Matrix Analysis - Science topic

Hossein Safar Yousefifard, your answer is interesting. I have the same issue with the extracted matrix. My primary concern is to find the center of rigidity since SAP2000 doesn't have this option. Is there any workaround you suggest?

Thanks in advance.

I assume by writing A ≥ (scalar) you mean that the matrix A-(scalar)*(Identity) is positive semi-definite. I also assume the ||X||_2 should be squared on both sides for the inequality to make sense. The upper bound is true because it follows trivially from the operator norm inequality ||AB|| ≤ ||A|| ||B|| and we also have ||A|| ≤ ||A||_2. However the lower bound is false because one could take X = diag(0,1) and Q = diag(1,1) = Identity.

Here's a follow up Camille,

Weight Matrix w in TRF Analysis:

The weight matrix w is a fundamental output of Temporal Response Function (TRF) analysis, providing insights into how different aspects of the stimulus relate to neural activity.

Mathematical Representation:

- Each row of w corresponds to a specific time window in the stimulus, denoted as t=1, t=2, t=3, and so on.

- Each column of w corresponds to a neural activity channel, represented as Channel 1, Channel 2, and so forth.

- The values in the weight matrix w are calculated using the formula:

w = (STS)^-1STr

Example:

Suppose we have a simplified weight matrix w, where rows represent different time windows and columns represent neural channels:

| w1, Channel 1 w1, Channel 2 ... w1, Channel N |

| w2, Channel 1 w2, Channel 2 ... w2, Channel N |

| w3, Channel 1 w3, Channel 2 ... w3, Channel N |

In this matrix:

- w1, Channel 1 represents the weight or correlation between the first time window (t=1) of the stimulus and neural Channel 1.

- w2, Channel 2 represents the weight or correlation between the second time window (t=2) of the stimulus and neural Channel 2.

- Each value w captures how strongly a specific time window influences the activity in a particular neural channel.

Interpretation:

- Larger positive values of w indicate that a particular time window has a strong positive influence on the neural activity in a given channel.

- Smaller positive values indicate a positive but weaker influence.

- Negative values suggest a negative correlation, meaning that the time window has an inhibitory effect on neural activity in that channel.

Practical Use:

By examining the weight matrix w, researchers can pinpoint which temporal aspects of the stimulus are most relevant for explaining neural responses. This information is crucial for understanding how auditory stimuli are processed in the brain and aids in the decoding of auditory speech perception.

The psychological matrix of digital marketing refers to the psychological factors that influence consumer behavior and decision-making in the context of digital marketing. These factors include:

Perception: This refers to how consumers perceive digital marketing messages, including the visual design, messaging, and branding of digital content.

Motivation: Consumers' motivations for engaging with digital marketing can vary depending on their needs, goals, and desires. Understanding these motivations can help marketers create content that resonates with their target audience.

Attitude: Consumers' attitudes towards digital marketing can be influenced by various factors such as trust, credibility, and perceived value. A positive attitude towards digital marketing can increase the likelihood of engagement and conversion.

Learning: Consumers' learning and knowledge acquisition through digital marketing channels can influence their decision-making process. Marketers can use various methods such as educational content, tutorials, and reviews to enhance consumer learning.

Memory: Consumers' memory of digital marketing messages and brand impressions can affect their long-term brand loyalty and repeat purchase behavior. Repetition, consistency, and emotional appeal can help create lasting brand memories.

Emotion: Consumers' emotional responses to digital marketing messages can play a significant role in their decision-making process. Emotional appeals such as humor, fear, or joy can increase engagement and conversion.

Understanding and leveraging these psychological factors can help digital marketers create more effective and impactful campaigns that resonate with their target audience.

It would help if you show what you measure. I cant see any meaning to the y-axis on the right as a «system order». The height of the peaks is a measure of the modal mass and damping of your measured object.

Can you show the mathematical expression of what you do? Then I can help

Dear Kamal Berahmand ,

In machine learning projects we often run into curse of dimensionality problem where the number of records of data are not a substantial factor of the number of features. This often leads to a problems since it means training a lot of parameters using a scarce data set, which can easily lead to overfitting and poor generalization. High dimensionality also means very large training times. So, dimensionality reduction techniques are commonly used to address these issues. It is often true that despite residing in high dimensional space, feature space has a low dimensional structure.

