Lecture 2 - Matlab Building Blocks - Arrays and Matrices

Eran Eden, Weizmann 2008 © 1

Introduction to Matlab & Data Analysis

Topic #2 (continued): The Matlab Building Blocks –

Arrays and Matrices

Announcements Apostrophe or Single quotes ‘ ‘

Parenthesis ()

HW Lab assistance:

2

3

Previously…

Matlab is a high level language Easy to learn

Easy to debug

Easy visualization

Many useful “toolboxes”

The Matlab working environment Our first program

help!

Variables… example:

count = 5;

4

Variables

Tip #1: Give your variable meaningful names.

salary = 9000;

hours = 5;

And NOT…

a = 9000;

b = 5;

5

Variables

Tip #2: Don’t make variable names too long

salary_I_got_for_my_work_at_the_gasoline_station = 9000;

salary_I_got_for_my_work_in_the_bakery = salary_I_got_for_my_work_at_the_gasoline_station * 3;

disp(salary_I_got_for_my_work_in_the_bakery);

Very bad choice of variable name!!!

Tip #3: Whatever you do - be consistent.

6

Variables

Several functions in a row. Should be separated by a comma

But:

7

Today on the menu

Operations on variables

1D Arrays

Creating

Indexing

Operating on

2D Arrays (matrices)

3D Arrays

Operations on variables

8

Round up

>> x = 2.2;

>> ceil(x)

ans =

3

Round down

>> x = 2.6;

>> floor(x)

ans =

2

Round

>> x = 2.6;

>> round(x)

ans =

3

>> x = 2.2;

>> round(x)

ans =

2

>> x = 1.5;

>> round(x)

ans =

2

Operations on variables I want to get the absolute value of a variable but don’t know

the command… what should I do?

… Google it!

9

Absolute value of a number

>> x = -2

>> abs(x)

ans =

2

10

Data is stored in data structures

A Data Structure is a set of values that are organized in a predefined manner

Examples:

Tree (parent)

Stack (LIFO)

Queue (FIFO)

Array (index)

And more…

11

Array is the main data structure used in Matlab

Almost everything we build in Matlab - we build from arrays.

The variables that contain one element that we learned so far can be thought of as a degenerate case of a 1 x 1 array.

12

Examples of 1D and 2D arrays An 1D array containing student grades

An 1D array containing protein levels

An 1D array of characters

A 2D array, also called matrix, of distances between cities

Note that all the elements within an array are of the same variable type

100 100 100 98 100 50 99

0.54 0.01 0.0 1.2 1.24

0 23 120 21

23 0 56 10

120 56 0 8

D o n ‘ t w o r r y , b e h a p p y

13

Creating 1D Arrays

The simplest way to create an array

>> x = [1 2 3 8 12 40];

>> disp(x)

1 2 3 8 12 40

Adding commas to separate the numbers gives the same result:

>> x = [1, 2, 3, 8, 12, 40];

>> disp(x)

1 2 3 8 12 40

No problem! This is how you do it: >>x = [ 10 11 12 13 14 15 16 ... 17 18 19 20 21 22 23 24 25 ... 26 27 28 29 30 31 32 33 34 ... 35 36 37 38 39 40 41 42 43 ... 44 45 46 47 48 49 50 51 52 ... 53 54 55 56 57 58 59 60 61 ... 62 63 64 65 66 67 68 69 70 ... 71 72 73 74 75 76 77 78 79 ... 80 81 82 83 84 85 86 87 88 ... 89 90 91 92 93 94 95 96 97 ... 98 99 100];

14

Creating 1D Arrays

How do we create an array containing all the numbers between 10 and 100 ?

15

Creating 1D Arrays

Alternatively we can do this:

>>x = 10 : 1 : 100;

>> disp(x) Columns 1 through 19

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Columns 20 through 38

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47


48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66


67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85


86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

Start at 10 Stop when you get to 100

count up by 1

16

Creating 1D Arrays

What is the value of x?

>> x = 0 : 0.1 : 2;

>> disp(x)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

2

How can you create an array with numbers from 100 to 1 in decreasing order?

17

Creating 1D Arrays

Initializing a vector to zeros >> x = zeros(1, 5)

x =

0 0 0 0 0

Initializing a vector to ones >> x = ones(1, 6)

x =

1 1 1 1 1 1

Initializing a vector to random numbers >> x = rand(1, 4)

x =

0.9501 0.2311 0.6068 0.4860

Generates a Uniformly distributed pseudo-random number between 0 and 1

1 row, 6 columns

18

Concatenating 1D Arrays

Example:

x = 1 : 5;

y = 100 : 105;

z = [x, y, x];

disp(z);

1 2 3 4 5 100 101 102 103 104 105 1 2 3 4 5

x y x

x, y and x are concatenated into one array

19

Indexing 1D Arrays Question: How do we retrieve/manipulate the content

of a specific element inside an array?

