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Notes of Andrew Ng’s Machine Learning —— (5) Octave Tutorial

GNU Octave 是一种采用高级编程语言的主要用于数值分析的软件。Octave有助于以数值方式解决线性和非线性问题，并使用与MATLAB兼容的语言进行其他数值实验。它也可以作为面向批处理的语言使用。因为它是GNU计划的一部分，所以它是GNU通用公共许可证条款下的自由软件。

Octave 是 MATLAB 的主要自由替代品之一。

—— Wikipedia

Octave 官网：www.gnu.org/software/oc…

官网给出的定义：Scientific Programming Language

Powerful mathematics-oriented syntax with built-in 2D/3D plotting and visualization tools
Free software, runs on GNU/Linux, macOS, BSD, and Microsoft Windows
Drop-in compatible with many Matlab scripts

Basic Operations

Elementary Operations

+, -, *, /, ^.

>> 5 + 6
ans =  11
>> 20 - 1
ans =  19
>> 3 * 4
ans =  12
>> 8 / 2
ans =  4
>> 2 ^ 8
ans =  256
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Logical Operations

==, ~=, &&, ||, xor().

Note that a not equal sign is ~=, and not !=.

>> 1 == 0
ans = 0
>> 1 ~= 0
ans = 1
>> 1 && 0
ans = 0
>> 1 || 0
ans = 1
>> xor(1, 0)
ans = 1
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Change the Prompt

We can change the prompt via PS1():

>> PS1("octave: > ")
octave: > PS1(">> ")
>> PS1("octave: > ")
octave: > PS1("SOMETHING > ")
SOMETHING > PS1(">> ")
>> % Prompt changed
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Variables

>> a = 3
a =  3
>> a = 3;    % semicolon supressing output
>> c = (3 >= 1);
>> c
c = 1
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Display variables

>> a = pi;
>> a
a =  3.1416
>> disp(a)
 3.1416
>> disp(sprintf('2 decimals: %0.2f', a))
2 decimals: 3.14
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We can also set the default length of decimal places by entering format short/long:

>> a
a =  3.1416
>> format long
>> a
a =  3.141592653589793
>> format short
>> a
a =  3.1416
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Create Matrices

>> A = [1, 2, 3; 4, 5, 6]
A =

   1   2   3
   4   5   6

>> B = [1 3 5; 7 9 11]
B =

    1    3    5
    7    9   11

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

   1   2   3
   4   5   6
   7   8   9

>> C = [1, 2, 4, 8]
C =

   1   2   4   8

>> D = [1; 2; 3; 4]
D =

   1
   2
   3
   4

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There are some useful methods to generate matrices:

Generate vector of a range

>> v = 1:10    % start:end
v =

    1    2    3    4    5    6    7    8    9   10

>> v = 1:0.1:2    % start:step:end
v =

 Columns 1 through 8:

    1.0000    1.1000    1.2000    1.3000    1.4000    1.5000    1.6000    1.7000

 Columns 9 through 11:

    1.8000    1.9000 
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Generate matrices of all ones/zeros

>> ones(2, 3)
ans =

   1   1   1
   1   1   1

>> zeros(3, 2)
ans =

   0   0
   0   0
   0   0

>> C = 2 * ones(4, 5)
C =

   2   2   2   2   2
   2   2   2   2   2
   2   2   2   2   2
   2   2   2   2   2

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Generate identity matrices

>> eye(3)
ans =

Diagonal Matrix

   1   0   0
   0   1   0
   0   0   1

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Generate matrices of random values

Uniform distribution between 0 and 1:

>> D = rand(1, 3)
D =

   0.14117   0.81424   0.83745

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Gaussian random:

>> D = randn(1, 3)
D =

   0.22133  -2.00002   1.61025


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We can generate a gaussian random vector with 10000 elements, and plot a histogram:

>> randn(1, 10000);
>> hist(w)
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Output figure:

We can also plot a histogram with more buckets, 50 bins for example:

>> hist(w, 50)
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Get Help

>> help

  For help with individual commands and functions type

    help NAME
    
......

>> help eye
'eye' is a built-in function from the file libinterp/corefcn/data.cc
 
......

>> help help

......

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Moving Data Around

Size of matrix

size(): get the size of a matrix, return [rows, columns].

>> A = [1, 2; 3, 4; 5, 6]
A =

   1   2
   3   4
   5   6

>> size(A)    % get the size of A
ans =

   3   2

>> sz = size(A);    % actually, size return a 1x2 matrix
>> size(sz)
ans =

   1   2

>> size(A, 1)    % get the first dimension of A (i.e. the number of rows)
ans =  3
>> size(A, 2)    % the number of columns
ans =  2
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length(): return the size of the longest dimension.

