If PCA also helps to normalize the data, how a normalized data is going to be improved by PCA. Thanks
PCA does more than normalize the data (in fact, it only normalizes if whiten=True). It also projects into a different space using the n_components eigenvectors with maximal variation (largest eignevalues) - this can provide better performance for your classifier/clustering if your data is "stretched" along a particular dimension. See this example for more information.
Related
Is it valid to run a PCA on data that is comprised of proportions? For example, I have data on the proportion of various food items in the diet of different species. Can I run a PCA on this type of data or should I transform the data or do something else beforehand?
I had a similar question. You should search for "compositional data analysis". There are transformation to apply to proportions in order to analyze them with multivariate tecniques such as PCA. You can find also "robust" PCA algorithms to run your analysis in R. Let us know if you find an appropriate solution to your specific problem.
I don't think so.
PCA will give you "impossible" answers. You might get principal components with values that proportions can't have, like negative values or values greater than 1. How would you interpret this component?
In technical terms, the support of your data is a subset of the support of PCA. Say you have $k$ classes. Then:
the support for PCA vectors is $\R^k$
the support for your proportion vectors is the $k$- dimensional simplex. By simplex I mean the set of $p$ vectors of length $k$ such that:
$0 \le p_i \le 1$ where $i = 1, ..., k$
$\sum_{i=1}^k{p_i} = 1$
One way around this is if there's a one to one mapping between the $k$-simplex to all of $\R^k$. If so, you could map from your proportions to $\R^k$, do PCA there, then map the PCA vectors to the simplex.
But I'm not sure the simplex is a self-contained linear space. If you add two elements of the simplex, you don't get an element of the simplex :/
A better approach, I think, is clustering, eg with Gaussian mixtures, or spectral clustering. This is related to PCA. But a nice property of clustering is you can express any element of your data as a "convex combination" of the clusters. If you analyze your proportion data and find clusters, they (unlike PCA vectors) will be within the simplex space, and any mixture of them will be, too.
I also recommend looking into nonnegative matrix factorization. This is like PCA but, as the name suggests, avoids negative components and also negative eigenvectors. It's very useful for inferring structure in strictly positive data, like proportions. But nmf does not give you a basis for simplex space.
In CNN, the filters are usually set as 3x3, 5x5 spatially. Can the sizes be comparable to the image size? One reason is for reducing the number of parameters to be learnt. Apart from this, is there any other key reasons? for example, people want to detect edges first?
You answer a point of the question. Another reason is that most of these useful features may be found in more than one place in an image. So, it makes sense to slide a single kernel all over the image in the hope of extracting that feature in different parts of the image using the same kernel. If you are using big kernel, the features could be interleaved and not concretely detected.
In addition to yourself answer, reduction in computational costs is a key point. Since we use the same kernel for different set of pixels in an image, the same weights are shared across these pixel sets as we convolve on them. And as the number of weights are less than a fully connected layer, we have lesser weights to back-propagate on.
I am using t-SNE for exploratory data analysis. I am using this instead of PCA because PCA is linear and t-SNE is non-linear.
It's really straight-forward to know how many dimensions are required to capture the necessary variance with PCA.
How do I know how many dimensions are required for my data using t-SNE?
I have read a popular website of very useful information, but it doesn't discuss dimensionality.
https://distill.pub/2016/misread-tsne/
I am wondering if PCA widget was performing centred and/or normalized PCA. I did'nt find any corresponding option in the widget.
Does anyone know the answer and if there is plan to add these options?
Thanks alot. Best regards,
Yes, the 'PCA' widget both centers and standardizes the data.
Usually, PCA is defined to be on standardized data.
The use of the term PCA for performing SVD on other matrixes (such as non-central correlation matrixes) is much less common - because there is a proper term for that, too: SVD.
Furthermore, using it on non-centered data means that the transformation is sensitive to data translation; an aspect one usually does not want.
So unless explicitly specified, you should assume that PCA is implemented by
Centering the data
Computing the covariance matrix
Applying SVD on the covariance matrix to decompose it as desired
I have 2D data (I have a zero mean normalized data). I know the covariance matrix, eigenvalues and eigenvectors of it. I want to decide whether to reduce the dimension to 1 or not (I use principal component analysis, PCA). How can I decide? Is there any methodology for it?
I am looking sth. like if you look at this ratio and if this ratio is high than it is logical to go on with dimensionality reduction.
PS 1: Does PoV (Proportion of variation) stands for it?
PS 2: Here is an answer: https://stats.stackexchange.com/questions/22569/pca-and-proportion-of-variance-explained does it a criteria to test it?
PoV (Proportion of variation) represents how much information of data will remain relatively to using all of them. It may be used for that purpose. If POV is high than less information will be lose.
You want to sort your eigenvalues by magnitude then pick the highest 1 or 2 values. Eigenvalues with a very small relative value can be considered for exclusion. You can then translate data values and using only the top 1 or 2 eigenvectors you'll get dimensions for plotting results. This will give a visual representation of the PCA split. Also check out scikit-learn for more on PCA. Precisions, recalls, F1-scores will tell you how well it works
from http://sebastianraschka.com/Articles/2014_pca_step_by_step.html...
Step 1: 3D Example
"For our simple example, where we are reducing a 3-dimensional feature space to a 2-dimensional feature subspace, we are combining the two eigenvectors with the highest eigenvalues to construct our d×kd×k-dimensional eigenvector matrix WW.
matrix_w = np.hstack((eig_pairs[0][1].reshape(3,1),
eig_pairs[1][1].reshape(3,1)))
print('Matrix W:\n', matrix_w)
>>>Matrix W:
[[-0.49210223 -0.64670286]
[-0.47927902 -0.35756937]
[-0.72672348 0.67373552]]"
Step 2: 3D Example
"
In the last step, we use the 2×32×3-dimensional matrix WW that we just computed to transform our samples onto the new subspace via the equation
y=W^T×x
transformed = matrix_w.T.dot(all_samples)
assert transformed.shape == (2,40), "The matrix is not 2x40 dimensional."