I commonly need to summarize a time series with irregular timing with a given aggregation function (i.e., sum, average, etc.). However, the current solution that I have seems inefficient and slow.
Take the aggregation function:
function aggArray = aggregate(array, groupIndex, collapseFn)
groups = unique(groupIndex, 'rows');
aggArray = nan(size(groups, 1), size(array, 2));
for iGr = 1:size(groups,1)
grIdx = all(groupIndex == repmat(groups(iGr,:), [size(groupIndex,1), 1]), 2);
for iSer = 1:size(array, 2)
aggArray(iGr,iSer) = collapseFn(array(grIdx,iSer));
end
end
end
Note that both array and groupIndex can be 2D. Every column in array is an independent series to be aggregated, but the columns of groupIndex should be taken together (as a row) to specify a period.
Then when we bring an irregular time series to it (note the second period is one base period longer), the timing results are poor:
a = rand(20006,10);
b = transpose([ones(1,5) 2*ones(1,6) sort(repmat((3:4001), [1 5]))]);
tic; aggregate(a, b, #sum); toc
Elapsed time is 1.370001 seconds.
Using the profiler, we can find out that the grpIdx line takes about 1/4 of the execution time (.28 s) and the iSer loop takes about 3/4 (1.17 s) of the total (1.48 s).
Compare this with the period-indifferent case:
tic; cumsum(a); toc
Elapsed time is 0.000930 seconds.
Is there a more efficient way to aggregate this data?
Timing Results
Taking each response and putting it in a separate function, here are the timing results I get with timeit with Matlab 2015b on Windows 7 with an Intel i7:
original | 1.32451
felix1 | 0.35446
felix2 | 0.16432
divakar1 | 0.41905
divakar2 | 0.30509
divakar3 | 0.16738
matthewGunn1 | 0.02678
matthewGunn2 | 0.01977
Clarification on groupIndex
An example of a 2D groupIndex would be where both the year number and week number are specified for a set of daily data covering 1980-2015:
a2 = rand(36*52*5, 10);
b2 = [sort(repmat(1980:2015, [1 52*5]))' repmat(1:52, [1 36*5])'];
Thus a "year-week" period are uniquely identified by a row of groupIndex. This is effectively handled through calling unique(groupIndex, 'rows') and taking the third output, so feel free to disregard this portion of the question.
Method #1
You can create the mask corresponding to grIdx across all
groups in one go with bsxfun(#eq,..). Now, for collapseFn as #sum, you can bring in matrix-multiplication and thus have a completely vectorized approach, like so -
M = squeeze(all(bsxfun(#eq,groupIndex,permute(groups,[3 2 1])),2))
aggArray = M.'*array
For collapseFn as #mean, you need to do a bit more work, as shown here -
M = squeeze(all(bsxfun(#eq,groupIndex,permute(groups,[3 2 1])),2))
aggArray = bsxfun(#rdivide,M,sum(M,1)).'*array
Method #2
In case you are working with a generic collapseFn, you can use the 2D mask M created with the previous method to index into the rows of array, thus changing the complexity from O(n^2) to O(n). Some quick tests suggest this to give appreciable speedup over the original loopy code. Here's the implementation -
n = size(groups,1);
M = squeeze(all(bsxfun(#eq,groupIndex,permute(groups,[3 2 1])),2));
out = zeros(n,size(array,2));
for iGr = 1:n
out(iGr,:) = collapseFn(array(M(:,iGr),:),1);
end
Please note that the 1 in collapseFn(array(M(:,iGr),:),1) denotes the dimension along which collapseFn would be applied, so that 1 is essential there.
Bonus
By its name groupIndex seems like would hold integer values, which could be abused to have a more efficient M creation by considering each row of groupIndex as an indexing tuple and thus converting each row of groupIndex into a scalar and finally get a 1D array version of groupIndex. This must be more efficient as the datasize would be 0(n) now. This M could be fed to all the approaches listed in this post. So, we would have M like so -
dims = max(groupIndex,[],1);
agg_dims = cumprod([1 dims(end:-1:2)]);
[~,~,idx] = unique(groupIndex*agg_dims(end:-1:1).'); %//'
m = size(groupIndex,1);
M = false(m,max(idx));
M((idx-1)*m + [1:m]') = 1;
Mex Function 1
HAMMER TIME: Mex function to crush it:
The base case test with original code from the question took 1.334139 seconds on my machine. IMHO, the 2nd fastest answer from #Divakar is:
groups2 = unique(groupIndex);
aggArray2 = squeeze(all(bsxfun(#eq,groupIndex,permute(groups,[3 2 1])),2)).'*array;
Elapsed time is 0.589330 seconds.
