpretation of the rôle of the different parameters.     

Assessment of repeatability of surface electromyography signals by singular value decomposition

Singular value decomposition (SVD) of full-wave rectified surface electromyography (sEMG) signals was investigated for repeatability indices of sEMG linear envelopes (LE) during biceps curl. The SVD based repeatability indices were compared with a well-known method, the variance ratio (VR). The usefulness of the offered indices was examined by a simulation and it was applied to the sEMG LEs. The results have shown that the VRs were correlated with the SVD based indices significantly. The usefulness of the offered indices on real world EMG signals practically comes from decreasing amplitudes of the first few singular values of EMG LE matrix. If repeatability is high, singular values decay fast and vice versa.If justified by further researches, the offered indices may be used practically for repeatability measurement of sEMG LEs.     

Analysis of self-associating proteins by singular value decomposition of solution scattering data

We describe a method by which a single experiment can reveal both association model (pathway and constants) and low-resolution structures of a self-associating system. Small-angle scattering data are collected from solutions at a range of concentrations. These scattering data curves are mass-weighted linear combinations of the scattering from each oligomer. Singular value decomposition of the data yields a set of basis vectors from which the scattering curve for each oligomer is reconstructed using coefficients that depend on the association model. A search identifies the association pathway and constants that provide the best agreement between reconstructed and observed data. Using simulated data with realistic noise, our method finds the correct pathway and association constants. Depending on the simulation parameters, reconstructed curves for each oligomer differ from the ideal by 0.05-0.99% in median absolute relative deviation. The reconstructed scattering curves are fundamental to further analysis, including interatomic distance distribution calculation and low-resolution ab initio shape reconstruction of each oligomer in solution. This method can be applied to x-ray or neutron scattering data from small angles to moderate (or higher) resolution. Data can be taken under physiological conditions, or particular conditions (e.g., temperature) can be varied to extract fundamental association parameters (ΔH_ass, ΔS_ass).     

Systemic metabolic reactions are obtained by singular value decomposition of genome-scale stoichiometric matrices

Genome-scale metabolic networks can be reconstructed. The systemic biochemical properties of these networks can now be studied. Here, genome-scale reconstructed metabolic networks were analysed using singular value decomposition (SVD). All the individual biochemical conversions contained in a reconstructed metabolic network are described by a stoichiometric matrix (S). SVD of S led to the definition of the underlying modes that characterize the overall biochemical conversions that take place in a network and rank-ordered their importance. The modes were shown to correspond to systemic biochemical reactions and they could be used to identify the groups and clusters of individual biochemical reactions that drive them. Comparative analysis of the Escherichia coli, Haemophilus influenzae, and Helicobacter pylori genome-scale metabolic networks showed that the four dominant modes in all three networks correspond to: (1) the conversion of ATP to ADP, (2) redox metabolism of NADP, (3) proton-motive force, and (4) inorganic phosphate metabolism. The sets of individual metabolic reactions deriving these systemic conversions, however, differed among the three organisms. Thus, we can now define systemic metabolic reactions, or eigen-reactions, for the study of systems biology of metabolism and have a basis for comparing the overall properties of genome-specific metabolic networks.     

Observation of multiple intermediates in alpha-synuclein fibril formation by singular value decomposition analysis

One of the most well known characteristics for Parkinson's disease (PD) is a polymerization of wild-type or mutant alpha-synuclein into aggregates and fibrils, commonly observed as Lewy bodies and Lewy neuritis in PD patients. Although numerous studies on alpha-synuclein fibrillation have been reported, the molecular mechanisms of aggregation and fibrillation are not well understood yet. In the present study, structural properties and propensities to form fibrils of wild-type, A30P, E46K, and A53T alpha-synucleins were investigated using fluorescence and circular dichroism (CD) methods. The results from these studies were analyzed using singular value decomposition (SVD) method which estimates a number of conformationally independent species for a given process. The time-dependent CD spectra of the wild-type alpha-synuclein indicated a multi-step process in the fibril formation, and SVD analysis using the time-dependent CD spectra revealed that five or nine intermediates were formed at the early stage of fibrillation.     

