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PROFESSOR: So you may recall
that, in the last lecture, we

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more or less solved the drunken
student problem, looking

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at a random walk.


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I now want to move on and
discuss some variants of the

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random walk problem that are
collectively known as

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biased random walks.


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So the notion here is, the walk
is still stochastic but there

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is some bias in the direction,
so the movements are not

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uniformly distributed or
equally distributed

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in all directions.


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As I go through this, I want
you to sort of, think in

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advance about what the
take-home message should be.

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One, and this is probably the
most important part for today,

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is I want to illustrate how by
designing our programs in a

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nice way around classes, we can
change them to do something

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else writing a minimal
amount of code.

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And the idea here is that
classes are not there just to

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provide some syntax that makes
life complicated, but to

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actually provide a mechanism
that lets us structure our

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programs in a way that we
can modify them later.

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I also want to give you a bit
more experience looking at

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data, and understanding how,
when you get the results of a

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simulation or anything else
that provides you with data,

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you can plot, them study the
plots, and try and use those

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to develop an understanding
about what's going on.

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And finally, I want to just
get you thinking more

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about random behavior.


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Because we're going to talk a
lot more this semester and

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00:02:20 --> 00:02:24
in general throughout your
careers, you'll see a lot of

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uses of stochastic things.


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00:02:26 --> 00:02:31
All right, so let's think
about these biased walks.

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So for example, assume that the
drunk grew up in South Florida,

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and really hates the
New England winter.

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Well there might be a bias that
even though the drunk is not in

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total control, she's kind
of wandering southward.

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So there's a bias when taking
a step to more likely go

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south than certainly north.


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Or imagine that the drunk is
photosensitive and moves either

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towards the sun or away from
the sun, or some such thing.

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So we're going to look at
different things of that nature

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and think about, I'll come back
to Chutes and Ladders later in

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the lecture, you can speculate
on why this is here.

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And we'll look at that.


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So, this is on the front
page of your handout here.

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I've taken the random walk
program we looked at last

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time and changed it a bit.


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And in particular, I've changed
the way the drunk works.

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So let's look at the code.


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Wow, haven't even done anything
it's -- see how this works.

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So now, drunk itself is a very
small class, one I don't

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actually intend to
ever instantiate.

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I've pared it down, it has
only two methods in it.

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It has an init, which is
the same as before, and a

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move which is the same as
before, almost the same.

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Now I'm going to have
the class usual drunk.

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That's the drunk we
looked at the last time.

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And this usual drunk will
behave just like the drunk

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we looked at last time.


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In here, I'm going to override
the move method of the

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inherited superclass, and what
this is going to do is, it's

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going to make the same random
choice it made before and then

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it's going to call drunk dot
move with the new

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compass point.


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So all I've done is, I've taken
a couple of lines of code out

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of the previous implementation
and moved it to the subclass.

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So it's now, excuse me, in
the subclass I'm making

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the decision about which
direction to move.

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Once I've made that decision,
then I can call the

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drunk in the superclass.


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Notice here, when I've made
the call, drunk dot move.

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So usually you're expecting
me to write something

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like d dot move.


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The object.


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Here, instead I'm specifying
which class to get the move

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from, So this is saying, don't
call my local move, call the

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move from the superclass.


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There are a number of other
syntaxes I could've used to

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accomplish the same thing.


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I used this because it seemed
the most straightforward.

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Now let's compare the usual
drunk to the cold drunk.

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So the cold drunk, again, I've
overridden the move, and now

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it does something different.


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It gets the random compass
point as before, but if it

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happens to be south, it
calls drunk dot move

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with twice the distance.


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So whenever I get the notion
I'm moving south, because I

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love the warmth, I actually
move twice as fast.

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So instead of taking one step,
which I do to the other compass

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points, if I happen to be going
south I'll take two steps.

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So I'm biased towards
moving southward.

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And then I call drunk dot move.


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Otherwise, I call drunk dot
move with the distances before.

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00:07:30 --> 00:07:34
So I've isolated into the
subclass this notion

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of being heat-seeking.


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We've got a third sub
class, ew drunk, known

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for east west drunk.


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00:07:47 --> 00:07:52
And here, what I've done is, I
choose to compass point and

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then while it's not either
east or west, I choose again.

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So this drunk can only move
eastward or westward.

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Can't move north or south.


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00:08:07 --> 00:08:12
So again you'll see I'm able to
have three kinds of drunks.

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Who as we will see, exhibit
rather different behavior from

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each other with a minimal
amount of change to the code.

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If we look down at the bottom,
we'll see that the code

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is pretty much exactly
what it was last time.

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00:08:37 --> 00:08:40
By the way, this doesn't quite
match the code in your handout.

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00:08:40 --> 00:08:44
I decided to make the plots a
little bit prettier with better

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00:08:44 --> 00:08:47
titles and things, but when I
tried to put these in your

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handouts it wouldn't fit.


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So I have a little more compact
things, but this is, I think,

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a better way to write it.


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00:08:55 --> 00:09:01
But essentially, perform trial,
perform sim, answer quest, are

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all identical to what
they were before.

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00:09:05 --> 00:09:11
So because I structured my
program in such a way that

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the drunk's behavior was
independent of all the code for

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the generating the trials and
plotting the results, I didn't

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have to change any of that.


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And this is the key
to good design.

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Is to isolate decisions in
small parts of your program, so

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when as inevitable happens, you
want to enhance the program,

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or you discover you've done
something wrong, the number

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changes you make is not large.


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00:09:48 --> 00:09:53
OK, one thing we do need to
look at down here is, it's not

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quite the same as before,
because we'll see that there's

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an extra parameter to these
things called drunk type.

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00:10:06 --> 00:10:10
So if we start with answer
question, it takes as before,

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the time, the number of trials,
a title, and this one extra

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thing called the drunk type.


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And then when it goes on and
it calls perform sim, it

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passes the drunk type.


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00:10:27 --> 00:10:38
And this is the key line here.


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00:10:38 --> 00:10:43
Before, you may recall we
just said d equals drunk

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00:10:43 --> 00:10:45
with the arguments.


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00:10:45 --> 00:10:51
Here, I say d
equals drunk type.

