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,
we'll be these source of computational
operations to implement our learning
algorithms. Let's get started. Here's my
active window. Let me just quickly
initialize some variables to use for our
examples and 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 wanna
compute A times C. I just type A times C.
So it's a three by two matrix times a two
by2 matrix. This gives me this three by
two matrix. You can also do elements-wise
operations, and do A. Times B. And what
this will do, is, it'll take each element
of A, and multiply it by the corresponding
elements of B. So that's A times B, that's
A. Times B. So, for example, the first
el-, first element. Is one times eleven
which gives eleven the second element
gives two times twelve which gives 24 and
so on, so the element wise multiplication
of two matrices and in general the P rate
tends to is usually used to denote element
wise operations in octave so here's a
matrix A and if I do A thought to carry
two this gives me the multi-. The element
wise squaring of A so you know one squared
is one, two squared is four, and so on.
Let's set V to a vector. We'll set v as
1,2,3 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.
This [inaudible] matrices. So one dot over
V gives me the reciprocal of one over A.
And once again the period here gives us a
clue that this is an elements wise
operation. It also do things like log b
this is a element wise logarithm. The
element of the b, e to the b is base e
exponentiation of these elements. So this
is e, this is e squared [inaudible], this
is b. And I can also do abs b to take the
element wise absolute value of b. So here
you know, b was all positive abs -one to
say -three. The element wise absent value
get's me back these non-negative values
and. Negative E gives me the minus of E
this is the same as negative one times B
but usually just for a negative B so a
negative one times B and What else can you
do? Here's another neat trick. So, let's
see. Let's say I want to take v and
increment each other's elements by one.
Well, one way to do it is by constructing
a 3-by-1. Vector is all 1's and adding
that to D so they do that this increments
D by from one two three to two three four
the way I did that was length. Of V is V
so 1's macro V by one this is let one V by
one so let's one V by one. Alright and
what I did was v plus one three by one,
which is adding this vector of all ones to
v. So this increments v by one. And you,
another simpler way to do that is to
actually type v plus one such as v and v
plus one also is to add one otherwise to
each of my elements of v. Now. Let's talk
about, more operations. So here's my
matrix A. If you want to write A
transpose, the way to do that is to write
A prime. That's the, apostrophe symbol, is
the left quote. So in, in, on your
keyboard, you probably have a left quote,
and a right quote. So this is, a, a,
excuse me, this is actually a standard
quotation mark, 'cause, you just type A
transpose. This gives me the, you know,
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
11520.5. So it's a you know one-fourth
matrix. Let's VAL = max of A this returns
the maximum value of A which in this case
is fifteen and I can do VAL end max A and
this returns VAL and end which are going
to be the maximum value of A which is
fifteen as well as the index so is the
element number two of A that is fifteen so
end 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 a column wise maximum. But say a
little bit more about this in a second. So
using this example, the variable lowercase
a, if I do a less than three, this does
the element-wise operation, element-wise
comparison, so, the first element of a is
less than three, [inaudible] it's a one.
The second element of A is not less than
three, so this evaluates as zero, cuz it's
false. The third and fourth element of a.
Are of this I mean less than three
[inaudible] less than three so it's just
one, one. So this does the element wise
comparison of all four elements of
[inaudible] lower case a to three and it
returns true or false depend on, on
whether or not it's less than three. Now
if I can find a less than three this will
tell me which are the elements of A.