Pathway level analysis of gene expression using singular value decomposition

Background  

A promising direction in the analysis of gene expression focuses on the changes in expression of specific predefined sets of genes that are known in advance to be related (e.g., genes coding for proteins involved in cellular pathways or complexes). Such an analysis can reveal features that are not easily visible from the variations in the individual genes and can lead to a picture of expression that is more biologically transparent and accessible to interpretation. In this article, we present a new method of this kind that operates by quantifying the level of 'activity' of each pathway in different samples. The activity levels, which are derived from singular value decompositions, form the basis for statistical comparisons and other applications.     

Comparing functional connectivity via thresholding correlations and singular value decomposition

We compare two common methods for detecting functional connectivity: thresholding correlations and singular value decomposition (SVD). We find that thresholding correlations are better at detecting focal regions of correlated voxels, whereas SVD is better at detecting extensive regions of correlated voxels. We apply these results to resting state networks in an fMRI dataset to look for connectivity in cortical thickness.     

Metabolic flux estimation--a self-adaptive evolutionary algorithm with singular value decomposition

Metabolic flux analysis is important for metabolic system regulation and intracellular pathway identification. A popular approach for intracellular flux estimation involves using ¹³C tracer experiments to label states that can be measured by nuclear magnetic resonance spectrometry or gas chromatography mass spectrometry. However, the bilinear balance equations derived from ¹³C tracer experiments and the noisy measurements require a nonlinear optimization approach to obtain the optimal solution. In this paper, the flux quantification problem is formulated as an error-minimization problem with equality and inequality constraints through the ¹³C balance and stoichiometric equations. The stoichiometric constraints are transformed to a null space by singular value decomposition. Self-adaptive evolutionary algorithms are then introduced for flux quantification. The performance of the evolutionary algorithm is compared with ordinary least squares estimation by the simulation of the central pentose phosphate pathway. The proposed algorithm is also applied to the central metabolism of Corynebacterium glutamicum under lysine-producing conditions. A comparison between the results from the proposed algorithm and data from the literature is given. The complexity of a metabolic system with bidirectional reactions is also investigated by analyzing the fluctuations in the flux estimates when available measurements are varied.     

Mapping gene expression quantitative trait loci by singular value decomposition and independent component analysis

Background  

The combination of gene expression profiling with linkage analysis has become a powerful paradigm for mapping gene expression quantitative trait loci (eQTL). To date, most studies have searched for eQTL by analyzing gene expression traits one at a time. As thousands of expression traits are typically analyzed, this can reduce power because of the need to correct for the number of hypothesis tests performed. In addition, gene expression traits exhibit a complex correlation structure, which is ignored when analyzing traits individually.     

Resampling methods for variance estimation of singular value decomposition analyses from microarray experiments

Microarray experiments offer the ability to generate gene expression measurements for thousands of genes simultaneously. Work has begun recently on attempting to reconstruct genetic networks based on analyses of microarray experiments in time-course studies. An important tool in these analyses has been the singular value decomposition method. However, little work has been done on assessing the variability associated with singular value decomposition analyses. In this report, we discuss use of the bootstrap as a method of obtaining standard errors for singular value decomposition analyses. We consider use of this method both when there are replicates and when no replicates exist. The proposed methods are illustrated with an application to two datasets: one involving a human foreskin study, the other involving yeast. Electronic Publication     

Application of singular value decomposition to the analysis of time-resolved macromolecular x-ray data

Singular value decomposition (SVD) is a technique commonly used in the analysis of spectroscopic data that both acts as a noise filter and reduces the dimensionality of subsequent least-squares fits. To establish the applicability of SVD to crystallographic data, we applied SVD to calculated difference Fourier maps simulating those to be obtained in a time-resolved crystallographic study of photoactive yellow protein. The atomic structures of one dark state and three intermediates were used in qualitatively different kinetic mechanisms to generate time-dependent difference maps at specific time points. Random noise of varying levels in the difference structure factor amplitudes, different extents of reaction initiation, and different numbers of time points were all employed to simulate a range of realistic experimental conditions. Our results show that SVD allows for an unbiased differentiation between signal and noise; a small subset of singular values and vectors represents the signal well, reducing the random noise in the data. Due to this, phase information of the difference structure factors can be obtained. After identifying and fitting a kinetic mechanism, the time-independent structures of the intermediates could be recovered. This demonstrates that SVD will be a powerful tool in the analysis of experimental time-resolved crystallographic data.     

