Machine Learning in R - Part 1: Correlation and Regression (Basics)
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This document discusses correlation and regression analysis. It defines concepts like covariance, correlation, and regression. Examples are provided to illustrate covariance, correlation coefficients from scatter plots, and linear regression analysis in R using the mtcars dataset. The regression case study finds a negative correlation between weight and miles per gallon for cars, with a slope of -5.3 and intercept of 37 for the best fit line. In conclusion, the document covers key concepts for understanding the relationship between variables using statistical techniques like correlation and regression.
Machine Learning in R - Part 1: Correlation and Regression (Basics)
- 1. Correlation and Regression in R Hamid Reza Bolhasani PhD, Data Scientist Jan 2020 1
- 2. Table of contents - Covariance - Correlation - Examples - Regression - Case Study in R - Conclusion 2
- 3. Covariance 1 ))(( ),cov( 1 ï ïï ïœ ï„ïœ n yyxx yx i n i i 1 )( 2 12 ï ï ïœ ï„ïœ n xx S n i i x Variance Gives information of a single variable Covariance Gives information on the degree to which two variables vary together. ïź When X and Y cov (x,y) = pos. ïź When X and Y cov (x,y) = neg. ïź When no constant relationship: cov (x,y) = 0 3
- 4. Covariance Example High variance data Low variance data Subject x y x error * y error x y X error * y error 1 101 100 2500 54 53 9 2 81 80 900 53 52 4 3 61 60 100 52 51 1 4 51 50 0 51 50 0 5 41 40 100 50 49 1 6 21 20 900 49 48 4 7 1 0 2500 48 47 9 Mean 51 50 51 50 Sum of x error * y error : 7000 Sum of x error * y error : 28 Covariance: 1166.67 Covariance: 4.67 4
- 5. Correlation & Regression Correlation - Is there any relationship between 2 variables (x,y)? - X is independent (Explanatory) and Y is dependent (Response) - Correlation â Causation Regression How well a certain independent variable predict dependent variable? ïš ï©ïš ï© ïš ï©ïš ï©2 2 2 2 N XY X Y N X X N Y Y xyr ï ï ïï„ ï„ ï„ ïœ ï„ ï„ ï„ ï„ xy xy x y s r s s ïœ 5
- 6. Correlation in Scatter Diagrams x y Strong negative correlation x y Weak positive correlation x y Strong positive correlation x y Nonlinear Correlation r = ï0.91 r = 0.88 r = 0.42 r = 0.07 6
- 7. Regression Example Smoking vs Lung Capacity N Cigarettes (X ) Lung Capacity (Y ) 1 0 45 2 5 42 3 10 33 4 15 31 5 20 29 7
- 8. Example Analysis Smoking vs Lung Capacity 20 25 30 35 40 45 50 -5 0 5 10 15 20 25 LungCapacity Smoking (yrs) Lung Capacity (Y ) 8 1 ( 215) 53.75 4 xyS ïœ ï ïœ ï 0.96xyr ïœ ï When smoking is above its group means, lung capacity tends to be below its group mean. Greater smoking exposure implies greater likelihood of lung damage.
- 9. Regression 9 - The process of predicting variable Y using variable X. - Tells us how values in Y changes as a function of changes in value X. - Calculates the âbest-fitâ line for a certain of data. x y d1 d2 d3 Observed y-value Predicted y-value intercept Δ Ć· = ax + b Δ = residual error = y i , true value slope = Ć·, predicted value
- 10. Regression: Case Study in R 10 library(ggplot2) ggplot(data=mtcars, aes(x=wt, y=mpg))+geom_point() - Data = mtcars - wt: weight - Mpg: miles per gallon - S(x,y) = -5.11 - r(x,y) = -0.86
- 11. Regression: Case Study in R 11 - Intercept = 37 - Slope = -5.3 ggplot(data=mtcars, aes(x=wt, y=mpg))+geom_point()+geom_smooth(method="lm",se=FALSE) lm(data=mtcars, mpg ~ wt)