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EVALUATION METRICS FOR REGRESSION

Hello guys, in my previous blog that is Mathematical implementation of Linear Regression I explained the internal operations of the algorithm, in this blog I will explain you the evaluation metrics that are used to evaluate the model.

Before going to the Evaluation metrics let me show you a graph.    

  • (x,y) is an Actual value.
  • (x,ŷ) is an predicted value.
  • y-ŷ= Difference between the actual and predicted value /also called as Residual.

EVALUATION METRICS

  • MEAN SQUARED ERROR(MSE)
  • MEAN ABSOLUTE ERROR(MAE)
  • R-SQUARED
  • ROOT MEAN SQUARE(RMSE)

MEAN SQUARED ERROR (MSE)

  • MSE is nothing but the squared difference between the actual and the predicted value.
  • Let me take an example


  • Here y is the actual value and yhat is the predicted value.
  • Let's begin the process:
  • Calculate the difference between actual(Y) and predicted(YHAT).


  • Here Error is the difference between Y and Yhat
  • Error=Y - Yhat
  • So, our next step is to square this Error


  • We have got the squares of the error now, now we have to calculate ∑ of squared error which means adding all the elements in the Squared error column
  • Adding up all the elements in the squared error we get 3.6.
  • The total no element present in our dataset is  5 or just calculate the no of rows.


  • Now let's calculate the Mean squared error
  • Formula to caluclate Mean squared error is,


  • No of data points(n) =5
  • The sum of Squared error /= 3.6
  • we have got the value now let; caluclate
    • MSE=3.6/5  =0.72
    • MSE=0.72
  • Smaller the Mean Squared error(MSE) closer we are to the best-fit line/Regression line.
    • This indicates the model is performing well  and the distance between the actual and predicted value is less.

MEAN ABSOLUTE ERROR

  • Mean absolute error is another loss function used for Regression models.
  • MAE is the sum of absolute differences between our actual(y) and predicted(yhat) values.
    • The absolute difference between the two numbers 4 and 2 is |4-2|=2
    • The absolute difference between the two numbers -4 and -2 is |-4-2|=|-6|=6
  • It measures the average magnitude/(size) of errors in a set of predictions.
  • So, let's apply this to our dataset now

  • As I have mentioned above y is the actual value and yhat is the predicted value.
  • In MAE we will take the absolute difference, though the difference might be negative, the absolute difference will be positive because the modulus of the no is always positive.
  • Now let's calculate ∑ Squared errors

  • Absolute error / ∑  =3.2
  • Total no data points= 5
  • Formula to find Mean Absolute error is,


  • Let's apply the calculated values on the formula,
    • MAE=3.2 / 5 = 0.64
    • MAE= 0.64

ROOT MEAN SQUARED ERROR(RMSE)

  • RMSE is the standard deviation of  the residuals/loss/error.
  • RMSE is a measure of how spread out the residuals are.
  • RMSE is calculated by taking the square root of MSE.
  • Formula to caluclate RMSE is,

  • Formula inside the square root is the formula of MSE.
  • Taking the square root of it √MSE, we will get RMSE
  • MSE=0.72(from the above calculation)
  • √0.72 = 0.848
  • RMSE=0.848
  • Lower values of RMSE indicates better fit

R-SQUARED


  • R-squared is a statistical measure of how close the data are to the fitted regression line.
  • R-squared is always between 0 - 100%
  • R-squared of 0% reveals us that 0% of data fits the regression model.
  • R-squared of 100% reveals us that 100% of data fits the regression model.
  • In general, the higher the R-squared, the better the model fits your data
  • Formula to calculate R_squared are:


  • Here,
    • SSRES = Sum of square of residuals
    • SSTOT=Sum of squares of total
    • SSRES we have calculated while calculating the MSE, you can go above and refer.
    • SSTOT is nothing but square of Actual value - Average of Actual value.


  • ∑ Squared error /     = 3.6
  • =5.2


  • so let's implemented the above calculated values in the formula
    • 1- 3.6 / 5.2 = 0.3076
    • R_squared = 0.3076  0.3076*100= 30.76%
  • In most of the cases R-squared threshold will be 50% .
  • If the calculated R-squared value is less than the threshold(50%) it is not considered as good fit.
  • If the calculated R-squared value is more than the threshold(50%) it is considered as good fit.
  • our model indicates only 30.076% of data fit the regression model.
  • so, our model is not a good fit.
  • So, the R-squared value tells us how well the model fits our data.
  • Higher the R-squared score, better the model fits your's data.

 THANK YOU



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