PSY8219 : Week 2. Homework 1 Due Today. Homework 2 Due September 12. Readings for Today Attaway Chapters 2, 7, and 8

Transcription

1 PSY8219 : Week 2 Homework 1 Due Today (homework solutions will be posted on the web site after class the day the assignment is due or two+ days after if anyone is late turning it in) Homework 2 Due September 12 Readings for Today Attaway Chapters 2, 7, and 8 Readings for Next Week Attaway Chapters 3, 4, and 5

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3 Scripts in Matlab

4 Best Practices Use folders/directories, subfolders/subdirectories Make a copy of a working program before you make new edits to it. Keep copies of experiment programs and analysis programs with any data you collect in an experiment.

5 Running a Script Green Arrow Save and Run Debugging setting breakpoints in code

6 Defining and Executing Sections %% a comment and a start of a section sections should be self-contained units of code single-click / left click to select a section two-finger click / right click to select Evaluate Current Section or Evaluate Current Selection* * on a MacBook trackpad, there is a known two-finger click bug - try using control-click instead

7 Homework Assignments Turn in.m file(s) on Brightspace; if there are multiple files, please turn in as a single ZIP file Use comments to indicate which part of a question you are answering If I ask a question that requires a short answer, you can usually answer it in a comment within the script For a longer answer you might need to submit a pdf file or a word file

8 Homework Assignments include all files needed to run your code when you submit your solutions, even files that I have supplied if you have more than 1 or 2 files, please submit them as a ZIP file clearly inform me which file I should run to test your program

9 Homework Assignments Unless required by the assignment to do otherwise, make sure your lines end with a ; to suppress output - in the class.m files I sometimes leave the ; off so that output is generated in the Command Window please use %% to separate logical sections of your code both as a matter of style, and so I can run subsections of your code when grading note other Best Practices that I will be looking for

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11 QUIZ Quiz.xlm MathWorks version of Jupyter notebooks

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13 Characters and Strings

14 Character Types >> x = 'A' >> whos x >> x = 'ABC' >> whos x

15 String Types >> x = "A" >> whos x >> x = "ABC" >> whos x There are lots of things you can do with characters

16 Character Arrays vs. Strings >> "abc" + "1234" >> 'abc' + '1234' >> 'abc' + '123'

17 ASCII Code

18 Character Arrays vs. Strings >> "abc" + "1234" >> 'abc' + '1234' >> 'abc' + '123' >> 'abc' - '123' >> "abc" - "123"

19 >> strcmp(a,c) Character / String Operations strcmp is case sensitivity >> strcmpi(a,c) strcmpi is not

20 >> a = 'abc' >> b = 'abc ' >> strcmp(a,b) Character / String Operations string comparison is not intelligent - we re used to search engines like google that take care of misspellings and extra spaces gracefully

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22 Data Structures

23 Data Structures one key to successful programming is using the right kind of data structure and using it the right way for the right task

24 Data Structures in Matlab Arrays, Vector, Matrices Structures Cell Arrays Tables Categorical Arrays Matlab does not so easily support more sophisticated data structures (queues, trees, hash tables, etc.), which is why some people use Python or C++

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26 Arrays

27 Why use a data structure? Imagine we have 10 subjects and each subject answers 10 true/false questions. 100 data points. We could create 100 variables, s1q1, s2q2, s2q1, s2q2, s10q9, s10q10. Why do we use a data structure instead?

28 Why use a data structure? Imagine we have 10 subjects and each subject answers 10 true/false questions. 100 data points. We could create 100 variables, s1q1, s2q2, s2q1, s2q2, s10q9, s10q10. Why do we use a data structure instead? - access data more easily and more efficiently - access data dynamically - all the data is in the same place

29 Arrays Arrays are the most common data structure used in just about every language. creating a one-dimensional array in Matlab >> a = [ ] referencing an element >> a(2) changing the value of an element >> a(2) = 10

30 Arrays remove an element from a one-dimensional array >> a(2) = [ ] adding an element to an array >> a(5) = 5 but note this >> a(10) = 10 regular arrays, by definition are contiguous there are advanced data structures called sparse matrices too:

31 Arrays >> a = [1 2 3] >> a(10) = 1 One nice thing about Matlab is that it lets you dynamically add new elements to an array. Other languages force you to say how big an array is at the outset (e.g., C++) and you have to resize it explicitly to add more elements. If you go past the boundaries of the array, the program crashes. but this can also make it harder to debug a program

32 Two-dimensional Arrays >> a = [1 2 3 ; 4 5 6] >> size(a) 2 rows 3 columns Now the meaning of ; changes. Within [ ] it creates a new row in an array.

