Loren on the Art of MATLAB

Acting on Specific Elements in a Matrix 32

Posted by Loren Shure,

Using MATLAB, there are several ways to identify elements from an array for which you wish to perform some action. Depending on how you've chosen the elements, you may either have the list of elements to toss or the list if elements to retain. And you might not have much if any control yourself how the list gets presented to you since the list could be passed to you from another calculation. The lists might be indices, subscripts, or logical arrays (often referred to as masks). Let's look at how you might arrive at such a situation and see what the code looks like to perform one particular action, setting the desired element values to 0.

Contents

Note: I am not discussing efficiency in this article. It is highly dependent on the number of elements in the original array and how many will be retained or thrown out. This article focuses on specifying what to keep or replace.

General Setup

Here's the setup for this investigation. I will use a fixed matrix for all the methods and always end up with the same final output. The plan is to show you multiple ways to get the result, since different methods may be appropriate under different circumstances.

A = magic(17);
Result = A;
Result( A < mean(A(:)) ) = 0;

Let's look at the nonzero pattern of Result using spy.

spy(Result)

Method #1 - Using Subscripts of Keepers

Here's a list of the subscripts for the elements to keep unchanged.

[rA,cA] = find(A > (17^2)/2);

Next we convert the subscripts to indices.

Result1 = zeros(size(A));
indices = sub2ind(size(A),rA,cA);
Result1(indices) = A(indices);
isequal(Result, Result1)
ans =
     1

Why did I convert subscripts to indices? Let me illustrate with a very small example.

matrix = [ -1 1 0; 2 0 -2; 0 3 -3]
[rows,cols] = find(matrix==0)
matrix =
    -1     1     0
     2     0    -2
     0     3    -3
rows =
     3
     2
     1
cols =
     1
     2
     3

Now let's see what I get if I use the subscripts to address the selected elements:

matrix(rows,cols)
ans =
     0     3    -3
     2     0    -2
    -1     1     0

I get the full matrix back, even though I selected only 3 elements. This definitely surprised me when I first encountered this. What's happening?

MATLAB matches each row element with each column element. matrix([1 2 3],2) returns the elements from rows 1 through 3 in column 1.

matrix(1:3,2)
ans =
     1
     0
     3

To learn more about indexing in general, you might want to read these posts or search the MATLAB documentation.

Method #2 - Using Indices of Keepers

Here we used the single output form of find which returns indices instead of subscripts.

indA = find(A > (17^2)/2);
Result2 = zeros(size(A));
Result2(indA) = A(indA);
isequal(Result, Result2)
ans =
     1

Method #3 - Using Logical Keepers

We'll try keeping about half of the elements unchanged.

keepA = (A > (17^2)/2);
Result3 = zeros(size(A));
Result3(keepA) = A(keepA);
isequal(Result, Result3)
ans =
     1

keepA is a logical matrix the same size as A. I use logical indexing to populate Result3 with the chosen values from A.

Method #4 - Subscripts for Elements to Set to Zero

If instead we have a list of candidates to set to 0, we have an easier time since we don't need to start off with a matrix of zeros. Instead we start with a copy of A.

Result4 = A;
[rnotA,cnotA] = find(A <= (17^2)/2);

Convert indices to subscripts, as in method #1.

indices = sub2ind(size(A),rnotA,cnotA);

Now zero out the selected matrix elements.

Result4(indices) = 0;
isequal(Result, Result4)
ans =
     1

Method #5 - Indices for Elements to Set to Zero

If we're instead given indices, we simply skip the step of converting subscripts and follow similar logic to that in method #4.

Result5 = A;
indnotA = find(A <= (17^2)/2);
Result5(indnotA) = 0;
isequal(Result, Result5)
ans =
     1

Method #6 - Using Logical Arrays to Specify Zero Elements

Finally, if we have a mask for the values to set to 0, we simply use it to select and set elements.

Result6 = A;
keepnotA = (A <= (17^2)/2);
Result6(keepnotA) = 0;
isequal(Result, Result6)
ans =
     1

Which Method(s) Do You Prefer?

Which method or methods do you naturally find yourself using? Do you ever invert the logic of your algorithm to fit your way of thinking about addressing the data (the ins or the outs)? Please post your thoughts here. I look forward to seeing them.


