Let's suppose we have a matrix and a list of indexes:
adj_mat = np.array([[1,2,3],
[4,5,6],
[7,8,9]])
indexes = [0,2]
What I want is to sum the rows and columns corresponding to the sub matrix we get by the intersection of the rows and columns of the indexes list. In this case it would be:
sub_matrix = ([[1,3]
[7,9]])
result_rows = [4,16]
result_columns = [8,12]
However, I do this calculation rather a lot of times with the same original matrix and different indexes lists, so I am looking for an efficent solution without creating the sub matrix each iteration. My solution so far is (and for columns respectively):
def sum_rows(matrix, indexes):
sum_r = [0]*len(indexes)
for i in range(len(indexes)):
for j in indexes:
sum_r[i] += matrix.item(indexes[i], j)
return sum_r
I'm looking for a more efficient algorithm as I remember there is a method which looks like this that sums all rows (or columns?) in the indexes:
matrix.sum(:, indexes)
matrix.sum(indexes, indexes)
I assume what I need is the second line, if it exists. I tried to google it, with or without numpy, but couldn't find the right syntax.
Is there a solution as I described here but I'm just using the wrong syntax? Or any other suggestions for improvement?
IIUC:
import numpy as np
adj_mat = np.array([[1,2,3],
[4,5,6],
[7,8,9]])
indexes = np.array([1, 3]) - 1
sub_matrix = adj_mat[np.ix_(indexes, indexes)]
result_rows, result_columns = sub_matrix.sum(axis=1), sub_matrix.sum(axis=0)
Result:
array([ 4, 16]) # result_rows
array([ 8, 12]) # result_columns
So assuming you made a mistake and you meant indexes = [0,2] and sub_matrix = [[1,3], [7,9]], then this should do what you want
def sum_sub(matrix, indices):
"""
Returns the sum of each row and column (as a tuple)
for each index in indices (as an array)
"""
# note that this sub matrix does not copy any data from matrix,
# it is a "view" which simply holds a reference to matrix
sub_mat = matrix[np.ix_(indices, indices)]
return sub_mat.sum(axis=1), sub_mat.sum(axis=0)
sum_row, sum_col = sum_sub(np.arange(1,10).reshape((3,3)), [0,2])
The results of this are
sum_col # --> [ 8 12]
sum_row # --> [ 4 16]
Since the point of efficiency was brought up in the question, a little further analysis should probably be done.
First and foremost, the code looks like code to find a matrix inverse using the adjoint matrix. Unless that particular method is important to the project, the standard np.linalg.inv() is almost certainly going to be faster than anything we cook up here. Moreover, in many applications you can get away with solving a system of linear equations rather than finding an inverse and multiplying by it, cutting run times in half or more again.
Second, any discussion of efficient numpy code needs to address views as opposed to copies. Memory allocation, writing to memory, and memory deallocation are all extremely expensive operations when compared with standard floating point arithmetic. That's not to say that they're slow, but you can notice an order of magnitude or two of difference in the speed of code memory efficient code vs nearly anything else. That's the entire premise behind the fastest implementation of persistent homology calculations I know of, among other things.
All of the other answers (at the time of writing) create a copy of the data they're working with, explicitly storing that information in a new variable sub_matrix. It isn't possible to create every fancy-indexed matrix with a copy, but oftentimes equivalent operations can be performed.
For example, if this really is a set of computations on adjoint matrices so that your indexes variable consists of all but one of the available indices (in your example, all but the middle index), then instead of explicitly summing over all the intended indices, we can sum over all indices and subtract the one we don't care about. The effect is that all the intermediate matrices are views rather than copies, preventing the expensive memory allocations. On my machine, this is twice as fast for the tiny 3x3 example given and 10x as fast for 500x500 matrices.
bad_row = 1
bad_col = 1
result_rows = (np.sum(adj_mat, axis=1)-adj_mat[:,bad_col])[np.arange(adj_mat.shape[0])!=bad_row]
result_cols = (np.sum(adj_mat, axis=0)-adj_mat[bad_row,:])[np.arange(adj_mat.shape[1])!=bad_col]
Of course, it's even faster if you can use slices to represent whatever you're doing and you don't have to work around the problem with extra operations as I did, but the example you gave doesn't easily permit slices.
Related
I have two arrays that are related to each other via a mapping operation. I will call them S(fk,fq) and Z(fi,αj). The arguments are all sampling frequencies. The mapping rule is fairly straightforward:
fi = 0.5 · (fk - fq)
αj = fk + fq
S is the result of several FFTs and complex multiplications and is defined on a rectangular grid. However, Z is defined on a diamond-shaped grid and it is not clear to me how best to store this. The image below is an attempt at visualizing the operation for a simple example of a 4×4 array, but in general the dimensions are not equal and are much larger (maybe 64×16384, but this is user-selectable). Blue points are the resulting values of fi and αj and the text describes how these are related to fk, fq, and the discrete indices.