33 Two-dimensional Arrays >> b = [1 2 ; 3 4 ; 5 6] How many columns and rows will this have?

34 Two-dimensional Arrays >> b = [1 2 ; 3 4 ; 5 6] How many columns and rows will this have? >> size(b) Note: It is entirely up to you to keep straight what the rows and columns MEAN. Rows could be subjects. Columns could be subjects. It s up to you.* * other data structures impose more "structure"

35 Two-dimensional Arrays >> a = [1 2 3 ; 4 5 6] What will happen here? >> a(4,1) Why?

36 Here? >> a(4.1, 1.2) Why? Two-dimensional Arrays

37 Here? >> a(4.1, 1.2) Why? Two-dimensional Arrays What if you referenced an array like this >> a(i,j) You need to make sure i and j are whole numbers.

38

39 >> x = 3; >> F(x) = x^2; A beautiful misconception

40 >> x = 3; >> F(x) = x^2; A beautiful misconception >> F(3) >> F(4) >> x = 3.1; >> F(x) = x^2;

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42 >> x = 1 >> size(x) We re already been using arrays 1 row 1 column there are no true "scalars" in Matlab (unlike other languages)

43 What does [ ] actually do? It s a concatenation operator. concatenate = link together in a chain [ ] = >> x = 1 >> y = [x x x]

44 What does [ ] actually do? What do you think this will do? >> x = 1 >> y = [x x x] >> z = [y y]

45 What does [ ] actually do? What do you think this will do? >> x = 1 >> y = [x x x] >> z = [y y] [ ] =

46 How about this? >> x = [2 ; 3] >> y = [4 ; 5] >> z = [6 ; 7] >> [x y z] What does [ ] actually do?

47 How about this? >> x = [2 ; 3] >> y = [4 ; 5] >> z = [6 ; 7] >> [x y z] What does [ ] actually do? [ ] =

48 This? >> x = [1 2 3] >> y = [4 5] >> z = [6 7 8] >> [x y z] What does [ ] actually do?

49 This? >> x = [1 2 3] >> y = [4 5] >> z = [6 7 8] >> [x y z] What does [ ] actually do? [ ] =

50 What does [ ] actually do? >> w = [1 1 ; 1 1] >> x = [2 2 ; 2 2] >> y = [3 3 ; 3 3] >> z = [4 4 ; 4 4] This? >> [w x y z]

51 What does [ ] actually do? >> w = [1 1 ; 1 1] >> x = [2 2 ; 2 2] >> y = [3 3 ; 3 3] >> z = [4 4 ; 4 4] This? >> [w ; x ; y ; z]

52 What does [ ] actually do? >> w = [1 1 ; 1 1] >> x = [2 2 ; 2 2] >> y = [3 3 ; 3 3] >> z = [4 4 ; 4 4] This? >> [w x ; y z]

53 What does [ ] actually do? >> w = [1 1 ; 1 1] >> x = [2 2 ; 2 2] >> y = [3 3 ; 3 3] >> z = [4 4 ; 4 4] This? >> [w x y ; z]

54 This? >> x = [2 ; 3] >> y = [4 5] >> [x y] What does [ ] actually do?

55 What does [ ] actually do? This? >> x = [2 3 ; 4 5] >> y = [6 ; 7] >> [x y]

56 >> a = 'apple' >> size(a) character arrays >> b = 'banana' >> size(b) What do you think this does? >> [a b]

57 >> a = 'apple' >> size(a) character arrays >> b = 'banana' >> size(b) These? >> [a ' ' b] >> a(4)

58 vs. strings >> a = "apple" >> size(a) >> b = "banana" >> size(b) These? >> [a b] >> a(4)

59

60 and versus and two differences (1) >> A B C calculates A, B, C to figure out logical expression >> A B C If A is false, it does not evaluate B and C short-circuit behavior

61 and versus and two differences (1) >> A B C calculates A, B, C to figure out logical expression >> A B C If A is true, it does not evaluate B and C short-circuit behavior

62 and versus and two differences (2) >> [1 0 1] [0 1 1] what does this result in? >> [1 0 1] [0 1 1] what does this result in?

63 and versus and two differences (2) >> [1 0 1] [0 1 1] what does this result in? >> [1 0 1] [0 1 1] what does this result in?

