I have the following piece of code, which runs through two nested for-loops and has an if condition in the middle:
N=1e4;
cond_array = [0:(N/2-1) -N/2+(0:(N/2-1))]/(N);
condition = 0.1;
arr_one = zeros(N, 1);
arr_two = zeros(N, 1);
for m=1:N
for k=1:N
if(abs(cond_array(k)) <= condition)
arr_one(m) = arr_one(m) + m*k;
else
arr_two(m) = arr_two(m) + m*k;
end
end
end
I'd like to optimize this code as I potentially need to use very large N (>1e4) and from my own experience, for-loops in MATLAB are often very CPU-consuming and not very efficient.
Is there a way to optimize this piece of code, maybe by using vectorized functions that work on entire arrays?
Here is a much faster (and readable) way to get the same result:
N=1e4;
cond_array = [0:(N/2-1) -N/2+(0:(N/2-1))]/(N);
condition = 0.1;
% this is so you don't check the same condition multiple times:
cond = abs(cond_array)<=condition;
% a vector of the positions in 'arr_one' and 'arr_two':
pos = (1:N).';
% instead of m*k(1)+m*k(2)... use: m*sum(k):
k1 = sum(pos(cond)); % for arr_one
k2 = sum(pos(~cond));% for arr_two
% the final result:
arr_one = pos.*k1;
arr_two = pos.*k2;
For the second case you mention in the comments, where m*k becomes exp((m-1)*(k-1)), we can again compute the sum of exp((m(1)-1)*(k(1)-1)) +...+ exp((m(1)-1)*(k(N)-1))... using vectorization, and then use some minimal loop to go over all ms:
% we define new vectors of constants:
k1 = pos(cond);
k2 = pos(~cond);
% define new functions for each of the arrays:
ek1 = #(m) sum(exp((m-1).*(k1-1)));
ek2 = #(m) sum(exp((m-1).*(k2-1)));
% and we use 'arrayfun' for the final result:
arr_one = arrayfun(ek1,1:N).';
arr_two = arrayfun(ek2,1:N).';
arrayfun is not faster than a for loop, just more compact. Using a for loop will be like this:
arr_one = zeros(N,1);
arr_two = zeros(N,1);
for k = 1:N
arr_one(k) = ek1(k);
arr_two(k) = ek2(k);
end
And here is another, more general, option using bsxfun:
ek = #(m,k) exp((m-1).*(k-1));
arr_one = sum(bsxfun(ek,1:N,k1)).';
arr_two = sum(bsxfun(ek,1:N,k2)).';
Look into the use of logical indexing.
Without understanding your code too well, I'd imagine that you can remove the loops through array functions similar to the below.
arr_one(abs(cond_array)<condition)
This will remove the need for the comparison function, which will allow you to use an anonymous array function in order to create your computation.
However, if I understand your code correctly, you're just adding the product of the position of the condition index and the position of the array index to the array.
If you do that, it will be much easier to just do something similar to the below.
position=(1:N)'
arr_one(abs(cond_array)<condition)=arr_one(abs(cond_array)<condition)+position(abs(cond_array)<condition)
In this statement, you're looking for all the items where the cond_array is less than the condition, and adding the position (effectively a proxy for the m and k in your previous example).
In terms of runtime, I think you have to be 1 to 3 orders of magnitude larger than 1e4 for this code to make a significant difference in runtimes.
I am interested in writing some matrix elements as functions which can take value as per my convenience and then the required matrix operations can be applied on top of that. More precisely, I am trying to integrate over x, the trace of a matrix which has matrix elements as functions of x (which are unknown analytically as they come through products of matrices dependent on x) .
When I try to write the matrix elements as functions, then I obviously get the error- Conversion to double from function_handle is not possible. Is there an easy way to write the matrix elements as functions ?
Thanks. Please ask if my question is not clear.