Then my MEX function:
[groups3, aggArray3] = mg_aggregate(array, groupIndex, #(x) sum(x, 1));
Elapsed time is 0.079725 seconds.
Testing that we get the same answer: norm(groups2-groups3) returns 0 and norm(aggArray2 - aggArray3) returns 2.3959e-15. Results also match original code.
Code to generate the test conditions:
array = rand(20006,10);
groupIndex = transpose([ones(1,5) 2*ones(1,6) sort(repmat((3:4001), [1 5]))]);
For pure speed, go mex. If the thought of compiling c++ code / complexity is too much of a pain, go with Divakar's answer. Another disclaimer: I haven't subject my function to robust testing.
Mex Approach 2
Somewhat surprising to me, this code appears even faster than the full Mex version in some cases (eg. in this test took about .05 seconds). It uses a mex function mg_getRowsWithKey to figure out the indices of groups. I think it may be because my array copying in the full mex function isn't as fast as it could be and/or overhead from calling 'feval'. It's basically the same algorithmic complexity as the other version.
[unique_groups, map] = mg_getRowsWithKey(groupIndex);
results = zeros(length(unique_groups), size(array,2));
for iGr = 1:length(unique_groups)
array_subset = array(map{iGr},:);
%// do your collapse function on array_subset. eg.
results(iGr,:) = sum(array_subset, 1);
end
When you do array(groups(1)==groupIndex,:) to pull out array entries associated with the full group, you're searching through the ENTIRE length of groupIndex. If you have millions of row entries, this will totally suck. array(map{1},:) is far more efficient.
There's still unnecessary copying of memory and other overhead associated with calling 'feval' on the collapse function. If you implement the aggregator function efficiently in c++ in such a way to avoid copying of memory, probably another 2x speedup can be achieved.
A little late to the party, but a single loop using accumarray makes a huge difference:
function aggArray = aggregate_gnovice(array, groupIndex, collapseFn)
[groups, ~, index] = unique(groupIndex, 'rows');
numCols = size(array, 2);
aggArray = nan(numel(groups), numCols);
for col = 1:numCols
aggArray(:, col) = accumarray(index, array(:, col), [], collapseFn);
end
end
Timing this using timeit in MATLAB R2016b for the sample data in the question gives the following:
original | 1.127141
gnovice | 0.002205
Over a 500x speedup!
Doing away with the inner loop, i.e.
function aggArray = aggregate(array, groupIndex, collapseFn)
groups = unique(groupIndex, 'rows');
aggArray = nan(size(groups, 1), size(array, 2));
for iGr = 1:size(groups,1)
grIdx = all(groupIndex == repmat(groups(iGr,:), [size(groupIndex,1), 1]), 2);
aggArray(iGr,:) = collapseFn(array(grIdx,:));
end
and calling the collapse function with a dimension parameter
res=aggregate(a, b, #(x)sum(x,1));
gives some speedup (3x on my machine) already and avoids the errors e.g. sum or mean produce, when they encounter a single row of data without a dimension parameter and then collapse across columns rather than labels.
If you had just one group label vector, i.e. same group labels for all columns of data, you could speed further up:
function aggArray = aggregate(array, groupIndex, collapseFn)
ng=max(groupIndex);
aggArray = nan(ng, size(array, 2));
for iGr = 1:ng
aggArray(iGr,:) = collapseFn(array(groupIndex==iGr,:));
end
The latter functions gives identical results for your example, with a 6x speedup, but cannot handle different group labels per data column.
Assuming a 2D test case for the group index (provided here as well with 10 different columns for groupIndex:
a = rand(20006,10);
B=[]; % make random length periods for each of the 10 signals
for i=1:size(a,2)
n0=randi(10);
b=transpose([ones(1,n0) 2*ones(1,11-n0) sort(repmat((3:4001), [1 5]))]);
B=[B b];
end
tic; erg0=aggregate(a, B, #sum); toc % original method
tic; erg1=aggregate2(a, B, #(x)sum(x,1)); toc %just remove the inner loop
tic; erg2=aggregate3(a, B, #(x)sum(x,1)); toc %use function below
Elapsed time is 2.646297 seconds.