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00:10:51 --> 00:10:56
So what is drunk type?


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00:10:56 --> 00:11:00
Drunk type is itself a type.


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00:11:00 --> 00:11:04
We can use types, you know
variables can have as their

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values types, parameters can
have as their values types.

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00:11:09 --> 00:11:13
So I'm just passing the type
around so that here, when it

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comes time to get a drunk, I
get a drunk of the

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00:11:17 --> 00:11:21
appropriate type.


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00:11:21 --> 00:11:25
So I can now answer the
question about a usual drunk,

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00:11:25 --> 00:11:31
or a cold drunk, or a ew drunk.


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00:11:31 --> 00:11:36
This is very common kind
of programming paradigm.

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00:11:36 --> 00:11:41
We've talked about it before,
but this is yet another example

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00:11:41 --> 00:11:44
of how we use polymorphism.


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00:11:44 --> 00:11:47
Polymorphism.


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We write our code in such
a way that it works with

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00:11:51 --> 00:11:54
many types of objects.


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00:11:54 --> 00:11:57
Here we're writing code
that will work with

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00:11:57 --> 00:12:01
any subtype of drunk.


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00:12:01 --> 00:12:04
Now I'd better not
pass it a float.

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00:12:04 --> 00:12:08
You know, if I answer quest,
and instead of something like

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usual drunk or cold drunk or e
w drunk, I pass it float or

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int, the world will come
crashing down around my head.

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00:12:17 --> 00:12:22
But I haven't, so you'll see
down here, I pass it num steps,

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00:12:22 --> 00:12:27
number trials, and usual drunk.


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00:12:27 --> 00:12:28
Which is the name of?


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00:12:28 --> 00:12:33
A class.


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00:12:33 --> 00:12:38
Any questions about this?


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00:12:38 --> 00:12:39
Yes?


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00:12:39 --> 00:12:45
STUDENT: [INAUDIBLE]


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00:12:45 --> 00:12:46
PROFESSOR: Louder, please?


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00:12:46 --> 00:12:53
STUDENT: [INAUDIBLE]


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00:12:53 --> 00:12:54
PROFESSOR: Good, good question.


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00:12:54 --> 00:12:58
The question is, does a drunk
type make a new drunk?

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Now, I have so much candy
to choose from, I don't

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know what to throw you.


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00:13:02 --> 00:13:07
Do you like Tootsie Rolls,
Snickers, Tootsie Roll Pops,

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00:13:07 --> 00:13:11
Dots, what's your choice?


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00:13:11 --> 00:13:19
Snickers.


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00:13:19 --> 00:13:23
OK, so what is it?


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00:13:23 --> 00:13:29
Down here nothing is created
when that call was made.

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00:13:29 --> 00:13:34
It was if I had typed
the word float or int.

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00:13:34 --> 00:13:41
I'm merely saying, here is a
type, and the parameter should

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now be bound to that type.


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00:13:45 --> 00:13:49
Now any time you use the
parameter, it's just as if you

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00:13:49 --> 00:13:56
had written the string, in
this case, usual drunk.

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00:13:56 --> 00:14:08
So when I get up to here, think
about, it will take, the

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variable drunk type, evaluate
it, it will evaluate to a

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00:14:17 --> 00:14:23
type usual drunk, say,
in the first call.

190
00:14:23 --> 00:14:28
And then it will, just as if I
had written usual drunk, invoke

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the init method of drunk, and
create a new drunk, but this

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00:14:34 --> 00:14:40
drunk will be of type usual
drunk rather than a drunk.

193
00:14:40 --> 00:14:44
Does that make sense?


194
00:14:44 --> 00:14:50
All right, so this avoids my
having to write a whole bunch

195
00:14:50 --> 00:14:55
of tests, I could have passed a
string, and said if the string

196
00:14:55 --> 00:14:59
said usual drunk, then
d equals something.

197
00:14:59 --> 00:15:01
But here instead of that,
I'm actually passing

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00:15:01 --> 00:15:07
the type itself.


199
00:15:07 --> 00:15:12
OK?


200
00:15:12 --> 00:15:16
So other than the fact that I
have to pass the type around,

201
00:15:16 --> 00:15:21
this is exactly the
same as before.

202
00:15:21 --> 00:15:26
Now just for fun, we saw that
last time when I asked you what

203
00:15:26 --> 00:15:30
did usual drunk do, there
was some confusion.

204
00:15:30 --> 00:15:34
Not confusion, just a
bad guess about how the

205
00:15:34 --> 00:15:35
drunk would wander.


206
00:15:35 --> 00:15:38
So, let's try it again.


207
00:15:38 --> 00:15:44
What do you think
about the cold drunk?

208
00:15:44 --> 00:15:48
Will the cold drunk get
further from the origin

209
00:15:48 --> 00:15:50
than the usual drunk?


210
00:15:50 --> 00:15:51
Not further from the origin?


211
00:15:51 --> 00:15:55
Who thinks the cold drunk
will be further away

212
00:15:55 --> 00:15:56
the end of 500 steps?


213
00:15:56 --> 00:16:01
Who thinks the cold drunk
will not be further away?

214
00:16:01 --> 00:16:05
And I'll bet we can all agr --
got that right, I think -- and

215
00:16:05 --> 00:16:08
I'll bet you'll all agree that,
with higher probability, the

216
00:16:08 --> 00:16:12
cold drunk will be south of the
origin than north

217
00:16:12 --> 00:16:14
of the origin.


218
00:16:14 --> 00:16:19
More interesting, how
about the ew drunk?

219
00:16:19 --> 00:16:21
So this is a drunk that
only goes east or

220
00:16:21 --> 00:16:27
west, east or west.


221
00:16:27 --> 00:16:31
Will this drunk, well,
first of all, what's

222
00:16:31 --> 00:16:33
the expected distance?


223
00:16:33 --> 00:16:37
Do you think it should be close
to zero or not close to zero?

224
00:16:37 --> 00:16:41
Who thinks it should
be close to 0?

225
00:16:41 --> 00:16:44
Who thinks it won't
be close to 0?

226
00:16:44 --> 00:16:47
Well you guys have all fallen
for the same trick you

227
00:16:47 --> 00:16:49
fell for last time.