Regards,

Shafagat

Correlation AnalyzeR: functional predictions from gene co-expression correlations

BMC Bioinformatics volume 22, Article number: 206 (2021) Cite this article

Abstract Co-expression correlations provide the ability to predict gene functionality within specific biological contexts, such as different tissue and disease conditions. However, current gene co-expression databases generally do not consider biological context. In addition, these tools often implement a limited range of unsophisticated analysis approaches, diminishing their utility for exploring gene functionality and gene relationships. Furthermore, they typically do not provide the summary visualizations necessary to communicate these results, posing a significant barrier to their utilization by biologists without computational skills.

Background Almost two decades after the completion of the Human Genome Project, the functionality of many genes remains largely enigmatic [1]. Many such “enigmatic genes” have immense biological significance, exemplified by the associations of thousands with cancer outcome [2]. Even genes which are well-characterized often play unexpected roles in different biological contexts (e.g., EZH2 is both a tumor-suppressor and an oncogene in different cancers [3]). Gene co-expression correlations provide a robust methodology for predicting gene function, as genes which share a biological process are often co-regulated [4,5,6]. Similar insights can be gained from using protein interaction (for example STRING [7] and InterologFinder [8]), phenome data, or even the combination of both [9]. Irrespective, generating expression data remains a cost-effective approach and co-expression analysis remains a prominent tool for exploratory systemic evaluation, largely because it is capable of considering gene co-expression across the genome. However, the applications which have been developed for such inference are hampered by key limitations. Tools like COXPRESdb [10] and GeneFriends [11] calculate gene set over-representation on an arbitrary number of co-expressed genes. Alternatively, GeneMANIA [12] and GIANT [13] construct co-expression networks and calculate gene set over-representation on an arbitrary number of nodes. Neither approach is sensitive to the genome-wide distribution of co-expression correlations or, with the exception of GIANT, differences between tissue/disease conditions. Furthermore, these functional predictions are limited in scope and do not generate relevant, user-friendly visualizations, limiting their utility for biologists without bioinformatics skills.Recently, Lachmann et al. introduced ARCHS4, a database with thousands of standardized RNA-Seq datasets [14]. We re-processed these data, calculating co-expression correlations with respect to tissue and disease (cancer/normal) condition and provided the results in a publicly accessible database. We now present Correlation AnalyzeR, a user-friendly interface to this co-expression database with a suite of tools for de-novo prediction of gene function, gene–gene relationships, and biologically relevant gene subgroups to facilitate discovery of novel relationships within genes of interest.

You can get answers of all these questions at a time by wisely utilizing technique of regression analysis. Of course, it will include use of dummy variables, ANOVA, ANCOVA etc.

This is a good question.

I think the question is not well posted.

B should be square and is not an additional condition!!

Also, A is a square; otherwise, there is no meaning to ask about the inverse.

In all cases, A - B is a square matrix, and det(A-B) is not zero, ensures the invertibility of A - B.

See some approaches in this question:

Dear Sir Victor Palamodov,

You are right, it is not true for the matrix. But in the question, the order of B is n*1 and order of C is 1*n. So B*C is a n*n matrix and that is a rank-one matrix.

Dear Liming, at a first glance the problem seems not to have a unique solution in general. Assuming for example that all the matrices are nxn, you need to solve n² equations (corresponding to the elements of B) in 2n² unknowns (the elements of X and Y.

If what matters is to find a solution, even if not unique, a possible approach could be to look for a minimum of the norm ||XAY-B|| with respect to the entries x_ij and y_ij. To get a differentiable function, one could take the Frobenius norm of the residual XAY-B.

You need to apply binary models then. Logit and Probit models are good in this case.

Besides, you can apply some non-parametric techniques that may be good to compare between the groups you have. For example, you can apply kernel densities. This helps you to compare microsites.

You can also categorize your microsites by defining: Small, Medium, large....and here you can apply ordered logit model

I am following up on this thread because I have a question that is somehow relevant to what you discussed.

