Regression

Regression & Regression Line With Scatter Diagram
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1. Presentation of Statistics Submitted to Mam Uzma Submitted By Ali Raza Ansari 13054156-070 Department Information Technology UNIVERSITY OF GUJRAT (PAKISTAN)

2. Presentation Topic Is “Regression”

3. Objectives About Regression Definition Types of Regression Types of Dependent Variable Assumptions Deterministic Model Probalistic Model Method of Least Square Scatter Diagram Regression Lines Purposes of Regression Properties of Regression Line Examples

4. Definition : The dependence of one variable upon variables is called Regression. “Y = a + bx ”

5. Types of regression There are two types of Regression Variables , which are following. Independent Variable i -e X Dependent Variables i -e Y

6. Types of Dependence of variables Simple Regression Multiple Regression Linear Regression

7. Simple regression Regression is said to be Simple, if one variable depends upon other Variable. Examples : Heater depends upon Gas . Motor depends upon Electricity. Multiple Regression Regression is said to be Multiple, if two or more variables depends upon other Variable . Examples : All Electronic things depends upon Electricity. All Vehicles depends upon Petroleum .

8. Linear regression Definition : When the dependency of one variable upon other variable is represented by a straight line, then Regression is called Linear Regression. Example: Current is directly proportional to voltage

9. A simple linear regression model Y = α+βx+ Є “Y” is dependent variable. “x” is independent variable. “a” is intercept on Y-axis. “ Є ” is Error Component. “ β ” is R egression Coefficient or Slop it is also called coefficient of regression.

10. Assumptions : Observations are selected (dependent or independent variable). Regression function is Linear as , “ α + β x + Є “. Expected values of Error term is zero, “ E ( Є ) = 0”. Variance of Error term is Constant , “ V ( Є ) = δ 2 ”. Error terms are independent of each other, “ E ( Є i Є j ) = 0 ” I ≠ j. Error term and x are also zero, “ E ( Є x) = 0 “ . Error term is normally distributed with mean zero and Variance sigma is zero Є~N(0 , δ 2 ). Dependent variable is normally distributed with mean, α + βx and δ 2 Є ~( α + βx , δ 2 )

11. Deterministic Model Definition: A model in which we can determine a unique value of dependent variable for each value of independent variable is called deterministic model. Formula is “Y = a + bx “. Examples : 1. Celsius Temperature is “ C = 5 / 9(F – 32) ”. 2. Average is “ Sum of total numbers / Total number (Sum of n / n)”. (Not Error Include)

12. Probabilistic Model (error Include) Definition: A model in which we can’t determine a unique value of dependent variable for each value of independent variable is called probabilistic model . Formula is “Y = α + βx + Є “.

13. Mothod of Least square Definition : Method of least square consists of determining the value of unknown parameters that minimizes the sum of square of residuals and it is denoted by, S.S.R = Σ (Y i – Ŷ) 2 Ŷ = a + bx HERE, b = n Σ xy – Σx Σ y / n Σ x 2 – ( Σ x) 2 and, a = Σ y – b Σ x / n Range : - ∞ to + ∞

14. Regression Line There are two regression lines : Y= a+bx ( y on x) x= c+dy (x on y)

15. Scatter Diagram Definition : Scatter diagram is a graphical picture of sample data. Consider a random sample of “n” pair of observation as (x1, y1) ; (x2, y2) …….( xn , yn ). These points are plotted on graph paper taking independent variables on x-axis and dependent variable on y-axis. Graphical picture so, obtain is called scatter diagram. USES: Scatter diagram is used to judge the relation or regression such as positive or negative ..

16. Purposes of regression Estimation of unknown parameters (a, b). Prediction of dependent variable , “ Y = a + bx ”. Testing of Hypnosis about α and β . Confidence interval of, about α and β . Best procedure available in regression analysis. For Future Prediction .

17. Properties of regression line Regression lines always passes through points (X, Y) . Regression Coefficient independent of origin , b yx = b vu if u = x ± a and v = y ± b . Sum of deviation observe Y i and estimated Ŷ is zero Σ (Y i – Ŷ ) = 0. Sum of square of deviation of observed Y i and estimated Ŷ is Minimum Σ (Y i – Ŷ) 2 = min. Sum of observed values is equal to sum of estimated values , Σ y i = ΣŶ . Range of Regression coefficient is - ∞ to + ∞ .

18. Example Q # Computer Speed Depends upon Processer SR # Processer speed –’X’ Computer Speed –’Y’ 1 128 GHz 20 MB/s 2 768 100 3 378 55 4 1024 185 5 512 80 6 1280 198 X Y x 2 y 2 xy 128 20 16384 400 2560 768 100 589824 10000 76800 378 55 142884 3025 20790 1024 185 1048576 34225 189440 512 80 262144 6400 40960 1280 198 1638400 39204 253440 4090 638 3698212 93254 583990 Total

19. Formula: Y on X Y = a + bx a = ў - b хˉ ў = Σ y / n => 638 / 6 = 106.3333 x ˉ = Σ x / n => 4090 / 6 = 681.6666 b = n Σ xy – Σ х Σ y / n Σ х 2 – ( Σ х ) 2 b = 6 (583990) – (638)(4090) / 6(3698212)-(4090) 2 b = 3503940– 2609420 / 22189272 - 16728100 b=0.1638 a = 106.3333-(0.1638)(681.6666) a= -5.3237 Prediction: The estimated regression co-efficient y on x b=0.1638 ,which indicates that the value of y is increase 0.1638 units for a unit increase in x. Ŷ = -5.3237+ (0.1638)X Ŷ = -5.3237+(0.1638)(2048) Ŷ = 330.1387

20. Formula: X on Y x = c + d y c = ў - b хˉ ў = Σ y / n => 638 / 6 = 106.3333 x ˉ = Σ x / n => 4090 / 6 = 681.6666 d = n Σ xy – Σ х Σ y / n Σ y 2 – ( Σ y ) 2 d = 6 (583990) – (638)(4090) / 6(93254 )-(638) 2 d = 3503940– 2609420 / 559524 - 407044 d = 1.7241 C = 681.6666-(1.7241)( 106.3333) C = 498.3374 Prediction: The estimated regression co-efficient x on y b=1.7241 ,which indicates that the value of x is increase 1.7241 units for a unit increase in y. x = 498.3374+ (1.7241)Y x = 498.3374+(1.7241)(200) x = 843.1574

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