How to fill NaN values in numeric array to apply SVD?

问题

I am combined two data-frames that have some common columns, however there are some different columns. I would like to apply Singular Value Decomposition (SVD) on the combined data-frame. However, filling NaN values will affect the results, even filling the data with zeros will be wrong in my case since there are some columns have zero values. Here's an example. Is there any ways to address this issue ?.

>>> df1 = pd.DataFrame(np.random.rand(6, 4), columns=['A', 'B', 'C', 'D'])
>>> df1
          A         B         C         D
0  0.763144  0.752176  0.601228  0.290276
1  0.632144  0.202513  0.111766  0.317838
2  0.494587  0.318276  0.951354  0.051253
3  0.184826  0.429469  0.280297  0.014895
4  0.236955  0.560095  0.357246  0.302688
5  0.729145  0.293810  0.525223  0.744513
>>> df2 = pd.DataFrame(np.random.rand(6, 4), columns=['A', 'B', 'C', 'E'])
>>> df2
          A         B         C         E
0  0.969758  0.650887  0.821926  0.884600
1  0.657851  0.158992  0.731678  0.841507
2  0.923716  0.524547  0.783581  0.268123
3  0.935014  0.219135  0.152794  0.433324
4  0.327104  0.581433  0.474131  0.521481
5  0.366469  0.709115  0.462106  0.416601
>>> df3 = pd.concat([df1,df2], axis=0)
>>> df3
          A         B         C         D         E
0  0.763144  0.752176  0.601228  0.290276       NaN
1  0.632144  0.202513  0.111766  0.317838       NaN
2  0.494587  0.318276  0.951354  0.051253       NaN
3  0.184826  0.429469  0.280297  0.014895       NaN
4  0.236955  0.560095  0.357246  0.302688       NaN
5  0.729145  0.293810  0.525223  0.744513       NaN
0  0.969758  0.650887  0.821926       NaN  0.884600
1  0.657851  0.158992  0.731678       NaN  0.841507
2  0.923716  0.524547  0.783581       NaN  0.268123
3  0.935014  0.219135  0.152794       NaN  0.433324
4  0.327104  0.581433  0.474131       NaN  0.521481
5  0.366469  0.709115  0.462106       NaN  0.416601
>>> U, s, V = np.linalg.svd(df3.values, full_matrices=True)

Traceback (most recent call last):
  File "<input>", line 1, in <module>
  File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/site-packages/numpy-1.11.0b3-py3.4-macosx-10.6-intel.egg/numpy/linalg/linalg.py", line 1359, in svd
    u, s, vt = gufunc(a, signature=signature, extobj=extobj)
  File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/site-packages/numpy-1.11.0b3-py3.4-macosx-10.6-intel.egg/numpy/linalg/linalg.py", line 99, in _raise_linalgerror_svd_nonconvergence
    raise LinAlgError("SVD did not converge")
numpy.linalg.linalg.LinAlgError: SVD did not converge

Note: I can't apply interpolation because i want to preserve that some records don't have some columns information, but other records have

回答1:

It's possible to approximate the SVD of a matrix with missing values using an iterative procedure:

Fill in the missing values with a rough approximation (e.g. replace them with the column means)
Perform SVD on the filled-in matrix
Reconstruct the data matrix from the SVD in order to get a better approximation of the missing values
Repeat steps 2-3 until convergence

This is a form of expectation maximization (EM) algorithm, where the E step updates the estimates of the missing values from the SVD, and the M step computes the SVD on the updated estimate of the data matrix (see Section 1.3 here for more details).

import numpy as np
from scipy.sparse.linalg import svds
from functools import partial


def emsvd(Y, k=None, tol=1E-3, maxiter=None):
    """
    Approximate SVD on data with missing values via expectation-maximization

    Inputs:
    -----------
    Y:          (nobs, ndim) data matrix, missing values denoted by NaN/Inf
    k:          number of singular values/vectors to find (default: k=ndim)
    tol:        convergence tolerance on change in trace norm
    maxiter:    maximum number of EM steps to perform (default: no limit)

    Returns:
    -----------
    Y_hat:      (nobs, ndim) reconstructed data matrix
    mu_hat:     (ndim,) estimated column means for reconstructed data
    U, s, Vt:   singular values and vectors (see np.linalg.svd and 
                scipy.sparse.linalg.svds for details)
    """

    if k is None:
        svdmethod = partial(np.linalg.svd, full_matrices=False)
    else:
        svdmethod = partial(svds, k=k)
    if maxiter is None:
        maxiter = np.inf

    # initialize the missing values to their respective column means
    mu_hat = np.nanmean(Y, axis=0, keepdims=1)
    valid = np.isfinite(Y)
    Y_hat = np.where(valid, Y, mu_hat)

    halt = False
    ii = 1
    v_prev = 0

    while not halt:

        # SVD on filled-in data
        U, s, Vt = svdmethod(Y_hat - mu_hat)

        # impute missing values
        Y_hat[~valid] = (U.dot(np.diag(s)).dot(Vt) + mu_hat)[~valid]

        # update bias parameter
        mu_hat = Y_hat.mean(axis=0, keepdims=1)

        # test convergence using relative change in trace norm
        v = s.sum()
        if ii >= maxiter or ((v - v_prev) / v_prev) < tol:
            halt = True
        ii += 1
        v_prev = v

    return Y_hat, mu_hat, U, s, Vt

来源：https://stackoverflow.com/questions/35577553/how-to-fill-nan-values-in-numeric-array-to-apply-svd

标签

python

python-3.x

numpy

svd