Regression Analysis

REGRESSION synopsis correlativity co competent except indicates the phase and direction of kin surrounded by both covariant stars. It does non, necessarily con n unrivaled a cause-effect kindred. Even when there ar railyard to mean the causative blood exits, coefficient of coefficient of correlation does non tell us which shifting is the cause and which, the effect. For example, the subscribe to for a commodity and its price bequeath chiefly be found to be correlated, but the question whether demand depends on price or vice-versa will not be answered by correlation. The vocabulary meaning of the regress is the act of the returning or exhalation back.The shape retroflexion was first employ by Francis Galton in 1877 sequence examine the relationship amidst the heights of fathers and sons. ingenuous regression is the beatnik of the just relationship in the midst of cardinal or more variable quantitys in ground of the genuine units of data. The ocellus of lapsing is the barrier, which gives the scoop cast to the determine of nonp aril variable for any specific places of some other variables. For two variables on turnaround epitome, there ar two turnab stunned ocelluss. One line as the infantile fixation of x on y and other is for fixing of y on x. These two relapse line show the modal(a) relationship between the two variables.The relapsing line of y on x gives the most presumptive revalue of y for given value of x and the relapse line of x and y gives the most probable values of x for the given value of y. For perfect correlation, positive or controvert i. e. for r= , the two lines coincide i. e. we will father only champion straight line. If r=0, i. e. both the unevenness argon in reliant whence the two lines will cut each other at a right angle. In this case the two lines will be to x and y axis. The Graph is given below- We restrict our intervention to bi bi one-dimensional relationships on ly that is the comparisons to be considered ar 1- y=a+bx x=a+by In comp atomic number 18 first x is inviteed the self-governing variable and y the dependent variable. Conditional on the x value, the comparisons gives the renewal of y. In other words ,it means that corresponding to each value of x ,there is whole conditional probability distribution of y. Similar tidings holds for the equation second, where y acts as supreme variable and x as dependent variable. What purpose does regression line serve? 1- The first purpose is to estimate the dependent variable from known values of self-governing variable. This is achievable from regression line. The next objective is to obtain a measure of the delusion assume-to doe with in using regression line for estimation. 3- With the help of regression coefficients we house calculate the correlation coefficient. The square of correlation coefficient (r), is send fored coefficient of determination, measure the degree of stand of correlation that exits between two variables. What is the difference between correlation and linear regression? Correlation and linear regression are not the corresponding. backvas these differences Correlation quantifies the degree to which two variables are related.Correlation does not finda best-fit line (that is regression). You simply are computing a correlation coefficient (r) that tells you how much one variable tends to change when the other one does. With correlation you dont have to regard nigh cause and effect. You simply limit how swell two variables relate to each other. With regression, you do have to think about cause and effect as the regression line is obstinate as the best way to predict Y from X. With correlation,it doesnt matter which of the two variables you call X and which you call Y.Youll get the aforementioned(prenominal) correlation coefficient if you exchange the two. With linear regression, the decision of which variable you call X and w hich you call Y matters a lot, as youll get a different best-fit line if you exchange the two. The line that best predicts Y from X is not the same as the line that predicts X from Y. Correlation is almost always apply when you measure both variables. It rarely is appropriate when one variable is or sothing you by experimentation manipulate. With linear regression, the X variable is often something you experimental manipulate (time, niggardness and the Y variable is something you measure. statistical regression outline is widely used for presage(including prodigyoftime-seriesdata). Use of regression psychoanalysis for expectation has substantial convergence with the matter ofmachine learning. Regression analysis is also used to generalize which among the independent variables are related to the dependent variable, and to explore the forms of these relationships. In limit circumstances, regression analysis can be used to infercausal relationshipsbetween the independent and dependent variables.A large body of techniques for carrying out regression analysis has been developed. Familiar methods such aslinear regressionand popular least squaresregression areparametric, in that the regression exploit is defined in terms of a finite number of unknown quantityparametersthat are estimated from thedata. Nonparametric regressionrefers to techniques that allow the regression function to deceit in a specified set offunctions, which whitethorn beinfinite-dimensional. The performance of regression analysis methods in practice depends on the form of the data-generating process, and how it relates to the regression draw close world used.Since the true form of the data-generating process is not known, regression analysis depends to some extent on making assumptions about this process. These assumptions are sometimes (but not always) testable if a large amount of data is available. Regression modellings for prediction are often useful even when the assumpti ons are moderately