I have a task where I need to train a machine learning model to predict a set of outputs from multiple inputs. My inputs are 1000 iterations of a set of 3x 1 vectors, a set of 3x3 covariance matrices and a set of scalars, while my output is just a set of scalars. I cannot use regression learner app because these inputs need to have the same dimensions, any idea on how to unify them?
One possible way to solve this is to flatten the covariance matrix into a vector. Once you did that, you can construct a 1000xN matrix where 1000 refers to the number of samples in your dataset and N is the number of features. For example if your features consist of a 3x1 vector, a 3x3 covariance matrix and lets say 5 other scalars, N could be 3+3*3+5=17. You then use this matrix to train an arbitrary model such as a linear regressor or more advanced models like a tree or the like.
When training machine learning models it is important to understand your data and exploit its structure to help the learning algorithms. For example we could use the fact that a covariance matrix is symmetric and positive semi-definite and thus lives in a closed convex cone. Symmetry of the matrix implies that it lives in a subspace of the set of all 3x3 matrices. In fact the dimension of the space of 3x3 symmetric matrices is only 6. You can use that knowledge to reduce redundancy in your data.
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I have a correlation matrix for N random variables. Each of them is uniformly distributed within [0,1]. I am trying to simulate these random variables, how can I do that? Note N > 2. I was trying to using Cholesky Decomposition and below is my steps:
get the lower triangle of the correlation matrix (L=N*N)
independently sample 10000 times for each of the N uniformly distributed random variables (S=N*10000)
multiply the two: L*S, and this gives me correlated samples but the range of them is not within [0,1] anymore.
How can I solve the problem?
I know that if I only have 2 random variables I can do something like:
1*x1+sqrt(1-tho^2)*y1
to get my correlated sample y. But if you have more than two variables correlated, not sure what should I do.
You can get approximate solutions by generating correlated normals using the Cholesky factorization, then converting them to U(0,1)'s using the normal CDF. The solution is approximate because the normals have the desired correlation, but converting to uniforms is a non-linear transformation and only linear xforms preserve correlation.
There's a transformation available which will give exact solutions if the transformed Var/Cov matrix is positive semidefinite, but that's not always the case. See the abstract at https://www.tandfonline.com/doi/abs/10.1080/03610919908813578.
I want to use adjacency matrix in MATLAB to solve a maximum flow problem formulated in Linear Programming. I want to try with adjacency matrix because using incidence matrix gives huge matrices due to the number of edges.
I am simulating a physical system, where I need to calculate the eigenvalues and vectors of a very large (~10000x10000) matrix.
So far I have used the in-built Eig algorithm in MATLAB but it is very slow for large matrices. Is there other algorithms in MATLAB that would do a better job or can I somehow improve the performance of Eig? Specifically it turns out that I only need the first ~100 eigenvectors of the matrix starting from the smallest numerical eigenvalue. Is there a way to get the algorithm to calculate only the first N eigenvectors and eigenvalues to save computation time? Of course this would only work if the eigenvectors come out sorted but they seem to do so, because of the symmetry of the Matrix I am using.
Your matrix has mostly zeros, so you should make it a sparse matrix. You'll then be able to use EIGS to calculate a smaller number of eigenvalues and eigenvectors.
http://www.mathworks.com/help/matlab/ref/eigs.html
I have a doubt about SVD. in the literature that i had read, it's written that we have to convert our input matrix into covariance matrix first, and then SVD function from matlab (SVD) is used.
But, in Mathworks website we can use SVD function directly to the input matrix (no need to convert it into covariance matrix)..
[U,S,V]=svd(inImageD);
Which one is the true??
And if we want to do dimensionality reduction, we have to project our data into eigen vector.. But where is the eigen vector generated by SVD function..
I know that S is the eigen value.. But what is U and S??
To reduce our data dimensional, do we need to substract the input matrix with its mean and then multiply it with eigen vector?? or we can just multiply our input matrix with the eigen vector (no need to substract it first with its mean)..
EDIT
Suppose if I want to do classification using SIFT as the features and SVM as the classifier.
I have 10 images for training and I arrange them in a different row..
So 1st row for 1st images, 2nd row for second images and so on...