Answer: We use a mailbox like system called indexing

x(i) is the i’th element of x

1 2 3 4 5

20

Indexing 1D Arrays We will study different types of indexing through

examples: What is the output of the following program?

x = 10 : -1 : 1;

y = x(3);

disp(y);

8

What is the output of the following program?

x = 10 : -2 : 1;

y = x(5);

disp(y);

10 9 8 7 6 5 4 3 2 1

1 2 3 4 5 6 7 8 9 10

8

x

y

21

Indexing 1D Arrays


x = 10 : -1 : 1;

y = x(12);

disp(y);


x = 10 : -1 : 1;

y = x(3.2);

disp(y);

??? Index exceeds matrix dimensions.

??? Subscript indices must either be real positive integers or logicals.

22

Indexing 1D Arrays

We can retrieve more than one element at a time: What is the output of the following program?

x = 10 : -1 : 1;

y = x([3,9,4]);

disp(y);


x = 10 : -1 : 1;

y = x(2 : 2 : 8);

disp(y);

23

Indexing 1D Arrays

Indexing the last element in a vector...

What is the output of the following program? x = 10 : -1 : 1;

y = x(end)

disp(y);

1


x = 10 : -1 : 1;

z = x(end - 1)

disp(z);

2

24

Indexing 1D Arrays

The same indexing can be applied to arrays of chars.


str = 'If I was a rich man, Yaba dibi dibi ... dibi dibi di';

str2 = str([1 : 10, 21: 25]);

disp(str2);

If I was a Yaba

25

Using indexing to edit 1D arrays

We can manipulate the content of an array


x = 10 : -1 : 1;

x(2) = 10;

disp(x);


x = 10 : -1 : 1;

x(14) = 99;

disp(x);

10 9 8 7 6 5 4 3 2 1

10 10 8 7 6 5 4 3 2 1

Remark: Notice that adding an element beyond the array boundary is NOT an error!

10 9 8 7 6 5 4 3 2 1 0 0 0 99

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

11 12 13 14

26


We can erase parts of an array


x = 10 : -1 : 1;

x(2) = [];

disp(x);


x = 10 : -1 : 1;

x(2 : 4) = [];

disp(x);

10 9 8 7 6 5 4 3 2 1

10 8 7 6 5 4 3 2 1

10 9 8 7 6 5 4 3 2 1

10 6 5 4 3 2 1

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7

27

1D Array Orientation

So far we’ve seen row vectors (one row and multiple columns), How can one create a column vector (one column and multiple rows) ?

Use semicolon seperator:

x = [1; 2; 3; 4; 5];

disp(x);

1

2

3

4

5

Use transpose operator x = [1, 2, 3, 4, 5]; y = x'; disp(y)

1 2 3 4 5

Simple operations on 1D arrays


>> c = 1 : 0.1 : 2

>> round(c)

ans =

1 1 1 1 1 2 2 2 2 2 2


>> c = 1 : 0.1 : 2

>> ceil(c)

1 2 2 2 2 2 2 2 2 2


>>abs(floor(-0.3 : 0.1 : 0.3))

28

All the operations that were applied on variables containing one element can be applied to 1D arrays


1 1 1 0 0 0 0

29

30


Finding the maximal number in a vector

>> x = 1 : 50;

>> max(x)

50

Finding the minimal number in a vector

>> min(x)

1

Finding the mean of a vector

>> mean(x)

25.500

Finding the standard diviation of a vector

>> std(x)

14.5774

31


Finding the size of a vector

>> x = 1 : 50;

>> size(x)

ans =

1 50

>> y = x'

>> size(y)

ans =

50 1

What is length then?

max(size(x))

Finding the length of a matrix

>> x = 1 : 50;

>> length(x)

ans =

50

>> y = x'

>> length(y)

ans =

50

Returns the maximal dimension

32


x= 1 : 50

What is the difference between:

y = x(length(x));

and

y = x(end);

“Same same…”

33


The rand(m,n) function is a pseudo randon number generator. The

function starts from a number (seed) which is operated on in a

complicated non linear formula. Each pass produces a number

that is apperantly uncorrelated to the previous number or to the

following number. It is “pseudo” in the sense that if we start

from the same seed we will always get the same series of

numbers (deteministic). However, it is random enough for the

purpose of simulations. In the Matlab environement the seed

changes as well in the default settings.