Elapsed time is 1.214365 seconds.
Elapsed time is 0.039678 seconds (!!!!).
function aggArray = aggregate3(array, groupIndex, collapseFn)
[groups,ix1,jx] = unique(groupIndex, 'rows','first');
[groups,ix2,jx] = unique(groupIndex, 'rows','last');
ng=size(groups,1);
aggArray = nan(ng, size(array, 2));
for iGr = 1:ng
aggArray(iGr,:) = collapseFn(array(ix1(iGr):ix2(iGr),:));
end
I think this is as fast as it gets without using MEX. Thanks to the suggestion of Matthew Gunn!
Profiling shows that 'unique' is really cheap here and getting out just the first and last index of the repeating rows in groupIndex speeds things up considerably. I get 88x speedup with this iteration of the aggregation.
Well I have a solution that is almost as quick as the mex but only using matlab.
The logic is the same as most of the above, creating a dummy 2D matrix but instead of using #eq I initialize a logical array from the start.
Elapsed time for mine is 0.172975 seconds.
Elapsed time for Divakar 0.289122 seconds.
function aggArray = aggregate(array, group, collapseFn)
[m,~] = size(array);
n = max(group);
D = false(m,n);
row = (1:m)';
idx = m*(group(:) - 1) + row;
D(idx) = true;
out = zeros(m,size(array,2));
for ii = 1:n
out(ii,:) = collapseFn(array(D(:,ii),:),1);
end
end
Related
I have a 3D Mesh grid, X, Y, Z. I want to create a new 3D array that is a function of X, Y, & Z. That function comprises the sum of several 3D Gaussians located at different points. Currently, I have a for loop that runs over the different points where I have my gaussians, and I have an array of center locations r0(nGauss, 1:3)
[X,Y,Z]=meshgrid(-10:.1:10);
Psi=0*X;
for index = 1:nGauss
Psi = Psi + Gauss3D(X,Y,Z,[r0(index,1),r0(index,2),r0(index,3)]);
end
where my 3D gaussian function is
function output=Gauss3D(X,Y,Z,r0)
output=exp(-(X-r0(1)).^2 + (Y-r0(2)).^2 + (Z-r0(3)).^2);
end
I'm happy to redesign the function, which is the slowest part of my code and has to happen many many time, but I can't figure out how to vectorize this so that it will run faster. Any suggestions would be appreciated
*****NB the Original function had a square root in it, and has been modified to make it an actual gaussian***
NOTE! I've modified your code to create a Gaussian, which was:
output=exp(-sqrt((X-r0(1)).^2 + (Y-r0(2)).^2 + (Z-r0(3)).^2));
That does not make a Gaussian. I changed this to:
output = exp(-((X-r0(1)).^2 + (Y-r0(2)).^2 + (Z-r0(3)).^2));
(note no sqrt). This is a Gaussian with sigma = sqrt(1/2).
If this is not what you want, then this answer might not be very useful to you, because your function does not go to 0 as fast as the Gaussian, and therefore is harder to truncate, and it is not separable.
Vectorizing this code is pointless, as the other answers attest. MATLAB's JIT is perfectly capable of running this as fast as it'll go. But you can reduce the amount of computation significantly by noting that the Gaussian goes to almost zero very quickly, and is separable:
Most of the exp evaluations you're doing here yield a very tiny number. You don't need to compute those, just fill in 0.
exp(-x.^2-y.^2) is the same as exp(-x.^2).*exp(-y.^2), which is much cheaper to compute.
