1
00:00:00,080 --> 00:00:01,680
So this should be a short video.

2
00:00:01,680 --> 00:00:03,510
I want to show you how to simulate

3
00:00:03,510 --> 00:00:06,210
continuous time dynamics
the same way we did

4
00:00:06,210 --> 00:00:08,733
very quickly for discrete event.

5
00:00:11,880 --> 00:00:16,880
So in an earlier video, we
saw how to accept or reject

6
00:00:17,220 --> 00:00:22,220
discrete event with probability.

7
00:00:27,570 --> 00:00:31,350
Let's call it P, right?

8
00:00:31,350 --> 00:00:33,333
I said all we need to do,

9
00:00:34,440 --> 00:00:35,370
let me erase this.

10
00:00:35,370 --> 00:00:37,080
All we need to do is to be able

11
00:00:37,080 --> 00:00:39,783
to draw random numbers
between zero and one.

12
00:00:40,890 --> 00:00:42,870
If we can draw uniformly,

13
00:00:42,870 --> 00:00:45,660
meaning that any interval
of the same length,

14
00:00:45,660 --> 00:00:46,950
no matter where it's at here,

15
00:00:46,950 --> 00:00:49,800
has the same probability of being chosen

16
00:00:49,800 --> 00:00:52,590
by our random number generator.

17
00:00:52,590 --> 00:00:55,830
Then we can simulate

18
00:00:55,830 --> 00:00:58,080
discrete event with probability P.

19
00:00:58,080 --> 00:00:59,010
Why?

20
00:00:59,010 --> 00:01:02,100
Because any event here
is as likely to be drawn.

21
00:01:02,100 --> 00:01:05,550
Every number here is as
as likely to be drawn

22
00:01:05,550 --> 00:01:07,350
on this interval.

23
00:01:07,350 --> 00:01:09,390
So if I ask what's the probability

24
00:01:09,390 --> 00:01:12,330
or what's the fraction of draws

25
00:01:12,330 --> 00:01:14,340
that are gonna fall below some value,

26
00:01:14,340 --> 00:01:15,633
let's say 0.4,

27
00:01:18,150 --> 00:01:19,290
or it's gonna be 40%.

28
00:01:19,290 --> 00:01:22,770
It's gonna be 0.4 of my
draws are gonna fall below.

29
00:01:22,770 --> 00:01:25,320
So if I want to say, well do this event

30
00:01:25,320 --> 00:01:27,090
which occurs with probability,

31
00:01:27,090 --> 00:01:29,551
beta times delta T equals to 0.4,

32
00:01:29,551 --> 00:01:31,200
does it occur over this time step?

33
00:01:31,200 --> 00:01:32,850
I draw a number between zero and one.

34
00:01:32,850 --> 00:01:34,860
If it's smaller than 0.4, yes.

35
00:01:34,860 --> 00:01:36,330
If not, no.

36
00:01:36,330 --> 00:01:39,510
That's all we had to do.

37
00:01:39,510 --> 00:01:42,450
And in fact, I claim that
being able to draw uniformly

38
00:01:42,450 --> 00:01:44,460
between zero and one, which is easy to do.

39
00:01:44,460 --> 00:01:48,120
It's ran in math lab.

40
00:01:48,120 --> 00:01:53,120
It's ran over max end
or rend end and in C++.

41
00:01:54,120 --> 00:01:56,910
All languages have an easy way to draw

42
00:01:56,910 --> 00:02:01,860
pseudo random numbers
between zero and one.

43
00:02:01,860 --> 00:02:04,510
That's the only thing that
we need to know how to do.

44
00:02:06,420 --> 00:02:08,350
And that's what we're gonna use now

45
00:02:09,870 --> 00:02:11,703
for continuous time event.

46
00:02:16,680 --> 00:02:17,790
Well, now it's not just a matter

47
00:02:17,790 --> 00:02:19,740
of what's the probability of that event.

