0
1
00:00:00,370 --> 00:00:04,450
So, speaking of data, let's create some data for our example.
1

2
00:00:04,590 --> 00:00:10,240
I'll add a few more cells here so that I'm working more in the middle of the screen and then I'm gonna
2

3
00:00:10,260 --> 00:00:19,650
add a comment, say "Make sample data" and I'm going to create two variables here - an x_5 variable
3

4
00:00:20,310 --> 00:00:23,360
and a y_5 variable.
4

5
00:00:23,390 --> 00:00:24,490
That's our fifth example,
5

6
00:00:24,510 --> 00:00:35,820
after all. Our x_5 is going to be equal to a numpy array, so "np.array(
6

7
00:00:36,240 --> 00:00:37,570
[])",
7

8
00:00:37,950 --> 00:00:40,140
and I'm going to punch in here seven values.
8

9
00:00:40,230 --> 00:00:45,360
So if you type in something else, then we're gonna get different results.
9

10
00:00:45,450 --> 00:00:56,200
But I'm going to punch in 0.1, 1.2, 2.4, 3.2, 4.1,
10

11
00:00:57,130 --> 00:01:01,800
5.7 and 6.5.
11

12
00:01:03,010 --> 00:01:05,150
So these are gonna be our X values,
12

13
00:01:05,230 --> 00:01:09,540
seven of them in total. Our y values,
13

14
00:01:09,580 --> 00:01:11,110
so y_5
14

15
00:01:11,320 --> 00:01:15,160
is gonna be equal to also an np array,
15

16
00:01:15,250 --> 00:01:18,470
same thing parentheses, square brackets and here
16

17
00:01:18,470 --> 00:01:29,670
we're going to punch in 1.7, 2.4, 3.5, 3.0, 6.1
17

18
00:01:30,690 --> 00:01:36,130
9.4 and 8.2.
18

19
00:01:36,150 --> 00:01:43,410
So we have two arrays and let's print out the shape of these arrays below.
19

20
00:01:43,410 --> 00:01:55,040
So I'm going to say "print('Shape of x_5 array :', x_5.
20

21
00:01:55,620 --> 00:02:01,620
shape)" and we're gonna do the same thing for our y array.
21

22
00:02:01,810 --> 00:02:09,090
I'm going to fix my typo, shape and then I'm going to do the same thing for our y array - I'm going to say print and
22

23
00:02:09,090 --> 00:02:19,020
then add a string "'Shape of y_5 array: ', y_5.shape".
23

24
00:02:19,470 --> 00:02:22,130
So let's take a look at the shape of these arrays.
24

25
00:02:22,560 --> 00:02:29,500
So they're both flattened meaning they only have one dimension and they've got seven values.
25

26
00:02:29,810 --> 00:02:30,260
Brilliant.
26

27
00:02:30,270 --> 00:02:35,500
So there's our fake sample data and we've added our print statements.
27

28
00:02:35,610 --> 00:02:41,680
Now let's fit a linear regression to this sample data.
28

29
00:02:41,760 --> 00:02:47,640
That way we can calculate the intercept and the slope parameter and we can check our work that we're
29

30
00:02:47,640 --> 00:02:51,540
gonna do against what we know to be correct.
30

31
00:02:51,570 --> 00:02:59,490
So scroll to the very, very top of the file where you're adding your import statements and then in
31

32
00:02:59,490 --> 00:03:06,140
the cell add "from sklearn.linear_model
32

33
00:03:06,570 --> 00:03:13,190
import LinearRegression", capital L,
33

34
00:03:13,190 --> 00:03:25,120
capital R. Remember to hit Shift+Enter and then scroll back down to where you left off and let's run
34

35
00:03:25,180 --> 00:03:29,060
a quick regression. So I'm going to add a comment
35

36
00:03:29,110 --> 00:03:39,550
"Quick linear regression", we're gonna run it like so, we're gonna say "regr = LinearRegresson",
36

37
00:03:41,600 --> 00:03:50,450
capital L, capital R, parentheses at the end and then we're gonna see "regr.fit()" then
37

38
00:03:50,630 --> 00:03:57,500
we're gonna supply two arguments within the parentheses - x_5, comma and then 
38

39
00:03:57,560 --> 00:03:59,160
y_5.
39

40
00:03:59,930 --> 00:04:05,320
So these are two arrays, our two pieces of data.
40

41
00:04:05,420 --> 00:04:08,040
Now let's see what happens.
41

42
00:04:08,060 --> 00:04:16,300
Because I'm going tell you right now this isn't gonna work just like that. When we hit Shift+Enter,
42

43
00:04:16,340 --> 00:04:24,320
we can scroll down and take a look at the error - what we get is a value error - "Found input variables with
43