The diamond-shaped nature of Z means that in one "row" there will be "columns" that fall in between the "columns" of adjacent "rows". Another way to think of this is that fi can take on fractional index values!
Note that using zero's or nan's to fill in elements that don't exist in any given row has two drawbacks 1) it inflates the size of what may already be a very large 2-D array and 2) it does not really represent the true nature of Z (e.g. the array size will not really be correct).
Currently I am using a dictionary indexed on the actual values of αj to store the results:
import numpy as np
from collections import defaultdict
nrows = 64
ncolumns = 16384
fk = np.fft.fftfreq(nrows)
fq = np.fft.fftfreq(ncolumns)
# using random numbers here to simplify the example
# in practice S is the result of several FFTs and complex multiplications
S = np.random.random(size=(nrows,ncolumns)) + 1j*np.random.random(size=(nrows,ncolumns))
ret = defaultdict(lambda: {"fi":[],"Z":[]})
for k in range(-nrows//2,nrows//2):
for q in range(-ncolumns//2,ncolumns//2):
fi = 0.5*fk[k] - fq[q]
alphaj = fk[k] + fq[q]
Z = S[k,q]
ret[alphaj]["fi"].append(fi)
ret[alphaj]["Z"].append(Z)
I still find this a bit cumbersome to work with and wonder if anyone has suggestions for a better approach? "Better" here would be defined as more computationally and memory efficient and/or easier to interact with and visualize using something like matplotlib.
Note: This is related to another question about how to get rid of those nasty for-loops. Since this is about storing the results I thought it would be better to create two separate questions.
You can still view it as a straight two-dimensional array. But you can represent it as an array of rows, each row of which has a different number of items. For example, here's your 4x4 as a 2D array: (each 0 here is a unique data item)
xxx0xxx
xx0x0xx
x0x0x0x
0x0x0x0
x0x0x0x
xx0x0xx
xxx0xxx
Its sparse representation would be:
[
[0],
[0,0],
[0,0,0],
[0,0,0,0],
[0,0,0],
[0,0],
[0]
]
With this representation you eliminate the empty space. There's a little math involved in converting from Color Temperature to row, and from Spectral Frequency to column (and vice-versa), but that's tractable. You know the bounds and that items are evenly spaced out across each row. So it should be easy enough to do the translation.
Unless I'm missing something . . .
It turns out that the answer to a related question on optimization effectively solved my problem of how to better store the data. The new code returns 2-D arrays for fi, %alpha;j, and these can be used to directly index S. So to get all values of S for %alpha;j = 0, for example, one can do
S[alphaj == 0]
I can use this pretty effectively and it seems like the quickest way to create a reasonable data structure.
In my program code I've got numpy value arrays and numpy index arrays. Both are preallocated and predefined during program initialization.
Each part of the program has one array values on which calculations are performed, and three index arrays idx_from_exch, idx_values and idx_to_exch. There is on global value array to exchange the values of several parts: exch_arr.
The index arrays have between 2 and 5 indices most of the times, seldomly (most probably never) more indices are needed. dtype=np.int32, shape and values are constant during the whole program run. Thus I set ndarray.flags.writeable=False after initialization, but this is optional. The index values of the index arrays idx_values and idx_to_exch are sorted in numerical order, idx_source may be sorted, but there is no way to define that. All index arrays corresponding to one value array/part have the same shape.
The values arrays and also the exch_arr usually have between 50 and 1000 elements. shape and dtype=np.float64 are constant during the whole program run, the values of the arrays change in each iteration.
Here are the example arrays:
import numpy as np
import numba as nb
values = np.random.rand(100) * 100 # just some random numbers
exch_arr = np.random.rand(60) * 3 # just some random numbers
idx_values = np.array((0, 4, 55, -1), dtype=np.int32) # sorted but varying steps
idx_to_exch = np.array((7, 8, 9, 10), dtype=np.int32) # sorted and constant steps!
idx_from_exch = np.array((19, 4, 7, 43), dtype=np.int32) # not sorted and varying steps
The example indexing operations look like this:
values[idx_values] = exch_arr[idx_from_exch] # get values from exchange array
values *= 1.1 # some inplace array operations, this is just a dummy for more complex things
exch_arr[idx_to_exch] = values[idx_values] # pass some values back to exchange array
Since these operations are being applied once per iteration for several million iterations, speed is crucial. I've been looking into many different ways of increasing indexing speed in my previous question, but forgot to be specific enough considering my application (especially getting values by indexing with constant index arrays and passing them to another indexed array).
