Linear Regression PDF

Linear Regression Regression  A statistical measure that determine the strength of the relationship between the one dependent variable (y) and other independent variables (x1, x2, x3……)  This is done to gain information about one through knowing values of the others.  It is basically used fo...

Linear Regression Regression  A statistical measure that determine the strength of the relationship between the one dependent variable (y) and other independent variables (x1, x2, x3……)  This is done to gain information about one through knowing values of the others.  It is basically used for predicting and forecasting. Linear Regression  The simplest mathematical linear relationship between two variables x and y.  The change in one variable make the other variable change.  In other words a dependency of one variable to other. Linear Regression Mathematical Model y = mx + c 𝑌 = 𝑏0 + 𝑏1 ∗ 𝑋+ ∈ Y = Dependent Variable X = Independent Variable b0 = Y - Intercept b1 = Slope of the line ∈ = Error Variable Linear Regression Mathematical Model Linear Regression Mathematical Model Linear Regression Mathematical Model y = mx + c 3.6 = 0.2*3 + c → c=3 y = 0.2x + 3  Understanding Linear Regression y = mx + c x y x-ẋ y - ẏ (x − ẋ )2 (x - ẋ)(y - ẏ) 3.6 = 0.2*3 + c → c=3 1 3 -2 -0.6 4 1.2 6 2 4 -1 0.4 1 -0.4 5 3 4 0 0.4 0 0 4 2 1 -1.6 1 -1.6 4 5 5 2 1.4 4 2.8 Y - AXIS 3 Mean 3 3.6 ෍ = 10 ෍ = 2 2 σ x − ẋ (y − ẏ ) 2 1 m= = σ (x − ẋ )2 10 0 0 1 2 3 X - AXIS 4 5 6 y = 0.2x + 3 Linear Regression Mathematical Model Linear Regression Mathematical Model x y y-ẏ (y - ẏ)2 yp yp - ẏ (yp − ẏ )2 1 3 -0.6 0.36 3.2 -0.4 0.16 y = 0.2x + 3 2 4 0.4 0.16 3.4 -0.2 0.04 3 4 0.4 0.16 3.6 0 0 4 2 -1.6 2.56 3.8 0.2 0.04 5 5 1.4 1.96 4.0 0.4 0.16 Linear Regression Mathematical Model x y x - ẋ y - ẏ (x − ẋ (x - ẋ)(y - )2 ẏ) 1 3 -2 -0.6 4 1.2 2 4 -1 0.4 1 -0.4 3 4 0 0.4 0 0 4 2 1 -1.6 1 -1.6 5 5 2 1.4 4 2.8 y = 0.2x + 3  Mean Square Error 6 m = 0.2 c=3 y = 0.2x + 3 5 5 4 4 4 3.8 4 For these values, the predicted values 3.6 3.2 3.4 for y for x = (1,2,3,4,5) will be - 3 Line Of Regression Y - AXIS 3 y = 0.2 * 1 + 3 = 3.2 2 2 y = 0.2 * 2 + 3 = 3.4 y = 0.2 * 3 + 3 = 3.6 1 y = 0.2 * 4 + 3 = 3.8 0 y = 0.2 * 5 + 3 = 4.0 0 1 2 3 4 5 6 X - AXIS R - Square  R – Squared value is a measure of how close the data is to the fitted regression line.  It is also known as Coefficient of Determination. σ (yp − ẏ )2 R2 = σ (y − ẏ )2 Calculation 0f R2 x y y-ẏ (y - ẏ)2 yp yp - ẏ (yp − ẏ )2 1 3 -0.6 0.36 3.2 -0.4 0.16 2 4 0.4 0.16 3.4 -0.2 0.04 3 4 0.4 0.16 3.6 0 0 4 2 -1.6 2.56 3.8 0.2 0.04 5 5 1.4 1.96 4.0 0.4 0.16 Mean 3 3.6 ෍ = 5.2 ෍ = 0.4 𝟎. 𝟒 R2 = 𝟓. 𝟐 ≈ 0.08 Line drawn not accurate then by using equation y=mx+c , by changing m value we have to calculate accuracy for each m. Whenever you get the maximum accuracy that will be the best & final line Above example tell how leaner regression model works Practical approach(LR model) : predicating salary of employee according to the work experience. We are gating 0.08=8% its poor accuracy due less number sample of data Basically, R square model not good approach its used only once in the model Line drawn not accurate then by using equation y=mx+c , by changing m value we have to calculate accuracy for each m. Whenever you get the maximum accuracy that will be the best & final line Above example tell how leaner regression model works Practical approach(LR model) : predicating salary of employee according to the work experience. Line drawn not accurate then by using equation y=mx+c , by changing m value we have to calculate accuracy for each m. Whenever you get the maximum accuracy that will be the best & final line Above example tell how leaner regression model works This step of optimizing the model is known as gradient descent optimization >> Neural Network it will come in Deep Learning