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I have a matrix called vec with two columns, vec[:,0] and vec[:,1]. P contains two matrices, P[0,:,:] and P[1,:,:]. I want to mulitiply P[0,:,:] with the first column of vec and multiply P[1,:,:] with the second column of vec. However, the operation P#vec also gives me the matrix product of P[0,:,:] with the second column of vec and the matrix product of P[1,:,:] with the first column of vec, which slows my code.
Is it possible to directly compute the pairs column 1 to matrix 1 and column 2 to matrix 2 without the "off" products?
import numpy as np
P=np.arange(50).reshape(2, 5, 5)
vec=np.arange(10).reshape(5,2)
have=P#vec
want=np.column_stack((have[0,:,0],have[1,:,1]))
have,want
There is a very powerful function in numpy called np.einsum. It can perform all kind of tensor contractions, axis reordering and matrix multiplication. For your example you could use
res = np.einsum('nij,jn->in', P, vec)
after which res is exactly like want.
How does this work:
You give the np.einsum function both your arrays as well as a signature (that 'nij,jn->in' string) that tells the function how to multiply the arrays. In short, you want the third axis of the P tensor to be contracted with the first axis of vec. Therefore you choose the same index j in the signature string and leave it out in the part after the ->. A mere broadcast is done if indices appear on the left and right hand side of the ->, which is done here for the n and i indices.
A more complete explanation of this very powerful function with many examples of how to use it can be found at the corresponding numpy documentation.
#/matmul handles batches nicely, but the rules are that for 3d arrays, the first dimension is the batch, and dot is done on the last 2 dimensions, with the usual "last of A with the second to the last of B" pairing.
It took a bit of reading to decipher you description but it appears that you want the first of p to the batch, and last of vec to be the batch. That means vec needs to transformed to a (2,5,1) to work with the (2,5,5) p.
In [176]: P#vec.T[:,:,None]
Out[176]:
array([[[ 60],
[ 160],
[ 260],
[ 360],
[ 460]],
[[ 695],
[ 820],
[ 945],
[1070],
[1195]]])
The result is (2,5,1). We can squeeze out the the last to get (2,5), but apparently you want a (5,2)
In [179]: (P#vec.T[:,:,None])[...,0].T
Out[179]:
array([[ 60, 695],
[ 160, 820],
[ 260, 945],
[ 360, 1070],
[ 460, 1195]])
np.einsum('nij,jn->in', P, vec) does effectively the same, with the n as the batch dimension that is 'carried through' to the result, and sum-of-products on the shared j dimension.
I have two multidimensional arrays, which I want to multiply with each other. One has the shape N,N,3 and the other has the shape N,N.
Let me set the stage:
I have an array of atom positions of the shape N,3:
atom_positions = [[x1,y1,z1],
[x2,y2,z2],
[x3,y3,z3],
...
]
From these I calculate an upper triangular matrix of distance vectors so that the resulting N,N,3 matrix contains all unique pair distance vectors r_ij of the vectors inside atom_positions:
pair_distance_vectors = [[[0,0,0],[x2-x1,y2-y1,z2-z1],[x3-x1,y3-y1,z3-z1],...],
[[0,0,0],[0,0,0] ,[x3-x2,y3-y2,z3-z2],...],
...
]
Now I want to normalize each of these pair distance vectors. For that I want to use my N,N pair_distances array, which contains the length of every vector inside pair_distance_vectors.
The formula for a single vector is:
r_ij/|r_ij|
I want to do that by doing a matrix multiplication, where every entry in the N,N array becomes a scalar by which a vector inside the N,N,3 array is multiplied. I'm pretty sure that this can be achieved somehow with numpy by using numpy.dot() or a different function, but I just can't find the answer myself. Also, I'm afraid if I do find a transformation which allows for this, that my maths will be faulty.
Here's some demonstration code, which achieves what I want in a very inefficient fashion:
import numpy as np
pair_distance_vectors = np.ones(shape=(2,2,3))
pair_distances = np.array(((1,2),(3,4)))
normalized_pair_distance_vectors = np.zeros(shape=(2,2,3))
for i,vec_list in enumerate(pair_distance_vectors):
for j,vec in enumerate(vec_list):
normalized_pair_distance_vectors[i,j] = vec*pair_distances[i,j]
print(normalized_pair_distance_vectors)
Thanks in advance.
EDIT: Maybe this is clearer:
distance_vectors = [[[x11,y11,z11],[x12,y12,z12],[x13,y13,z13],...],
[[x21,y21,z21],[x22,y22,z22],[x23,y23,z23],...],
... ]
distance_matrix = [[r_11,r_12,r_13,...],
[r_21,r_22,r_23,...],
... ]
norm_distance_vectors = some_operation(distance_vectors,distance_matrix)
norm_distance_vectors = [[r_11*[x11,y11,z11],r_12*[x12,y12,z12],r_13*[x13,y13,z13],...],
[r_21*[x21,y21,z21],r_22*[x22,y22,z22],r_23*[x23,y23,z23],...],
... ]
You won't need a loop. Trick is to expand your pair_distance in the 3rd dimension by repeating it m times (m being the dimension of your vectors, here 3D) and then divide two arrays element wise (works for any m-dimensional vectors, replace 3 with m):
pair_distances = np.repeat(pair_distances[:,:,None], 3, axis=2)
normalized_pair_distance_vectors = np.nan_to_num(pair_distance_vectors/ pair_distances)
Output for your example inputs:
[[[1. 1. 1. ]
[0.5 0.5 0.5 ]]
[[0.33333333 0.33333333 0.33333333]
[0.25 0.25 0.25 ]]]
I am trying to plot a 4D array using as 4th dimension the color. Here is a sample of my matrix:
[[ 4.216 0. 1. 0. ]
[ 5.36 0. 1. 0. ]
[ 5.374 0. 2. 0. ]
...,
[ 0.294 0. 1. 0. ]
[ 0.314 0. 2. 0. ]
[ 0.304 0. 1. 0. ]]
4th column only contains values 0, 1 and 2.