Let's put these two things to the test. Here is the test code:
function gaussian_test
N = 100;
r0 = rand(N,3)*20 - 10;
% Original
tic
[X,Y,Z] = meshgrid(-10:.1:10);
Psi1 = zeros(size(X));
for index = 1:N
Psi1 = Psi1 + Gauss3D(X,Y,Z,r0(index,:));
end
t = toc;
fprintf('original, time = %f\n',t)
% Fast, large truncation
tic
[X,Y,Z] = deal(-10:.1:10);
Psi2 = zeros(numel(X),numel(Y),numel(Z));
for index = 1:N
Psi2 = Gauss3D_fast(Psi2,X,Y,Z,r0(index,:),5);
end
t = toc;
fprintf('tuncation = 5, time = %f\n',t)
fprintf('mean abs error = %f\n',mean(reshape(abs(Psi2-Psi1),[],1)))
fprintf('mean square error = %f\n',mean(reshape((Psi2-Psi1).^2,[],1)))
fprintf('max abs error = %f\n',max(reshape(abs(Psi2-Psi1),[],1)))
% Fast, smaller truncation
tic
[X,Y,Z] = deal(-10:.1:10);
Psi3 = zeros(numel(X),numel(Y),numel(Z));
for index = 1:N
Psi3 = Gauss3D_fast(Psi3,X,Y,Z,r0(index,:),3);
end
t = toc;
fprintf('tuncation = 3, time = %f\n',t)
fprintf('mean abs error = %f\n',mean(reshape(abs(Psi3-Psi1),[],1)))
fprintf('mean square error = %f\n',mean(reshape((Psi3-Psi1).^2,[],1)))
fprintf('max abs error = %f\n',max(reshape(abs(Psi3-Psi1),[],1)))
% DIPimage, same smaller truncation
tic
Psi4 = newim(201,201,201);
coords = (r0+10) * 10;
Psi4 = gaussianblob(Psi4,coords,10*sqrt(1/2),(pi*100).^(3/2));
t = toc;
fprintf('DIPimage, time = %f\n',t)
fprintf('mean abs error = %f\n',mean(reshape(abs(Psi4-Psi1),[],1)))
fprintf('mean square error = %f\n',mean(reshape((Psi4-Psi1).^2,[],1)))
fprintf('max abs error = %f\n',max(reshape(abs(Psi4-Psi1),[],1)))
end % of function gaussian_test
function output = Gauss3D(X,Y,Z,r0)
output = exp(-((X-r0(1)).^2 + (Y-r0(2)).^2 + (Z-r0(3)).^2));
end
function Psi = Gauss3D_fast(Psi,X,Y,Z,r0,trunc)
% sigma = sqrt(1/2)
x = X-r0(1);
y = Y-r0(2);
z = Z-r0(3);
mx = abs(x) < trunc*sqrt(1/2);
my = abs(y) < trunc*sqrt(1/2);
mz = abs(z) < trunc*sqrt(1/2);
Psi(my,mx,mz) = Psi(my,mx,mz) + exp(-x(mx).^2) .* reshape(exp(-y(my).^2),[],1) .* reshape(exp(-z(mz).^2),1,1,[]);
% Note! the line above uses implicit singleton expansion. For older MATLABs use bsxfun
end
This is the output on my machine, reordered for readability (I'm still on MATLAB R2017a):
| time(s) | mean abs | mean sq. | max abs
--------------+----------+----------+----------+----------
original | 5.035762 | | |
tuncation = 5 | 0.169807 | 0.000000 | 0.000000 | 0.000005
tuncation = 3 | 0.054737 | 0.000452 | 0.000002 | 0.024378
DIPimage | 0.044099 | 0.000452 | 0.000002 | 0.024378
As you can see, using these two properties of the Gaussian we can reduce time from 5.0 s to 0.17 s, a 30x speedup, with hardly noticeable differences (truncating at 5*sigma). A further 3x speedup can be gained by allowing a small error. The smallest the truncation value, the faster this will go, but the larger the error will be.
I added that last method, the gaussianblob function from DIPimage (I'm an author), just to show that option in case you need to squeeze that bit of extra time from your code. That function is implemented in C++. This version that I used you will need to compile yourself. Our current official release implements this function still in M-file code, and is not as fast.
Further chance of improvement is if the fractional part of the coordinates is always the same (w.r.t. the pixel grid). In this case, you can draw the Gaussian once, and shift it over to each of the centroids.
Another alternative involves computing the Gaussian once, at a somewhat larger scale, and interpolating into it to generate each of the 1D Gaussians needed to generate the output. I did not implement this, I have no idea if it will be faster or if the time difference will be significant. In the old days, exp was expensive, I'm not sure this is still the case.
So, I am building off of the answer above me #Durkee. I enjoy these kinds of problems, so I thought a little about how to make each of the expansions implicit, and I have the one-line function below. Using this function I shaved .11 seconds off of the call, which is completely negligible. It looks like yours is pretty decent. The only advantage of mine might be how the code scales on a finer mesh.
xLin = [-10:.1:10]';
tic
psi2 = sum(exp(-sqrt((permute(xLin-r0(:,1)',[3 1 4 2])).^2 ...