I am curious if it is possible to use the symbolic factorization of a less sparse matrix for a more sparse matrix? Specifically, A is a sparse matrix; I know that all the non-zero indices of B are also non-zero in A, but not vice-a-versa. If I know the symbolic factorization of A would it be possible to use that to obtain the numerical factorization of B?

It sounds to me that it should be possible but I am trying this with the Eigen library and if B is way more sparse than A (but with the constraint I mentioned above) this idea does not work anymore. Can someone explain why?

I am testing this with SparseLU and SparseQR solvers in Eigen 3.3.7

You should know all the required details including the boundary condition of the structural member. Depending on the conditions of your case, please study the very useful, brief and having full of example source entitled " BEAM ANALYSIS USING THE STIFFNESS METHOD" given via the following link:

https://www.erbakan.edu.tr/storage/files/department/insaatmuhendisligi/editor/DersSayfalari/YapSt2/06_Matrix_Beam.pdf

i hope it is useful for you.

All the best.

G. C, Rota wrote his thesis under Nelson Dunford that characterized the extensions of OD operators via the concept of boundary values. It appeared in Communications of Pure and Applied Mathematics, in 1958. Might what to check that out relative to this paper.

KLDIV Kullback-Leibler or Jensen-Shannon divergence between two distributions. KLDIV(X,P1,P2) returns the Kullback-Leibler divergence between two distributions specified over the M variable values in vector X. P1 is a length-M vector of probabilities representing distribution 1, and P2 is a length-M vector of probabilities representing distribution 2. Thus, the probability of value X(i) is P1(i) for distribution 1 and P2(i) for distribution 2. The Kullback-Leibler divergence is given by:

KL(P1(x),P2(x)) = sum[P1(x).log(P1(x)/P2(x))]

If X contains duplicate values, there will be an warning message, and these values will be treated as distinct values. (I.e., the actual values do not enter into the computation, but the probabilities for the two duplicate values will be considered as probabilities corresponding to two unique values.) The elements of probability vectors P1 and P2 must each sum to 1 +/- .00001.

A "log of zero" warning will be thrown for zero-valued probabilities. Handle this however you wish. Adding 'eps' or some other small value to all probabilities seems reasonable. (Renormalize if necessary.)

KLDIV(X,P1,P2,'sym') returns a symmetric variant of the Kullback-Leibler divergence, given by [KL(P1,P2)+KL(P2,P1)]/2. See Johnson and Sinanovic (2001).

KLDIV(X,P1,P2,'js') returns the Jensen-Shannon divergence, given by [KL(P1,Q)+KL(P2,Q)]/2, where Q = (P1+P2)/2. See the Wikipedia article for "Kullback–Leibler divergence". This is equal to 1/2 the so-called "Jeffrey divergence." See Rubner et al. (2000).

EXAMPLE: Let the event set and probability sets be as follow: X = [1 2 3 3 4]'; P1 = ones(5,1)/5; P2 = [0 0 .5 .2 .3]' + eps;

Note that the event set here has duplicate values (two 3's). These will be treated as DISTINCT events by KLDIV. If you want these to be treated as the SAME event, you will need to collapse their probabilities together before running KLDIV. One way to do this is to use UNIQUE to find the set of unique events, and then iterate over that set, summing probabilities for each instance of each unique event. Here, we just leave the duplicate values to be treated independently (the default): KL = kldiv(X,P1,P2); KL = 19.4899

Note also that we avoided the log-of-zero warning by adding 'eps' to all probability values in P2. We didn't need to renormalize because we're still within the sum-to-one tolerance.

REFERENCES: 1) Cover, T.M. and J.A. Thomas. "Elements of Information Theory," Wiley, 1991. 2) Johnson, D.H. and S. Sinanovic. "Symmetrizing the Kullback-Leibler distance." IEEE Transactions on Information Theory (Submitted). 3) Rubner, Y., Tomasi, C., and Guibas, L. J., 2000. "The Earth Mover's distance as a metric for image retrieval." International Journal of Computer Vision, 40(2): 99-121. 4) <a href="http://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence">Kullback–Leibler divergence</a>. Wikipedia, The Free Encyclopedia.