228
00:16:49 --> 00:16:50
Fooled you twice.


229
00:16:50 --> 00:16:54
Be careful, one more
and you're out.

230
00:16:54 --> 00:16:56
As we'll see.


231
00:16:56 --> 00:16:57
But it's interesting.


232
00:16:57 --> 00:17:12
So let's run this program
and see what happens.

233
00:17:12 --> 00:17:15
I can take a just a second
or two and we should

234
00:17:15 --> 00:17:22
get a bunch of plots.


235
00:17:22 --> 00:17:25
All right, so here's
the usual drunk.

236
00:17:25 --> 00:17:33
100 trials and just as
before, wandered away.

237
00:17:33 --> 00:17:36
Moderately smooth, but of
course little ups and downs,

238
00:17:36 --> 00:17:42
maybe suggesting averaging over
100 trials is not quite enough.

239
00:17:42 --> 00:17:45
I felt a little better if it
were smoother, but the trend

240
00:17:45 --> 00:17:52
is pretty clear here, I think
we probably don't doubt it.

241
00:17:52 --> 00:17:56
Well, look at cold drunk.


242
00:17:56 --> 00:18:00
Like a shot heading south.


243
00:18:00 --> 00:18:04
So here we see, there's bias,
that if you get south you go

244
00:18:04 --> 00:18:08
twice as far makes
a huge difference.

245
00:18:08 --> 00:18:12
Instead of being roughly
20 steps away, it's

246
00:18:12 --> 00:18:16
roughly 120 steps away.


247
00:18:16 --> 00:18:21
So we see, and this is
something to think about, a

248
00:18:21 --> 00:18:27
small bias can end up making
a huge difference over time.

249
00:18:27 --> 00:18:31
This is why casinos do so well.


250
00:18:31 --> 00:18:34
They don't have to have a very
much of a bias in their favor

251
00:18:34 --> 00:18:37
on each roll of the dice
in order to accumulate a

252
00:18:37 --> 00:18:40
lot of money over time.


253
00:18:40 --> 00:18:43
It's why if you just weigh your
dice a little bit, so that

254
00:18:43 --> 00:18:47
they're not fair, you'll
do very very well.

255
00:18:47 --> 00:18:52
I'm not recommending that.


256
00:18:52 --> 00:18:55
Kind of interesting.


257
00:18:55 --> 00:18:58
The e w drunk is pretty much
the same distance away

258
00:18:58 --> 00:19:04
as the usual drunk.


259
00:19:04 --> 00:19:09
This is why we run these
simulations, is because

260
00:19:09 --> 00:19:10
we learn things.


261
00:19:10 --> 00:19:13
Now, I have to confess,
it fooled me, too.

262
00:19:13 --> 00:19:16
The first time I ran this, I
looked at this and said, wait

263
00:19:16 --> 00:19:20
a minute, that's not
what I expected.

264
00:19:20 --> 00:19:24
So I scratched my head, and
thought about, all right, why

265
00:19:24 --> 00:19:27
is it behaving this way?


266
00:19:27 --> 00:19:32
This happens me a lot in my own
research, that I crunch some

267
00:19:32 --> 00:19:37
data, do some plots, look at
it, get surprised, and on the

268
00:19:37 --> 00:19:41
basis of that I learned
something about the underlying

269
00:19:41 --> 00:19:42
system I'm modeling.


270
00:19:42 --> 00:19:47
Because it provokes me to think
in ways I had not thought

271
00:19:47 --> 00:19:49
before I saw the data.


272
00:19:49 --> 00:19:50
This is an important lesson.


273
00:19:50 --> 00:19:54
You don't ignore the data, you
try and explain the data.

274
00:19:54 --> 00:19:58
So when I thought about what
happened, I realize that while

275
00:19:58 --> 00:20:02
I might have started at the
origin, and after the first

276
00:20:02 --> 00:20:07
step I'm equally likely
to be here or here.

277
00:20:07 --> 00:20:17
Once I've taken the first step,
now my expected location

278
00:20:17 --> 00:20:20
is not zero, right?


279
00:20:20 --> 00:20:25
The first step, my expected
location was zero.

280
00:20:25 --> 00:20:29
Because equally likely to
be plus 1 or minus 1.

281
00:20:29 --> 00:20:34
But once I've taken a step
and I happen to be here, my

282
00:20:34 --> 00:20:40
expected location, if I'm
here, is plus 1, it's not 0.

283
00:20:40 --> 00:20:43
Because half the time I'll
get back to 0, but half

284
00:20:43 --> 00:20:48
the time I'll get to 2.


285
00:20:48 --> 00:20:55
So once I have moved away from
where I've started, the

286
00:20:55 --> 00:21:03
expectation of where I'll
end up has changed.

287
00:21:03 --> 00:21:07
This is kind of a profound
thing to think about, because

288
00:21:07 --> 00:21:12
when we look at it, this kind
of behavior explains a lot of

289
00:21:12 --> 00:21:14
surprising results in life.


290
00:21:14 --> 00:21:21
Where, if you get a small run
of random events that take you

291
00:21:21 --> 00:21:28
far from where you started,
it's hard to get back.

292
00:21:28 --> 00:21:32
So if you think about the stock
market, for example, if you

293
00:21:32 --> 00:21:36
happen to get unlucky and have
a few days or weeks or months,

294
00:21:36 --> 00:21:42
as the case may be, where you
get a run of bad days, it's

295
00:21:42 --> 00:21:45
really hard to get back if all
the movements after

296
00:21:45 --> 00:21:46
that are random.


297
00:21:46 --> 00:21:50
Because you've established
a new baseline.

298
00:21:50 --> 00:21:53
And as we'll see later, when
you start doing things at

299
00:21:53 --> 00:21:57
random, it's likely you'll get
a few runs in one direction or

300
00:21:57 --> 00:22:02
another, and then you've got a
new baseline, and it's unlikely

301
00:22:02 --> 00:22:04
to get back to where
you were originally.

302
00:22:04 --> 00:22:07
We'll see a lot of
examples of this.

303
00:22:07 --> 00:22:13
But that's what's happening
here, and that's why it

304
00:22:13 --> 00:22:17
looks the way it does.


305
00:22:17 --> 00:22:20
That makes sense to people?


