1
00:00:00,633 --> 00:00:02,966
Hello and welcome back to the course
on Machine Learning.

2
00:00:02,966 --> 00:00:05,900
Today we're talking
about the polynomial regression.

3
00:00:05,900 --> 00:00:07,666
So let's get straight into it.

4
00:00:07,666 --> 00:00:09,866
We already know
a couple of types of regressions.

5
00:00:09,866 --> 00:00:12,600
We know the simple linear regression
which we can see over here.

6
00:00:12,600 --> 00:00:13,533
Then we've also discussed

7
00:00:13,533 --> 00:00:17,233
the multiple linear regression
which is written out over here.

8
00:00:17,866 --> 00:00:21,633
And finally we've got the polynomial
linear regression which is written out

9
00:00:21,633 --> 00:00:22,100
here.

10
00:00:22,100 --> 00:00:25,466
So notice how it's very similar
to the multiple linear regression.

11
00:00:25,700 --> 00:00:28,933
But at the same time
instead of the different variables

12
00:00:28,933 --> 00:00:32,200
like x2, x3, x4 and so on zn

13
00:00:32,533 --> 00:00:37,400
we have the same variable x1,
but it is in a different power.

14
00:00:37,400 --> 00:00:42,233
So instead of x2 we have x1 squared,
instead of x3 would have x1 cubed.

15
00:00:42,533 --> 00:00:45,866
And so instead of x n
we would have x1 to the power of n.

16
00:00:46,233 --> 00:00:48,100
So basically we're using one variable.

17
00:00:48,100 --> 00:00:51,166
But we're using the different powers
of that variable.

18
00:00:51,533 --> 00:00:52,266
So let's have a look at

19
00:00:52,266 --> 00:00:55,500
when you would use a polynomial regression
when it would come in handy.

20
00:00:56,100 --> 00:01:00,300
Let's say we've got a observation
a set of observations which is then

21
00:01:00,300 --> 00:01:05,333
the line that fits this data is obviously
a simple linear regression.

22
00:01:05,333 --> 00:01:07,366
As you can see,
it feels fitted quite well.

23
00:01:07,366 --> 00:01:11,200
But let's, for a change, say that the data
set looked something like this.

24
00:01:11,800 --> 00:01:14,633
So if we try to use a simple linear
regression here

25
00:01:14,633 --> 00:01:16,466
which is expressed like that.

26
00:01:16,466 --> 00:01:18,866
You'll see that it doesn't fit quite well.

27
00:01:18,866 --> 00:01:21,500
So in the middle
you've got data underneath.

28
00:01:21,500 --> 00:01:24,600
And then as you go further
the data will be above the line.

29
00:01:24,900 --> 00:01:26,566
So how can we correct that.

30
00:01:26,566 --> 00:01:30,100
Well we can try to correct that
by using a polynomial regression.

31
00:01:30,100 --> 00:01:30,800
Let's have a look.

32
00:01:30,800 --> 00:01:34,800
So instead of the linear regression we're
going to conduct a polynomial regression.

33
00:01:34,800 --> 00:01:37,800
And that in this case fits perfectly.

34
00:01:38,100 --> 00:01:39,533
And what is a formula.

35
00:01:39,533 --> 00:01:43,700
Well that is a formula
for this particular case y equals b0.

36
00:01:43,700 --> 00:01:46,666
So that's a constant plus b1 x1.
So that's a simple linear regression part.

37
00:01:46,666 --> 00:01:49,766
But then we're adding the b2 x1 squared.

38
00:01:50,000 --> 00:01:55,200
And the b2 x1 squared gives its
that parabolic effect so that the curve

39
00:01:55,200 --> 00:01:58,466
becomes parabolic and therefore
it will fit this data better.

40
00:01:58,900 --> 00:01:59,433
As you can see,

41
00:01:59,433 --> 00:02:03,300
polynomial regression is a bit different
to a simple linear regression.

42
00:02:03,600 --> 00:02:06,233
And at the same time
it has its own use cases.

43
00:02:06,233 --> 00:02:09,233
So it's all in comes on a case by case
basis. You.

44
00:02:09,500 --> 00:02:10,566
You have a problem.

45
00:02:10,566 --> 00:02:14,466
And then you might try a simple linear
regression or multiple linear regression

46
00:02:14,666 --> 00:02:15,600
if you have many variables.

47
00:02:15,600 --> 00:02:19,500
Or you might try a polynomial linear
regression and see what happens.

48
00:02:19,500 --> 00:02:22,566
And sometimes the polynomial regressions
do work better.

49
00:02:22,800 --> 00:02:27,033
For example, they're used to describe
how diseases spread

50
00:02:27,033 --> 00:02:32,466
or pandemics and epidemics are spread
across territory or across population.