44
00:04:24,380 --> 00:04:30,770
inconsistent number of samples: [1,7]".
44

45
00:04:30,860 --> 00:04:38,150
So that's a very cryptic error message, but let's take a look at the quick documentation for this fit
45

46
00:04:38,510 --> 00:04:41,690
method and see what we need.
46

47
00:04:41,690 --> 00:04:42,490
So.
47

48
00:04:42,890 --> 00:04:45,940
So our fit method takes an X and Y.
48

49
00:04:46,100 --> 00:04:53,310
So the X is our training data and the Y are our target values.
49

50
00:04:53,330 --> 00:05:02,270
Now the interesting thing to note here is that this method requires a two dimensional input.
50

51
00:05:02,320 --> 00:05:09,960
So at the moment our shape of our x ray and our y array is actually one dimensional it's just got seven
51

52
00:05:09,960 --> 00:05:11,010
values in it.
52

53
00:05:11,070 --> 00:05:15,780
But we can see here that this method requires a two dimensional inputs.
53

54
00:05:16,020 --> 00:05:20,070
It needs to know how many samples and how many features.
54

55
00:05:20,070 --> 00:05:22,130
Now we've got only one feature right.
55

56
00:05:22,140 --> 00:05:28,800
We've only got some x values but we still have to pay attention to the kind of array that we're giving.
56

57
00:05:29,020 --> 00:05:36,360
So even though we've only got one feature namely our Xs we still have to pass in a two dimensional array
57

58
00:05:37,290 --> 00:05:40,180
and the same is true for our wise.
58

59
00:05:40,260 --> 00:05:47,700
So even though we're only predicting one value namely the y value we still have to pass in a two dimensional
59

60
00:05:48,060 --> 00:05:48,910
array.
60

61
00:05:48,960 --> 00:05:54,500
So let's go back up here and take a look at where we're creating these arrays.
61

62
00:05:54,600 --> 00:06:02,100
We're going to have to add some additional Python code to reshape or transform our x's and our y array
62

63
00:06:02,700 --> 00:06:07,190
to meet this requirement from our fit method from our regression.
63

64
00:06:07,230 --> 00:06:11,000
I'll show you two possible ways we can solve this problem.
64

65
00:06:11,040 --> 00:06:13,430
Here's the first one - for our x's.
65

66
00:06:13,470 --> 00:06:24,540
If I add another pair of square brackets inside the parentheses and then I put a dot afterwards and
66

67
00:06:24,540 --> 00:06:32,520
write transpose, yeah with parentheses at the end and hit Shift+Enter,
67

68
00:06:33,240 --> 00:06:38,430
then you'll see in the print statement that our X array now is two dimensional.
68

69
00:06:38,430 --> 00:06:41,940
It's got seven rows and one column.
69

70
00:06:42,420 --> 00:06:49,950
So this is one way you can take a one dimensional array which was just seven values and make it into
70

71
00:06:50,100 --> 00:06:57,780
two dimensions even though it's only got one column - two pairs of square brackets and then calling the
71

72
00:06:57,840 --> 00:07:07,150
transpose method. Another way you can accomplish the very very same thing is by using a method called
72

73
00:07:07,360 --> 00:07:08,140
reshape.
73

74
00:07:08,140 --> 00:07:16,540
So writing ".reshape" and then in the parentheses I'm going to supply how many rows I want - 7
74

75
00:07:17,140 --> 00:07:25,410
comma and then how many columns I want - namely 1; 7 rows 1 column.
75

76
00:07:25,410 --> 00:07:31,220
Let me hit Shift+Enter and update the shape of our y array.
76

77
00:07:31,400 --> 00:07:32,010
Voila!
77

78
00:07:32,750 --> 00:07:34,850
I get exactly the same result.
78

79
00:07:34,920 --> 00:07:37,810
So two ways you can do it.
79

80
00:07:38,040 --> 00:07:43,650
Now with that in place I'm going to click into our cell below where we're doing our regression
80

81
00:07:44,310 --> 00:07:49,560
and I'm going to hit Shift+Enter and voila! Our error goes away.
81

82
00:07:49,560 --> 00:07:52,840
So we've managed to fit our regression perfectly.
82

83
00:07:52,920 --> 00:07:54,790
Now what are we actually interested in?
83

84
00:07:54,810 --> 00:07:57,220
We're interested in the theta values right - 
84

85
00:07:57,240 --> 00:08:02,220
the intercept and our slope parameter - our theta zero and our theta one.
85

86
00:08:02,340 --> 00:08:03,290
Let's print these out.
86

87
00:08:03,780 --> 00:08:16,030
I'm going to put a print statement here and write the string "'Theta 0:', regr.intercept_
87

88
00:08:16,060 --> 00:08:19,290
[0]".
88

89
00:08:19,420 --> 00:08:25,990
That's how we got our intercept, our theta zero; and then I'm going to add another print statement "'Theta
89