For example, its something like this:-
N_k = 10;
M_sigma = cell(N_k,1);
M_rho = cell(N_k,1);
for ii = 1:N_k
M_sigma {ii}(1,2) = #(sigma) sigma; %this kind of thing is not allowed in matlab
M_sigma {ii}(2,1) = #(sigma) -conj(sigma);
M_sigma {ii}(1,1) = 0;
M_sigma {ii}(2,2) = 0;
end
for ii = 1:N_k
M_rho {ii}(1,2) = #(rho) rho;
M_rho {ii}(2,1) = #(rho) -conj(rho);
M_rho {ii}(1,1) = 0;
M_rho {ii}(2,2) = 0;
end
M_tau = cell(N_k,1);
for ii = 1:N_k
M_tau {ii} = exp(M_sigma{ii})*exp(M_rho{ii});
end
% the following statement is wrong but I want to do something like
%:-write M_tau as a function of sigma and sum(integrate) the trace of M_tau for all values of sigma
integral(#(sigma) M_tau{1}(sigma), {0,1})
I don't think that you would be able to accomplish that with function handles. I think the best way you could do it would be to define a function as follows :
function x = myFunction(rowindex,colindex)
x = x * rowindex + colindex;
end
You would replace this function with your algorithm and then you could iterate over it doing the following:
for a=1:10
for b=1:10
x(a,b)=myFunction(a,b);
end
end
Say I have a long list A of values (say of length 1000) for which I want to compute the std in pairs of 100, i.e. I want to compute std(A(1:100)), std(A(2:101)), std(A(3:102)), ..., std(A(901:1000)).
In Excel/VBA one can easily accomplish this by writing e.g. =STDEV(A1:A100) in one cell and then filling down in one go. Now my question is, how could one accomplish this efficiently in Matlab without having to use any expensive for-loops.
edit: Is it also possible to do this for a list of time series, e.g. when A has dimensions 1000 x 4 (i.e. 4 time series of length 1000)? The output matrix should then have dimensions 901 x 4.
Note: For the fastest solution see Luis Mendo's answer
So firstly using a for loop for this (especially if those are your actual dimensions) really isn't going to be expensive. Unless you're using a very old version of Matlab, the JIT compiler (together with pre-allocation of course) makes for loops inexpensive.
Secondly - have you tried for loops yet? Because you should really try out the naive implementation first before you start optimizing prematurely.
Thirdly - arrayfun can make this a one liner but it is basically just a for loop with extra overhead and very likely to be slower than a for loop if speed really is your concern.
Finally some code:
n = 1000;
A = rand(n,1);
l = 100;
for loop (hardly bulky, likely to be efficient):
S = zeros(n-l+1,1); %//Pre-allocation of memory like this is essential for efficiency!
for t = 1:(n-l+1)
S(t) = std(A(t:(t+l-1)));
end
A vectorized (memory in-efficient!) solution:
[X,Y] = meshgrid(1:l)
S = std(A(X+Y-1))
A probably better vectorized solution (and a one-liner) but still memory in-efficient:
S = std(A(bsxfun(#plus, 0:l-1, (1:l)')))
Note that with all these methods you can replace std with any function so long as it is applies itself to the columns of the matrix (which is the standard in Matlab)
Going 2D:
To go 2D we need to go 3D
n = 1000;
k = 4;
A = rand(n,k);
l = 100;
ind = bsxfun(#plus, permute(o:n:(k-1)*n, [3,1,2]), bsxfun(#plus, 0:l-1, (1:l)')); %'
S = squeeze(std(A(ind)));
M = squeeze(mean(A(ind)));
%// etc...
OR
[X,Y,Z] = meshgrid(1:l, 1:l, o:n:(k-1)*n);
ind = X+Y+Z-1;
S = squeeze(std(A(ind)))
M = squeeze(mean(A(ind)))
%// etc...
OR
ind = bsxfun(#plus, 0:l-1, (1:l)'); %'
for t = 1:k
S = std(A(ind));
M = mean(A(ind));
%// etc...
end
OR (taken from Luis Mendo's answer - note in his answer he shows a faster alternative to this simple loop)
S = zeros(n-l+1,k);
M = zeros(n-l+1,k);
for t = 1:(n-l+1)
S(t,:) = std(A(k:(k+l-1),:));
M(t,:) = mean(A(k:(k+l-1),:));
%// etc...
end
What you're doing is basically a filter operation.
If you have access to the image processing toolbox,
stdfilt(A,ones(101,1)) %# assumes that data series are in columns
will do the trick (no matter the dimensionality of A). Note that if you also have access to the parallel computing toolbox, you can let filter operations like these run on a GPU, although your problem might be too small to generate noticeable speedups.
To minimize number of operations, you can exploit the fact that the standard deviation can be computed as a difference involving second and first moments,
and moments over a rolling window are obtained efficiently with a cumulative sum (using cumsum):
A = randn(1000,4); %// random data
N = 100; %// window size
c = size(A,2);
A1 = [zeros(1,c); cumsum(A)];
A2 = [zeros(1,c); cumsum(A.^2)];
S = sqrt( (A2(1+N:end,:)-A2(1:end-N,:) ...
