Probabilistic graphical models (PGMs) are a rich framework for encoding probability distributions over complex domains: joint (multivariate) distributions over large numbers of random variables that interact with each other. These representations sit at the intersection of statistics and computer science, relying on concepts from probability theory, graph algorithms, machine learning, and more. They are the basis for the state-of-the-art methods in a wide variety of applications, such as medical diagnosis, image understanding, speech recognition, natural language processing, and many, many more. They are also a foundational tool in formulating many machine learning problems. This course is the first in a sequence of three. It describes the two basic PGM representations: Bayesian Networks, which rely on a directed graph; and Markov networks, which use an undirected graph. The course discusses both the theoretical properties of these representations as well as their use in practice. The (highly recommended) honors track contains several hands-on assignments on how to represent some real-world problems. The course also presents some important extensions beyond the basic PGM representation, which allow more complex models to be encoded compactly.
Moving Data Around
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來自 Stanford University 的課程
Probabilistic Graphical Models 1: Representation
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從本節課中
Bayesian Network (Directed Models)
In this module, we define the Bayesian network representation and its semantics. We also analyze the relationship between the graph structure and the independence properties of a distribution represented over that graph. Finally, we give some practical tips on how to model a real-world situation as a Bayesian network.
與講師見面
Daphne KollerProfessor
School of Engineering
In the second tutorial video on Octave, I'd like to start to tell you how to move
data around in Octave. So, if you have data from machine
learning problem, how you load that data into Octave?
How do you put it into matrix? How do you manipulate these matrices?
How do you save the results? how do you move data around and operate
with data? Here's my Octave, window as before.
picking up from where we left off in the last video.
If I type A, that's the matrix that we generate, right,
with this command equals 1, 2, 3, 4, 5, 6 and the this is a 3 by 2
matrix. The size command in Octave lets you tells
you what is the size of the matrix, so size A returns 3, 2, it turns out the
size command itself is actually returning a 1 by 2 matrix. So you can actually set
SZ equals size of A and SZ is now a 1 by 2 matrix where the first element of this
is 3 and the second element of this is 2. So you can just type size of SZ SZ as a 1
by 2 matrix whose two elements contain the dimensions of the matrix A.
You can also type size(A,1) to give you back the first dimension of A, of,
size of the first dimension of A, so that's the number of rows and size(A,2)
to give you back 2 which is the number of columns in the matrix A.
If you have a vector V, so let's say V equals 1, 2, 3, 4, and you type length V,
what this does is it gives you the sides of the longest dimension.
So you can also type length A and because A is a 3 by 2 matrix, the longer
dimension is of size 3, so this should printout 3.
But usually we apply length only to vectors, so your length 1, 2, 3, 4, 5
rather than apply length to matrices, because that's, that's a little more
confusing. Now, let's look at how to load data and, find
data on a file system. when we start up Octave, we're usually,
we're often in a path that is near the location of where the Octave program is.
So the pwd command shows the current directory or the current path that Octave
is in. So right now, we're in this somewhat, maybe somewhat obscure
directory, the cd command that stands for change directory, so I can go to
c:\User\ang\Desktop and now in my desktop.
And if I type ls ls is a type that comes from a Unix for a Linux command, but ls
will list the directories on my desktop. And so, you know, these are the files
that are on my desktop right now. In fact, on my desktop are two files,
featuresX and priceY, that's maybe come from a machine learning problem were
going to solve. So, here's my desktop, here's featuresX.
And featuresX is this window, excuse me, is this file with two columns of data.
This is actually my housing prices data, so I think you know, I think I have 47
rows in this data sets and so the first house had size 2104 square feet, has
three bedrooms, second house has 1600 square feet, has three bedrooms and so on
and priceY is this file that has the prices of the data in my training set. So
[SOUND] featuresX and priceY are just text files with my data.
How do I load this data into Octave? Well, I just, type load featuresX.dat and
if I do that and load the featuresX X, I can load priceY.dat.
And, by the way, there are multiple ways to do this,
this command, if you put featuresX.dat in strings and load it like so.
