Linear regression attempts to model the relationship between two variables by fitting a linear equation to observed data. One variable is considered to be an explanatory variable, and the other is considered to be a dependent variable. The case of one explanatory variable is called simple linear regression. For more than one explanatory variable, the process is called multiple linear regression.

Linear regression models are often fitted using the least squares approach.

If there appears to be no association between the proposed explanatory and dependent variables (i.e., the scatterplot does not indicate any increasing or decreasing trends), then fitting a linear regression model to the data probably will not provide a useful model. A valuable numerical measure of association between two variables is the correlation coefficient, which is a value between -1 and 1 indicating the strength of the association of the observed data for the two variables.

There are many names for a regression’s dependent variable. It may be called an outcome variable, criterion variable, endogenous variable, or regressand. The independent variables can be called exogenous variables, predictor variables, or regressors.

Linear regression using python

Following are the ways to do linear regression using python

statsmodels

scikit-learn

scipy

Linear Regression using statsmodels

Here is sample code

and here is the output

Linear Regression using scikit-learn

Here is the code

and output of this code is as below

Linear Regression using scipy

Sample code

and output

If you look at code, it seems finding linear regression using scipy is shortest and easiest to understand.

Correlation is used to indicate dependence or association is any statistical relationship, whether causal or not, between two random variables or bivariate data. It is a measure of relationship between two mathematical variables or measured data values, which includes the Pearson correlation coefficient as a special case.Correlation is any of a broad class of statistical relationships involving dependence, though in common usage it most often refers to how close two variables are to having a linear relationship with each other.

The strength of the linear association between two variables is quantified by the correlation coefficient.

Formula for correlation is as below

The correlation coefficient always takes a value between -1 and 1,

Value of 1 or -1 indicating perfect correlation (all points would lie along a straight line in this case).

A correlation value close to 0 indicates no association between the variables.The closer the value of r to 0 the greater the variation around the line of best fit.

A positive correlation indicates a positive association between the variables (increasing values in one variable correspond to increasing values in the other variable),

while a negative correlation indicates a negative association between the variables (increasing values is one variable correspond to decreasing values in the other variable).

The square of the correlation coefficient, r², is a useful value in linear regression. This value represents the fraction of the variation in one variable that may be explained by the other variable. Thus, if a correlation of 0.8 is observed between two variables (say, height and weight, for example), then a linear regression model attempting to explain either variable in terms of the other variable will account for 64% of the variability in the data^{1}

the least-squares regression line will always pass through the means of x and y, the regression line may be entirely described by the means, standard deviations, and correlation of the two variables under investigation.

Pearson correlation coefficient

Pearsons correlation coefficient is a measure of the linear correlation between two variables X and Y. It has a value between +1 and −1 ^{2}.t is obtained by dividing the covariance of the two variables by the product of their standard deviations.

Formula for Pearson Correlation Coefficient

Rank correlation coefficients

Spearman’s rank correlation coefficient

The Spearman correlation coefficient is defined as the Pearson correlation coefficient between the ranked variables.^{3}

Kendall rank correlation coefficient

the Kendall correlation between two variables will be high when observations have a similar (or identical for a correlation of 1) rank (i.e. relative position label of the observations within the variable: 1st, 2nd, 3rd, etc.) between the two variables, and low when observations have a dissimilar (or fully different for a correlation of -1) rank between the two variables.^{4}

Goodman and Kruskal’s gamma

Goodman and Kruskal’s gamma is a measure of rank correlation, i.e., the similarity of the orderings of the data when ranked by each of the quantities. ^{5}

^{2}Regression is a set of statistical processes for estimating the relationships among variables. It includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable (target) and one or more independent variables (or ‘predictors’).

Regression analysis helps one understand how the typical value of the dependent variable (or ‘criterion variable’) changes when any one of the independent variables is varied, while the other independent variables are held fixed.

Usage

It is used in variety of places such as forecasting, time series analysis etc. across industries.

Regression analysis is used characterize the variation of the dependent variable around the prediction of the regression function using a probability distribution

A function of the independent variables called the regression function is to be estimated

Regression analysis can be used to infer causal relationships between the independent and dependent variables. However this can lead to illusions or false relationships, so caution is advisable as correlation does not prove causation.

Types of regression:

Linear Regression

Simple Linear Regression

multiple linear regression.

Logistic Regression

Simple Logistic Regression

Multiple Logistic Regression

Polynomial Regression

Stepwise Regression

Ridge Regression

Lasso Regression

ElasticNet Regression^{1}

1.

Ray S. 7 Types of Regression Techniques you should know! analyticsvidhya. https://www.analyticsvidhya.com. Accessed July 14, 2018.