64

65 How are arrays stored in memory? Create a variable >> x = 2 O/S Matlab Data Memory

66 How are arrays stored in memory? Create a variable >> x = 2 O/S Matlab Data Memory

67 How are arrays stored in memory? Create a variable >> x = 2 free filled

68 How are arrays stored in memory? Create a variable >> x = 2 x length 1

69 How are arrays stored in memory? Create a variable >> x = [2 3 4] x length 1

70 How are arrays stored in memory? Create a variable >> x = [2 3 4] x length 3

71 How are arrays stored in memory? Create a variable >> x(4) = 5 x length 3

72 How are arrays stored in memory? Create a variable >> x(4) = 5 x length 4

73 Preallocating arrays Creating and moving large arrays every time a new index is added can take a lot of time. Preallocate >> x = zeros(1,100) or >> x = ones(1,100) or >> x = zeros(5,20)

74 How are two-dimension arrays stored? >> x = [1 2 3; 4 5 6]

75 How are two-dimension arrays stored? >> x = [1 2 3; 4 5 6] x total length 6

76 How are two-dimension arrays stored? >> x = [1 2 3; 4 5 6] how is a 2D array stored? x total length 6 Column Major Order

77 How are two-dimension arrays stored? >> x = [1 2 3; 4 5 6] What will these spit out? >> x(1,1) >> x(1,2) >> x(1,3) These? >> x(1) >> x(2) >> x(3)

78 How are two-dimension arrays stored? >> x = [1 2 3; 4 5 6] What will these spit out? >> x(1,1) >> x(1,2) >> x(1,3) These? >> x(1) >> x(2) >> x(3) can be a dangerous way to reference a multidimensional array if you are not very careful can be a difficult "bug" to detect as well

79 size vs. length of an array >> clear all >> x = [ ] >> size(x) >> length(x) >> x = [1 2 ; 3 4 ; 5 6] >> size(x) >> length(x)

80 Extracting part of an array >> a = [ ; ; ; ; ]; How do we extract row 2? >> [a(2,1) a(2,2) a(2,3) a(2,4) a(2,5)]

81 Extracting part of an array >> a = [ ; ; ; ; ]; How do we extract row 2? >> [a(2,1) a(2,2) a(2,3) a(2,4) a(2,5)] Here s an easier way: >> a(2,[ ])

82 Using the colon : operator >> 1:5 creates a sequence within an array

83 Using the colon : operator >> 1:5 creates a sequence within an array Try these. >> a >> a(2, 1:5) >> a(1:5, 2)

84 Using the colon : operator >> 1:5 creates a sequence within an array Try these. >> a >> a(2, 1:5) >> a(1:5, 2) What will this give you? >> a(1:2, 1:2)

85 Using the colon : operator : by itself within an array >> a(2, :) >> a(:, 2) >> a(:, :)

86 Using the colon : operator Create a 10x10 array containing the integers between 0 and 99.

87 Using the colon : operator Create a 10x10 array containing the integers between 0 and 99. >> a = [0:9 ; 10:19; 20:29; 30:39; 40:49; 50:59; 60:69; 70:79; 80:89; 90:99];

88 Using the colon : operator Create a 10x10 array containing the integers between 0 and 99. >> a = [0:9 ; 10:19; 20:29; 30:39; 40:49; 50:59; 60:69; 70:79; 80:89; 90:99]; >> a = 0:99

89 Using the colon : operator Create a 10x10 array containing the integers between 0 and 99. >> a = [0:9 ; 10:19; 20:29; 30:39; 40:49; 50:59; 60:69; 70:79; 80:89; 90:99]; >> a = 0:99 >> a = reshape([0:99],10,10);

90 transpose (') swapping rows and columns >> x = [ ] >> x >> x' >> y = [1 2 ; 3 4 ; 5 6] >> y' (Note: Technically, if you just want to transpose rows and columns and keep the numbers exactly the same, you should use.' instead of ', but it only makes a difference if you have complex numbers.)

91 Multidimensional Arrays By default, all arrays in Matlab are 2-dimensional. You can create arrays with more than 2 dimensions.

92 Multidimensional Arrays e.g., imagine a within-subjects design rows are subjects columns are levels along IV 1 depth are levels along IV 2 each entry (x,y,z) within the array is the score for subject x, along level y of IV 1 and level z of IV 2

93 Creating a multidimensional array No way to create one directly using the [ ] operator. Remember, [ ] simply concatenates. While it looks like [ ] creates a two-dimensional array from onedimensional elements (e.g., [2 3 ; 4 5]), recall that even a number like 2 is technically a twodimensional array (1 row, 1 column) already. So, while [ ] can concatenate three-dimensional arrays, it cannot create them from scratch.