This is oops, typo there. This is a equivalent command so you can
you know, but this way, I'm, I'm just putting the
file name of the string in the file name in, in a string and in Octave use single
quotes to you know, to represent strings like so.
So that's a string, and I can load the file whose name is given by that string.
Now, the who command now shows me what variables I have in my Octave workspace,
so who shows me what are the variables that Octaves has in memory currently,
featuresX and priceY are among them, as well as the variables that,
you know, we created earlier in this session.
So I can type featuresX to display featuresX and there's my data.
And I can type size featuresX and that's my 47 by 2 matrix. And sum of the size
priceY, that gives me my 47 by 1 vector for, this
is a 47-dimensional vector, this tall column vector that has all the
pricesY in my training set. Now, the who function shows you what are
the variables that in the current workspace.
There's also the whos variable that gives you the detail view, and so, this also,
with an S at the end, this also lists my variables, except that it now lists the
sizes as well. So A is a 3 by 2 matrix and featuresX is
a 47 by 2 matrix price, priceY is a 47 by 1 matrix, meaning this is just a vector
and it shows, you know, how many bites of memory it's taking up as well as what
type of data this is. Double means double precision, 48 points,
so that just means these are real values of floating point numbers.
Now, if you want to get rid of a variable, you can use the clear command.
So clear featuresX and type whos again, you notice that the featuresX variable
has now disappeared. And how do we save data? Let's see, let's
take the variable V and set it to priceY(1:10), this sets V to be the first
ten elements of of the vector Y. So let's type who or whose,
whereas Y was a 47 by 1 vector. V is now 10 by 1,
because Vprice equals priceY(1:10), this sets into the, just the first ten
elements of Y. Let's say I want to save this to data, to
disc. The command, save hello.matv,
this will save the variable V into a file called hello.mat.
So let's do that. And, now, a file has appeared on my
desktop, you know, called hello, hello.mat I happened to have MATLAB
installed in this Windows, which is why, you know, this this icon looks like this,
because Windows is recognizing this is a MATLAB file. But don't worry about it if
this file looks like it has a different icon on your machine.
And let's say I clear all my variables, if I, if you type clear without anything,
then this actually deletes all the variables on your workspace. so I type
whos, there's now nothing left in my workspace.
And if I load hello.mat, I can now load back my variable V, which
is my the, the data that I previously saved
into the hello.mat file. So, hello.mat or what, what we did just
now, the save hello.mat to V, this saves the data in the binary format
in a somewhat more compressed binary format.
So that if V is a lot of data, this, you know, will be somewhat more compressed
and will take up less of the space. And if you want to save your data in a
human-readable format, then you type save hello.text, the variable V and then
-ascii. So this will save it.
As text, or as ASCII formatted text. And now, once I've done that, I have this
file, hello.txt is just appeared on my desktop.
And if I open this up, you see that this is a text file with my, with my data
saved away. So that's how you load and save data.
Now, let's talk a bit about how to manipulate data.
Let's set A equal to that matrix again. So there's my 3 by 2 matrix.
Let's start by indexing, so if I type A(3,2), this indexes into
the three comma two element of the matrix A.
So this is what a, this is, you know, normally we would write this as A
subscript 3, 2 or A subscript, you know, 3,
2. And so, that's the element in the third
row and second column of A, which is the element 6.
And I can also type A(2, ): to fetch everything in the second row. So the
colon means every element along that row or column.
So, A(2, ): is this second row of A. Right?
And similarly, if I do A(:, 2), then this means get everything in the second column
of A. So, this gives me 2, 4, 6 by this means
of A, everything comma second column,
so this is my second column of A which is 2, 4, 6.
Now, you can also use somewhat most sophisticated indexing operations. So,
so, I'll just quickly show you an example.
You, you do this maybe less often, but let me do A([1 3], ),: this means, get
all the elements of A, whose first index is 1 or 3.
This means get everything from the first and third rows of A and from all
columns. So this was the matrix A, and so A([1 3], ): means get everything from the
first row and from the second row and, and from the third row.
