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Linear Regression

                                            Linear Regression


 
As you can see from the image above, a line of best fit is drawn on a scatter plot. The line of best fit is essentially the Linear Regression model and the blue data points is the training data.  

What we need to do in order to implement the linear regression model to the data points/training data is to calculate the line of best fit.

Linear Regression Model:
output = weight * input + b   (where "weight" and "input" are vectors)

When looking at how the linear regression model works, you need to have an understanding of what the terms "weight" and "bias" mean. You can think of the "weight" vector and "bias" as terms that optimize the line to be a line of best fit on the training data points (like a gradient the y-intercept in a linear function).      

Loss Function: Mean Squared Error 

In order to calculate a line of best fit, we will need to find the weight and the bias of the model. 

Before we get into an optimization algorithm to calculate the weight and the bias of our model. We will need to take a look at what a loss function is. Loss function or cost function is a function that measures the performance of your model; it's able to calculate how inaccurate your model is. 

This is a loss function called "Mean Squared Error"





This function squares the difference between the target output and the output obtained from the current model. It sums up the output for each of the data points and calculates the average loss. 

Gradient Decent:

This is the algorithm that is used to help find the optimal weight and bias of the model to the training data points.








As you can see from the image above, we can see how the cost in the y-axis changes with respect to the "weight" on the x-axis.

Gradient Decent works by finding the derivative of the cost function with respect to the current weight or bias.


EQUATION: Where theta can be replaced with the weight and the bias of the model.

Theta(new weight or bias)  = Theta(current weight or bias) - (leaning rate) x (derivative of cost function with respect to weight or bias)

This is how the gradient descent algorithm is applied to optimize the weight and the bias.

Learning Rate: 
The learning rate is a tuning hyperparameter that controls how rapidly your weight or bias gets corrected. 
If your learning rate is too big, it would be taking large steps each time you try and optimize it. As you can see from the image above. It wouldn't be very effective to get to the local minimum(point with lowest cost). On the other hand, if your learning rate is too small, it would be very time consuming as it takes small incremental steps to reach the local minimum. Therefore, it's important to make sure that you have the just right learning rate to be efficient.  
Also, using a small learning rate could cause it to get stuck in the local minimum. preventing the model from getting the optimal weight and bias to minimize the 'error' as much as possible.


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