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Published with MATLAB® 7.6

32 CommentsOldest to Newest

I definitely prefer the use of logical index, because the it’s more explicit. I think the code stay more readable. It’s something that I miss on other lenguajes.

It depends on the algorithm whether logical or numerical indexing is most suitable.

When I need a list of indices several times because I don’t want to create new variables, a list of numerical indices is shorter than a logical matrix. Also, when the positions of selected elements are of interest, a list of numerical indices is better, because subscripts are easy to calculate from numerical indices.

In other cases there is the warning “logical indexing is usually faster than find”. Then I use logical indexing.

Personally, I have started to use logical indexing as much as possible. What I like about logical indexing is that I can plot the data and the mask in the same plot and clearly see what I am removing and what I am not removing.

–DA

Thanks to all of you. For a long time, it seemed like people weren’t learning about logical indexing but now people are. I think that’s good

(but, as OkinawaDolphin says, not always optimal when the indices need to hang around for several operations and the array is large compared to the number of values of interest).

–Loren

Loren,

nowadays I do use logical indexing (with the exception that OkinawaDolphin pointed out). But the ONE thing which prevented me from using logical indexing from the beginning is the following.

When I saw the result of the following operations
> a = [1 2 3 4 5]
> i1 = (a > 3)
ans =
0 0 0 1 1
and when I realized that I can use this array in the following way (for logical indexing):
> a(i1)
ans =
4 5
I thought I can create the logical arrays by hand and use them:
> i2 = [0 0 0 1 1]
> a(i2)
??? Subscript indices must be real positive integers or logicals.

Of course, now I know that i2 created that way is real array which cannot be used as a mask and that the right way is
> a(logical(i2))
ans =
4 5

Trivial. But for me, this was the reason why I tended to use indices and ‘find’ instead of logicals for a long time.

Petr

Petr-

Thanks for clarifying that the array must be of logical type for logical indexing.

–Loren

I started using logical arrays a LOT more often after heeding the warnings and recommendations issued by ‘mlint’. Initially, I didn’t really believe logical arrays were that much more efficient, but profiling my code before and after has shown direct performance improvements.

I’ve always enjoyed using array indexing of all sorts (logical or otherwise) because it can be a very compact way of expressing an operation.

Here, for example, is a “logical array” method for removing row R and column C (both scalars) from a matrix X…

X(R~=1:end, C~=1:end)

How cool is that ?

…how would you do the same thing if R and C were vectors ?

-Tony

Actually, to answer Loren’s original question “Which Method(s) Do You Prefer?”, I have to say “none of the above”.

I often rely on the rather naughty mix of arithmetic and logical operations:

A .* (A > (17^2)/2)

…because it doesn’t require an assignment statement and can be used inline as part of a longer expression.

-Tony

For myself, generally the answer to my preference depends on what I’m going to do with the selected elements. If I’m going to be doing something that can be vectorized (is “vectorizable”?) then I try to use logical arrays. If not, I tend to need the actual indices of the selected elements. As a related question, is there any way to determine which of the following will be faster (or even a rule of thumb)?
A= rand(10000,100);
A(5001:end,:)=[];
B=A(1:5000,:);
A=A(1:5000,:);
A=A(1:5000,1:100); %Explicityly stating the number of columns
B=zeros(5000,100);
for n=1:size(A,2)
B(:,n)=A(1:5000,n);
end
some other method?

When I’m dealing with very large data sets, this is one of the most frustrating slow points, since it can exceed the time requirements of many of the calculations.
Dan

Tony,

With regard to removing rows R and columns C from a matrix, you can use:

A = reshape(1:49, 7, 7);
B = A; % for comparison
R = 1:2:7; % Rows to remove
C = 2:2:7;
B(R, :) = [];
B(:, C) = [];

or:

A = reshape(1:49, 7, 7);
R = 1:2:7; % Rows to remove
C = 2:2:7;
B = A(setxor(R, 1:7), setxor(C, 1:7));
% or
B = A(setdiff(1:7, R), setdiff(1:7, C));

As for the arithmetic/logical combination, that will work … as long as all your elements of A are finite.