The average accuracy of each class is equal to the accuracy of the full confusion matrix only if the number of instances is identical across classes. If some classes occur more frequently (more data loaded into one column than another) then the two numbers will be different.

https://www.mathworks.com/matlabcentral/answers/333135-what-is-a-confusion-matrix

I recommend the book by Prof. Schiehlen:

SCHIEHLEN, Werner; EBERHARD, Peter. Applied dynamics. Berlin: Springer, 2014.

The kinematic differential equations with the rotational Jacobian Matrix will solve your problem.

Dear Gyan,

I suggest you to see links in topic.

-Solving Linear-Quadratic Optimal Control Problems on Parallel ...

https://pdfs.semanticscholar.org/.../1d665cefbf9bf0adb9ad2b487e...

-New results in singular linear quadratic optimal control (PDF ...

https://www.researchgate.net/.../259561371_New_results_in_singul...

-The Generalised Discrete Algebraic Riccati Equation arising in LQ ...

https://www.researchgate.net/.../259533086_The_Generalised_Disc...

-SR and SZ algorithms for the symplectic (butterfly ... - Science Direct

https://www.sciencedirect.com/science/article/pii/.../pdf?...

-Linear Matrix Inequalities in System and Control ... - Stanford University

Best regards

Dear Dr. Cherdieu

I will read it as soon as I can and then reply you here

This may help --

Dear Nazari

Please down the manuscript from the following address

Feng Qi, Xiao-Long Qin, and Yong-Hong Yao, The generating function of the Catalan numbers and lower triangular integer matrices, Preprints 2017, 2017110120, 12 pages; Available online at https://doi.org/10.20944/preprints201711.0120.v1

Sorry, could you formulate better the question?

Given P.x=-x, in matrix analysis P.P^-1 = I. Maybe I do not understand your nomenclature...

GIlles, although this doesn't answer your question, just for your information, SIMPER and ANOSIM have serious issues since it is very difficult to determine whether differences are attributed to within-group or between-group variation, which may provide misleading results. You may want to look Warton et al. 2012. Distance-based multivariate analyses confound location and dispersion effects. Methods in Ecology and Evolution, 3, 89--101.

Sometimes, you might need to apply other modal decomposition technique instead of POD. For example, DMD would be proper answer for some cases. 

You can find open source codes here:

and I found this comparison useful:

Comparison of optimized Dynamic Mode Decomposition vs POD for the shallow water equations model reduction with large-time-step observations

 HI GAYS 

I would like a code from lumeric FDTD WITH PLASMONICS TO FIND E&h?

thank you so much

You need to understand the meaning of inverse and where it exists.

thanks dear Wiwat and Hanifa nice answers..

Dear Pedro, for the group inverse, yes. just  P has to be  left invertible and Q right invertible, and of course rank A= rank A² (the condition of existence). But for the Moore-Penrose, we need orthogonality or which is equivalent for the 3rd and the 4 th conditions. So if we take P just left invertible, that means it has a full column rank, we write PAQ=P(AQ) , then, we need  (AQ) to has full row rank to have (PAQ)⁺ =Q'A⁺ P', where P' and Q' are left and right inverses of P and Q respectively.

Excuse me, I just checked, for the group inverse, No. because there is no commutativity, i.e. the 3rd equation in the definition of the group inverse.

This problem proposed by Professor Qi Feng from the book《MARSHALL A W, OLKIN I, ARNOLD B C. Inequalities: theory of majorization and its application (Second Edition) [M]. New York : Springer Press, 2011.》pp.295-296.

whats the problem with EVD, or why are u calling it 'complex'?

As the previous comment stated, it is literally the easiest and fastest way to obtain an eigenvalue decomposition. If you need a speed increase than try a hermitian matrix  EVD solver.

Dear Viera

Thank you very much for your detailed answers to my questions. Now I have increased some knowledge about pronounciation in Russian and the like. I will cite the reference by [1] above.

Moreover, if I could use the Cauchy-Binet (Binet-Cauchy) formula to complete a previous draft on the logarithmic concavity for the Riemann Zeta function, I would acknowledge you and others, who ever gave me a hand, in the final work.

Best regards

Feng

I think you can find accurent data for test  in this website: http://www.bgu.ac.il/~bargera/tntp/

You have to complete the missing values. Try to avg the values and complete the missing

Dear  Francois

You can use transformation to a percentage of the mean instead of Z score , for example, to compare variance in two groups different in the units we use the coefficient of variation ( C. V. % ), this approach cancel the role of the units and use the percentage in the calculation.

