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Descriptive Analysis and Bivariate Data

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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.
Out of three variables, two variables are qualitative (both attribute) and one is qualitative (attribute) and another variable is quantitative or (numerical), bivariate data analysis is done using one qualitative and one quantitative variable. The quantitative values are seen as a separate sample, every Set is identified by qualitative variable. When the bivariate data are obtained by the two quantitative variables, then it is represented by the data mathematically as order pair (p, q), where ‘p’ represents the input variables (sometime it is called as independent variable) and ‘q’ is output variable (sometime it is called dependent variable). The bivariate data is called Ordered Pair because one value of ordered i.e. ‘p’ is always written first and the data is called as paired because for every ‘p’ value there is a corresponding value ‘q’ is present from the same source.
Like if there is an ordered pair (p, q), if ‘p’ is the height and ‘q’ is the weight, then the height value and the corresponding weight are recorded for each person. The input variable ‘p’ is measured as the output variable ‘q’. The basic bivariate Descriptive Statistics or univariate data description is used to define the distribution of the values of a single variable. This method is used to represent the joint distribution of the pairs, in a way in which the values of the variables are distributed among the group. There are some steps for conducting bivariate analysis.
1. When we define the relationship in the independent variables and the dependent variable then we see that the values of the independent variables and the dependent variables relate each other or not, then after that we can identify the types and direction of variables which is applicable or not of the given relationship. Then we determine the variables in the relationship which is statistically significant to the population. Then we determine the strength of the relationship i.e. the degree of independent variable determine the variation in the dependent variable. Common form in the bivariate analysis includes a Percentage table, scatter graph and the simple correlation coefficient. First step of bivariate descriptive analysis is analysis of your data whatever you have and we cannot assume that the data is representative. The raw Numbers do not reflect the relative strength of the relationship. It is also used in Mean, median and Mode. Each variable is either dependent or independent.

Linear Regression

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Linear Regression is a part of Statistics. Linear regression statistics is basically an approach which helps us in determining the relationship between two variables; one is a Scalar variable say y and another is explanatory variable say x. The explanatory variable can be one or more for modeling the relation with the scalar variable. If there is only a single explanatory variable then this is known as Simple Linear regression. If more than one explanatory variable is present then this is called a Multiple Linear Regression.

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.