Inverse Of Coefficient Matrix Research Articles

Analysis of an image noise sensitivity mechanism for matrix-operation-mode-decomposition and a strong anti-noise method.

Mode decomposition (MD) based on the matrix operation (MDMO) is one of the fastest mode decomposition methods in fiber laser which has great potential for optical communications, nonlinear optics and spatial characterization applications. However, we found that the image noise sensitivity is the main limit to the accuracy of the original MDMO method, but improving the decomposition accuracy by using conventional image filtering methods is almost ineffective. By using the norm theory of matrices, the analysis result shows that both the image noise and the coefficient matrix condition number determine the total upper-bound error of the original MDMO method. Besides, the greater the condition number, the more sensitive of MDMO method is to noise. In addition, it is found that the local error of each mode information solution in the original MDMO method is different, which depends on the L2-norm of each row vector of the inverse coefficient matrix. Moreover, a more noise-insensitive MD method is achieved by screening out the information corresponding to large L2-norm. In particular, selecting the higher accuracy among the original MDMO method and such noise-insensitive method as the result in a single MD process, a strong anti-noise MD method was proposed in this paper, which displays high MD accuracy in strong noise for both near-filed and far-filed MD cases.

Incremental holistic controller

The trace of inverse coefficient matrix method in state space is used to implement and simplify an iterative design of a control system with main interest in increasing dynamical performance quality or to check if stability danger exists. The tau -locus is a helpful design element. Various applications and examples are presented for illustration.

Indirect-Inversion Algorithm via Precise Integration for Ill-Conditioned Matrix in Ambiguity Resolution

Global navigation satellite system (GNSS) positioning depends on the correct integer ambiguity resolution (AR). If the double difference equation for solving the float solution remains ill-conditioned, often happening due to the environment complexity and the equipment mobility, the correct AR is difficult to achieve. Concerning the ill-conditioned problem, methods of modifying the equation coefficient matrix are widely applied, whose effects are heavily dependent on modifying parameters. Besides, the direct-inversion of the ill-conditioned coefficient matrix can lead to a reduction in the accuracy and stability of the float solution. To solve the problem of ill-conditioned matrix inversion and further improve the accuracy, the present study for the first time proves the positive definite symmetry of the coefficient matrix in AR model and employs precise integration method to the indirect inverse of coefficient matrix. AR model for the GNSS positioning and the general resolving strategies introduction are briefly introduced. An indirect-inversion algorithm via precise integration for ill-conditioned coefficient matrix is proposed. According to the simulations and comparisons, the proposed strategy has higher precision and stability on float solution, and less dependence on modifying parameters.

Genomic selection through single-step genomic best linear unbiased prediction improves the accuracy of evaluation in Hanwoo cattle.

ObjectiveGenomic selection (GS) is becoming popular in animals’ genetic development. We, therefore, investigated the single-step genomic best linear unbiased prediction (ssGBLUP) as tool for GS, and compared its efficacy with the traditional pedigree BLUP (pedBLUP) method.MethodsA total of 9,952 males born between 1997 and 2018 under Hanwoo proven-bull selection program was studied. We analyzed body weight at 12 months and carcass weight (kg), backfat thickness, eye muscle area, and marbling score traits. About 7,387 bulls were genotyped using Illumina 50K BeadChip Arrays. Multiple-trait animal model analyses were performed using BLUPF90 software programs. Breeding value accuracy was calculated using two methods: i) Pearson’s correlation of genomic estimated breeding value (GEBV) with EBV of all animals (rM1) and ii) correlation using inverse of coefficient matrix from the mixed-model equations (rM2). Then, we compared these accuracies by overall population, info-type (PHEN, phenotyped-only; GEN, genotyped-only; and PH+GEN, phenotyped and genotyped), and bull-types (YBULL, young male calves; CBULL, young candidate bulls; and PBULL, proven bulls).ResultsThe rM1 estimates in the study were between 0.90 and 0.96 among five traits. The rM1 estimates varied slightly by population and info-type, but noticeably by bull-type for traits. Generally average rM2 estimates were much smaller than rM1 (pedBLUP, 0.40 to 0.44; ssGBLUP, 0.41 to 0.45) at population level. However, rM2 from both BLUP models varied noticeably across info-types and bull-types. The ssGBLUP estimates of rM2 in PHEN, GEN, and PH+ GEN ranged between 0.51 and 0.63, 0.66 and 0.70, and 0.68 and 0.73, respectively. In YBULL, CBULL, and PBULL, the rM2 estimates ranged between 0.54 and 0.57, 0.55 and 0.62, and 0.70 and 0.74, respectively. The pedBLUP based rM2 estimates were also relatively lower than ssGBLUP estimates. At the population level, we found an increase in accuracy by 2.0% to 4.5% among traits. Traits in PHEN were least influenced by ssGBLUP (0% to 2.0%), whereas the highest positive changes were in GEN (8.1% to 10.7%). PH+GEN also showed 6.5% to 8.5% increase in accuracy by ssGBLUP. However, the highest improvements were found in bull-types (YBULL, 21% to 35.7%; CBULL, 3.3% to 9.3%; PBULL, 2.8% to 6.1%).ConclusionA noticeable improvement by ssGBLUP was observed in this study. Findings of differential responses to ssGBLUP by various bulls could assist in better selection decision making as well. We, therefore, suggest that ssGBLUP could be used for GS in Hanwoo proven-bull evaluation program.