>> length(A)    % get the size of the longest dimension. Confusing, not recommend
ans =  3
>> v = [1, 2, 3, 4];
>> length(v)    % We often length() to get the length of a vector
ans =  4
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Load data

We can use basic shell commands to find data that we want.

>> pwd
ans = /Users/c
>> cd MyProg/octave/
>> pwd
ans = /Users/c/MyProg/octave
>> ls
featureX.dat featureY.dat
>> ls -l
total 16
-rw-r--r--  1 c  staff  188 Sep  8 10:00 featureX.dat
-rw-r--r--  1 c  staff  135 Sep  8 10:00 featureY.dat
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load command can load data from a file.

>> load featureX.dat
>> load('featureY.dat')
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The data from file is now comed into matrices after load

>> featureX
featureX =

   2104      3
   1600      3
   2400      3
   1416      2
......

>> size(featureX)
ans =

   27    2

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Show variables

who/whos: show variables in memory currently.

>> who
Variables in the current scope:

A         ans       featureX  featureY  sz        v         w

>> whos    % for more details
Variables in the current scope:

   Attr Name          Size                     Bytes  Class
   ==== ====          ====                     =====  ===== 
        A             3x2                         48  double
        ans           1x2                         16  double
        featureX     27x2                        432  double
        featureY     27x1                        216  double
        sz            1x2                         16  double
        v             1x4                         32  double
        w             1x10000                  80000  double

Total is 10095 elements using 80760 bytes

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Clear variables

clear command can help us to clear variables that are no longer useful.

>> who
Variables in the current scope:

A         ans       featureX  featureY  sz        v         w

>> clear A    % clear a variable
>> clear sz v w    % clear variables
>> whos
Variables in the current scope:

   Attr Name          Size                     Bytes  Class
   ==== ====          ====                     =====  ===== 
        ans           1x2                         16  double
        featureX     27x2                        432  double
        featureY     27x1                        216  double

Total is 83 elements using 664 bytes
>> clear    % clear all variables
>> whos
>> 
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Save data

Take a part of a vector.

>> v = featureY(1:5)
v =

   3999
   3299
   3690
   2320
   5399
   
>> whos
Variables in the current scope:

   Attr Name          Size                     Bytes  Class
   ==== ====          ====                     =====  ===== 
        featureX     27x2                        432  double
        featureY     27x1                        216  double
        v             5x1                         40  double

Total is 86 elements using 688 bytes

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Save data to disk: save file_name variable [-ascii]

>> save hello.mat v    % save as a binary format
>> ls
featureX.dat featureY.dat hello.mat
>> save hello.txt v -ascii;    % save as a ascii txt
>> 
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Then we can clear it from memory and load v back from disk:

>> clear v
>> whos
Variables in the current scope:

   Attr Name          Size                     Bytes  Class
   ==== ====          ====                     =====  ===== 
        featureX     27x2                        432  double
        featureY     27x1                        216  double

Total is 81 elements using 648 bytes

>> load hello.mat 
>> whos
Variables in the current scope:

   Attr Name          Size                     Bytes  Class
   ==== ====          ====                     =====  ===== 
        featureX     27x2                        432  double
        featureY     27x1                        216  double
        v             5x1                         40  double

Total is 86 elements using 688 bytes

>> 
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Manipulate data

Get element from a matrix:

>> A = [1, 2; 3, 4; 5, 6]
A =

   1   2
   3   4
   5   6

>> A(3, 2)    % get a element of matrix
ans =  6
>> A(2, :)    % ":" means every element along that row/column
ans =

   3   4

>> A(:, 1)
ans =

   1
   3
   5

>> A([1, 3], :)    % get the elements along row 1 & 3
ans =

   1   2
   5   6

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Change the elements of a matrix:

>> A = [1, 2; 3, 4; 5, 6]
A =

   1   2
   3   4
   5   6

>> A(:, 2) = [10, 11, 12]
A =

    1   10
    3   11
    5   12

>> A(1, 1) = 0
A =

    0   10
    3   11
    5   12

>> A = [A, [100; 101; 102]]    % append another column vector to right
A =

     0    10   100
     3    11   101
     5    12   102

>> A = [1, 2; 3, 4; 5, 6]
A =

   1   2
   3   4
   5   6

>> B = A + 10
B =

   11   12
   13   14
   15   16

>> C = [A, B]
C =

    1    2   11   12
    3    4   13   14
    5    6   15   16

>> D = [A; B];
>> size(D)
ans =

   6   2

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Put all elements of a matrix into a single column vector:

>> A
A =

     0    10   100
     3    11   101
     5    12   102

>> A(:)    % put all elements of A into a single vector
ans =

     0
     3
     5
    10
    11
    12
   100
   101
   102
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Computing on Data

Element-wise operations

Use .<operator> instead of <operator> for element-wise operations (i.e. operations between elements).

