Python | Mean Squared Error

The Mean Squared Error (MSE) or Mean Squared Deviation (MSD) of an estimator measures the average of error squares i.e. the average squared difference between the estimated values and true value. It is a risk function, corresponding to the expected value of the squared error loss. It is always non – negative and values close to zero are better. The MSE is the second moment of the error (about the origin) and thus incorporates both the variance of the estimator and its bias.

Steps to find the MSE

  1. Find the equation for the regression line.

    (1)   \begin{equation*} \hat{Y}_i = \hat{\beta}_0 + \hat{\beta}_1 X_i + \hat{\epsilon}_i \end{equation*}

  2. Insert X values in the equation found in step 1 in order to get the respective Y values i.e.

    (2)   \begin{equation*} \hat{Y}_i \end{equation*}

  3. Now subtract the new Y values (i.e. \hat{Y}_i) from the original Y values. Thus, found values are the error terms. It is also known as the vertical distance of the given point from the regression line.

    (3)   \begin{equation*} Y_i - \hat{Y}_i \end{equation*}

  4. Square the errors found in step 3.

    (4)   \begin{equation*} {(Y_i - \hat{Y}_i)}^2 \end{equation*}

  5. Sum up all the squares.

    (5)   \begin{equation*} \sum_{i=1}^{N}(Y_i - \hat{Y}_i)^2 \end{equation*}

  6. Divide the value found in step 5 by the total number of observations.

    (6)   \begin{equation*} MSE = \frac{1}{N}\sum_{i=1}^{N}(Y_i - \hat{Y}_i)^2 \end{equation*}

Example:
Consider the given data points: (1,1), (2,1), (3,2), (4,2), (5,4)
You can use this online calculator to find the regression equation / line.

Regression line equation: Y = 0.7X – 0.1

X Y \hat{Y}_i
1 1 0.6
2 1 1.29
3 2 1.99
4 2 2.69
5 4 3.4

Now, using formula found for MSE in step 6 above, we can get MSE = 0.21606

MSE using scikit – learn:

filter_none

edit
close

play_arrow

link
brightness_4
code

from sklearn.metrics import mean_squared_error
  
# Given values
Y_true = [1,1,2,2,4]  # Y_true = Y (original values)
  
# calculated values
Y_pred = [0.6,1.29,1.99,2.69,3.4]  # Y_pred = Y'
  
# Calculation of Mean Squared Error (MSE)
mean_squared_error(Y_true,Y_pred)

chevron_right






                        filter_none










Output: 0.21606


MSE using Numpy module:













                                    filter_none




                                    edit

                                    close




                                    play_arrow




                                    link

                                    brightness_4

                                    code
                                




















import numpy as np 


  


# Given values 


Y_true = [1,1,2,2,4]  # Y_true = Y (original values) 


  


# Calculated values 


Y_pred = [0.6,1.29,1.99,2.69,3.4]  # Y_pred = Y' 


  


# Mean Squared Error 


MSE = np.square(np.subtract(Y_true,Y_pred)).mean() 



















                        chevron_right







                        filter_none










Output: 0.21606



          
          
          
          

            

    

        My Personal Notes
        arrow_drop_up
    

    

        
        

            
            
        

    


Recommended Posts:
mkumarchaudhary06
Possess good Mathematical and Statistical Foundation Learning Data Science and Machine Learning Interested in Mathematical and Statistical Analysis
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on  the "Improve Article" button below.