48
00:02:19,740 --> 00:02:21,423
In continuous time,

49
00:02:22,770 --> 00:02:25,410
I have a distribution PFT

50
00:02:25,410 --> 00:02:27,600
for the time until the next event, right?

51
00:02:27,600 --> 00:02:29,160
If it's a plus all process,

52
00:02:29,160 --> 00:02:31,860
this distribution here would just be

53
00:02:31,860 --> 00:02:35,910
lambda times exponential
to the minus lambda,

54
00:02:35,910 --> 00:02:37,500
by definition of the plus all process.

55
00:02:37,500 --> 00:02:39,840
We solved for that in an earlier video,

56
00:02:39,840 --> 00:02:41,430
but it could also be something crazier.

57
00:02:41,430 --> 00:02:43,530
It could be something
that goes back to zero

58
00:02:43,530 --> 00:02:46,413
and goes back up, something
with multiple peaks.

59
00:02:48,750 --> 00:02:52,230
All that I know is that
the area under the curve,

60
00:02:52,230 --> 00:02:53,760
the integral of this is one,

61
00:02:53,760 --> 00:02:57,390
because it's a probability distribution.

62
00:02:57,390 --> 00:03:01,770
So it's PFT times DT is
a probability density

63
00:03:01,770 --> 00:03:03,240
and the event has to occur.

64
00:03:03,240 --> 00:03:05,493
So this has to integrate to one.

65
00:03:06,750 --> 00:03:07,650
That's all I know.

66
00:03:08,940 --> 00:03:12,150
Well, to be able to leverage
our random number generator

67
00:03:12,150 --> 00:03:13,950
between zero and one and draw

68
00:03:13,950 --> 00:03:17,400
from this distribution of
time to the next event,

69
00:03:17,400 --> 00:03:19,530
we're gonna fall back on our CDF trick.

70
00:03:19,530 --> 00:03:23,430
If you remember, CDF is a
cumulative distribution function,

71
00:03:23,430 --> 00:03:27,720
which is the integral of
the probability density

72
00:03:27,720 --> 00:03:28,553
to the left.

73
00:03:31,050 --> 00:03:33,090
And what that means is that

74
00:03:33,090 --> 00:03:36,930
in some cases, we'll be able
to do the integral by hand.

75
00:03:36,930 --> 00:03:40,080
So we saw the CDF for
the plus all distribution

76
00:03:40,080 --> 00:03:44,193
was one minus E to the minus lambda T.

77
00:03:46,260 --> 00:03:48,070
I'll use capital T here

78
00:03:49,230 --> 00:03:50,553
'cause it's integrated.

79
00:03:51,900 --> 00:03:53,250
But in most cases,

80
00:03:53,250 --> 00:03:55,830
maybe you have a crazy
distribution on the left

81
00:03:55,830 --> 00:03:58,350
because it's an input from some data.

82
00:03:58,350 --> 00:03:59,850
It's an input of your model.

83
00:03:59,850 --> 00:04:02,583
So you might want to do
your integral numerically.

84
00:04:03,420 --> 00:04:06,630
So basically you take very
small steps in your interval T

85
00:04:06,630 --> 00:04:11,630
and you sum the values that you get.

86
00:04:12,150 --> 00:04:14,730
So here my integral is
growing faster and faster.

87
00:04:14,730 --> 00:04:16,833
As I get bigger and bigger values,

88
00:04:18,330 --> 00:04:19,890
eventually it stops growing.

89
00:04:19,890 --> 00:04:22,380
So I get an inflection
point, but it keeps growing.

90
00:04:22,380 --> 00:04:23,213
This is the sum.

91
00:04:23,213 --> 00:04:25,560
So it's never gonna
decrease, keeps growing,

92
00:04:25,560 --> 00:04:28,770
eventually doesn't grow at all

93
00:04:28,770 --> 00:04:30,573
'cause I go back to zero here,

94
00:04:31,590 --> 00:04:33,750
and then start growing again

95
00:04:33,750 --> 00:04:35,790
and then stops growing forever.