violated, although they may not perform optimally. However when carrying out conclusionusing regression models, in particular involving small effectuateor questions ofcausalitybased on empiric data, regression methods must(prenominal) be used cautiously as they can easily give misleading results.Underlying assumptions Classical assumptions for regression analysis include ? The sample must be representative of the universe of discourse for the inference prediction. ? The illusion is assumed to be arandom variablewith a mean of zero conditional on the explanatory variables. ? The variables are error-free. If this is not so, modeling may be done usingerrors-in-variables modeltechniques. ? The predictors must belinearly independent, i. e. it must not be executable to stub out any predictor as a linear junto of the others. SeeMulticollinearity. The errors areuncorrelated, that is, thevariance-covariance matrixof the errors isdiagonaland each non-zero element is the variance of the error. ? The variance of the error is constant across observations (homoscedasticity). If not,weighted least squaresor other methods big businessman be used. These are sufficient (but not all necessary) conditions for the least-squares computer to induce desirable properties, in particular, these assumptions imply that the parameter estimates will beunbiased,consistent, andefficientin the class of linear unbiased estimators.Many of these assumptions may be relaxed in more advanced treatments. Basic Formula of Regression abstract- X=a+by (Regression line x on y) Y=a+bx (Regression line y on x) 1st Regression equation of x on y- 2nd Regression equation of y on x- Regression Coefficient- Case 1st when x on y means regression coefficient is bxy Case 2nd when y on x means regression coefficient is byx Least Square bringing close together- The master(prenominal) object of constructing statistical relationship is to predict or inform the effects on one depend ent variable resulting from changes in one or more explanatory variables.Under the least square criteria, the line of best fit is said to be that which minimizes the sum of the squared residuals between the points of the interpret and the points of straight line. The least squares method is the most widely used outgrowth for developing estimates of the model parameters. The graph of the estimated regression equation for simple linear regression is a straight line approximation to the relationship between y and x. When regression equations obtained directly that is without taking deviation from veridical or assumed mean then the two expression equations are to be solved simultaneously as followsFor Regression Equation of x on y i. e. x=a+by The two Normal Equations are- For Regression Equation of y on x i. e. y=a+bx The two Normal Equations are- Remarks- 1- It may be mention that both the regression coefficient ( x on y means bxy and y on x means byx ) cannot exceed 1. 2- Both t he regression coefficient shall either be positive + or negative -. 3- Correlation coefficient (r) will have same sign as that of regression coefficient.Regression AnalysisREGRESSION ANALYSIS Correlation only indicates the degree and direction of relationship between two variables. It does not, necessarily connote a cause-effect relationship. Even when there are grounds to believe the causal relationship exits, correlation does not tell us which variable is the cause and which, the effect. For example, the demand for a commodity and its price will generally be found to be correlated, but the question whether demand depends on price or vice-versa will not be answered by correlation. The dictionary meaning of the regression is the act of the returning or going back.The term regression was first used by Francis Galton in 1877 while studying the relationship between the heights of fathers and sons. Regression is the measure of the average relationship between two or more variables in t erms of the original units of data. The line of regression is the line, which gives the best estimate to the values of one variable for any specific values of other variables. For two variables on regression analysis, there are two regression lines. One line as the regression of x on y and other is for regression of y on x. These two regression line show the average relationship between the two variables.The regression line of y on x gives the most probable value of y for given value of x and the regression line of x and y gives the most probable values of x for the given value of y. For perfect correlation, positive or negative i. e. for r= , the two lines coincide i. e. we will find only one straight line. If r=0, i. e. both the variance are independent then the two lines will cut each other at a right angle. In this case the two lines will be to x and y axis. The Graph is given below- We restrict our discussion to linear relationships only that is the equations to be considered are 1- y=a+bx x=a+by In equation first x is called the independent variable and y the dependent variable. Conditional on the x value, the equations gives the variation of y. In other words ,it means that corresponding to each value of x ,there is whole conditional probability distribution of y. Similar discussion holds for the equation second, where y acts as independent variable and x as dependent variable. What purpose does regression line serve? 1- The first object is to estimate the dependent variable from known values of independent variable. This is possible from regression line. The next objective is to obtain a measure of the error involved in using regression line for estimation. 