from __future__ import absolute_import
from ...extern.six.moves import range
from ...extern.six import string_types
__all__ = [
'plot_corrcoef_matrix',
'corrcoef',
'cov',
]
[docs]def plot_corrcoef_matrix(matrix, names=None,
cmap=None, cmap_text=None,
fontsize=12, grid=False,
axes=None):
"""
This function will draw a lower-triangular correlation matrix
Parameters
----------
matrix : 2-dimensional numpy array/matrix
A correlation coefficient matrix
names : list of strings, optional (default=None)
List of the parameter names corresponding to the rows in ``matrix``.
cmap : matplotlib color map, optional (default=None)
Color map used to color the matrix cells.
cmap_text : matplotlib color map, optional (default=None)
Color map used to color the cell value text. If None, then
all values will be black.
fontsize : int, optional (default=12)
Font size of parameter name and correlation value text.
grid : bool, optional (default=False)
If True, then draw dashed grid lines around the matrix elements.
axes : matplotlib Axes instance, optional (default=None)
The axes to plot on. If None then use the global current axes.
Notes
-----
NumPy and matplotlib are required
Examples
--------
>>> matrix = corrcoef(data.T, weights=weights)
>>> plot_corrcoef_matrix(matrix, names)
"""
import numpy as np
from matplotlib import pyplot as plt
from matplotlib import cm
if axes is None:
axes = plt.gca()
matrix = np.asarray(matrix)
if matrix.ndim != 2:
raise ValueError("matrix is not a 2-dimensional array or matrix")
if matrix.shape[0] != matrix.shape[1]:
raise ValueError("matrix is not square")
if names is not None and len(names) != matrix.shape[0]:
raise ValueError("the number of names does not match the number of "
"rows/columns in the matrix")
# mask out the upper triangular matrix
matrix[np.triu_indices(matrix.shape[0])] = np.nan
if isinstance(cmap_text, string_types):
cmap_text = cm.get_cmap(cmap_text, 201)
if cmap is None:
cmap = cm.get_cmap('jet', 201)
elif isinstance(cmap, string_types):
cmap = cm.get_cmap(cmap, 201)
# make NaN pixels white
cmap.set_bad('w')
axes.imshow(matrix, interpolation='nearest',
cmap=cmap, origin='upper',
vmin=-1, vmax=1)
axes.set_frame_on(False)
plt.setp(axes.get_yticklabels(), visible=False)
plt.setp(axes.get_yticklines(), visible=False)
plt.setp(axes.get_xticklabels(), visible=False)
plt.setp(axes.get_xticklines(), visible=False)
if grid:
# draw grid lines
for slot in range(1, matrix.shape[0] - 1):
# vertical
axes.plot((slot - 0.5, slot - 0.5),
(slot - 0.5, matrix.shape[0] - 0.5), 'k:', linewidth=1)
# horizontal
axes.plot((-0.5, slot + 0.5),
(slot + 0.5, slot + 0.5), 'k:', linewidth=1)
if names is not None:
for slot in range(1, matrix.shape[0]):
# diagonal
axes.plot((slot - 0.5, slot + 1.5),
(slot - 0.5, slot - 2.5), 'k:', linewidth=1)
# label cell values
for row, col in zip(*np.tril_indices(matrix.shape[0], k=-1)):
value = matrix[row][col]
if cmap_text is not None:
color = cmap_text((value + 1.) / 2.)
else:
color = 'black'
axes.text(
col, row,
"{0:d}%".format(int(value * 100)),
color=color,
ha='center', va='center',
fontsize=fontsize)
if names is not None:
# write parameter names
for i, name in enumerate(names):
axes.annotate(
name, (i, i),
rotation=45,
ha='left', va='bottom',
transform=axes.transData,
fontsize=fontsize)
def cov(m, y=None, rowvar=1, bias=0, ddof=None, weights=None, repeat_weights=0):
"""
Estimate a covariance matrix, given data.
Covariance indicates the level to which two variables vary together.
If we examine N-dimensional samples, :math:`X = [x_1, x_2, ... x_N]^T`,
then the covariance matrix element :math:`C_{ij}` is the covariance of
:math:`x_i` and :math:`x_j`. The element :math:`C_{ii}` is the variance
of :math:`x_i`.
Parameters
----------
m : array_like
A 1-D or 2-D array containing multiple variables and observations.
Each row of `m` represents a variable, and each column a single
observation of all those variables. 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 : int, optional
If `rowvar` is non-zero (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 : int, optional
Default normalization is by ``(N - 1)``, where ``N`` is the number of
observations given (unbiased estimate). If `bias` is 1, then
normalization is by ``N``. These values can be overridden by using
the keyword ``ddof`` in numpy versions >= 1.5.
ddof : int, optional
.. versionadded:: 1.5
If not ``None`` normalization is by ``(N - ddof)``, where ``N`` is
the number of observations; this overrides the value implied by
``bias``. The default value is ``None``.
weights : array-like, optional
A 1-D array of weights with a length equal to the number of
observations.
repeat_weights : int, optional
The default treatment of weights in the weighted covariance is to first
normalize them to unit sum and use the biased weighted covariance
equation. If `repeat_weights` is 1 then the weights must represent an
integer number of occurrences of each observation and both a biased and
unbiased weighted covariance is defined because the total sample size
can be determined.