Good Luck

thank you very much Rodrigo Augusto Ricco and M. S. N. Murty 

What are the dimension of the short fibers, the volume content in the matrix and the fiber diameter?

Probably you have to check the memory status during execution of command as.matrix(), given your operational system (for windows leave open the task manager). What is your RAM in GB? Are you using 32 or 64 bit computer?

 First, let's be clear about what you can do and what you can't do.  The characteristic equation for an 8x8 matrix to determine the 8 eigenvalues is in general an eighth order polynomial.  First you may want to read about Abel's Impossibility Theorem ( see: http://mathworld.wolfram.com/AbelsImpossibilityTheorem.html ) for finding formulas for computing eigenvalues from polynomials. In particular see the link for quintic polynomials (http://mathworld.wolfram.com/QuinticEquation.html)  . So in general the only hope you have for a 8th order polynomial is if the substitution z=y^2 reduces the 8th order polynomial in y to a 4th order polynomial in z, which would then allow you to find an analytic expression for the eigenvalues.  In your case this is not possible as the coefficients (B, g1, g2, k1, k2, J) are not specified.

Nevertheless,  using a software packages ( here  I will discuss using Mathematica) you can readily explore the solution space for the eigenvalues. My preliminary calculations shows that the eigenvalues for random real numbers that I have selected  for parameter values between  (-2, 2) and  (0,5) give  eight real eigenvalues: 4 positive and 4 negative.

I have attached a PDF of a Mathematica notebook that illustrates these calculations. Enjoy 

Nils i got the answers, thanks. My mistake i didnt take 2*pi into consideration.

For symmetric matrices there are ways to show the similarity. One way is to start with an arrowhead matrix, and find its eigenvalues, then construct a tridiagonal matrix with those eigenvalues. The existence of the tridiagonal matrices with a given set of eigenvalues is shown in several papers. One best source is: A.L. Duarte, Construction of acyclic matrices from spectral data, Linear Algebra Appl., 113 (1989) 173–182.

Then the easiest way to show the similarity I think is by the diagonalization.

This is a good question.

A good place to start looking for an answer to this question is

HANS SCHNEIDER, Olga Taussky-Todd/s Influence on Matrix Theory and Matrix Theorists:

More to the point, consider

LENNY FUKSHANSKY, DIOPHANTINE APPROXIMATIONS AND GEOMETRY

OF NUMBERS, 2006:

See Section 4. Theorems of Blichfeldt and Minkowski, starting on page 26.   And, then, see Section 5. Successive minima, starting on page 33, especially Minkowski's successive minima theorem on page 34.   And Section 6. Inhomogeneous minimum, where Minkowski's successive minima theorem in terms of lattices is given, starting on page 39.

Write down a basis for the set of vectors generated by the two eigenvectors v_1, v_2.  

The matrix A represents a linear transformation L of R^4 to R^4, but relative to this basis of a two dimensional space, it maps v_1 to lambda_1 v_1 and v_2 to lambda_2 v_2.

Hence, *with respect to this basis for a 2-dimensional subspace* L is represented by a 2x2 matrix B, where 

B= [ lambda_1       0            ]

      [    0               lambda_2 ]

This is what I meant about regarding a matrix as representing the action of a linear transformation with respect to a basis or two bases.

@Abdur  Usually you vary the size of the matrix for performance analysis.  Typically, better parallel speedup is obtained for large problem sizes (i.e. bigger matrices).

Moslem,

I am sorry

There is  an obvious mistake in my proposed argument for $m>1$, however I think that  my argument for $m=1$ is correct.

My mistake for $m>1$ is that the complement S of full rank matrices is not convex, but in Hahn banach theorem we need convexity of both compact set K and closed set S. 

BTW, you called your question as a "conjecture". Where is  a  reference for this problem. Is it a well known conjecture?