306
00:22:20 --> 00:22:24
Well, so I had this theory,
as I often do when I have

307
00:22:24 --> 00:22:28
theories, I don't believe them,
so I decided I better write

308
00:22:28 --> 00:22:34
some more code to check it out.


309
00:22:34 --> 00:22:40
So, what I did is, I took the
previous simulation and now

310
00:22:40 --> 00:22:44
instead of looking at different
kinds of drunks, I got more

311
00:22:44 --> 00:22:50
aggressive in thinking about
my analysis of the data.

312
00:22:50 --> 00:22:53
So again, there's
a lesson here.

313
00:22:53 --> 00:22:58
I decided I wanted to analyze
my data in new ways.

314
00:22:58 --> 00:23:03
And the beauty of it was, I
could do that without changing

315
00:23:03 --> 00:23:07
either any of the code about
the drunk, or any they of

316
00:23:07 --> 00:23:10
the code about performing
the simulation, really.

317
00:23:10 --> 00:23:14
All I had to do was do a
different kind of analysis

318
00:23:14 --> 00:23:18
to a first approximation.


319
00:23:18 --> 00:23:26
So, let's look at
what I've done here.

320
00:23:26 --> 00:23:30
What I've decided to do here is
not only plot, and again, I

321
00:23:30 --> 00:23:37
think you'll see this code, not
only plot how far from the

322
00:23:37 --> 00:23:43
origin the drunk is over time,
but I'm going to actually

323
00:23:43 --> 00:23:46
plot two other things.


324
00:23:46 --> 00:23:51
I'm going to use a scatter
plot, I think I use scatter in

325
00:23:51 --> 00:23:55
this one, I do, to show where
the drunk is at each

326
00:23:55 --> 00:23:56
period of time.


327
00:23:56 --> 00:23:59
Actually, just all the
places the drunk ends

328
00:23:59 --> 00:24:00
up in each trial.


329
00:24:00 --> 00:24:04
This will give me a visualized
way just see, OK, in my, say,

330
00:24:04 --> 00:24:08
100 trials, how many the drunk
ended up near the origin,

331
00:24:08 --> 00:24:11
how many of them ended
up really far away?

332
00:24:11 --> 00:24:16
And just, allow me to
visualize what's going on.

333
00:24:16 --> 00:24:19
And then I'm going to also
use a histogram to sort

334
00:24:19 --> 00:24:21
of summarize some things.


335
00:24:21 --> 00:24:34
To get a look at the
distribution of the data.

336
00:24:34 --> 00:24:39
We've been just looking at
average results, but it's often

337
00:24:39 --> 00:24:41
the case that the average
does not really tell

338
00:24:41 --> 00:24:44
the whole story.


339
00:24:44 --> 00:24:46
You want to ask, well, how
many of the drunks ended

340
00:24:46 --> 00:24:48
up really far away?


341
00:24:48 --> 00:24:51
How many of the drunks ended
up where they started?

342
00:24:51 --> 00:24:55
You can't tell that
from the averages.

343
00:24:55 --> 00:24:59
So you use a distribution,
which tells you information

344
00:24:59 --> 00:25:03
about how many people
ended up where.

345
00:25:03 --> 00:25:07
So we again saw that, in a less
pleasant context, maybe, in

346
00:25:07 --> 00:25:10
looking at the quizzes,
remember I sent you some plots

347
00:25:10 --> 00:25:16
about the quizzes, and I didn't
just say, here was the average.

348
00:25:16 --> 00:25:20
I showed you the
distribution of grades.

349
00:25:20 --> 00:25:23
That allowed you to see, not
only where you were relative

350
00:25:23 --> 00:25:27
to the average, but whether
you were an outlier.

351
00:25:27 --> 00:25:30
Was your grade much
better than the average?

352
00:25:30 --> 00:25:33
Or were there a whole bunch of
people around where you are?

353
00:25:33 --> 00:25:37
So there's a lot of information
there, that will help us

354
00:25:37 --> 00:25:41
understand what's
really going on.

355
00:25:41 --> 00:25:45
So we'll run this now.


356
00:25:45 --> 00:25:48
Again, there's nothing very
interesting here about how I

357
00:25:48 --> 00:25:52
did it, I just used some of
these Pylab dot hist, which

358
00:25:52 --> 00:25:55
gives me a histogram, Pylab
dot scatter, which gives

359
00:25:55 --> 00:25:57
me a scatter plot.


360
00:25:57 --> 00:26:01
I'll take 500
steps, 400 trials.

361
00:26:01 --> 00:26:04
I've done some more trials
here, so we get some smoother

362
00:26:04 --> 00:26:07
lines, better distributions.


363
00:26:07 --> 00:26:10
And then I'm going to look at
the usual drunk and ew drunk

364
00:26:10 --> 00:26:24
only, just to save some time.


365
00:26:24 --> 00:26:31
At least I think I am.


366
00:26:31 --> 00:26:35
My computer runs faster
when it's plugged in.

367
00:26:35 --> 00:26:37
All right, but we have
our figures here,

368
00:26:37 --> 00:26:40
let's look at them.


369
00:26:40 --> 00:26:42
So here's figure one,
the usual drunk.

370
00:26:42 --> 00:26:46
Well, all right, as we've
seen this 1000 times,

371
00:26:46 --> 00:26:47
or at least six.


372
00:26:47 --> 00:26:55
So doing what it usually
does, not quite 20.

373
00:26:55 --> 00:26:56
This is kind of interesting.


374
00:26:56 --> 00:27:00
This is the final locations
of the 400 trials

375
00:27:00 --> 00:27:02
of the usual drunk.


376
00:27:02 --> 00:27:04
So we see some information
here, that we really couldn't

377
00:27:04 --> 00:27:08
get from the averages.


378
00:27:08 --> 00:27:12
That there are more of these
points near the middle,

379
00:27:12 --> 00:27:18
but there are certainly
plenty scattered around.

380
00:27:18 --> 00:27:21
And we get to see that,
they're pretty much symmetric

381
00:27:21 --> 00:27:24
around where we started.


382
00:27:24 --> 00:27:26
This is what we would
have hoped, right?