>> A = [1, 2; 3, 4; 5, 6];
>> B = [11, 12; 13, 14; 15, 16];
>> C = [1 1; 2 2];
>> v = [1, 2, 3];
>> A .* B    % element-wise multiplication (ans = [A(1,1)*B(1,1), A(1,2)*B(1,2); ...])
ans =

   11   24
   39   56
   75   96

>> A .^ 2    % squaring each element of A
ans =

    1    4
    9   16
   25   36

>> 1 ./ A
ans =

   1.00000   0.50000
   0.33333   0.25000
   0.20000   0.16667

>> v .+ 1    % equals to `v + 1` & `v + ones(1, length(v))`
ans =

   2   3   4

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Element-wise comparison:

>> a
a =

    1.00000   15.00000    2.00000    0.50000

>> a < 3
ans =

  1  0  1  1
  
>> find(a < 3)    % to find the elements that are less then 3 in a, return their indices
ans =

   1   3   4
   
>> A
A =

   1   2
   3   4
   5   6

>> [r, c] = find(A < 3)
r =

   1
   1

c =

   1
   2

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Functions are element-wise:

>> v = [1, 2, 3]
v =

   1   2   3

>> log(v)
ans =

   0.00000   0.69315   1.09861

>> exp(v)
ans =

    2.7183    7.3891   20.0855

>> abs([-1, 2, -3, 4])
ans =

   1   2   3   4

>> -v    % -1 * v
ans =

  -1  -2  -3

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Floor and Ceil of elements:

>> a
a =

    1.00000   15.00000    2.00000    0.50000

>> floor(a)
ans =

    1   15    2    0

>> ceil(a)
ans =

    1   15    2    1

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Matrix operations

Matrix multiplication:

>> A = [1, 2; 3, 4; 5, 6];
>> C = [1 1; 2 2];
>> A * C    % matrix multiplication
ans =

    5    5
   11   11
   17   17
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Transpose:

>> A = [1, 2; 3, 4; 5, 6];
>> A'    % transposed
ans =

   1   3   5
   2   4   6

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Get the max element of a vector | matrix:

>> a = [1 15 2 0.5];
>> A = [1, 2; 3, 4; 5, 6];
>> max_val = max(a)
max_val =  15
>> [val, index] = max(a)
val =  15
index =  2
>> max(A)    % `max(<Matrix>)` does a column-wise maximum
ans =

   5   6
>> max(A, [], 1)    % max per column
ans =

   5   6

>> max(A, [], 2)    % max per row
ans =

   2
   4
   6

>> max(max(A))    % the max element of whole matrix
ans =  6
>> max(A(:))
ans =  6
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Sum & prod of vector:

>> a
a =

    1.00000   15.00000    2.00000    0.50000
    
>> A
A =

   1   2
   3   4
   5   6

>> sum(a)
ans =  18.500
>> sum(A)
ans =

    9   12
    
>> sum(A, 1)
ans =

    9   12

>> sum(A, 2)
ans =

    3
    7
   11

>> prod(a)
ans =  15
>> prod(A)
ans =

   15   48

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Get the diagonal elements:

>> A = magic(4)
A =

   16    2    3   13
    5   11   10    8
    9    7    6   12
    4   14   15    1

>> A .* eye(4)
ans =

   16    0    0    0
    0   11    0    0
    0    0    6    0
    0    0    0    1

>> sum(A .* eye(4))
ans =

   16   11    6    1

>> flipud(eye(4))    % flip up down
ans =

Permutation Matrix

   0   0   0   1
   0   0   1   0
   0   1   0   0
   1   0   0   0

>> sum(A .* flipud(eye(4)))
ans =

    4    7   10   13

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Inverse:

>> A = magic(3)
A =

   8   1   6
   3   5   7
   4   9   2

>> pinv(A)
ans =

   0.147222  -0.144444   0.063889
  -0.061111   0.022222   0.105556
  -0.019444   0.188889  -0.102778

>> pinv(A) * A    % get identity matrix
ans =

   1.0000e+00   2.0817e-16  -3.1641e-15
  -6.1062e-15   1.0000e+00   6.2450e-15
   3.0531e-15   4.1633e-17   1.0000e+00

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Plotting Data

Plotting a function

>> clear
>> t = [0:0.01:0.98];
>> size(t)
ans =

    1   99

>> y1 = sin(2*pi*4*t);
>> plot(t, y1);
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It will show you a figure like this:

>> y2 = cos(2*pi*4*t);
>> plot(t, y2);
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? This will replace the sin figure with a new cos figure.