The best way to do it seems to be fancy indexing so far. I'm currently also experimenting with numba guvectorize, but it seems that it is not worth the effort since my arrays are quite small.
memoryviews would be nice, but since the index arrays do not necessarily have consistent steps, I know of no way to use memoryviews.
So is there any faster way to do repeated indexing? Some way of predefining memory address arrays for each indexing operation, as dtype and shape are always constant? ndarray.__array_interface__ gave me a memory address, but I wasn't able to use it for indexing. I thought about something like:
stride_exch = exch_arr.strides[0]
mem_address = exch_arr.__array_interface__['data'][0]
idx_to_exch = idx_to_exch * stride_exch + mem_address
Is that feasible?
I've also been looking into using strides directly with as_strided, but as far as I know only consistent strides are allowed and my problem would require inconsistent strides.
Any help is appreciated!
Thanks in advance!
edit:
I just corrected a massive error in my example calculation!
The operation values = values * 1.1 changes the memory address of the array. All my operations in the program code are layed out to not change the memory address of the arrays, because alot of other operations rely on using memoryviews. Thus I replaced the dummy operation with the correct in-place operation: values *= 1.1
One solution to getting round expensive fancy indexing using numpy boolean arrays is using numba and skipping over the False values in your numpy boolean array.
Example implementation:
#numba.guvectorize(['float64[:], float64[:,:], float64[:]'], '(n),(m,n)->(m)', nopython=True, target="cpu")
def test_func(arr1, arr2, inds, res):
for i in range(arr1.shape[0]):
if not inds[i]:
continue
for j in range(arr2.shape[0]):
res[j, i] = arr1[i] + arr2[j, i]
Of course, play around with the numpy data types (smaller byte sizes will run faster) and target being "cpu" or "parallel".
I have a very large 400x300x60x27 array (lets call it 'A'). I took the maximum values which is now a 400x300x60 array called 'B'. Basically I need to find the index in 'A' of each value in 'B'. I have converted them both to lists and set up a for loop to find the indices, but it takes an absurdly long time to get through it because there are over 7 million values. This is what I have:
B=np.zeros((400,300,60))
C=np.zeros((400*300*60))
B=np.amax(A,axis=3)
A=np.ravel(A)
A=A.tolist()
B=np.ravel(B)
B=B.tolist()
for i in range(0,400*300*60):
C[i]=A.index(B[i])
Is there a more efficient way to do this? Its taking hours and hours and the program is still stuck on the last line.
You don't need amax, you need argmax. In case of argmax, the array will only contain the indices rather than values, the computational efficiency of finding the values using indices are much better than vice versa.
So, I would recommend you to store only the indices. Before flattening the array.
instead of np.amax, run A.argmax, this will contain the indices.
But before you're flattening it to 1D, you will need to use a mapping function that causes the indices to 1D as well. This is probably a trivial problem, as you'd need to just use some basic operations to achieve this. But that would also consume some time as it needs to be executed quite some times. But it won't be a searching probem and would save you quite some time.
You are getting those argmax indices and because of the flattening, you are basically converting to linear index equivalents of those.
Thus, a solution would be to add in the proper offsets into the argmax indices in steps leveraging broadcasting at each one of them, like so -
m,n,r,s = A.shape
idx = A.argmax(axis=3)
idx += s*np.arange(r)
idx += r*s*np.arange(n)[:,None]
idx += n*r*s*np.arange(m)[:,None,None] # idx is your C output
Alternatively, a compact way to put it would be like so -
m,n,r,s = A.shape
I,J,K = np.ogrid[:m,:n,:r]
idx = n*r*s*I + r*s*J + s*K + A.argmax(axis=3)
I have an 2D-array (array1), which has an arbitrary number of rows and in the first column I have strictly monotonic increasing numbers (but not linearly), which represent a position in my system, while the second one gives me a value, which represents the state of my system for and around the position in the first column.
Now I have a second array (array2); its range should usually be the same as for the first column of the first array, but does not matter to much, as you will see below.
I am now interested for every element in array2:
1. What is the argument in array1[:,0], which has the closest value to the current element in array2?
2. What is the value (array1[:,1]) of those elements.
As usually array2 will be longer than the number of rows in array1 it is perfectly fine, if I get one argument from array1 more than one time. In fact this is what I expect.
The value from 2. is written in the second and third column, as you will see below.