Influencing the oscillation properties of a control system

The operation of shifting the poles to the left in the complex plane is transferred to the method of reducing the main oscillation frequency of a control system. Influencing the oscillation properties is realized in an unstructured manner without using solution details of the characteristic equation. The holistic design procedures using the trace of the inverse coefficient matrix and the Michailow distance are applied to find a gradient-based algorithm for improving the controller, after having executed a double-turning or a single-turning and shifting of the closed-loop coefficient matrix.

Connectedness among test series in mixed linear models of genetic evaluation for forest trees

In forest tree species with large natural ranges, there are usually several to many separate breeding populations, each designed to capture elite material suited to a particular geographic region. Separate test series are often dedicated to each population. Because the aim is to optimise gain in the meta-population, it is important to ensure that test series are linked so that individuals can be compared across test series as well as within. Computer simulation was used to determine the most efficient strategy for obtaining linkage. The average accuracy of a genetic value contrast between individuals in the same and in different test series was used as the criterion for assessing the optimal level of linkage. Accuracy is a function of the elements of the inverse coefficient matrix for a mixed linear model within a best linear unbiased prediction framework (BLUP). Material used to link test series was either common test families, common check-lots such as seed orchard bulks, or families generated by inter-crossing parents from different test series. Use of common test families was the most efficient strategy for the scenarios tested, which included having 50 parents crossed to produce 50 test families in each of three populations. For a low-heritability scenario, the amount of linkage material, relative to test material, needed to be 8 and 12 %, for progeny and parents, respectively, in order for a contrast between individuals in different test series to have equivalent accuracy as a contrast between individuals in the same test series. Other strategies were less efficient in terms of the amount of linkage material needed to obtain this equivalency.

Dynamic simulation of multi-unit air conditioners based on two-phase fluid network model

Multi-unit air conditioners (MUACs) are widely used in light commercial buildings and residential buildings due to their higher thermal comfort and energy efficiency. To investigate the transient characteristics of MUACs, a dynamic simulation model with the framework of two-phase fluid network is developed. The state-space forms are used to model the system and components, and the component submodels are embedded in the fluid network model, which makes it possible to update the system model and components submodels independently. In the model of state-space form, the differentials are obtained by taking the inverse of coefficient matrix, and then the state parameters are calculated by integrating the differentials with time. The simulation outputs are compared with the experimental data in the step changes of the compressor speed and electronic expansion valve openings. The comparison shows that the proposed model can catch the dynamic characteristics of MUACs with high accuracy. Therefore, it can be used as an effective tool to analyze the transient performance and optimize the control algorithm of MUACs.

