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How to compare the predictive ability of PCA and NMF

I would like to compare the output of the algorithm with various pre-processed data: NMF and PCA. In order to get some comparable result, instead of choosing only the same number of components for each PCA and NMF, I would like to choose a sum that explains, for example, 95% of the stored variance.

I was wondering if it is possible to determine the variance stored in each component of NMF.

For example, using a PCA, this will be given: retainedVariance(i) = eigenvalue(i) / sum(eigenvalue)

Any ideas?

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TL DR

n_components explained_variance_score X. , , 95% .

, .

PCA NMF

NMF PCA, , , :

  • X H;
  • H X', X.

:

  • PCA NMF: X' = dot(H, W), W .
  • . PCA : H = dot(X, V), V . NMF H = argmin(loss(X, H, W)) ( H), loss - X dot(H, W), . , X.
  • . PCA : MSE , k - MSE, . NMF loss(X, H, W), , H, W.

/, :

  • + X_train
  • , X_train'=decode(encode(X_train)) X_train (, MAE, RMSE )
  • ( ) , 2 X_test.

PCA NMF!

from sklearn import decomposition, datasets, model_selection, preprocessing, metrics
# use the well-known Iris dataset
X, _ = datasets.load_iris(return_X_y=True)
# split the dataset, to measure overfitting
X_train, X_test = model_selection.train_test_split(X, test_size=0.5, random_state=1)
# I scale the data in order to give equal importance to all its dimensions
# NMF does not allow negative input, so I don't center the data
scaler = preprocessing.StandardScaler(with_mean=False).fit(X_train)
X_train_sc = scaler.transform(X_train)
X_test_sc = scaler.transform(X_test)
# train the both decomposers
pca = decomposition.PCA(n_components=2).fit(X_train_sc)
nmf = decomposition.NMF(n_components=2).fit(X_train_sc)
print(sum(pca.explained_variance_ratio_))

0.9536930834362043 - PCA, . - "" :

def get_score(model, data, scorer=metrics.explained_variance_score):
    """ Estimate performance of the model on the data """
    prediction = model.inverse_transform(model.transform(data))
    return scorer(data, prediction)

print('train set performance')
print(get_score(pca, X_train_sc))
print(get_score(nmf, X_train_sc))

print('test set performance')
print(get_score(pca, X_test_sc))
print(get_score(nmf, X_test_sc))


train set performance
0.9536930834362043 # same as before!
0.937291711378812 
test set performance
0.9597828443047842
0.9590555069007827

, PCA , NMF, . , NMF :

  • H W ( )
  • H ( L1 L2)
  • W ( L1 L2)

NMF , , , , .

PCA , h_1, h_2, ... h_k . h_(k+1), k . , , . PCA explained_variance_ratio_ .

NMF , , . , k+1 th, k , ( ).

NMF :

ks = [1,2,3,4]
perfs_train = []
perfs_test = []
for k in ks:
    nmf = decomposition.NMF(n_components=k).fit(X_train_sc)
    perfs_train.append(get_score(nmf, X_train_sc))
    perfs_test.append(get_score(nmf, X_test_sc))
print(perfs_train)
print(perfs_test)


[0.3236945680665101, 0.937291711378812, 0.995459457205891, 0.9974027602663655]
[0.26186701106012833, 0.9590555069007827, 0.9941424954209546, 0.9968456603914185]

, 95% ( ) ( ). , : , .

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