Hello, Today I am going to share with you the basics of linear regression. For this, I haven’t planned to use Gradient Descent as I want the model to be a little fast and quick.
So, I am going to apply Closed-Form(Normal Equation) on Linear Regression with RMSE to get θ (model’s parameter vector). we will see this as we go on.
Ok let’s start with Linear regression
The linear model predicts by simple computing a weighted sum of the input features, plus a constant called bias term.
θ: model’s parameter vector, containing the bias term theta-0 =1 and the feature weights theta-1 to theta-n
x is the instance’s feature vector, containing x0 to xn, with x0 always equal to 1.
θ. x: is the dot product of the vectors θand X.
h(θ): hypothesis function, using the model parameter θ.
import numpy as np# input X
y=4+3*X+np.random.rand(100,1) # I have used y = Gaussian Noise
Apply MSE to reduce Loss
now apply mse on the above equation to reduce the error and fit the model.
Let’s calculate: best θ (Normal Equation)
By applying the Above equation we got a normal equation
Now we have to find the value of Theta that minimizes the cost function, there is a closed-form solution also known as Normal Equation which can give you direct results.
# concatinate x0 =1 to each instance with X
theta_cap = np.lialg.inv(X_.T.dot(X_)).dot(X_.T).dot(y)
Yes now we got our theta_cap now we can use this theta_cap for our prediction.
read here for derviation .
Now Let’s move for prediction
X_new=np.array([,]) # create input
X_new_b=np.c_[np.ones(2,1)),X_new) # add x0=1 to each instance
y_predict will give you output as theta value
The computational complexity of inverting such a matrix is typically about O(n³) depending on implemetation.
once you have trained your Linear Regression model, prediction are very fast.
I hope you enjoyed my post flow me for more updates.