A period is the number of passes between two identical numbers,

in which the numbers do not repeat. The period is usually set to

be extermely long. (Wikipedia).

34

Sub-array searching

The “find” operation returns indices

>> x = [2 8 7 6 4 2 3];

>> find(x == 2)

1 6

>> find(x > 3)

2 3 4 5

How do we get the values in x that are larger than 3?

How do we get the values in x that are larger than 3 that is most to the right?

Finds indices of all values equal to 2

Finds indices of all values larger than 3

35

Sub-array searching

The “find” operation How do we get the values in x that are larger than 3? >>y=x(find(x>3))

y =

8 7 6 4

How do we get the values in x that are larger than 3 that is most to the right?

>>v=x(max(find(x>3)))

v =

4

More functions on vectors

Intersection of two vectors >> doron_free_days = 1 : 2 : 31;

>> tal_free_days = [ 6 7 8 12 14 18 19 20];

>> meeting_days = intersect(doron_free_days, tal_free_days)

meeting_days =

7 19

Union of two vectors >> dog_eating_days = union(doron_free_days, tal_free_days)

dog_eating_days =

1 3 5 6 7 8 9 11 12 13 14 15

2 17 18 19 20 21 23 25 27 29 31

36

37

Arithmetic operations on 1D arrays

The plus (+) and minus (-) operations are naturaly extended to arrays

Example 1:

x =[ 2 4 6];

y = x – 1

y =

1 3 5

Example 2:

x = x + x

x =

4 8 12

Notice that the operation is applied to all the elements in the array

(1) Scalar multiplication

price = [10 20 30];

new_price = price * 2

new_price =

20 40 60

You can use paranthesis to change order of multiplication (just like in regular math) >> (price + 1) * 2

ans =

22 42 62

>> price + 1 * 2

ans =

12 22 32

38


The multiplication * is a bit more tricky...

(2) Vector multiplication

price = [10 20 30];

quantity = [3 2 1]';

price * quantity

ans =

100

Vector dimensions should match (exactly like in mathemtics)

39


3

2

1

10 20 30

price

quantity'


price = [10 20 30];

quantity = [3 2 1];

price * quantity

??? Error using ==> mtimes

Inner matrix dimensions must agree.

10 20 30

price quantity

3 2 1 *

*

(3) Element by element multiplication

>> price = [10 20 30];

quantity = [3 2 1];

price .* quantity

ans =

30 40 30

40


10 20 30

price


quantity 3 2 1 .*

Summary Example of 1D arrays: The Casino Fraud...

Simulate the throw of two dice (i.e. randomly choose two numbers between 1 and 6)

>> dice = ceil(6 * rand(1,2))

dice =

6 5

Create loaded two dice where the chances of getting a 6 is ½ and the chances of getting the other numbers is 1/10 for each number.

>> dice = ceil(10 * rand(1,2));

>> dice(find(dice > 6)) = 6

dice =

6 6

41

Summary Example of 1D arrays: A small warm-up on plotting arrays...

What will the following program plot?

x = 0 : 0.1 : 2;

x = x * pi;

y = sin(x);

plot(x, y);

Radians or degrees?

42

0 1 2 3 4 5 6 7-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Matrices (2D arrays)

43

44

Matrices (2D arrays)

Why do we need Matrices?

1D Arrays that we learned so far can be thought of as a degenerate case of a 1 x N or N x 1 2D array.

10 21 10 21

73 21 18 21

10 4 8 21

3 21 10 45

8 21 2 21

Employee Salary Vacation days

Sickness days

Yosi 10000 18 4

Dudu 12000 8 10

Ruti 30000 10 12

Pinhas 60000 2 1

45

Creating matrices

One way to create a matrix:

x = [1 2 3 4

5 6 7 8]

x =

1 2 3 4

5 6 7 8

Another way:

x = [1 2 3 4; 5 6 7 8]

x =

1 2 3 4

5 6 7 8

46

Creating Matrices x = ones(3,4)

x = 1 1 1 1 1 1 1 1 1 1 1 1

x = zeros(3,4)

x = 0 0 0 0 0 0 0 0 0 0 0 0

Number of rows

Number of columns

open inetrval does not include ends

47

Creating Matrices Initiating a matrix with random values: x = rand(3,4)

x =

0.0832 0.0018 0.0143 0.6034

0.0539 0.8238 0.3402 0.2905

0.4206 0.8405 0.6103 0.3517

Simulate a random choice of numbers in a roulette game (0, 00, 1...36). ’00’ should be represented by -1. Store result s in a 3 by 3 matrix.

x = ceil(rand(3,3)* 38) – 2

x =

13 15 29

25 25 35

19 22 18

http://upload.wikimedia.org/wikipedia/en/1/11/Rwheel.jpg

48

Indexing 2D arrays

Indexing 2D arrays is done the similarly to 1D arrays

x(h,w) is the h row and the w column

what is the value of x(3,4)?

