Vectors in AI plays a very crucial role for fundamental operations for different AI Based applications. Most of the time the underline calculation are hidden to the naked eyes and has been done using python bases libraries.
Let’s try to understand the concept of vectors in simple regression problem.
Dataset :https://www.kaggle.com/datasets/abhishek14398/salary-dataset-simple-linear-regression/data
Consist of three columns , ID, YearsExperience,Salary
| ID | YearsExperience | Salary |
| 0 | 1.2 | 39344.0 |
| 1 | 1.4 | 46206.0 |
| 2 | 1.6 | 37732.0 |
| 3 | 2.1 | 43526.0 |
| 4 | 2.3 | 39892.0 |
Now we need to predict the salary based on years of experience
ID Column does not make any contribution for predicting the salary so we are going to drop it , now the input data looks like below
| YearsExperience | Salary |
| 1.2 | 39344.0 |
| 1.4 | 46206.0 |
| 1.6 | 37732.0 |
| 2.1 | 43526.0 |
| 2.3 | 39892.0 |
Where YearsExperience I.e X is input feature and Salary I.e is Output feature
Here is a way how Y is related to X , and relations seems linear in nature I.e. with year of experience the salary is increasing.
Also lets now check statistically how the X and Y are related to each other
As we can see there is a strong correlation between X and Y , so this problem is linear problem.
A linear problem can be solved using linear equation
y=ax+b
Where,
y=salary
x=year of experience
a=slope I.e. change in value of y ,when 1 unit of X changes
b=intercept I.e. the value of y when x=0
The same equation for Machine learning and deep learning can be written as
y=Wx+b
We need to find a or W and b , which can be find is least square method without using vectors
As per our dataset , if we put the values W comes out as -109.91 and b comes out as 41529.04 which graphically comes as
Lets now calculate Y for some value of X is 2.3
y= 41529.04*2.3+-109.91
y=41,501.06
Vectors
Lets try to solve the same problem using vectorised form
Understand what would be the calculation points involved here , lets revisit our equation
y=wx+b
Where w is the weight
And b is intercept
Let’s see now how can we put below data into linearised equation and then into vector forms.
| YearsExperience | Salary |
| 1.2 | 39344.0 |
| 1.4 | 46206.0 |
| 1.6 | 37732.0 |
| 2.1 | 43526.0 |
| 2.3 | 39892.0 |
39344=w*1.2+b
46206=w*1.4+b
————————
39892=w*2.3+b
Writing the above equation in matrix form
[39344]=[1.2]*w+[1]*b
[46206]=[1.4]*w+[1]*b
[37732]=[1.6]*w+[1]*b
[43526]=[2.1]*w+[1]*b
[39892]=[2.3]*w+[1]*b
Putting in matrix form containing vectors
Now lets calculate a I.e W and b
The formula is
Putting the values in place a and b comes as -109.91. and b=41529.04
Lets try to calculate Y for x 2.3
Y=2.3*(-109.91)+41529.04
y≈41276.25 (predicted value)
Why Vectors Then?
Now the question is that if we can solve the problem without vectors , then what is the need of vectors.
Here is a time comparison for finding W and b both using traditional computation and vector calculations for large dataset I.e for 10k records
We can see that using vectorise form we can have faster calculation which increases the efficiency
References
Kaggle : https://www.kaggle.com/code/mishrapankaj26/traditional-vs-vector-approach-for-ml-tasks/edit