383
00:27:26 --> 00:27:31
If we had looked at it, and we
had seen all the points up

384
00:27:31 --> 00:27:34
here, or most of them up here,
we would have said, wait a

385
00:27:34 --> 00:27:38
minute, there's something
wrong with my code.

386
00:27:38 --> 00:27:40
This was supposed to not
be a biased walk, but

387
00:27:40 --> 00:27:43
an unbiased walk.


388
00:27:43 --> 00:27:47
And if I'd seen a bias in this
population, I should have

389
00:27:47 --> 00:27:50
gotten nervous that I had some
problems with my

390
00:27:50 --> 00:27:51
implementation.


391
00:27:51 --> 00:27:53
My simulation.


392
00:27:53 --> 00:27:56
So in addition to helping me
understand results, this gives

393
00:27:56 --> 00:27:59
me some confidence that
I really am doing an

394
00:27:59 --> 00:28:03
unbiased walk, right?


395
00:28:03 --> 00:28:04
Some over here, some over here.


396
00:28:04 --> 00:28:09
Not exactly the same, but
close enough that I feel

397
00:28:09 --> 00:28:19
pretty comfortable.


398
00:28:19 --> 00:28:25
And if I look at the
distribution, which probably I

399
00:28:25 --> 00:28:27
could've if I worked really
hard, gotten from the scatter

400
00:28:27 --> 00:28:34
plot, but I really didn't want
to, we'll see that most of the

401
00:28:34 --> 00:28:39
drunks actually do end up
pretty close to the origin.

402
00:28:39 --> 00:28:43
But that there are a few
that are pretty far away.

403
00:28:43 --> 00:28:49
And again, it's close to
symmetric around zero.

404
00:28:49 --> 00:28:52
Which gives me a good feeling
that things are working

405
00:28:52 --> 00:28:54
the way they should.


406
00:28:54 --> 00:29:00
This, by the way, is what's
called a normal distribution.

407
00:29:00 --> 00:29:04
Because once upon a time,
people believed this was the

408
00:29:04 --> 00:29:07
way things usually happened.


409
00:29:07 --> 00:29:16
This is called either
normal or Gaussian.

410
00:29:16 --> 00:29:18
The mathematician Gauss was one
of the first people to write

411
00:29:18 --> 00:29:20
about this distribution.


412
00:29:20 --> 00:29:23
We'll come back to this later,
and among other things,

413
00:29:23 --> 00:29:27
we'll try and see whether
normal is really normal.

414
00:29:27 --> 00:29:29
But you see it a lot of times.


415
00:29:29 --> 00:29:34
And basically what it says
is that most of the values

416
00:29:34 --> 00:29:38
hang out around the
average, the middle.

417
00:29:38 --> 00:29:42
And there are a few outliers,
and as we get further and

418
00:29:42 --> 00:29:49
further out, fewer and
fewer points occur.

419
00:29:49 --> 00:29:52
So you'll see these kind of
curves, of distributions,

420
00:29:52 --> 00:29:53
occurring a lot.


421
00:29:53 --> 00:29:56
Again, we'll come back to
distributions in a week, or

422
00:29:56 --> 00:29:58
actually in a lecture or two.


423
00:29:58 --> 00:30:03
See a little bit more today,
and then more later on.

424
00:30:03 --> 00:30:13
All right, here's our east
west drunk, again as before,

425
00:30:13 --> 00:30:18
drifting off to around 18.


426
00:30:18 --> 00:30:23
Well, look at these dots.


427
00:30:23 --> 00:30:26
Well, this makes me feel, that
at least I know that the drunk

428
00:30:26 --> 00:30:30
is only moving east and west.


429
00:30:30 --> 00:30:32
And again you'll see,
pretty dense in the middle

430
00:30:32 --> 00:30:35
and a few outliers.


431
00:30:35 --> 00:30:37
Managed to get a little
further east than west,

432
00:30:37 --> 00:30:41
but that happens.


433
00:30:41 --> 00:30:47
East is always more attractive.


434
00:30:47 --> 00:30:49
What we might have guessed.


435
00:30:49 --> 00:30:56
And if we look at the
distribution, again we can look

436
00:30:56 --> 00:31:00
at it, and I'm just giving you
the east west values here, and

437
00:31:00 --> 00:31:07
again it's about the same.


438
00:31:07 --> 00:31:11
So, what do I want you to
take away from these graphs?

439
00:31:11 --> 00:31:15
Different plots show
you different things.

440
00:31:15 --> 00:31:18
So I'm not going to try and
claim that of these three

441
00:31:18 --> 00:31:22
different plots we did, one
is the right one to do.

442
00:31:22 --> 00:31:24
What I'm going to assert
is that you want

443
00:31:24 --> 00:31:26
to do all of them.


444
00:31:26 --> 00:31:31
Try and visualize the
data in multiple ways.

445
00:31:31 --> 00:31:36
And sometimes it's much easier
to, even if different plots

446
00:31:36 --> 00:31:41
contain exactly the same
information, and in some sense

447
00:31:41 --> 00:31:45
of the scatter plot and the
histogram do contain exactly

448
00:31:45 --> 00:31:47
the same information.


449
00:31:47 --> 00:31:49
In fact, there's more
information in the scatter

450
00:31:49 --> 00:31:53
plot, in some sense, because
I see each one individually.

451
00:31:53 --> 00:31:57
But it's, for me, a
lot harder to see.

452
00:31:57 --> 00:32:00
If I'd ask you from the
scatter plot, are they

453
00:32:00 --> 00:32:02
normally distributed?


454
00:32:02 --> 00:32:04
You might have had a
hard time answering it.

455
00:32:04 --> 00:32:08
Particularly for the usual
drunk, where you just

456
00:32:08 --> 00:32:10
saw a lot of points.


457
00:32:10 --> 00:32:14
So the idea of some summarizing
it in the histogram makes it

458
00:32:14 --> 00:32:19
much easier to make sense of
the data, and see what you're

459
00:32:19 --> 00:32:32
doing, and to try and learn the
lesson you're trying to learn.

460
00:32:32 --> 00:32:37
OK, this makes sense to people?


461
00:32:37 --> 00:32:40
All right, let's move on.


462
00:32:40 --> 00:32:48
Now we'll go back to this
lovely picture we had before.

