Our first learning algorithm will be linear regression. In this video, you'll see
what the model looks like and more importantly you'll see what the overall
process of supervised learning looks like. Let's use some motivating example of predicting
housing prices. We're going to use a data set of housing prices from the city of
Portland, Oregon. And here I'm gonna plot my data set of a number of houses
that were different sizes that were sold for a range of different prices. Let's say
that given this data set, you have a friend that's trying to sell a house and
let's see if friend's house is size of 1250 square feet and you want to tell them
how much they might be able to sell the house for. Well one thing you could do is
fit a model. Maybe fit a straight line to this data. Looks something like that and based
on that, maybe you could tell your friend that let's say maybe he can sell the
house for around $220,000. So this is an example of a
supervised learning algorithm. And it's supervised learning because we're given
the, quotes, "right answer" for each of our examples. Namely we're told what was
the actual house, what was the actual price of each of the houses in our data
set were sold for and moreover, this is an example of a regression problem where
the term regression refers to the fact that we are predicting a real-valued output
namely the price. And just to remind you the other most common type of supervised
learning problem is called the classification problem where we predict
discrete-valued outputs such as if we are looking at cancer tumors and trying to
decide if a tumor is malignant or benign. So that's a zero-one valued discrete output. More
formally, in supervised learning, we have a data set and this data set is called a
training set. So for housing prices example, we have a training set of
different housing prices and our job is to learn from this data how to predict prices
of the houses. Let's define some notation that we're using throughout this course.
We're going to define quite a lot of symbols. It's okay if you don't remember
all the symbols right now but as the course progresses it will be useful
[inaudible] convenient notation. So I'm gonna use lower case m throughout this course to
denote the number of training examples. So in this data set, if I have, you know,
let's say 47 rows in this table. Then I have 47 training examples and m equals 47.
Let me use lowercase x to denote the input variables often also called the
features. That would be the x is here, it would the input features. And I'm gonna
use y to denote my output variables or the target variable which I'm going to
predict and so that's the second column here. [inaudible] notation, I'm
going to use (x, y) to denote a single training example. So, a single row in this
table corresponds to a single training example and to refer to a specific
training example, I'm going to use this notation x(i) comma gives me y(i) And, we're
going to use this to refer to the ith training example. So this superscript i
over here, this is not exponentiation right? This (x(i), y(i)), the superscript i in
parentheses that's just an index into my training set and refers to the ith row in
this table, okay? So this is not x to the power of i, y to the power of i. Instead
(x(i), y(i)) just refers to the ith row of this table. So for example, x(1) refers to the
input value for the first training example so that's 2104. That's this x in the first
row. x(2) will be equal to 1416 right? That's the second x
and y(1) will be equal to 460. The first, the y value for my first
training example, that's what that (1) refers to. So as mentioned, occasionally I'll ask you a
question to let you check your understanding and a few seconds in this
video a multiple-choice question will pop up in the video. When it does,
please use your mouse to select what you think is the right answer. What defined by
the training set is. So here's how this supervised learning algorithm works.
We saw that with the training set like our training set of housing prices and we feed
that to our learning algorithm. Is the job of a learning algorithm to then output a
function which by convention is usually denoted lowercase h and h
stands for hypothesis And what the job of the hypothesis is, is, is a function that
takes as input the size of a house like maybe the size of the new house your friend's
trying to sell so it takes in the value of x and it tries to output the estimated
value of y for the corresponding house. So h is a function that maps from x's
to y's. People often ask me, you know, why is this function called
hypothesis. Some of you may know the meaning of the term hypothesis, from the
dictionary or from science or whatever. It turns out that in machine learning, this
is a name that was used in the early days of machine learning and it kinda stuck. 'Cause
maybe not a great name for this sort of function, for mapping from sizes of
houses to the predictions, that you know.... I think the term hypothesis, maybe isn't
the best possible name for this, but this is the standard terminology that people use in
machine learning. So don't worry too much about why people call it that. When
designing a learning algorithm, the next thing we need to decide is how do we
represent this hypothesis h. For this and the next few videos, I'm going to choose
our initial choice , for representing the hypothesis, will be the following. We're going to
represent h as follows. And we will write this as htheta(x) equals theta0
plus theta1 of x. And as a shorthand, sometimes instead of writing, you
know, h subscript theta of x, sometimes there's a shorthand, I'll just write as a h of x.
But more often I'll write it as a subscript theta over there. And plotting
this in the pictures, all this means is that, we are going to predict that y is a linear
function of x. Right, so that's the data set and what this function is doing,
is predicting that y is some straight line function of x. That's h of x equals theta 0
plus theta 1 x, okay? And why a linear function? Well, sometimes we'll want to
fit more complicated, perhaps non-linear functions as well. But since this linear
case is the simple building block, we will start with this example first of fitting
linear functions, and we will build on this to eventually have more complex
models, and more complex learning algorithms. Let me also give this
particular model a name. This model is called linear regression or this, for
example, is actually linear regression with one variable, with the variable being
x. Predicting all the prices as functions of one variable X. And another name for
this model is univariate linear regression. And univariate is just a
fancy way of saying one variable. So, that's linear regression. In the next
video we'll start to talk about just how we go about implementing this model.