linear regression

Author

chandan v

ABSTRACT:

Linear-regression models are relatively simple and provide an easy-to-interpret a mathematical formula that can generate predictions. Linear regression can be applied to various areas of business and academic study. You’ll find that linear regression is used in everything from biological, behavioral, environmental, and social sciences to business. Linear regression models have become a proven way to scientifically and reliably predict the future. Because linear regression is a long-established statistical procedure, the properties of linear regression models are well understood and can be trained very quickly.

INTRODUCTION:

Linear regression is an algorithm used to predict or visualize, a relationship between two different features/variables. In linear regression tasks, there are two kinds of variables being examined: the dependent variable and the independent variable. The independent variable is the variable that stands by itself, not impacted by the other variable. As the independent variable is adjusted, the levels of the dependent variable will fluctuate. The dependent variable is the variable that is being studied, and it is what the regression model solves for/attempts to predict. In linear regression tasks, every observation/instance is comprised of both the dependent variable value and the independent variable value.

LINEAR REGRESSION FORMULA:

As we know, linear regression shows the linear relationship between two variables. The equation of linear regression is similar to that of the slope formula. We have learned this formula before in earlier classes such as a linear equation in two variables. Linear Regression Formula is given by the equation

Y= a + bX

We will find the value of a and b by using the below formula.

a= ((∑Y)(∑X2)−(∑X)(∑XY))/n(∑x2)−(∑x)2

b= (n(∑XY)−(∑X)(∑Y))/n(∑x2)−(∑x)2

REAL-LIFE APPLICATION OF LINEAR REGRESSION:

1)Analyze pricing elasticity

Changes in pricing often impact consumer behavior and linear regression can help you analyze how. For instance, if the price of a particular product keeps changing, you can use regression analysis to see whether consumption drops as the price increases. What if consumption does not drop significantly as the price increases? At what price point do buyers stop purchasing the product? This information would be very helpful for leaders in a retail business.

2)Assess risk in an insurance company

Linear regression techniques can be used to analyze risk. For example, an insurance the company might have limited resources with which to investigate homeowners’ insurance claims; with linear regression, the company’s team can build a model for estimating claims costs. The analysis could help company leaders make important business decisions about what risks to take.

PROBLEMS AND SOLUTIONS WITH RESPECT TO LINEAR REGRESSION:

1)Consider the following set of points: {(-2, -1), (1, 1), (3, 2)}

a) Find the least square regression line for the given data points.

b) Plot the given points and the regression line in the same rectangular system of axes.

Solution:

a) Let us organize the data in a table.

x       y       x y     x\^2

-2      -1      2       4

1       1       1       1

3       2       6       9

Σx = 2  Σy = 2  Σxy = 9 Σx2 = 14

We now use the above formula to calculate a and b as follows

a = (nΣx y - ΣxΣy) / (nΣx2 - (Σx)2) = (3*9 - 2*2) / (3*14 - 22) = 23/38

b = (1/n)(Σy - a Σx) = (1/3)(2 - (23/38)*2) = 5/19

b) We now graph the regression line given by y = ax + b and the given points.

2)a) Find the least square regression line for the following set of data

{(-1 , 0),(0 , 2),(1 , 4),(2 , 5)}

b) Plot the given points and the regression line in the same rectangular system of axes.

Solution:

a) We use a table as follows

x       y       x y     x\^2

-1      0       0       1

0       2       0       0

1       4       4       1

2       5       10      4

Σx = 2  Σy = 11 Σx y = 14   Σx2 = 6

We now use the above formula to calculate a and b as follows

a = (nΣx y - ΣxΣy) / (nΣx2 - (Σx)2) = (4*14 - 2*11) / (4*6 - 22) = 17/10 = 1.7

b = (1/n)(Σy - a Σx) = (1/4)(11 - 1.7*2) = 1.9

b) We now graph the regression line given by y = ax + b and the given points.

CONCLUSION:

The function of a regression model is to determine a linear function between the X and Y variables that best describe the relationship between the two variables. In linear regression, it’s assumed that Y can be calculated from some combination of the input variables.

REFERENCES:

David A. Freedman (2009). Statistical Models: Theory and Practice. Cambridge University Press. p. 26. A simple regression equation has on the right hand side an intercept and an explanatory variable with a slope coefficient. A multiple regression e right hand side, each with its own slope coefficient
Hilary L. Seal (1967). “The historical development of the Gauss linear model”. Biometrika. 54 (1/2): 1–24. doi:10.1093/biomet/54.1-2.1. JSTOR 2333849.