And colon means, you know, I want both the first and the second columns and so
this gives me this 1, 2, 5, 6. Although you, you, use these sorts of
more sophisticated indexing operations, maybe somewhat less often.
To show you what else we can do. Here's the A matrix,
and let's, this was a colon comma to give me the second column.
And you can also use this to do assignments.
So I can take the second column of A and assign that to ten, eleven, twelve.
And if I do that, I'm now, you know, taking the second column with A and I'm
assigning this column vector 10, 11, 12 to it.
So now A is this matrix that's 1, 3, 5, and the second column has been replaced
by 10, 11, 12. And here's another operation, it's A
let's set A, that's A to be equal to A comma, 100, 101, 102, like so, and, what
this will do, is it appends another column n vector to the right.
So now oops, I think I made a mistake. Should I put semicolons there?
And now, A is equal to this. Okay? I hope that makes sense.
So this 100, 101, 102, this is a column vector.
And what we did was, we said AA take A and set it to the original definition and
then we put that column vector to the right and so we ended up taking the
matrix A and which was this, these six elements on the left.
So we took the matrix A and we appended another column vector to the right, which
is now Y, now A is a 3 by 3 matrix that looks like that.
And finally, when we click that I sometimes use do A and then just a colon
like so, this is a somewhat special case syntax.
what this means is that, puts all elements of A into a single column vector
and this gives me a 9 by 1 vector that just has all the algorithms of A
[INAUDIBLE] together. just, a couple more examples.
I can also, let's see.
Let's say I set A to be equal to 1, 2, 3, 4, 5, 6, again. And, let's see I
said B to be equal to 11, 12, 13, 14, 15, 16.
I can create a new matrix C as [A B] and this just means, so here's my matrix A,
here's my matrix B, and off a set C to be equal to AB.
What I'm doing is I'm taking these two matrices and just concatenating them onto
each other. And so the left, [INAUDIBLE] matrix A on
the left. And I have the matrix B on the right.
And that's how I form this, you know? this matrix C, by putting them together.
I can also do C = [A; B]. The semi colon notation means that means,
it, means go put the next thing at the bottom. So we do sequence A sem colons B
it also puts the matrices A and B together, except that it now puts them on
top of each other, so now you have A on top and B at the bottom and C here, is
now a 6 by 2 matrix. So, so just saying, the semi colon thing
usually means you go to the next line. So C is comprised by A and then go to the
bottom of that and then put B, the bottom and by the way, this AB is the
same as A, B and so, you know, either of these gives you the same result.
So, with that, hopefully you now know how to construct matrices and hopefully this
starts to show you some of the commands that you can use to quickly put together
matrices and take matrices and, you know, slam them together to form big, bigger
matrices. And with just a few lines of code, octave
is very convenient in terms of how quickly we can assemble complex matrices
and move data around. So that's it for moving data around, in
the next video we'll start talk about how to actually do complex computations on
this, on, on, on our data. So hopefully that gives you a sense of
how, with just a few commands, you can very quickly move data around in Octave.
You know, load and save vectors and matrices or load and save data put
together matrices create bigger matrices indexed into or select specific elements
of the matrices. I know I went through a lot of commands,
so I think the best thing for you to do is afterwards to look at the transcript
of the things I was typing, you know, look at the, look at the
coursework sign and download the transcript of the session from there.
And look for the transcript and type some of those commands into Octave yourself,
so that you can start to play with these commands and get it to work.
And obviously, you know, there's no point at all to try to memorize all these
commands, it's just, but what you should do is hopefully from
this video you have gotten a sense of the sorts of things you can do, so that when,
later on, when you're trying to program a learning algorithms system yourself.
If you are trying to find a specific amount that maybe you think Octave can do
because you think you might see it here you should refer to the, to the
transcript of the session and look through that in order to find the
commands you want to use. So, that's it for moving data around and
in the next video, what I'd l like to do is start to tell you how to actually do
complex computations on our data and how to compute some data and how to actually
start to implement learning algorithms.
