Apply dynamical function to each point in phase sp

Question

I have a matrix of integers, phase_space of shape (n,n), where each entry represents the number of points in that location in space. I also have two update matrices u_x, u_y also of shape (n,n), with integers in the range 0,...,n specifying where my dynamical system takes each corresponding point in space. I want to "apply" the update matrices to the phase space iteratively.

For example, if

>>>u_x
array([[1, 2, 1],
       [0, 1, 2],
       [0, 0, 0]])
>>>u_y
array([[2, 1, 2],
       [1, 0, 1],
       [2, 2, 0]])
>>>phase_space 
array([[1, 1, 1],
       [1, 1, 1],
       [1, 1, 1]])

I want

>>>new_phase_space
array([[1., 1., 2.],
       [1., 0., 2.],
       [0., 2., 0.]])

My current (working) solution is to loop as follows

for i in range(n):
    for j in range(n):
        new_phase_space[u_x[i, j], u_y[i, j]] += phase_space[i,j] 

Is there any way to vectorize this?

Answer 1

We can use np.bincount -

M,N = u_x.max()+1,u_y.max()+1
ids = u_x*N+u_y
out = np.bincount(ids.ravel(),phase_space.ravel(),minlength=M*N).reshape(M,N)

Sample run on a more generic setup -

In [14]: u_x
Out[14]: 
array([[1, 2, 1],
       [0, 1, 4],
       [0, 0, 0]])

In [15]: u_y
Out[15]: 
array([[2, 1, 2],
       [6, 0, 1],
       [2, 6, 0]])

In [17]: phase_space
Out[17]: 
array([[1, 1, 1],
       [5, 1, 1],
       [1, 1, 1]])

In [18]: out
Out[18]: 
array([[1., 0., 1., 0., 0., 0., 6.],
       [1., 0., 2., 0., 0., 0., 0.],
       [0., 1., 0., 0., 0., 0., 0.],
       [0., 0., 0., 0., 0., 0., 0.],
       [0., 1., 0., 0., 0., 0., 0.]])

We could also make use of sparse matrices, especially if memory is a concern -

from scipy.sparse import csr_matrix,coo_matrix

out = coo_matrix( (phase_space.ravel(), (u_x.ravel(), u_y.ravel())), shape = (M,N))

Output would be a sparse matrix. To convert to a dense one, use out.toarray().

Answer 2

You can use pandas.DataFrame.groupby() to accumulate all moves with same coordinates in phase_space:

new_phase_space + (pd.DataFrame(phase_space)
           .stack()
           .groupby([u_x.ravel(), u_y.ravel()])
           .sum()
           .unstack(fill_value=0)
           .values
)

Output:

array([[2., 2., 4.],
       [2., 0., 4.],
       [0., 4., 0.]])