A note on estimation of dynamics of multiple gene expression based on singular value decomposition

Recently, data on multiple gene expression at sequential time points were analyzed, using singular value decomposition (SVD) as a means to capture dominant trends, called characteristic modes, followed by fitting of a linear discrete-time dynamical system in which the expression values at a given time point are linear combinations of the values at a previous time point. We attempt to address several aspects of the method. To obtain the model we formulate a non-linear optimization problem and present how to solve it numerically using standard MATLAB procedures. We use publicly available data to test the approach. For reader's convenience, we provide a straightforward, ready-to-use, procedure in MATLAB, which employs its standard features to analyze data of this kind. Then, we investigate the sensitivity of the method to missing measurements and its possibilities to reconstruct missing data. Also, we discuss the possible consequences of data regularization, called sometimes 'polishing', on the outcome of analysis, especially when model is to be used for prediction purposes. Summarizing we point out that approximation of multiple gene expression data preceded by SVD provides some insight into the dynamics but may also lead to unexpected difficulties, like overfitting problems.     

A higher-order generalized singular value decomposition for comparison of global mRNA expression from multiple organisms

The number of high-dimensional datasets recording multiple aspects of a single phenomenon is increasing in many areas of science, accompanied by a need for mathematical frameworks that can compare multiple large-scale matrices with different row dimensions. The only such framework to date, the generalized singular value decomposition (GSVD), is limited to two matrices. We mathematically define a higher-order GSVD (HO GSVD) for N≥2 matrices D(i)∈R(m(i) × n), each with full column rank. Each matrix is exactly factored as D(i)=U(i)Σ(i)V(T), where V, identical in all factorizations, is obtained from the eigensystem SV=VΛ of the arithmetic mean S of all pairwise quotients A(i)A(j)(-1) of the matrices A(i)=D(i)(T)D(i), i≠j. We prove that this decomposition extends to higher orders almost all of the mathematical properties of the GSVD. The matrix S is nondefective with V and Λ real. Its eigenvalues satisfy λ(k)≥1. Equality holds if and only if the corresponding eigenvector v(k) is a right basis vector of equal significance in all matrices D(i) and D(j), that is σ(i,k)/σ(j,k)=1 for all i and j, and the corresponding left basis vector u(i,k) is orthogonal to all other vectors in U(i) for all i. The eigenvalues λ(k)=1, therefore, define the "common HO GSVD subspace." We illustrate the HO GSVD with a comparison of genome-scale cell-cycle mRNA expression from S. pombe, S. cerevisiae and human. Unlike existing algorithms, a mapping among the genes of these disparate organisms is not required. We find that the approximately common HO GSVD subspace represents the cell-cycle mRNA expression oscillations, which are similar among the datasets. Simultaneous reconstruction in the common subspace, therefore, removes the experimental artifacts, which are dissimilar, from the datasets. In the simultaneous sequence-independent classification of the genes of the three organisms in this common subspace, genes of highly conserved sequences but significantly different cell-cycle peak times are correctly classified.     

Seven-color fluorescence imaging of tissue samples based on Fourier spectroscopy and singular value decomposition.

