Now that you know how to load and save data in Octave, put your data into matrices and so on. In this video, I'd like to show you how to do computational operations on data. And later on, we'll be using these source of computational operations to implement our learning algorithms. Let's get started. Here's my Octave window. Let me just quickly initialize some variables to use for our example. So set A to be a three by two matrix, and set B to a three by two matrix, and let's set C to a two by two matrix like so. Now let's say I want to multiply two of my matrices. So let's say I want to compute A*C, I just type A*C, so it's a three by two matrix times a two by two matrix, this gives me this three by two matrix. You can also do element wise operations and do A.* B and what this will do is it'll take each element of A and multiply it by the corresponding elements B, so that's A, that's B, that's A .* B. So for example, the first element gives 1 times 11, which gives 11. The second element gives 2 time 12 Which gives 24, and so on. So this is element-wise multiplication of two matrices. And in general, the period tends to, is usually used to denote element-wise operations in Octave. So here's a matrix A, and if I do A .^ 2, this gives me the element wise squaring of A. So 1 squared is 1, 2 squared is 4, and so on. Let's set v as a vector. Let's set v as one, two, three as a column vector. You can also do one dot over v to do the element-wise reciprocal of v, so this gives me one over one, one over two, and one over three, and this is where I do the matrices, so one dot over a gives me the element wise inverse of a. And once again, the period here gives us a clue that this an element-wise operation. We can also do things like log(v), this is a element-wise logarithm of the v E to the V is base E exponentiation of these elements, so this is E, this is E squared EQ, because this was V, and I can also do abs V to take the element-wise absolute value of V. So here, V was our positive, abs, minus one, two minus 3, the element-wise absolute value gives me back these non-negative values. And negative v gives me the minus of v. This is the same as negative one times v, but usually you just write negative v instead of -1*v. And what else can you do? Here's another neat trick. So, let's see. Let's say I want to take v an increment each of its elements by one. Well one way to do it is by constructing a three by one vector that's all ones and adding that to v. So if I do that, this increments v by from 1, 2, 3 to 2, 3, 4. The way I did that was, length(v) is 3, so ones(length(v),1), this is ones of 3 by 1, so that's ones(3,1) on the right and what I did was v plus ones v by one, which is adding this vector of our ones to v, and so this increments v by one, and another simpler way to do that is to type v plus one. So she has v, and v plus one also means to add one element wise to each of my elements of v. Now, let's talk about more operations. So here's my matrix A, if you want to buy A transposed, the way to do that is to write A prime, that's the apostrophe symbol, it's the left quote, so it's on your keyboard, you have a left quote and a right quote. So this is actually the standard quotation mark. Just type A transpose, this gives me the transpose of my matrix A. And, of course, A transpose, if I transpose that again, then I should get back my matrix A. Some more useful functions. Let's say lower case a is 1 15 2 0.5, so it's 1 by 4 matrix. Let's say val equals max of A this returns the maximum value of A which in this case is 15 and I can do val, ind max(a) and this returns val and ind which are going to be the maximum value of A which is 15, as well as the index. So it was the element number two of A that was 15 so ind is my index into this. Just as a warning, if you do max(A), where A is a matrix, what this does is this actually does the column wise maximum. But say a little more about this in a second. Still using this example that there for lowercase a. If I do a < 3, this does the element wise operation. Element wise comparison, so the first element of A is less than three so this one. Second element of A is not less than three so this value says zero cuz it's false. The third and fourth elements of A are less than three, so that's just 1 1. So that's the element-wise comparison of all four elements of the variable a < 3. And it returns true or false depending on whether or not there's less than three. Now, if I do find(a < 3), this will tell me which are the elements of a, the variable a, that are less than 3, and in this case, the first, third and fourth elements are less than 3. For our next example, let me set a to be equal to magic(3). The magic function returns, let's type help magic. The magic function returns these matrices called magic squares. They have this, you know, mathematical property that all of their rows and columns and diagonals sum up to the same thing. So, you know, it's not actually useful for machine learning as far as I know, but I'm just using this as a convenient way to generate a three by three matrix. And these magic squares have the property that each row, each column, and the diagonals all add up to the same thing, so it's kind of a mathematical construct. I use this magic function