Bivariate Data Analysis includes or comparison between two variables, if there is a relationship between the variables. Mostly we can see that the independent variables are represented by the columns and the dependent variable is represented by the rows or we can say that it is the Combination of two variables which are used for analysis. If we have different variables, and the values of the different variables which are obtained from the same element is known as bivariate data. Out of these two variables, it may be either qualitative or quantitative and in case of three combinations of variables, we can obtain its type from bivariate data.
Linear RegressionBack to Top
In statistics linear regression the data is modeled by using a special function called linear function. The unknown model parameters are derived from the data. These models are known as Linear Models. The linear regression always indicates a model in which a conditional Mean is taken of a scalar variable and this conditional mean is the value of the explanatory variable x that is an affine function of x. The linear regression always gives extra attention to the Conditional Probability distribution of the scalar variable given x, rather than on the joint Probability Distribution of y and x.
Linear Regression is a type of regression analysis, and it is widely used in practical applications. Models which linearly depends on their respective parameters (which are unknown); are easier to fit. But the other models which are non-linearly dependent on their unknown parameters are not easy to fit. Also we can easily get the statistical properties of the estimators for the models which are linearly dependent.
It has many applications and practical uses; mainly there are two broad categories:
1. In prediction and Forecasting
2. To quantify the strength of any relation between the scalar variable and one or more explanatory variables
For linear regression models we may also use the least Square approach to fit them; but we may also use other ways for the same purpose like by least absolute deviation regression and Loss Functions in ridge regression etc. The least square and linear model; these two terms are not identical. The least square term is used for the fitting purpose of the models which are nonlinear.
Consider a given Set of data yi, xi1. . . xip where i is 1 to n units. A linear regression model considers that the relationship between the dependent variable and p vector of regression xi is modeled and designed is a linear. We use an unknown Random Variable that is εi for modeling the relationship. This random variable εi mix some noise to the linear relationship between the repressors and the dependent scalar variable.
This takes a form:
yi = α1 xi1 + α1 xi2 + . . . + αp xip + εi = x’i α + εi
Here i is 1, 2, 3. . . . . . , n. and denotes the transpose of the variable so here x’i α is the inner product of vector xi and α.
Also these n equations can be written in a vector form as:
y = xα + ε
Where y, x, ε and α are the matrices with their n values.