If we want to have both the sin and cos plots, the hold on command will help:

>> plot(t, y1);
>> hold on;
>> plot(t, y2, 'r');
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We can set some text on thw figure:

>> xlabel("time");
>> ylabel("value");
>> legend('sin', 'cos');    % Show what the 2 lines are
>> title('my plot');
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Now, we get this:

myPlot

Then, we save it and close the plotting window:

>> print -dpng 'myPlot.png'    % save it to $(pwd)
>> close
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We can show two figures at the same time:

>> figure(1); plot(t, y1);
>> figure(2); plot(t, y2);
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Then, we can also generate figures like this:

What we need to do is using a subplot:

>> subplot(1, 2, 1);    % Divides plot a 1x2 grid, access first element
>> plot(t, y1);
>> subplot(1, 2, 2);
>> plot(t, y2);
>> axis([0.5, 1, -1, 1])    % change the range of axis
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Use clf to clear a figure:

>> clf;
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Showing a matrix

>> A = magic(5)
A =

   17   24    1    8   15
   23    5    7   14   16
    4    6   13   20   22
   10   12   19   21    3
   11   18   25    2    9

>> imagesc(A), colorbar
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It gives us a figure like this:

The different colors correspond to the different values.

Another example:

>> B = magic(10);
>> imagesc(B), colorbar, colormap gray;
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Output:

Contriol Statements

`for`

>> v = zeros(10, 1)
v =

   0
   0
   0
   0
   0
   0
   0
   0
   0
   0

>> for i = 1: 10,
>     v(i) = 2^i;
> end;
>> v'
ans =

      2      4      8     16     32     64    128    256    512   1024

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`while`

>> i = 1;
>> while i <= 5,
>     v(i) = 100;
>     i = i + 1;
> end;
>> v'
ans =

    100    100    100    100    100     64    128    256    512   1024

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`if`

>> for i = 1: 10,
>     if v(i) > 100,
>         disp(v(i));
>     end;
> end;
 128
 256
 512
 1024
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Or, we can program like this,

x = 1;
if (x == 1)
    disp ("one");
elseif (x == 2)
    disp ("two");
else
    disp ("not one or two");
endif
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`break` & `continue`

i = 1;
while true,
    v(i) = 999;
    i = i + 1;
    if i == 6,
        break;
    end;
end;
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Output:

Function

Create a Function

To create a function, type the function code in a text editor (e.g. gedit or notepad), and save the file as functionName.m

Example function:

function y = squareThisNumber(x)

y = x^2;

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To call this function in Octave, do either:

cd to the directory of the functionName.m file and call the function:

% Navigate to directory:
cd /path/to/function

% Call the function:
functionName(args)
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Add the directory of the function file to the load path:

% To add the path for the current session of Octave:
addpath('/path/to/function/')

% To remember the path for future sessions of Octave, after executing addpath above, also do:
    savepath
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Function with multiple return values

Octave’s functions can return more than one value:

function [square, cube] = squareAndCubeThisNumber(x)

square = x^2;
cube = x^3;

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>> [s, c] = squareAndCubeThisNumber(5)
s =  25
c =  125
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Practice

Let’s say I have a data set that looks like this, with data points at (1, 1), (2, 2), (3, 3). And what I’d like to do is to define an octave function to compute the cost function J of theta for different values of theta.

First, put the data into octave:

X = [1, 1; 1, 2; 1, 3]    % Design matrix
y = [1; 2; 3]
theta = [0; 1]
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Output:

Then define the cost function:

% costFunctionJ.m

function J = costFunctionJ(X, y, theta)

% X is the *design matrix* containing our training examples.
% y is the class labels

m = size(X, 1);    % number of training examples
predictions = X * theta;    % predictions of hypothesis on all m examples
sqrErrors = (predictions - y) .^ 2;    % squared erroes

J = 1 / (2*m) * sum(sqrErrors);

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Now, use the costFunctionJ:

>> j = costFunctionJ(X, y, theta)
j = 0
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Got j = 0 because we set theta as [0; 1] which is fitting our data set perfectly.

Vectorization

Vectorization is the process of taking code that relies on loops and converting it into matrix operations. It is more efficient, more elegant, and more concise.

As an example, let’s compute our prediction from a hypothesis. Theta is the vector of fields for the hypothesis and x is a vector of variables.

With loops:

prediction = 0.0;
for j = 1:n+1,
  prediction += theta(j) * x(j);
end;
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With vectorization:

prediction = theta' * x;
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If you recall the definition multiplying vectors, you’ll see that this one operation does the element-wise multiplication and overall sum in a very concise notation.

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