Linear Regression ( Fun with Mathematics )

So in this Series of Machine  Learning , I plan to teach machine learning in a fun as  well as mathematically.

This is the starting point in any Machine  learning Engineer career.

Linear Regression...........



y = a*x  + b

Everyone one Linear Regressions , a Regression technique to fit  the best line in a given data sets.

So here in this article we will derive a Linear Regression Formula using the help of our beloved Calculus....

So , what is the point of doing this linear regression ?
Yes , Obviously to predict. We fit the best line in our datasets.

The equation looks like this

y = a * x + b

Here "y" is the dependent variable or anything we are going to predict  , "x" is independent variable , or a predictor  , and "a" and "b" are the slopes and y-intercept of the lines , in general , the values of a and b are responsible on making how good the line fits on data.
So our main concern is to find the optimal values of a and b which fits the data.

There  are many technique that can be used to find values of a and b ,
Here in this article we will be going through the simplest method using SSE(Sum of Squared Error)  method, where our main concern is to  minimize the error.

Terminologies used

y(i) = Actual Value
y^(i) = Predicted Value

and our SSE ( E ) (equation ......a )  is the differ of sum of Squares between real(actual) and predicted value.

a , b are to be determined,

As there are two unknown variables "a" and "b" but we got only one equation that is  SSE , so to solve for "a" and "b" , we perform partial derivative on E( equation ...... a ) with respect to a and b simultaneously.

We get two equations ( equation.......... b and equation........ c ) as in picture below.

Solving for a and b from these two equations as

equation..........b = 0
and
equation.........c  = 0

we get the following two equations marked inside rectangular box.






Now we can easily solve these two equations for a and b ,which i leave for you and you get the solved equations as follows...

Now ,
we have our parameters a and b , our regression model looks like y = a*x + b, plug in x variable and get y , easy peasyyyyyyyy


In next Article we will implement this Formula in Python and also Visualize Linear Regression.