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Subtitle preview

First 50 subtitle cues from .().eng.srt

50 cues

#100:00:00,000 --> 00:00:05,039
In this video we'll define something called the cost function. This will let us
#200:00:05,039 --> 00:00:10,068
figure out how to fit the best possible straight line to our data. In linear
#300:00:10,068 --> 00:00:16,075
regression we have a training set like that shown here. Remember our notation M
#400:00:16,075 --> 00:00:21,097
was the number of training examples. So maybe M=47. And the form of the
#500:00:21,097 --> 00:00:27,073
hypothesis, which we use to make predictions, is this linear function. To
#600:00:27,073 --> 00:00:33,072
introduce a little bit more terminology, these theta zero and theta one, right,
#700:00:33,072 --> 00:00:39,075
these theta i's are what I call the parameters of the model. What we're
#800:00:39,075 --> 00:00:44,057
going to do in this video is talk about how to go about choosing these two
#900:00:44,057 --> 00:00:49,065
parameter values, theta zero and theta one. With different choices of parameters
#1000:00:49,065 --> 00:00:54,040
theta zero and theta one we get different hypotheses, different hypothesis
#1100:00:54,040 --> 00:00:59,035
functions. I know some of you will probably be already familiar with what I'm
#1200:00:59,035 --> 00:01:04,036
going to do on this slide, but just to review here are a few examples. If theta
#1300:01:04,036 --> 00:01:09,037
zero is 1.5 and theta one is 0, then the hypothesis function will look like
#1400:01:09,037 --> 00:01:15,070
this. Right, because your hypothesis function will be h( x) equals 1.5 plus
#1500:01:15,070 --> 00:01:22,064
0 times x which is this constant value function, this is flat at 1.5. If
#1600:01:22,064 --> 00:01:29,033
theta zero equals 0 and theta one equals 0.5, then the hypothesis will look
#1700:01:29,033 --> 00:01:35,053
like this. And it should pass through this point (2, 1), says you now have h(x) or
#1800:01:35,053 --> 00:01:40,066
really some htheta(x) but sometimes I'll just omit theta for
#1900:01:40,066 --> 00:01:46,051
brevity. So, h(x) will be equal to just 0.5 times x which looks like that. And
#2000:01:46,051 --> 00:01:52,044
finally if theta zero equals 1 and theta one equals 0.5 then we end up with the
#2100:01:52,044 --> 00:01:58,059
hypothesis that looks like this. Let's see, it should pass through the (2, 2)
#2200:01:58,059 --> 00:02:04,046
point like so. And this is my new h(x) or my new htheta(x). All right? Well
#2300:02:04,046 --> 00:02:09,097
you remember that this is htheta(x) but as a shorthand
#2400:02:09,097 --> 00:02:16,058
sometimes I just write this as h(x). In linear regression we have a training set,
#2500:02:16,058 --> 00:02:22,043
like maybe the one I've plotted here. What we want to do is come up with values for
#2600:02:22,043 --> 00:02:28,029
the parameters theta zero and theta one. So that the straight line we get out
#2700:02:28,029 --> 00:02:33,079
of this corresponds to a straight line that somehow fits the data well. Like
#2800:02:33,079 --> 00:02:39,075
maybe that line over there. So how do we come up with values theta zero, theta one
#2900:02:39,075 --> 00:02:45,034
that corresponds to a good fit to the data? The idea is we're going to choose
#3000:02:45,034 --> 00:02:51,016
our parameters theta zero, theta one so that h(x), meaning the value we predict
#3100:02:51,016 --> 00:02:56,033
on input x, that this at least close to the values y for the examples in our
#3200:02:56,033 --> 00:03:01,013
training set, for our training examples. So, in our training set we're given a
#3300:03:01,013 --> 00:03:06,050
number of examples where we know x decides the house and we know the actual price of
#3400:03:06,050 --> 00:03:11,079
what it's sold for. So let's try to choose values for the parameters so that
#3500:03:11,079 --> 00:03:17,030
at least in the training set, given the x's in the training set, we make
#3600:03:17,030 --> 00:03:23,050
reasonably accurate predictions for the y values. Let's formalize this. So linear
#3700:03:23,050 --> 00:03:29,040
regression, what we're going to do is that I'm going to want to solve a minimization
#3800:03:29,040 --> 00:03:38,078
problem. So I'm going to write minimize over theta zero, theta one. And, I want this to be
#3900:03:38,078 --> 00:03:44,037
small, right, I want the difference between h(x) and y to be small. And one
#4000:03:44,037 --> 00:03:50,049
thing I'm gonna do is try to minimize the square difference between the output of
#4100:03:50,049 --> 00:03:56,015
the hypothesis and the actual price of the house. Okay? So let's fill in some
#4200:03:56,015 --> 00:04:01,037
details. Remember that I was using the notation (x(i), y(i)) to represent the
#4300:04:01,037 --> 00:04:07,041
ith training example. So what I want really is to sum over my training
#4400:04:07,041 --> 00:04:13,020
set. Sum from i equals 1 to M of the square difference between
#4500:04:13,020 --> 00:04:18,089
this is the prediction of my hypothesis when it is input the size of house number
#4600:04:18,089 --> 00:04:24,037
i, right, minus the actual price that house number i will sell for and I want to
#4700:04:24,037 --> 00:04:29,058
minimize the sum of my training set sum from i equals 1 through M of the
#4800:04:29,058 --> 00:04:35,028
difference of this squared error, square difference between the predicted
#4900:04:35,028 --> 00:04:41,009
price of the house and the price that it will actually sell for. And just
#5000:04:41,009 --> 00:04:47,072
remind you of your notation M here was the, the size of my training set, right,