## Monday, April 1, 2013

### Salty Residuals

Washington, this Winter, has not been wearing "on his smiling face a dream of Spring". On the contrary, Spring has been continually cold and damp much like the movie Groundhog Day. Just last week we had an unusual mid-March snowstorm. Groundskeepers spread salt along campus walkways to speed the snow melt. Their spreader sprayed the salt left and right as they drove down the path. After melt, the salt remained attracting moisture, absorbing - not reflecting - the light that fell on it, leaving a dark, wet residual on the asphalt.

At every point down the walk we can see the horizontal spread of the dark residuals, much concentrated near the center of the walk with lesser concentrations to the right and left. Horizontally, what remains is a bell-shaped distribution of salt deposition. The distribution has this same, consistent shape at each place down the walk. In this angled, perspective view the distributions look more skewed to the right. But the symmetry can be seen more accurately in the rotated image below.
This rotated image now shows the distributions in vertical slices as we move horizontally along the walk. The distributions are centered along the same horizontal line with the same shape and degree of vertical spread.

This is exactly the image of ideal residuals from a simple linear regression fit to data plotted against an explanatory variable from a uniform design. Of course, a different design placement of the explanatory variable would vary the pattern horizontally, but not so vertically. Other design patterns could arise from the spreader moving faster or slower down path leaving a more uneven, erratic deposition of salt - more at some steps along the walk than at others. But the assumptions for such a regression model still require identical, vertical normal distributions of scatter around a straight line of means irrespective of the horizontal position down the path. For an even, uniform walk down the path, our salty residuals model and reflect the ideal behavior of regression residuals.