Accuracy of genomic breeding values in multi-breed dairy cattle populations

BackgroundTwo key findings from genomic selection experiments are 1) the reference population used must be very large to subsequently predict accurate genomic estimated breeding values (GEBV), and 2) prediction equations derived in one breed do not predict accurate GEBV when applied to other breeds. Both findings are a problem for breeds where the number of individuals in the reference population is limited. A multi-breed reference population is a potential solution, and here we investigate the accuracies of GEBV in Holstein dairy cattle and Jersey dairy cattle when the reference population is single breed or multi-breed. The accuracies were obtained both as a function of elements of the inverse coefficient matrix and from the realised accuracies of GEBV.MethodsBest linear unbiased prediction with a multi-breed genomic relationship matrix (GBLUP) and two Bayesian methods (BAYESA and BAYES_SSVS) which estimate individual SNP effects were used to predict GEBV for 400 and 77 young Holstein and Jersey bulls respectively, from a reference population of 781 and 287 Holstein and Jersey bulls, respectively. Genotypes of 39,048 SNP markers were used. Phenotypes in the reference population were de-regressed breeding values for production traits. For the GBLUP method, expected accuracies calculated from the diagonal of the inverse of coefficient matrix were compared to realised accuracies.ResultsWhen GBLUP was used, expected accuracies from a function of elements of the inverse coefficient matrix agreed reasonably well with realised accuracies calculated from the correlation between GEBV and EBV in single breed populations, but not in multi-breed populations. When the Bayesian methods were used, realised accuracies of GEBV were up to 13% higher when the multi-breed reference population was used than when a pure breed reference was used. However no consistent increase in accuracy across traits was obtained.ConclusionPredicting genomic breeding values using a genomic relationship matrix is an attractive approach to implement genomic selection as expected accuracies of GEBV can be readily derived. However in multi-breed populations, Bayesian approaches give higher accuracies for some traits. Finally, multi-breed reference populations will be a valuable resource to fine map QTL.

Microwave Subsurface Imaging Using Direct Finite-Difference Frequency-Domain-Based Inversion

We have developed a new algorithm for electromagnetic inverse scattering problems in inhomogeneous media using finite-difference frequency-domain (FDFD) forward modeling, referred to as the FDFD-based inversion method. The key issue of this method is to build a linear expression for the inverse problem from an FDFD forward model by using Born approximation to neglect mutual coupling between scattered pixels and to then solve for the inverse coefficient matrix. An important advantage of this matrix-based method is that there is no need to specify a Green's function. As such, this inverse scattering algorithm is easily implemented and is robust to the heterogeneity in the background. We test the algorithm with a microwave subsurface object detection application using cross-well radar. The new method is compared with conventional inversion using Green's function-based Born approximation. Numerical experiments are presented for a 2-D borehole geometry for buried object detection in uniform soil and in multilayered soil backgrounds.

A study of surface roughness in drilling using mathematical analysis and neural networks

Drilling is one of the most common and fundamental machining processes. It is most frequently performed in material removal and is used as a preliminary step for many operations, such as reaming, tapping and boring. Because of their importance in nearly all production operations, twist drills have been the subject of numerous investigations. The aim of this study is to identify suitable parameters for the prediction of surface roughness. Back propagation neural networks are used for the detection of surface roughness. Drill diameter, cutting speed, feed and machining time are given as inputs to the neural network structure and surface roughness was estimated. Drilling experiments with 12 mm drills are performed at three cutting speeds and feeds. The number of neurons are selected from 1,2,3, ..., 20. The learning rate was selected as 0.01, and no smoothing factor was used. The best structure of neural network was selected based on a criteria including the minimum of sum of squares with the actual value of surface roughness. For mathematical analysis, an inverse coefficient matrix method was used for calculating the estimated values of surface roughness. Comparative analysis was performed between actual values and estimated values obtained by mathematical analysis and neural network structures.

On Fast Modal Frequency Response Analysis of Large Structural System with Gyroscopic Effect

A new algorithm, fast frequency response analysis (FFRA), for the modal frequency response analysis (FRA) with gyroscopic effects is presented for a large and complex structural system. Because the modal gyroscopic matrix is low rank matrix compared to other modal matrices, the FFRA algorithm uses the Sherman-Morrison-Woodbury formula for computing the inverse of coefficient matrix efficiently. The new method, FFRA algorithm, dramatically improve the performance of the modal frequency response analysis with gyroscopic matrix compared to conventional methods in industry with the same accuracy.

COMPARATIVE ANALYSIS OF VARIOUS METHODS OF SURFACE ROUGHNESS IN DRILLING

Drilling is one of the most common and fundamental machining processes. Since approximately 40% of all the cutting operations are drilling in industry. It is most frequently performed, material removing process and is used as a preliminary step for many operations, such as reaming, tapping and boring. Because of their importance in nearly all production operations twist drills have been the subject of numerous investigations. Surface finish quality of a machined work piece is an issue of main concern to the manufacturing industry. The aim of the present work is to identify suitable parameters for the prediction of surface roughness. Back propagation neural networks were used for detection of surface roughness. Drill diameter, cutting speed, feed, and machining time were given as inputs to the neural network structure and surface roughness was estimated. Drilling experiments with 10 mm drill size were performed at three cutting speeds and feeds. The number of neurons were selected from 1,2,3,…, 20. The learning rate was selected as 0.01 and no smoothing factor was used. The best structure of neural networks were selected based on the criteria as the minimum of summation of square with the actual value of surface roughness. For statistical analysis, it was assumed that surface roughness depends on cutting speed, feed and machining time. For the mathematical analysis inverse coefficient matrix method was used for calculating the estimated values of surface roughness. Comparative analysis has been done between the actual values and the estimated values obtained by statistical analysis, mathematical analysis and neural network structure.

