I'm wondering if someone is familiar with the NLMEFITSA algorithm in matlab. This algorithm gives me as result the fixed effects parameters (beta) for a mixed effects model as well as their covariance matrix (psi), but it also gives me, in the "stats" struct, one covariance matrix called "covb" I know this one has to do with the standard errors and it is important for calculating confidence intervals, but to be honest I don't know what is exactly the difference between this "covb" and the "psi " covariance matrix. And how could I use "covb" when simulating new data?
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I have computed colour descriptors of a dataset of images and generated a 152×320 matrix (152 samples and 320 features). I would like to use PCA to reduce the dimensionality of my image descriptors space. I know that I could implement this using Matlab PCA built-in function but as I have just started learning about this concept I would like to implement the Matlab code without the built-in function so I can have a clear understanding how the function works. I tried to find how to do that online but all I could find is the either the general concept of PCA or the implementation of it with the built-in functions without explaining clearly how it works. Anyone could help me with a step by step instructions or a link that could explain a simple way on how to implement PCA for dimensionality reduction. The reason why I'm so confused is because there are so many uses for PCA and methods to implement it and the more I read about it the more confused I get.
PCA is basically taking the dominant eigen vectors of the data (Or better yet their projection of the dominant Eigen Vectors of the covariance matrix).
What you can do is use the SVD (Singular Value Decomposition).
To imitate MATLAB's pca() function here what you should do:
Center all features (Each column of your data should have zero mean).
Apply the svd() function on your data.
Use the V Matrix (Its columns) as your vectors to project your data on. Chose the number of columns to use according to the dimension of the data you'd like to have.
The projected data is now you new dimensionality reduction data.
I know a Gaussian Process Regression model is mainly specified by its covariance matrix and the free hyper-parameters act as the 'weights'of the model. But could anyone explain what do the 2 hyper-parameters (length-scale & amplitude) in the covariance matrix represent (since they are not 'real' parameters)? I'm a little confused on the 'actual' meaning of these 2 parameters.
Thank you for your help in advance. :)
First off I would like to point out that there are infinite number of kernels that could be used in a gaussian process. One of the most common however is the RBF (also referred to as squared exponential, the expodentiated quadratic, etc). This kernel is of the following form:
The above equation is of course for the simple 1D case. Here l is the length scale and sigma is the variance parameter (note they go under different names depending on the source). Effectively the length scale controls how two points appear to be similar as it simply magnifies the distance between x and x'. The variance parameter controls how smooth the function is. These are related but not the same.
The Kernel Cookbook give a nice little description and compares RBF kernels to other commonly used kernels.
I am using PCA for face recognition. I have obtained the eigenvectors / eigenfaces for each image, which is a colomn matrix. I want to know if selecting the first three eigenvectors , since their corresponding eigenvalues amount to 70% of total variance, will be sufficient for face recognition?
Firstly, lets be clear about a few things. The eigenvectors are computed from the covariance matrix formed from the entire dataset i.e., you reshape each grayscale image of a face into a single column and treat it as a point in R^d space, compute the covariance matrix from them and compute the eigenvectors of the covariance matrix. These eigenvectors become a new basis for your space of face images. You do not have eigenvectors for each image. Instead, you represent each face image in terms of the eigenvectors by projecting onto (a possibly subset) of them.
Limitations of eigenfaces
As to whether the representation of your face images under this new basis good enough for face recognition depends on many factors. But in general, the eigenfaces method does not perform well for real world unconstrained faces. It only works for faces which are pixel-wise aligned, facing frontal, and has fairly uniform illumination conditions across the images.
More is not necessarily better
While it is commonly believed (when using PCA) that retaining more variance is better than less, things are much more complicated than that because of two factors: 1) Noise in real world data and 2) dimensionality of data. Sometimes projecting to a lower dimension and losing variance can actually produce better results.
Conclusion
Hence, my answer is it is difficult to say whether retaining a certain amount of variance is enough beforehand. The number of dimensions (and hence the number of eigenvectors to keep and the associated variance retained) should be determined by cross-validation. But ultimately, as I have mentioned above, eigenfaces is not a good method for face recognition unless you have a "nice" dataset. You might be slightly better off using "Fisherfaces" i.e., LDA on the face images or combine these methods with Local Binary Pattern (LBP) as features (instead of raw face pixels). But seriously, face recognition is a difficult problem and in general the state-of-the-art has not reached a stage where it can be deployed in real world systems.
It's not impossible, but a little rare to me that only 3 eigenvalues can achieve 70% variance. How many training samples do you have (what is the total dimension)? Make sure you are reshape each image from the database into a vector, normalize the vector data then align them into a matrix. The eigenvalues/eigenvectors are obtained from the covariance of the matrix.
In theory, 70% variance should be enough to form a human-recognizable face with the corresponding eigenvectors. However, the optimal number of eigenvalues is better to get from cross-validation: you can try to increase 1 eigenvector each time, observe the face formation and the recognition accuracy. You can even plot the cross validation accuracy curve, there may be a sharp corner on the curve, then the corresponding eigenvector number is hopefully applied in your test.
I understand the concept of PCA, and what it's doing, but trying to apply the concept to my application is proving difficult.
I have a 1 by X matrix of a physiological signal (it's not EMG, but very similar, so think of it as EMG if it helps) which contains various noise and artefacts. What I've noticed of the noise is that some of it is very large and I would assume after PCA this would be the largest principal component, thus my idea of using PCA for some dimensional reduction.
My problem is that with a 1 by X matrix there is no covariance matrix, only the variance, and thus eigenvectors and all of PCA falls through.
I know I need to rearrange my data into a matrix more than 1D, but this is where I need some suggestions. Do I split my data into windows of equal length to create a large dimensional matrix which I can apply PCA to? Do I perform several trials of the same action so I have lots of data sets (this would be impractical for my application)?
Any suggestions or examples would be helpful. I'm using MATLAB to perform this task.
I have a problem where I am fitting a high-order polynomial to (not very) noisy data using linear least squares. Currently I'm using polynomial orders around 15 - 25, which work surprisingly well: The dependence is very nearly linear, but the accuracy of modelling the 'very nearly' is critical. I'm using Matlab's polyfit() function, and (obviously) normalising the x-data. This generally works fine, but I have come across an issue with some recent datasets. The fitted polynomial has extrema within the x-data interval. For the application I'm working on this is a non-no. The polynomial model must have no stationary points over the x-interval.
So I need to add a constraint to the least-squares problem: the derivative of the fitted polynomial must be strictly positive over a known x-range (or strictly negative - this depends on the data but a simple linear fit will quickly tell me which it is.) I have had a quick look at the available optimisation toolbox functions, but I admit I'm at a loss to know how to go about this. Does anyone have any suggestions?
[I appreciate there are probably better models than polynomials for this data, but in the short term it isn't feasible to change the form of the model]
[A closing note: I have finally got the go-ahead to replace this awful polynomial model! I am going to adopt a nonparametric approach, spline smoothing, using the excellent SPLINEFIT code by Jonas Lundgren. This has the advantage that I'm already using a spline model in the end-user application, so I already have C# code available to evaluate a spline model]
You could use cftool and use the exclude data points option.