10 21 10 4

73 21 18 10

10 4 8 21

3 21 10 45

8 21 2 21

3

4

what is the value of x(1 : 3, 4)?

10 21 10 4

73 21 18 10

10 4 8 21

3 21 10 45

8 21 2 21

3

4

2 1

what is the value of x(2 : 3, 3 : 4)?

10 21 10 4

73 21 18 10

10 4 8 21

3 21 10 45

8 21 2 21

49

Indexing 2D arrays

x(h,:) is the h row x(:,w) is the w column

what is the value of x(3,:)?

10 21 10 4

73 21 18 10

10 4 8 21

3 21 10 45

8 21 2 21

what is the value of x(:, 1:2)?

10 21 10 4

73 21 18 10

10 4 8 21

3 21 10 45

8 21 2 21

what is the value of x(:, :)?

10 21 10 4

73 21 18 10

10 4 8 21

3 21 10 45

8 21 2 21

50

Indexing 2D arrays

>> a = [1 2 3; 4 5 6; 7 8 9];

>> disp(a)

1 2 3

4 5 6

7 8 9

>> a(1)

ans =

1

>> a(2)

ans =

4

>> a(3)

ans =

7

It’s possible to index a 2D array using a single coordinate

>> a(5)

ans =

5

>> a(6)

ans =

8

>> a' ans = 1 4 7 2 5 8 3 6 9

This is the order matlab reads matrix elements

51


y = zeros(3,3);

y(1,3) = 4

y =

0 0 4

0 0 0

0 0 0

What does the matrix y look like?

The matrix size on the left and right of the assignment should match in size.

y = zeros(3,3);

x = ones(2,2);

y(1 : 2, 1 : 2) = x

y =

1 1 0

1 1 0

0 0 0


Size 1 x 1 Size 1 x 1

Size 2 x 2 Size 2 x 2 = =

52


y = zeros(3,3);

y(1:2,1:2) = 4

y =

4 4 0

4 4 0

0 0 0


The matrix size on the left and right of the assignment should match in size,

unless the right matrix has a size of 1 x 1


53


y = zeros(5,5);

x = ones(2,2);

y = x

y =

1 1

1 1


The matrix size on the left and right of the assignment should match in size,

unless the right matrix has a size of 1 x 1 OR

unless you are overwriting a matrix


54


y = zeros(3,3)

y([1, 3] , [1, 3]) = 4

y =

4 0 4

0 0 0

4 0 4


y = zeros(3,3);

x = ones(2,2);

y(2 : 3, 2 : 3) = x

y =

0 0 0

0 1 1

0 1 1


y = zeros(3,3);

x = ones(4,4);

y(2 : 3, 2 : 3) = x

??? Subscripted assignment dimension mismatch.


55


>> y = [1 2 3; 4 5 6; 7 8 9];

>> disp(y)

1 2 3

4 5 6

7 8 9

>> y(2, :) = [];

>> disp(y);

1 2 3

7 8 9

>> y(: , [1 3]) = [];

>> disp(y);

disp(y);

2

8

Erasing a part of a matrix


>> y = [1 2 3; 4 5 6; 7 8 9];

>> disp(y)

1 2 3

4 5 6

7 8 9

>> y(2, 2) = []

??? Indexed empty matrix

assignment is not allowed


56


Finding the size of a matrix

x =[ 2 4 6

3 6 9];

size(x)

2 3

Finding the length of a matrix

length(x)

3

Number of rows Number of columns

All the operations that were applied on 1D arrays can be applied to 2D arrays

transpose

x =[ 2 4 6

3 6 9];

x'

2 3

4 6

6 9

57


Finding the maximal numbers in each matrix column

>> x = [1 2 3

7 8 9

4 5 6];

>> max(x)

ans =

7 8 9

Think of two ways to get the maximal element in the entire matrix…

>>max(max(x))

Finding the mean of each matrix column

>> mean(x)

ans =

4 5 6

How do we get the mean of each matrix row?

>> mean(x,2)

58


Finding elements within 2D arrays is similar to 1D arrays.

Finding using 2 coordinates indexing:

>> x = [1 2 3

7 8 9

4 5 6];

>> [h, w] = find(x > 5)

h =

2

2

2

3

w =

1

2

3

3

59


Finding things within 2D arrays is similar to 1D arrays.