463
00:32:48 --> 00:32:51
Any of you ever play the
game Chutes and Ladders?

464
00:32:51 --> 00:32:53
Raise your hand?


465
00:32:53 --> 00:32:57
OK, some things never
go out of style.

466
00:32:57 --> 00:33:04
Well, imagine your poor drunk
wandering through this board,

467
00:33:04 --> 00:33:09
and every once in a while,
because I'm kind of a, not a

468
00:33:09 --> 00:33:12
nice person, I've eliminated
all the ladders, and

469
00:33:12 --> 00:33:13
have only some chutes.


470
00:33:13 --> 00:33:16
And every once in a while
the drunk hits a chute

471
00:33:16 --> 00:33:18
and goes whoosh.


472
00:33:18 --> 00:33:27
And in fact, let's, for the
sake of simplicity here, assume

473
00:33:27 --> 00:33:32
that the chutes are going to,
every once in a while, the

474
00:33:32 --> 00:33:37
drunk will hit a bad spot, and
go right back to the origin.

475
00:33:37 --> 00:33:40
So this poor, say, heat-seeking
drunk who's trying to head

476
00:33:40 --> 00:33:45
south, does it for a while and
maybe gets zipped right back

477
00:33:45 --> 00:33:48
to where he or she started.


478
00:33:48 --> 00:33:54
All right, I've now told
you I want to put some,

479
00:33:54 --> 00:33:55
make this happen.


480
00:33:55 --> 00:34:01
What part of the code do
you think I should change?

481
00:34:01 --> 00:34:02
Somebody?


482
00:34:02 --> 00:34:04
STUDENT: The field.


483
00:34:04 --> 00:34:11
PROFESSOR: The
field, absolutely.

484
00:34:11 --> 00:34:14
Those are good to throw
short distances.

485
00:34:14 --> 00:34:17
The field, right, because what
we're talking about here is not

486
00:34:17 --> 00:34:23
a property of the drunk, or a
property of the simulation, but

487
00:34:23 --> 00:34:30
a property of the space in
which the drunk is wandering.

488
00:34:30 --> 00:34:34
So again, we see that by
structuring the initial program

489
00:34:34 --> 00:34:40
around the natural abstractions
that occur in the problem,

490
00:34:40 --> 00:34:46
natural kinds of changes
can be localized.

491
00:34:46 --> 00:34:52
So let's go and
change the field.

492
00:34:52 --> 00:34:57
So I'm going to have the
field I had before plus this

493
00:34:57 --> 00:35:03
thing called an odd field.


494
00:35:03 --> 00:35:06
Odd field will be a
subclass of field.

495
00:35:06 --> 00:35:08
This is not odd as in
odd or even, this is

496
00:35:08 --> 00:35:12
odd as in strange.


497
00:35:12 --> 00:35:19
So I'm going to first define
where my chutes are.

498
00:35:19 --> 00:35:30
So is chute will take, will get
the coordinates, and assign it

499
00:35:30 --> 00:35:34
to x and y, so it gets the
coordinates of a place, and

500
00:35:34 --> 00:35:39
then will return abs x minus
abs y is equal to zero.

501
00:35:39 --> 00:35:45
So, where are my chutes
going to occur?

502
00:35:45 --> 00:35:53
Somebody?


503
00:35:53 --> 00:36:00
Yeah, kind of radiating
out from the origin.

504
00:36:00 --> 00:36:06
Any place, so I'll
have my origin, here.

505
00:36:06 --> 00:36:08
Got my graph.


506
00:36:08 --> 00:36:13
And all along here, any
place x and y are equal,

507
00:36:13 --> 00:36:18
there'll be a chute.


508
00:36:18 --> 00:36:21
Now, I have to be a little bit
careful that it doesn't happen

509
00:36:21 --> 00:36:27
here, or we won't get anywhere.


510
00:36:27 --> 00:36:32
So, and then move, in odd
field, will call field dot

511
00:36:32 --> 00:36:38
move, which is unchanged
from what it was before.

512
00:36:38 --> 00:36:42
And then it will say, if
self dot is chute, set

513
00:36:42 --> 00:36:45
the location back to 0,0.


514
00:36:45 --> 00:36:50
So the poor drunk will move as
before, and if he or she hits

515
00:36:50 --> 00:36:58
this wormhole, get instantly
teleported back to the origin.

516
00:36:58 --> 00:37:01
Very small change, but it will
give us rather different

517
00:37:01 --> 00:37:08
behaviors, at least
I think it will.

518
00:37:08 --> 00:37:17
And then if we look at how we
use it, not much changes.

519
00:37:17 --> 00:37:20
Except, what do you think is
going to have to change?

520
00:37:20 --> 00:37:23
Where will I have to make
the change, somebody?

521
00:37:23 --> 00:37:27
You can actually
see the code here.

522
00:37:27 --> 00:37:31
I'll have to get a
different field, right?

523
00:37:31 --> 00:37:35
So, at the place where I
instantiate a field, instead of

524
00:37:35 --> 00:37:40
instantiating a field, I'll
instantiate an odd field.

525
00:37:40 --> 00:37:44
Now if I'd had 16 different
kinds of odd fields, or even

526
00:37:44 --> 00:37:49
two, I might well have done
what I did with drunk.

527
00:37:49 --> 00:37:53
But in order to sort of
minimize things, I've done

528
00:37:53 --> 00:37:55
something much smaller.


529
00:37:55 --> 00:37:57
So let's see, where
did we get the field?

530
00:37:57 --> 00:37:59
I don't even remember.


531
00:37:59 --> 00:38:03
So, here's what I do when I'm
looking at my program and I

532
00:38:03 --> 00:38:06
want to see where something is.


533
00:38:06 --> 00:38:13
I use the text editor, and
I'm going to search for it.

534
00:38:13 --> 00:38:24
Let's see.


535
00:38:24 --> 00:38:31
Well, it's not there.


536
00:38:31 --> 00:38:33
There it is.


537
00:38:33 --> 00:38:37
So here when I get my field, I
get my drunk, and then I get a

538
00:38:37 --> 00:38:44
field which is an odd field.


