Lecture 4 amended and corrected

1
Lecture 4 (amended and corrected)

• Divide and conquer
• Dot product

2
Last time yahoo/google correlation

• Start with 'yahoo.csv' and 'google.csv', stock
  prices per day for 2 stocks
• Data is in dollars
• Want to find what days both prices rose

3
Strategy

• 1. Convert the Google data into positive and
  negative percent changes.
• 2. Convert the Yahoo data into positive and
  negative percent changes.
• 3. Find out when Google rose.
• 4. Find out when Yahoo rose.
• 5. Find out when they both rose.

4
Steps

• Really only three steps compute percent changes,
  find rises, and match-up rises.

5
Google percentage price change

• change 100 (today-yesterday) / yesterday
• In matlab
• googleChange
• 100(google(2end) google(1end-1))
• ./ google(1end-1)
• Changes for Yahoo will look very similar.

6
Write a procedure!

• function res percentChange(pricelist)
• for a list of n prices, compute the n-1
  percentage
• changes in price from previous day.
• res 100(priceList(2end) priceList(1end-1))
  
• ./ priceList(1end-1)
• Alternative
• today pricelist(2end)
• Yesterday pricelist(1end-1)
• res 100(today yesterday) ./ yesterday

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Strategy

• Convert the Google data into positive and
  negative percent changes.
• function compareStocks()
• yahoo csvread(yahoo.csv)
• google csvread(google.csv)
• yahooChanges percentChange(yahoo)
• googleChanges percentChange(google)
• more to come

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Strategy

• 1. Convert the Google data into positive and
  negative percent changes.
• 2. Convert the Yahoo data into positive and
  negative percent changes.
• 3. Find out when Google rose.
• 4. Find out when Yahoo rose.
• 5. Find out when they both rose.

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When did google rise?

• Google price rose on days when googleRises gt 0.
• Lets get a list of 1s and 0s for rise or
  didnt.
• Easy
• googleRises (googleChanges gt 0)

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Write another function

• function res findRises(data)
• report which elements of data represent price
• rises
• res (data gt 0)

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Main procedure, revised

• function compareStocks()
• yahoo csvread(yahoo.csv)
• google csvread(google.csv)
• yahooChanges percentChange(yahoo)
• googleChanges percentChange(google)
• yahooRises findRises(yahooChanges)
• googleRises findRises(googleChanges)
• rises (yahooRises googleRises) gets printed
• days find(rises) gets printed indexes of
  days.
• Note the single-ampersand () is elementwise
  and, which you should use is something
  slightly different, which we wont need.

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Logical operations

• If p and q are logicals, then AND and
  OR.
• P Q is True only of p is True AND q is True.
• P Q is true if
• p is True or
• Q is True or
• Both are true
• p is not p not true gt false not false gt
  true.
• , , and all operate elementwise on arrays.

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Example of multiple returns

• Compute both the greatest common divisor and the
  least common multiple of n and k.
• function g, m both(n, k)
• compute gcd and lcm of n and k.
• g gcd(n,k)
• m lcm(n,k)

14
New topic matrix products

• Useful in economic computations, all sorts of
  scientific modeling, etc.
• Surprisingly easy
• Impress your friends by saying that you learned
  to compute tensor products for tensors of type
  (1,1). ?

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First step an example

• Read in two files
• Number of computers sold by each of several
  manufacturers
• Average computer price for each manufacturer
• computers csvread('computers.csv')
• prices csvread('prices.csv')
• What is total revenue?
• totalmoney1 sum(computers . prices)
• Can also be written
• totalmoney1 dot(computers, prices)

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Dot product

• 10 20 60
• 200 500 100
• 200010000 6000 18000
• The dot product mutliplies corresponding
  elements of two vectorsand then sums up
• Vectors must have same number of elements!
• Also called inner product

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Exercises

• Whats the dot product of ones(1,12) and
  ones(1,12)? (ans 12)
• The dot product of 2 4 5 and -1 0 1? (ans 3)
• The dot product of 1 1 and -1 1? (ans 0)

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Vectors and arrows

• We can interpret a vector like 2 3 as
  representing an arrow from the origin (0,0) to
  the point (2, 3)

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Deep Geometric Truth

• If the dot product of u and v is zero, and both u
  and v are nonzero, then theyre perpendicular!

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Algebraic fact

• Dot product follows the same rules as ordinary
  products, when they make sense.
• Example
• v DOT (u w) v DOT u v DOT w.

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Recall our arrow picture

• We can interpret a vector like 2 3 as
  representing an arrow from the origin (0,0) to
  the point (2, 3)

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Length of a vector

• Pythagoras says h2 22 32, so
• h sqrt(4 9).
• In general
• length of v sqrt(dot(v,v))

h
3
2
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Matrix Product

• If A is n x k, and B is k x p, then AB is an n
  x p matrix.

p
B
k
A
C
n
k
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Matrix Product

• (i,j) entry of AB is dot product of ith row of A
  with jth column of B

Col j
p
B
k
Row i
C
n
k
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Example
0 1 1 4 2 0 1 1 0 0 1 1
B
2 1 3 2 0 7
A
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Properties of matrix multiply

• A(BC) AB AC
• (AB)C AC BC
• (AB)C A(BC)
• The identity matrix I ( eye(n)) has the
  property that I A A (where A is n x k)
• For some n x n matrices, A, theres an inverse
  a matrix B with AB BA I.
• For others, theres not.
• Matlab can find B inv(A).
• Never use this! (or rarely)

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Systems of equations and matrix multiplication

• The equations
• 2x 3y 6
• x 4y 5
• can be rewritten as
• 2 3x 6
• 1 -4y 5
• i.e., in the form Ax b.

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Solving systems

• Matlab can solve systems like Ax b
• Write x A\b.
• Alternative x inv(A) b MUCH
  SLOWER

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Other cases

• You might have more equations than unknowns
• 3 2x 5
• 1 -3y 2
• 4 7 3
• Matlab will find x and y with Ax b as small a
  vector as possible.

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Other cases (II)

• You might have FEWER equations than unknowns
• Matlab will find the smallest possible solution
  vector.

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Other cases (III)

• You might want to solve Ax b, and A is not
  invertiblematlab will tell you so with an
  error/warning.

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New topic LOOPS

• Waited this long because we generally avoid them
  in Matlab code
• Some of you have seen a for-loop in lab already

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for loops

• u 0
• for t 15
• u u t
• end
• U will end up being 1 2 3 4 5.
• Execute the steps inside the loop over and over,
  each time with t having a new value from the list
  of possibilities

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Problem

• Sum up the primes from 1 to 100.
• Solution (one of MANY solutions)
• total 0
• for i 1100
• if (isprime(i))
• total total i
• end
• end
• total do this to make the total print out