Octave is a high-level language that can be used for performing complex mathematical calculations and also creating prototypes for Machine Learning (ML) projects quickly and easily, and I mean it. It is one of the best languages to start your journey into the world of Machine Learning.
ML projects have a low success rate, as per Gartner, 87% of ML projects fail due to various reasons. It is therefore very important not to invest huge resources and money during initial analysis and POC for an ML-based project. That's were Octave can be useful
Below are some of the advantages of using Octave
Easy to learn with a low learning curve
Excellent CLI for directly defining variables and executing commands
Powerful mathematical and matrix operations
Powerful 2D and 3D plotting and visualization tool
Free and open-source and can run on multiple platforms including Windows, Mac, Linux, and more
Instead of implementing initial level ML prototypes in a language like Python or a library like Tensorflow, Octave can be used to quickly implement a working prototype, plot graphs, and analyze the nature of your ML data.
Matrices are building blocks of the data used in machine learning programs, they are an almost inseparable part of ML programs. Most ML libraries come with built-in support for Matrix operations like addition, subtraction, division, inverse, and so on.
Octave Operations
You can download and install Octave using the following link https://www.gnu.org/software/octave/download
Installation is easy and quick. You can run octave commands via Octave-cli or by writing functions inside a file. You can directly evaluate an expression, assign variables, print results, plot graphs even with CLI. Click on the Octave-cli icon to open Octave-cli. This will give you a prompt like below
Try some simple mathematical equations like 7 + 4 -2 and hit enter, octave CLI will provide you results and store it in a variable ans,
You can use any expression for evaluation as well ex, 1==2, which will returns 0 (false). Assign a variable with a value ex a=10.
Let us see what all basic operations we can perform in Octave
Basic Operations
octave:2> a =10
a=10
octave:3> 1==2
ans=0
octave:4> 1==1
ans=1
octave:5> b= 20
b = 20
octave:6> who
Variables in the current scope:
a ans b
octave:7> disp(a)
10
octave:8> x=pi
x = 3.1416
octave:9> r=10
r=10
octave:9> area=pi*r^2
area = 78.540Initializing Matrix
Octave is really cool when we try to perform matrix operations. Let us see how to create a matrix and perform some operations on the matrix.
You can define the matrix using [] brackets. Elements in a column should be separated by space and semicolon (;) separates rows in a matrix.
octave:10> A=[1 2 3] % Creates a 1x3 Matrix, 1 Row & 3 Columns
A =
1 2 3
octave:11> A = [1 2; 3 4] % Creates a 2x2 Matrix
A =
1 2
3 4
octave:15> B = [5 6;7 8]
B =
5 6
7 8
octave:16> X = [1:1:10] % Starting 1 with difference of 1 upto 10
X =
1 2 3 4 5 6 7 8 9 10
octave:17> X = [1:2:10] % Starting 1 with difference of 2 upto 10
X =
1 3 5 7 9
octave:18> X = [1:0.1:2]
X =
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 2.0000Matrix Operations
Now let us try to perform some basic matrix operations
octave:16> A + B
ans = 6 8
10 12
octave:17> A * B
ans = 19 22
43 50
octave:17> B * A
ans = 23 34
31 46
octave:17> C = A - B % Store result in Matrix C
C = 19 22
43 50
octave:17> C = A - B % Store result in Matrix C
C = 19 22
43 50
octave:18> A = A.*2 % . Signifies element wise operator
A = 2 4
6 8
octave:19> A = A./2
A = 1 2
3 4
octave:20> A = ones(3,2) % Creates 3x2 matrix of 1
A =
1 1
1 1
1 1
octave:21> B = zeros(3,2) % Creates 3x2 matrix of 0
B =
0 0
0 0
0 0
octave:22> C = eye(3,3) % Identity 3 x 3 matrix (Diagonal 1 and rest 0
C =
Diagonal Matrix
1 0 0
0 1 0
0 0 1
octave:22> D = rand(2,2) % 2x2 Matrix of random numbers
ans =
0.96599 0.56881
0.43081 0.71877
I am sure you all are aware of rules related to operations that we can perform on Matix
For example, you can multiple matrices only if matrix A has m x n columns and matrix B has n x q columns. This means the number of columns in the first matrix should be equal to the number of rows in the second matrix.
Consider some examples
If you are not comfortable with Matrix operations, I recommend you to try out various combinations using pen and paper and verify the same using Octave CLI.
Here are some details on how to get the element out of Matrix
octave:23> A = [1 2 3;4 5 6]
A =
1 2 3
4 5 6
octave:24> size(A) % Returns matrix with details on A rows and columns
ans =
2 3
octave:25> C = size(A) % 2x3 Matrix C, C(1) = 2 & C(2) = 3
C =
2 3
octave:26> size(A,1) % Returns total Rows
ans = 2
octave:27> size(A,2) % Returns total Columns
ans = 3
octave:28> length(A) % Returns longest dimension
ans = 3 * - A one-dimensional matrix is also called a Vector, matrix can be a row vector or a column vector.
octave:29> A
A =
1 2 3
4 5 6
octave:30> A(1,1) % Returns element from 1st row and 1st column
ans = 1
octave:31> A(1,2) % Returns element from 1st row and 2nd column
ans = 2
octave:32> A(3,3) % Returns element from 3rd row and 3rd column
ans = 3
octave:33> A(:,1) % Returns all Rows and 1st Column
ans =
1
4
octave:34> A(2,:) % Returns all columns and 2nd row
ans =
4 5 6
octave:35> A(:,:) % Returns all rows and all columns
ans =
1 2 3
4 5 6
octave:36> A(:,[1 3],:) % Returns all rows and 1st and 3rd columns
ans =
1 3
4 6
octave:37> A(:,2)=[22; 55] % Replace 2nd column with new values
A =
1 22 3
4 55 6
octave:38> A(1,:)=[9 10 11] % Replace 1st row with with new values
A =
9 10 11
4 55 6
octave:39> A = [A,[7;8]] % Add one column vector to right
A =
9 10 11 7
4 55 6. 8
octave:40> A = [[9;10], A] % Add one column vector to left
A =
9 10 11 7
4 55 6 8
octave:41> A(:) % Put all elements in a single column vector
A =
9
10
11
7
4
55
6
8Inverse & Transpose
The inverse of a matrix is a matrix which when multiplied with the original matrix produces an identity matrix. The transpose of a matrix is an operation that flips the matrix over its diagonal.