Seven-color analyses of immunofluorescence-stained tissue samples were accomplished using Fourier spectroscopy-based hyperspectral imaging and singular value decomposition. This system consists of a combination of seven fluorescent dyes, three filtersets, an epifluorescence microscope, a spectral imaging system, a computer for data acquisition, and data analysis software. The spectra of all pixels in a multicolor image were taken simultaneously using a Sagnac type interferometer. The spectra were deconvolved to estimate the contribution of each component dye, and individual dye images were constructed based on the intensities of assigned signals. To obtain mixed spectra, three filter sets, i.e., Bl, Gr, and Rd for Alexa488 and Alexa532, for Alexa546, Alexa568, and Alexa594, and for Cy5 and Cy5.5, respectively, were used for simultaneous excitation of two or three dyes. These fluorophores have considerable spectral overlap which precludes their separation by conventional analysis. We resolved their relative contributions to the fluorescent signal by a method involving linear unmixing based on singular value decomposition of the matrices consisting of dye spectra. Analyses of mouse thymic tissues stained with seven different fluorescent dyes provided clear independent images, and any combination of two or three individual dye images could be used for constructing multicolor images.     

Improving the efficiency of multidimensional scaling in the analysis of high-dimensional data using singular value decomposition

MOTIVATION: Multidimensional scaling (MDS) is a well-known multivariate statistical analysis method used for dimensionality reduction and visualization of similarities and dissimilarities in multidimensional data. The advantage of MDS with respect to singular value decomposition (SVD) based methods such as principal component analysis is its superior fidelity in representing the distance between different instances specially for high-dimensional geometric objects. Here, we investigate the importance of the choice of initial conditions for MDS, and show that SVD is the best choice to initiate MDS. Furthermore, we demonstrate that the use of the first principal components of SVD to initiate the MDS algorithm is more efficient than an iteration through all the principal components. Adding stochasticity to the molecular dynamics simulations typically used for MDS of large datasets, contrary to previous suggestions, likewise does not increase accuracy. Finally, we introduce a k nearest neighbor method to analyze the local structure of the geometric objects and use it to control the quality of the dimensionality reduction. RESULTS: We demonstrate here the, to our knowledge, most efficient and accurate initialization strategy for MDS algorithms, reducing considerably computational load. SVD-based initialization renders MDS methodology much more useful in the analysis of high-dimensional data such as functional genomics datasets.     

Application of linear prediction singular value decomposition for processingin vivo NMR data with low signal-to-noise ratio

Summary Low sensitivity of nuclear magnetic resonance (NMR) measurements of living cell composition by conventional methods requires samples with high cell density compared to that in growing cultures. Reasonably accurate intracellular concentration estimates from lower cell density samples can be obtained by treating the time-domain NMR data by linear prediction singular value decomposition (LPSVD) prior to Fourier transformation. Alternatively, application of LPSVD enables intracellular concentration estimates in less NMR acquisition time, improving time resolution in NMR measurements of intracellular transients.     

Reliability of ultrasound speckle tracking with singular value decomposition for quantifying displacement in the carpal tunnel

Inhibited movement patterns of carpal tunnel structures have been found in carpal tunnel syndrome (CTS) patients. Motion analysis on ultrasound images allows us to non-invasively study the (relative) movement of carpal tunnel structures and recently a speckle tracking method using singular value decomposition (SVD) has been proposed to optimize this tracking. This study aims to assess the reliability of longitudinal speckle tracking with SVD in both healthy volunteers and patients with CTS.Images from sixteen healthy volunteers and twenty-two CTS patients were used. Ultrasound clips of the third superficial flexor tendon and surrounding subsynovial connective tissue (SSCT) were acquired during finger flexion-extension. A custom made tracking algorithm was used for the analysis. Intra-class correlation coefficients (ICCs) were calculated using a single measure, two-way random model with absolute agreement and Bland-Altman plots were added for graphical representation.ICC values varied between 0.73 and 0.95 in the control group and 0.66–0.98 in the CTS patients, with the majority of the results classified as good to excellent. Tendon tracking showed higher reliability values compared to the SSCT, but values between the control and CTS groups were comparable.Speckle tracking with SVD can reliably be used to analyze longitudinal movement of anatomical structures with different sizes and compositions within the context of the carpal tunnel in both a healthy as well as a pathological state. Based on these results, this technique also holds relevant potential for areas where ultrasound based dynamic imaging requires quantification of motion.