Therefore [inaudible] in this case the
first third and fourth elements are less
than three. For my next example let me set
A to be equal to magic three the magic
function returns let's type help magic,
the magic function returns functions
called magic, returns functions called
matrices squared, they have this your
mathematical property that all of the rows
and columns and diagonals sum up to the
same thing so you know, is not actually
useful for machine learning as far as I
know but I'm, I'm just using this as a
convenient way. You know, to generate a
three by three matrix. And, and these
magic squares had prop-, had 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 magic, I use this magic function only
when I'm doing demos, or when I'm teaching
Octave like this. And I don't actually use
it for any you know, useful machines or
any application. But let's see if I type
RC=Find A greater than=7, this finds All
the elements of a that I grid to equals
seven. And so, ASI sets our row in
columns. So the one, one element is going
to seven, the three two elements go into
seven, and the two three elements is going
into seven. So, let's see. The two three
element, for example, is a, two, three is
seven, is this elements. Out here and that
is indeed greater than seven. By the way I
actually don't even memorize myself what
these fine functions do and what all of
these things do all myself. And when I use
the fine function sometimes I forget
myself exactly what it does and you know,
I type help find to look up. Okay, just
two more [inaudible] I'll briefly show
you. One is the sum function. So here's my
A and type sum A. This adds up all the
elements of A and to multiply them
together I take prod A [inaudible] and
this returns the product of these four
elements of A. [inaudible] these elements
of a. So 0.5 gets rounded down to zero and
ceiling A gets rounded up, 0.5 rounded up
to nearest integer, so 0.5 gets rounded up
to one. You can also let's see let me type
rand V this generates a 3x3 matrix if I
type max rand three rand three what this
does is it takes the element of Y maximum
of two random 3x3 matrices. So you notice
all of these numbers tend to be a bit on
the large size because each of these is
actually the max of a randomly of an
element wise max of two randomly generated
matrices. This is this is my matrix number
this is my matrix square 3x3. A, let's say
I type max A Then this with. Open closed
square brackets comma one. What this does
is [inaudible] column wise maximum. So
maximum of first column is eight, maximum
of second column is nine, the maximum of
third column is seven. This one
[inaudible] max along the first dimension
of A. In contrast if I were to take max A
this plenty 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 max's either per
row or per column. And if you want,
remember it defaults to column [inaudible]
wise elements. If you want to find the
maximum element in the entire matrix A,
you can type max of max of A, like so,
which is nine, or you can turn A into a
vector and type max of A colon, like so,
and this treats this as a vector and takes
the max elements of our vector. Finally.
Let's set A to be a nine by nine magic
square. So remember the magic square has
this property that every column and every
row sums the same thing and also the
diagonal. So here is a nine by nine matri
magic square. So let me do sum A one so
this does a per column sum. So I want to
take each column of A and add them up. And
this, you know, lesson's verified that
indeed for a nine by nine magic square
every column adds up to 369, adds up to
the same thing. Now lets do the row wide
sum so the sum A [inaudible]. Two, and
this. Sums up each row of A and indeed
each row of A also sums up 2639. Now lets
sum the diagonal elements of A and make
sure that they that, that also sums up to
the same thing so what I'm going to do is.
Constructing 9x9 identity matrix that's
I9. And gonna take A and construct you
know multiply A elements wise, so here's
my matrix A. And to A times I9 and 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 going to
do sum, sum A of that and this gives me
the Some of this these [inaudible]
elements and indeed the 3,6,9 you can sum
up the other diagonal as well so it's top
left to bottom right you can sum up the
opposite diagonal from bottom left to top
right the sum of the command for this is
somewhat more cryptic you don't really
need to know this I am just showing you
this in case any of you are curious but
let's see. So if you so if you [inaudible]
stands up down but if you do that, that
turns out to sum up the elements in the
opposite of the other. Diagonal feed that
also sums up to [inaudible] here I'll show
you where as I9 is this matrix, that.
Uptown of I9. You know takes the identity
matrix and flips it vertically so that you
end up with excuse me [inaudible] and that
with 1's on this opposite diagonal as
well. Just one last command and then
that's it and then that'll be it for
[inaudible]. Lesser a to be the magic, the
magic square game and you want to invert a
matrix you type p is a. This is
[inaudible] called a pseudo inference, but
it doesn't matter. Just think of it as
basically inverse of a and that's the
inverse of a. And so you can set, you
know, temp equals pn of a and of a temp
times a. This is indeed the identity
matrix where essentially ones on the
diagonal and zeros on the off diagonal up
to a numerical round off. So, that's it
for how to do, different computational
operations on the 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 it's a
plot, a visualizer result. And in the next
video, I'm going to very quickly show you
how, again, with one or two lines of code,
using [inaudible], you can quickly
visualize your data or plot your data.
And, use that to better understand, you
know, what your learning algorithms are