Matlab - higher dimensions

Introduction

Matlab supports the use of 1D vectors and 2D matrices. It also copes with higher dimensioned matrices ("N-D arrays"), though there are a few surprises. Just as a vector with 4 elements is often described as a 4x1, so a 4x3 matrix could be described as a 4x3x1 (a slice of a 4x3xn block) or even a 4x3x1x1. These single-thickness dimensions are called "singleton dimensions".

The ndims command returns how many dimensions a matrix has. It never returns less than 2. After

   m=ones(4,1)

ndims(m) return 2. After

   m=ones(4,2,3)

(which creates a 4x2x3 matrix full of 1s) ndims(m) returns 3, though after

   m=ones(4,1,1)

ndims(m) returns 2 because it ignores trailing singleton dimensions. If you create m in the following way

   m=ones(1,1,4)

m is essentially the same shape as before, but this time ndims(m) returns 3. You can do

   m=squeeze(m)

which removes singleton dimensions after which ndims(m) returns 2.

Another complication is that

   m=ones(4,0)

is legal - it means that m is empty along one dimension.

Creating N-D arrays

We've already seen how ones can be used to create N-D arrays. zeros and randn can be used in the same way. Also

If you have a 2D matrix you can add dimensions to it.
```
   A=randn(3,4)
   A(:,:,2) = 5; 
```
makes A (a 3x4 matrix) into a 3D matrix whose 1st layer is the same as the original A and whose 2nd layer is full of 5s.
repmat can replicate through several dimensions. The following creates 24 copies of a 2x2 matrix spread through 3 dimensions.
```
   m=repmat([1 2;3 4], [2, 3, 4])
```
The cat command ("cat" stands for "concatenate") joins a list of arrays along a specified dimension. So
```
   A=randn(3,4)
   B=cat(3,A,A)
```
creates a 3D matrix with 2 layers each holding a copy of A.
ndgrid - does in N dimensions what meshgrid does in 2 dimensions. It's useful in combination with interpn

Manipulating N-D arrays

There are some new commands to manipulate N-D arrays. Often they're generalisations of commands to deal with 2-D arrays.

Changing the shape

reshape works with N-D arrays. Also

shiftdim shifts dimensions, so that (for example) the 1st dimension becomes the 2nd and so on. For example
```
   newm=shiftdim(m,1)
```
shifts the dimensions to the left and wraps the dimensions round so that the first becomes the last, and
```
   newm=shiftdim(m,-1)
```
shifts the dimensions to the right, this time padding with singletons.
permute rearranges the dimensions as specified by the vector provided -
```
   newm=permute(m,[3 2 1 4])
```
You can use this to transpose matrices.
shiftdata (there's an inverse, unshiftdata) shifts data so that it's in the right dimension for certain functions.

Changing the contents

circshift circularly shifts the values in the array. so
```
   v=[1:4;5:8]
   newv=circshift(v,1)
```
shifts the values along the 1st dimension. Different dimensions can be selectively shifted
```
   v=[1:4;5:8]
   newv=circshift(v,[1 2])
```
shifts the values down 1 and right 2. The following would shift data in the first 3 dimensions of m by 0, -1 and 2 respectively.
```
   newm=circshift(m,[0,-1,2])
```
flipdim is a generalisation of flipup and fliplr. flipdim(m,1) is equivalent to flipup(m), and flipdim(m,2) is equivalent to fliplr(m).

Taking slices

If you want to take 2-D slices from a 3-D matrix A you can use something like A(:,:,1).

Processing N-D arrays

Some old commands have been rewritten to cope with N-D arrays. For example, sum by default sums along the first non-singleton dimension. You can make it sum along other dimensions. So

   A=randn(3,4,2)
   sum(A,3)

sums along the 3rd dimension (i.e. "depth") giving 12 values.

Note that you can't have sparse N-D arrays.

An example

Suppose you wanted to calculate the possible outcomes of rolling 2 dice (a red one and a green one, say). You could use

   for red=1:6
      for green=1:6
         outcomes(red,green)=red+green;
      end
   end

To add another (blue) die, you could do

   oldoutcomes=outcomes;
   for blue=1:6
      outcomes(:,:,blue)=oldoutcomes(:,:)+blue;
   end

We can now try some simple commands just to check that all is as expected

   ndims(outcomes)  
   size(outcomes)
   numel(outcomes)
   min (outcomes(:))
   max (outcomes(:))
   lessthansix=outcomes<6
   sum(lessthansix(:))/numel(outcomes) % the chance of getting <6
   % Now find the dice values corresponding to these throws.
   % Note that for N-D arrays you need to use ind2sub as well
   lessthansixindices=find(outcomes<6)
   [red,green,blue]=ind2sub(size(outcomes),lessthansixindices)
   % Plot them
   scatter3(red,green,blue)

See also Mathworks' Multidimensional Arrays and volume visualization pages.

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