So when I try to plot it using this script:
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.scatter(data[:,0],data[:,1],data[:,2], c=data[:,3], cmap=plt.hot())
plt.show()
I am getting this error:
TypeError: can't multiply sequence by non-int of type 'float'
This isn't a 4D array. It's a 2D array with 4 columns (the 2 dimensions could be referred to as "rows" and "columns"). But I see what you're trying to say—each row could be interpreted as describing a point in 4-dimensional space, with the fourth "dimension" being colour.
Two-dimensionality is actually the key to the problem. I suspect your data variable is a numpy.matrix rather than a vanilla numpy.array. A matrix is particular class of 2D-array that has various special properties, including the fact that a slice of it (for example, data[:, 0]) is still a 2-dimensional matrix object, whereas .scatter() expects each argument to be a 1-D array.
The fix is to say:
data = numpy.asarray(data)
to convert your data from a matrix to a normal array whose column slices will be 1-dimensional.
BTW: you probably meant to say cmap='hot'. The call to plt.hot() sets the default colormap (so your figure may look right, but there's a side effect) but it actually returns None.
I have a problem to understand the matrix multiplication in numpy.
For example I have the following matrix (2d numpy array):
a = [ [ 1. 1. ]
[ 1. 2. ]
[ 1. 3. ] ]
And the following row vector theta:
theta = [ 1. 1. ]
The only way to multiply a with theta would be to transform
theta in a column vector first and then I would get the result:
result = [ [ 2. ]
[ 3. ]
[ 4. ] ]
When I multiply the matrix and the row vector (without transforming)
result = np.dot(a,theta)
I get this:
result = [ 2. 3. 4. ]
How is this even possible? I mean, I didn't transform the matrix.
Can you please tell me how this numpy multiplication works?
Thank you for your attention.
No, you're multiplying numpy array with another numpy array (not a matrix with a vector), although it looks like that. This is because, in essence, numpy arrays are not the same as matrices. And the dot product treats it that way as well.
If you write out the array and multiply, then you will see why. It's just the dot product (element-wise multiplication) of each row in the array 'a' with the vector 'theta'.
PS: (matrices are 2-D while arrays are not limited to any dimension)
Also, please take a look at this answer and this excellent answer
I have a program in numpy utf8, which allows me to calculate the coordinates of a parabolic shot from the ground. I need to create a function which returns the coordinates (#1), create the different arrays of values to work with (#2), and finally use the function to generate the different coordinates for each pack of values
#1
def coordenadas(v,a,t,g=9.81):
rad=deg2rad(a)
x=v*cos(a)*t
y=v*sin(a)*t-(1./2.)*g*(t**2)
xy=array([x,y]).T
return xy
#2
v=arange(50,100,10) #m/s
adegree=arange(45,90,5) #degrees
a=deg2rad(adegree) #rads
t=linspace(0,10,50) #segundos
#3
v.shape=(5,1,1)
a.shape=(1,9,1)
t.shape=(1,1,50)
#5
XY=coordenadas(v,a,t,g=9.81)
print shape(XY)
print XY
#4
My question is that shape(XY) returns
(50L, 9L, 5L, 2L)
And XY (only a bit, is too long)
[[[[ 0. 0. ]
[ 0. 0. ]
[ 0. 0. ]
[ 0. 0. ]
[ 0. 0. ]]
And more boxes of this shape
What this really means(big boxes, boxes, small boxes, rows, columns) ???
(50L, 9L, 5L, 2L) means a 4D array.
You can visualize as a 50x9 matrix and each cell of this matrix contains a 5x2 matrix
Numpy arrays are basically matrices, where each box [] represents the start of a new dimension. As an easy example the matrix
11
23
could be written in numpy as:
a = numpy.array([[1,1],[2,3]])
which then would be printed as
array([[1, 1],
[2, 3]])
As this is a two-dimensional matrix, the outer "box" marks the edges of the matrix, whereas the inner boxes are the rows of the matrix with the , separating the entries. Calling a.shape without an argument gives the shape of the 2x2 matrix:
(2, 2)
Calling the shape method with argument reshapes the matrix given to the shape defined in the argument. But to further help you with the code:
1
Your function definition seems to be totally fine, except I don't see a reason, why you export x and y in an array, rather than just returning two different values.
2
The initialization of your arrays seem to be fine as well.
3
There is totally no reason to reshape the arrays you just created, just leave them as they are.
4
You have to call the function separately with each set of values to create the coordinates. Do that by using an itteration over the arrays you just created.