Let us see some examples
octave:43> A = [1 2 3; 4 5 6]
A =
1 2 3
4 5 6
octave:44> B = pinv(A)
B =
-0.94444 0.44444
-0.11111 0.11111
0.72222 -0.22222
octave:45> A * B
ans =
1.00000 0.00000
0.00000 1.00000
octave:46> A' % Creates transpose of matrix A
ans =
1 4
2 5
3 6
octave:47> (A')'
A =
1 2 3
4 5 6
octave:48> find(A>2) % Get all elements greater than 2
A =
0 2 3
4 5 6
octave:49> A>2 % Element wise comparision and returns 1 or 0 (True of False)
A=
0 0 1
1 1 1
If Statement
if (condition)
then-body
else
else-body
endif
Practical Examples
Let us try to solve some mathematical equations using Matrices in Octave
Ex. 1 Consider an example where housing price data where you have a size in sq. ft and price in $
Suppose this data is available in a text file, like the one below
Let us write some Octave commands to load this data in the matrix. In order to load data, your file should be either present in the root of your working directory or you need to use the full path.
You can use Octave CLI or create a function inside the file and make it run. Please note that your file name and function name should be the same. To run this file you simply need to write the name of the file in Octave CLI, this will automatically execute the function inside this file. If you do not want to print statements inside the file, please use a semicolon (;) after each statement. If you want to debug your program and check variable values, then simply do not use a semicolon at the end of each statement.
octave:1> data = load('housingdata.txt')
data =
260 1000
280 1200
340 1500
450 2000
550 2800
Octave loads the data into a 2x2 matrix. Let us move this data into 2 different matrix x and y.
In CLI you can execute two commands using,
octave:1> x = data(:,1),y=data(:,2)
x =
260
280
340
450
550
y =
1000
1200
1500
2000
2800Let us see how we can solve the below equation using normal as well as vectorized methods.
Hypothesis Function
# Example
Let us calculate the hypothesis first using the normal method. To calculate the hypothesis for all the examples in matrix x, we need to perform a loop. Let us try this out using Octave however you can use any other language of your choice as far as normal method is concerned. For matrix operations, you need to choose a language that supports such complex matrix operations.
Let us load the data first
function [] = lg()
data = load('housingdata.txt');
x = data(:,1);
y = data(:,2);
m = size(x,1);
theta = [0.5,0.5];
endNow loop through the data to calculate hypothesis, so let us first define a data structure to hold hypothesis values and create a function to calculate hypothesis
function [] = lg()
data = load('housingdata.txt');
x = data(:,1);
y = data(:,2);
m = size(x,1);
theta = [0.5,0.5];
hypothesis = calculateHypothesis(x,y,m,theta(1),theta(2))
end
function[hypothesis] = calculateHypothesis(x,y,m,theta0,theta1)
hypothesis = zeros(m,1);
for(i=1:m)
hypothesis(i) = theta0 + theta1 * x(i);
end
end
You will get the following output in the form of a column vector
hypothesis =
130.50
140.50
170.50
225.50
275.50Now consider we have millions of examples and we need to calculate the hypothesis for million records, looping through millions of options will be a time-consuming operation. Let us evaluate the matrix multiplication option. Consider modifying the above hypothesis equation as
So let us add one more column to our matrix x with value 1
tempx = [ones(m,1),x]
Matrix tempx will look like this
tempx =
1 260
1 280
1 340
1 450
1 550
Now to calculate the hypothesis in one go, you can apply the formula
hypothesis = x*theta' Please use a pen and paper to evaluate this function and see how this works. Now adding one more function for vectorized implementation
function [] = lg()
data = load('housingdata.txt');
x = data(:,1);
y = data(:,2);
m = size(x,1);
theta = [0.5,0.5];
%hypothesis = calculateHypothesis(x,y,m,theta(1),theta(2))
hypothesis_vector = calculateHypothesisVectorize(x,y,m,theta)
end
function[hypothesis] = calculateHypothesis(x,y,m,theta0,theta1)
hypothesis = zeros(m,1);
for(i=1:m)
hypothesis(i) = theta0 + theta1 * x(i);
end
end
function[hypothesis_vector] calculateHypothesisVectorize(x,y,m,theta)
tempx = [ones(m,1),x]
hypothesis_vector = zeros(m,1);
hypothesis_vector = tempx*theta';
endWe have calculated the hypothesis using two different approaches. In ML we rely on vectorized implementation in order to perform the task quickly and efficiently.
Plotting Graph
Octave also comes with a powerful 2D/3D plotting engine, let us see how to draw graphs using Octave. Plotting data is also very simple, to plot graph for x and y use below code
plotData(x, y);
xlabel('\Size (Sq.Ft)'); ylabel('\Price ($) in 10,000s');
The following graph will be displayed
I would strongly recommend you to try out various octave operations on your own. You can also go through Octave documentation available on octave website https://www.gnu.org/software/octave/index
Happy Reading...
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