94 Creating a multidimensional array Preallocating >> x = zeros(20,10,5) You need to remember what the dimensions mean. And this would be impossible to understand if you saw it in someone s Matlab script.

95 Creating a multidimensional array This is better style >> Nsubj = 8; >> NIV1 = 5; >> NIV2 = 3; >> data = zeros(nsubj, NIV1, NIV2);

96 The only number (2, 125, 25.1) you should EVER see in a computer program is one that is an intrinsic part of a calculation or a formula. e.g., Best Practices f = (1/(sig*sqrt(2*pi))) * exp(-((x-mu).^2)/(2*sig.^2)) for i=1:size(data,1) end

97 Best Practices You NEVER EVER want to see 64 in a loop if you have 64 trials, or 3 in a loop if you have 3 blocks, or 2.1 in a formula if that s some scaling factor you picked to adjust image contrast. Make those variables with meaningful names and comment them. Even if this is a "one-off" analysis program, you might need to go back to it in 6 months or more after a couple rounds of reviews through a journal.

98 Best Practices In all homework assignments, I will expect that you use good programming style.

99 Referencing a multidimensional array Back to our data example

100 Referencing a multidimensional array How would you pull out the data for subject 3?

101 Referencing a multidimensional array How would you pull out the data for subject 3? >> x = data(3,:,:) >> size(x)

102 Referencing a multidimensional array How would you pull out the data for subject 3? >> x = data(3,:,:) >> size(x) You might want to reduce this to two dimensions. >> x = squeeze(data(3,:,:))

103 Referencing a multidimensional array Matlab is mind-numbingly arbitrary sometimes. Try this now >> x = data(:,:,1) >> size(x)

104 Removing data? Can you remove subject three?

105 Removing data? Can you remove subject three? >> data(3,:,:) = [ ]

106 Removing data? Can you remove subject three? >> data(3,:,:) = [ ] Can you remove level 2 of IV 1?

107 Removing data? Can you remove subject three? >> data(3,:,:) = [ ] Can you remove level 2 of IV 1? >> data(:,2,:) = [ ]

108 Removing data? Can you remove subject three? >> data(3,:,:) = [ ] Can you remove level 2 of IV 1? >> data(:,2,:) = [ ] Now what if you wanted to remove subject four?

109 Removing data? Can you remove subject three? >> data(3,:,:) = [ ] Can you remove level 2 of IV 1? >> data(:,2,:) = [ ] Now what if you wanted to remove subject four? Is this right? >> data(4,:,:) = []

110 Removing data? Can you remove subject three? >> data(3,:,:) = [ ] Can you remove level 2 of IV 1? >> data(:,2,:) = [ ] Now what if you wanted to remove subject four? Is this right? >> data(4,:,:) = [] Which row has subject 4 s data now?

111 Removing data? Can you remove subject three? >> data(3,:,:) = [ ] Can you remove level 2 of IV 1? >> data(:,2,:) = [ ] Now what if you wanted to remove subject four? Is this right? >> data(4,:,:) = [] Which row has subject 4 s data now? >> data(3,:,:) = []

112 Removing data? Suppose you think that the data for subject 1 is suspect, but only for level 1 on IV1 and level 3 on IV2. Can you remove that one data point?

113 Removing data? Suppose you think that the data for subject 1 is suspect, but only for level 1 on IV1 and level 3 on IV2. Can you remove that one data point? >> data(1,1,3) = [ ] Will this work?

114 Removing data? Suppose you think that the data for subject 1 is suspect, but only for level 1 on IV1 and level 3 on IV2. Can you remove that one data point? >> data(1,1,3) = [ ] Will this work? No. But people sometimes do this. >> data(1,1,3) = NaN

115

116 Best Practices A common (and nasty) problem people run into is that they don t realize they are reusing a variable, especially an array that s already been set up. What s especially troubling is that they try to restart Matlab and the problem goes away, so they think it was just a Matlab hiccup. Why does it go away when you restart Matlab?

117 Best Practices A common (and nasty) problem people run into is that they don t realize they are reusing a variable, especially an array that s already been set up. What s especially troubling is that they try to restart Matlab and the problem goes away, so they think it was just a Matlab hiccup. Why does it go away when you restart Matlab? The Workspace is clear.