90
00:08:25,990 --> 00:08:37,900

91
1: ', regr.coef_" and this was two dimensional remember.
90

92
00:08:37,900 --> 00:08:43,780
So we had s[0] and then another square brackets with a zero.
91

93
00:08:43,900 --> 00:08:54,070
So first column first row of this two dimensional coefficient array. I'm going to hit Shift+Enter we can see
92

94
00:08:54,190 --> 00:09:01,600
what the values. Our intercept is at 0.85 approximately and our theta one, our
93

95
00:09:01,600 --> 00:09:07,120
slope has the value 1.2 approximately.
94

96
00:09:07,200 --> 00:09:15,270
So now that we have the parameters that describe our best fit line let's throw them up on screen because
95

97
00:09:15,300 --> 00:09:19,580
I'm a very, very visual person and I'm sure it'll help you as well.
96

98
00:09:19,590 --> 00:09:28,950
So I'm going to add a few more cells, scroll down and now let's plot this stuff. First let's plot our data points.
97

99
00:09:29,050 --> 00:09:29,790
So this
98

100
00:09:29,790 --> 00:09:37,490
we can do with a scatter plot so "plt.scatter" then in the parentheses we'll have "x_5,
99

101
00:09:37,500 --> 00:09:46,400
y_5", and then size for these points maybe at 50.
100

102
00:09:46,420 --> 00:09:52,270
So that's our scatter plot and now we'll add our best fit line between these points.
101

103
00:09:52,300 --> 00:10:03,360
So I'll write "plt.plot(x_5, )", our x values and then for our y values, the predicted
102

104
00:10:03,360 --> 00:10:10,470
values, I'm going to write "regr", which is our linear regression,
103

105
00:10:10,470 --> 00:10:17,420
then ".predict()" and then in the parentheses I'm going to supply again the x values.
104

106
00:10:17,460 --> 00:10:21,420
So I'm going to predict the y values based on the x values.
105

107
00:10:21,590 --> 00:10:22,830
I'm going to this line orange.
106

108
00:10:22,830 --> 00:10:36,180
So I'm going to say "color='orange'" and maybe 'linewidth=3'.
107

109
00:10:36,280 --> 00:10:38,280
Let's add some labels as well.
108

110
00:10:38,280 --> 00:10:42,070
So "plt.xlabel()",
109

111
00:10:42,250 --> 00:10:48,270
it's gonna be the string "x values" and "plt.ylabel()",
110

112
00:10:48,280 --> 00:10:51,760
is gonna be the string "y values".
111

113
00:10:52,060 --> 00:10:57,360
Last but not least "plt.show()".
112

114
00:10:57,420 --> 00:11:00,160
Now let's take a gander what this looks like.
113

115
00:11:01,570 --> 00:11:08,560
Voila! Brilliant. Here we can clearly see our data points and the best fit line.
114

116
00:11:08,710 --> 00:11:17,950
Nothing new so far but I quickly want to mention something quite important and that's to do with a difference
115

117
00:11:17,980 --> 00:11:22,880
in notation that we're gonna be using in this example than in the previous examples.
116

118
00:11:23,380 --> 00:11:26,200
You see this isn't the cost function that we're looking at.
117

119
00:11:26,260 --> 00:11:34,390
This is the best fit line and the data. The plots that we've been running previously have been the cost
118

120
00:11:34,390 --> 00:11:40,210
functions and we're going to be plotting the mean square error shortly.
119

121
00:11:40,260 --> 00:11:46,110
Now the cost functions plot the relationship between the parameters that we're trying to estimate and
120

122
00:11:46,110 --> 00:11:49,720
how far off we were from the actual data.
121

123
00:11:49,920 --> 00:11:55,980
So the cost functions don't plot the actual data itself.
122

124
00:11:56,130 --> 00:11:57,960
That regression line right here,
123

125
00:11:58,080 --> 00:12:00,750
this graph - that's plotting the data.
124

126
00:12:00,750 --> 00:12:09,060
It's a function of x and y and not the theta parameters, theta zero and theta one, that we're gonna be
125

127
00:12:09,120 --> 00:12:13,710
interested in and looking at for our cost function.
126

128
00:12:13,890 --> 00:12:20,760
And the reason that I'm mentioning this is that previously we've been using x and y for our cost function
127

129
00:12:20,760 --> 00:12:26,020
parameters because that's how we tend to learn about mathematics in school and functions.
128

130
00:12:26,160 --> 00:12:32,610
And it's quite a comfortable sort of notation but when looking at this cost function that's coming up.
129

131
00:12:32,610 --> 00:12:40,500
It's going depend on the theta values and the theta values are not our x's and y's so just keep that
130

132
00:12:40,500 --> 00:12:43,560
in mind, x_5 and y_5
131

133
00:12:43,560 --> 00:12:49,620
in this example have a different purpose than the data that we generated in the previous lesson.