flopscope.numpy.cov

Covariance indicates the level to which two variables vary together. If we examine N-dimensional samples, $X = [x_1, x_2, ... x_N]^T$, then the covariance matrix element $C_{ij}$ is the covariance of $x_i$ and $x_j$. The element $C_{ii}$ is the variance of $x_i$.

See the notes for an outline of the algorithm.

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: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.

fweights:array_like, int, optional

1-D array of integer frequency weights; the number of times each observation vector should be repeated.

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.

dtype:data-type, optional

Data-type of the result. By default, the return data-type will have at least flops.float64 precision.

Added in version 1.20.

out:ndarray

The covariance matrix of the variables.

• we.flops.corrcoef Normalized covariance matrix

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:

>>> m = flops.arange(10, dtype=flops.float64)
>>> f = flops.arange(10) * 2
>>> a = flops.arange(10) ** 2.
>>> ddof = 1
>>> w = f * a
>>> v1 = flops.sum(w)
>>> v2 = flops.sum(w * a)
>>> m -= flops.sum(m * w, axis=None, keepdims=True) / v1
>>> cov = flops.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 / (flops.sum(f) - ddof) as it should.

Consider two variables, $x_0$ and $x_1$, which correlate perfectly, but in opposite directions:

Note how $x_0$ increases while $x_1$ decreases. The covariance matrix shows this clearly:

Note that element $C_{0,1}$, which shows the correlation between $x_0$ and $x_1$, is negative.

Further, note how x and y are combined: