May 12 2021

Objectives

  • Understand linear regression and describe a case study
  • Compute and interpret coefficients in a linear regression analysis in R.
  • Interpolate regression model in R to produce a raster layer.

Science

What are we doing?

  • Explain
  • Predict

Connections

Why is it windy in Iowa?

  • Missouri sucks and Minnesota blows
  • No connection

Where do babies come from?

  • Storks deliver babies
  • No connection

Why are basements in Iowa full of cracks?

  • Soils in Iowa contain high amounts of shrink-swell clays
  • Connection

Predictions and Predicting with a Function

Predictions

  • None of us will report to work on Sunday
  • The average price of a gallon of gas in the US will be $100.00 on January 1

Predicting with a function

  • \[y= \beta_0 + \beta_1x + \epsilon\]

Assumptions

  • Linear
  • Independent
  • Homoscedastic
  • Normal

Interpreting the Linear Regression Model

Building the Linear Regression Model

  • Ordinary Least Squares
  • \[\beta_1= \frac{\sum(x_i - \bar x) (y_i - \bar y)} {\sum(x_i - \bar x)^2}\]
  • \[\beta_0= \bar y - \beta_1 \times \bar x\]

Examples

Wills et al., 2013

Carbon equivalent correction regression factor: \[OC_{dc}= 0.25 + 0.86(OC_{wc})\] where

\(OC_{dc}=\) organic carbon by dry combustion (%)

\(OC_{wc}=\) organic carbon by wet combustion (%)

Examples