539
00:38:44 --> 00:38:46
All right, anyone want to
speculate what will happen

540
00:38:46 --> 00:38:54
when I run this one?


541
00:38:54 --> 00:38:57
Think the drunk will be
closer or further from the

542
00:38:57 --> 00:39:00
origin when we're done?


543
00:39:00 --> 00:39:03
Who thinks closer?


544
00:39:03 --> 00:39:05
Who thinks further?


545
00:39:05 --> 00:39:07
Who thinks no difference?


546
00:39:07 --> 00:39:10
Well, you're right, I mean it's
sort of logical if we think

547
00:39:10 --> 00:39:12
about it, that if every once in
a while you get to zipped, back

548
00:39:12 --> 00:39:15
it's going to be harder
to get further away.

549
00:39:15 --> 00:39:17
We'll see different behaviors,
by the way, it depends

550
00:39:17 --> 00:39:19
on the drunk, right?


551
00:39:19 --> 00:39:23
That's true for the usual
drunk, but maybe not for

552
00:39:23 --> 00:39:27
the east west drunk.


553
00:39:27 --> 00:39:29
We can see what happens.


554
00:39:29 --> 00:39:33
So let's try some.


555
00:39:33 --> 00:39:36
We'll do it for the usual
drunk, which is the more

556
00:39:36 --> 00:39:58
interesting case to start with.


557
00:39:58 --> 00:40:01
So you'll see whereas before we
were up just shy of 20 most of

558
00:40:01 --> 00:40:06
the time, here we don't get
very much much, further.

559
00:40:06 --> 00:40:13
We do get steadily further,
but at a lot slower pace.

560
00:40:13 --> 00:40:16
What do you think it means
that, before you remember, we

561
00:40:16 --> 00:40:17
saw a pretty smooth line.


562
00:40:17 --> 00:40:21
Here we see what may look
you like a fat line, but

563
00:40:21 --> 00:40:25
is in fact a line with
a lot of wiggles in it.

564
00:40:25 --> 00:40:30
What does that imply, about
what's going on in the

565
00:40:30 --> 00:40:35
simulation, the fact that
line is jagged rather than

566
00:40:35 --> 00:40:41
smooth as it was before?


567
00:40:41 --> 00:40:43
Pardon?


568
00:40:43 --> 00:40:44
Can't hear you?


569
00:40:44 --> 00:40:44
STUDENT:


570
00:40:44 --> 00:40:44
That it jumps.


571
00:40:44 --> 00:40:50
PROFESSOR: Well, certainly it
happens because of the jumps,

572
00:40:50 --> 00:40:51
but what else is going on?


573
00:40:51 --> 00:40:54
But what is it, sort of, imply
more, whoa, sorry about

574
00:40:54 --> 00:40:57
that, more generally?


575
00:40:57 --> 00:41:01
Right, what it's saying, is
that because of the jumps,

576
00:41:01 --> 00:41:07
there's a lot more variation
from trial to trial.

577
00:41:07 --> 00:41:09
Not surprising, right, because
you sometimes you hit

578
00:41:09 --> 00:41:12
these wormholes and
sometimes you don't.

579
00:41:12 --> 00:41:15
So let's look at
this other figure.

580
00:41:15 --> 00:41:18
This is kind of interesting.


581
00:41:18 --> 00:41:24
So here's the scatter plot.


582
00:41:24 --> 00:41:37
And if we zoom in on it, what
we see here is, there are

583
00:41:37 --> 00:41:41
essentially holes, you know,
these alleys where

584
00:41:41 --> 00:41:46
no points lie.


585
00:41:46 --> 00:41:50
Just as we might have
guessed from here.

586
00:41:50 --> 00:41:53
The only way you would get a
point lying on these alleys, if

587
00:41:53 --> 00:41:55
it happened to be the very last
step of the simulation, not

588
00:41:55 --> 00:41:58
even then, because the
last thing we do is go

589
00:41:58 --> 00:42:00
back to the origin.


590
00:42:00 --> 00:42:04
So we can look at this, and
say, well, sure enough, this

591
00:42:04 --> 00:42:15
is keeping people off of a
certain part of the field.

592
00:42:15 --> 00:42:20
All right, some of you will be
happy to know we are, for the

593
00:42:20 --> 00:42:22
moment, actually maybe for
the whole rest of the term,

594
00:42:22 --> 00:42:24
leaving the notion of drunks.


595
00:42:24 --> 00:42:26
Though not of random walks.


596
00:42:26 --> 00:42:30
All right, any questions
about what's going on here?

597
00:42:30 --> 00:42:32
And again, I would urge you to
sort of study the code and the

598
00:42:32 --> 00:42:35
way it's been factored to
accomplish these things.

599
00:42:35 --> 00:42:38
And play with it, and look
at what you get from

600
00:42:38 --> 00:42:40
the different plots.


601
00:42:40 --> 00:42:44
I now want to pull back from
this specific example, and

602
00:42:44 --> 00:42:50
spend a few minutes talking
about simulation in general.

603
00:42:50 --> 00:42:55
Computer simulation really
grew hand in hand with the

604
00:42:55 --> 00:42:59
development of the computers,
from the very beginning.

605
00:42:59 --> 00:43:02
The first large-scale
deployment of computer

606
00:43:02 --> 00:43:15
simulation was as part of
the Manhattan Project.

607
00:43:15 --> 00:43:19
This was done during the
war to model the process

608
00:43:19 --> 00:43:29
of nuclear detonation.


609
00:43:29 --> 00:43:33
And, in fact, what they did was
a simulation of 12 hard

610
00:43:33 --> 00:43:37
spheres, and what would happen
when they would bump

611
00:43:37 --> 00:43:38
into each other.


612
00:43:38 --> 00:43:43
It was a huge step forward to
do it with simulation, since

613
00:43:43 --> 00:43:46
the whole project had been
stalled by their attempt

614
00:43:46 --> 00:43:48
to do this analytically.


615
00:43:48 --> 00:43:52
They were unable to actually
solve the problem analytically.

616
00:43:52 --> 00:43:55
And it was only when they hit
upon the notion of using a

617
00:43:55 --> 00:43:59
computer, a relatively new tool
in those days, to simulate it,

618
00:43:59 --> 00:44:02
that they were able to get
the answers they needed.

