ML | Mathematical explanation of standard deviation and R-squared error

Michael Zippo

RMSE: root mean square error — it is a measure of how well the regression line fits the data points. RMSE can also be interpreted as the standard deviation of the residual.
Consider these points: (1, 1), (2, 2), (2, 3), (3, 6).
We split the above data points into the 1st list.
Login :

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

Code: Regression Plot

import matplotlib.pyplot as plt

import math

# dots
plt. plot (x, y)

# x-axis title

plt.xlabel ( ’x - axis’ )

# Y-axis name

plt.ylabel ( ’y - axis’ )

# giving a title to my graphic

plt.title ( ’Regression Graph’ )

# plot show function
plt.show ()

Code: Average

       # in the next step we will find the equation of the line of best fit  
  # we will use the slope of a linear algebra point to find the equation of the regression line  
  # the shape of the slope is represented by y = mx + c  
  # where m mean s slope (change in y) / (change in x)  
  # c is a constant, it represents where the line will cross the Y-axis  
  # Slope m can be formulated like this:  
  "" " 
     N  
   m =? (xi - X mean ) (yi - Y mean ) /? (xi - X mean ) ^ 2  
   i = 1  
   "" " 
  # calculate X mean and Y mean  
   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 x mean and y_ mean

Output:

 Value of X  mean  2.0 value of Y  mean  3.0

Code: linear equation

# below is the process of finding a linear equation in mathematical terms
# our line’s slope is 2.5
# evaluate c to find the equation

m = 2.5

c = y_ mean - m * x_ mean

print ( ’Intercept’ , c)

Output:

 Intercept -2.0

Code: Medium square error

Our regression line equation looks like this:
# y_pred = 2.5x-2.0
# we name the string y_pred
# insert a regression line plot

from sklearn.metrics import mean _squared_error

# y_pred for our exhaustive data points as shown below

y = [ 1 , 2 , 3 , 6 ]

y_pred = [ 0.5 , 3 , 3 , 5.5 ]

# sklearn root mean square

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

# let’s see how the mean square is calculated mathematically
# let’s introduce a term called residuals
# remainder - it is basically the distance from the data point to the regression line
# the residuals are indicated by the red line in the graph below
# RMS and residuals are calculated as shown below
# we have 4 data points
"" "
r = 1, ri = yi-y_pred
y_pred is mx + c
ri = yi- (mx + c)
eg x = 1, we have the y value as 1
we want to evaluate exactly what our model 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 )

# on top of the calculation we have residual values

residuals = [ 0.5 , - 1 , 0 ,. 5 ]

# now calculate the root mean square error
# N = 4 tons data points

N = 4

rmse = math. sqrt ((r1 * * 2 + r2 * * 2 + r3 * * 2 + r4 * * 2 ) / N)

print ( ’Root Mean square error using maths’ , rmse)

# RMS value actually calculated using math
# both RMSEs are calculated the same

Output:

 Root Mean square error using maths 0.6123724356957945

R-squared error or coefficient of determination
Error R2 answers the following question.
How much y changes with change in x. Basically, the percentage change in y when changing from x