- (A1(1+N:end,:)-A1(1:end-N,:)).^2/N) / (N-1) ); %// result
Benchmarking
Here's a comparison against a loop based solution, using timeit. The loop approach is as in Dan's solution but adapted to the 2D case, exploting the fact that std works along each column in a vectorized manner.
%// File loop_approach.m
function S = loop_approach(A,N);
[n, p] = size(A);
S = zeros(n-N+1,p);
for k = 1:(n-N+1)
S(k,:) = std(A(k:(k+N-1),:));
end
%// File bsxfun_approach.m
function S = bsxfun_approach(A,N);
[n, p] = size(A);
ind = bsxfun(#plus, permute(0:n:(p-1)*n, [3,1,2]), bsxfun(#plus, 0:n-N, (1:N).')); %'
S = squeeze(std(A(ind)));
%// File cumsum_approach.m
function S = cumsum_approach(A,N);
c = size(A,2);
A1 = [zeros(1,c); cumsum(A)];
A2 = [zeros(1,c); cumsum(A.^2)];
S = sqrt( (A2(1+N:end,:)-A2(1:end-N,:) ...
- (A1(1+N:end,:)-A1(1:end-N,:)).^2/N) / (N-1) );
%// Benchmarking code
clear all
A = randn(1000,4); %// Or A = randn(1000,1);
N = 100;
t_loop = timeit(#() loop_approach(A,N));
t_bsxfun = timeit(#() bsxfun_approach(A,N));
t_cumsum = timeit(#() cumsum_approach(A,N));
disp(' ')
disp(['loop approach: ' num2str(t_loop)])
disp(['bsxfun approach: ' num2str(t_bsxfun)])
disp(['cumsum approach: ' num2str(t_cumsum)])
disp(' ')
disp(['bsxfun/loop gain factor: ' num2str(t_loop/t_bsxfun)])
disp(['cumsum/loop gain factor: ' num2str(t_loop/t_cumsum)])
Results
I'm using Matlab R2014b, Windows 7 64 bits, dual core processor, 4 GB RAM:
4-column case:
loop approach: 0.092035
bsxfun approach: 0.023535
cumsum approach: 0.0002338
bsxfun/loop gain factor: 3.9106
cumsum/loop gain factor: 393.6526
Single-column case:
loop approach: 0.085618
bsxfun approach: 0.0040495
cumsum approach: 8.3642e-05
bsxfun/loop gain factor: 21.1431
cumsum/loop gain factor: 1023.6236
So the cumsum-based approach seems to be the fastest: about 400 times faster than the loop in the 4-column case, and 1000 times faster in the single-column case.
Several functions can do the job efficiently in Matlab.
On one side, you can use functions such as colfilt or nlfilter, which performs computations on sliding blocks. colfilt is way more efficient than nlfilter, but can be used only if the order of the elements inside a block does not matter. Here is how to use it on your data:
S = colfilt(A, [100,1], 'sliding', #std);
or
S = nlfilter(A, [100,1], #std);
On your example, you can clearly see the difference of performance. But there is a trick : both functions pad the input array so that the output vector has the same size as the input array. To get only the relevant part of the output vector, you need to skip the first floor((100-1)/2) = 49 first elements, and take 1000-100+1 values.
S(50:end-50)
But there is also another solution, close to colfilt, more efficient. colfilt calls col2im to reshape the input vector into a matrix on which it applies the given function on each distinct column. This transforms your input vector of size [1000,1] into a matrix of size [100,901]. But colfilt pads the input array with 0 or 1, and you don't need it. So you can run colfilt without the padding step, then apply std on each column and this is easy because std applied on a matrix returns a row vector of the stds of the columns. Finally, transpose it to get a column vector if you want. In brief and in one line:
S = std(im2col(X,[100 1],'sliding')).';
Remark: if you want to apply a more complex function, see the code of colfilt, line 144 and 147 (for v2013b).
If your concern is speed of the for loop, you can greatly reduce the number of loop iteration by folding your vector into an array (using reshape) with the columns having the number of element you want to apply your function on.
This will let Matlab and the JIT perform the optimization (and in most case they do that way better than us) by calculating your function on each column of your array.
You then reshape an offseted version of your array and do the same. You will still need a loop but the number of iteration will only be l (so 100 in your example case), instead of n-l+1=901 in a classic for loop (one window at a time).
When you're done, you reshape the array of result in a vector, then you still need to calculate manually the last window, but overall it is still much faster.
