# ML | Mathematical explanation of RMSE and R-squared error

RMSE: Root Mean Square Error is the measure of how well a regression line fits the data points. RMSE can also be construed as Standard Deviation in the residuals.
Consider the given data points: (1, 1), (2, 2), (2, 3), (3, 6).
Lets break the above data points into 1-d lists.
Input :

```x = [1, 2, 2, 3]
y = [1, 2, 3, 6]```

Code : Regression Graph

 `import` `matplotlib.pyplot as plt  ` `import` `math ` ` `  `# plotting the points   ` `plt.plot(x, y)  ` ` `  `# naming the x axis  ` `plt.xlabel(``'x - axis'``)  ` ` `  `# naming the y axis  ` `plt.ylabel(``'y - axis'``)  ` ` `  `# giving a title to my graph  ` `plt.title(``'Regression Graph'``)  ` ` `  `# function to show the plot  ` `plt.show()  ` Code: Mean Calculation

 `# in the next step we will find the equation of the best fit line ` `# we will use Linear algebra's Point slope form to find regression line equation ` `# point-slope form is represented by y = mx + c ` `# where m is slope means (change in y) / (change in x) ` `# c is constant, it represents at which point line will intercept y-axis ` `# slope m can be formulated as below: ` `''' ` `   ``n ` `m =? (xi - Xmean) (yi - Ymean)/?(xi - Xmean)^2 ` ` ``i = 1 ` `'''` `# calculate Xmean and Ymean ` `ct ``=` `len``(x) ` `sum_x ``=` `0` `sum_y ``=` `0` ` `  `for` `i ``in` `x: ` `    ``sum_x ``=` `sum_x ``+` `i ` `x_mean ``=` `sum_x ``/` `ct ` `print``(``'Value of X mean'``, x_mean) ` ` `  `for` `i ``in` `y: ` `    ``sum_y ``=` `sum_y ``+` `i ` `y_mean ``=` `sum_y ``/` `ct ` `print``(``'value of Y mean'``, y_mean) ` ` `  `# we have the values of x mean and y_mean `

Output :

```Value of X mean 2.0
value of Y mean 3.0
```

Code : Line Equation

 `# below is the process of finding line equation in mathematical terms ` `# slope of our line is 2.5 ` `# calculate c to find out the equation ` ` `  `m ``=` `2.5` `c ``=` `y_mean ``-` `m ``*` `x_mean ` `print``(``'Intercept'``, c) `

Output :

`Intercept -2.0`

Code : Mean Squared Error

 `# equation of our Regression line comes out to be as below: ` `# y_pred = 2.5x-2.0 ` `# we call the line y_pred ` `# paste regression line graph ` `from` `sklearn.metrics ``import` `mean_squared_error  ` `# y_pred for our exusting data points is as below ` ` `  `y ``=``[``1``, ``2``, ``3``, ``6``] ` `y_pred ``=``[``0.5``, ``3``, ``3``, ``5.5``] ` `# root mean square calculated by sklearn package ` `mse ``=` `math.sqrt(mean_squared_error(y, y_pred)) ` `print``(``'Root mean square error'``, mse) `

Output :

