Assume we have following set of data

X Y
20 23
21 21
22 26
23 22
24 25
25 24

For this data, we get this linear regression using this calculator

y = 15.14 + 0.37x

We can solve this equation and get the value of $\hat y$

X Y $ \hat Y$
20 23 22.54
21 21 22.91
22 26 23.28
23 22 23.65
24 25 24.02
25 24 24.39

We can evaluate this regression line in terms of different error metrics :-

MAE (Mean Absolute Error) :-

$$MAE = (\frac{1}{n})\sum_{i=1}^{n}\left | y_{i} - \hat y_{i} \right |$$

where y = actual value in the data set ; $\hat y$ = value computed by solving the regression equation

  1. Calculate the difference between Y and $\hat Y$
  2. Get the absolute values
  3. Take the mean/average i.e. divide by number of elements
X = [20, 21, 22, 23, 24, 25]
Y = [23, 21, 26, 22, 25, 24]
Y_BAR = [22.54, 22.91, 23.28, 23.65, 24.02, 24.39]

# Core Python

n = len(X) # Use length of either X or Y to get number of elements
s = 0
for i in range(0,n):
    s += abs(Y[i] - Y_BAR[i])
MAE = s/n
print ("MAE using Python: %", MAE)

# Using Scikit-Learn Library

from sklearn.metrics import mean_absolute_error

MAE_sci = mean_absolute_error(Y, Y_BAR)
print ("MAE using Sklearn: % ", MAE_sci)

# Using Numpy

import numpy as np

MAE_numpy = np.mean(np.abs(np.subtract(Y,Y_BAR)))
print ("MAE using Numpy: % ", MAE_numpy)

MAE using Python: % 1.3516666666666666
MAE using Sklearn: %  1.3516666666666666
MAE using Numpy: %  1.3516666666666666

MSE (Mean Square Error) :-

$$MSE = (\frac{1}{n})\sum_{i=1}^{n}\left ( y_{i} - \hat y_{i} \right )^2$$

where y = actual value in the data set ; $\hat y$ = value computed by solving the regression equation

  1. Calculate the difference between Y and $\hat Y$
  2. Take a square
  3. Take the mean/average i.e. divide by number of elements
X = [20, 21, 22, 23, 24, 25]
Y = [23, 21, 26, 22, 25, 24]
Y_BAR = [22.54, 22.91, 23.28, 23.65, 24.02, 24.39]

# Core Python

n = len(X) # Use length of either X or Y to get number of elements
s = 0
for i in range(0,n):
    s += (Y[i] - Y_BAR[i])**2
MSE = s/n
print ("MSE using Python: %", MSE)

# Using Scikit-Learn Library

from sklearn.metrics import mean_squared_error

MSE_sci = mean_squared_error(Y, Y_BAR)
print ("MSE using Sklearn: % ", MSE_sci)

# Using Numpy

import numpy as np

MSE_numpy = np.mean(np.square(np.subtract(Y,Y_BAR)))
print ("MSE using Numpy: % ", MSE_numpy)

MSE using Python: % 2.5155166666666653
MSE using Sklearn: %  2.5155166666666653
MSE using Numpy: %  2.5155166666666653

RMSE (Root Mean Square Error) :-

$$RMSE = \sqrt{(\frac{1}{n})\sum_{i=1}^{n}\left ( y_{i} - \hat y_{i} \right )^2}$$

where y = actual value in the data set ; $\hat y$ = value computed by solving the regression equation

  1. Calculate the difference between Y and $\hat Y$
  2. Take a square
  3. Take the mean/average i.e. divide by number of elements
  4. Take the square root
X = [20, 21, 22, 23, 24, 25]
Y = [23, 21, 26, 22, 25, 24]
Y_BAR = [22.54, 22.91, 23.28, 23.65, 24.02, 24.39]

# Core Python
from math import sqrt

n = len(X) # Use length of either X or Y to get number of elements
s = 0
for i in range(0,n):
    s += (Y[i] - Y_BAR[i])**2
RMSE = sqrt(s/n)
print ("RMSE using Python: %", RMSE)

# Using Scikit-Learn Library

from sklearn.metrics import mean_squared_error

RMSE_sci = sqrt(mean_squared_error(Y, Y_BAR))
print ("RMSE using Sklearn: % ", RMSE_sci)

# Using Numpy

import numpy as np

RMSE_numpy = np.sqrt(np.mean(np.square(np.subtract(Y,Y_BAR))))
print ("RMSE using Numpy: % ", RMSE_numpy)

RMSE using Python: % 1.5860380407375685
RMSE using Sklearn: %  1.5860380407375685
RMSE using Numpy: %  1.5860380407375685

RAE (Relative Absolute Error) :-

$$RAE = \frac{\sum_{i=1}^{n}\left | y_{i} - \hat y_{i} \right |}{\sum_{i=1}^{n}\left | y_{i} - \bar y \right |}$$

where y = actual value in the data set ; $\hat y$ = value computed by solving the regression equation ; $\bar y$ is mean value of y

  1. Calculate the difference between Y and $\hat Y$ for each row, take absolute value and sum it all
  2. Calculate the mean of Y denoted by $\bar Y$
  3. Calculate the difference between Y and $\bar Y$ for each row, take absolute value and sum it all
  4. Divide value obtained in step1 by step3
X = [20, 21, 22, 23, 24, 25]
Y = [23, 21, 26, 22, 25, 24]
Y_BAR = [22.54, 22.91, 23.28, 23.65, 24.02, 24.39]

# Using Numpy

import numpy as np

RAE_numpy = np.sum(np.abs(np.subtract(Y,Y_BAR))) / np.sum(np.abs(np.subtract(Y, np.mean(Y))))
print ("RAE using Numpy: % ", RAE_numpy)

RAE using Numpy: %  0.9011111111111111

RSE (Relative Squared Error) :-

$$RSE = \frac{\sum_{i=1}^{n}\left ( y_{i} - \hat y_{i} \right )^2}{\sum_{i=1}^{n}\left ( y_{i} - \bar y \right )^2}$$

where y = actual value in the data set ; $\hat y$ = value computed by solving the regression equation ; $\bar y$ is mean value of y

  1. Calculate the difference between Y and $\hat Y$ for each row, square it and sum it all
  2. Calculate the mean of Y denoted by $\bar Y$
  3. Calculate the difference between Y and $\bar Y$ for each row, square it and sum it all
  4. Divide value obtained in step1 by step3
X = [20, 21, 22, 23, 24, 25]
Y = [23, 21, 26, 22, 25, 24]
Y_BAR = [22.54, 22.91, 23.28, 23.65, 24.02, 24.39]

# Using Numpy

import numpy as np

RSE_numpy = np.sum(np.square(np.subtract(Y,Y_BAR))) / np.sum(np.square(np.subtract(Y, np.mean(Y))))
print ("RSE using Numpy: % ", RSE_numpy)

RSE using Numpy: %  0.8624628571428568