r squared and adj r squared added

r squared and adjusted r squared have been added as regression evaluation metrics. More will be updated later.

Author: purnesh

01-Regression/00-RegressionModelsComparison.md

@@ -1,4 +1,26 @@
 # Performance Evaluation
 
-## R-Squared Intuition
+## R-Squared 
+
+There are a few terms that need to be understood before we can calculate the R-squared value for any regression model and are as follows:
+
+1. **Sum of squared residuals** ![ssres](http://mathurl.com/ycattv9v.png) : It is exactly what the name suggests. A residual is the difference between actual values and the predicted values. If we add the squares of all residuals, that would give us the value of ![ssres](http://mathurl.com/ycattv9v.png), shown as follows in mathematical terms:
+    ![](http://mathurl.com/yawhk57s.png)
+
+2. **Total sum of squares** ![sstot](http://mathurl.com/y7hyn3w4.png):  Instead of calculating the residuals, we calculate the difference between the average **y** values and the actual **y** values present in the dataset. We then add the squares of these values, represented mathematically as :
+    ![](http://mathurl.com/yazstja5.png)
+
+R-sqaure is given by the following formula ![rsq](http://mathurl.com/ybf2xwp2.png) 
+
+The idea is to minimize ![ssres](http://mathurl.com/ycattv9v.png) so as to keep the value of ![rsq](http://mathurl.com/3kwwdyh.png) as close as possible to 1. The calculation essentially gives us a numeric value as to how good is our regression line, as compared to the average value.
+
+## Adjusted R-Squared
+
+The value of ![rsq](http://mathurl.com/3kwwdyh.png) is considered to be better as it gets closer to 1, but there's a catch to this statement. The ![rsq](http://mathurl.com/3kwwdyh.png) value can be artifically inflated by simply adding more variables. This is a problem because the complexity of the model would increase due to this and would result in overfitting. The formulae for Adjusted R-squared is mathematically given as:
+
+![](http://mathurl.com/yclkhq5z.png) 
+
+where **p** is the number of regressors and **n** is the sample size. Adjusted R-squared has a penalizing factor that reduces it's value when a non-significant variable is added to the model.
+
+> **p-value** based backward elimination can be useful in removing non-significant variables that we might have added in our model initially. (full process explained in **Linear Regression** section of the book.