118 Another example How would we create a multidimensional array to hold the data for a mixed design where there could be unequal numbers of subjects in the betweensubjects levels? One B/S variable with 3 levels 10 subjects in level 1 12 subjects in level 2 8 subjects in level 3 One W/S variable with 2 levels One W/S variable with 5 levels

119 >> NBet = 3 >> NWith1 = 2 >> NWith2 = 5 >> NMaxSubj = 12 Another example >> Nsubj = zeros(1,nbet) >> Nsubj(1) = 10 >> Nsubj(2) = 12 >> Nsubj(3) = 8 >> data = different from how you might code data in Excel or SPSS zeros(nbet,nmaxsubj,nwith1,nwith2)

120

121 Simple I/O in Matlab For Homework Assignment (read these notes, the book, and online help) save, load, input, fprintf

122 Simple I/O in Matlab We ll talk about more complex I/O later Files or User Input

123 Simple I/O in Matlab Save data in Matlab (.mat) format: >> save filename.mat >> save('filename.mat') saves all variables in the current workspace in file named filename.mat >> save filename.mat X a b >> save('filename.mat', 'X', 'a', 'b') saves variables X, a, and b in filename.mat

124 Simple I/O in Matlab Load data in Matlab (.mat) format: >> load filename.mat >> load('filename.mat') loads everything within filename.mat into workspace You can also load specific variables >> load filename.mat X >> load('filename.mat', 'X') Only loads variable X

125 Simple I/O in Matlab User input >> age = input('enter age : '); >> name = input('enter name : ', 's');

126 Simple I/O in Matlab Screen output >> X We ve already used this. Often we want it formatted. fprintf() command (borrowed from C)

127 fprintf() >> ans = 42 >> fprintf(['answer to the Ultimate Question' 'of Life, the Universe, and Everything' 'is %d\n'], ans) First, what am I doing with the [ ] and why?

128 fprintf() >> ans = 42 >> fprintf(['answer to the Ultimate Question' 'of Life, the Universe, and Everything' 'is %d\n'], ans) %d inserts an integer at that location in the string %f inserts a floating point number %c inserts a single character %s inserts a string

129 fprintf() >> ans = 42 >> fprintf(['answer to the Ultimate Question' 'of Life, the Universe, and Everything' 'is %d\n'], ans) \n new line \t tab put a quote in the string \\ put a \ in the string

130 fprintf() What will this print? >> ans = 10 >> str = 'hello' >> fprintf('say \'%s\' and give me $%d\n', str, ans)

131 What will this print? >> fprintf('pi = \n', pi) fprintf()

132 fprintf() What will this print? >> fprintf('pi = %f\n', pi)

133 fprintf() What will this print? >> fprintf('pi = %9.9f\n', pi) >> fprintf('pi = %12.9f\n', pi) >> fprintf('pi = %15.9f\n', pi) >> fprintf('pi = %15.1f\n', pi)

134 fprintf() What will this print? >> fprintf('pi = %9.9f\n', pi) >> fprintf('pi = %12.9f\n', pi) >> fprintf('pi = %15.9f\n', pi) >> fprintf('pi = %15.1f\n', pi) >> ans = 1000*pi >> fprintf('pi = %4.4f\n', pi)

135 fprintf() What will this print? >> ans = >> fprintf('ans = %d\n', ans) >> fprintf('ans = %10d\n', ans)

136 fprintf() What will this print? >> ans = >> fprintf('ans = %d\n', ans) >> fprintf('ans = %10d\n', ans) Please use fprintf() statements where appropriate in your homework assignments

137

138 Mathematical Operations on Arrays Basic Linear Algebra

139 Mathematical Operations on Arrays >> clear all >> data = [1 2 ; 3 4] add a number to a particular element in an array >> data(1,2) = data(1,2) + 10

140 Mathematical Operations on Arrays >> clear all >> data = [1 2 ; 3 4] add or subtract a number to/from EVERY element >> data = data + 10 >> data = data 10

141 Mathematical Operations on Arrays >> clear all >> data = [1 2 ; 3 4] multiple or divide by the same number for every element in the array >> data = data * 10 >> data = data / 10

142 Mathematical Operations on Arrays >> clear all >> data = [1 2 ; 3 4] >> x = repmat(10, 2, 2) >> x = repmat(10, size(data)) what does repmat do?