My striped code looks like this:
from numpy import arange, zeros, absolute, argmin, mod, newaxis, ones
ysize1 = 50
array1 = zeros((ysize1+1,2))
array1[:,0] = arange(ysize1+1)**2
# can be any strictly monotonic increasing array
array1[:,1] = mod(arange(ysize1+1),2)
# in my current case, but could also be something else
ysize2 = (ysize1)**2
array2 = zeros((ysize2+1,3))
array2[:,0] = arange(0,ysize2+1)
# is currently uniformly distributed over the whole range, but does not necessarily have to be
a = 0
for i, array2element in enumerate(array2[:,0]):
a = argmin(absolute(array1[:,0]-array2element))
array2[i,1] = array1[a,1]
It works, but takes quite a lot time to process large arrays. I then tried to implement broadcasting, which seems to work with the following code:
indexarray = argmin(absolute(ones(array2[:,0].shape[0])[:,newaxis]*array1[:,0]-array2[:,0][:,newaxis]),1)
array2[:,2]=array1[indexarray,1] # just to compare the results
Unfortunately now I seem to run into a different problem: I get a memory error on the sizes of arrays I am using in the line of code with the broadcasting.
For small sizes it works, but for larger ones where len(array2[:,0]) is something like 2**17 (and could be even larger) and len(array1[:,0]) is about 2**14. I get, that the size of the array is bigger than the available memory. Is there an elegant way around that or to speed up the loop?
I do not need to store the intermediate array(s), I am just interested in the result.
Thanks!
First lets simplify this line:
argmin(absolute(ones(array2[:,0].shape[0])[:,newaxis]*array1[:,0]-array2[:,0][:,newaxis]),1)
it should be:
a = array1[:, 0]
b = array2[:, 0]
argmin(abs(a - b[:, newaxis]), 1)
But even when simplified, you're creating two large temporary arrays. If a and b have sizes M and N, b - a and abs(...) each create a temporary array of size (M, N). Because you've said that a is monotonically increasing, you can avoid the issue all together by using a binary search (sorted search) which is much faster anyways. Take a look at the answer I wrote to this question a while back. Using the function from this answer, try this:
closest = find_closest(array1[:, 0], array2[:, 0])
array2[:, 2] = array1[closest, 1]
I will keep it simple.I have a loop that appends new row to a numpy array...what is the efficient way to do this.
n=np.zeros([1,2])
for x in [[2,3],[4,5],[7,6]]
n=np.append(n,x,axis=1)
Now the thing is there is a [0,0] sticking to it so I have to remove it by
del n[0]
Which seems dumb...So please tell me an efficient way to do this.
n=np.empty([1,2])
is even worse it creates an uninitialised value.
A bit of technical explanation for the "why lists" part.
Internally, the problem for a list of unknown length is that it needs to fit in memory somehow regardless of its length. There are essentially two different possibilities:
Use a data structure (linked list, some tree structure, etc.) which makes it possible to allocate memory separately for each new element in a list.
Store the data in a contiguous memory area. This area has to be allocated when the list is created, and it has to be larger than what we initially need. If we get more stuff into the list, we need to try to allocate more memory, preferably at the same location. If we cannot do it at the same location, we need to allocate a bigger block and move all data.
The first approach enables all sorts of fancy insertion and deletion options, sorting, etc. However, it is slower in sequential reading and allocates more memory. Python actually uses the method #2, the lists are stored as "dynamic arrays". For more information on this, please see:
Size of list in memory
What this means is that lists are designed to be very efficient with the use of append. There is very little you can do to speed things up if you do not know the size of the list beforehand.
If you know even the maximum size of the list beforehand, you are probably best off allocating a numpy.array using numpy.empty (not numpy.zeros) with the maximum size and then use ndarray.resize to shrink the array once you have filled in all data.
For some reason numpy.array(l) where l is a list is often slow with large lists, whereas copying even large arrays is quite fast (I just tried to create a copy of a 100 000 000 element array; it took less than 0.5 seconds).
This discussion has more benchmarking on different options:
Fastest way to grow a numpy numeric array
I have not benchmarked the numpy.empty + ndarray.resize combo, but both should be rather microsecond than millisecond operations.
There are three ways to do this, if you already have everything in a list:
data = [[2, 3], [4, 5], [7, 6]]
n = np.array(data)
If you know how big the final array will be:
exp = np.array([2, 3])
n = np.empty((3, 2))
for i in range(3):
n[i, :] = i ** exp
If you don't know how big the final array will be:
exp = np.array([2, 3])
n = []
i = np.random.random()
while i < .9:
n.append(i ** exp)
i = np.random.random()
n = np.array(n)
Just or the record you can start with n = np.empty((0, 2)) but I would not suggest appending to that array in a loop.
You might want to try:
import numpy as np
n = np.reshape([], (0, 2))
for x in [[2,3],[4,5],[7,6]]:
n = np.append(n, [x], axis=0)
Instead of np.append you can also use n = np.vstack([n,x]). I also agree with #Bi Rico that I also would use a list, if n does not need to accessed within the loop.