+ (permute(xLin-r0(:,2)',[1 3 4 2])).^2 ...
+ (permute(xLin-r0(:,3)',[3 4 1 2])).^2)),4);
toc
The relative run times on my computer were (all things kept the same):
Original - 1.234085
Other - 2.445375
Mine - 1.120701
So this is a bit of an unusual problem where on my computer the unvectorized code actually works better than the vectorized code, here is my script
clear
[X,Y,Z]=meshgrid(-10:.1:10);
Psi=0*X;
nGauss = 20; %Sample nGauss as you didn't specify
r0 = rand(nGauss,3); % Just make this up as it doesn't really matter in this case
% Your original code
tic
for index = 1:nGauss
Psi = Psi + Gauss3D(X,Y,Z,[r0(index,1),r0(index,2),r0(index,3)]);
end
toc
% Vectorize these functions so we can use implicit broadcasting
X1 = X(:);
Y1 = Y(:);
Z1 = Z(:);
tic
val = [X1 Y1 Z1];
% Change the dimensions so that r0 operates on the right elements
r0_temp = permute(r0,[3 2 1]);
% Perform the gaussian combination
out = sum(exp(-sqrt(sum((val-r0_temp).^2,2))),3);
toc
% Check to make sure both functions match
sum(abs(vec(Psi)-vec(out)))
function output=Gauss3D(X,Y,Z,r0)
output=exp(-sqrt((X-r0(1)).^2 + (Y-r0(2)).^2 + (Z-r0(3)).^2));
end
function out = vec(in)
out = in(:);
end
As you can see, this is probably about as vectorized as you can get. The whole function is done using broadcasting and vectorized operations which normally improve performance ten-one hundredfold. However, in this case, this is not what we see
Elapsed time is 1.876460 seconds.
Elapsed time is 2.909152 seconds.
This actually shows the unvectorized version as being faster.
There could be a few reasons for this of which I am by no means an expert.
MATLAB uses a JIT compiler now which means that for loops are no longer inefficient.
Your code is already reasonably vectorized, you are operating at 8 million elements at once
Unless nGauss is 1000 or something, you're not looping through that much, and at that point, vectorization means you will run out of memory
I could be hitting some memory threshold where I am using too much memory and that is making my code inefficient, I noticed that when I lowered the resolution on the meshgrid the vectorized version worked better
As an aside, I tested this on my GTX 1060 GPU with single precision(single precision is 10x faster than double precision on most GPUs)
Elapsed time is 0.087405 seconds.
Elapsed time is 0.241456 seconds.
Once again the unvectorized version is faster, sorry I couldn't help you out but it seems that your code is about as good as you are going to get unless you lower the tolerances on your meshgrid.
I am writing an Octave script to calculate the price of an European option.
The first part uses Monte Carlo to simulate the underlying asset price over n number of time periods. This is repeated nIter number of times.
Octave makes it very easy to setup initial matrices. But I haven't found the way to complete the task in a vectorized way, avoiding FOR loops:
%% Octave simplifies creation of 'e', 'dlns', and 'Prices'
e = norminv(rand(nIter,n));
dlns = cat(2, ones(nIter,1), exp((adj_r+0.5*sigma^2)*dt+sigma*e.*sqrt(dt)));
Prices = zeros(nIter, n+1);
for i = 1:nIter % IS THERE A WAY TO VECTORIZE THESE FOR LOOPS?
for j = 1:n+1
if j == 1
Prices(i,j)=S0;
else
Prices(i,j)=Prices(i,j-1)*dlns(i,j);
end
endfor
endfor
Note that the price in n is equal to price in n-1 times a factor, hence the following does not work...
Prices(i,:) = S0 * dlns(i,:)
...since it takes S0 and multiplies it by all the factors, yielding different results than the expected random walk.