```Root mean square error 0.6123724356957945
```

Code : RMSE Calculation

 `# lets check how the Root mean square is calculated mathematically ` `# lets introduce a term called residuals ` `# residual are basically the distance of data point from the regression line ` `# residuals are denoted by red marked line in below graph ` `# root mean square and residuals are calculated as below ` `# we have 4 data points  ` `''' ` `r = 1, ri = yi-y_pred ` `y_pred is mx + c ` `ri = yi-(mx + c)  ` `e.g. x = 1, we have value of y as 1 ` `we want to evaluate what exactly our model has predicted for x = 1 ` `(1, 1)r1 = 1, x = 2  ` `'''` `# y_pred1 = 1-(2.5 * 1-2.0)= 0.5 ` `r1 ``=` `1``-``(``2.5` `*` `1``-``2.0``) ` ` `  `#(2, 2) r2 = 2, x = 2  ` `# y_pred2 = 2-(2.5 * 2-2.0)=-1 ` `r2 ``=` `2``-``(``2.5` `*` `2``-``2.0``) ` ` `  `#(2, 3) r3 = 3, x = 2  ` `# y_pred3 = 3-(2.5 * 2-2.0)= 0 ` `r3 ``=` `3``-``(``2.5` `*` `2``-``2.0``) ` ` `  `#(3, 6) r4 = 4, x = 3  ` `# y_pred4 = 6-(2.5 * 3-2.0)=.5 ` `r4 ``=` `6``-``(``2.5` `*` `3``-``2.0``) ` ` `  `# from above calculation we have values of residuals ` `residuals ``=``[``0.5``, ``-``1``, ``0``, .``5``] ` ` `  `# now calculate root mean square error ` `# N = 4 data points ` `N ``=` `4` `rmse ``=` `math.sqrt((r1``*``*``2` `+` `r2``*``*``2` `+` `r3``*``*``2` `+` `r4``*``*``2``)``/``N) ` `print``(``'Root Mean square error using maths'``, rmse) ` ` `  `# root mean square actually calculated using mathematics ` `# both of RMSE calculated are same ` Output :

`Root Mean square error using maths 0.6123724356957945`

R-squared Error or Coefficient of Determination
R2 error answers the below question.
How much y varies with variation in x.Basically the % variation of y on variation with x Code : R-Squared Error

 `# SEline =(y1-(mx1 + b)**2 + y2-(mx2 + b)**2...+yn-(mxn + b)**2) ` `# SE_line =(1-(2.5 * 1+(-2))**2 + (2-(2.5 * 2+(-2))**2) +(3-(2.5*(2)+(-2))**2) + (6-(2.5*(3)+(-2))**2)) ` ` `  `val1 ``=``(``1``-``(``2.5` `*` `1``+``(``-``2``)))``*``*``2` `val2 ``=``(``2``-``(``2.5` `*` `2``+``(``-``2``)))``*``*``2` `val3 ``=``(``3``-``(``2.5` `*` `2``+``(``-``2``)))``*``*``2` `val4 ``=``(``6``-``(``2.5` `*` `3``+``(``-``2``)))``*``*``2` `SE_line ``=` `val1 ``+` `val2 ``+` `val3 ``+` `val4 ` `print``(``'val'``, val1, val2, val3, val4) ` ` `  `# next to calculate total variation in Y from mean value ` `# variation in y is calcualted as ` `# y_var =(y1-ymean)**2+(y2-ymean)**2...+(yn-ymean)2 ` ` `  `y ``=``[``1``, ``2``, ``3``, ``6``] ` ` `  `y_var ``=``(``1``-``3``)``*``*``2``+``(``2``-``3``)``*``*``2``+``(``3``-``3``)``*``*``2``+``(``6``-``3``)``*``*``2` `SE_mean ``=` `y_var ` ` `  `# by calculating y_var we are calculating the distance ` `# between y data points and mean value of y ` `# so answer to our question, % of the total variation ` `# of wrt x is denoted as below: ` `r_squared ``=` `1``-``(SE_line ``/` `SE_mean) ` ` `  `# [SE_line / SE_mean] -->tells us the what % of variation ` `# in y is not described by regression line ` `# 1-(SE_line / SE_mean) --> gives us the exact value of  ` `# how much % y varies with variation in x ` `print``(``'Rsquared error'``, r_squared) `

Output :

```Rsquared error 0.8928571428571429
```

Code : R-Squared Error with sklearn

 `from` `sklearn.metrics ``import` `r2_score ` ` `  `# r2 error calculated by sklearn is similar  ` `# to ours mathematically calculated r2 error ` `# calculate r2 error using sklearn ` `r2_score(y, y_pred) `

Output :

`0.8928571428571429`

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