Flow of Control

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Flow of Control

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Minimize sum of SQUARES of errors. Called 'least squares fitting' ... Provides a UNIQUE best-fit line. Corresponds to the 'short error vector' ... – PowerPoint PPT presentation

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Title: Flow of Control

1
Lecture 6

Flow of Control
Regression

2
Control flow

Telling what should happen next in a program
Invoking a function is an example
for loops are another example
ifthen is the other important one

3
ifthen

Two-sided, unfair, coin
x rand(1)
if (x gt 0.6)
red
else
blue
end

4
Three-sided coin

Three-sided, unfair, coin
x rand(1)
if (x lt 0.3)
red
else if (x lt 0.7)
blue
else
green
end
end

5
Better

x rand(1)
if (x lt 0.3)
red
elseif (x lt 0.7)
blue
else
green
end
Avoids excess indentation

6
Concrete example

function res countRoots(a,b,c)
how many real roots does ax2 bx c 0
have?
disc bb 4ac
if (disc gt 0))
res 2
elseif (disc 0)
res 1
else
res 0
end

7
For loops, again

What does this program do?
array
for i 140000
array array,i
end

8
Confusion

During class, students were puzzled about array
construction.
Note 3, 4 concatenates the 1x1 array 3
with the 1x1 array 4 to get
3 4
Remember that EVERYTHING in matlab is an array,
even if it doesnt look like one

9
Again

array
for i 140000
array array,i
end
Compare with
array 140000
Second form is MUCH faster!

10
Gauss-in-a-box model

Suppose that size(a) 1 n
a a, i
The right hand side evaluates to a 1 x (n1)
array
Gauss asks the operating system for room to write
it
He writes it down, and then labels it a
He crosses out the previous value of a.

11
How many steps?

array
for i 140000
array array,i
end
First time 1 number must be copied
Second time 2 numbers must be copied
40000th time 40,000 numbers must be copied
Total 1 2 3 40,000 .8 billion steps!

12
While loops

Repeats a set of steps while some condition is
true
function powerTwo(n)
find a number k with 2k gt n
k 0
while (2k lt n)
k k1
end

13
Vectors and If

Suppose you write a procedure to implement this
if x gt 10, y 4
if x lt 10, y 1

14
Code

function y foo(x)
if x gt 10
y 4
else
y 1
end

15
Now vectorize

Youd like to be able to use foo on vectors,
too, so that
foo(1 15 6) gt 1 4 1
Wont work

16
Solution

function y foo(x)
vectorized version of foo
y 4(x gt 10) 1(x lt 10)
x 1 15 6
(xgt10) 0 1 0
4(xgt10) 0 4 0
(x lt 10) 1 0 1
1 (x lt 10) 1 0 1
y 1 4 1

17
Improved code

function y foo(x)
vectorized version of foo
select x gt 10
y 4select 1(select)

18
Ive got some data

Fifty x-y pairs, from an experiment I ran
I THINK they all fit on a line

19
Given these xs and ys, find line

Lets start with a simpler case
x 1 2 3
y 2 6 7

20
Red lines errors
21
Best line?

Solution 1 minimize the sum of the errors.
Sum of blues sum of reds!

22
Best line, version 2

Minimize sum of SQUARES of errors.
Called least squares fitting
If you think of the list of errors as entries in
a vector, youre making this vector as short as
possible
Rationale
Corresponds to what we do instinctively
Provides a UNIQUE best-fit line
Corresponds to the short error vector

23
A little math (in matlab notation)