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想了解numpy 协方差矩阵 numpy.cov的新动态吗？本文将为您提供详细的信息，我们还将为您解答关于numpy 协方差矩阵的相关问题，此外，我们还将为您介绍关于Anaconda Numpy 错误“Importing the Numpy C Extension Failed”是否有另一种解决方案、cvxpy 和 numpy 之间的版本冲突：“针对 API 版本 0xe 编译的模块，但此版本的 numpy 是 0xd”、Jupyter 中的 Numpy 在打印时出错（Python 版本 3.8.8）：TypeError: 'numpy.ndarray' object is not callable、ML基础：协方差矩阵！的新知识。

• numpy 协方差矩阵 numpy.cov（numpy 协方差矩阵）
• Anaconda Numpy 错误“Importing the Numpy C Extension Failed”是否有另一种解决方案
• cvxpy 和 numpy 之间的版本冲突：“针对 API 版本 0xe 编译的模块，但此版本的 numpy 是 0xd”
• Jupyter 中的 Numpy 在打印时出错（Python 版本 3.8.8）：TypeError: 'numpy.ndarray' object is not callable
• ML基础：协方差矩阵！

numpy 协方差矩阵 numpy.cov（numpy 协方差矩阵）

numpy.cov(m, y=None, rowvar=True, bias=False, ddof=None, fweights=None, aweights=None)[source]

Estimate a covariance matrix, given data and weights.

Covariance indicates the level to which two variables vary together. If we examine N-dimensional samples, , then the covariance matrix element  is the covariance of  and . The element  is the variance of .

See the notes for an outline of the algorithm.

Parameters:

m : array_like A 1-D or 2-D array containing multiple variables and observations. Each row （行） of m represents a variable（变量）, and each column（列） a single observation of all those variables（样本）. Also see rowvar below. y : array_like, optional An additional set of variables and observations. y has the same form as that of m. rowvar : bool, optional If rowvar is True (default), then each row represents a variable, with observations in the columns. Otherwise, the relationship is transposed: each column represents a variable, while the rows contain observations. bias : bool, optional Default normalization (False) is by (N - 1), where N is the number of observations given (unbiased estimate). If bias is True, then normalization is by N. These values can be overridden by using the keyword ddof in numpy versions >= 1.5. ddof : int, optional If not None the default value implied by bias is overridden. Note that ddof=1 will return the unbiased estimate, even if both fweights and aweights are specified, and ddof=0 will return the simple average. See the notes for the details. The default value is None. New in version 1.5. fweights : array_like, int, optional 1-D array of integer freguency weights; the number of times each observation vector should be repeated. New in version 1.10. aweights : array_like, optional 1-D array of observation vector weights. These relative weights are typically large for observations considered “important” and smaller for observations considered less “important”. If ddof=0 the array of weights can be used to assign probabilities to observation vectors. New in version 1.10.

Returns:

out : ndarray The covariance matrix of the variables.

See also

corrcoef

Normalized covariance matrix

Notes

Assume that the observations are in the columns of the observation array m and let f = fweights and a = aweights for brevity. The steps to compute the weighted covariance are as follows:

>>> w = f * a >>> v1 = np.sum(w) >>> v2 = np.sum(w * a) >>> m -= np.sum(m * w, axis=1, keepdims=True) / v1 >>> cov = np.dot(m * w, m.T) * v1 / (v1**2 - ddof * v2) 

Note that when a == 1, the normalization factor v1 / (v1**2 - ddof * v2) goes over to 1 / (np.sum(f) - ddof) as it should.

Examples

Consider two variables,  and , which correlate perfectly, but in opposite directions:

>>> x = np.array([[0, 2], [1, 1], [2, 0]]).T >>> x array([[0, 1, 2],  [2, 1, 0]]) 

Note how  increases while  decreases. The covariance matrix shows this clearly:

>>> np.cov(x) array([[ 1., -1.],  [-1., 1.]]) 

Note that element , which shows the correlation between  and , is negative.