only when I'm doing demos or when I'm teaching octave like those in, I don't actually use it for any useful machine learning application. But let's see, if I type RC = find(A > 7) this finds All the elements of A that are greater than equal to seven, and so r, c stands for row and column. So the 1,1 element is greater than 7, the 3,2 element is greater than 7, and the 2,3 element is greater than 7. So let's see. The 2,3 element, for example, is A(2,3), is 7 is this element out here, and that is indeed greater than equal seven. By the way, I actually don't even memorize myself what these find functions do and what all of these things do myself. And whenever I use the find function, sometimes I forget myself exactly what it does, and now I would type help find to look at the document. Okay, just two more things that I'll quickly show you. One is the sum function, so here's my a, and then type sum(a). This adds up all the elements of a, and if I want to multiply them together, I type prod(a) prod sends the product, and this returns the product of these four elements of A. Floor(a) rounds down these elements of A, so 0.5 gets rounded down to 0. And ceil, or ceiling(A) gets rounded up to the nearest integer, so 0.5 gets rounded up to 1. You can also, let's see. Let me type rand(3), this generates a three by three matrix. If i type max(rand(3), what this does is it takes the element-wise maximum of 3 random 3 by 3 matrices. So you notice all of these numbers tend to be a bit on the large side because each of these is actually the max of a element wise max of two randomly generated matrices. This is my magic number. This is my magic square, three by three A. Let's say I type max A, and then this will be a [], 1, what this does is this texts the column wise maximum. So the max of the first column is 8, max of second column is 9, the max of the third column is 7. This 1 means to take the max among the first dimension of 8. In contrast, if I were to type max A, this funny notation, two, then this takes the per row maximum. So the max of the first row is eight, max of second row is seven, max of the third row is nine, and so this allows you to take maxes either per row or per column. And remember the default's to a column wise element. So if you want to find the maximum element in the entire matrix A, you can type max(max(A)) like so, which is 9. Or you can turn A into a vector and type max(A(:)) like so and this treats this as a vector and takes the max element of that vector. Finally let's set A to be a 9 by 9 magic square. So remember the magic square has this property that every column and every row sums the same thing, and also the diagonals, so just a nine by nine matrix square. So let me just sum(A, 1). So this does a per column sum, so we'll take each column of A and add them up and this is verified that indeed for a nine by nine matrix square, every column adds up to 369, adds up to the same thing. Now let's do the row wide sum. So the sum(A,2), and this sums up each row of A, and indeed each row of A also sums up to 369. Now, let's sum the diagonal elements of A and make sure that also sums up to the same thing. So what I'm gonna do is construct a nine by nine identity matrix, that's eye nine. And let me take A and construct, multiply A element wise, so here's my matrix A. I'm going to do A .^ eye(9). What this will do is take the element wise product of these two matrices, and so this should Wipe out everything in A, except for the diagonal entries. And now, I'm gonna do sum sum of A of that and this gives me the sum of these diagonal elements, and indeed that is 369. You can sum up the other diagonals as well. So this top left to bottom left, you can sum up the opposite diagonal from bottom left to top right. The commands for this is somewhat more cryptic, you don't really need to know this. I'm just showing you this in case any of you are curious. But let's see. Flipud stands for flip up down. But if you do that, that turns out to sum up the elements in the opposite. So the other diagram, that also sums up to 369. Here, let me show you. Whereas eye(9) is this matrix. Flipup(eye(9)), takes the identity matrix, and flips it vertically, so you end up with, excuse me, flip UD, end up with ones on this opposite diagonal as well. Just one last command and then that's it, and then that'll be it for this video. Let's set A to be the three by three magic square game. If you want to invert a matrix, you type pinv(A). This is typically called the pseudo-inverse, but it does matter. Just think of it as basically the inverse of A, and that's the inverse of A. And so I can set temp = pinv(A) and temp times A, this is indeed the identity matrix, where it's essentially ones on the diagonals, and zeroes on the off-diagonals, up to a numeric round off. So, that's it for how to do different computational operations on data and matrices. And after running a learning algorithm, often one of the most useful things is to be able to look at your results, so to plot or visualize your result. And in the next video, I'm going to very quickly show you how again with one or two lines of code using Octave. You can quickly visualize your data or plot your data and use that to better understand what you're learning algorithms are doing.