143 Mathematical Operations on Arrays >> clear all >> data = [1 2 ; 3 4] >> x = repmat(10, 2, 2) >> x = repmat(10, size(data)) >> data + 10 >> data + x

144 Mathematical Operations on Arrays >> clear all >> data = [1 2 ; 3 4] When you add or subtract two arrays, the corresponding elements are added or subtracted. >> data2 = [8 7 ; 6 5] >> data + data2 >> data data2

145 Mathematical Operations on Arrays >> clear all >> data = [1 2 ; 3 4] What about these? >> x = repmat(10, 2, 2) >> data * 10 >> data * x >> data / 10 >> data / x

146 Arrays vs. Matrices Arrays are data structures with rows and columns used to organize and use data. Matrices are mathematical entities used in linear algebra. Unfortunately, in Matlab, arrays and matrices are defined in exactly the same way. While they are distinct computationally and mathematically, Matlab treats them as the very same type. Watch out.

147 Mathematical Operations on Arrays >> clear all >> data = [1 2 ; 3 4] What about these? >> x = repmat(10, 2, 2) >> data * 10 >> data.* x >> data / 10 >> data./ x Element-by-element

148 Array vs. Matrix multiplication and division Best Practices Get in the habit, within Matlab, of always using.* or./ when you multiply or divide, even if you do not need it (unless you really intend matrix operations)

149 Some more operations on arrays >> clear all >> data = [1 2 ; 3 4] What does this do? >> data.^ 2 Versus this? >> data ^ 2 Best Practices Also always use.^

150 Some more operations on arrays >> log10(data) >> exp(data) >> sin(data)

151 Some more operations on arrays >> mean(data) What s it doing? How could you find out? >> help mean >> doc mean

152 Some more operations on arrays >> mean(data) What s it doing? How could you find out? >> help mean >> doc mean >> mean(data,1) >> mean(data,2)

153 Evaluating Mathematical Formulas in Matlab One thing that Matlab does really well is calculate (and plot) lots of values for some function

154 Evaluating Mathematical Formulas in Matlab One thing that Matlab does really well is calculate (and plot) lots of values for some function f (x) = What is this? 1 2πσ $ (x µ) 2 exp 2 % 2σ 2 ' ) (

155 Evaluating Mathematical Formulas in Matlab One thing that Matlab does really well is calculate (and plot) lots of values for some function f (x) = What is this? 1 2πσ $ (x µ) 2 exp 2 % 2σ 2 ' ) ( probability density function for a normal distribution

156 Evaluating Mathematical Formulas in Matlab How would we code this in Matlab? f (x) = 1 2πσ $ (x µ) 2 exp 2 % 2σ 2 ' ) (

157 Evaluating Mathematical Formulas in Matlab How would we code this in Matlab? sig2 = 1 mu = 0 x = 1 f (x) = 1 2πσ $ (x µ) 2 exp 2 % 2σ 2 % variance % mean % a particular value f = (1/sqrt(2*pi*sig2)) * exp(-((x-mu)^2)/(2*sig2)) ' ) (

158 Evaluating Mathematical Formulas in Matlab How would we code this in Matlab? sig2 = 1 mu = 0 x = 1 f (x) = 1 2πσ $ (x µ) 2 exp 2 % 2σ 2 % variance % mean % a particular value f = (1/sqrt(2*pi*sig2)) * exp(-((x-mu)^2)/(2*sig2)) This calculates only one value at a time. ' ) (

159 Evaluating Mathematical Formulas in Matlab How would we code this in Matlab? sig2 = 1 mu = 0 x = -4:.01:4 % variance % mean % a particular value f = (1/sqrt(2*pi*sig2)) * exp(-((x-mu)^2)/(2*sig2)) plot(x,f) f (x) = 1 2πσ $ (x µ) 2 exp 2 % 2σ 2 What is this doing? ' ) (

160 Evaluating Mathematical Formulas in Matlab How would we code this in Matlab? sig2 = 1 mu = 0 x = -4:.01:4 % variance % mean % a particular value f = (1/sqrt(2*pi*sig2)) * exp(-((x-mu)^2)/(2*sig2)) plot(x,f) f (x) = 1 2πσ $ (x µ) 2 exp 2 % 2σ 2 What's wrong with this? ' ) (

161 Evaluating Mathematical Formulas in Matlab How would we code this in Matlab? sig2 = 1 mu = 0 x = -4:.01:4 % variance % mean % a particular value f = (1/sqrt(2*pi*sig2)).* exp(-((x-mu).^2)/(2*sig2)) plot(x,f) f (x) = 1 2πσ $ (x µ) 2 exp 2 % 2σ 2 ' ) (

162

163 Vectors, Matrices, and Linear Algebra

164 Vectors, Matrices, and Linear Algebra Arrays are merely containers that hold numeric data in an organized way. Vectors (1-dimensional) and Matrix (2-dimensional) are mathematical entities with mathematical operators that act on them. In Matlab, a 1-dimensional vector and 1-dimensional array, and a 2-dimensional matrix and a 2- dimensional array are defined exactly the same way, but they need to be thought of differently.