Returns
-------
out : ndarray
The covariance matrix of the variables.
See Also
--------
corrcoef : Normalized covariance matrix
Examples
--------
Consider two variables, :math:`x_0` and :math:`x_1`, 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 :math:`x_0` increases while :math:`x_1` decreases. The covariance
matrix shows this clearly:
>>> np.cov(x)
array([[ 1., -1.],
[-1., 1.]])
Note that element :math:`C_{0,1}`, which shows the correlation between
:math:`x_0` and :math:`x_1`, is negative.
Further, note how `x` and `y` are combined:
>>> x = [-2.1, -1, 4.3]
>>> y = [3, 1.1, 0.12]
>>> X = np.vstack((x,y))
>>> 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
"""
import numpy as np
# Check inputs
if ddof is not None and ddof != int(ddof):
raise ValueError(
"ddof must be integer")
X = np.array(m, ndmin=2, dtype=float)
if X.size == 0:
# handle empty arrays
return np.array(m)
if X.shape[0] == 1:
rowvar = 1
if rowvar:
axis = 0
tup = (slice(None), np.newaxis)
else:
axis = 1
tup = (np.newaxis, slice(None))
if y is not None:
y = np.array(y, copy=False, ndmin=2, dtype=float)
X = np.concatenate((X, y), axis)
if ddof is None:
if bias == 0:
ddof = 1
else:
ddof = 0
if weights is not None:
weights = np.array(weights, dtype=float)
weights_sum = weights.sum()
if weights_sum <= 0:
raise ValueError(
"sum of weights is non-positive")
X -= np.average(X, axis=1-axis, weights=weights)[tup]
if repeat_weights:
# each weight represents a number of repetitions of an observation
# the total sample size can be determined in this case and we have
# both an unbiased and biased weighted covariance
fact = weights_sum - ddof
else:
# normalize weights so they sum to unity
weights /= weights_sum
# unbiased weighted covariance is not defined if the weights are
# not integral frequencies (repeat-type)
fact = (1. - np.power(weights, 2).sum())
else:
weights = 1
X -= X.mean(axis=1-axis)[tup]
if rowvar:
N = X.shape[1]
else:
N = X.shape[0]
fact = float(N - ddof)
if not rowvar:
return (np.dot(weights * X.T, X.conj()) / fact).squeeze()
else:
return (np.dot(weights * X, X.T.conj()) / fact).squeeze()
def corrcoef(x, y=None, rowvar=1, bias=0, ddof=None, weights=None,
repeat_weights=0):
"""
Return correlation coefficients.
Please refer to the documentation for `cov` for more detail. The
relationship between the correlation coefficient matrix, `P`, and the
covariance matrix, `C`, is
.. math:: P_{ij} = \\frac{ C_{ij} } { \\sqrt{ C_{ii} * C_{jj} } }
The values of `P` are between -1 and 1, inclusive.
Parameters
----------
x : array_like
A 1-D or 2-D array containing multiple variables and observations.
Each row of `m` represents a variable, and each column a single
observation of all those variables. Also see `rowvar` below.
y : array_like, optional
An additional set of variables and observations. `y` has the same
shape as `m`.
rowvar : int, optional
If `rowvar` is non-zero (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 : int, optional
Default normalization is by ``(N - 1)``, where ``N`` is the number of
observations (unbiased estimate). If `bias` is 1, then
normalization is by ``N``. These values can be overridden by using
the keyword ``ddof`` in numpy versions >= 1.5.
ddof : {None, int}, optional
.. versionadded:: 1.5
If not ``None`` normalization is by ``(N - ddof)``, where ``N`` is
the number of observations; this overrides the value implied by
``bias``. The default value is ``None``.
weights : array-like, optional
A 1-D array of weights with a length equal to the number of
observations.
repeat_weights : int, optional
The default treatment of weights in the weighted covariance is to first
normalize them to unit sum and use the biased weighted covariance
equation. If `repeat_weights` is 1 then the weights must represent an
integer number of occurrences of each observation and both a biased and
unbiased weighted covariance is defined because the total sample size
can be determined.
Returns
-------
out : ndarray
The correlation coefficient matrix of the variables.
See Also
--------
cov : Covariance matrix
"""
import numpy as np
c = cov(x, y, rowvar, bias, ddof, weights, repeat_weights)
if c.size == 0:
# handle empty arrays
return c
try:
d = np.diag(c)
except ValueError: # scalar covariance
return 1
return c / np.sqrt(np.multiply.outer(d, d))