Further, note how x and y are combined:

>>> x = [-2.1, -1, 4.3] >>> y = [3, 1.1, 0.12] >>> X = np.stack((x, y), axis=0) >>> print(np.cov(X)) [[ 11.71 -4.286 ]  [ -4.286 2.14413333]] >>> print(np.cov(x, y)) [[ 11.71 -4.286 ]  [ -4.286 2.14413333]] >>> print(np.cov(x)) 11.71

总结

理解协方差矩阵的关键就在于牢记它的计算是不同维度之间的协方差，而不是不同样本之间。拿到一个样本矩阵，最先要明确的就是一行是一个样本还是一个维度，心中明确整个计算过程就会顺流而下，这么一来就不会迷茫了。

Anaconda Numpy 错误“Importing the Numpy C Extension Failed”是否有另一种解决方案

  \Traceback (most recent call last):
File "C:\Conda\envs\gcp\lib\site-packages\numpy\core\__init__.py",line 22,in <module>
from . import multiarray
File "C:\Conda\envs\gcp\lib\site-packages\numpy\core\multiarray.py",line 12,in <module>
from . import overrides
File "C:\Conda\envs\gcp\lib\site-packages\numpy\core\overrides.py",line 7,in <module>
from numpy.core._multiarray_umath import (
ImportError: DLL load Failed while importing _multiarray_umath: The specified module Could not be found.

During handling of the above exception,another exception occurred:

Traceback (most recent call last):
File "c:\API\citi-bike.py",line 4,in <module>
import numpy as np
File "C:\Conda\envs\gcp\lib\site-packages\numpy\__init__.py",line 150,in <module>
from . import core
File "C:\Conda\envs\gcp\lib\site-packages\numpy\core\__init__.py",line 48,in <module>
raise ImportError(msg)
ImportError:

IMPORTANT: PLEASE READ THIS FOR ADVICE ON HOW TO SOLVE THIS ISSUE!

Importing the numpy C-extensions Failed. This error can happen for
many reasons,often due to issues with your setup or how NumPy was
installed.

We have compiled some common reasons and troubleshooting tips at:

https://numpy.org/devdocs/user/troubleshooting-importerror.html

Please note and check the following:

* The Python version is: python3.9 from "C:\Conda\envs\gcp\python.exe"
* The NumPy version is: "1.21.1"

and make sure that they are the versions you expect.
Please carefully study the documentation linked above for further help.

Original error was: DLL load Failed while importing _multiarray_umath: The specified module Could not be found.

RuntimeError: module compiled against API version 0xe but this version of numpy is 0xd
Traceback (most recent call last):
  File "test.py",line 1,in <module>
    import cvxpy
  File "c:\Projekte\algo5\venv\lib\site-packages\cvxpy\__init__.py",line 18,in <module>
    from cvxpy.atoms import *
  File "c:\Projekte\algo5\venv\lib\site-packages\cvxpy\atoms\__init__.py",line 20,in <module>
    from cvxpy.atoms.geo_mean import geo_mean
  File "c:\Projekte\algo5\venv\lib\site-packages\cvxpy\atoms\geo_mean.py",in <module>
    from cvxpy.utilities.power_tools import (fracify,decompose,approx_error,lower_bound,File "c:\Projekte\algo5\venv\lib\site-packages\cvxpy\utilities\power_tools.py",in <module>
    from cvxpy.atoms.affine.reshape import reshape
  File "c:\Projekte\algo5\venv\lib\site-packages\cvxpy\atoms\affine\reshape.py",in <module>
    from cvxpy.atoms.affine.hstack import hstack
  File "c:\Projekte\algo5\venv\lib\site-packages\cvxpy\atoms\affine\hstack.py",in <module>
    from cvxpy.atoms.affine.affine_atom import AffAtom
  File "c:\Projekte\algo5\venv\lib\site-packages\cvxpy\atoms\affine\affine_atom.py",line 22,in <module>
    from cvxpy.cvxcore.python import canonInterface
  File "c:\Projekte\algo5\venv\lib\site-packages\cvxpy\cvxcore\python\__init__.py",line 3,in <module>
    import _cvxcore
ImportError: numpy.core.multiarray Failed to import

cvxpy 
numpy~=1.19.2 # tensorflow 2.4.1 requires this version

import cvxpy

if __name__ == ''__main__'':
    pass

import numpy as np
    for i in range (5):
        n = np.random.rand ()
    print (n)

Results
0.6604903457995978
0.8236300859753154
0.16067650689842816
0.6967868357083673
0.4231597934445466

import numpy as np
    for i in range (5):
        n = np.random.rand ()
    print (n)

-------------------------------------------------- ------
TypeError Traceback (most recent call last)
<ipython-input-78-0c6a801b3ea9> in <module>
       2 for i in range (5):
       3 n = np.random.rand ()
---->  4 print (n)

       TypeError: ''numpy.ndarray'' object is not callable