619
00:44:02 --> 00:44:05
And they did it using
something called a

620
00:44:05 --> 00:44:19
Monte Carlo simulation.


621
00:44:19 --> 00:44:22
Now in fact, that's just what
we've been doing with the

622
00:44:22 --> 00:44:24
random walk, the Monte Carlo
simulation, and I'll come

623
00:44:24 --> 00:44:27
back to that in a minute.


624
00:44:27 --> 00:44:32
In general, the thing to think
about, is we use simulation

625
00:44:32 --> 00:44:39
when we really can't get a
closed form analytic solution.

626
00:44:39 --> 00:44:42
If you can put down a system
of equations and easily solve

627
00:44:42 --> 00:44:47
it to get an answer, that's
usually the right thing to do.

628
00:44:47 --> 00:44:50
But when we can't easily do
that, the right thing to do

629
00:44:50 --> 00:44:54
is typically to fall
back on simulation.

630
00:44:54 --> 00:44:57
Sometimes even when we can do
it analytically, simulation

631
00:44:57 --> 00:45:05
has some advantages, as we'll
be seeing as we go forward.

632
00:45:05 --> 00:45:16
What we're typically doing when
we're simulating anything, is

633
00:45:16 --> 00:45:35
we're attempting to generate a
sample of representative

634
00:45:35 --> 00:45:43
scenarios.


635
00:45:43 --> 00:45:47
Because an exhaustive
enumeration of all possible

636
00:45:47 --> 00:45:50
states would be impossible.


637
00:45:50 --> 00:45:53
So again, if sometimes you
can't solve it analytically,

638
00:45:53 --> 00:45:56
you can exhaustively
enumerate the space and

639
00:45:56 --> 00:45:59
then see what's going on.


640
00:45:59 --> 00:46:01
But again, usually you can't.


641
00:46:01 --> 00:46:04
And so we look for a sample,
and the key is, it's

642
00:46:04 --> 00:46:07
gotta be representative.


643
00:46:07 --> 00:46:11
Of what we would
get in reality.

644
00:46:11 --> 00:46:14
We'll see an example
of that shortly.

645
00:46:14 --> 00:46:20
So simulation attempts to build
an experimental device that

646
00:46:20 --> 00:46:25
will act like the real system
in important aspects.

647
00:46:25 --> 00:46:29
So I always think of
a simulation as an

648
00:46:29 --> 00:46:32
experimental device.


649
00:46:32 --> 00:46:37
And every time I run it, I'm
running an experiment designed

650
00:46:37 --> 00:46:41
to give me some information
about the real world.

651
00:46:41 --> 00:46:44
These things are enormously
popular, so if you were

652
00:46:44 --> 00:46:52
to get on Google and
look at simulation.

653
00:46:52 --> 00:47:00
So for, example, we could
Google simulation finance.

654
00:47:00 --> 00:47:06
You know, we see we get
about 3,000,000 hits.

655
00:47:06 --> 00:47:13
We can do biology.


656
00:47:13 --> 00:47:16
We get twice as many hits,
showing the relative

657
00:47:16 --> 00:47:21
importance in the world
of biology and finance.

658
00:47:21 --> 00:47:25
For all you Course 7 majors, we
should make sure that should

659
00:47:25 --> 00:47:27
compare biology to physics.


660
00:47:27 --> 00:47:31
Oh dear, we'll see that physics
is even more important

661
00:47:31 --> 00:47:35
than biology.


662
00:47:35 --> 00:47:38
And I know we have a lot of
Course 2 students in here,

663
00:47:38 --> 00:47:43
so let's try mechanical.


664
00:47:43 --> 00:47:45
Oh, lot of those, too.


665
00:47:45 --> 00:47:48
And if I did baseball,
or football, we'd get

666
00:47:48 --> 00:47:50
more than any of them.


667
00:47:50 --> 00:47:54
Showing the real importance
of things in the world.

668
00:47:54 --> 00:48:00
As we look at simulations, I
want you to keep in mind that

669
00:48:00 --> 00:48:12
they are typically descriptive
not prescriptive.

670
00:48:12 --> 00:48:18
So by that I mean, they
describe a situation,

671
00:48:18 --> 00:48:22
they don't tell you
what the answer is.

672
00:48:22 --> 00:48:28
So another way to think about
this is, a simulation is not

673
00:48:28 --> 00:48:31
an optimization procedure.


674
00:48:31 --> 00:48:35
When we looked at optimization,
that was prescriptive.

675
00:48:35 --> 00:48:41
We ran an optimization
algorithm, and it gave us

676
00:48:41 --> 00:48:45
the best possible solution.


677
00:48:45 --> 00:48:48
A simulation typically doesn't
give you this best possible

678
00:48:48 --> 00:48:52
solution, but if you give
it the starting point,

679
00:48:52 --> 00:48:58
it will tell you the
consequences of that.

680
00:48:58 --> 00:49:01
Now, can we use simulation
to do optimization?

681
00:49:01 --> 00:49:03
Absolutely.


682
00:49:03 --> 00:49:07
For example, we can use it
to do guess and check.

683
00:49:07 --> 00:49:10
Where we guess a, probably not
to get a truly optimal

684
00:49:10 --> 00:49:15
solution, but to get a good
solution, we can guess various

685
00:49:15 --> 00:49:18
possibilities, simulate
them, and see.

686
00:49:18 --> 00:49:21
People do that all the time.


687
00:49:21 --> 00:49:27
If they want to see what's the
right number of checkout

688
00:49:27 --> 00:49:30
counters at the supermarket, or
what's the right airline

689
00:49:30 --> 00:49:32
schedule to use?


690
00:49:32 --> 00:49:35
They'll make a guess, they'll
simulate it, and they'll say

691
00:49:35 --> 00:49:39
all right, this was better than
this, so we'll choose it.

692
00:49:39 --> 00:49:42
All right, I'm going
to stop here.

693
00:49:42 --> 00:49:45
Next time we're going to get
more deeply into the actual

694
00:49:45 --> 00:49:49
stochastics, the probability
distributions, and start

695
00:49:49 --> 00:49:52
understanding what's going
on under the covers.

696
00:49:52 --> 00:49:52