96
00:04:35,790 --> 00:04:40,790
And as I said, we know
that it sums to one.

97
00:04:41,370 --> 00:04:45,063
So this CDF is between
zero and one by definition.

98
00:04:46,440 --> 00:04:47,290
Well that's cool.

99
00:04:48,570 --> 00:04:52,290
So now what I'm gonna
say is that if we draw,

100
00:04:52,290 --> 00:04:57,290
here we were drawing on the
X axis visually and asking,

101
00:04:58,320 --> 00:05:00,840
what's the probability
that it falls below a line?

102
00:05:00,840 --> 00:05:04,650
Here we're gonna draw on the Y axis.

103
00:05:04,650 --> 00:05:07,920
So you draw a random number,
again, between zero and one.

104
00:05:07,920 --> 00:05:09,603
So it falls like somewhere here.

105
00:05:11,580 --> 00:05:15,630
And I'm gonna say that the places

106
00:05:15,630 --> 00:05:17,920
where it crosses this line

107
00:05:21,300 --> 00:05:23,790
or the chances that it
crosses at any point

108
00:05:23,790 --> 00:05:26,193
is proportional to this PFT.

109
00:05:27,480 --> 00:05:28,740
And you can think of it this way.

110
00:05:28,740 --> 00:05:30,120
Where is it gonna cross?

111
00:05:30,120 --> 00:05:34,170
Well, if the derivative, like here,

112
00:05:34,170 --> 00:05:36,690
is really, really small,

113
00:05:36,690 --> 00:05:38,910
in fact here it's almost zero,

114
00:05:38,910 --> 00:05:43,910
then it's almost impossible
for a point to cross here

115
00:05:44,340 --> 00:05:45,840
'cause it's flat.

116
00:05:45,840 --> 00:05:50,550
So you can't like cross a flat
line during this interval T.

117
00:05:50,550 --> 00:05:55,550
So while my probability
distribution here was zero

118
00:05:55,980 --> 00:05:56,883
on the left,

119
00:05:58,470 --> 00:06:01,440
I end up here with a region
where I can't really cross.

120
00:06:01,440 --> 00:06:02,823
It's very unlikely.

121
00:06:05,340 --> 00:06:06,780
And basically what that means

122
00:06:06,780 --> 00:06:11,070
is that as I ask what's the probability

123
00:06:11,070 --> 00:06:13,560
of crossing at any given point,

124
00:06:13,560 --> 00:06:15,610
what matters is this cross section

125
00:06:18,450 --> 00:06:23,450
and this cross section by
construction is PFT times DT,

126
00:06:24,450 --> 00:06:27,330
where DT is the time step that we use

127
00:06:27,330 --> 00:06:28,630
to integrate the function.

128
00:06:31,140 --> 00:06:33,210
So that's it all over.

129
00:06:33,210 --> 00:06:34,950
You just do that all over the space

130
00:06:34,950 --> 00:06:37,020
and then you keep your CDF in memory.

131
00:06:37,020 --> 00:06:38,040
So that would be,

132
00:06:38,040 --> 00:06:42,000
if my time interval that I care about

133
00:06:42,000 --> 00:06:45,810
is in days, from zero to a thousand days

134
00:06:45,810 --> 00:06:48,390
and I have little DT of half a day,

135
00:06:48,390 --> 00:06:51,510
and then I have 2,000 values that I store.

136
00:06:51,510 --> 00:06:53,280
So those values go from zero to one

137
00:06:53,280 --> 00:06:56,970
It's my CDF over all my increments DT.

138
00:06:56,970 --> 00:07:00,150
And then whenever I draw a random number

139
00:07:00,150 --> 00:07:03,210
between zero and one, I can
do a smart, binary search

140
00:07:03,210 --> 00:07:08,010
and find where does my
CDF cross that value?

141
00:07:08,010 --> 00:07:12,450
And then that interval becomes my time T

142
00:07:12,450 --> 00:07:14,433
at which that event happens.