51
00:02:32,700 --> 00:02:35,400
Polynomial linear
regressions can be handy there.

52
00:02:35,400 --> 00:02:37,200
And they also have other use cases.

53
00:02:37,200 --> 00:02:39,200
So it's a matter of what works best.

54
00:02:39,200 --> 00:02:42,200
So it's always good
to have more tools in your arsenal.

55
00:02:42,533 --> 00:02:44,833
And we have one final question left.

56
00:02:44,833 --> 00:02:48,100
The question is
why is it called linear still.

57
00:02:48,133 --> 00:02:48,366
Right.

58
00:02:48,366 --> 00:02:53,033
So we saw those different powers squared
cube two to the power of n and so on.

59
00:02:53,066 --> 00:02:55,733
Why is it still called linear.
And I'll show you what I mean.

60
00:02:55,733 --> 00:02:59,066
If you look on the left here it says
polynomial linear regression.

61
00:02:59,800 --> 00:03:03,633
So why is it still called
a linear regression if it's a polynomial

62
00:03:03,633 --> 00:03:04,466
regression?

63
00:03:04,466 --> 00:03:08,366
Well, the trick here is that when we're
talking about linear and non-linear,

64
00:03:08,366 --> 00:03:11,566
we're not actually talking
about the x variables.

65
00:03:11,766 --> 00:03:12,000
Right.

66
00:03:12,000 --> 00:03:15,800
So even though they're non linear here
the relationship between y and x is.

67
00:03:16,733 --> 00:03:20,600
When you're talking about the class
of a regression you're talking.

68
00:03:20,600 --> 00:03:25,000
So whether it's linear non-linear you're
talking about the coefficients here.

69
00:03:25,033 --> 00:03:26,433
So that's the interesting part.

70
00:03:26,433 --> 00:03:29,900
So whether or not this function
which we have here.

71
00:03:29,900 --> 00:03:32,400
So y is a function of x right.

72
00:03:32,400 --> 00:03:37,133
And so the question is can
this function be expressed as a linear

73
00:03:37,133 --> 00:03:42,400
combination of these coefficients that
because ultimately they are the unknowns.

74
00:03:42,400 --> 00:03:42,600
Right.

75
00:03:42,600 --> 00:03:46,500
So your goal when you're building a
regression is to find these coefficients,

76
00:03:46,733 --> 00:03:49,733
find out their actual values
so that then further down the track

77
00:03:49,800 --> 00:03:54,166
you can use those coefficients
to then plug in x and predict y

78
00:03:54,166 --> 00:03:58,166
whether it's a linear
well it's a simple linear multiple linear

79
00:03:58,166 --> 00:03:59,766
regression
or polynomial linear regression.

80
00:03:59,766 --> 00:04:02,200
That's your goal
to find these B coefficients.

81
00:04:02,200 --> 00:04:06,000
And that's why linear
non linear refers to the coefficients.

82
00:04:06,633 --> 00:04:09,033
So an example of a nonlinear regression

83
00:04:09,033 --> 00:04:14,666
would be
if the equation was y equals b0 plus b1 x1

84
00:04:14,666 --> 00:04:18,333
divided by b2 plus x2 or something,

85
00:04:18,566 --> 00:04:22,033
or a b0 divided by b1 plus x1.

86
00:04:22,033 --> 00:04:26,300
So a situation where you really cannot
replace the coefficients

87
00:04:26,300 --> 00:04:29,400
with other coefficients
to turn the equation into a linear one

88
00:04:29,700 --> 00:04:33,266
in regards to the coefficients,
not the x values.

89
00:04:33,866 --> 00:04:34,500
So there you go.

90
00:04:34,500 --> 00:04:38,366
That's why a polynomial regression
is still called a linear regression.

91
00:04:38,366 --> 00:04:39,866
And that's your fun fact for the day.

92
00:04:39,866 --> 00:04:41,700
And maybe you can show off
to your colleagues.

93
00:04:41,700 --> 00:04:45,733
And also because of that,
the polynomial linear regression

94
00:04:45,733 --> 00:04:50,633
is actually a special case
of the multiple linear regression.

95
00:04:51,066 --> 00:04:54,666
So that's just something
to also kind of note that

96
00:04:54,666 --> 00:04:57,900
this is a version of the multiple linear
regression.

97
00:04:58,233 --> 00:05:01,533
Rather than a standalone
absolutely new type of regression.

98
00:05:01,900 --> 00:05:06,233
So I hope you enjoyed today's tutorial and
I look forward to seeing you next time.

99
00:05:06,266 --> 00:05:08,200
Until then, enjoy machine learning.