I have two 3d column vectors B_mu
and B_nu
that vary as a function of time:
import numpy as np
N = 5 # 5 time-steps
B_mu = np.array(
[[5, 5, 8],
[4, 8, 7],
[2, 3, 1],
[5, 7, 8],
[6, 2, 7]]
)
B_nu = np.array(
[[3, 2, 9],
[9, 8, 8],
[4, 9, 9],
[4, 9, 6],
[1, 9, 1]]
)
For every index i
in the first vector, and every index j
in the second vector, I want to compute the difference between the time-average of the product < B_mu[i] B_nu[j] >
and the product of the time-averages <B_mu[i]> <B_nu[j]>.
In other words, I want to construct the matrix M
such that:
M[i,j] = 1/N sum(B_mu[i] * B_nu[j]) - 1/N**2 * sum(B_mu[i]) * sum(B_nu[j])
where the sums are taken over the time parameter.
Here is the equation:
And an explicit, expanded version:
How can I express this equation in python?
CodePudding user response:
Manipulation of matrices is relatively easy with module numpy. Here we're looking for:
- averaging (or summing) over one dimension (time);
- taking the outer product of two column vectors.
We're going to use:
- method
array.sum
with the optionalaxis
parameter to sum over the time dimension; - function
outer
.
These two functions combine straightforwardly to compute the product of averages. The average of the products, on the other hand, is straightforward to compute with python builtins sum
and map
; I don't know how to do it in pure numpy.
import numpy as np
def diff_avgprod_prodavg(B_mu, B_nu):
N = B_mu.shape[0]
avg_of_prod = 1/N * sum(map(np.outer, B_mu, B_nu)) # not pure numpy
prod_of_avg = 1/(N*N) * np.outer(B_mu.sum(axis=0), B_nu.sum(axis=0))
return avg_of_prod - prod_of_avg
Testing:
B_mu = np.array(
[[5, 5, 8],
[4, 8, 7],
[2, 3, 1],
[5, 7, 8],
[6, 2, 7]]
)
B_nu = np.array(
[[3, 2, 9],
[9, 8, 8],
[4, 9, 9],
[4, 9, 6],
[1, 9, 1]]
)
print( diff_avgprod_prodavg(B_mu, B_nu) )
# [[-1.48 -0.76 -2.84]
# [ 4.8 -0.6 3. ]
# [-0.04 -2.68 -2.52]]
print( diff_avgprod_prodavg(B_mu, B_mu) )
# [[1.84 0. 3.12]
# [0. 5.2 2.8 ]
# [3.12 2.8 6.96]]
print( diff_avgprod_prodavg(B_nu, B_nu) )
# [[ 6.96 0.72 4.28]
# [ 0.72 7.44 -3.64]
# [ 4.28 -3.64 9.04]]