Thanks for responses, and sorry for my bad English!!, Yes I have already tested "optimtool" functions on Matlab (may be applied on any non linear function), however, the results are mostly local optimum. I was hopping for more specific solution to the problem with numericals approaches, here I have downed the problem as, above there is a little mistake (between sx and sy):

[Xi Yi]  and [Xi' Yi'] are sets of centred 2d point, so we do not bother with translation (tx, ty)

[Xi' Yi'] = [Xi Yi] * |Sx    0| * | cos(r)  -sin(r)|   = [Xi Yi] * Sc * Rot

                           |0    Sy|   | sin(r)    cos(r)|     

The result may be threshold after, to satisfy constrains on r, sx and sy.

Rot matrix is orthonormal, and "pinv" is pseudo inverse matrix

Sc*Rot=pinv([Xi Yi])*[Xi' Yi']

use of SVD decomposition:

Sc*Rot=V*S*D    , where V and D are orthonormal matrix:

Here we can assume :  Rot=D  and Sc=V*S, however  Sc is diagonal matrix,

Then, is there any approximation that may estimate Sc (diagonal matrix) to minimize error between [Xi' Yi'] and transformed [Xi Yi].

Thanks :)

I agree with the method provided above. But I wonder why in the first place the author got the t values and the coordinates in an xls file to start with. If this is fMRI imaging processing result, I suggest to use SPM to analyze the data and the final result (with clusters) would be easy to get, which would make your life easier. 

Dear Hanifa,

Thank you for your kindness.

Regards,

Wiwat Wanicharpichat

@Axel - the construction of Bourgain et al. is deterministic and is an amazing theoretical breakthrough. Unfortunately it relies on results from additive combinatorics which require the size of the matrices constructed to be too large to fit on a computer. 

@I. E. Karporin - In the original work of Candes-Romberg-Tao they show that in a random ensemble there is a very high probability that EVERY column column sub matrix with n/log n columns is close to an isometry. So the probability of writing down a truly random matrix with the property you describe is exponentially small.

You may refer to Prof. V. Simoncini's personal webpage for her survey on matrix eqs. http://www.dm.unibo.it/~simoncin/list.html

Dear Alaa Eleyan

Thanks for your reply

Kindly find the attachment .

Already I tried the function cell2mat(C), but my problem is different with that variable C values.

I have a cell consists of 18 X 9 matrix.After opening the matrix it will again show the 9 columns with 18 rows (with cell array format).

I want to convert the 18 rows with 9 columns (with cell array format) to matrix of each individual row and column of the data.

regards

manoj kumar

The word random implies unpredictable, but does not specify the distribution.  There must be a dozen ways to select a vector "at random" each with a different distribution.  To illustrate with a simpler example, suppose we select a random chord in a circle of radius 1.  What is the probability that the length exceeds sqrt(3)?  If we find our chord by choosing two points uniformly on the circle, the answer is 1/3.  But we could determine our chord by selecting one point uniformly in the interior of the circle, and then drawing the unique radius through this point, and then constructing the perpendicular line through our point to the radius.  This gives the unique chord having our point as its midpoint.  The probability that this is longer than sort(3) is 1/4.  Or a third method:  Select the radius at random, select a point on the radius uniformly, and then construct the perpendicular.  Now the probability is 1/2.  

The mechanism chosen for the random vector selection determines the distribution.

Type the code bellow in the command window:

net(target_data)

I appreciate your assist Jose

You can check image compression algorithms with SVD, or check moore penrose inverse for the approached resolution of an inverse.

For smallest eigenvalues you can check the structural dynamics fields (modal truncation), or also linear bucking analysis.

J.R. Schott, Matrix Analysis for Statistics, Second Edition,

          John Wiley & Sons, Inc., New Jersey, 2005.

Theorem 7.28. Let A and B be m×m nonnegative definite matrices (with ordered eigenvalues λ₁≥…≥ λ_{m}). Then

                              λ_{m}(A∘B) ≥ λ_{m}(AB).

On p.315

Problem 35. Let A, B, and C he m×n matrices. Show that

                              tr{(A′∘B′)C} = tr{A′(B∘C)}.

If the n'th column of your matrix is all zeros, then it means that the n'th independent variable in a system of equations does not play any role in determining the solution of the system of equations. So...simply remove all zero columns and insignificant variables from the system of equations. remove them because they have nothing to do with the solution of the system of equations. 