The Optimal Radius of the Support of Radial Weights Used in Moving Least Squares Approximation

Owing to the meshless and local character- istics, moving least squares (MLS) methods have been used extensivelytoapproximatetheunknownfunctionof partial differential equation initial boundary value prob- lem. In this paper, based on matrix analysis, a sufficient and necessary condition for the existence of inverse of coefficient matrix used in MLS methods is developed firstly. Then in the light of approximate theory, a prac- tical mathematics model is posed to obtain the optimal radius of support of radial weights used in MLS meth- ods. As an example, while uniform distributed particles and the 4 th order spline weight function are adopted in MLS method in two dimension domain and two kinds of norms are used to measure error, optimal results for lin- ear and quadratic basis are gained. Finally, the test data verify that the optimal values are correct. The research idea can be used in 3-dimension problems too.

Predicting breeding values and accuracies from group in comparison to individual observations

Individual observations are routinely used in livestock evaluations. In some cases, pooled data representing the joint but not individual performance of a group of animals may be available. For example, pooled feed intake may be measured on a pen of livestock. The usual mixed model approach to genetic evaluation can still be applied as an exact method in this setting, provided incidence and residual variance-covariance matrices are suitably modified to account for the pooling. Approximate evaluations may be achieved by treating average performance as if it pertained to each individual in the pool. Theoretical accuracies can be obtained as a function of elements of the inverse coefficient matrix. A 3-generation data set representing 1,000 animals with feed intake observations from 49 sires and 200 maternal grand sires was simulated with heritability of 0.34. Individual records were pooled to represent circumstances in which animals with records were collectively measured. Animals were allocated into pens at random, by sire, or by maternal grand sire. Simulation was replicated with unique fixed effects for each pen. Following evaluation from each method, the empirical accuracy or product-moment correlation between true (simulated) and estimated merit could be quantified. The analysis of individual observations resulted in empirical accuracy of 0.63 for animals on test and 0.77 for their sires. Pooling the observations in pens of 2, 4, or 12 animals reduced empirical accuracies for animals on test to 0.50, 0.41, and 0.21 when pooling was at random and 0.53, 0.47, and 0.34 when pooling was by sire. Simulating a fixed pen effect representing 10% phenotypic variation, but ignoring that effect in the evaluation minimally reduced empirical accuracies to 0.52, 0.46, and 0.33 when pooling by sire. Theoretical accuracies were in close agreement with empirical accuracies when the exact method was used. The approximate method that treated averages of pooled data as if they were individually observed overstated accuracy and should not be used. Selection on the basis of pooled observations can be almost as effective as using individual observations when pool sizes are small. The exact method to account for pooled data is no more complex than conventional procedures.

Approximate accuracy of genetic evaluation under an animal model

A procedure is described to approximate the accuracy of genetic evaluation by Best Linear Unbiased Prediction under an Individual Animal Model. This involved adjusting the diagonal elements of the coefficient matrix in the Mixed Model Equations for connections to parents, progeny and fixed effects. The reciprocals of the resulting coefficients then provided a good approximation to the diagonal elements of the inverse of the coefficient matrix, and, hence, approximate prediction error variances and accuracies could be calculated. Simulation was used to check the adequacy of the procedure. Type scores for 282 030 Canadian Holstein heifers were replaced by simulated records, maintaining the original data and relationship structure. Analysing this data and grouping animals according to the amount of information available, empirical accuracies, i.e. correlations between true and estimated genetic merit, were calculated within groups. Predicted, approximate accuracies were consistently higher than their empirical counterparts. Differences were small for animals in the data, amounting to 1 or 2% only. For parents not in the data, empirical accuracies were 92% or more of predicted, except for sires with 6–20 daughters. Diagonal elements of the inverse coefficient matrix were obtained iteratively for a subset of the data, and showed a correlation close to unity with the approximate values. Overall, the approximation procedure performed well and can be recommended for use in conjunction with the genetic evaluation of large populations under an Individual Animal Model.