How to get the number of components needed in PCA with all extreme variance?

Question

I am trying to get the number of components needed to be used for classification. I have read a similar question Finding the dimension with highest variance using scikit-learn PCA and the scikit documents about this:

http://scikit-learn.org/dev/tutorial/statistical_inference/unsupervised_learning.html#principal-component-analysis-pca

However, this still did not solve my question. All of my PCA components are super big and of cause I could select all of them but if I do so PCA will be useless.

I also read the PCA library in scikit learn http://scikit-learn.org/stable/modules/generated/sklearn.decomposition.PCA.html It indicates thatL:

if n_components == ‘mle’, Minka’s MLE is used to guess the dimension if 0 < n_components < 1, select the number of components such that the amount of variance that needs to be explained is greater than the percentage specified by n_components

However I cannot find any more information about use this techniques for analysis n_components of PCA

Here is my code of PCA analysis:

from sklearn.decomposition import PCA
    pca = PCA()
    pca.fit(x_array_train)
    print(pca.explained_variance_)

result:

   [  6.58902714e+50   6.23266555e+49   2.93568652e+49   2.25418736e+49
       1.10063872e+49   3.25107359e+40   4.72113817e+39   1.40411862e+39
       4.03270198e+38   1.60662882e+38   3.20028861e+28   2.35570241e+27
       1.54944915e+27   8.05181151e+24   1.42231553e+24   5.05155955e+23
       2.90909468e+23   2.60339206e+23   1.95672973e+23   1.22987336e+23
       9.67133111e+22   7.07208772e+22   4.49067983e+22   3.57882593e+22
       3.03546737e+22   2.38077950e+22   2.18424235e+22   1.79048845e+22
       1.50871735e+22   1.35571453e+22   1.26605081e+22   1.04851395e+22
       8.88191944e+21   6.91581346e+21   5.43786989e+21   5.05544020e+21
       4.33110823e+21   3.18309135e+21   3.06169368e+21   2.66513522e+21
       2.57173046e+21   2.36482212e+21   2.32203521e+21   2.06033130e+21
       1.89039408e+21   1.51882514e+21   1.29284842e+21   1.26103770e+21
       1.22012185e+21   1.07857244e+21   8.55143095e+20   4.82321416e+20
       2.98301261e+20   2.31336276e+20   1.31712446e+20   1.05253795e+20
       9.84992112e+19   8.27574150e+19   4.66007620e+19   4.09687463e+19
       2.89855823e+19   2.79035170e+19   1.57015298e+19   1.39218538e+19
       1.00594159e+19   7.31960049e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.29043685e+18   5.29043685e+18   5.29043685e+18
       5.29043685e+18   5.24952686e+18   2.09685699e+18   4.16588190e+17]

I tried PCA(n_components = 'mle') however I got these errors ..

    Traceback (most recent call last):
  File "xx", line 166, in <module>
    pca.fit(x_array_train)
  File "xx", line 225, in fit
    self._fit(X)
  File "/Users/lib/python2.7/site-packages/sklearn/decomposition/pca.py", line 294, in _fit
    n_samples, n_features)
  File "/Users/lib/python2.7/site-packages/sklearn/decomposition/pca.py", line 98, in _infer_dimension_
    ll[rank] = _assess_dimension_(spectrum, rank, n_samples, n_features)
  File "/Users/lib/python2.7/site-packages/sklearn/decomposition/pca.py", line 83, in _assess_dimension_
    (1. / spectrum_[j] - 1. / spectrum_[i])) + log(n_samples)
ValueError: math domain error

Really appreciate for any helps...

Answer 1

I am not using Python, but I did something you need in C++ & opencv. Hope you succeed in converting it to whatever language.

// choose how many eigenvectors you want:
int nEigensOfInterest = 0;
float sum = 0.0;
for (int i = 0; i < mEiVal.rows; ++i)
{
    sum += mEiVal.at<float>(i, 0);
    if (((sum * 100) / (sumOfEigens)) > 80)
    {
        nEigensOfInterest = i;
        break;
    }
}
logfile << "No of Eigens of interest: " << nEigensOfInterest << std::endl << std::endl;

The basic idea is to decide "whatever %" components of you need to go ahead with. I chose those to be 80. mEiVal is column matrix of eigen values sorted in descending order. sumOfEigens is sum of all the eigen values.

I have no experience with scikit-learn, please let me know, I'll delete the answer.

Answer 2

There are several surveys available on the subject of finding the number of relevant eigenvalues in a pca. I like the broken-stick method and parallel analysis. Google them or have a look at the tutorial of this lecture.

Answer 3

I'm just learning this myself, but it seems to me that the reference to using 0 < n_components < 1 suggests that you could set n_components to, say, 0.85, and the exact number of components you need to explain 85% of the variance will be used. You can verify that the right number of components is selected by also printing sum(pca.explained_variance_). You should get the smallest sum of variance percentages over 0.85 (or whatever value you chose) that's possible for your data.

There are more sophisticated ways to choose a number of components, of course, but a rule of thumb of 70% - 90% is a reasonable start.

Answer 4

I assume your train array (x_array_train) is standardized like so:

from sklearn.preprocessing import StandardScaler
x_array_train = StandardScaler().fit_transform(x_array)

nevertheless my solution is following:

from sklearn.decomposition import PCA as pca
your_pca = pca(n_components = "mle", svd_solver ="full")
your_pca.fit_transform(x_array_train)
print(your_pca.explained_variance_)

This way you should get as few principal components as mle algorithm lets you to