Taking the same input notation than Dan:
n = 1000;
A = rand(n,1);
l = 100;
It will take this shape:
width = (n/l)-1 ; %// width of each line in the temporary result array
tmp = zeros( l , width ) ; %// preallocation never hurts
for k = 1:l
tmp(k,:) = std( reshape( A(k:end-l+k-1) , l , [] ) ) ; %// calculate your stat on the array (reshaped vector)
end
S2 = [tmp(:) ; std( A(end-l+1:end) ) ] ; %// "unfold" your results then add the last window calculation
If I tic ... toc the complete loop version and the folded one, I obtain this averaged results:
Elapsed time is 0.057190 seconds. %// windows by window FOR loop
Elapsed time is 0.016345 seconds. %// "Folded" FOR loop
I know tic/toc is not the way to go for perfect timing but I don't have the timeit function on my matlab version. Besides, the difference is significant enough to show that there is an improvement (albeit not precisely quantifiable by this method). I removed the first run of course and I checked that the results are consistent with different matrix sizes.
Now regarding your "one liner" request, I suggest your wrap this code into a function like so:
function out = foldfunction( func , vec , nPts )
n = length( vec ) ;
width = (n/nPts)-1 ;
tmp = zeros( nPts , width ) ;
for k = 1:nPts
tmp(k,:) = func( reshape( vec(k:end-nPts+k-1) , nPts , [] ) ) ;
end
out = [tmp(:) ; func( vec(end-nPts+1:end) ) ] ;
Which in your main code allows you to call it in one line:
S = foldfunction( #std , A , l ) ;
The other great benefit of this format, is that you can use the very same sub function for other statistical function. For example, if you want the "mean" of your windows, you call the same just changing the func argument:
S = foldfunction( #mean , A , l ) ;
Only restriction, as it is it only works for vector as input, but with a bit of rework it could be made to take arrays as input too.
This question is from chegg.com.
Given a vector a of N elements a_{n},n =1,2,...,N, the simple moving average of m sequential elements of this vector is defined as
mu(j) = mu(j-1) + (a(m+j-1)-a(j-1))/m for j = 2,3,...,(N-m+1)
where
mu(1) = sum(a(k))/m for k = 1,2,...,m
Write a script that computes these moving averages when a is given by a=5*(1+rand(N,1)), where rand generates uniformly distributed random numbers. Assume that N=100 and m=6. Plot the results using plot(j,mu(j)) for j=1,2,...,N-m+1.
My current code is below, but I'm not sure where to go from here or if it's even right.
close all
clear all
clc
N = 100;
m = 6;
a = 5*(1+rand(N,1));
mu = zeros(N-m+1,1);
mu(1) = sum(a(1:m));
for j=2
mu(j) = mu(j-1) + (a-a)/m
end
plot(1:N-m+1,mu)
I'll step you through the modifications.
Firstly, mu(1) was not fully defined. The equation given is slightly incorrect, but this is what it should be:
mu(1) = sum(a(1:m))/m;
then the for loop has to go from j=2 to j=N-m+1
for j=2:N-m+1
and at each step, mu(j) is given by this formula, the same as given in the question
mu(j) = mu(j-1) + (a(m+j-1)-a(j-1))/m
And that's all you need to change!
I have a probability matrix(glcm) of size 256x256x20. I have reshaped the matrix to
65536x20, so that I can eliminate one loop (along the 3rd dimension).
I want to do the following calculation.
for y = 1:256
for x = 1:256
if (ismember((x + y),(2:2*256)))
p_xplusy((x+y),:) = p_xplusy((x+y),:) + glcm(((y-1)*256+x),:);
end
end
end
So the p_xplusy will be a 511x20 matrix which each element is the sum of the diagonal of nxn sub matrix (where n belongs to 1:256) of the original 256x256x20 matrix.
This code block is making my program inefficient and I want to vectorize this loop. Any help would be appreciated.
Since your if statement is just checking whether x+y is less than or equal to 256, just force it to always be, and remove excess loops:
for y = 1:256
for x = 1:256-y
p_xplusy((x+y),:) = p_xplusy((x+y),:) + glcm(((y-1)*256+x),:);
end
end
This should provide a noticeable speed-up to your code.
You can reduce the complexity from O(n^2) to O(2*n) and thus improve runtime efficiency -
N = 256;
for k1 = 1:N
idx_glcm = k1:N-1:N*(k1-1)+1;
p_xplusy(k1+1,:) = p_xplusy(k1+1,:) + sum(glcm(idx_glcm,:),1);
end
for k1 = 2:N
idx_glcm = k1*N:N-1:N*(N-1)+k1;
p_xplusy(N+k1,:) = p_xplusy(N+k1,:) + sum(glcm(idx_glcm,:),1);
end
Some quick runtime tests seem to confirm our efficiency theory too.