165 Vectors

166 Vectors Vectors have a magnitude and direction. Terminology can be a bit confusing in that a vector is one-dimensional in one sense >> a = [1 2] >> b = [3 2 1] >> c = [ ]

167 Vectors Vectors have a magnitude and direction. But we ll also illustrate plotting a vector in a multidimensional space >> a = [1 2] % vector 2D space >> b = [3 2 1] % vector 3D space >> c = [ ] % vector 4D space

168 Vectors >> a = [1 2] % vector 2D space >> b = [3 2] >> a + b y x

169 Vectors >> a + b y x

170 Vectors >> a + b y x

171 Vectors >> a = [1 2] >> b = 2*a y x

172 Vectors >> a = [1 2] >> b = 2*a y x

173 Vectors >> a = [1 2] >> b = 2*a y x

174 Vectors >> norm(b) % vector length (norm) Euclidian norm is found using the Pythagorean Theorem (also called 2-norm or an L 2 norm) y norm(b) x

175 Vectors angle between two vectors >> a = [1 3] >> b = [2 1] y θ x

176 Vectors angle between two vectors Implement in Matlab? cos(θ) = a b a b dot product norm y θ x

177 Vectors >> theta = radtodeg(acos( dot(a,b) / (norm(a)*norm(b)))) a b = n i=1 a i b i y cos(θ) = a b a b θ x

178

179 Matrices

180 Solving systems of linear algebra x + 2y = 2 x + y = 3 How would you view these equations in Matlab?

181 Solving systems of linear algebra How would you view these equations in Matlab? x1 = -5:5; x2 = -5:5; y1 = -0.5*x1 + 1 y2 = -x2 + 3 plot(x1,y1,x2,y2) x + 2y = 2 x + y = 3 What s the solution to the system of equations?

182 Solving systems of linear algebra x + 2y = 2 x + y = 3 Rewrite in matrix notation:! " $! % " x y $! = 2 % " 3 $ %

183 Multiplying Matrices! " a 11 a 12! a 1n a 21 a 22! a 2n " " " a m1 a m2! a mn $ %! " b 11 b 12! b 1p b 21 b 22! b 2p " " " b n1 b n2! b np $ % =! " c 11 c 12! c 1p c 21 c 22! c 2p " " " c m1 c m2! c mp mxn matrix nxp matrix mxp matrix $ %

184 Multiplying Matrices! " a 11 a 12! a 1n a 21 a 22! a 2n " " " a m1 a m2! a mn $ %! " b 11 b 12! b 1p b 21 b 22! b 2p " " " b n1 b n2! b np $ % =! " c 11 c 12! c 1p c 21 c 22! c 2p " " " c m1 c m2! c mp mxn matrix nxp matrix mxp matrix $ % n k =1 c i j = a i k b kj

185 Multiplying Matrices! " a 11 a 12! a 1n a 21 a 22! a 2n " " " a m1 a m2! a mn $ %! " b 11 b 12! b 1p b 21 b 22! b 2p " " " b n1 b n2! b np $ % =! " c 11 c 12! c 1p c 21 c 22! c 2p " " " c m1 c m2! c mp mxn matrix nxp matrix mxp matrix $ % n k =1 c i j = a i k b kj

186 Multiplying Matrices! " a 11 a 12! a 1n a 21 a 22! a 2n " " " a m1 a m2! a mn $ %! " b 11 b 12! b 1p b 21 b 22! b 2p " " " b n1 b n2! b np $ % =! " c 11 c 12! c 1p c 21 c 22! c 2p " " " c m1 c m2! c mp mxn matrix nxp matrix mxp matrix $ % n k =1 c i j = a i k b kj

187 Multiplying Matrices! " a 11 a 12! a 1n a 21 a 22! a 2n " " " a m1 a m2! a mn $ %! " b 11 b 12! b 1p b 21 b 22! b 2p " " " b n1 b n2! b np $ % =! " c 11 c 12! c 1p c 21 c 22! c 2p " " " c m1 c m2! c mp mxn matrix nxp matrix mxp matrix $ % n k =1 c i j = a i k b kj