A = 1:1000;
B = A.*(A > (17^2)/2);
all(B == A | B == 0)

A = 1:1000;
A([5 982]) = NaN;
A([7 562]) = Inf;
A([324 870]) = -Inf;
B = A.*(A > (17^2)/2);
all(B == A | B == 0)

Note that not only are elements 5 and 982 of the second B NaN (as you would expect, since any arithmetic operation on a NaN results in a NaN result) but so are elements 324 and 870. This is because 0*Inf and 0*(-Inf) also both return NaN.

Steve-

Nice reply to Tony’s challenge. You can use end’s in the expressions of your first solutions as well:

  A = reshape(1:49, 7, 7);
  R = 1:2:7; % Rows to remove
  C = 2:2:7;
  B = A(setxor(R, 1:end), setxor(C, 1:end));
  % or
  B = A(setdiff(1:end, R), setdiff(1:end, C));

–Loren

I definitely prefer, and use as often as possible, logical indexing, possibly keeping the indices stored for future reference only if really needed.
It makes the code much more readable and, as I used to be programming with SQL languages, it exploits the semantic power of the WHERE clauses which are what make DB programming so powerful.

I tend to use all methods mentioned based on my need.

What I find interesting is how common a problem it is for (not necessarily beginner) users to run into the issue mentioned in Method #1. Often users are using subscripting and don’t realize they need to convert to linear indices to solve a specific problem.

I find myself having to repeatedly explain the difference and am glad you have addressed the topic here.

Steve, Loren,

Thanks very much for this solution…

B = A(setdiff(1:end, R), setdiff(1:end, C));

Steve – very neat,
Loren – I had no idea I could use “end” in a function call !

…where’s that documented ?

Hey Loren,

Logical indexing is nice, but not always practical. We produce hundreds of GB of data every day a subsequently load it into MATLAB. It’s possible to construct a short array of numerical indices, which contains the subset of the data I may be interested in but keeping a logical index around is not really feasible, either in MATLAB or on disk.

Further, to the comparison of logical vs. numerical indexing (on a smaller footprint example)

x = rand(2 * 1e9, 1); %16GB of data

tic; l = x > 0.5; toc
Elapsed time is 4.771542 seconds.

tic; i = find(l); toc
Elapsed time is 29.321260 seconds.

>> tic; y=x(l); toc
Elapsed time is 49.782177 seconds.

>> tic; z=x(i); toc
Elapsed time is 51.138457 seconds.

As you can see, there is very little in it for a large set, since most of the computation is likely to be memory bound. I have to say that I’m disappointed with the implementation of the numeric subscript. It is not surprising that a logical index must grow the resulting array, since it doesn’t know in advance how many elements may be in the result. However, to see reallocations on my system monitors during the final subscript call seems unnecessary, since I know *exactly* how many elements there’ll be. Memory reallocation would actually be the reason why, for large datasets, I’d expect the numerical index to outperform the logical index.

On a similar note, here are two features I’d like to see in MATLAB:
1) Don’t construct the logical array if you don’t need to: For example,

i = find(logical expression);

or even

b = logical expression;
i = find(b);
% b no longer used after here

There seems no reason why you have to do construct the temporary logical array first and then follow the computation by another pass over the logical array. This is slow and wastes an unnecessary amount of memory for a temporary.

2) Again, suppose I have a large dataset (200+ GB in size). I want to perform a computation on a subset of it (say 50GB). I have to make a copy to use the data in the computation, which is a problem, as I’m already riding hard against the amount of available memory. If I have a list of the indices already, I’d like to use an “indirect” array – an object that holds both indices and the original data and allows me to subscript it as a regular array. To preempt objections: a) Yes, I could write a class to do it but that’s not very fast. b) The object could be read only, so that you wouldn’t have to worry about multiple numerical indices with the same value.

Many thanks,

Tom

Thank you very much for Method #4.

I’ve been trying hard to find a way for vectorizing relational matrix manipulation in Matlab.
Maybe there is an even simpler way that I am not aware of, but your #4 works.
I used it with
indices = sub2ind(size(A),rnotA+1,cnotA+1);

Note +1. I basically needed to manipulate the i+1,j+1 neighbors of elements satisfying a condition without relying on a for loop.

Thanks again!

