1. Linear Regression

2. Example data and scatter plot

3. 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

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

5. 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

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

7. 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?

8. Correlation Coefficients

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

10. 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)…

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

12. Polynomial Curve fitting used in forestry