Finding using 1 coordinate indexing

>> x = [1 2 3

7 8 9

4 5 6];

>> inds = find(x > 5)

inds =

2

5

8

9

60


The plus (+) and minus (-) operations are naturaly extended and 2D arrays

Example 1:

x =[ 2 4 6

3 6 9];

y = x – 1

y =

1 3 5

2 5 8

Example 2:

x = x + x

x =

4 8 12

6 12 18

y =

1 3 4

2 4 7

61


Question: write a commmand that subtracts 1 from all the values in y that are larger than 4 and stores it back into y

y = [1 3 5

2 5 8];

Answer

y(find(y > 4)) = y(find(y > 4)) - 1

(1) Scalar multiplication

price = [10 20 30; 2 3 4];

new_price = price * 2

new_price =

20 40 60

4 6 8

62


63


(2) Matrix multiplicaiton

price = [2 3 4; 2 4 5];

quantity = [4 1; 0 2; 2 1];

price * quantity

ans =

16 12

18 15

4 1

0 2

2 1

2 3 4

2 4 5

* = 16 12

18 15

64


(3) element by element multiplicaiton

price = [2 3 4; 2 4 5];

quantity = [4 0 2; 1 2 1];

price .* quantity

ans =

8 0 8

2 8 5

2 3 4

2 4 5 .* = 4 0 2

1 2 1

8 0 8

2 8 5

65

“Flattening” 2D arrays We can turn a matrix into a vector

(ordered according to columns)

x = [1 2 3

4 5 6

7 8 9];

y= [x(:, 1); x(:, 2); x(:, 3)]

y =

1

4

7

2

5

8

3

6

9

Alternatively the following command will do the trick…

y = x(:)

y =

1

4

7

2

5

8

3

6

9

66

Final small example In the matrix vac_days each column represents the number of vacations days an employee

took in the last 3 months.

In the matrix salaries each column represents that employee’s salary in the last 3 months.

Correct the salaries so that employees that took less then 3 days of vacation in a certain month get a 10% bonus to their salary.

>> vac_days = [1 5 10 0; 0 0 1 10 ; 0 0 5 7 ]

vac_days =

1 5 10 0

0 0 1 10

0 0 5 7

>> salaries = [6000 6000 7000 20000; 6000 6000 7000 20000; 7000 7000 7000 20000;]

salaries =

6000 6000 7000 20000

6000 6000 7000 20000

7000 7000 7000 20000

67

Final small example

Answer:

>> inds = find(vac_days < 3);

>> salaries(inds) = salaries(inds) * 110/100;

>> disp(salaries)

6600 6000 7000 22000

6600 6600 7700 20000

7700 7700 7000 20000

3D arrays…

68

10 21 10 21

73 21 18 21

10 4 8 21

3 21 10 45

8 21 2 21

10 21 10 21

73 21 18 21

10 4 8 21

3 21 10 45

8 21 2 21

10 21 10 21

73 21 18 21

10 4 8 21

3 21 10 45

8 21 2 21

Color images are 3D arrays (We will learn this at the end of the course)

Why the heck do we need 3D arrays ?#@!?

R

G

B

69

3D arrays

All the (initialization / indexing / operations) that we learnt on 2D arrays can be applied to 3D arrays.

In fact, they can be applied to any N dimensional array.

70

3D arrays - example >> x = zeros(3,3,3)

x(:,:,1) =

0 0 0

0 0 0

0 0 0

x(:,:,2) =

0 0 0

0 0 0

0 0 0

x(:,:,3) =

0 0 0

0 0 0

0 0 0

71

3D arrays (example continued)

>> x(:, :, 3) = ones(3,3)

x(:,:,1) =

0 0 0

0 0 0

0 0 0

x(:,:,2) =

0 0 0

0 0 0

0 0 0

x(:,:,3) =

1 1 1

1 1 1

1 1 1

72

3D arrays (example continued) >> x(2, 2, :) = 3

x(:,:,1) =

0 0 0

0 3 0

0 0 0

x(:,:,2) =

0 0 0

0 3 0

0 0 0

x(:,:,3) =

1 1 1

1 3 1

1 1 1

73

Tip of the day… improving your code readbility

Use spaces and tabs to make your code more clear

This code is

Better than this one...

age=10; weight=50; height= 1:0.1:4; hair= 2:0.1:5;

age = 10; weight = 50; height = 1 : 0.1 : 4; hair = 2 : 0.1 : 5;

Lecture 2 - Matlab Building Blocks - Arrays and Matrices

Documents

Transcript of Lecture 2 - Matlab Building Blocks - Arrays and Matrices