-Nickitas

hi i would like to ask something…I want to remove an especific NaN element from a matrix, I do not know how to do it. All the solutions that I’ve read explain how to remove rows where the NaN exists, but not specific elements..Could you please help me? I appreciate your help

Marcela-

If you have a vector, you can remove NaNs like so:

a(isnan(a)) = [];

But if you have a matrix, you aren’t guaranteed that the same number of NaNs are in each row/column so you may not end up with a rectangular array.

–Loren

i have a question about removing elements from an array based on their indices, not their values. please consider the following code snippet showing 3 methods for doing such an operation:

n=100;          % square nxn matrices
n2=n^2;
A0=rand(n,n,n); % generate matrices
for i=1:n       % for each matrix
  
  % method 1 of dropping a matrix
  A1=A0;         
  A1(:,:,i)=[];  
  
  % method 2 of dropping a matrix
  A2=A0;
  A2(1:n2)=[];    

  % method 3 of dropping a matrix
  A3=A0(:,:,2:n); 
  
  % mean computation for comparison purposes 
  mean(A0,3);   
end

note that upon running profiler, method 3 takes about half the time of method 1, and about 1/3 of the time of method 2. but the mean computation takes far less time than any!

this quite surprising result (to me) leads me to have the following questions:

1) why does it take so long?
2) is there a way to make it go *much* faster?

many thanks,
joshua

Joshua-

mean takes less time since there’s no need to copy the array at all before acting on it.

What exactly is taking so long that you want to go faster? I doubt you can do better than the 3rd method, which I think makes the fewest temporary intermediate arrays (none). It only makes a copy of the array to keep and nothing more.

–loren

hey loren, thanks for the response. so, the first line of the first two methods, where the copy is made, takes nearly no time. it is the second line, where i only discard elements of the matrix, that takes time. that might have been confusing. in practice, what i do now is something like:

mean(A(:,:,s),3)

where ‘s’ is a vector listing the indices that i care about.

and the reason i care about all this, is because i am doing leave-one-out cross-validation. i have a bunch of training data, and i am fitting my model, but want to ensure that i have not overfit. so, i loop through every data point, discard it to fit the model, and then check the model accuracy on it. the part of the code that takes by far the longest was dropping a matrix from the array. now, i skip that by doing it all in one line, as shown above. but now that line just takes about as long as dropping the matrix did, so i didn’t get any speed up.

specifically, my code now looks like this:

n=100;          % square nxn matrices
n2=n^2;
A0=rand(n,n,n); % generate matrices
for i=1:n       % for each matrix
  
  % method 1 of getting mean
  A1=A0(:,:,2:n); 
  M1=mean(A3,3);   
  
  % method 2 of getting mean
  M2=mean(A0(:,:,2:n),3);  % mean computation for comparison purposes 
end

running the profiler shows that the line to compute M2 takes equally long as the line to compute A1 and M1 combined. and, if i look carefully, i see that actually computing M2, matlab only spends a fraction of the time computing the mean, the rest is passing the variable to the mean function.

so my question remains: is there anyway for matlab to not take a long time when dropping elements in an array? trying to trick it by putting it all on one line did not seem to work, sadly ;(

Joshua-

The first line takes no time since the 2 arrays are identical and MATLAB only does lazy copy or copy on write (see blogs on memory to learn more about this). Once an element is changing, MATLAB makes a full copy of the original array to be sure it doesn’t get modified improperly.

Again, the best way to drop elements is to create the new array and fill it, you’re 3rd way – only the copy needed for the output is created in addition to the original, I believe. I don’t think you can do better than that.

–Loren

Hi Loren,

i have red this post, but didn’t find what I was looking for. Is there a easy way to make a logical matrix from subscript indices like the function find does but the other way round. Under method number 2 in your post you gave a small example what happens when you use subscripts to address the selected elements of a matrix. You get the full matrix back and not only the selected elements. I think the easiest way would be a logical to adress and get back only the selected elements. Using the function sub2ind is too much bother in my opinion.

-Jan

Jan-

As far as I know, there is not a simple 1-liner to go from subscripts to logicals without using sub2ind. Sorry you don’t care for that solution.

–Loren

These postings are the author's and don't necessarily represent the opinions of MathWorks.