143
00:07:17,460 --> 00:07:19,920
It's useful to draw a few example

144
00:07:19,920 --> 00:07:21,780
and play with it to convince yourself

145
00:07:21,780 --> 00:07:23,310
that what matters is indeed

146
00:07:23,310 --> 00:07:25,800
sort of a cross section of that CDF,

147
00:07:25,800 --> 00:07:28,950
which is by definition its slope

148
00:07:28,950 --> 00:07:31,770
and therefore the the probability density.

149
00:07:31,770 --> 00:07:33,090
But that's a simple trick.

150
00:07:33,090 --> 00:07:35,980
And a lot of models to be computationally

151
00:07:39,600 --> 00:07:43,290
easy to run will either
keep a file of the CDF,

152
00:07:43,290 --> 00:07:45,000
so you might have a lot of different CDFs

153
00:07:45,000 --> 00:07:46,980
for different type of event.

154
00:07:46,980 --> 00:07:49,560
You might want to keep them
on file, just load them once

155
00:07:49,560 --> 00:07:52,290
to not have to construct or
do this numerical integration

156
00:07:52,290 --> 00:07:53,640
time and time again.

157
00:07:53,640 --> 00:07:55,200
But otherwise, it's pretty easy.

158
00:07:55,200 --> 00:07:56,880
You just draw between zero and one

159
00:07:56,880 --> 00:07:58,620
and now you run a binary search

160
00:07:58,620 --> 00:08:01,650
to ask what is the time T

161
00:08:01,650 --> 00:08:04,080
at which my CDF crosses that value.

162
00:08:04,080 --> 00:08:05,010
And then you're done.

163
00:08:05,010 --> 00:08:07,980
You have your time T at
which the event happened.

164
00:08:07,980 --> 00:08:09,840
So just to make things a little clearer,

165
00:08:09,840 --> 00:08:12,030
if on the left this was
a funny distribution

166
00:08:12,030 --> 00:08:13,770
of time to recovery, right?

167
00:08:13,770 --> 00:08:16,230
There's a lot of people
that have an easy time

168
00:08:16,230 --> 00:08:18,190
and recover on average in a week

169
00:08:19,080 --> 00:08:20,280
and some people are unlucky

170
00:08:20,280 --> 00:08:22,410
and recover on average in two weeks,

171
00:08:22,410 --> 00:08:25,533
which is the integral of
this second bump here.

172
00:08:28,980 --> 00:08:32,370
Well, basically I just do this step

173
00:08:32,370 --> 00:08:34,710
and now I draw between zero and one

174
00:08:34,710 --> 00:08:35,970
and for one person,

175
00:08:35,970 --> 00:08:39,900
the intersection between the
number that I drew in my CDF

176
00:08:39,900 --> 00:08:43,920
gives me the time to
recovery for that person.

177
00:08:43,920 --> 00:08:47,100
So you have to do this process
of finding the intersection

178
00:08:47,100 --> 00:08:50,280
any time you want to draw from
the distribution of the left,

179
00:08:50,280 --> 00:08:52,950
but it's still very easy to do

180
00:08:52,950 --> 00:08:55,410
and doesn't take too
much computational power.

181
00:08:55,410 --> 00:08:57,450
So we'll go over that in class eventually

182
00:08:57,450 --> 00:09:00,780
as we start doing more
advanced numerical model.

183
00:09:00,780 --> 00:09:03,360
We'll go over all of these things again

184
00:09:03,360 --> 00:09:07,290
and show sort of a complete
numerical recipe for modeling.

185
00:09:07,290 --> 00:09:08,910
But this will be a key tool

186
00:09:08,910 --> 00:09:11,850
in our numerical toolbox
or computational toolbox.

187
00:09:11,850 --> 00:09:13,770
So I'll see you in class
for some discussion

188
00:09:13,770 --> 00:09:15,900
about continuous time models.

189
00:09:15,900 --> 00:09:16,733
Bye.