188 Multiplying Matrices! " a 11 a 12! a 1n a 21 a 22! a 2n " " " a m1 a m2! a mn $ %! " b 11 b 12! b 1p b 21 b 22! b 2p " " " b n1 b n2! b np $ % =! " c 11 c 12! c 1p c 21 c 22! c 2p " " " c m1 c m2! c mp mxn matrix nxp matrix mxp matrix $ % n k =1 c i j = a i k b kj

189 Multiplying Matrices! " a 11 a 12! a 1n a 21 a 22! a 2n " " " a m1 a m2! a mn $ %! " b 11 b 12! b 1p b 21 b 22! b 2p " " " b n1 b n2! b np $ % =! " c 11 c 12! c 1p c 21 c 22! c 2p " " " c m1 c m2! c mp mxn matrix nxp matrix mxp matrix $ % n k =1 c i j = a i k b kj

190 Multiplying Matrices! " a 11 a 12! a 1n a 21 a 22! a 2n " " " a m1 a m2! a mn $ %! " b 11 b 12! b 1p b 21 b 22! b 2p " " " b n1 b n2! b np $ % =! " c 11 c 12! c 1p c 21 c 22! c 2p " " " c m1 c m2! c mp mxn matrix nxp matrix mxp matrix $ % n k =1 c i j = a i k b kj

191 Multiplying Matrices! " a 11 a 12! a 1n a 21 a 22! a 2n " " " a m1 a m2! a mn $ %! " b 11 b 12! b 1p b 21 b 22! b 2p " " " b n1 b n2! b np $ % =! " c 11 c 12! c 1p c 21 c 22! c 2p " " " c m1 c m2! c mp mxn matrix nxp matrix mxp matrix $ % n k =1 c i j = a i k b kj

192 Multiplying Matrices! " a 11 a 12! a 1n a 21 a 22! a 2n " " " a m1 a m2! a mn $ %! " b 11 b 12! b 1p b 21 b 22! b 2p " " " b n1 b n2! b np $ % =! " c 11 c 12! c 1p c 21 c 22! c 2p " " " c m1 c m2! c mp mxn matrix nxp matrix mxp matrix $ % n k =1 c i j = a i k b kj

193 Multiplying Matrices! " a 11 a 12! a 1n a 21 a 22! a 2n " " " a m1 a m2! a mn $ %! " b 11 b 12! b 1p b 21 b 22! b 2p " " " b n1 b n2! b np $ % =! " c 11 c 12! c 1p c 21 c 22! c 2p " " " c m1 c m2! c mp mxn matrix nxp matrix mxp matrix $ % n k =1 c i j = a i k b kj

194 Multiplying Matrices! " a 11 a 12! a 1n a 21 a 22! a 2n " " " a m1 a m2! a mn $ %! " b 11 b 12! b 1p b 21 b 22! b 2p " " " b n1 b n2! b np $ % =! " c 11 c 12! c 1p c 21 c 22! c 2p " " " c m1 c m2! c mp mxn matrix nxp matrix mxp matrix $ % n k =1 c i j = a i k b kj

195 Multiplying Matrices! " a 11 a 12! a 1n a 21 a 22! a 2n " " " a m1 a m2! a mn $ %! " b 11 b 12! b 1p b 21 b 22! b 2p " " " b n1 b n2! b np $ % =! " c 11 c 12! c 1p c 21 c 22! c 2p " " " c m1 c m2! c mp mxn matrix nxp matrix mxp matrix $ % n k =1 c i j = a i k b kj

196 Solving systems of linear algebra x + 2y = 2 x + y = 3 Rewrite in matrix notation:! " $! % " x y $! = 2 % " 3 $ % 2x2 2x1

197 Solving systems of linear algebra x + 2y = 2 x + y = 3 Rewrite in matrix notation:! " $! % " x y $! = 2 % " 3 $ %! " x + 2 y x + y $! = 2 % " 3 $ %

198 Solving systems of linear algebra x + 2x = x + x = Rewrite :! " $! % " x 1 x 2 $! = 2 % " 3 $ % A x = b matrix vector vector

199 Mathematical Operations on Matrices A + B = B + A c(a+b) = ca + cb Convince yourself in Matlab >> A = [2 1 ; 2 3] >> B = [1 2 ; 0 1] >> c = 2 >> (A + B) (B + A) >> c.*(a+b) (c.*a + c.*b)

200 Mathematical Operations on Matrices How about this? Try it in Matlab. AB =? BA