Because of the dependency between iterations to obtain results for each new column with respect to the previous column, it seems you would need at least one loop there, but do all operations within a column in a vectorized fashion and that might speed it up for you. The vectorized replacement for the two nested loops would look something like this -
Prices(:,1)=S0;
for j = 2:n+1
Prices(:,j) = Prices(:,j-1).*dlns(:,j);
endfor
It just occurred to me that the dependency can be taken care of with cumprod that gets us cumulative product which is essentially being done here and thus would lead to a no-loop solution! Here's the implementation -
Prices = [repmat(S0,nIter,1) cumprod(dlns(:,2:end),2)*S0]
Benchmarking on MATLAB
Benchmarking Code -
%// Parameters as told by OP and then create the inputs
nIter= 100000;
n = 100;
adj_r = 0.03;
sigma = 0.2;
dt = 1/n;
S0 = 60;
e = norminv(rand(nIter,n));
dlns = cat(2, ones(nIter,1), exp((adj_r+0.5*sigma^2)*dt+sigma*e.*sqrt(dt)));
disp('-------------------------------------- With Original Approach')
tic
Prices = zeros(nIter, n+1);
for i = 1:nIter
for j = 1:n+1
if j == 1
Prices(i,j)=S0;
else
Prices(i,j)=Prices(i,j-1)*dlns(i,j);
end
end
end
toc, clear Prices
disp('-------------------------------------- With Proposed Approach - I')
tic
Prices2(nIter, n+1)=0; %// faster pre-allocation scheme
Prices2(:,1)=S0;
for j = 2:n+1
Prices2(:,j)=Prices2(:,j-1).*dlns(:,j);
end
toc, clear Prices2
disp('-------------------------------------- With Proposed Approach - II')
tic
Prices3 = [repmat(S0,nIter,1) cumprod(dlns(:,2:end),2)*S0];
toc, clear Prices3
Runtimes results -
-------------------------------------- With Original Approach
Elapsed time is 0.259054 seconds.
-------------------------------------- With Proposed Approach - I
Elapsed time is 0.020566 seconds.
-------------------------------------- With Proposed Approach - II
Elapsed time is 0.067292 seconds.
Now, the runtimes do suggest that the first proposed approach might be a better fit here!
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.
yesterday I implemented my first bootstrap in MATLab. (and yes, I know, for loops are evil.):
%data is an mxn matrix where the data should be sampled per column but there
can be a NaNs Elements
%from the array (a column of data) n values are sampled nReps times
function result = bootstrap_std(data, n, nReps,quantil)
result = zeros(1,size(data,2));
for i=1:size(data,2)
bootstrap_data = zeros(n,nReps);
values = find(~isnan(data(:,i)));
if isempty(values)
bootstrap_data(:,:) = NaN;
else
for k=1:nReps
bootstrap_data(:,k) = datasample(data(values,i),n);
end
end
stat = zeros(1,nReps);
for k=1:nReps
stat(k) = nanstd(bootstrap_data(:,k));
end
sort(stat);
result(i) = quantile(stat,quantil);
end
end
As one can see, this version works columnwise. The algorithm does what it should but is really slow when the data size increaes. My question is now: Is it possible to implement this logic without using for loops? My problem is here that I could not find a version of datasample which does the sampling columnwise. Or is there a better function to use?
I am happy for any hint or idea how I can speed up this implementation.
Thanks and best regards!
stephan
The bottlenecks in your implementation are
The function spends a lot of time inside nanstd which is unnecessary since you exclude NaN values from your sample anyway.
There are a lot of functions that operate column-wise, but you spend time looping over the columns and calling them many times.
You make many calls to datasample which is a relatively slow function. It's much faster to create a random vector of indices using randi and use that instead.
Here's how I would write the function (actually I probably wouldn't put in this many comments, and I wouldn't use so many temp variables, but I'm doing it now so you can see what all the steps of the computation are).
function result = bootstrap_std_new(data, n, nRep, quantil)
result = zeros(1, size(data,2));
for i = 1:size(data,2)
isbad = isnan(data(:,i)); %// Vector of NaN values
if all(isbad)
result(i) = NaN;
else
data0 = data(~isbad, i); %// Temp copy of this column for indexing
index = randi(size(data0,1), n, nRep); %// Create the indexing vector
bootstrapdata = data0(index); %// Sample the data
stdevs = std(bootstrapdata); %// Stdev of sampled data
result(i) = quantile(stdevs, quantil); %// Find the correct quantile
end
end
end
Here are some timings
>> data = randn(100,10);
>> data(randi(1000, 50, 1)) = NaN;
>> tic, bootstrap_std(data, 50, 1000, 0.5); toc
Elapsed time is 1.359529 seconds.
>> tic, bootstrap_std_new(data, 50, 1000, 0.5); toc
Elapsed time is 0.038558 seconds.
So this gives you about a 35x speedup.
