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Заметки о машинном обучении Эндрю Нг - (5) Учебник по октавам
GNU OctaveЭто программное обеспечение в основном используется для численного анализа с использованием языка программирования высокого уровня. Octave помогает численно решать линейные и нелинейные задачи и проводить другие численные эксперименты, используя язык, совместимый с MATLAB. Он также доступен как пакетно-ориентированный язык. Поскольку это часть проекта GNU, это бесплатное программное обеспечение в соответствии с условиями Стандартной общественной лицензии GNU.
Octave — одна из ведущих бесплатных альтернатив MATLAB.
Официальный сайт Октавы:Woohoo.Bowbow.org/software/OC…
Определение, данное на официальном сайте: Научный язык программирования
- 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
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
Change the Prompt
We can change the prompt via PS1():
>> PS1("octave: > ")
octave: > PS1(">> ")
>> PS1("octave: > ")
octave: > PS1("SOMETHING > ")
SOMETHING > PS1(">> ")
>> % Prompt changed
Variables
>> a = 3
a = 3
>> a = 3; % semicolon supressing output
>> c = (3 >= 1);
>> c
c = 1
Display variables
>> a = pi;
>> a
a = 3.1416
>> disp(a)
3.1416
>> disp(sprintf('2 decimals: %0.2f', a))
2 decimals: 3.14
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
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
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
- 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
- Generate identity matrices
>> eye(3)
ans =
Diagonal Matrix
1 0 0
0 1 0
0 0 1
- Generate matrices of random values
Uniform distribution between 0 and 1:
>> D = rand(1, 3)
D =
0.14117 0.81424 0.83745
Gaussian random:
>> D = randn(1, 3)
D =
0.22133 -2.00002 1.61025
We can generate a gaussian random vector with 10000 elements, and plot a histogram:
>> randn(1, 10000);
>> hist(w)
Output figure:
We can also plot a histogram with more buckets, 50 bins for example:
>> hist(w, 50)
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
......
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
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
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
load command can load data from a file.
>> load featureX.dat
>> load('featureY.dat')
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
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
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
>>
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
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
>>
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
>>
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
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
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
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
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
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
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
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
Transpose:
>> A = [1, 2; 3, 4; 5, 6];
>> A' % transposed
ans =
1 3 5
2 4 6
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
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
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
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
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);
It will show you a figure like this:
>> y2 = cos(2*pi*4*t);
>> plot(t, y2);
? Это заменит фигуру греха на новую фигуру кос.
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');
We can set some text on thw figure:
>> xlabel("time");
>> ylabel("value");
>> legend('sin', 'cos'); % Show what the 2 lines are
>> title('my plot');
Now, we get this:
Then, we save it and close the plotting window:
>> print -dpng 'myPlot.png' % save it to $(pwd)
>> close
We can show two figures at the same time:
>> figure(1); plot(t, y1);
>> figure(2); plot(t, y2);
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
Use clf to clear a figure:
>> clf;
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
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;
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
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
if
>> for i = 1: 10,
> if v(i) > 100,
> disp(v(i));
> end;
> end;
128
256
512
1024
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
break & continue
i = 1;
while true,
v(i) = 999;
i = i + 1;
if i == 6,
break;
end;
end;
Output:
v =
999
999
999
999
999
64
128
256
512
1024
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;
To call this function in Octave, do either:
-
cdto the directory of the functionName.m file and call the function:
% Navigate to directory:
cd /path/to/function
% Call the function:
functionName(args)
- 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
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;
>> [s, c] = squareAndCubeThisNumber(5)
s = 25
c = 125
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]
Output:
X =
1 1
1 2
1 3
y =
1
2
3
theta =
0
1
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);
Now, use the costFunctionJ:
>> j = costFunctionJ(X, y, theta)
j = 0
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;
With vectorization:
prediction = theta' * x;
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.