If your matrix was a square matrix to begin with, however, then you will be left with a non-square matrix that has more rows than columns. In other words, the system of equations is over determined.

To invert a matrix that has more rows than columns (an over determined system of equations), use singular value decomposition (SVD). There are standard algorithms for applying SVD. You can also use least square methods, but I prefer SVD.

SVD also works nicely for under determined systems of equations, when you have more columns than rows, in which case the solution of a system of equations is not unique. But I suspect that this is not your situation.

There are SVD routines that work on matrices of complex numbers (with real and imaginary parts).

As I think of it, I have not tried SVD on matrices with zero columns left in the matrix. SVD may also work if you leave the zero columns in the matrix. I am unsure. I would remove the columns, and the corresponding independent variables, however, because the matrix is telling you that those independent variables play no role in determining the solution of the system of equations.. 

Ronald

Apparently it is not an easy task. Everybody knows that if you consider a product of two square matrices GH, the inverse matrix is given by H^-1G^-1. But the problem of calculating the inverse of the sum is more difficult. However, it has been solved and a recursive formula for (G+H)^-1 has been found. You can find details in the article by Kenneth S. Miller "On the Inverse of the Sum of Matrices". Find link to the article in the attachment.

A system of linear equation Ax = b is consistent if and only if b in column space of A.

Please see the attachment.

Clerc, M. (2013). Particle Swarm Optimization, Wiley.

Any algorithms that rely on pseudo-random number generators can, in principle, be forced to output the same results in every run. The only thing that's needed is to set the seed value of the generator to the same value before running the algorithm. That's also stated in MATLAB help (http://se.mathworks.com/help/matlab/ref/rand.html): "The sequence of numbers produced by rand is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn. You can control that shared random number generator using rng." See http://se.mathworks.com/help/matlab/ref/rng.html for help on the rng function.

Genetic algorithm maybe? It would consume time but eventually, put you in the right place...

Look at this:

Please refer to "MATLAB Programming for Engineers" by Lynch. You can also take a look at this toolbox:

you can import your 3D geometry model with designed dimensions  to simulink/SimMechanics.

I would recommend buying two books: one that is simple such that you aren't lost from the start because the text assumes you are very experienced with MATLAB and/or image processing, and another which is almost at that level to use as a reference guide. For the simple book, you might wish to check out

Marques, O. (2011). Practical image and video processing using MATLAB. Wiley.

For a more comprehensive source to use as reference material, I would suggest

Solomon, C., & Breckon, T. (2011). Fundamentals of Digital Image Processing: A practical approach with examples in Matlab. Wiley.

or

Gonzalez, R. C., Woods, R. E., & Eddins, S. L. (2009). Digital image processing using MATLAB (2nd Ed.). Gatesmark Publishing.

Dear Kirti Bihade ,

it is Still not clear.........

hi

see this link

You can get the similar or apropriate answer by searching the keyword in the GOOGLE SCHOLAR page. Usually you will get the first paper similar to your keyword.

From my experience, this way will help you a lot. If you still have a problem, do not hasitate to let me know.

Kind regards, Dr ZOL BAHRI - Universiti Malaysia Perlis, MALAYSIA

BNT-BKT-BT. Available from: https://www.researchgate.net/post/BNT-BKT-BT#view=54f68badd3df3ee77c8b4638 [accessed May 7, 2015].

try fitting a mixture distribution

Francesco - Yes, using Y=-X+max(X) [or even just Y=-X] does the trick. But I was wondering if perfcurve is able to do this by itself, without the need to manipulate the inputs, or the outputs.

Peng - If I change the value of posclass I can get the correct value for AUC, but the values for the ROC plot are (somewhat) swapped around. Of course, this can be fixed manually, but again I don't find it ideal.

@Abeyou: I suggest you to use the re-gridded data from CATDS: http://catds.ifremer.fr/Products/Available-products-from-CPDC

After login you can download all the SMOS data.

1.Read an image

2. Extract the sift descriptor and store it in some variable name such as siftpoints and then add this code

figure;

imshow(Image);

title('100 Strongest Feature Points from Image');

hold on;

plot(selectStrongest(scenePoints, 100));

selectStrongest is an inbuilt function in matlab 2014b