Your main issue seems to be that you may have varying numbers/positions of NaN in each column, so can't work on the full matrix unless you're okay with also sampling NaNs. However, some of the inner loops could be simplified.
for k=1:nReps
bootstrap_data(:,k) = datasample(data(values,i),n);
end
Since you're sampling with replacement, you should be able to just do:
bootstrap_data = datasample(data(values,i), n*nReps);
bootstrap_data = reshape(bootstrap_data, [n nReps]);
Also nanstd can work on a full matrix so no need to loop:
stat = nanstd(bootstrap_data); % or nanstd(x,0,2) to change dimension
It would also be worth just looking over your code with profile to see where the bottlenecks are.
here is the input data:
% #param Landmarks:
% Landmarks should be 1*m struct.
% m is the number of training set.
% Landmark(i).data is a n*2 matrix
old function:
function Landmarks=CenterOfGravity(Landmarks)
% align center of gravity
for i=1 : length(Landmarks)
Landmarks(i).data=Landmarks(i).data - ones(size(Landmarks(i).data,1),1)...
*mean(Landmarks(i).data);
end
end
new function which use arrayfun:
function [Landmarks] = center_to_gravity(Landmarks)
Landmarks = arrayfun(#(struct_data)...
struct('data', struct_data.data - repmat(mean(struct_data.data), [size(struct_data.data, 1), 1]))...
,Landmarks);
end %function center_to_gravity
when using profiler, I find the usage of time is NOT what I expected:
Function Total Time Self Time*
CenterOfGravity 0.011s 0.004 s
center_to_gravity 0.029s 0.001 s
Can someone tell me why?
BTW...I can't add "arrayfun" as a new tag for my reputation.
Using arrayfun does not count as "vectorizing your code" as described in every Matlab performance blog post ever written.
If your .data field is the same length for all entries of landmark, your could vectorize this code by first placing all of the data into a single DATASIZE-BY-LANDMARKSIZE martix, and then running this command
meanRemovedData = bsxfun(#minus, data, mean(data,1));
But you lose an awful lot of code clarity that way. (I'm pretty sure that bsxfun usually has vectorization-like speed advantages, but I haven't done any time testing this morning.)
In terms of why, I'm not really the right guy to ask. But many of the advantages of vectorization are dependent on performing simple operations of contiguous blocks of memory. Data stored in an array of structures is (I believe) stored as an array of pointers to disparate memory locations, which is why you can change the size or class of Landmarks(i).data without reallocating the whole structure array.
Thanks for Amro and Pursuit's enthusiastic to my question.
I get the best solution at Matlab answers from Jan Simon:
why arrayfun does NOT improve my struct array operation performance
There are some points that do improve the performance:
It is surprisingly that SUM/LENGTH is faster than MEAN
timeit can give more accurate result.
The fastest approach use tricks like this:
m = sum(data, 1) / size(data, 1);
data(:, 1) = data(:, 1) - m(1);
Consider the following three implementations (all vectorized using BSXFUN):
function s = func1(s)
for i=1:numel(s)
s(i).data = bsxfun(#minus, s(i).data, mean(s(i).data));
end
end
function v = func2(s)
v = arrayfun(#(ss) bsxfun(#minus,ss.data,mean(ss.data)), ...
s, 'UniformOutput',false);
v = struct('data',v);
end
function v = func3(s)
v = arrayfun(#(ss) struct('data',bsxfun(#minus,ss.data,mean(ss.data))), ...
s, 'UniformOutput',true);
end
Explanation:
First uses a for-loop to iterate over the array of structs.
Second uses ARRAYFUN to return a cell array of the data matrices, which are then passed to STRUCT to build the array of structures.
The last one uses ARRAYFUN and builds a structure directly at each iteration.
Here is a simple test to compare the timings:
function testArrayStruct()
%# sample array of structures
s = struct('data',[]);
for i=5000:-1:1
s(i).data = rand(randi(1000),2);
end
%# timing
tic; v1 = func1(s); toc
tic; v2 = func2(s); toc
tic; v3 = func3(s); toc
%# check all have the same output
assert(isequal(v1,v2,v3))
end
The results:
Elapsed time is 0.357796 seconds. %# func1
Elapsed time is 0.427568 seconds. %# func2
Elapsed time is 0.537971 seconds. %# func3
So you can see the loop-based solution is actually the fastest..