Feat=[1 2 5 6 7 >> Images1
2 9 0 6 5 >> Images2
3 4 7 8 2 >> Images3
2 3 6 3 1 >> Images4
..
.
so on. . ]
To do dimensionality reduction (from my 10x5 matrix), we have yo do A*EigenVector
And from what U had explained (#Sam Roberts), I can compute it by using EIGS function from the covariance matrix (instead of using SVD function).
And as I arrange the feat of images in different row, so I need to do A'*A
So it becomes:
Matrix=A'*A
MAT_Cov=Cov(Matrix)
[EigVector EigValue] = eigs (MAT_Cov);
is that right??
Eigenvector decomposition (EVD) and singular value decomposition (SVD) are closely related.
Let's say you have some data a = rand(3,4);. Note that this not a square matrix - it represents a dataset of observations (rows) and variables (columns).
Do the following:
[u1,s1,v1] = svd(a);
[u2,s2,v2] = svd(a');
[e1,d1] = eig(a*a');
[e2,d2] = eig(a'*a);
Now note a few things.
Up to the sign (+/-), which is arbitrary, u1 is the same as v2. Up to a sign and an ordering of the columns, they are also equal to e1. (Note that there may be some very very tiny numerical differences as well, due to slight differences in the svd and eig algorithms).
Similarly, u2 is the same as v1 and e2.
s1 equals s2, and apart from some extra columns and rows of zeros, both also equal sqrt(d1) and sqrt(d2). Again, there may be some very tiny numerical differences as well just due to algorithmic issues (they'll be on the order of something to the -10 or so).
Note also that a*a' is basically the covariances of the rows, and a'*a is basically the covariances of the columns (that's not quite true - a would need to be centred first by subtracting the column or row mean for them to be equal, and there might be a multiplicative constant difference as well, but it's basically pretty similar).
Now to answer your questions, I assume that what you're really trying to do is PCA. You can do PCA either by taking the original data matrix and applying SVD, or by taking its covariance matrix and applying EVD. Note that Statistics Toolbox has two functions for PCA - pca (in older versions princomp) and pcacov.
Both do essentially the same thing, but from different starting points, because of the above equivalences between SVD and EVD.
Strictly speaking, u1, v1, u2 and v2 above are not eigenvectors, they are singular vectors - and s1 and s2 are singular values. They are singular vectors/values of the matrix a. e1 and d1 are the eigenvectors and eigenvalues of a*a' (not a), and e2 and d2 are the eigenvectors and eigenvalues of a'*a (not a). a does not have any eigenvectors - only square matrices have eigenvectors.
Centring by subtracting the mean is a separate issue - you would typically do that prior to PCA, but there are situations where you wouldn't want to. You might also want to normalise by dividing by the standard deviation but again, you wouldn't always want to - it depends what the data represents and what question you're trying to answer.
My aim is to classify types of cars (Sedans,SUV,Hatchbacks) and earlier I was using corner features for classification but it didn't work out very well so now I am trying Gabor features.
code from here
Now the features are extracted and suppose when I give an image as input then for 5 scales and 8 orientations I get 2 [1x40] matrices.
1. 40 columns of squared Energy.
2. 40 colums of mean Amplitude.
Problem is I want to use these two matrices for classification and I have about 230 images of 3 classes (SUV,sedan,hatchback).
I do not know how to create a [N x 230] matrix which can be taken as vInputs by the neural netowrk in matlab.(where N be the total features of one image).
My question:
How to create a one dimensional image vector from the 2 [1x40] matrices for one image.(should I append the mean Amplitude to square energy matrix to get a [1x80] matrix or something else?)
Should I be using these gabor features for my purpose of classification in first place? if not then what?
Thanks in advance
In general, there is nothing to think about - simple neural network requires one dimensional feature vector and does not care about the ordering, so you can simply concatenate any number of feature vectors into one (and even do it in random order - it does not matter). In particular if you have same feature matrices you also concatenate each of its row to create a vectorized format.
The only exception is when your data actually has some underlying geometrical dependicies, for example - matrix is actualy a pixels matrix. In such case architectures like PyraNet, Convolutional Neural Networks and others, which apply some kind of receptive fields based on this 2d structure - should be better. But those implementations simply accept 2d feature vector as an input.