swetak20 / ML-Multivariate-Linear-Regression

Linear Regression is implemented on a dataset to predict the "Chance of admit" based on various factors

ML-Multivariate-Linear-Regression

Linear Regression is implemented on a dataset to predict the "Chance of admit" based on various factors as mentioned in the .csv file.

Linear Regression is implemented in two difeerent ways:

1. Using the library - sklearn
2. From scratch, by obtaining matrices and using the least square solution formula.

Later the results are verified by obtaining the prediction for the same test case.

1. Using scikit-learn

Using functions of scikit-learn library, the data is used to train the regression and co-effecients are obtained and hence prediction for test case is obtained.

The following functions are used here:

• LinearRegression()
• fit()
• .coef_
• .intercept_

2. Least Square Solution

For linear system Ax = b of m equations in n unknowns that are inconsistent, one looks for vectors that come as close as possible to being solutions in the sense that they minimize b − Ax with respect to the Euclidean inner product.

If A is an m × n matrix with linearly independent column vectors, then for every m × 1 matrix b, the linear system Ax = b has a unique least squares solution. This solution is given by $$x = inv(((\mathbf{A}^\intercal)A ))\mathbf{A}^\intercal\ b\ $$

c + $a_{0}$$x_{0}$ + $a_{0}$$x_{0}$ + $a_{0}$$x_{0}$ +.....+ $a_{0}$$x_{0}$ = b

where :

• c = constant
• $x_{0}$ ,$x_{1}$ ,$x_{2}$,....,$x_{n}$ are the independent variables
• b is the dependent variable

To convert the equation to the form Ax = b:

1. Matrix A contains all the columns of dependent variable and an additional column of with value 1( to obtain the constant c), here columns are: 'const', 'GRE Score', 'TOEFL Score', 'University Rating', 'SOP', 'LOR', 'CGPA', 'Research'
2. Matrix b contains the values of dependent variables, here the column is : Chance_of_admit
3. Once the above matrices are obtained, we plug them in the formula of least square solution to get the matrix x and hence obtain the co-effecients.
4. Once x is obtained, b is calculated for a test case by matrix multiplication of A and x