Linear Regression

Example data and scatter plot

Regression Line Best fitting straight line that minimizes the sum of the squared errors of prediction The colored lines show the amount of the error of prediction

Parts of the regression equation Y = (slope times X) + intercept

Measure 2 variables, plot results Data for Scots Pine Logically a tree with zero diameter would have a zero crown size. If a linear relationship exists, it should pass through the origin. This makes the formula Y = slope times X

Find out if relationship exists and how good the fit is y = mx + b In Excel…

Regression and Correlation The relationship between the value of X and it’s corresponding value of Y can be inverse or negatively correlated (higher values of X result in lower values of Y). Example might be X equals poison ingested and Y is survivors. It can be positive where higher values of X result in higher values of Y. Example might be x equals fuel added and Y equals heat given off by a fire. If you plot your data points and the line is flat, that means regardless of the level of X, Y stays the same. This means there is no correlation between X and Y. So the closer your line is to horizontal the weaker the correlation. Can you imagine a situation where this would occur?

Correlation Coefficients

Problems with linear regression Outliers, typos Non-linear data Curvilinear Point cloud

Multiple Regression Taking simple linear regression one step further Predicting the value of a dependent variable based on the value of 2 or more independent variables. Y = intercept + (coefficient1 X Variable1) + (coefficient2 X Variable2)…

Polynomial Regression Intercept = constant +/- (coefficient1 X variable1) +/- (coefficent2 X variable1 squared)

Polynomial Curve fitting used in forestry