3- With the help of regression coefficients we can calculate the correlation coefficient. The square of correlation coefficient (r), is called coefficient of determination, measure the degree of association of correlation that exits between two variables. What is the difference between correlatio n and linear regression? Correlation and linear regression are not the same. Consider these differences Correlation quantifies the degree to which two variables are related.Correlation does not finda best-fit line (that is regression). You simply are computing a correlation coefficient (r) that tells you how much one variable tends to change when the other one does. With correlation you dont have to think about cause and effect. You simply quantify how well two variables relate to each other. With regression, you do have to think about cause and effect as the regression line is determined as the best way to predict Y from X. With correlation,it doesnt matter which of the two variables you call X and which you call Y.Youll get the same correlation coefficient if you swap the two. With linear regression, the decision of which variable you call X and which you call Y matters a lot, as youll get a different best-fit line if you swap the two. The line that best predicts Y from X is no t the same as the line that predicts X from Y. Correlation is almost always used when you measure both variables. It rarely is appropriate when one variable is something you experimentally manipulate. With linear regression, the X variable is often something you experimental manipulate (time, concentration and the Y variable is something you measure. Regression analysis is widely used forprediction(includingforecastingoftime-seriesdata). Use of regression analysis for prediction has substantial overlap with the field ofmachine learning. Regression analysis is also used to understand which among the independent variables are related to the dependent variable, and to explore the forms of these relationships. In restricted circumstances, regression analysis can be used to infercausal relationshipsbetween the independent and dependent variables.A large body of techniques for carrying out regression analysis has been developed. Familiar methods such aslinear regressionandordinary least squaresregression areparametric, in that the regression function is defined in terms of a finite number of unknownparametersthat are estimated from thedata. Nonparametric regressionrefers to techniques that allow the regression function to lie in a specified set offunctions, which may beinfinite-dimensional. The performance of regression analysis methods in practice depends on the form of the data-generating process, and how it relates to the regression approach being used.Since the true form of the data-generating process is not known, regression analysis depends to some extent on making assumptions about this process. These assumptions are sometimes (but not always) testable if a large amount of data is available. Regression models for prediction are often useful even when the assumptions are moderately violated, although they may not perform optimally. However when carrying outinferenceusing regression models, especially involving smalleffectsor questions ofcausalitybased onobser vational data, regression methods must be used cautiously as they can easily give misleading results.Underlying assumptions Classical assumptions for regression analysis include ? The sample must be representative of the population for the inference prediction. ? The error is assumed to be arandom variablewith a mean of zero conditional on the explanatory variables. ? The variables are error-free. If this is not so, modeling may be done usingerrors-in-variables modeltechniques. ? The predictors must belinearly independent, i. e. it must not be possible to express any predictor as a linear combination of the others. SeeMulticollinearity. The errors areuncorrelated, that is, thevariance-covariance matrixof the errors isdiagonaland each non-zero element is the variance of the error. ? The variance of the error is constant across observations (homoscedasticity). If not,weighted least squaresor other methods might be used. These are sufficient (but not all necessary) conditions for the l east-squares estimator to possess desirable properties, in particular, these assumptions imply that the parameter estimates will beunbiased,consistent, andefficientin the class of linear unbiased estimators.Many of these assumptions may be relaxed in more advanced treatments. Basic Formula of Regression Analysis- X=a+by (Regression line x on y) Y=a+bx (Regression line y on x) 1st Regression equation of x on y- 2nd Regression equation of y on x- Regression Coefficient- Case 1st when x on y means regression coefficient is bxy Case 2nd when y on x means regression coefficient is byx Least Square Estimation- The main object of constructing statistical relationship is to predict or explain the effects on one dependent variable resulting from changes in one or more explanatory variables.Under the least square criteria, the line of best fit is said to be that which minimizes the sum of the squared residuals between the points of the graph and the points of straight line. The least squa res method is the most widely used procedure for developing estimates of the model parameters. The graph of the estimated regression equation for simple linear regression is a straight line approximation to the relationship between y and x. When regression equations obtained directly that is without taking deviation from actual or assumed mean then the two Normal equations are to be solved simultaneously as followsFor Regression Equation of x on y i. e. x=a+by The two Normal Equations are- For Regression Equation of y on x i. e. y=a+bx The two Normal Equations are- Remarks- 1- It may be noted that both the regression coefficient ( x on y means bxy and y on x means byx ) cannot exceed 1. 2- Both the regression coefficient shall either be positive + or negative -. 3- Correlation coefficient (r) will have same sign as that of regression coefficient.